Optimal. Leaf size=463 \[ -\frac{154 a^4 c^3 (g \cos (e+f x))^{5/2}}{585 f g \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{22 a^3 c^3 \sqrt{a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{195 f g \sqrt{c-c \sin (e+f x)}}-\frac{2 a^2 c^3 (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{39 f g \sqrt{c-c \sin (e+f x)}}+\frac{154 a^4 c^3 g \sqrt{\cos (e+f x)} E\left (\left .\frac{1}{2} (e+f x)\right |2\right ) \sqrt{g \cos (e+f x)}}{195 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{14 a c^3 (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{585 f g \sqrt{c-c \sin (e+f x)}}+\frac{22 c^2 (a \sin (e+f x)+a)^{7/2} \sqrt{c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{195 f g}+\frac{14 c^3 (a \sin (e+f x)+a)^{7/2} (g \cos (e+f x))^{5/2}}{195 f g \sqrt{c-c \sin (e+f x)}}+\frac{2 c (a \sin (e+f x)+a)^{7/2} (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{15 f g} \]
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Rubi [A] time = 2.38991, antiderivative size = 463, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 4, integrand size = 42, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2851, 2842, 2640, 2639} \[ -\frac{154 a^4 c^3 (g \cos (e+f x))^{5/2}}{585 f g \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{22 a^3 c^3 \sqrt{a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{195 f g \sqrt{c-c \sin (e+f x)}}-\frac{2 a^2 c^3 (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{39 f g \sqrt{c-c \sin (e+f x)}}+\frac{154 a^4 c^3 g \sqrt{\cos (e+f x)} E\left (\left .\frac{1}{2} (e+f x)\right |2\right ) \sqrt{g \cos (e+f x)}}{195 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{14 a c^3 (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{585 f g \sqrt{c-c \sin (e+f x)}}+\frac{22 c^2 (a \sin (e+f x)+a)^{7/2} \sqrt{c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{195 f g}+\frac{14 c^3 (a \sin (e+f x)+a)^{7/2} (g \cos (e+f x))^{5/2}}{195 f g \sqrt{c-c \sin (e+f x)}}+\frac{2 c (a \sin (e+f x)+a)^{7/2} (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{15 f g} \]
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))^{5/2} \, dx &=\frac{2 c (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{3/2}}{15 f g}+\frac{1}{15} (11 c) \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{22 c^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{7/2} \sqrt{c-c \sin (e+f x)}}{195 f g}+\frac{2 c (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{3/2}}{15 f g}+\frac{1}{195} \left (77 c^2\right ) \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^3 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{7/2}}{195 f g \sqrt{c-c \sin (e+f x)}}+\frac{22 c^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{7/2} \sqrt{c-c \sin (e+f x)}}{195 f g}+\frac{2 c (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{3/2}}{15 f g}+\frac{1}{65} \left (7 c^3\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^3 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{585 f g \sqrt{c-c \sin (e+f x)}}+\frac{14 c^3 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{7/2}}{195 f g \sqrt{c-c \sin (e+f x)}}+\frac{22 c^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{7/2} \sqrt{c-c \sin (e+f x)}}{195 f g}+\frac{2 c (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{3/2}}{15 f g}+\frac{1}{39} \left (7 a c^3\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{2 a^2 c^3 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{39 f g \sqrt{c-c \sin (e+f x)}}-\frac{14 a c^3 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{585 f g \sqrt{c-c \sin (e+f x)}}+\frac{14 c^3 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{7/2}}{195 f g \sqrt{c-c \sin (e+f x)}}+\frac{22 c^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{7/2} \sqrt{c-c \sin (e+f x)}}{195 f g}+\frac{2 c (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{3/2}}{15 f g}+\frac{1}{39} \left (11 a^2 c^3\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{22 a^3 c^3 (g \cos (e+f x))^{5/2} \sqrt{a+a \sin (e+f x)}}{195 f g \sqrt{c-c \sin (e+f x)}}-\frac{2 a^2 c^3 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{39 f g \sqrt{c-c \sin (e+f x)}}-\frac{14 a c^3 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{585 f g \sqrt{c-c \sin (e+f x)}}+\frac{14 c^3 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{7/2}}{195 f g \sqrt{c-c \sin (e+f x)}}+\frac{22 c^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{7/2} \sqrt{c-c \sin (e+f x)}}{195 f g}+\frac{2 c (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{3/2}}{15 f g}+\frac{1}{195} \left (77 a^3 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{154 a^4 c^3 (g \cos (e+f x))^{5/2}}{585 f g \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}-\frac{22 a^3 c^3 (g \cos (e+f x))^{5/2} \sqrt{a+a \sin (e+f x)}}{195 f g \sqrt{c-c \sin (e+f x)}}-\frac{2 a^2 c^3 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{39 f g \sqrt{c-c \sin (e+f x)}}-\frac{14 a c^3 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{585 f g \sqrt{c-c \sin (e+f x)}}+\frac{14 c^3 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{7/2}}{195 f g \sqrt{c-c \sin (e+f x)}}+\frac{22 c^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{7/2} \sqrt{c-c \sin (e+f x)}}{195 f g}+\frac{2 c (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{3/2}}{15 f g}+\frac{1}{195} \left (77 a^4 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{154 a^4 c^3 (g \cos (e+f x))^{5/2}}{585 f g \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}-\frac{22 a^3 c^3 (g \cos (e+f x))^{5/2} \sqrt{a+a \sin (e+f x)}}{195 f g \sqrt{c-c \sin (e+f x)}}-\frac{2 a^2 c^3 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{39 f g \sqrt{c-c \sin (e+f x)}}-\frac{14 a c^3 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{585 f g \sqrt{c-c \sin (e+f x)}}+\frac{14 c^3 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{7/2}}{195 f g \sqrt{c-c \sin (e+f x)}}+\frac{22 c^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{7/2} \sqrt{c-c \sin (e+f x)}}{195 f g}+\frac{2 c (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{3/2}}{15 f g}+\frac{\left (77 a^4 c^3 g \cos (e+f x)\right ) \int \sqrt{g \cos (e+f x)} \, dx}{195 \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}\\ &=-\frac{154 a^4 c^3 (g \cos (e+f x))^{5/2}}{585 f g \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}-\frac{22 a^3 c^3 (g \cos (e+f x))^{5/2} \sqrt{a+a \sin (e+f x)}}{195 f g \sqrt{c-c \sin (e+f x)}}-\frac{2 a^2 c^3 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{39 f g \sqrt{c-c \sin (e+f x)}}-\frac{14 a c^3 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{585 f g \sqrt{c-c \sin (e+f x)}}+\frac{14 c^3 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{7/2}}{195 f g \sqrt{c-c \sin (e+f x)}}+\frac{22 c^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{7/2} \sqrt{c-c \sin (e+f x)}}{195 f g}+\frac{2 c (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{3/2}}{15 f g}+\frac{\left (77 a^4 c^3 g \sqrt{\cos (e+f x)} \sqrt{g \cos (e+f x)}\right ) \int \sqrt{\cos (e+f x)} \, dx}{195 \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}\\ &=-\frac{154 a^4 c^3 (g \cos (e+f x))^{5/2}}{585 f g \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}+\frac{154 a^4 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 )}{195 f \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}-\frac{22 a^3 c^3 (g \cos (e+f x))^{5/2} \sqrt{a+a \sin (e+f x)}}{195 f g \sqrt{c-c \sin (e+f x)}}-\frac{2 a^2 c^3 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{39 f g \sqrt{c-c \sin (e+f x)}}-\frac{14 a c^3 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{585 f g \sqrt{c-c \sin (e+f x)}}+\frac{14 c^3 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{7/2}}{195 f g \sqrt{c-c \sin (e+f x)}}+\frac{22 c^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{7/2} \sqrt{c-c \sin (e+f x)}}{195 f g}+\frac{2 c (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{3/2}}{15 f g}\\ \end{align*}
Mathematica [A] time = 3.68715, size = 226, normalized size = 0.49 \[ -\frac{a^3 c^2 (\sin (e+f x)-1)^2 (\sin (e+f x)+1)^3 \sqrt{a (\sin (e+f x)+1)} \sqrt{c-c \sin (e+f x)} (g \cos (e+f x))^{3/2} \left (\sqrt{\cos (e+f x)} (-3794 \sin (2 (e+f x))-800 \sin (4 (e+f x))-90 \sin (6 (e+f x))+1365 \cos (e+f x)+819 \cos (3 (e+f x))+273 \cos (5 (e+f x))+39 \cos (7 (e+f x)))-14784 E\left (\left .\frac{1}{2} (e+f x)\right |2\right )\right )}{18720 f \cos ^{\frac{3}{2}}(e+f x) \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^5 \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^7} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.464, size = 392, normalized size = 0.9 \begin{align*} -{\frac{2}{585\,f \left ( 1+\sin \left ( fx+e \right ) \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{7}\sin \left ( fx+e \right ) } \left ( -c \left ( -1+\sin \left ( fx+e \right ) \right ) \right ) ^{{\frac{5}{2}}} \left ( 39\,\sin \left ( fx+e \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{8}+45\, \left ( \cos \left ( fx+e \right ) \right ) ^{8}+10\, \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 ) \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 ) +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{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^{3} c^{2} g \cos \left (f x + e\right )^{5} \sin \left (f x + e\right ) + a^{3} c^{2} g \cos \left (f x + e\right )^{5}\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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