Optimal. Leaf size=138 \[ \frac{22 e^3 (e \cos (c+d x))^{7/2}}{21 a^3 d}+\frac{22 e^5 \sin (c+d x) (e \cos (c+d x))^{3/2}}{15 a^3 d}+\frac{22 e^6 E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{e \cos (c+d x)}}{5 a^3 d \sqrt{\cos (c+d x)}}+\frac{4 e (e \cos (c+d x))^{11/2}}{3 a d (a \sin (c+d x)+a)^2} \]
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Rubi [A] time = 0.152068, antiderivative size = 138, 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, 2682, 2635, 2640, 2639} \[ \frac{22 e^3 (e \cos (c+d x))^{7/2}}{21 a^3 d}+\frac{22 e^5 \sin (c+d x) (e \cos (c+d x))^{3/2}}{15 a^3 d}+\frac{22 e^6 E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{e \cos (c+d x)}}{5 a^3 d \sqrt{\cos (c+d x)}}+\frac{4 e (e \cos (c+d x))^{11/2}}{3 a d (a \sin (c+d x)+a)^2} \]
Antiderivative was successfully verified.
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Rule 2680
Rule 2682
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))^3} \, dx &=\frac{4 e (e \cos (c+d x))^{11/2}}{3 a d (a+a \sin (c+d x))^2}+\frac{\left (11 e^2\right ) \int \frac{(e \cos (c+d x))^{9/2}}{a+a \sin (c+d x)} \, dx}{3 a^2}\\ &=\frac{22 e^3 (e \cos (c+d x))^{7/2}}{21 a^3 d}+\frac{4 e (e \cos (c+d x))^{11/2}}{3 a d (a+a \sin (c+d x))^2}+\frac{\left (11 e^4\right ) \int (e \cos (c+d x))^{5/2} \, dx}{3 a^3}\\ &=\frac{22 e^3 (e \cos (c+d x))^{7/2}}{21 a^3 d}+\frac{22 e^5 (e \cos (c+d x))^{3/2} \sin (c+d x)}{15 a^3 d}+\frac{4 e (e \cos (c+d x))^{11/2}}{3 a d (a+a \sin (c+d x))^2}+\frac{\left (11 e^6\right ) \int \sqrt{e \cos (c+d x)} \, dx}{5 a^3}\\ &=\frac{22 e^3 (e \cos (c+d x))^{7/2}}{21 a^3 d}+\frac{22 e^5 (e \cos (c+d x))^{3/2} \sin (c+d x)}{15 a^3 d}+\frac{4 e (e \cos (c+d x))^{11/2}}{3 a d (a+a \sin (c+d x))^2}+\frac{\left (11 e^6 \sqrt{e \cos (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx}{5 a^3 \sqrt{\cos (c+d x)}}\\ &=\frac{22 e^3 (e \cos (c+d x))^{7/2}}{21 a^3 d}+\frac{22 e^6 \sqrt{e \cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 a^3 d \sqrt{\cos (c+d x)}}+\frac{22 e^5 (e \cos (c+d x))^{3/2} \sin (c+d x)}{15 a^3 d}+\frac{4 e (e \cos (c+d x))^{11/2}}{3 a d (a+a \sin (c+d x))^2}\\ \end{align*}
Mathematica [C] time = 0.228969, size = 66, normalized size = 0.48 \[ -\frac{2\ 2^{3/4} (e \cos (c+d x))^{15/2} \, _2F_1\left (\frac{1}{4},\frac{15}{4};\frac{19}{4};\frac{1}{2} (1-\sin (c+d x))\right )}{15 a^3 d e (\sin (c+d x)+1)^{15/4}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.662, size = 216, normalized size = 1.6 \begin{align*}{\frac{2\,{e}^{7}}{105\,{a}^{3}d} \left ( -240\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{9}-504\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{6}\cos \left ( 1/2\,dx+c/2 \right ) +480\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{7}+504\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}\cos \left ( 1/2\,dx+c/2 \right ) +200\, \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 ) -126\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}\cos \left ( 1/2\,dx+c/2 \right ) -440\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{3}+125\,\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 \frac{\left (e \cos \left (d x + c\right )\right )^{\frac{13}{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^{6} \cos \left (d x + c\right )^{6}}{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(-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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