公式：
J_(n) = ∫ sec^n(x) dx
J_(n) = [sec^(n - 1)(x) sin(x)]/(n - 1) + (n - 2)/(n - 1) J_(n - 2)
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∫ tan^4(x) secx dx
= ∫ (sec²x - 1)² secx dx
= ∫ (sec^4(x) - 2sec²x + 1) secx dx
= ∫ sec^5(x) dx - 2∫ sec³x dx + ∫ secx dx
= J_(5) - 2J_(3) + J
= (1/4) sec^4(x) sinx + 3/4 J_(3)
= (1/4) sec^4(x) sinx - (5/4)J_(3) + J
= (1/4) sec^4(x) sinx - (5/4)[(1/2) sec²x sinx + 1/2 J] + J
= (1/4) sec^4(x) sinx - (5/8) sec²x sinx + (3/8)J
= (1/4) sec^4(x) sinx - (5/8) sec²x sinx + (3/8)ln|secx + tanx| + C敢不敢不用那个坑爹的reduction formula...0.0呵呵，就慢慢积分吧。J = ∫ sec^5(x) dx= ∫ sec³x d(tanx)= sec³x tanx - ∫ tanx 3sec²x secx tanx dx= sec³x tanx - 3∫ sec³x tan²x dx= sec³x tanx - 3∫ sec³x (sec²x - 1) dx= sec³x tanx - 3J + 3∫ sec³x dxJ = 1/4 sec³x tanx + 3/4 ∫ sec³x dxK = ∫ sec³x dx= ∫ secx dtanx= secx tanx - ∫ tanx secx tanx dx= secx tanx - ∫ secx (sec²x - 1) dx= secx tanx - K + ∫ secx dxK = 1/2 secx tanx + 1/2 ln|secx + tanx|原式 = 1/4 sec³x tanx + 3/4 [1/2 secx tanx + 1/2 ln|secx + tanx|]- [secx tanx + ln|secx + tanx|] + ln|secx + tanx| + C= (1/8) secx tanx (2sec²x - 5) + (3/8) ln|secx + tanx| + C