Optimal. Leaf size=118 \[ -\frac{2 e^2 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 a^3 d \sqrt{e \cos (c+d x)}}+\frac{2 e \sqrt{e \cos (c+d x)}}{21 d \left (a^3 \sin (c+d x)+a^3\right )}-\frac{4 e \sqrt{e \cos (c+d x)}}{7 a d (a \sin (c+d x)+a)^2} \]
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Rubi [A] time = 0.133818, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {2680, 2683, 2642, 2641} \[ -\frac{2 e^2 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 a^3 d \sqrt{e \cos (c+d x)}}+\frac{2 e \sqrt{e \cos (c+d x)}}{21 d \left (a^3 \sin (c+d x)+a^3\right )}-\frac{4 e \sqrt{e \cos (c+d x)}}{7 a d (a \sin (c+d x)+a)^2} \]
Antiderivative was successfully verified.
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Rule 2680
Rule 2683
Rule 2642
Rule 2641
Rubi steps
\begin{align*} \int \frac{(e \cos (c+d x))^{3/2}}{(a+a \sin (c+d x))^3} \, dx &=-\frac{4 e \sqrt{e \cos (c+d x)}}{7 a d (a+a \sin (c+d x))^2}-\frac{e^2 \int \frac{1}{\sqrt{e \cos (c+d x)} (a+a \sin (c+d x))} \, dx}{7 a^2}\\ &=-\frac{4 e \sqrt{e \cos (c+d x)}}{7 a d (a+a \sin (c+d x))^2}+\frac{2 e \sqrt{e \cos (c+d x)}}{21 d \left (a^3+a^3 \sin (c+d x)\right )}-\frac{e^2 \int \frac{1}{\sqrt{e \cos (c+d x)}} \, dx}{21 a^3}\\ &=-\frac{4 e \sqrt{e \cos (c+d x)}}{7 a d (a+a \sin (c+d x))^2}+\frac{2 e \sqrt{e \cos (c+d x)}}{21 d \left (a^3+a^3 \sin (c+d x)\right )}-\frac{\left (e^2 \sqrt{\cos (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{21 a^3 \sqrt{e \cos (c+d x)}}\\ &=-\frac{2 e^2 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 a^3 d \sqrt{e \cos (c+d x)}}-\frac{4 e \sqrt{e \cos (c+d x)}}{7 a d (a+a \sin (c+d x))^2}+\frac{2 e \sqrt{e \cos (c+d x)}}{21 d \left (a^3+a^3 \sin (c+d x)\right )}\\ \end{align*}
Mathematica [C] time = 0.0659085, size = 66, normalized size = 0.56 \[ -\frac{(e \cos (c+d x))^{5/2} \, _2F_1\left (\frac{5}{4},\frac{11}{4};\frac{9}{4};\frac{1}{2} (1-\sin (c+d x))\right )}{5\ 2^{3/4} a^3 d e (\sin (c+d x)+1)^{5/4}} \]
Antiderivative was successfully verified.
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Maple [B] time = 1.905, size = 401, normalized size = 3.4 \begin{align*}{\frac{2\,{e}^{2}}{21\,{a}^{3}d} \left ( 8\,\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 ) \sqrt{2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1} \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{6}-12\,\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 ) \sqrt{2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1} \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}+8\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{6}\cos \left ( 1/2\,dx+c/2 \right ) +6\,\sqrt{2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}{\it EllipticF} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) \sqrt{ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}} \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-8\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}\cos \left ( 1/2\,dx+c/2 \right ) -28\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{5}-\sqrt{2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}\sqrt{ \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}}{\it EllipticF} \left ( \cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) ,\sqrt{2} \right ) -22\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}\cos \left ( 1/2\,dx+c/2 \right ) +28\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{3}+5\,\sin \left ( 1/2\,dx+c/2 \right ) \right ) \left ( 8\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{6}-12\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}+6\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1 \right ) ^{-1} \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 \frac{\left (e \cos \left (d x + c\right )\right )^{\frac{3}{2}}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{3}}\,{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 (-\frac{\sqrt{e \cos \left (d x + c\right )} e \cos \left (d x + c\right )}{3 \, a^{3} \cos \left (d x + c\right )^{2} - 4 \, a^{3} +{\left (a^{3} \cos \left (d x + c\right )^{2} - 4 \, a^{3}\right )} \sin \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 \frac{\left (e \cos \left (d x + c\right )\right )^{\frac{3}{2}}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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