Optimal. Leaf size=156 \[ \frac{2 a^4 \sin (c+d x)}{117 d e^5 (e \sec (c+d x))^{3/2}}-\frac{4 i \left (a^4+i a^4 \tan (c+d x)\right )}{117 d e^2 (e \sec (c+d x))^{9/2}}+\frac{2 a^4 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{39 d e^6 \sqrt{\cos (c+d x)} \sqrt{e \sec (c+d x)}}-\frac{4 i a (a+i a \tan (c+d x))^3}{13 d (e \sec (c+d x))^{13/2}} \]
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Rubi [A] time = 0.146584, antiderivative size = 156, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3496, 3769, 3771, 2639} \[ \frac{2 a^4 \sin (c+d x)}{117 d e^5 (e \sec (c+d x))^{3/2}}-\frac{4 i \left (a^4+i a^4 \tan (c+d x)\right )}{117 d e^2 (e \sec (c+d x))^{9/2}}+\frac{2 a^4 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{39 d e^6 \sqrt{\cos (c+d x)} \sqrt{e \sec (c+d x)}}-\frac{4 i a (a+i a \tan (c+d x))^3}{13 d (e \sec (c+d x))^{13/2}} \]
Antiderivative was successfully verified.
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Rule 3496
Rule 3769
Rule 3771
Rule 2639
Rubi steps
\begin{align*} \int \frac{(a+i a \tan (c+d x))^4}{(e \sec (c+d x))^{13/2}} \, dx &=-\frac{4 i a (a+i a \tan (c+d x))^3}{13 d (e \sec (c+d x))^{13/2}}+\frac{a^2 \int \frac{(a+i a \tan (c+d x))^2}{(e \sec (c+d x))^{9/2}} \, dx}{13 e^2}\\ &=-\frac{4 i a (a+i a \tan (c+d x))^3}{13 d (e \sec (c+d x))^{13/2}}-\frac{4 i \left (a^4+i a^4 \tan (c+d x)\right )}{117 d e^2 (e \sec (c+d x))^{9/2}}+\frac{\left (5 a^4\right ) \int \frac{1}{(e \sec (c+d x))^{5/2}} \, dx}{117 e^4}\\ &=\frac{2 a^4 \sin (c+d x)}{117 d e^5 (e \sec (c+d x))^{3/2}}-\frac{4 i a (a+i a \tan (c+d x))^3}{13 d (e \sec (c+d x))^{13/2}}-\frac{4 i \left (a^4+i a^4 \tan (c+d x)\right )}{117 d e^2 (e \sec (c+d x))^{9/2}}+\frac{a^4 \int \frac{1}{\sqrt{e \sec (c+d x)}} \, dx}{39 e^6}\\ &=\frac{2 a^4 \sin (c+d x)}{117 d e^5 (e \sec (c+d x))^{3/2}}-\frac{4 i a (a+i a \tan (c+d x))^3}{13 d (e \sec (c+d x))^{13/2}}-\frac{4 i \left (a^4+i a^4 \tan (c+d x)\right )}{117 d e^2 (e \sec (c+d x))^{9/2}}+\frac{a^4 \int \sqrt{\cos (c+d x)} \, dx}{39 e^6 \sqrt{\cos (c+d x)} \sqrt{e \sec (c+d x)}}\\ &=\frac{2 a^4 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 a^4 \sin (c+d x)}{117 d e^5 (e \sec (c+d x))^{3/2}}-\frac{4 i a (a+i a \tan (c+d x))^3}{13 d (e \sec (c+d x))^{13/2}}-\frac{4 i \left (a^4+i a^4 \tan (c+d x)\right )}{117 d e^2 (e \sec (c+d x))^{9/2}}\\ \end{align*}
Mathematica [C] time = 6.92916, size = 450, normalized size = 2.88 \[ \frac{\sec ^3(c+d x) (a+i a \tan (c+d x))^4 \left (\left (-\frac{59 \sin (c)}{468}-\frac{59}{468} i \cos (c)\right ) \cos (3 d x)+\left (\frac{37 \sin (c)}{468}-\frac{37}{468} i \cos (c)\right ) \cos (5 d x)+\left (\frac{1}{52} \sin (3 c)-\frac{1}{52} i \cos (3 c)\right ) \cos (7 d x)+\left (\frac{55}{468} \cos (3 c)-\frac{55}{468} i \sin (3 c)\right ) \sin (d x)+\left (\frac{59 \cos (c)}{468}-\frac{59}{468} i \sin (c)\right ) \sin (3 d x)+\left (\frac{37 \cos (c)}{468}+\frac{37}{468} i \sin (c)\right ) \sin (5 d x)+\left (\frac{1}{52} \cos (3 c)+\frac{1}{52} i \sin (3 c)\right ) \sin (7 d x)+\csc (c) (24 \cos (c)+31 i \sin (c)) \left (-\frac{1}{468} \cos (3 c)+\frac{1}{468} i \sin (3 c)\right ) \cos (d x)\right )}{d (\cos (d x)+i \sin (d x))^4 (e \sec (c+d x))^{13/2}}-\frac{2 i \sqrt{2} e^{-i (3 c+d x)} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \sec ^{\frac{5}{2}}(c+d x) \left (\left (-1+e^{2 i c}\right ) e^{2 i d x} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{3}{4},\frac{7}{4},-e^{2 i (c+d x)}\right )-3 \sqrt{1+e^{2 i (c+d x)}}\right ) (a+i a \tan (c+d x))^4}{117 \left (-1+e^{2 i c}\right ) d (\cos (d x)+i \sin (d x))^4 (e \sec (c+d x))^{13/2}} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.357, size = 380, normalized size = 2.4 \begin{align*}{\frac{2\,{a}^{4}}{117\,d\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{7}} \left ( -72\,i\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{7}-72\, \left ( \cos \left ( dx+c \right ) \right ) ^{8}+52\,i\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{5}+3\,i{\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 ) \cos \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}}}-3\,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 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 ) +88\, \left ( \cos \left ( dx+c \right ) \right ) ^{6}+3\,i{\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 ) \sqrt{ \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}-3\,i\sin \left ( dx+c \right ){\it EllipticE} \left ({\frac{i \left ( \cos \left ( dx+c \right ) -1 \right ) }{\sin \left ( dx+c \right ) }},i \right ) \sqrt{ \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}-17\, \left ( \cos \left ( dx+c \right ) \right ) ^{4}-2\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}+3\,\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(-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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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\sqrt{2}{\left (-9 i \, a^{4} e^{\left (8 i \, d x + 8 i \, c\right )} + 9 i \, a^{4} e^{\left (7 i \, d x + 7 i \, c\right )} - 37 i \, a^{4} e^{\left (6 i \, d x + 6 i \, c\right )} + 37 i \, a^{4} e^{\left (5 i \, d x + 5 i \, c\right )} - 59 i \, a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} + 59 i \, a^{4} e^{\left (3 i \, d x + 3 i \, c\right )} - 55 i \, a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} + 31 i \, a^{4} e^{\left (i \, d x + i \, c\right )} - 24 i \, a^{4}\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 )} + 468 \,{\left (d e^{7} e^{\left (i \, d x + i \, c\right )} - d e^{7}\right )}{\rm integral}\left (\frac{\sqrt{2}{\left (-i \, a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} - 2 i \, a^{4} e^{\left (i \, d x + i \, c\right )} - i \, a^{4}\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 )}{468 \,{\left (d e^{7} e^{\left (i \, d x + i \, c\right )} - d e^{7}\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 )}^{4}}{\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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