Optimal. Leaf size=183 \[ \frac{22 e^7 \sin (c+d x) \sqrt{e \sec (c+d x)}}{15 a^2 d}+\frac{22 e^5 \sin (c+d x) (e \sec (c+d x))^{5/2}}{45 a^2 d}+\frac{22 e^3 \sin (c+d x) (e \sec (c+d x))^{9/2}}{63 a^2 d}-\frac{4 i e^2 (e \sec (c+d x))^{11/2}}{7 d \left (a^2+i a^2 \tan (c+d x)\right )}-\frac{22 e^8 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 a^2 d \sqrt{\cos (c+d x)} \sqrt{e \sec (c+d x)}} \]
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Rubi [A] time = 0.127274, antiderivative size = 183, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3500, 3768, 3771, 2639} \[ \frac{22 e^7 \sin (c+d x) \sqrt{e \sec (c+d x)}}{15 a^2 d}+\frac{22 e^5 \sin (c+d x) (e \sec (c+d x))^{5/2}}{45 a^2 d}+\frac{22 e^3 \sin (c+d x) (e \sec (c+d x))^{9/2}}{63 a^2 d}-\frac{4 i e^2 (e \sec (c+d x))^{11/2}}{7 d \left (a^2+i a^2 \tan (c+d x)\right )}-\frac{22 e^8 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 a^2 d \sqrt{\cos (c+d x)} \sqrt{e \sec (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 3500
Rule 3768
Rule 3771
Rule 2639
Rubi steps
\begin{align*} \int \frac{(e \sec (c+d x))^{15/2}}{(a+i a \tan (c+d x))^2} \, dx &=-\frac{4 i e^2 (e \sec (c+d x))^{11/2}}{7 d \left (a^2+i a^2 \tan (c+d x)\right )}+\frac{\left (11 e^2\right ) \int (e \sec (c+d x))^{11/2} \, dx}{7 a^2}\\ &=\frac{22 e^3 (e \sec (c+d x))^{9/2} \sin (c+d x)}{63 a^2 d}-\frac{4 i e^2 (e \sec (c+d x))^{11/2}}{7 d \left (a^2+i a^2 \tan (c+d x)\right )}+\frac{\left (11 e^4\right ) \int (e \sec (c+d x))^{7/2} \, dx}{9 a^2}\\ &=\frac{22 e^5 (e \sec (c+d x))^{5/2} \sin (c+d x)}{45 a^2 d}+\frac{22 e^3 (e \sec (c+d x))^{9/2} \sin (c+d x)}{63 a^2 d}-\frac{4 i e^2 (e \sec (c+d x))^{11/2}}{7 d \left (a^2+i a^2 \tan (c+d x)\right )}+\frac{\left (11 e^6\right ) \int (e \sec (c+d x))^{3/2} \, dx}{15 a^2}\\ &=\frac{22 e^7 \sqrt{e \sec (c+d x)} \sin (c+d x)}{15 a^2 d}+\frac{22 e^5 (e \sec (c+d x))^{5/2} \sin (c+d x)}{45 a^2 d}+\frac{22 e^3 (e \sec (c+d x))^{9/2} \sin (c+d x)}{63 a^2 d}-\frac{4 i e^2 (e \sec (c+d x))^{11/2}}{7 d \left (a^2+i a^2 \tan (c+d x)\right )}-\frac{\left (11 e^8\right ) \int \frac{1}{\sqrt{e \sec (c+d x)}} \, dx}{15 a^2}\\ &=\frac{22 e^7 \sqrt{e \sec (c+d x)} \sin (c+d x)}{15 a^2 d}+\frac{22 e^5 (e \sec (c+d x))^{5/2} \sin (c+d x)}{45 a^2 d}+\frac{22 e^3 (e \sec (c+d x))^{9/2} \sin (c+d x)}{63 a^2 d}-\frac{4 i e^2 (e \sec (c+d x))^{11/2}}{7 d \left (a^2+i a^2 \tan (c+d x)\right )}-\frac{\left (11 e^8\right ) \int \sqrt{\cos (c+d x)} \, dx}{15 a^2 \sqrt{\cos (c+d x)} \sqrt{e \sec (c+d x)}}\\ &=-\frac{22 e^8 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 a^2 d \sqrt{\cos (c+d x)} \sqrt{e \sec (c+d x)}}+\frac{22 e^7 \sqrt{e \sec (c+d x)} \sin (c+d x)}{15 a^2 d}+\frac{22 e^5 (e \sec (c+d x))^{5/2} \sin (c+d x)}{45 a^2 d}+\frac{22 e^3 (e \sec (c+d x))^{9/2} \sin (c+d x)}{63 a^2 d}-\frac{4 i e^2 (e \sec (c+d x))^{11/2}}{7 d \left (a^2+i a^2 \tan (c+d x)\right )}\\ \end{align*}
Mathematica [C] time = 2.28387, size = 302, normalized size = 1.65 \[ \frac{(\cos (d x)+i \sin (d x))^2 (e \sec (c+d x))^{15/2} \left (\frac{22 i \sqrt{2} e^{3 i c-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \left (\left (-1+e^{2 i c}\right ) e^{2 i d x} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{3}{4},\frac{7}{4},-e^{2 i (c+d x)}\right )-3 \sqrt{1+e^{2 i (c+d x)}}\right )}{-1+e^{2 i c}}+\frac{1}{56} \csc (c) (\cos (2 c)+i \sin (2 c)) \sec ^{\frac{9}{2}}(c+d x) (-720 i \sin (2 c+d x)+1050 \cos (2 c+d x)+1078 \cos (2 c+3 d x)+77 \cos (4 c+3 d x)+231 \cos (4 c+5 d x)+720 i \sin (d x)+1260 \cos (d x))\right )}{45 d \sec ^{\frac{11}{2}}(c+d x) (a+i a \tan (c+d x))^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.312, size = 384, normalized size = 2.1 \begin{align*}{\frac{2\, \left ( \cos \left ( dx+c \right ) +1 \right ) ^{2} \left ( \cos \left ( dx+c \right ) -1 \right ) ^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{315\,{a}^{2}d \left ( \sin \left ( dx+c \right ) \right ) ^{5}} \left ( 231\,i \left ( \cos \left ( dx+c \right ) \right ) ^{5}\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 ) \sin \left ( dx+c \right ) -231\,i\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{5}\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 ) +231\,i\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{4}\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 ) -231\,i\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{4}\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 ) -231\, \left ( \cos \left ( dx+c \right ) \right ) ^{5}+154\, \left ( \cos \left ( dx+c \right ) \right ) ^{4}-90\,i\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) +112\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}-35 \right ) \left ({\frac{e}{\cos \left ( dx+c \right ) }} \right ) ^{{\frac{15}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \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 (-462 i \, e^{7} e^{\left (9 i \, d x + 9 i \, c\right )} - 2156 i \, e^{7} e^{\left (7 i \, d x + 7 i \, c\right )} - 3960 i \, e^{7} e^{\left (5 i \, d x + 5 i \, c\right )} - 3540 i \, e^{7} e^{\left (3 i \, d x + 3 i \, c\right )} - 154 i \, e^{7} e^{\left (i \, d x + i \, c\right )}\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 )} + 315 \,{\left (a^{2} d e^{\left (8 i \, d x + 8 i \, c\right )} + 4 \, a^{2} d e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, a^{2} d e^{\left (4 i \, d x + 4 i \, c\right )} + 4 \, a^{2} d e^{\left (2 i \, d x + 2 i \, c\right )} + a^{2} d\right )}{\rm integral}\left (\frac{11 i \, \sqrt{2} e^{7} \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 )}}{15 \, a^{2} d}, x\right )}{315 \,{\left (a^{2} d e^{\left (8 i \, d x + 8 i \, c\right )} + 4 \, a^{2} d e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, a^{2} d e^{\left (4 i \, d x + 4 i \, c\right )} + 4 \, a^{2} d e^{\left (2 i \, d x + 2 i \, c\right )} + a^{2} 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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (e \sec \left (d x + c\right )\right )^{\frac{15}{2}}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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