Optimal. Leaf size=101 \[ \frac{2 e^3 \sin (c+d x) \sqrt{e \cos (c+d x)}}{3 a d}+\frac{2 e^4 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 a d \sqrt{e \cos (c+d x)}}+\frac{2 e (e \cos (c+d x))^{5/2}}{5 a d} \]
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Rubi [A] time = 0.0949842, antiderivative size = 101, 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 = {2682, 2635, 2642, 2641} \[ \frac{2 e^3 \sin (c+d x) \sqrt{e \cos (c+d x)}}{3 a d}+\frac{2 e^4 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 a d \sqrt{e \cos (c+d x)}}+\frac{2 e (e \cos (c+d x))^{5/2}}{5 a d} \]
Antiderivative was successfully verified.
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Rule 2682
Rule 2635
Rule 2642
Rule 2641
Rubi steps
\begin{align*} \int \frac{(e \cos (c+d x))^{7/2}}{a+a \sin (c+d x)} \, dx &=\frac{2 e (e \cos (c+d x))^{5/2}}{5 a d}+\frac{e^2 \int (e \cos (c+d x))^{3/2} \, dx}{a}\\ &=\frac{2 e (e \cos (c+d x))^{5/2}}{5 a d}+\frac{2 e^3 \sqrt{e \cos (c+d x)} \sin (c+d x)}{3 a d}+\frac{e^4 \int \frac{1}{\sqrt{e \cos (c+d x)}} \, dx}{3 a}\\ &=\frac{2 e (e \cos (c+d x))^{5/2}}{5 a d}+\frac{2 e^3 \sqrt{e \cos (c+d x)} \sin (c+d x)}{3 a d}+\frac{\left (e^4 \sqrt{\cos (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{3 a \sqrt{e \cos (c+d x)}}\\ &=\frac{2 e (e \cos (c+d x))^{5/2}}{5 a d}+\frac{2 e^4 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 a d \sqrt{e \cos (c+d x)}}+\frac{2 e^3 \sqrt{e \cos (c+d x)} \sin (c+d x)}{3 a d}\\ \end{align*}
Mathematica [C] time = 0.0788619, size = 66, normalized size = 0.65 \[ -\frac{4 \sqrt [4]{2} (e \cos (c+d x))^{9/2} \, _2F_1\left (-\frac{1}{4},\frac{9}{4};\frac{13}{4};\frac{1}{2} (1-\sin (c+d x))\right )}{9 a d e (\sin (c+d x)+1)^{9/4}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.516, size = 181, normalized size = 1.8 \begin{align*} -{\frac{2\,{e}^{4}}{15\,da} \left ( 24\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{7}+20\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}\cos \left ( 1/2\,dx+c/2 \right ) -36\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{5}+5\,\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 ) -10\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}\cos \left ( 1/2\,dx+c/2 \right ) +18\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{3}-3\,\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{7}{2}}}{a \sin \left (d x + c\right ) + a}\,{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^{3} \cos \left (d x + c\right )^{3}}{a \sin \left (d x + c\right ) + a}, 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{7}{2}}}{a \sin \left (d x + c\right ) + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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