Optimal. Leaf size=147 \[ \frac{10 a^2 \sin (c+d x)}{33 d e^5 \sqrt{e \sec (c+d x)}}+\frac{2 a^2 \sin (c+d x)}{11 d e^3 (e \sec (c+d x))^{5/2}}+\frac{10 a^2 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{e \sec (c+d x)}}{33 d e^6}-\frac{4 i \left (a^2+i a^2 \tan (c+d x)\right )}{11 d (e \sec (c+d x))^{11/2}} \]
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Rubi [A] time = 0.105537, antiderivative size = 147, 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, 2641} \[ \frac{10 a^2 \sin (c+d x)}{33 d e^5 \sqrt{e \sec (c+d x)}}+\frac{2 a^2 \sin (c+d x)}{11 d e^3 (e \sec (c+d x))^{5/2}}+\frac{10 a^2 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{e \sec (c+d x)}}{33 d e^6}-\frac{4 i \left (a^2+i a^2 \tan (c+d x)\right )}{11 d (e \sec (c+d x))^{11/2}} \]
Antiderivative was successfully verified.
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Rule 3496
Rule 3769
Rule 3771
Rule 2641
Rubi steps
\begin{align*} \int \frac{(a+i a \tan (c+d x))^2}{(e \sec (c+d x))^{11/2}} \, dx &=-\frac{4 i \left (a^2+i a^2 \tan (c+d x)\right )}{11 d (e \sec (c+d x))^{11/2}}+\frac{\left (7 a^2\right ) \int \frac{1}{(e \sec (c+d x))^{7/2}} \, dx}{11 e^2}\\ &=\frac{2 a^2 \sin (c+d x)}{11 d e^3 (e \sec (c+d x))^{5/2}}-\frac{4 i \left (a^2+i a^2 \tan (c+d x)\right )}{11 d (e \sec (c+d x))^{11/2}}+\frac{\left (5 a^2\right ) \int \frac{1}{(e \sec (c+d x))^{3/2}} \, dx}{11 e^4}\\ &=\frac{2 a^2 \sin (c+d x)}{11 d e^3 (e \sec (c+d x))^{5/2}}+\frac{10 a^2 \sin (c+d x)}{33 d e^5 \sqrt{e \sec (c+d x)}}-\frac{4 i \left (a^2+i a^2 \tan (c+d x)\right )}{11 d (e \sec (c+d x))^{11/2}}+\frac{\left (5 a^2\right ) \int \sqrt{e \sec (c+d x)} \, dx}{33 e^6}\\ &=\frac{2 a^2 \sin (c+d x)}{11 d e^3 (e \sec (c+d x))^{5/2}}+\frac{10 a^2 \sin (c+d x)}{33 d e^5 \sqrt{e \sec (c+d x)}}-\frac{4 i \left (a^2+i a^2 \tan (c+d x)\right )}{11 d (e \sec (c+d x))^{11/2}}+\frac{\left (5 a^2 \sqrt{\cos (c+d x)} \sqrt{e \sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{33 e^6}\\ &=\frac{10 a^2 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{e \sec (c+d x)}}{33 d e^6}+\frac{2 a^2 \sin (c+d x)}{11 d e^3 (e \sec (c+d x))^{5/2}}+\frac{10 a^2 \sin (c+d x)}{33 d e^5 \sqrt{e \sec (c+d x)}}-\frac{4 i \left (a^2+i a^2 \tan (c+d x)\right )}{11 d (e \sec (c+d x))^{11/2}}\\ \end{align*}
Mathematica [A] time = 1.24247, size = 155, normalized size = 1.05 \[ \frac{a^2 \sqrt{e \sec (c+d x)} (\cos (2 (c+2 d x))+i \sin (2 (c+2 d x))) \left (-6 \sin (2 (c+d x))+7 \sin (4 (c+d x))-24 i \cos (2 (c+d x))+4 i \cos (4 (c+d x))+40 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) (\cos (2 (c+d x))-i \sin (2 (c+d x)))-28 i\right )}{132 d e^6 (\cos (d x)+i \sin (d x))^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.283, size = 205, normalized size = 1.4 \begin{align*} -{\frac{2\,{a}^{2}}{33\,d \left ( \cos \left ( dx+c \right ) \right ) ^{6}} \left ( 6\,i \left ( \cos \left ( dx+c \right ) \right ) ^{6}-6\, \left ( \cos \left ( dx+c \right ) \right ) ^{5}\sin \left ( dx+c \right ) -5\,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 ) -5\,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 ) -3\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}\sin \left ( dx+c \right ) -5\,\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2}}{\left (e \sec \left (d x + c\right )\right )^{\frac{11}{2}}}\,{d x} \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 (264 \, d e^{6} e^{\left (2 i \, d x + 2 i \, c\right )}{\rm integral}\left (-\frac{5 i \, \sqrt{2} a^{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 )}}{33 \, d e^{6}}, x\right ) + \sqrt{2}{\left (-3 i \, a^{2} e^{\left (8 i \, d x + 8 i \, c\right )} - 18 i \, a^{2} e^{\left (6 i \, d x + 6 i \, c\right )} - 56 i \, a^{2} e^{\left (4 i \, d x + 4 i \, c\right )} - 30 i \, a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + 11 i \, a^{2}\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 )}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{264 \, d e^{6}} \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 )}^{2}}{\left (e \sec \left (d x + c\right )\right )^{\frac{11}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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