Optimal. Leaf size=186 \[ \frac{12 i e^2}{55 d \left (a^3+i a^3 \tan (c+d x)\right ) (e \sec (c+d x))^{7/2}}+\frac{2 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{e \sec (c+d x)}}{11 a^3 d e^2}+\frac{6 e \sin (c+d x)}{55 a^3 d (e \sec (c+d x))^{5/2}}+\frac{2 \sin (c+d x)}{11 a^3 d e \sqrt{e \sec (c+d x)}}+\frac{2 i}{15 d (a+i a \tan (c+d x))^3 (e \sec (c+d x))^{3/2}} \]
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Rubi [A] time = 0.18169, antiderivative size = 186, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {3502, 3500, 3769, 3771, 2641} \[ \frac{12 i e^2}{55 d \left (a^3+i a^3 \tan (c+d x)\right ) (e \sec (c+d x))^{7/2}}+\frac{2 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{e \sec (c+d x)}}{11 a^3 d e^2}+\frac{6 e \sin (c+d x)}{55 a^3 d (e \sec (c+d x))^{5/2}}+\frac{2 \sin (c+d x)}{11 a^3 d e \sqrt{e \sec (c+d x)}}+\frac{2 i}{15 d (a+i a \tan (c+d x))^3 (e \sec (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 3502
Rule 3500
Rule 3769
Rule 3771
Rule 2641
Rubi steps
\begin{align*} \int \frac{1}{(e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^3} \, dx &=\frac{2 i}{15 d (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^3}+\frac{3 \int \frac{1}{(e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^2} \, dx}{5 a}\\ &=\frac{2 i}{15 d (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^3}+\frac{12 i e^2}{55 d (e \sec (c+d x))^{7/2} \left (a^3+i a^3 \tan (c+d x)\right )}+\frac{\left (21 e^2\right ) \int \frac{1}{(e \sec (c+d x))^{7/2}} \, dx}{55 a^3}\\ &=\frac{6 e \sin (c+d x)}{55 a^3 d (e \sec (c+d x))^{5/2}}+\frac{2 i}{15 d (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^3}+\frac{12 i e^2}{55 d (e \sec (c+d x))^{7/2} \left (a^3+i a^3 \tan (c+d x)\right )}+\frac{3 \int \frac{1}{(e \sec (c+d x))^{3/2}} \, dx}{11 a^3}\\ &=\frac{6 e \sin (c+d x)}{55 a^3 d (e \sec (c+d x))^{5/2}}+\frac{2 \sin (c+d x)}{11 a^3 d e \sqrt{e \sec (c+d x)}}+\frac{2 i}{15 d (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^3}+\frac{12 i e^2}{55 d (e \sec (c+d x))^{7/2} \left (a^3+i a^3 \tan (c+d x)\right )}+\frac{\int \sqrt{e \sec (c+d x)} \, dx}{11 a^3 e^2}\\ &=\frac{6 e \sin (c+d x)}{55 a^3 d (e \sec (c+d x))^{5/2}}+\frac{2 \sin (c+d x)}{11 a^3 d e \sqrt{e \sec (c+d x)}}+\frac{2 i}{15 d (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^3}+\frac{12 i e^2}{55 d (e \sec (c+d x))^{7/2} \left (a^3+i a^3 \tan (c+d x)\right )}+\frac{\left (\sqrt{\cos (c+d x)} \sqrt{e \sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{11 a^3 e^2}\\ &=\frac{2 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{e \sec (c+d x)}}{11 a^3 d e^2}+\frac{6 e \sin (c+d x)}{55 a^3 d (e \sec (c+d x))^{5/2}}+\frac{2 \sin (c+d x)}{11 a^3 d e \sqrt{e \sec (c+d x)}}+\frac{2 i}{15 d (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^3}+\frac{12 i e^2}{55 d (e \sec (c+d x))^{7/2} \left (a^3+i a^3 \tan (c+d x)\right )}\\ \end{align*}
Mathematica [A] time = 0.582232, size = 151, normalized size = 0.81 \[ \frac{\sec ^5(c+d x) \left (-114 i \sin (c+d x)-81 i \sin (3 (c+d x))+33 i \sin (5 (c+d x))-332 \cos (c+d x)-154 \cos (3 (c+d x))+22 \cos (5 (c+d x))+240 i \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) (\cos (3 (c+d x))+i \sin (3 (c+d x)))\right )}{1320 a^3 d (\tan (c+d x)-i)^3 (e \sec (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.411, size = 261, normalized size = 1.4 \begin{align*}{\frac{2\, \left ( \cos \left ( dx+c \right ) -1 \right ) ^{2} \left ( \cos \left ( dx+c \right ) +1 \right ) ^{2}\cos \left ( dx+c \right ) }{165\,d{a}^{3}{e}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{4}} \left ({\frac{e}{\cos \left ( dx+c \right ) }} \right ) ^{{\frac{3}{2}}} \left ( 44\,i \left ( \cos \left ( dx+c \right ) \right ) ^{8}+44\,\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{7}-15\,i \left ( \cos \left ( dx+c \right ) \right ) ^{6}+7\, \left ( \cos \left ( dx+c \right ) \right ) ^{5}\sin \left ( dx+c \right ) +15\,i\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}}}{\it EllipticF} \left ({\frac{i \left ( \cos \left ( dx+c \right ) -1 \right ) }{\sin \left ( dx+c \right ) }},i \right ) +9\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}\sin \left ( dx+c \right ) +15\,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 ) +15\,\cos \left ( dx+c \right ) \sin \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{{\left (2640 \, a^{3} d e^{2} e^{\left (8 i \, d x + 8 i \, c\right )}{\rm integral}\left (-\frac{i \, \sqrt{2} \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 )}}{11 \, a^{3} d e^{2}}, x\right ) + \sqrt{2} \sqrt{\frac{e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}{\left (-55 i \, e^{\left (10 i \, d x + 10 i \, c\right )} + 235 i \, e^{\left (8 i \, d x + 8 i \, c\right )} + 446 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 218 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 73 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 11 i\right )} e^{\left (\frac{1}{2} i \, d x + \frac{1}{2} i \, c\right )}\right )} e^{\left (-8 i \, d x - 8 i \, c\right )}}{2640 \, a^{3} d e^{2}} \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{3}{2}}{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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