Optimal. Leaf size=409 \[ -\frac{14 a^4 c^2 (g \cos (e+f x))^{5/2}}{39 f g \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{2 a^3 c^2 \sqrt{a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{13 f g \sqrt{c-c \sin (e+f x)}}-\frac{10 a^2 c^2 (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{143 f g \sqrt{c-c \sin (e+f x)}}+\frac{14 a^4 c^2 g \sqrt{\cos (e+f x)} E\left (\left .\frac{1}{2} (e+f x)\right |2\right ) \sqrt{g \cos (e+f x)}}{13 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{14 a c^2 (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{429 f g \sqrt{c-c \sin (e+f x)}}+\frac{14 c^2 (a \sin (e+f x)+a)^{7/2} (g \cos (e+f x))^{5/2}}{143 f g \sqrt{c-c \sin (e+f x)}}+\frac{2 c (a \sin (e+f x)+a)^{7/2} \sqrt{c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{13 f g} \]
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Rubi [A] time = 2.04229, antiderivative size = 409, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 4, integrand size = 42, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2851, 2842, 2640, 2639} \[ -\frac{14 a^4 c^2 (g \cos (e+f x))^{5/2}}{39 f g \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{2 a^3 c^2 \sqrt{a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{13 f g \sqrt{c-c \sin (e+f x)}}-\frac{10 a^2 c^2 (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{143 f g \sqrt{c-c \sin (e+f x)}}+\frac{14 a^4 c^2 g \sqrt{\cos (e+f x)} E\left (\left .\frac{1}{2} (e+f x)\right |2\right ) \sqrt{g \cos (e+f x)}}{13 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{14 a c^2 (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{429 f g \sqrt{c-c \sin (e+f x)}}+\frac{14 c^2 (a \sin (e+f x)+a)^{7/2} (g \cos (e+f x))^{5/2}}{143 f g \sqrt{c-c \sin (e+f x)}}+\frac{2 c (a \sin (e+f x)+a)^{7/2} \sqrt{c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{13 f g} \]
Antiderivative was successfully verified.
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Rule 2851
Rule 2842
Rule 2640
Rule 2639
Rubi steps
\begin{align*} \int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{3/2} \, dx &=\frac{2 c (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{7/2} \sqrt{c-c \sin (e+f x)}}{13 f g}+\frac{1}{13} (7 c) \int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2} \sqrt{c-c \sin (e+f x)} \, dx\\ &=\frac{14 c^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{7/2}}{143 f g \sqrt{c-c \sin (e+f x)}}+\frac{2 c (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{7/2} \sqrt{c-c \sin (e+f x)}}{13 f g}+\frac{1}{143} \left (21 c^2\right ) \int \frac{(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2}}{\sqrt{c-c \sin (e+f x)}} \, dx\\ &=-\frac{14 a c^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{429 f g \sqrt{c-c \sin (e+f x)}}+\frac{14 c^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{7/2}}{143 f g \sqrt{c-c \sin (e+f x)}}+\frac{2 c (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{7/2} \sqrt{c-c \sin (e+f x)}}{13 f g}+\frac{1}{143} \left (35 a c^2\right ) \int \frac{(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{5/2}}{\sqrt{c-c \sin (e+f x)}} \, dx\\ &=-\frac{10 a^2 c^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{143 f g \sqrt{c-c \sin (e+f x)}}-\frac{14 a c^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{429 f g \sqrt{c-c \sin (e+f x)}}+\frac{14 c^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{7/2}}{143 f g \sqrt{c-c \sin (e+f x)}}+\frac{2 c (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{7/2} \sqrt{c-c \sin (e+f x)}}{13 f g}+\frac{1}{13} \left (5 a^2 c^2\right ) \int \frac{(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{3/2}}{\sqrt{c-c \sin (e+f x)}} \, dx\\ &=-\frac{2 a^3 c^2 (g \cos (e+f x))^{5/2} \sqrt{a+a \sin (e+f x)}}{13 f g \sqrt{c-c \sin (e+f x)}}-\frac{10 a^2 c^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{143 f g \sqrt{c-c \sin (e+f x)}}-\frac{14 a c^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{429 f g \sqrt{c-c \sin (e+f x)}}+\frac{14 c^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{7/2}}{143 f g \sqrt{c-c \sin (e+f x)}}+\frac{2 c (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{7/2} \sqrt{c-c \sin (e+f x)}}{13 f g}+\frac{1}{13} \left (7 a^3 c^2\right ) \int \frac{(g \cos (e+f x))^{3/2} \sqrt{a+a \sin (e+f x)}}{\sqrt{c-c \sin (e+f x)}} \, dx\\ &=-\frac{14 a^4 c^2 (g \cos (e+f x))^{5/2}}{39 f g \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}-\frac{2 a^3 c^2 (g \cos (e+f x))^{5/2} \sqrt{a+a \sin (e+f x)}}{13 f g \sqrt{c-c \sin (e+f x)}}-\frac{10 a^2 c^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{143 f g \sqrt{c-c \sin (e+f x)}}-\frac{14 a c^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{429 f g \sqrt{c-c \sin (e+f x)}}+\frac{14 c^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{7/2}}{143 f g \sqrt{c-c \sin (e+f x)}}+\frac{2 c (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{7/2} \sqrt{c-c \sin (e+f x)}}{13 f g}+\frac{1}{13} \left (7 a^4 c^2\right ) \int \frac{(g \cos (e+f x))^{3/2}}{\sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}} \, dx\\ &=-\frac{14 a^4 c^2 (g \cos (e+f x))^{5/2}}{39 f g \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}-\frac{2 a^3 c^2 (g \cos (e+f x))^{5/2} \sqrt{a+a \sin (e+f x)}}{13 f g \sqrt{c-c \sin (e+f x)}}-\frac{10 a^2 c^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{143 f g \sqrt{c-c \sin (e+f x)}}-\frac{14 a c^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{429 f g \sqrt{c-c \sin (e+f x)}}+\frac{14 c^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{7/2}}{143 f g \sqrt{c-c \sin (e+f x)}}+\frac{2 c (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{7/2} \sqrt{c-c \sin (e+f x)}}{13 f g}+\frac{\left (7 a^4 c^2 g \cos (e+f x)\right ) \int \sqrt{g \cos (e+f x)} \, dx}{13 \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}\\ &=-\frac{14 a^4 c^2 (g \cos (e+f x))^{5/2}}{39 f g \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}-\frac{2 a^3 c^2 (g \cos (e+f x))^{5/2} \sqrt{a+a \sin (e+f x)}}{13 f g \sqrt{c-c \sin (e+f x)}}-\frac{10 a^2 c^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{143 f g \sqrt{c-c \sin (e+f x)}}-\frac{14 a c^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{429 f g \sqrt{c-c \sin (e+f x)}}+\frac{14 c^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{7/2}}{143 f g \sqrt{c-c \sin (e+f x)}}+\frac{2 c (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{7/2} \sqrt{c-c \sin (e+f x)}}{13 f g}+\frac{\left (7 a^4 c^2 g \sqrt{\cos (e+f x)} \sqrt{g \cos (e+f x)}\right ) \int \sqrt{\cos (e+f x)} \, dx}{13 \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}\\ &=-\frac{14 a^4 c^2 (g \cos (e+f x))^{5/2}}{39 f g \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}+\frac{14 a^4 c^2 g \sqrt{\cos (e+f x)} \sqrt{g \cos (e+f x)} E\left (\left .\frac{1}{2} (e+f x)\right |2\right )}{13 f \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}-\frac{2 a^3 c^2 (g \cos (e+f x))^{5/2} \sqrt{a+a \sin (e+f x)}}{13 f g \sqrt{c-c \sin (e+f x)}}-\frac{10 a^2 c^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{143 f g \sqrt{c-c \sin (e+f x)}}-\frac{14 a c^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{429 f g \sqrt{c-c \sin (e+f x)}}+\frac{14 c^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{7/2}}{143 f g \sqrt{c-c \sin (e+f x)}}+\frac{2 c (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{7/2} \sqrt{c-c \sin (e+f x)}}{13 f g}\\ \end{align*}
Mathematica [A] time = 3.14435, size = 212, normalized size = 0.52 \[ \frac{a^3 c (\sin (e+f x)-1) (\sin (e+f x)+1)^3 \sqrt{a (\sin (e+f x)+1)} \sqrt{c-c \sin (e+f x)} (g \cos (e+f x))^{3/2} \left (\sqrt{\cos (e+f x)} (-1507 \sin (2 (e+f x))-88 \sin (4 (e+f x))+33 \sin (6 (e+f x))+1560 \cos (e+f x)+780 \cos (3 (e+f x))+156 \cos (5 (e+f x)))-7392 E\left (\left .\frac{1}{2} (e+f x)\right |2\right )\right )}{6864 f \cos ^{\frac{3}{2}}(e+f x) \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^3 \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^7} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.41, size = 404, normalized size = 1. \begin{align*} -{\frac{2}{429\,f \left ( - \left ( \cos \left ( fx+e \right ) \right ) ^{2}+2\,\sin \left ( fx+e \right ) +2 \right ) \sin \left ( fx+e \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{5}} \left ( -c \left ( -1+\sin \left ( fx+e \right ) \right ) \right ) ^{{\frac{3}{2}}} \left ( -33\, \left ( \cos \left ( fx+e \right ) \right ) ^{8}+78\, \left ( \cos \left ( fx+e \right ) \right ) ^{6}\sin \left ( fx+e \right ) +231\,i{\it EllipticE} \left ({\frac{i \left ( -1+\cos \left ( fx+e \right ) \right ) }{\sin \left ( fx+e \right ) }},i \right ) \sin \left ( fx+e \right ) \cos \left ( fx+e \right ) \sqrt{ \left ( \cos \left ( fx+e \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( fx+e \right ) }{\cos \left ( fx+e \right ) +1}}}-231\,i\sin \left ( fx+e \right ) \cos \left ( fx+e \right ) \sqrt{ \left ( \cos \left ( fx+e \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( fx+e \right ) }{\cos \left ( fx+e \right ) +1}}}{\it EllipticF} \left ({\frac{i \left ( -1+\cos \left ( fx+e \right ) \right ) }{\sin \left ( fx+e \right ) }},i \right ) +88\, \left ( \cos \left ( fx+e \right ) \right ) ^{6}+231\,i{\it EllipticE} \left ({\frac{i \left ( -1+\cos \left ( fx+e \right ) \right ) }{\sin \left ( fx+e \right ) }},i \right ) \sin \left ( fx+e \right ) \sqrt{ \left ( \cos \left ( fx+e \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( fx+e \right ) }{\cos \left ( fx+e \right ) +1}}}-231\,i\sin \left ( fx+e \right ) \sqrt{ \left ( \cos \left ( fx+e \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( fx+e \right ) }{\cos \left ( fx+e \right ) +1}}}{\it EllipticF} \left ({\frac{i \left ( -1+\cos \left ( fx+e \right ) \right ) }{\sin \left ( fx+e \right ) }},i \right ) +22\, \left ( \cos \left ( fx+e \right ) \right ) ^{4}+154\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}-231\,\cos \left ( fx+e \right ) \right ) \left ( g\cos \left ( fx+e \right ) \right ) ^{{\frac{3}{2}}} \left ( a \left ( 1+\sin \left ( fx+e \right ) \right ) \right ) ^{{\frac{7}{2}}}} \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 \left (g \cos \left (f x + e\right )\right )^{\frac{3}{2}}{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac{7}{2}}{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac{3}{2}}\,{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 (-{\left (a^{3} c g \cos \left (f x + e\right )^{5} - 2 \, a^{3} c g \cos \left (f x + e\right )^{3} \sin \left (f x + e\right ) - 2 \, a^{3} c g \cos \left (f x + e\right )^{3}\right )} \sqrt{g \cos \left (f x + e\right )} \sqrt{a \sin \left (f x + e\right ) + a} \sqrt{-c \sin \left (f x + e\right ) + c}, 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(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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