Optimal. Leaf size=154 \[ \frac{2 e^2 E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{e \cos (c+d x)}}{15 a^4 d \sqrt{\cos (c+d x)}}+\frac{2 e (e \cos (c+d x))^{3/2}}{15 d \left (a^4 \sin (c+d x)+a^4\right )}+\frac{2 e (e \cos (c+d x))^{3/2}}{15 d \left (a^2 \sin (c+d x)+a^2\right )^2}-\frac{4 e (e \cos (c+d x))^{3/2}}{9 a d (a \sin (c+d x)+a)^3} \]
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Rubi [A] time = 0.182129, antiderivative size = 154, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2680, 2681, 2683, 2640, 2639} \[ \frac{2 e^2 E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{e \cos (c+d x)}}{15 a^4 d \sqrt{\cos (c+d x)}}+\frac{2 e (e \cos (c+d x))^{3/2}}{15 d \left (a^4 \sin (c+d x)+a^4\right )}+\frac{2 e (e \cos (c+d x))^{3/2}}{15 d \left (a^2 \sin (c+d x)+a^2\right )^2}-\frac{4 e (e \cos (c+d x))^{3/2}}{9 a d (a \sin (c+d x)+a)^3} \]
Antiderivative was successfully verified.
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Rule 2680
Rule 2681
Rule 2683
Rule 2640
Rule 2639
Rubi steps
\begin{align*} \int \frac{(e \cos (c+d x))^{5/2}}{(a+a \sin (c+d x))^4} \, dx &=-\frac{4 e (e \cos (c+d x))^{3/2}}{9 a d (a+a \sin (c+d x))^3}-\frac{e^2 \int \frac{\sqrt{e \cos (c+d x)}}{(a+a \sin (c+d x))^2} \, dx}{3 a^2}\\ &=-\frac{4 e (e \cos (c+d x))^{3/2}}{9 a d (a+a \sin (c+d x))^3}+\frac{2 e (e \cos (c+d x))^{3/2}}{15 d \left (a^2+a^2 \sin (c+d x)\right )^2}-\frac{e^2 \int \frac{\sqrt{e \cos (c+d x)}}{a+a \sin (c+d x)} \, dx}{15 a^3}\\ &=-\frac{4 e (e \cos (c+d x))^{3/2}}{9 a d (a+a \sin (c+d x))^3}+\frac{2 e (e \cos (c+d x))^{3/2}}{15 d \left (a^2+a^2 \sin (c+d x)\right )^2}+\frac{2 e (e \cos (c+d x))^{3/2}}{15 d \left (a^4+a^4 \sin (c+d x)\right )}+\frac{e^2 \int \sqrt{e \cos (c+d x)} \, dx}{15 a^4}\\ &=-\frac{4 e (e \cos (c+d x))^{3/2}}{9 a d (a+a \sin (c+d x))^3}+\frac{2 e (e \cos (c+d x))^{3/2}}{15 d \left (a^2+a^2 \sin (c+d x)\right )^2}+\frac{2 e (e \cos (c+d x))^{3/2}}{15 d \left (a^4+a^4 \sin (c+d x)\right )}+\frac{\left (e^2 \sqrt{e \cos (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx}{15 a^4 \sqrt{\cos (c+d x)}}\\ &=\frac{2 e^2 \sqrt{e \cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 a^4 d \sqrt{\cos (c+d x)}}-\frac{4 e (e \cos (c+d x))^{3/2}}{9 a d (a+a \sin (c+d x))^3}+\frac{2 e (e \cos (c+d x))^{3/2}}{15 d \left (a^2+a^2 \sin (c+d x)\right )^2}+\frac{2 e (e \cos (c+d x))^{3/2}}{15 d \left (a^4+a^4 \sin (c+d x)\right )}\\ \end{align*}
Mathematica [C] time = 0.087732, size = 66, normalized size = 0.43 \[ -\frac{(e \cos (c+d x))^{7/2} \, _2F_1\left (\frac{7}{4},\frac{13}{4};\frac{11}{4};\frac{1}{2} (1-\sin (c+d x))\right )}{14 \sqrt [4]{2} a^4 d e (\sin (c+d x)+1)^{7/4}} \]
Antiderivative was successfully verified.
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Maple [B] time = 2.808, size = 514, normalized size = 3.3 \begin{align*} \text{result too large to display} \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{5}{2}}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{4}}\,{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^{2} \cos \left (d x + c\right )^{2}}{a^{4} \cos \left (d x + c\right )^{4} - 8 \, a^{4} \cos \left (d x + c\right )^{2} + 8 \, a^{4} - 4 \,{\left (a^{4} \cos \left (d x + c\right )^{2} - 2 \, a^{4}\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{5}{2}}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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