Optimal. Leaf size=352 \[ \frac{14 a^2 c^3 (g \cos (e+f x))^{5/2}}{45 f g \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{2 a^2 c^2 \sqrt{c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{15 f g \sqrt{a \sin (e+f x)+a}}+\frac{14 a^2 c^3 g \sqrt{\cos (e+f x)} E\left (\left .\frac{1}{2} (e+f x)\right |2\right ) \sqrt{g \cos (e+f x)}}{15 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{2 a^2 c (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{33 f g \sqrt{a \sin (e+f x)+a}}-\frac{14 a^2 (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{99 f g \sqrt{a \sin (e+f x)+a}}-\frac{2 a \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{11 f g} \]
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Rubi [A] time = 1.72694, antiderivative size = 352, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 42, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2851, 2842, 2640, 2639} \[ \frac{14 a^2 c^3 (g \cos (e+f x))^{5/2}}{45 f g \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{2 a^2 c^2 \sqrt{c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{15 f g \sqrt{a \sin (e+f x)+a}}+\frac{14 a^2 c^3 g \sqrt{\cos (e+f x)} E\left (\left .\frac{1}{2} (e+f x)\right |2\right ) \sqrt{g \cos (e+f x)}}{15 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{2 a^2 c (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{33 f g \sqrt{a \sin (e+f x)+a}}-\frac{14 a^2 (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{99 f g \sqrt{a \sin (e+f x)+a}}-\frac{2 a \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{11 f g} \]
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))^{3/2} (c-c \sin (e+f x))^{5/2} \, dx &=-\frac{2 a (g \cos (e+f x))^{5/2} \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}{11 f g}+\frac{1}{11} (7 a) \int (g \cos (e+f x))^{3/2} \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2} \, dx\\ &=-\frac{14 a^2 (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{5/2}}{99 f g \sqrt{a+a \sin (e+f x)}}-\frac{2 a (g \cos (e+f x))^{5/2} \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}{11 f g}+\frac{1}{33} \left (7 a^2\right ) \int \frac{(g \cos (e+f x))^{3/2} (c-c \sin (e+f x))^{5/2}}{\sqrt{a+a \sin (e+f x)}} \, dx\\ &=\frac{2 a^2 c (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}}{33 f g \sqrt{a+a \sin (e+f x)}}-\frac{14 a^2 (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{5/2}}{99 f g \sqrt{a+a \sin (e+f x)}}-\frac{2 a (g \cos (e+f x))^{5/2} \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}{11 f g}+\frac{1}{3} \left (a^2 c\right ) \int \frac{(g \cos (e+f x))^{3/2} (c-c \sin (e+f x))^{3/2}}{\sqrt{a+a \sin (e+f x)}} \, dx\\ &=\frac{2 a^2 c^2 (g \cos (e+f x))^{5/2} \sqrt{c-c \sin (e+f x)}}{15 f g \sqrt{a+a \sin (e+f x)}}+\frac{2 a^2 c (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}}{33 f g \sqrt{a+a \sin (e+f x)}}-\frac{14 a^2 (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{5/2}}{99 f g \sqrt{a+a \sin (e+f x)}}-\frac{2 a (g \cos (e+f x))^{5/2} \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}{11 f g}+\frac{1}{15} \left (7 a^2 c^2\right ) \int \frac{(g \cos (e+f x))^{3/2} \sqrt{c-c \sin (e+f x)}}{\sqrt{a+a \sin (e+f x)}} \, dx\\ &=\frac{14 a^2 c^3 (g \cos (e+f x))^{5/2}}{45 f g \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}+\frac{2 a^2 c^2 (g \cos (e+f x))^{5/2} \sqrt{c-c \sin (e+f x)}}{15 f g \sqrt{a+a \sin (e+f x)}}+\frac{2 a^2 c (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}}{33 f g \sqrt{a+a \sin (e+f x)}}-\frac{14 a^2 (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{5/2}}{99 f g \sqrt{a+a \sin (e+f x)}}-\frac{2 a (g \cos (e+f x))^{5/2} \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}{11 f g}+\frac{1}{15} \left (7 a^2 c^3\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^2 c^3 (g \cos (e+f x))^{5/2}}{45 f g \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}+\frac{2 a^2 c^2 (g \cos (e+f x))^{5/2} \sqrt{c-c \sin (e+f x)}}{15 f g \sqrt{a+a \sin (e+f x)}}+\frac{2 a^2 c (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}}{33 f g \sqrt{a+a \sin (e+f x)}}-\frac{14 a^2 (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{5/2}}{99 f g \sqrt{a+a \sin (e+f x)}}-\frac{2 a (g \cos (e+f x))^{5/2} \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}{11 f g}+\frac{\left (7 a^2 c^3 g \cos (e+f x)\right ) \int \sqrt{g \cos (e+f x)} \, dx}{15 \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}\\ &=\frac{14 a^2 c^3 (g \cos (e+f x))^{5/2}}{45 f g \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}+\frac{2 a^2 c^2 (g \cos (e+f x))^{5/2} \sqrt{c-c \sin (e+f x)}}{15 f g \sqrt{a+a \sin (e+f x)}}+\frac{2 a^2 c (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}}{33 f g \sqrt{a+a \sin (e+f x)}}-\frac{14 a^2 (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{5/2}}{99 f g \sqrt{a+a \sin (e+f x)}}-\frac{2 a (g \cos (e+f x))^{5/2} \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}{11 f g}+\frac{\left (7 a^2 c^3 g \sqrt{\cos (e+f x)} \sqrt{g \cos (e+f x)}\right ) \int \sqrt{\cos (e+f x)} \, dx}{15 \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}\\ &=\frac{14 a^2 c^3 (g \cos (e+f x))^{5/2}}{45 f g \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}+\frac{14 a^2 c^3 g \sqrt{\cos (e+f x)} \sqrt{g \cos (e+f x)} E\left (\left .\frac{1}{2} (e+f x)\right |2\right )}{15 f \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}+\frac{2 a^2 c^2 (g \cos (e+f x))^{5/2} \sqrt{c-c \sin (e+f x)}}{15 f g \sqrt{a+a \sin (e+f x)}}+\frac{2 a^2 c (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}}{33 f g \sqrt{a+a \sin (e+f x)}}-\frac{14 a^2 (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{5/2}}{99 f g \sqrt{a+a \sin (e+f x)}}-\frac{2 a (g \cos (e+f x))^{5/2} \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}{11 f g}\\ \end{align*}
Mathematica [A] time = 1.38476, size = 193, normalized size = 0.55 \[ \frac{c^2 (\sin (e+f x)-1)^2 (a (\sin (e+f x)+1))^{3/2} \sqrt{c-c \sin (e+f x)} (g \cos (e+f x))^{3/2} \left (3696 E\left (\left .\frac{1}{2} (e+f x)\right |2\right )+\sqrt{\cos (e+f x)} (836 \sin (2 (e+f x))+110 \sin (4 (e+f x))+450 \cos (e+f x)+225 \cos (3 (e+f x))+45 \cos (5 (e+f x)))\right )}{3960 f \cos ^{\frac{3}{2}}(e+f x) \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^5 \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^3} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.386, size = 382, normalized size = 1.1 \begin{align*}{\frac{2}{495\,f \left ( -1+\sin \left ( fx+e \right ) \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{5}{2}}} \left ( -45\, \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 ) +55\, \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{3}{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{3}{2}}{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac{5}{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 c^{2} g \cos \left (f x + e\right )^{3} \sin \left (f x + e\right ) - a c^{2} 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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