Optimal. Leaf size=155 \[ \frac{14 a^3 \sin (c+d x)}{117 d e^5 (e \sec (c+d x))^{3/2}}-\frac{28 i \left (a^3+i a^3 \tan (c+d x)\right )}{117 d e^2 (e \sec (c+d x))^{9/2}}+\frac{14 a^3 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{39 d e^6 \sqrt{\cos (c+d x)} \sqrt{e \sec (c+d x)}}-\frac{2 i (a+i a \tan (c+d x))^3}{13 d (e \sec (c+d x))^{13/2}} \]
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Rubi [A] time = 0.147979, antiderivative size = 155, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {3497, 3496, 3769, 3771, 2639} \[ \frac{14 a^3 \sin (c+d x)}{117 d e^5 (e \sec (c+d x))^{3/2}}-\frac{28 i \left (a^3+i a^3 \tan (c+d x)\right )}{117 d e^2 (e \sec (c+d x))^{9/2}}+\frac{14 a^3 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{39 d e^6 \sqrt{\cos (c+d x)} \sqrt{e \sec (c+d x)}}-\frac{2 i (a+i a \tan (c+d x))^3}{13 d (e \sec (c+d x))^{13/2}} \]
Antiderivative was successfully verified.
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Rule 3497
Rule 3496
Rule 3769
Rule 3771
Rule 2639
Rubi steps
\begin{align*} \int \frac{(a+i a \tan (c+d x))^3}{(e \sec (c+d x))^{13/2}} \, dx &=-\frac{2 i (a+i a \tan (c+d x))^3}{13 d (e \sec (c+d x))^{13/2}}+\frac{(7 a) \int \frac{(a+i a \tan (c+d x))^2}{(e \sec (c+d x))^{9/2}} \, dx}{13 e^2}\\ &=-\frac{2 i (a+i a \tan (c+d x))^3}{13 d (e \sec (c+d x))^{13/2}}-\frac{28 i \left (a^3+i a^3 \tan (c+d x)\right )}{117 d e^2 (e \sec (c+d x))^{9/2}}+\frac{\left (35 a^3\right ) \int \frac{1}{(e \sec (c+d x))^{5/2}} \, dx}{117 e^4}\\ &=\frac{14 a^3 \sin (c+d x)}{117 d e^5 (e \sec (c+d x))^{3/2}}-\frac{2 i (a+i a \tan (c+d x))^3}{13 d (e \sec (c+d x))^{13/2}}-\frac{28 i \left (a^3+i a^3 \tan (c+d x)\right )}{117 d e^2 (e \sec (c+d x))^{9/2}}+\frac{\left (7 a^3\right ) \int \frac{1}{\sqrt{e \sec (c+d x)}} \, dx}{39 e^6}\\ &=\frac{14 a^3 \sin (c+d x)}{117 d e^5 (e \sec (c+d x))^{3/2}}-\frac{2 i (a+i a \tan (c+d x))^3}{13 d (e \sec (c+d x))^{13/2}}-\frac{28 i \left (a^3+i a^3 \tan (c+d x)\right )}{117 d e^2 (e \sec (c+d x))^{9/2}}+\frac{\left (7 a^3\right ) \int \sqrt{\cos (c+d x)} \, dx}{39 e^6 \sqrt{\cos (c+d x)} \sqrt{e \sec (c+d x)}}\\ &=\frac{14 a^3 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{39 d e^6 \sqrt{\cos (c+d x)} \sqrt{e \sec (c+d x)}}+\frac{14 a^3 \sin (c+d x)}{117 d e^5 (e \sec (c+d x))^{3/2}}-\frac{2 i (a+i a \tan (c+d x))^3}{13 d (e \sec (c+d x))^{13/2}}-\frac{28 i \left (a^3+i a^3 \tan (c+d x)\right )}{117 d e^2 (e \sec (c+d x))^{9/2}}\\ \end{align*}
Mathematica [C] time = 6.36057, size = 155, normalized size = 1. \[ -\frac{a^3 e^{-4 i (c+d x)} (\tan (c+d x)-i)^3 \left (112 e^{2 i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{3}{4},\frac{7}{4},-e^{2 i (c+d x)}\right )-34 e^{2 i (c+d x)}+124 e^{4 i (c+d x)}+50 e^{6 i (c+d x)}+9 e^{8 i (c+d x)}-117\right )}{936 d e^4 (e \sec (c+d x))^{5/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.37, size = 380, normalized size = 2.5 \begin{align*}{\frac{2\,{a}^{3}}{117\,d \left ( \cos \left ( dx+c \right ) \right ) ^{7}\sin \left ( dx+c \right ) } \left ( -36\,i \left ( \cos \left ( dx+c \right ) \right ) ^{7}\sin \left ( dx+c \right ) -36\, \left ( \cos \left ( dx+c \right ) \right ) ^{8}+13\,i \left ( \cos \left ( dx+c \right ) \right ) ^{5}\sin \left ( dx+c \right ) +21\,i\sqrt{ \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}{\it EllipticF} \left ({\frac{i \left ( \cos \left ( dx+c \right ) -1 \right ) }{\sin \left ( dx+c \right ) }},i \right ) \cos \left ( dx+c \right ) \sin \left ( dx+c \right ) -21\,i{\it EllipticE} \left ({\frac{i \left ( \cos \left ( dx+c \right ) -1 \right ) }{\sin \left ( dx+c \right ) }},i \right ) \cos \left ( dx+c \right ) \sin \left ( dx+c \right ) \sqrt{ \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}+31\, \left ( \cos \left ( dx+c \right ) \right ) ^{6}+21\,i\sqrt{ \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}{\it EllipticF} \left ({\frac{i \left ( \cos \left ( dx+c \right ) -1 \right ) }{\sin \left ( dx+c \right ) }},i \right ) \sin \left ( dx+c \right ) -21\,i{\it EllipticE} \left ({\frac{i \left ( \cos \left ( dx+c \right ) -1 \right ) }{\sin \left ( dx+c \right ) }},i \right ) \sin \left ( dx+c \right ) \sqrt{ \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}-2\, \left ( \cos \left ( dx+c \right ) \right ) ^{4}-14\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}+21\,\cos \left ( dx+c \right ) \right ) \left ({\frac{e}{\cos \left ( dx+c \right ) }} \right ) ^{-{\frac{13}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{3}}{\left (e \sec \left (d x + c\right )\right )^{\frac{13}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\sqrt{2}{\left (-9 i \, a^{3} e^{\left (9 i \, d x + 9 i \, c\right )} + 9 i \, a^{3} e^{\left (8 i \, d x + 8 i \, c\right )} - 50 i \, a^{3} e^{\left (7 i \, d x + 7 i \, c\right )} + 50 i \, a^{3} e^{\left (6 i \, d x + 6 i \, c\right )} - 124 i \, a^{3} e^{\left (5 i \, d x + 5 i \, c\right )} + 124 i \, a^{3} e^{\left (4 i \, d x + 4 i \, c\right )} - 302 i \, a^{3} e^{\left (3 i \, d x + 3 i \, c\right )} - 34 i \, a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} - 219 i \, a^{3} e^{\left (i \, d x + i \, c\right )} - 117 i \, a^{3}\right )} \sqrt{\frac{e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (\frac{1}{2} i \, d x + \frac{1}{2} i \, c\right )} + 936 \,{\left (d e^{7} e^{\left (2 i \, d x + 2 i \, c\right )} - d e^{7} e^{\left (i \, d x + i \, c\right )}\right )}{\rm integral}\left (\frac{\sqrt{2}{\left (-7 i \, a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} - 14 i \, a^{3} e^{\left (i \, d x + i \, c\right )} - 7 i \, a^{3}\right )} \sqrt{\frac{e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (\frac{1}{2} i \, d x + \frac{1}{2} i \, c\right )}}{39 \,{\left (d e^{7} e^{\left (3 i \, d x + 3 i \, c\right )} - 2 \, d e^{7} e^{\left (2 i \, d x + 2 i \, c\right )} + d e^{7} e^{\left (i \, d x + i \, c\right )}\right )}}, x\right )}{936 \,{\left (d e^{7} e^{\left (2 i \, d x + 2 i \, c\right )} - d e^{7} e^{\left (i \, d x + i \, c\right )}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{3}}{\left (e \sec \left (d x + c\right )\right )^{\frac{13}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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