Optimal. Leaf size=92 \[ -\frac{2 \cos ^{\frac{5}{2}}(c+d x) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 a^2 d (e \cos (c+d x))^{5/2}}+\frac{4 i \cos ^2(c+d x)}{3 d \left (a^2+i a^2 \tan (c+d x)\right ) (e \cos (c+d x))^{5/2}} \]
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Rubi [A] time = 0.14994, antiderivative size = 92, 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 = {3515, 3500, 3771, 2641} \[ -\frac{2 \cos ^{\frac{5}{2}}(c+d x) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 a^2 d (e \cos (c+d x))^{5/2}}+\frac{4 i \cos ^2(c+d x)}{3 d \left (a^2+i a^2 \tan (c+d x)\right ) (e \cos (c+d x))^{5/2}} \]
Antiderivative was successfully verified.
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Rule 3515
Rule 3500
Rule 3771
Rule 2641
Rubi steps
\begin{align*} \int \frac{1}{(e \cos (c+d x))^{5/2} (a+i a \tan (c+d x))^2} \, dx &=\frac{\int \frac{(e \sec (c+d x))^{5/2}}{(a+i a \tan (c+d x))^2} \, dx}{(e \cos (c+d x))^{5/2} (e \sec (c+d x))^{5/2}}\\ &=\frac{4 i \cos ^2(c+d x)}{3 d (e \cos (c+d x))^{5/2} \left (a^2+i a^2 \tan (c+d x)\right )}-\frac{e^2 \int \sqrt{e \sec (c+d x)} \, dx}{3 a^2 (e \cos (c+d x))^{5/2} (e \sec (c+d x))^{5/2}}\\ &=\frac{4 i \cos ^2(c+d x)}{3 d (e \cos (c+d x))^{5/2} \left (a^2+i a^2 \tan (c+d x)\right )}-\frac{\cos ^{\frac{5}{2}}(c+d x) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{3 a^2 (e \cos (c+d x))^{5/2}}\\ &=-\frac{2 \cos ^{\frac{5}{2}}(c+d x) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 a^2 d (e \cos (c+d x))^{5/2}}+\frac{4 i \cos ^2(c+d x)}{3 d (e \cos (c+d x))^{5/2} \left (a^2+i a^2 \tan (c+d x)\right )}\\ \end{align*}
Mathematica [A] time = 0.368917, size = 116, normalized size = 1.26 \[ \frac{2 \sqrt{\cos (c+d x)} (\cos (d x)+i \sin (d x))^2 \left (2 \sqrt{\cos (c+d x)} (\sin (c-d x)-i \cos (c-d x))+(\cos (2 c)+i \sin (2 c)) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )\right )}{3 a^2 d (\tan (c+d x)-i)^2 (e \cos (c+d x))^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 2.291, size = 170, normalized size = 1.9 \begin{align*}{\frac{2}{3\,{e}^{2}{a}^{2}d} \left ( 8\,i \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}-8\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}\cos \left ( 1/2\,dx+c/2 \right ) -8\,i \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}+\sqrt{ \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}}\sqrt{2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}{\it EllipticF} \left ( \cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) ,\sqrt{2} \right ) +4\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}\cos \left ( 1/2\,dx+c/2 \right ) +2\,i\sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}{\frac{1}{\sqrt{-2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}e+e}}}} \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 (3 \, a^{2} d e^{3} e^{\left (i \, d x + i \, c\right )}{\rm integral}\left (\frac{2 i \, \sqrt{\frac{1}{2}} \sqrt{e e^{\left (2 i \, d x + 2 i \, c\right )} + e} e^{\left (-\frac{1}{2} i \, d x - \frac{1}{2} i \, c\right )}}{3 \,{\left (a^{2} d e^{3} e^{\left (2 i \, d x + 2 i \, c\right )} + a^{2} d e^{3}\right )}}, x\right ) + 4 i \, \sqrt{\frac{1}{2}} \sqrt{e e^{\left (2 i \, d x + 2 i \, c\right )} + e} e^{\left (-\frac{1}{2} i \, d x - \frac{1}{2} i \, c\right )}\right )} e^{\left (-i \, d x - i \, c\right )}}{3 \, a^{2} d e^{3}} \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 \cos \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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