Optimal. Leaf size=149 \[ -\frac{154 e^5 \sin (c+d x) (e \cos (c+d x))^{3/2}}{15 a^4 d}-\frac{44 e^3 (e \cos (c+d x))^{7/2}}{3 d \left (a^4 \sin (c+d x)+a^4\right )}-\frac{154 e^6 E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{e \cos (c+d x)}}{5 a^4 d \sqrt{\cos (c+d x)}}-\frac{4 e (e \cos (c+d x))^{11/2}}{a d (a \sin (c+d x)+a)^3} \]
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Rubi [A] time = 0.153125, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {2680, 2635, 2640, 2639} \[ -\frac{154 e^5 \sin (c+d x) (e \cos (c+d x))^{3/2}}{15 a^4 d}-\frac{44 e^3 (e \cos (c+d x))^{7/2}}{3 d \left (a^4 \sin (c+d x)+a^4\right )}-\frac{154 e^6 E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{e \cos (c+d x)}}{5 a^4 d \sqrt{\cos (c+d x)}}-\frac{4 e (e \cos (c+d x))^{11/2}}{a d (a \sin (c+d x)+a)^3} \]
Antiderivative was successfully verified.
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Rule 2680
Rule 2635
Rule 2640
Rule 2639
Rubi steps
\begin{align*} \int \frac{(e \cos (c+d x))^{13/2}}{(a+a \sin (c+d x))^4} \, dx &=-\frac{4 e (e \cos (c+d x))^{11/2}}{a d (a+a \sin (c+d x))^3}-\frac{\left (11 e^2\right ) \int \frac{(e \cos (c+d x))^{9/2}}{(a+a \sin (c+d x))^2} \, dx}{a^2}\\ &=-\frac{4 e (e \cos (c+d x))^{11/2}}{a d (a+a \sin (c+d x))^3}-\frac{44 e^3 (e \cos (c+d x))^{7/2}}{3 d \left (a^4+a^4 \sin (c+d x)\right )}-\frac{\left (77 e^4\right ) \int (e \cos (c+d x))^{5/2} \, dx}{3 a^4}\\ &=-\frac{154 e^5 (e \cos (c+d x))^{3/2} \sin (c+d x)}{15 a^4 d}-\frac{4 e (e \cos (c+d x))^{11/2}}{a d (a+a \sin (c+d x))^3}-\frac{44 e^3 (e \cos (c+d x))^{7/2}}{3 d \left (a^4+a^4 \sin (c+d x)\right )}-\frac{\left (77 e^6\right ) \int \sqrt{e \cos (c+d x)} \, dx}{5 a^4}\\ &=-\frac{154 e^5 (e \cos (c+d x))^{3/2} \sin (c+d x)}{15 a^4 d}-\frac{4 e (e \cos (c+d x))^{11/2}}{a d (a+a \sin (c+d x))^3}-\frac{44 e^3 (e \cos (c+d x))^{7/2}}{3 d \left (a^4+a^4 \sin (c+d x)\right )}-\frac{\left (77 e^6 \sqrt{e \cos (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx}{5 a^4 \sqrt{\cos (c+d x)}}\\ &=-\frac{154 e^6 \sqrt{e \cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 a^4 d \sqrt{\cos (c+d x)}}-\frac{154 e^5 (e \cos (c+d x))^{3/2} \sin (c+d x)}{15 a^4 d}-\frac{4 e (e \cos (c+d x))^{11/2}}{a d (a+a \sin (c+d x))^3}-\frac{44 e^3 (e \cos (c+d x))^{7/2}}{3 d \left (a^4+a^4 \sin (c+d x)\right )}\\ \end{align*}
Mathematica [C] time = 0.219982, size = 66, normalized size = 0.44 \[ -\frac{2^{3/4} (e \cos (c+d x))^{15/2} \, _2F_1\left (\frac{5}{4},\frac{15}{4};\frac{19}{4};\frac{1}{2} (1-\sin (c+d x))\right )}{15 a^4 d e (\sin (c+d x)+1)^{15/4}} \]
Antiderivative was successfully verified.
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Maple [A] time = 1.056, size = 190, normalized size = 1.3 \begin{align*} -{\frac{2\,{e}^{7}}{15\,{a}^{4}d} \left ( -24\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{6}\cos \left ( 1/2\,dx+c/2 \right ) +24\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}\cos \left ( 1/2\,dx+c/2 \right ) +80\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{5}+231\,\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 ) -246\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}\cos \left ( 1/2\,dx+c/2 \right ) -80\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{3}+140\,\sin \left ( 1/2\,dx+c/2 \right ) \right ){\frac{1}{\sqrt{-2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}e+e}}} \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [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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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{e \cos \left (d x + c\right )} e^{6} \cos \left (d x + c\right )^{6}}{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(-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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