Optimal. Leaf size=290 \[ -\frac{22 a^3 c (g \cos (e+f x))^{5/2}}{45 f g \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{22 a^2 c \sqrt{a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{105 f g \sqrt{c-c \sin (e+f x)}}+\frac{22 a^3 c g \sqrt{\cos (e+f x)} E\left (\left .\frac{1}{2} (e+f x)\right |2\right ) \sqrt{g \cos (e+f x)}}{15 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{2 a c (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{21 f g \sqrt{c-c \sin (e+f x)}}+\frac{2 c (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{9 f g \sqrt{c-c \sin (e+f x)}} \]
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Rubi [A] time = 1.42011, antiderivative size = 290, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 42, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2851, 2842, 2640, 2639} \[ -\frac{22 a^3 c (g \cos (e+f x))^{5/2}}{45 f g \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{22 a^2 c \sqrt{a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{105 f g \sqrt{c-c \sin (e+f x)}}+\frac{22 a^3 c g \sqrt{\cos (e+f x)} E\left (\left .\frac{1}{2} (e+f x)\right |2\right ) \sqrt{g \cos (e+f x)}}{15 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{2 a c (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{21 f g \sqrt{c-c \sin (e+f x)}}+\frac{2 c (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{9 f g \sqrt{c-c \sin (e+f x)}} \]
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))^{5/2} \sqrt{c-c \sin (e+f x)} \, dx &=\frac{2 c (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{9 f g \sqrt{c-c \sin (e+f x)}}+\frac{1}{3} c \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 c (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{21 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))^{5/2}}{9 f g \sqrt{c-c \sin (e+f x)}}+\frac{1}{21} (11 a c) \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^2 c (g \cos (e+f x))^{5/2} \sqrt{a+a \sin (e+f x)}}{105 f g \sqrt{c-c \sin (e+f x)}}-\frac{2 a c (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{21 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))^{5/2}}{9 f g \sqrt{c-c \sin (e+f x)}}+\frac{1}{15} \left (11 a^2 c\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{22 a^3 c (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{22 a^2 c (g \cos (e+f x))^{5/2} \sqrt{a+a \sin (e+f x)}}{105 f g \sqrt{c-c \sin (e+f x)}}-\frac{2 a c (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{21 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))^{5/2}}{9 f g \sqrt{c-c \sin (e+f x)}}+\frac{1}{15} \left (11 a^3 c\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{22 a^3 c (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{22 a^2 c (g \cos (e+f x))^{5/2} \sqrt{a+a \sin (e+f x)}}{105 f g \sqrt{c-c \sin (e+f x)}}-\frac{2 a c (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{21 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))^{5/2}}{9 f g \sqrt{c-c \sin (e+f x)}}+\frac{\left (11 a^3 c 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{22 a^3 c (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{22 a^2 c (g \cos (e+f x))^{5/2} \sqrt{a+a \sin (e+f x)}}{105 f g \sqrt{c-c \sin (e+f x)}}-\frac{2 a c (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{21 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))^{5/2}}{9 f g \sqrt{c-c \sin (e+f x)}}+\frac{\left (11 a^3 c 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{22 a^3 c (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{22 a^3 c 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{22 a^2 c (g \cos (e+f x))^{5/2} \sqrt{a+a \sin (e+f x)}}{105 f g \sqrt{c-c \sin (e+f x)}}-\frac{2 a c (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{21 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))^{5/2}}{9 f g \sqrt{c-c \sin (e+f x)}}\\ \end{align*}
Mathematica [C] time = 2.48427, size = 281, normalized size = 0.97 \[ \frac{a^3 g e^{-4 i (e+f x)} \left (e^{i (e+f x)}+i\right ) \left (\sqrt{1+e^{2 i (e+f x)}} \left (180 i e^{i (e+f x)}+238 e^{2 i (e+f x)}+540 i e^{3 i (e+f x)}+3696 e^{4 i (e+f x)}+540 i e^{5 i (e+f x)}-238 e^{6 i (e+f x)}+180 i e^{7 i (e+f x)}+35 e^{8 i (e+f x)}-35\right )-2464 e^{6 i (e+f x)} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (e+f x)}\right )\right ) \sqrt{c-c \sin (e+f x)} \sqrt{g \cos (e+f x)}}{2520 f \left (e^{i (e+f x)}-i\right ) \sqrt{1+e^{2 i (e+f x)}} \sqrt{a (\sin (e+f x)+1)}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.365, size = 394, normalized size = 1.4 \begin{align*}{\frac{2}{315\,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 ) ^{3}}\sqrt{-c \left ( -1+\sin \left ( fx+e \right ) \right ) } \left ( 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 ) \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}}}+35\, \left ( \cos \left ( fx+e \right ) \right ) ^{6}+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 ) -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}}}-90\,\sin \left ( fx+e \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{4}-112\, \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{5}{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{5}{2}} \sqrt{-c \sin \left (f x + e\right ) + c}\,{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^{2} g \cos \left (f x + e\right )^{3} - 2 \, a^{2} g \cos \left (f x + e\right ) \sin \left (f x + e\right ) - 2 \, a^{2} g \cos \left (f x + e\right )\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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