Optimal. Leaf size=176 \[ \frac{2 (g \cos (e+f x))^{5/2}}{a f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac{2 (g \cos (e+f x))^{5/2}}{f g (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{3/2}}-\frac{2 g \sqrt{\cos (e+f x)} E\left (\left .\frac{1}{2} (e+f x)\right |2\right ) \sqrt{g \cos (e+f x)}}{a c f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}} \]
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Rubi [A] time = 0.878735, antiderivative size = 176, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 42, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2852, 2842, 2640, 2639} \[ \frac{2 (g \cos (e+f x))^{5/2}}{a f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac{2 (g \cos (e+f x))^{5/2}}{f g (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{3/2}}-\frac{2 g \sqrt{\cos (e+f x)} E\left (\left .\frac{1}{2} (e+f x)\right |2\right ) \sqrt{g \cos (e+f x)}}{a c f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 2852
Rule 2842
Rule 2640
Rule 2639
Rubi steps
\begin{align*} \int \frac{(g \cos (e+f x))^{3/2}}{(a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{3/2}} \, dx &=-\frac{2 (g \cos (e+f x))^{5/2}}{f g (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{3/2}}+\frac{\int \frac{(g \cos (e+f x))^{3/2}}{\sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}} \, dx}{a}\\ &=-\frac{2 (g \cos (e+f x))^{5/2}}{f g (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{3/2}}+\frac{2 (g \cos (e+f x))^{5/2}}{a f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}-\frac{\int \frac{(g \cos (e+f x))^{3/2}}{\sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}} \, dx}{a c}\\ &=-\frac{2 (g \cos (e+f x))^{5/2}}{f g (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{3/2}}+\frac{2 (g \cos (e+f x))^{5/2}}{a f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}-\frac{(g \cos (e+f x)) \int \sqrt{g \cos (e+f x)} \, dx}{a c \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}\\ &=-\frac{2 (g \cos (e+f x))^{5/2}}{f g (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{3/2}}+\frac{2 (g \cos (e+f x))^{5/2}}{a f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}-\frac{\left (g \sqrt{\cos (e+f x)} \sqrt{g \cos (e+f x)}\right ) \int \sqrt{\cos (e+f x)} \, dx}{a c \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}\\ &=-\frac{2 (g \cos (e+f x))^{5/2}}{f g (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{3/2}}+\frac{2 (g \cos (e+f x))^{5/2}}{a f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}-\frac{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 )}{a c f \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}\\ \end{align*}
Mathematica [A] time = 0.75021, size = 92, normalized size = 0.52 \[ -\frac{2 (g \cos (e+f x))^{5/2} \left (\sin (e+f x)-\sqrt{\cos (e+f x)} E\left (\left .\frac{1}{2} (e+f x)\right |2\right )\right )}{c f g (\sin (e+f x)-1) (a (\sin (e+f x)+1))^{3/2} \sqrt{c-c \sin (e+f x)}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.353, size = 361, normalized size = 2.1 \begin{align*} 2\,{\frac{ \left ( \cos \left ( fx+e \right ) +1 \right ) ^{2} \left ( g\cos \left ( fx+e \right ) \right ) ^{3/2} \left ( -1+\cos \left ( fx+e \right ) \right ) ^{2} \left ( 1+\sin \left ( fx+e \right ) \right ) \left ( -1+\sin \left ( fx+e \right ) \right ) }{f \left ( a \left ( 1+\sin \left ( fx+e \right ) \right ) \right ) ^{3/2} \left ( -c \left ( -1+\sin \left ( fx+e \right ) \right ) \right ) ^{3/2} \left ( \sin \left ( fx+e \right ) \right ) ^{5}\cos \left ( fx+e \right ) } \left ( i{\it EllipticF} \left ({\frac{i \left ( -1+\cos \left ( fx+e \right ) \right ) }{\sin \left ( fx+e \right ) }},i \right ) \sqrt{ \left ( \cos \left ( fx+e \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( fx+e \right ) }{\cos \left ( fx+e \right ) +1}}}\sin \left ( fx+e \right ) \cos \left ( fx+e \right ) -i{\it EllipticE} \left ({\frac{i \left ( -1+\cos \left ( fx+e \right ) \right ) }{\sin \left ( fx+e \right ) }},i \right ) \sqrt{ \left ( \cos \left ( fx+e \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( fx+e \right ) }{\cos \left ( fx+e \right ) +1}}}\sin \left ( fx+e \right ) \cos \left ( fx+e \right ) +i\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 ) \sin \left ( fx+e \right ) -i\sin \left ( fx+e \right ){\it EllipticE} \left ({\frac{i \left ( -1+\cos \left ( fx+e \right ) \right ) }{\sin \left ( fx+e \right ) }},i \right ) \sqrt{ \left ( \cos \left ( fx+e \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( fx+e \right ) }{\cos \left ( fx+e \right ) +1}}}+\cos \left ( fx+e \right ) -1 \right ) } \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 (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{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 (\frac{\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} g}{a^{2} c^{2} \cos \left (f x + e\right )^{3}}, 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 (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{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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