Optimal. Leaf size=150 \[ \frac{a^3 c^2 \cos ^5(e+f x)}{f (c-c \sin (e+f x))^{7/2}}-\frac{10 \sqrt{2} a^3 \tanh ^{-1}\left (\frac{\sqrt{c} \cos (e+f x)}{\sqrt{2} \sqrt{c-c \sin (e+f x)}}\right )}{c^{3/2} f}+\frac{5 a^3 \cos ^3(e+f x)}{3 f (c-c \sin (e+f x))^{3/2}}+\frac{10 a^3 \cos (e+f x)}{c f \sqrt{c-c \sin (e+f x)}} \]
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Rubi [A] time = 0.320458, antiderivative size = 150, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {2736, 2680, 2679, 2649, 206} \[ \frac{a^3 c^2 \cos ^5(e+f x)}{f (c-c \sin (e+f x))^{7/2}}-\frac{10 \sqrt{2} a^3 \tanh ^{-1}\left (\frac{\sqrt{c} \cos (e+f x)}{\sqrt{2} \sqrt{c-c \sin (e+f x)}}\right )}{c^{3/2} f}+\frac{5 a^3 \cos ^3(e+f x)}{3 f (c-c \sin (e+f x))^{3/2}}+\frac{10 a^3 \cos (e+f x)}{c f \sqrt{c-c \sin (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 2736
Rule 2680
Rule 2679
Rule 2649
Rule 206
Rubi steps
\begin{align*} \int \frac{(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^{3/2}} \, dx &=\left (a^3 c^3\right ) \int \frac{\cos ^6(e+f x)}{(c-c \sin (e+f x))^{9/2}} \, dx\\ &=\frac{a^3 c^2 \cos ^5(e+f x)}{f (c-c \sin (e+f x))^{7/2}}-\frac{1}{2} \left (5 a^3 c\right ) \int \frac{\cos ^4(e+f x)}{(c-c \sin (e+f x))^{5/2}} \, dx\\ &=\frac{a^3 c^2 \cos ^5(e+f x)}{f (c-c \sin (e+f x))^{7/2}}+\frac{5 a^3 \cos ^3(e+f x)}{3 f (c-c \sin (e+f x))^{3/2}}-\left (5 a^3\right ) \int \frac{\cos ^2(e+f x)}{(c-c \sin (e+f x))^{3/2}} \, dx\\ &=\frac{a^3 c^2 \cos ^5(e+f x)}{f (c-c \sin (e+f x))^{7/2}}+\frac{5 a^3 \cos ^3(e+f x)}{3 f (c-c \sin (e+f x))^{3/2}}+\frac{10 a^3 \cos (e+f x)}{c f \sqrt{c-c \sin (e+f x)}}-\frac{\left (10 a^3\right ) \int \frac{1}{\sqrt{c-c \sin (e+f x)}} \, dx}{c}\\ &=\frac{a^3 c^2 \cos ^5(e+f x)}{f (c-c \sin (e+f x))^{7/2}}+\frac{5 a^3 \cos ^3(e+f x)}{3 f (c-c \sin (e+f x))^{3/2}}+\frac{10 a^3 \cos (e+f x)}{c f \sqrt{c-c \sin (e+f x)}}+\frac{\left (20 a^3\right ) \operatorname{Subst}\left (\int \frac{1}{2 c-x^2} \, dx,x,-\frac{c \cos (e+f x)}{\sqrt{c-c \sin (e+f x)}}\right )}{c f}\\ &=-\frac{10 \sqrt{2} a^3 \tanh ^{-1}\left (\frac{\sqrt{c} \cos (e+f x)}{\sqrt{2} \sqrt{c-c \sin (e+f x)}}\right )}{c^{3/2} f}+\frac{a^3 c^2 \cos ^5(e+f x)}{f (c-c \sin (e+f x))^{7/2}}+\frac{5 a^3 \cos ^3(e+f x)}{3 f (c-c \sin (e+f x))^{3/2}}+\frac{10 a^3 \cos (e+f x)}{c f \sqrt{c-c \sin (e+f x)}}\\ \end{align*}
Mathematica [C] time = 0.798989, size = 173, normalized size = 1.15 \[ -\frac{a^3 \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right ) \left (50 \sin \left (\frac{1}{2} (e+f x)\right )-25 \sin \left (\frac{3}{2} (e+f x)\right )+\sin \left (\frac{5}{2} (e+f x)\right )+50 \cos \left (\frac{1}{2} (e+f x)\right )+25 \cos \left (\frac{3}{2} (e+f x)\right )+\cos \left (\frac{5}{2} (e+f x)\right )-(120+120 i) \sqrt [4]{-1} (\sin (e+f x)-1) \tan ^{-1}\left (\left (\frac{1}{2}+\frac{i}{2}\right ) \sqrt [4]{-1} \left (\tan \left (\frac{1}{4} (e+f x)\right )+1\right )\right )\right )}{6 c f (\sin (e+f x)-1) \sqrt{c-c \sin (e+f x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.599, size = 189, normalized size = 1.3 \begin{align*}{\frac{2\,{a}^{3}}{3\,f\cos \left ( fx+e \right ) } \left ( 15\,\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{c \left ( 1+\sin \left ( fx+e \right ) \right ) }\sqrt{2}}{\sqrt{c}}} \right ) \sin \left ( fx+e \right ){c}^{2}-12\,\sqrt{c \left ( 1+\sin \left ( fx+e \right ) \right ) }{c}^{3/2}\sin \left ( fx+e \right ) - \left ( c \left ( 1+\sin \left ( fx+e \right ) \right ) \right ) ^{{\frac{3}{2}}}\sqrt{c}\sin \left ( fx+e \right ) -15\,\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{c \left ( 1+\sin \left ( fx+e \right ) \right ) }\sqrt{2}}{\sqrt{c}}} \right ){c}^{2}+18\,\sqrt{c \left ( 1+\sin \left ( fx+e \right ) \right ) }{c}^{3/2}+ \left ( c \left ( 1+\sin \left ( fx+e \right ) \right ) \right ) ^{{\frac{3}{2}}}\sqrt{c} \right ) \sqrt{c \left ( 1+\sin \left ( fx+e \right ) \right ) }{c}^{-{\frac{7}{2}}}{\frac{1}{\sqrt{c-c\sin \left ( fx+e \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 (a \sin \left (f x + e\right ) + a\right )}^{3}}{{\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 [B] time = 1.12119, size = 844, normalized size = 5.63 \begin{align*} \frac{\frac{15 \, \sqrt{2}{\left (a^{3} c \cos \left (f x + e\right )^{2} - a^{3} c \cos \left (f x + e\right ) - 2 \, a^{3} c +{\left (a^{3} c \cos \left (f x + e\right ) + 2 \, a^{3} c\right )} \sin \left (f x + e\right )\right )} \log \left (-\frac{\cos \left (f x + e\right )^{2} +{\left (\cos \left (f x + e\right ) - 2\right )} \sin \left (f x + e\right ) - \frac{2 \, \sqrt{2} \sqrt{-c \sin \left (f x + e\right ) + c}{\left (\cos \left (f x + e\right ) + \sin \left (f x + e\right ) + 1\right )}}{\sqrt{c}} + 3 \, \cos \left (f x + e\right ) + 2}{\cos \left (f x + e\right )^{2} +{\left (\cos \left (f x + e\right ) + 2\right )} \sin \left (f x + e\right ) - \cos \left (f x + e\right ) - 2}\right )}{\sqrt{c}} - 2 \,{\left (a^{3} \cos \left (f x + e\right )^{3} + 13 \, a^{3} \cos \left (f x + e\right )^{2} + 18 \, a^{3} \cos \left (f x + e\right ) + 6 \, a^{3} +{\left (a^{3} \cos \left (f x + e\right )^{2} - 12 \, a^{3} \cos \left (f x + e\right ) + 6 \, a^{3}\right )} \sin \left (f x + e\right )\right )} \sqrt{-c \sin \left (f x + e\right ) + c}}{3 \,{\left (c^{2} f \cos \left (f x + e\right )^{2} - c^{2} f \cos \left (f x + e\right ) - 2 \, c^{2} f +{\left (c^{2} f \cos \left (f x + e\right ) + 2 \, c^{2} f\right )} \sin \left (f x + e\right )\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*} \mathit{sage}_{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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