Optimal. Leaf size=150 \[ \frac{4 i e^2}{13 d \left (a^2+i a^2 \tan (c+d x)\right ) (e \sec (c+d x))^{9/2}}+\frac{42 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{65 a^2 d e^2 \sqrt{\cos (c+d x)} \sqrt{e \sec (c+d x)}}+\frac{2 e \sin (c+d x)}{13 a^2 d (e \sec (c+d x))^{7/2}}+\frac{14 \sin (c+d x)}{65 a^2 d e (e \sec (c+d x))^{3/2}} \]
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Rubi [A] time = 0.109295, antiderivative size = 150, 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 = {3500, 3769, 3771, 2639} \[ \frac{4 i e^2}{13 d \left (a^2+i a^2 \tan (c+d x)\right ) (e \sec (c+d x))^{9/2}}+\frac{42 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{65 a^2 d e^2 \sqrt{\cos (c+d x)} \sqrt{e \sec (c+d x)}}+\frac{2 e \sin (c+d x)}{13 a^2 d (e \sec (c+d x))^{7/2}}+\frac{14 \sin (c+d x)}{65 a^2 d e (e \sec (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 3500
Rule 3769
Rule 3771
Rule 2639
Rubi steps
\begin{align*} \int \frac{1}{(e \sec (c+d x))^{5/2} (a+i a \tan (c+d x))^2} \, dx &=\frac{4 i e^2}{13 d (e \sec (c+d x))^{9/2} \left (a^2+i a^2 \tan (c+d x)\right )}+\frac{\left (9 e^2\right ) \int \frac{1}{(e \sec (c+d x))^{9/2}} \, dx}{13 a^2}\\ &=\frac{2 e \sin (c+d x)}{13 a^2 d (e \sec (c+d x))^{7/2}}+\frac{4 i e^2}{13 d (e \sec (c+d x))^{9/2} \left (a^2+i a^2 \tan (c+d x)\right )}+\frac{7 \int \frac{1}{(e \sec (c+d x))^{5/2}} \, dx}{13 a^2}\\ &=\frac{2 e \sin (c+d x)}{13 a^2 d (e \sec (c+d x))^{7/2}}+\frac{14 \sin (c+d x)}{65 a^2 d e (e \sec (c+d x))^{3/2}}+\frac{4 i e^2}{13 d (e \sec (c+d x))^{9/2} \left (a^2+i a^2 \tan (c+d x)\right )}+\frac{21 \int \frac{1}{\sqrt{e \sec (c+d x)}} \, dx}{65 a^2 e^2}\\ &=\frac{2 e \sin (c+d x)}{13 a^2 d (e \sec (c+d x))^{7/2}}+\frac{14 \sin (c+d x)}{65 a^2 d e (e \sec (c+d x))^{3/2}}+\frac{4 i e^2}{13 d (e \sec (c+d x))^{9/2} \left (a^2+i a^2 \tan (c+d x)\right )}+\frac{21 \int \sqrt{\cos (c+d x)} \, dx}{65 a^2 e^2 \sqrt{\cos (c+d x)} \sqrt{e \sec (c+d x)}}\\ &=\frac{42 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{65 a^2 d e^2 \sqrt{\cos (c+d x)} \sqrt{e \sec (c+d x)}}+\frac{2 e \sin (c+d x)}{13 a^2 d (e \sec (c+d x))^{7/2}}+\frac{14 \sin (c+d x)}{65 a^2 d e (e \sec (c+d x))^{3/2}}+\frac{4 i e^2}{13 d (e \sec (c+d x))^{9/2} \left (a^2+i a^2 \tan (c+d x)\right )}\\ \end{align*}
Mathematica [C] time = 2.02242, size = 149, normalized size = 0.99 \[ \frac{(\cos (2 (c+d x))-i \sin (2 (c+d x))) \left (-\frac{224 i e^{4 i (c+d x)} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{3}{4},\frac{7}{4},-e^{2 i (c+d x)}\right )}{\sqrt{1+e^{2 i (c+d x)}}}-356 \sin (2 (c+d x))+18 \sin (4 (c+d x))+416 i \cos (2 (c+d x))-8 i \cos (4 (c+d x))+88 i\right )}{520 a^2 d e^2 \sqrt{e \sec (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.723, size = 386, normalized size = 2.6 \begin{align*}{\frac{2\, \left ( \cos \left ( dx+c \right ) \right ) ^{2} \left ( \cos \left ( dx+c \right ) -1 \right ) ^{2} \left ( \cos \left ( dx+c \right ) +1 \right ) ^{2}}{65\,{a}^{2}d{e}^{5} \left ( \sin \left ( dx+c \right ) \right ) ^{5}} \left ({\frac{e}{\cos \left ( dx+c \right ) }} \right ) ^{{\frac{5}{2}}} \left ( 10\,i \left ( \cos \left ( dx+c \right ) \right ) ^{7}\sin \left ( dx+c \right ) -10\, \left ( \cos \left ( dx+c \right ) \right ) ^{8}+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\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 ) +5\, \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 ) } \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} \sqrt{\frac{e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}{\left (-13 i \, e^{\left (11 i \, d x + 11 i \, c\right )} + 13 i \, e^{\left (10 i \, d x + 10 i \, c\right )} - 299 i \, e^{\left (9 i \, d x + 9 i \, c\right )} - 373 i \, e^{\left (8 i \, d x + 8 i \, c\right )} - 198 i \, e^{\left (7 i \, d x + 7 i \, c\right )} - 474 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 118 i \, e^{\left (5 i \, d x + 5 i \, c\right )} - 118 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 35 i \, e^{\left (3 i \, d x + 3 i \, c\right )} - 35 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 5 i \, e^{\left (i \, d x + i \, c\right )} - 5 i\right )} e^{\left (\frac{1}{2} i \, d x + \frac{1}{2} i \, c\right )} + 1040 \,{\left (a^{2} d e^{3} e^{\left (8 i \, d x + 8 i \, c\right )} - a^{2} d e^{3} e^{\left (7 i \, d x + 7 i \, c\right )}\right )}{\rm integral}\left (\frac{\sqrt{2} \sqrt{\frac{e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}{\left (-21 i \, e^{\left (2 i \, d x + 2 i \, c\right )} - 42 i \, e^{\left (i \, d x + i \, c\right )} - 21 i\right )} e^{\left (\frac{1}{2} i \, d x + \frac{1}{2} i \, c\right )}}{65 \,{\left (a^{2} d e^{3} e^{\left (3 i \, d x + 3 i \, c\right )} - 2 \, a^{2} d e^{3} e^{\left (2 i \, d x + 2 i \, c\right )} + a^{2} d e^{3} e^{\left (i \, d x + i \, c\right )}\right )}}, x\right )}{1040 \,{\left (a^{2} d e^{3} e^{\left (8 i \, d x + 8 i \, c\right )} - a^{2} d e^{3} e^{\left (7 i \, d x + 7 i \, c\right )}\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{1}{\left (e \sec \left (d x + c\right )\right )^{\frac{5}{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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