Optimal. Leaf size=106 \[ \frac{2 a^3 c^2 \cos ^5(e+f x)}{5 f (c-c \sin (e+f x))^5}+\frac{2 a^3 \cos (e+f x)}{f \left (c^3-c^3 \sin (e+f x)\right )}-\frac{a^3 x}{c^3}-\frac{2 a^3 \cos ^3(e+f x)}{3 f (c-c \sin (e+f x))^3} \]
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Rubi [A] time = 0.188967, antiderivative size = 106, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {2736, 2680, 8} \[ \frac{2 a^3 c^2 \cos ^5(e+f x)}{5 f (c-c \sin (e+f x))^5}+\frac{2 a^3 \cos (e+f x)}{f \left (c^3-c^3 \sin (e+f x)\right )}-\frac{a^3 x}{c^3}-\frac{2 a^3 \cos ^3(e+f x)}{3 f (c-c \sin (e+f x))^3} \]
Antiderivative was successfully verified.
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Rule 2736
Rule 2680
Rule 8
Rubi steps
\begin{align*} \int \frac{(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^3} \, dx &=\left (a^3 c^3\right ) \int \frac{\cos ^6(e+f x)}{(c-c \sin (e+f x))^6} \, dx\\ &=\frac{2 a^3 c^2 \cos ^5(e+f x)}{5 f (c-c \sin (e+f x))^5}-\left (a^3 c\right ) \int \frac{\cos ^4(e+f x)}{(c-c \sin (e+f x))^4} \, dx\\ &=\frac{2 a^3 c^2 \cos ^5(e+f x)}{5 f (c-c \sin (e+f x))^5}-\frac{2 a^3 \cos ^3(e+f x)}{3 f (c-c \sin (e+f x))^3}+\frac{a^3 \int \frac{\cos ^2(e+f x)}{(c-c \sin (e+f x))^2} \, dx}{c}\\ &=\frac{2 a^3 c^2 \cos ^5(e+f x)}{5 f (c-c \sin (e+f x))^5}-\frac{2 a^3 \cos ^3(e+f x)}{3 f (c-c \sin (e+f x))^3}+\frac{2 a^3 \cos (e+f x)}{f \left (c^3-c^3 \sin (e+f x)\right )}-\frac{a^3 \int 1 \, dx}{c^3}\\ &=-\frac{a^3 x}{c^3}+\frac{2 a^3 c^2 \cos ^5(e+f x)}{5 f (c-c \sin (e+f x))^5}-\frac{2 a^3 \cos ^3(e+f x)}{3 f (c-c \sin (e+f x))^3}+\frac{2 a^3 \cos (e+f x)}{f \left (c^3-c^3 \sin (e+f x)\right )}\\ \end{align*}
Mathematica [B] time = 0.455322, size = 249, normalized size = 2.35 \[ \frac{(a \sin (e+f x)+a)^3 \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right ) \left (48 \sin \left (\frac{1}{2} (e+f x)\right )-15 (e+f x) \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^5+92 \sin \left (\frac{1}{2} (e+f x)\right ) \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^4-44 \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^3-88 \sin \left (\frac{1}{2} (e+f x)\right ) \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^2+24 \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )\right )}{15 f (c-c \sin (e+f x))^3 \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^6} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.092, size = 143, normalized size = 1.4 \begin{align*} -{\frac{64\,{a}^{3}}{5\,f{c}^{3}} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) -1 \right ) ^{-5}}-32\,{\frac{{a}^{3}}{f{c}^{3} \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{4}}}-{\frac{80\,{a}^{3}}{3\,f{c}^{3}} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) -1 \right ) ^{-3}}-8\,{\frac{{a}^{3}}{f{c}^{3} \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{2}}}-4\,{\frac{{a}^{3}}{f{c}^{3} \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) }}-2\,{\frac{{a}^{3}\arctan \left ( \tan \left ( 1/2\,fx+e/2 \right ) \right ) }{f{c}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 2.66346, size = 1060, normalized size = 10. \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.35291, size = 551, normalized size = 5.2 \begin{align*} \frac{60 \, a^{3} f x -{\left (15 \, a^{3} f x - 46 \, a^{3}\right )} \cos \left (f x + e\right )^{3} - 24 \, a^{3} -{\left (45 \, a^{3} f x + 2 \, a^{3}\right )} \cos \left (f x + e\right )^{2} + 6 \,{\left (5 \, a^{3} f x - 12 \, a^{3}\right )} \cos \left (f x + e\right ) -{\left (60 \, a^{3} f x + 24 \, a^{3} -{\left (15 \, a^{3} f x + 46 \, a^{3}\right )} \cos \left (f x + e\right )^{2} + 6 \,{\left (5 \, a^{3} f x - 8 \, a^{3}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{15 \,{\left (c^{3} f \cos \left (f x + e\right )^{3} + 3 \, c^{3} f \cos \left (f x + e\right )^{2} - 2 \, c^{3} f \cos \left (f x + e\right ) - 4 \, c^{3} f -{\left (c^{3} f \cos \left (f x + e\right )^{2} - 2 \, c^{3} f \cos \left (f x + e\right ) - 4 \, c^{3} 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 [A] time = 2.35508, size = 150, normalized size = 1.42 \begin{align*} -\frac{\frac{15 \,{\left (f x + e\right )} a^{3}}{c^{3}} + \frac{4 \,{\left (15 \, a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} - 30 \, a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} + 100 \, a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - 50 \, a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 13 \, a^{3}\right )}}{c^{3}{\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) - 1\right )}^{5}}}{15 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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