Optimal. Leaf size=169 \[ \frac{26 e^3 (e \cos (c+d x))^{9/2}}{45 a^3 d}+\frac{26 e^7 \sin (c+d x) \sqrt{e \cos (c+d x)}}{21 a^3 d}+\frac{26 e^5 \sin (c+d x) (e \cos (c+d x))^{5/2}}{35 a^3 d}+\frac{26 e^8 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 a^3 d \sqrt{e \cos (c+d x)}}+\frac{4 e (e \cos (c+d x))^{13/2}}{5 a d (a \sin (c+d x)+a)^2} \]
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Rubi [A] time = 0.179051, antiderivative size = 169, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2680, 2682, 2635, 2642, 2641} \[ \frac{26 e^3 (e \cos (c+d x))^{9/2}}{45 a^3 d}+\frac{26 e^7 \sin (c+d x) \sqrt{e \cos (c+d x)}}{21 a^3 d}+\frac{26 e^5 \sin (c+d x) (e \cos (c+d x))^{5/2}}{35 a^3 d}+\frac{26 e^8 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 a^3 d \sqrt{e \cos (c+d x)}}+\frac{4 e (e \cos (c+d x))^{13/2}}{5 a d (a \sin (c+d x)+a)^2} \]
Antiderivative was successfully verified.
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Rule 2680
Rule 2682
Rule 2635
Rule 2642
Rule 2641
Rubi steps
\begin{align*} \int \frac{(e \cos (c+d x))^{15/2}}{(a+a \sin (c+d x))^3} \, dx &=\frac{4 e (e \cos (c+d x))^{13/2}}{5 a d (a+a \sin (c+d x))^2}+\frac{\left (13 e^2\right ) \int \frac{(e \cos (c+d x))^{11/2}}{a+a \sin (c+d x)} \, dx}{5 a^2}\\ &=\frac{26 e^3 (e \cos (c+d x))^{9/2}}{45 a^3 d}+\frac{4 e (e \cos (c+d x))^{13/2}}{5 a d (a+a \sin (c+d x))^2}+\frac{\left (13 e^4\right ) \int (e \cos (c+d x))^{7/2} \, dx}{5 a^3}\\ &=\frac{26 e^3 (e \cos (c+d x))^{9/2}}{45 a^3 d}+\frac{26 e^5 (e \cos (c+d x))^{5/2} \sin (c+d x)}{35 a^3 d}+\frac{4 e (e \cos (c+d x))^{13/2}}{5 a d (a+a \sin (c+d x))^2}+\frac{\left (13 e^6\right ) \int (e \cos (c+d x))^{3/2} \, dx}{7 a^3}\\ &=\frac{26 e^3 (e \cos (c+d x))^{9/2}}{45 a^3 d}+\frac{26 e^7 \sqrt{e \cos (c+d x)} \sin (c+d x)}{21 a^3 d}+\frac{26 e^5 (e \cos (c+d x))^{5/2} \sin (c+d x)}{35 a^3 d}+\frac{4 e (e \cos (c+d x))^{13/2}}{5 a d (a+a \sin (c+d x))^2}+\frac{\left (13 e^8\right ) \int \frac{1}{\sqrt{e \cos (c+d x)}} \, dx}{21 a^3}\\ &=\frac{26 e^3 (e \cos (c+d x))^{9/2}}{45 a^3 d}+\frac{26 e^7 \sqrt{e \cos (c+d x)} \sin (c+d x)}{21 a^3 d}+\frac{26 e^5 (e \cos (c+d x))^{5/2} \sin (c+d x)}{35 a^3 d}+\frac{4 e (e \cos (c+d x))^{13/2}}{5 a d (a+a \sin (c+d x))^2}+\frac{\left (13 e^8 \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{26 e^3 (e \cos (c+d x))^{9/2}}{45 a^3 d}+\frac{26 e^8 \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{26 e^7 \sqrt{e \cos (c+d x)} \sin (c+d x)}{21 a^3 d}+\frac{26 e^5 (e \cos (c+d x))^{5/2} \sin (c+d x)}{35 a^3 d}+\frac{4 e (e \cos (c+d x))^{13/2}}{5 a d (a+a \sin (c+d x))^2}\\ \end{align*}
Mathematica [C] time = 0.369593, size = 66, normalized size = 0.39 \[ -\frac{4 \sqrt [4]{2} (e \cos (c+d x))^{17/2} \, _2F_1\left (-\frac{1}{4},\frac{17}{4};\frac{21}{4};\frac{1}{2} (1-\sin (c+d x))\right )}{17 a^3 d e (\sin (c+d x)+1)^{17/4}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.658, size = 251, normalized size = 1.5 \begin{align*} -{\frac{2\,{e}^{8}}{315\,{a}^{3}d} \left ( -1120\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{11}-2160\,\cos \left ( 1/2\,dx+c/2 \right ) \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{8}+2800\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{9}+3240\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{6}\cos \left ( 1/2\,dx+c/2 \right ) -784\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{7}-840\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}\cos \left ( 1/2\,dx+c/2 \right ) -1624\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{5}+195\,\sqrt{2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}\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 ) -120\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}\cos \left ( 1/2\,dx+c/2 \right ) +1162\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{3}-217\,\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{15}{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^{7} \cos \left (d x + c\right )^{7}}{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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