Optimal. Leaf size=94 \[ \frac{2 a^3 c^2 \cos ^5(e+f x)}{f (c-c \sin (e+f x))^3}+\frac{15 a^3 \cos (e+f x)}{2 c f}+\frac{5 a^3 \cos ^3(e+f x)}{2 f (c-c \sin (e+f x))}-\frac{15 a^3 x}{2 c} \]
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Rubi [A] time = 0.182553, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {2736, 2680, 2679, 2682, 8} \[ \frac{2 a^3 c^2 \cos ^5(e+f x)}{f (c-c \sin (e+f x))^3}+\frac{15 a^3 \cos (e+f x)}{2 c f}+\frac{5 a^3 \cos ^3(e+f x)}{2 f (c-c \sin (e+f x))}-\frac{15 a^3 x}{2 c} \]
Antiderivative was successfully verified.
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Rule 2736
Rule 2680
Rule 2679
Rule 2682
Rule 8
Rubi steps
\begin{align*} \int \frac{(a+a \sin (e+f x))^3}{c-c \sin (e+f x)} \, dx &=\left (a^3 c^3\right ) \int \frac{\cos ^6(e+f x)}{(c-c \sin (e+f x))^4} \, dx\\ &=\frac{2 a^3 c^2 \cos ^5(e+f x)}{f (c-c \sin (e+f x))^3}-\left (5 a^3 c\right ) \int \frac{\cos ^4(e+f x)}{(c-c \sin (e+f x))^2} \, dx\\ &=\frac{2 a^3 c^2 \cos ^5(e+f x)}{f (c-c \sin (e+f x))^3}+\frac{5 a^3 \cos ^3(e+f x)}{2 f (c-c \sin (e+f x))}-\frac{1}{2} \left (15 a^3\right ) \int \frac{\cos ^2(e+f x)}{c-c \sin (e+f x)} \, dx\\ &=\frac{15 a^3 \cos (e+f x)}{2 c f}+\frac{2 a^3 c^2 \cos ^5(e+f x)}{f (c-c \sin (e+f x))^3}+\frac{5 a^3 \cos ^3(e+f x)}{2 f (c-c \sin (e+f x))}-\frac{\left (15 a^3\right ) \int 1 \, dx}{2 c}\\ &=-\frac{15 a^3 x}{2 c}+\frac{15 a^3 \cos (e+f x)}{2 c f}+\frac{2 a^3 c^2 \cos ^5(e+f x)}{f (c-c \sin (e+f x))^3}+\frac{5 a^3 \cos ^3(e+f x)}{2 f (c-c \sin (e+f x))}\\ \end{align*}
Mathematica [A] time = 0.50658, size = 153, normalized size = 1.63 \[ \frac{a^3 (\sin (e+f x)+1)^3 \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right ) \left (\cos \left (\frac{1}{2} (e+f x)\right ) (30 (e+f x)-\sin (2 (e+f x))-16 \cos (e+f x))+\sin \left (\frac{1}{2} (e+f x)\right ) (\sin (2 (e+f x))+16 \cos (e+f x)-30 e-30 f x-64)\right )}{4 c f (\sin (e+f x)-1) \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^6} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.08, size = 181, normalized size = 1.9 \begin{align*} -16\,{\frac{{a}^{3}}{cf \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) }}-{\frac{{a}^{3}}{cf} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{3} \left ( 1+ \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{2} \right ) ^{-2}}+8\,{\frac{{a}^{3} \left ( \tan \left ( 1/2\,fx+e/2 \right ) \right ) ^{2}}{cf \left ( 1+ \left ( \tan \left ( 1/2\,fx+e/2 \right ) \right ) ^{2} \right ) ^{2}}}+{\frac{{a}^{3}}{cf}\tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \left ( 1+ \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{2} \right ) ^{-2}}+8\,{\frac{{a}^{3}}{cf \left ( 1+ \left ( \tan \left ( 1/2\,fx+e/2 \right ) \right ) ^{2} \right ) ^{2}}}-15\,{\frac{{a}^{3}\arctan \left ( \tan \left ( 1/2\,fx+e/2 \right ) \right ) }{cf}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 2.1269, size = 585, normalized size = 6.22 \begin{align*} -\frac{6 \, a^{3}{\left (\frac{\frac{\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac{\sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - 2}{c - \frac{c \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac{c \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac{c \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}} + \frac{\arctan \left (\frac{\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )}{c}\right )} + a^{3}{\left (\frac{\frac{\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac{5 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac{3 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} - \frac{3 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - 4}{c - \frac{c \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac{2 \, c \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac{2 \, c \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac{c \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - \frac{c \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}} + \frac{3 \, \arctan \left (\frac{\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )}{c}\right )} + 6 \, a^{3}{\left (\frac{\arctan \left (\frac{\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )}{c} - \frac{1}{c - \frac{c \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}}\right )} - \frac{2 \, a^{3}}{c - \frac{c \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}}}{f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.409, size = 312, normalized size = 3.32 \begin{align*} \frac{a^{3} \cos \left (f x + e\right )^{3} - 15 \, a^{3} f x + 8 \, a^{3} \cos \left (f x + e\right )^{2} + 16 \, a^{3} -{\left (15 \, a^{3} f x - 23 \, a^{3}\right )} \cos \left (f x + e\right ) +{\left (15 \, a^{3} f x + a^{3} \cos \left (f x + e\right )^{2} - 7 \, a^{3} \cos \left (f x + e\right ) + 16 \, a^{3}\right )} \sin \left (f x + e\right )}{2 \,{\left (c f \cos \left (f x + e\right ) - c f \sin \left (f x + e\right ) + c f\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 17.9005, size = 1170, normalized size = 12.45 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.29381, size = 158, normalized size = 1.68 \begin{align*} -\frac{\frac{15 \,{\left (f x + e\right )} a^{3}}{c} + \frac{32 \, a^{3}}{c{\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) - 1\right )}} + \frac{2 \,{\left (a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} - 8 \, a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) - 8 \, a^{3}\right )}}{{\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} + 1\right )}^{2} c}}{2 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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