Optimal. Leaf size=95 \[ \frac{2 a e^2 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d \sqrt{e \cos (c+d x)}}-\frac{2 a (e \cos (c+d x))^{5/2}}{5 d e}+\frac{2 a e \sin (c+d x) \sqrt{e \cos (c+d x)}}{3 d} \]
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Rubi [A] time = 0.0649123, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {2669, 2635, 2642, 2641} \[ \frac{2 a e^2 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d \sqrt{e \cos (c+d x)}}-\frac{2 a (e \cos (c+d x))^{5/2}}{5 d e}+\frac{2 a e \sin (c+d x) \sqrt{e \cos (c+d x)}}{3 d} \]
Antiderivative was successfully verified.
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Rule 2669
Rule 2635
Rule 2642
Rule 2641
Rubi steps
\begin{align*} \int (e \cos (c+d x))^{3/2} (a+a \sin (c+d x)) \, dx &=-\frac{2 a (e \cos (c+d x))^{5/2}}{5 d e}+a \int (e \cos (c+d x))^{3/2} \, dx\\ &=-\frac{2 a (e \cos (c+d x))^{5/2}}{5 d e}+\frac{2 a e \sqrt{e \cos (c+d x)} \sin (c+d x)}{3 d}+\frac{1}{3} \left (a e^2\right ) \int \frac{1}{\sqrt{e \cos (c+d x)}} \, dx\\ &=-\frac{2 a (e \cos (c+d x))^{5/2}}{5 d e}+\frac{2 a e \sqrt{e \cos (c+d x)} \sin (c+d x)}{3 d}+\frac{\left (a e^2 \sqrt{\cos (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{3 \sqrt{e \cos (c+d x)}}\\ &=-\frac{2 a (e \cos (c+d x))^{5/2}}{5 d e}+\frac{2 a e^2 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d \sqrt{e \cos (c+d x)}}+\frac{2 a e \sqrt{e \cos (c+d x)} \sin (c+d x)}{3 d}\\ \end{align*}
Mathematica [A] time = 0.363893, size = 75, normalized size = 0.79 \[ -\frac{a (e \cos (c+d x))^{3/2} \left (\sqrt{\cos (c+d x)} (-10 \sin (c+d x)+3 \cos (2 (c+d x))+3)-10 F\left (\left .\frac{1}{2} (c+d x)\right |2\right )\right )}{15 d \cos ^{\frac{3}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.352, size = 179, normalized size = 1.9 \begin{align*} -{\frac{2\,a{e}^{2}}{15\,d} \left ( -24\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{7}+20\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}\cos \left ( 1/2\,dx+c/2 \right ) +36\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{5}+5\,\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}}{\it EllipticF} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) -10\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}\cos \left ( 1/2\,dx+c/2 \right ) -18\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{3}+3\,\sin \left ( 1/2\,dx+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 (a \sin \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*}{\rm integral}\left ({\left (a e \cos \left (d x + c\right ) \sin \left (d x + c\right ) + a e \cos \left (d x + c\right )\right )} \sqrt{e \cos \left (d x + c\right )}, x\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{3}{2}}{\left (a \sin \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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