Optimal. Leaf size=124 \[ -\frac{4 i \left (a^3+i a^3 \tan (c+d x)\right )}{21 d e^2 (e \sec (c+d x))^{3/2}}-\frac{2 a^3 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{e \sec (c+d x)}}{21 d e^4}-\frac{2 i (a+i a \tan (c+d x))^3}{7 d (e \sec (c+d x))^{7/2}} \]
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Rubi [A] time = 0.126506, antiderivative size = 124, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3497, 3496, 3771, 2641} \[ -\frac{4 i \left (a^3+i a^3 \tan (c+d x)\right )}{21 d e^2 (e \sec (c+d x))^{3/2}}-\frac{2 a^3 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{e \sec (c+d x)}}{21 d e^4}-\frac{2 i (a+i a \tan (c+d x))^3}{7 d (e \sec (c+d x))^{7/2}} \]
Antiderivative was successfully verified.
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Rule 3497
Rule 3496
Rule 3771
Rule 2641
Rubi steps
\begin{align*} \int \frac{(a+i a \tan (c+d x))^3}{(e \sec (c+d x))^{7/2}} \, dx &=-\frac{2 i (a+i a \tan (c+d x))^3}{7 d (e \sec (c+d x))^{7/2}}+\frac{a \int \frac{(a+i a \tan (c+d x))^2}{(e \sec (c+d x))^{3/2}} \, dx}{7 e^2}\\ &=-\frac{2 i (a+i a \tan (c+d x))^3}{7 d (e \sec (c+d x))^{7/2}}-\frac{4 i \left (a^3+i a^3 \tan (c+d x)\right )}{21 d e^2 (e \sec (c+d x))^{3/2}}-\frac{a^3 \int \sqrt{e \sec (c+d x)} \, dx}{21 e^4}\\ &=-\frac{2 i (a+i a \tan (c+d x))^3}{7 d (e \sec (c+d x))^{7/2}}-\frac{4 i \left (a^3+i a^3 \tan (c+d x)\right )}{21 d e^2 (e \sec (c+d x))^{3/2}}-\frac{\left (a^3 \sqrt{\cos (c+d x)} \sqrt{e \sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{21 e^4}\\ &=-\frac{2 a^3 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{e \sec (c+d x)}}{21 d e^4}-\frac{2 i (a+i a \tan (c+d x))^3}{7 d (e \sec (c+d x))^{7/2}}-\frac{4 i \left (a^3+i a^3 \tan (c+d x)\right )}{21 d e^2 (e \sec (c+d x))^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.932838, size = 133, normalized size = 1.07 \[ -\frac{a^3 \sqrt{e \sec (c+d x)} (\cos (2 c+5 d x)+i \sin (2 c+5 d x)) \left (-\sin (2 (c+d x))+5 i \cos (2 (c+d x))+2 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) (\cos (2 (c+d x))-i \sin (2 (c+d x)))+5 i\right )}{21 d e^4 (\cos (d x)+i \sin (d x))^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.231, size = 199, normalized size = 1.6 \begin{align*} -{\frac{2\,{a}^{3}}{21\,d \left ( \cos \left ( dx+c \right ) \right ) ^{4}} \left ( 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 ) +12\,i \left ( \cos \left ( dx+c \right ) \right ) ^{4}+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 ) -12\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}\sin \left ( dx+c \right ) -7\,i \left ( \cos \left ( dx+c \right ) \right ) ^{2}+\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) \right ) \left ({\frac{e}{\cos \left ( dx+c \right ) }} \right ) ^{-{\frac{7}{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 )}^{3}}{\left (e \sec \left (d x + c\right )\right )^{\frac{7}{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{21 \, d e^{4}{\rm integral}\left (\frac{i \, \sqrt{2} a^{3} \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 )}}{21 \, d e^{4}}, x\right ) + \sqrt{2}{\left (-3 i \, a^{3} e^{\left (4 i \, d x + 4 i \, c\right )} - 5 i \, a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} - 2 i \, a^{3}\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 )}}{21 \, d e^{4}} \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 )}^{3}}{\left (e \sec \left (d x + c\right )\right )^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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