Optimal. Leaf size=119 \[ -\frac{7 i \sec ^5(c+d x)}{15 a^3 d}+\frac{7 \tanh ^{-1}(\sin (c+d x))}{8 a^3 d}+\frac{7 \tan (c+d x) \sec ^3(c+d x)}{12 a^3 d}+\frac{7 \tan (c+d x) \sec (c+d x)}{8 a^3 d}-\frac{2 i \sec ^7(c+d x)}{3 a d (a+i a \tan (c+d x))^2} \]
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Rubi [A] time = 0.114983, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3500, 3501, 3768, 3770} \[ -\frac{7 i \sec ^5(c+d x)}{15 a^3 d}+\frac{7 \tanh ^{-1}(\sin (c+d x))}{8 a^3 d}+\frac{7 \tan (c+d x) \sec ^3(c+d x)}{12 a^3 d}+\frac{7 \tan (c+d x) \sec (c+d x)}{8 a^3 d}-\frac{2 i \sec ^7(c+d x)}{3 a d (a+i a \tan (c+d x))^2} \]
Antiderivative was successfully verified.
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Rule 3500
Rule 3501
Rule 3768
Rule 3770
Rubi steps
\begin{align*} \int \frac{\sec ^9(c+d x)}{(a+i a \tan (c+d x))^3} \, dx &=-\frac{2 i \sec ^7(c+d x)}{3 a d (a+i a \tan (c+d x))^2}+\frac{7 \int \frac{\sec ^7(c+d x)}{a+i a \tan (c+d x)} \, dx}{3 a^2}\\ &=-\frac{7 i \sec ^5(c+d x)}{15 a^3 d}-\frac{2 i \sec ^7(c+d x)}{3 a d (a+i a \tan (c+d x))^2}+\frac{7 \int \sec ^5(c+d x) \, dx}{3 a^3}\\ &=-\frac{7 i \sec ^5(c+d x)}{15 a^3 d}+\frac{7 \sec ^3(c+d x) \tan (c+d x)}{12 a^3 d}-\frac{2 i \sec ^7(c+d x)}{3 a d (a+i a \tan (c+d x))^2}+\frac{7 \int \sec ^3(c+d x) \, dx}{4 a^3}\\ &=-\frac{7 i \sec ^5(c+d x)}{15 a^3 d}+\frac{7 \sec (c+d x) \tan (c+d x)}{8 a^3 d}+\frac{7 \sec ^3(c+d x) \tan (c+d x)}{12 a^3 d}-\frac{2 i \sec ^7(c+d x)}{3 a d (a+i a \tan (c+d x))^2}+\frac{7 \int \sec (c+d x) \, dx}{8 a^3}\\ &=\frac{7 \tanh ^{-1}(\sin (c+d x))}{8 a^3 d}-\frac{7 i \sec ^5(c+d x)}{15 a^3 d}+\frac{7 \sec (c+d x) \tan (c+d x)}{8 a^3 d}+\frac{7 \sec ^3(c+d x) \tan (c+d x)}{12 a^3 d}-\frac{2 i \sec ^7(c+d x)}{3 a d (a+i a \tan (c+d x))^2}\\ \end{align*}
Mathematica [A] time = 0.37638, size = 113, normalized size = 0.95 \[ \frac{\sec ^8(c+d x) (\cos (3 (c+d x))+i \sin (3 (c+d x))) \left (-150 i \sin (2 (c+d x))+105 i \sin (4 (c+d x))+640 \cos (2 (c+d x))+1680 i \cos ^5(c+d x) \tanh ^{-1}\left (\cos (c) \tan \left (\frac{d x}{2}\right )+\sin (c)\right )+448\right )}{960 a^3 d (\tan (c+d x)-i)^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.102, size = 430, normalized size = 3.6 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.08053, size = 460, normalized size = 3.87 \begin{align*} \frac{\frac{16 \,{\left (-\frac{15 i \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{320 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{390 i \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac{400 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{960 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac{390 i \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac{360 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac{15 i \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - 136\right )}}{-120 i \, a^{3} + \frac{600 i \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac{1200 i \, a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{1200 i \, a^{3} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac{600 i \, a^{3} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac{120 i \, a^{3} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}}} + \frac{7 \, \log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a^{3}} - \frac{7 \, \log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a^{3}}}{8 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.59923, size = 836, normalized size = 7.03 \begin{align*} \frac{105 \,{\left (e^{\left (10 i \, d x + 10 i \, c\right )} + 5 \, e^{\left (8 i \, d x + 8 i \, c\right )} + 10 \, e^{\left (6 i \, d x + 6 i \, c\right )} + 10 \, e^{\left (4 i \, d x + 4 i \, c\right )} + 5 \, e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )} \log \left (e^{\left (i \, d x + i \, c\right )} + i\right ) - 105 \,{\left (e^{\left (10 i \, d x + 10 i \, c\right )} + 5 \, e^{\left (8 i \, d x + 8 i \, c\right )} + 10 \, e^{\left (6 i \, d x + 6 i \, c\right )} + 10 \, e^{\left (4 i \, d x + 4 i \, c\right )} + 5 \, e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )} \log \left (e^{\left (i \, d x + i \, c\right )} - i\right ) - 210 i \, e^{\left (9 i \, d x + 9 i \, c\right )} - 980 i \, e^{\left (7 i \, d x + 7 i \, c\right )} - 1792 i \, e^{\left (5 i \, d x + 5 i \, c\right )} - 1580 i \, e^{\left (3 i \, d x + 3 i \, c\right )} + 210 i \, e^{\left (i \, d x + i \, c\right )}}{120 \,{\left (a^{3} d e^{\left (10 i \, d x + 10 i \, c\right )} + 5 \, a^{3} d e^{\left (8 i \, d x + 8 i \, c\right )} + 10 \, a^{3} d e^{\left (6 i \, d x + 6 i \, c\right )} + 10 \, a^{3} d e^{\left (4 i \, d x + 4 i \, c\right )} + 5 \, a^{3} d e^{\left (2 i \, d x + 2 i \, c\right )} + a^{3} d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.23278, size = 224, normalized size = 1.88 \begin{align*} \frac{\frac{105 \, \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right )}{a^{3}} - \frac{105 \, \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right )}{a^{3}} + \frac{2 \,{\left (15 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 360 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} - 390 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 960 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 400 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 390 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 320 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 15 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 136 i\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1\right )}^{5} a^{3}}}{120 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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