Optimal. Leaf size=112 \[ \frac{a^2 c^2 (A+B) \cos ^5(e+f x)}{5 f (c-c \sin (e+f x))^5}+\frac{2 a^2 B \cos (e+f x)}{f \left (c^3-c^3 \sin (e+f x)\right )}-\frac{a^2 B x}{c^3}-\frac{2 a^2 B \cos ^3(e+f x)}{3 f (c-c \sin (e+f x))^3} \]
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Rubi [A] time = 0.277027, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {2967, 2859, 2680, 8} \[ \frac{a^2 c^2 (A+B) \cos ^5(e+f x)}{5 f (c-c \sin (e+f x))^5}+\frac{2 a^2 B \cos (e+f x)}{f \left (c^3-c^3 \sin (e+f x)\right )}-\frac{a^2 B x}{c^3}-\frac{2 a^2 B \cos ^3(e+f x)}{3 f (c-c \sin (e+f x))^3} \]
Antiderivative was successfully verified.
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Rule 2967
Rule 2859
Rule 2680
Rule 8
Rubi steps
\begin{align*} \int \frac{(a+a \sin (e+f x))^2 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^3} \, dx &=\left (a^2 c^2\right ) \int \frac{\cos ^4(e+f x) (A+B \sin (e+f x))}{(c-c \sin (e+f x))^5} \, dx\\ &=\frac{a^2 (A+B) c^2 \cos ^5(e+f x)}{5 f (c-c \sin (e+f x))^5}-\left (a^2 B c\right ) \int \frac{\cos ^4(e+f x)}{(c-c \sin (e+f x))^4} \, dx\\ &=\frac{a^2 (A+B) c^2 \cos ^5(e+f x)}{5 f (c-c \sin (e+f x))^5}-\frac{2 a^2 B \cos ^3(e+f x)}{3 f (c-c \sin (e+f x))^3}+\frac{\left (a^2 B\right ) \int \frac{\cos ^2(e+f x)}{(c-c \sin (e+f x))^2} \, dx}{c}\\ &=\frac{a^2 (A+B) c^2 \cos ^5(e+f x)}{5 f (c-c \sin (e+f x))^5}-\frac{2 a^2 B \cos ^3(e+f x)}{3 f (c-c \sin (e+f x))^3}+\frac{2 a^2 B \cos (e+f x)}{f \left (c^3-c^3 \sin (e+f x)\right )}-\frac{\left (a^2 B\right ) \int 1 \, dx}{c^3}\\ &=-\frac{a^2 B x}{c^3}+\frac{a^2 (A+B) c^2 \cos ^5(e+f x)}{5 f (c-c \sin (e+f x))^5}-\frac{2 a^2 B \cos ^3(e+f x)}{3 f (c-c \sin (e+f x))^3}+\frac{2 a^2 B \cos (e+f x)}{f \left (c^3-c^3 \sin (e+f x)\right )}\\ \end{align*}
Mathematica [B] time = 0.695633, size = 278, normalized size = 2.48 \[ \frac{a^2 (\sin (e+f x)+1)^2 \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right ) \left (24 (A+B) \sin \left (\frac{1}{2} (e+f x)\right )+2 (3 A+43 B) \sin \left (\frac{1}{2} (e+f x)\right ) \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^4-4 (3 A+8 B) \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^3-8 (3 A+8 B) \sin \left (\frac{1}{2} (e+f x)\right ) \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^2+12 (A+B) \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )-15 B (e+f x) \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^5\right )}{15 f (c-c \sin (e+f x))^3 \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^4} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.127, size = 249, normalized size = 2.2 \begin{align*} -2\,{\frac{A{a}^{2}}{f{c}^{3} \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) }}-2\,{\frac{B{a}^{2}}{f{c}^{3} \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) }}-{\frac{32\,A{a}^{2}}{5\,f{c}^{3}} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) -1 \right ) ^{-5}}-{\frac{32\,B{a}^{2}}{5\,f{c}^{3}} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) -1 \right ) ^{-5}}-16\,{\frac{A{a}^{2}}{f{c}^{3} \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{3}}}-{\frac{32\,B{a}^{2}}{3\,f{c}^{3}} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) -1 \right ) ^{-3}}-16\,{\frac{A{a}^{2}}{f{c}^{3} \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{4}}}-16\,{\frac{B{a}^{2}}{f{c}^{3} \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{4}}}-8\,{\frac{A{a}^{2}}{f{c}^{3} \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{2}}}-2\,{\frac{B{a}^{2}\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 = 1.57944, size = 1538, normalized size = 13.73 \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 = 1.42677, size = 655, normalized size = 5.85 \begin{align*} \frac{60 \, B a^{2} f x -{\left (15 \, B a^{2} f x -{\left (3 \, A + 43 \, B\right )} a^{2}\right )} \cos \left (f x + e\right )^{3} - 12 \,{\left (A + B\right )} a^{2} -{\left (45 \, B a^{2} f x -{\left (9 \, A - 11 \, B\right )} a^{2}\right )} \cos \left (f x + e\right )^{2} + 6 \,{\left (5 \, B a^{2} f x -{\left (A + 11 \, B\right )} a^{2}\right )} \cos \left (f x + e\right ) -{\left (60 \, B a^{2} f x + 12 \,{\left (A + B\right )} a^{2} -{\left (15 \, B a^{2} f x +{\left (3 \, A + 43 \, B\right )} a^{2}\right )} \cos \left (f x + e\right )^{2} + 6 \,{\left (5 \, B a^{2} f x +{\left (A - 9 \, B\right )} a^{2}\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 = 1.21705, size = 215, normalized size = 1.92 \begin{align*} -\frac{\frac{15 \,{\left (f x + e\right )} B a^{2}}{c^{3}} + \frac{2 \,{\left (15 \, A a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} + 15 \, B a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} - 60 \, B a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} + 30 \, A a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} + 170 \, B a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - 100 \, B a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 3 \, A a^{2} + 23 \, B a^{2}\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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