Optimal. Leaf size=90 \[ -\frac{2 i a (e \cos (c+d x))^{5/2}}{5 d}+\frac{6 a E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) (e \cos (c+d x))^{5/2}}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 a \tan (c+d x) (e \cos (c+d x))^{5/2}}{5 d} \]
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Rubi [A] time = 0.111353, 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, 2639} \[ -\frac{2 i a (e \cos (c+d x))^{5/2}}{5 d}+\frac{6 a E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) (e \cos (c+d x))^{5/2}}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 a \tan (c+d x) (e \cos (c+d x))^{5/2}}{5 d} \]
Antiderivative was successfully verified.
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Rule 3515
Rule 3486
Rule 3769
Rule 3771
Rule 2639
Rubi steps
\begin{align*} \int (e \cos (c+d x))^{5/2} (a+i a \tan (c+d x)) \, dx &=\left ((e \cos (c+d x))^{5/2} (e \sec (c+d x))^{5/2}\right ) \int \frac{a+i a \tan (c+d x)}{(e \sec (c+d x))^{5/2}} \, dx\\ &=-\frac{2 i a (e \cos (c+d x))^{5/2}}{5 d}+\left (a (e \cos (c+d x))^{5/2} (e \sec (c+d x))^{5/2}\right ) \int \frac{1}{(e \sec (c+d x))^{5/2}} \, dx\\ &=-\frac{2 i a (e \cos (c+d x))^{5/2}}{5 d}+\frac{2 a (e \cos (c+d x))^{5/2} \tan (c+d x)}{5 d}+\frac{\left (3 a (e \cos (c+d x))^{5/2} (e \sec (c+d x))^{5/2}\right ) \int \frac{1}{\sqrt{e \sec (c+d x)}} \, dx}{5 e^2}\\ &=-\frac{2 i a (e \cos (c+d x))^{5/2}}{5 d}+\frac{2 a (e \cos (c+d x))^{5/2} \tan (c+d x)}{5 d}+\frac{\left (3 a (e \cos (c+d x))^{5/2}\right ) \int \sqrt{\cos (c+d x)} \, dx}{5 \cos ^{\frac{5}{2}}(c+d x)}\\ &=-\frac{2 i a (e \cos (c+d x))^{5/2}}{5 d}+\frac{6 a (e \cos (c+d x))^{5/2} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 a (e \cos (c+d x))^{5/2} \tan (c+d x)}{5 d}\\ \end{align*}
Mathematica [C] time = 12.4758, size = 387, normalized size = 4.3 \[ \frac{(\cos (d x)-i \sin (d x)) (a+i a \tan (c+d x)) (e \cos (c+d x))^{5/2} \left (\frac{2 \sqrt{2} (\cot (c)-i) e^{-i d x} \left (e^{2 i d x} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right )+3 e^{2 i (c+d x)}-3 \sqrt{1-i e^{i (c+d x)}} \sqrt{e^{i (c+d x)} \left (e^{i (c+d x)}-i\right )} F\left (\left .\sin ^{-1}\left (\sqrt{\sin (c+d x)-i \cos (c+d x)}\right )\right |-1\right )+3 \sqrt{1-i e^{i (c+d x)}} \sqrt{e^{i (c+d x)} \left (e^{i (c+d x)}-i\right )} E\left (\left .\sin ^{-1}\left (\sqrt{\sin (c+d x)-i \cos (c+d x)}\right )\right |-1\right )+3\right )}{5 \sqrt{e^{-i (c+d x)} \left (1+e^{2 i (c+d x)}\right )}}+\frac{2}{5} \sin (c) \sqrt{\cos (c+d x)} \left ((1-i \cot (c)) \cos (2 d x)+\cot (c) (\sin (2 d x)+5 i)-6 \cot ^2(c)+i \sin (2 d x)-1\right )\right )}{2 d \cos ^{\frac{3}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Maple [B] time = 2.346, size = 205, normalized size = 2.3 \begin{align*}{\frac{2\,a{e}^{3}}{5\,d} \left ( 8\,i \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{7}+8\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{6}\cos \left ( 1/2\,dx+c/2 \right ) -12\,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 ) +6\,i \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}+3\,{\it EllipticE} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) \sqrt{2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}\sqrt{ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}+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{5}{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{\sqrt{\frac{1}{2}}{\left (-i \, a e^{2} e^{\left (3 i \, d x + 3 i \, c\right )} + i \, a e^{2} e^{\left (2 i \, d x + 2 i \, c\right )} - 7 i \, a e^{2} e^{\left (i \, d x + i \, c\right )} - 5 i \, a e^{2}\right )} \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 )} + 5 \,{\left (d e^{\left (i \, d x + i \, c\right )} - d\right )}{\rm integral}\left (\frac{\sqrt{\frac{1}{2}}{\left (-6 i \, a e^{2} e^{\left (2 i \, d x + 2 i \, c\right )} - 12 i \, a e^{2} e^{\left (i \, d x + i \, c\right )} - 6 i \, a e^{2}\right )} \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 )}}{5 \,{\left (d e^{\left (4 i \, d x + 4 i \, c\right )} - 2 \, d e^{\left (3 i \, d x + 3 i \, c\right )} + 2 \, d e^{\left (2 i \, d x + 2 i \, c\right )} - 2 \, d e^{\left (i \, d x + i \, c\right )} + d\right )}}, x\right )}{5 \,{\left (d e^{\left (i \, d x + i \, c\right )} - d\right )}} \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{5}{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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