Optimal. Leaf size=152 \[ \frac{14 e^5 \sin (c+d x) \sqrt{e \sec (c+d x)}}{5 a^2 d}+\frac{14 e^3 \sin (c+d x) (e \sec (c+d x))^{5/2}}{15 a^2 d}-\frac{4 i e^2 (e \sec (c+d x))^{7/2}}{3 d \left (a^2+i a^2 \tan (c+d x)\right )}-\frac{14 e^6 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 a^2 d \sqrt{\cos (c+d x)} \sqrt{e \sec (c+d x)}} \]
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Rubi [A] time = 0.107401, antiderivative size = 152, 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, 3768, 3771, 2639} \[ \frac{14 e^5 \sin (c+d x) \sqrt{e \sec (c+d x)}}{5 a^2 d}+\frac{14 e^3 \sin (c+d x) (e \sec (c+d x))^{5/2}}{15 a^2 d}-\frac{4 i e^2 (e \sec (c+d x))^{7/2}}{3 d \left (a^2+i a^2 \tan (c+d x)\right )}-\frac{14 e^6 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 a^2 d \sqrt{\cos (c+d x)} \sqrt{e \sec (c+d x)}} \]
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))^{11/2}}{(a+i a \tan (c+d x))^2} \, dx &=-\frac{4 i e^2 (e \sec (c+d x))^{7/2}}{3 d \left (a^2+i a^2 \tan (c+d x)\right )}+\frac{\left (7 e^2\right ) \int (e \sec (c+d x))^{7/2} \, dx}{3 a^2}\\ &=\frac{14 e^3 (e \sec (c+d x))^{5/2} \sin (c+d x)}{15 a^2 d}-\frac{4 i e^2 (e \sec (c+d x))^{7/2}}{3 d \left (a^2+i a^2 \tan (c+d x)\right )}+\frac{\left (7 e^4\right ) \int (e \sec (c+d x))^{3/2} \, dx}{5 a^2}\\ &=\frac{14 e^5 \sqrt{e \sec (c+d x)} \sin (c+d x)}{5 a^2 d}+\frac{14 e^3 (e \sec (c+d x))^{5/2} \sin (c+d x)}{15 a^2 d}-\frac{4 i e^2 (e \sec (c+d x))^{7/2}}{3 d \left (a^2+i a^2 \tan (c+d x)\right )}-\frac{\left (7 e^6\right ) \int \frac{1}{\sqrt{e \sec (c+d x)}} \, dx}{5 a^2}\\ &=\frac{14 e^5 \sqrt{e \sec (c+d x)} \sin (c+d x)}{5 a^2 d}+\frac{14 e^3 (e \sec (c+d x))^{5/2} \sin (c+d x)}{15 a^2 d}-\frac{4 i e^2 (e \sec (c+d x))^{7/2}}{3 d \left (a^2+i a^2 \tan (c+d x)\right )}-\frac{\left (7 e^6\right ) \int \sqrt{\cos (c+d x)} \, dx}{5 a^2 \sqrt{\cos (c+d x)} \sqrt{e \sec (c+d x)}}\\ &=-\frac{14 e^6 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 a^2 d \sqrt{\cos (c+d x)} \sqrt{e \sec (c+d x)}}+\frac{14 e^5 \sqrt{e \sec (c+d x)} \sin (c+d x)}{5 a^2 d}+\frac{14 e^3 (e \sec (c+d x))^{5/2} \sin (c+d x)}{15 a^2 d}-\frac{4 i e^2 (e \sec (c+d x))^{7/2}}{3 d \left (a^2+i a^2 \tan (c+d x)\right )}\\ \end{align*}
Mathematica [C] time = 0.870055, size = 123, normalized size = 0.81 \[ \frac{2 i e^5 e^{i (c+d x)} \left (7 \left (1+e^{2 i (c+d x)}\right )^{5/2} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{3}{4},\frac{7}{4},-e^{2 i (c+d x)}\right )-56 e^{2 i (c+d x)}-21 e^{4 i (c+d x)}-47\right ) \sqrt{e \sec (c+d x)}}{15 a^2 d \left (1+e^{2 i (c+d x)}\right )^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.284, size = 374, normalized size = 2.5 \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}}{15\,{a}^{2}d \left ( \sin \left ( dx+c \right ) \right ) ^{5}} \left ( 21\,i \left ( \cos \left ( dx+c \right ) \right ) ^{3}\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 \left ( \cos \left ( dx+c \right ) \right ) ^{3}\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}}}{\it EllipticE} \left ({\frac{i \left ( \cos \left ( dx+c \right ) -1 \right ) }{\sin \left ( dx+c \right ) }},i \right ) +21\,i \left ( \cos \left ( dx+c \right ) \right ) ^{2}\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 \left ( \cos \left ( dx+c \right ) \right ) ^{2}\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 ) +21\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}+10\,i\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) -24\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}+3 \right ) \left ({\frac{e}{\cos \left ( dx+c \right ) }} \right ) ^{{\frac{11}{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 (-42 i \, e^{5} e^{\left (5 i \, d x + 5 i \, c\right )} - 112 i \, e^{5} e^{\left (3 i \, d x + 3 i \, c\right )} - 94 i \, e^{5} 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 )} + 15 \,{\left (a^{2} d e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, a^{2} d e^{\left (2 i \, d x + 2 i \, c\right )} + a^{2} d\right )}{\rm integral}\left (\frac{7 i \, \sqrt{2} e^{5} \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 )}}{5 \, a^{2} d}, x\right )}{15 \,{\left (a^{2} d e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, 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{11}{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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