Optimal. Leaf size=90 \[ -\frac{2 i a (e \cos (c+d x))^{3/2}}{3 d}+\frac{2 a F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) (e \cos (c+d x))^{3/2}}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a \tan (c+d x) (e \cos (c+d x))^{3/2}}{3 d} \]
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Rubi [A] time = 0.109314, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {3515, 3486, 3769, 3771, 2641} \[ -\frac{2 i a (e \cos (c+d x))^{3/2}}{3 d}+\frac{2 a F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) (e \cos (c+d x))^{3/2}}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a \tan (c+d x) (e \cos (c+d x))^{3/2}}{3 d} \]
Antiderivative was successfully verified.
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Rule 3515
Rule 3486
Rule 3769
Rule 3771
Rule 2641
Rubi steps
\begin{align*} \int (e \cos (c+d x))^{3/2} (a+i a \tan (c+d x)) \, dx &=\left ((e \cos (c+d x))^{3/2} (e \sec (c+d x))^{3/2}\right ) \int \frac{a+i a \tan (c+d x)}{(e \sec (c+d x))^{3/2}} \, dx\\ &=-\frac{2 i a (e \cos (c+d x))^{3/2}}{3 d}+\left (a (e \cos (c+d x))^{3/2} (e \sec (c+d x))^{3/2}\right ) \int \frac{1}{(e \sec (c+d x))^{3/2}} \, dx\\ &=-\frac{2 i a (e \cos (c+d x))^{3/2}}{3 d}+\frac{2 a (e \cos (c+d x))^{3/2} \tan (c+d x)}{3 d}+\frac{\left (a (e \cos (c+d x))^{3/2} (e \sec (c+d x))^{3/2}\right ) \int \sqrt{e \sec (c+d x)} \, dx}{3 e^2}\\ &=-\frac{2 i a (e \cos (c+d x))^{3/2}}{3 d}+\frac{2 a (e \cos (c+d x))^{3/2} \tan (c+d x)}{3 d}+\frac{\left (a (e \cos (c+d x))^{3/2}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{3 \cos ^{\frac{3}{2}}(c+d x)}\\ &=-\frac{2 i a (e \cos (c+d x))^{3/2}}{3 d}+\frac{2 a (e \cos (c+d x))^{3/2} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a (e \cos (c+d x))^{3/2} \tan (c+d x)}{3 d}\\ \end{align*}
Mathematica [A] time = 0.342929, size = 100, normalized size = 1.11 \[ \frac{2 a e \sqrt{\cos (c+d x)} (\tan (c+d x)-i) (\cos (d x)-i \sin (d x)) \sqrt{e \cos (c+d x)} \left (\sqrt{\cos (c+d x)} (\cos (d x)+i \sin (d x))+(\sin (c)+i \cos (c)) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )\right )}{3 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 2.457, size = 168, normalized size = 1.9 \begin{align*} -{\frac{2\,a{e}^{2}}{3\,d} \left ( 4\,i \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}+4\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}\cos \left ( 1/2\,dx+c/2 \right ) -4\,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 ) -2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}\cos \left ( 1/2\,dx+c/2 \right ) +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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (e \cos \left (d x + c\right )\right )^{\frac{3}{2}}{\left (i \, a \tan \left (d x + c\right ) + a\right )}\,{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{-2 i \, \sqrt{\frac{1}{2}} \sqrt{e e^{\left (2 i \, d x + 2 i \, c\right )} + e} a e e^{\left (\frac{1}{2} i \, d x + \frac{1}{2} i \, c\right )} + 3 \, d{\rm integral}\left (-\frac{2 i \, \sqrt{\frac{1}{2}} \sqrt{e e^{\left (2 i \, d x + 2 i \, c\right )} + e} a e e^{\left (-\frac{1}{2} i \, d x - \frac{1}{2} i \, c\right )}}{3 \,{\left (d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}}, x\right )}{3 \, d} \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 \left (e \cos \left (d x + c\right )\right )^{\frac{3}{2}}{\left (i \, a \tan \left (d x + c\right ) + a\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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