Optimal. Leaf size=85 \[ \frac{4 a^4 (e \cos (c+d x))^{3/2}}{d e^3 \left (a^2-a^2 \sin (c+d x)\right )}-\frac{6 a^2 E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{e \cos (c+d x)}}{d e^2 \sqrt{\cos (c+d x)}} \]
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Rubi [A] time = 0.132392, antiderivative size = 85, 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 = {2670, 2680, 2640, 2639} \[ \frac{4 a^4 (e \cos (c+d x))^{3/2}}{d e^3 \left (a^2-a^2 \sin (c+d x)\right )}-\frac{6 a^2 E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{e \cos (c+d x)}}{d e^2 \sqrt{\cos (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 2670
Rule 2680
Rule 2640
Rule 2639
Rubi steps
\begin{align*} \int \frac{(a+a \sin (c+d x))^2}{(e \cos (c+d x))^{3/2}} \, dx &=\frac{a^4 \int \frac{(e \cos (c+d x))^{5/2}}{(a-a \sin (c+d x))^2} \, dx}{e^4}\\ &=\frac{4 a^4 (e \cos (c+d x))^{3/2}}{d e^3 \left (a^2-a^2 \sin (c+d x)\right )}-\frac{\left (3 a^2\right ) \int \sqrt{e \cos (c+d x)} \, dx}{e^2}\\ &=\frac{4 a^4 (e \cos (c+d x))^{3/2}}{d e^3 \left (a^2-a^2 \sin (c+d x)\right )}-\frac{\left (3 a^2 \sqrt{e \cos (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx}{e^2 \sqrt{\cos (c+d x)}}\\ &=-\frac{6 a^2 \sqrt{e \cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{d e^2 \sqrt{\cos (c+d x)}}+\frac{4 a^4 (e \cos (c+d x))^{3/2}}{d e^3 \left (a^2-a^2 \sin (c+d x)\right )}\\ \end{align*}
Mathematica [C] time = 0.061673, size = 64, normalized size = 0.75 \[ \frac{4\ 2^{3/4} a^2 \sqrt [4]{\sin (c+d x)+1} \, _2F_1\left (-\frac{3}{4},-\frac{1}{4};\frac{3}{4};\frac{1}{2} (1-\sin (c+d x))\right )}{d e \sqrt{e \cos (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.483, size = 120, normalized size = 1.4 \begin{align*} -2\,{\frac{ \left ( 3\,\sqrt{ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}{\it EllipticE} \left ( \cos \left ( 1/2\,dx+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\,\sin \left ( 1/2\,dx+c/2 \right ) \right ){a}^{2}}{e\sqrt{-2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}e+e}\sin \left ( 1/2\,dx+c/2 \right ) d}} \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 (a \sin \left (d x + c\right ) + a\right )}^{2}}{\left (e \cos \left (d x + c\right )\right )^{\frac{3}{2}}}\,{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{{\left (a^{2} \cos \left (d x + c\right )^{2} - 2 \, a^{2} \sin \left (d x + c\right ) - 2 \, a^{2}\right )} \sqrt{e \cos \left (d x + c\right )}}{e^{2} \cos \left (d x + c\right )^{2}}, 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 (a \sin \left (d x + c\right ) + a\right )}^{2}}{\left (e \cos \left (d x + c\right )\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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