Optimal. Leaf size=233 \[ \frac{a^2 \left (-44 c^2 d^2-10 c^3 d+c^4-40 c d^3-12 d^4\right ) \cos (e+f x)}{10 d f}+\frac{a^2 \left (c^2-10 c d-12 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{20 d f}+\frac{a^2 \left (-20 c^2 d+2 c^3-57 c d^2-30 d^3\right ) \sin (e+f x) \cos (e+f x)}{40 f}+\frac{3}{8} a^2 x (2 c+d) \left (2 c^2+3 c d+2 d^2\right )-\frac{a^2 \cos (e+f x) (c+d \sin (e+f x))^4}{5 d f}+\frac{a^2 (c-10 d) \cos (e+f x) (c+d \sin (e+f x))^3}{20 d f} \]
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Rubi [A] time = 0.308547, antiderivative size = 233, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {2763, 2753, 2734} \[ \frac{a^2 \left (-44 c^2 d^2-10 c^3 d+c^4-40 c d^3-12 d^4\right ) \cos (e+f x)}{10 d f}+\frac{a^2 \left (c^2-10 c d-12 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{20 d f}+\frac{a^2 \left (-20 c^2 d+2 c^3-57 c d^2-30 d^3\right ) \sin (e+f x) \cos (e+f x)}{40 f}+\frac{3}{8} a^2 x (2 c+d) \left (2 c^2+3 c d+2 d^2\right )-\frac{a^2 \cos (e+f x) (c+d \sin (e+f x))^4}{5 d f}+\frac{a^2 (c-10 d) \cos (e+f x) (c+d \sin (e+f x))^3}{20 d f} \]
Antiderivative was successfully verified.
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Rule 2763
Rule 2753
Rule 2734
Rubi steps
\begin{align*} \int (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^3 \, dx &=-\frac{a^2 \cos (e+f x) (c+d \sin (e+f x))^4}{5 d f}+\frac{\int \left (9 a^2 d-a^2 (c-10 d) \sin (e+f x)\right ) (c+d \sin (e+f x))^3 \, dx}{5 d}\\ &=\frac{a^2 (c-10 d) \cos (e+f x) (c+d \sin (e+f x))^3}{20 d f}-\frac{a^2 \cos (e+f x) (c+d \sin (e+f x))^4}{5 d f}+\frac{\int (c+d \sin (e+f x))^2 \left (3 a^2 d (11 c+10 d)-3 a^2 \left (c^2-10 c d-12 d^2\right ) \sin (e+f x)\right ) \, dx}{20 d}\\ &=\frac{a^2 \left (c^2-10 c d-12 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{20 d f}+\frac{a^2 (c-10 d) \cos (e+f x) (c+d \sin (e+f x))^3}{20 d f}-\frac{a^2 \cos (e+f x) (c+d \sin (e+f x))^4}{5 d f}+\frac{\int (c+d \sin (e+f x)) \left (3 a^2 d \left (31 c^2+50 c d+24 d^2\right )-3 a^2 \left (2 c^3-20 c^2 d-57 c d^2-30 d^3\right ) \sin (e+f x)\right ) \, dx}{60 d}\\ &=\frac{3}{8} a^2 (2 c+d) \left (2 c^2+3 c d+2 d^2\right ) x+\frac{a^2 \left (c^4-10 c^3 d-44 c^2 d^2-40 c d^3-12 d^4\right ) \cos (e+f x)}{10 d f}+\frac{a^2 \left (2 c^3-20 c^2 d-57 c d^2-30 d^3\right ) \cos (e+f x) \sin (e+f x)}{40 f}+\frac{a^2 \left (c^2-10 c d-12 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{20 d f}+\frac{a^2 (c-10 d) \cos (e+f x) (c+d \sin (e+f x))^3}{20 d f}-\frac{a^2 \cos (e+f x) (c+d \sin (e+f x))^4}{5 d f}\\ \end{align*}
Mathematica [A] time = 0.909665, size = 204, normalized size = 0.88 \[ -\frac{a^2 \cos (e+f x) \left (30 \left (8 c^2 d+4 c^3+7 c d^2+2 d^3\right ) \sin ^{-1}\left (\frac{\sqrt{1-\sin (e+f x)}}{\sqrt{2}}\right )+\sqrt{\cos ^2(e+f x)} \left (8 d \left (5 c^2+10 c d+3 d^2\right ) \sin ^2(e+f x)+5 \left (24 c^2 d+4 c^3+21 c d^2+6 d^3\right ) \sin (e+f x)+8 \left (25 c^2 d+10 c^3+20 c d^2+6 d^3\right )+10 d^2 (3 c+2 d) \sin ^3(e+f x)+8 d^3 \sin ^4(e+f x)\right )\right )}{40 f \sqrt{\cos ^2(e+f x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 329, normalized size = 1.4 \begin{align*}{\frac{1}{f} \left ({a}^{2}{c}^{3} \left ( -{\frac{\sin \left ( fx+e \right ) \cos \left ( fx+e \right ) }{2}}+{\frac{fx}{2}}+{\frac{e}{2}} \right ) -{a}^{2}{c}^{2}d \left ( 2+ \left ( \sin \left ( fx+e \right ) \right ) ^{2} \right ) \cos \left ( fx+e \right ) +3\,{a}^{2}c{d}^{2} \left ( -1/4\, \left ( \left ( \sin \left ( fx+e \right ) \right ) ^{3}+3/2\,\sin \left ( fx+e \right ) \right ) \cos \left ( fx+e \right ) +3/8\,fx+3/8\,e \right ) -{\frac{{a}^{2}{d}^{3}\cos \left ( fx+e \right ) }{5} \left ({\frac{8}{3}}+ \left ( \sin \left ( fx+e \right ) \right ) ^{4}+{\frac{4\, \left ( \sin \left ( fx+e \right ) \right ) ^{2}}{3}} \right ) }-2\,{a}^{2}{c}^{3}\cos \left ( fx+e \right ) +6\,{a}^{2}{c}^{2}d \left ( -1/2\,\sin \left ( fx+e \right ) \cos \left ( fx+e \right ) +1/2\,fx+e/2 \right ) -2\,{a}^{2}c{d}^{2} \left ( 2+ \left ( \sin \left ( fx+e \right ) \right ) ^{2} \right ) \cos \left ( fx+e \right ) +2\,{a}^{2}{d}^{3} \left ( -1/4\, \left ( \left ( \sin \left ( fx+e \right ) \right ) ^{3}+3/2\,\sin \left ( fx+e \right ) \right ) \cos \left ( fx+e \right ) +3/8\,fx+3/8\,e \right ) +{a}^{2}{c}^{3} \left ( fx+e \right ) -3\,{a}^{2}{c}^{2}d\cos \left ( fx+e \right ) +3\,{a}^{2}c{d}^{2} \left ( -1/2\,\sin \left ( fx+e \right ) \cos \left ( fx+e \right ) +1/2\,fx+e/2 \right ) -{\frac{{a}^{2}{d}^{3} \left ( 2+ \left ( \sin \left ( fx+e \right ) \right ) ^{2} \right ) \cos \left ( fx+e \right ) }{3}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.17888, size = 429, normalized size = 1.84 \begin{align*} \frac{120 \,{\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} a^{2} c^{3} + 480 \,{\left (f x + e\right )} a^{2} c^{3} + 480 \,{\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} a^{2} c^{2} d + 720 \,{\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} a^{2} c^{2} d + 960 \,{\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} a^{2} c d^{2} + 45 \,{\left (12 \, f x + 12 \, e + \sin \left (4 \, f x + 4 \, e\right ) - 8 \, \sin \left (2 \, f x + 2 \, e\right )\right )} a^{2} c d^{2} + 360 \,{\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} a^{2} c d^{2} - 32 \,{\left (3 \, \cos \left (f x + e\right )^{5} - 10 \, \cos \left (f x + e\right )^{3} + 15 \, \cos \left (f x + e\right )\right )} a^{2} d^{3} + 160 \,{\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} a^{2} d^{3} + 30 \,{\left (12 \, f x + 12 \, e + \sin \left (4 \, f x + 4 \, e\right ) - 8 \, \sin \left (2 \, f x + 2 \, e\right )\right )} a^{2} d^{3} - 960 \, a^{2} c^{3} \cos \left (f x + e\right ) - 1440 \, a^{2} c^{2} d \cos \left (f x + e\right )}{480 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.79178, size = 474, normalized size = 2.03 \begin{align*} -\frac{8 \, a^{2} d^{3} \cos \left (f x + e\right )^{5} - 40 \,{\left (a^{2} c^{2} d + 2 \, a^{2} c d^{2} + a^{2} d^{3}\right )} \cos \left (f x + e\right )^{3} - 15 \,{\left (4 \, a^{2} c^{3} + 8 \, a^{2} c^{2} d + 7 \, a^{2} c d^{2} + 2 \, a^{2} d^{3}\right )} f x + 80 \,{\left (a^{2} c^{3} + 3 \, a^{2} c^{2} d + 3 \, a^{2} c d^{2} + a^{2} d^{3}\right )} \cos \left (f x + e\right ) - 5 \,{\left (2 \,{\left (3 \, a^{2} c d^{2} + 2 \, a^{2} d^{3}\right )} \cos \left (f x + e\right )^{3} -{\left (4 \, a^{2} c^{3} + 24 \, a^{2} c^{2} d + 27 \, a^{2} c d^{2} + 10 \, a^{2} d^{3}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{40 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.25343, size = 729, normalized size = 3.13 \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 = 1.38968, size = 437, normalized size = 1.88 \begin{align*} -\frac{a^{2} d^{3} \cos \left (5 \, f x + 5 \, e\right )}{80 \, f} + \frac{a^{2} d^{3} \cos \left (3 \, f x + 3 \, e\right )}{12 \, f} - \frac{3 \, a^{2} c d^{2} \sin \left (2 \, f x + 2 \, e\right )}{4 \, f} + \frac{1}{8} \,{\left (4 \, a^{2} c^{3} + 24 \, a^{2} c^{2} d + 9 \, a^{2} c d^{2} + 6 \, a^{2} d^{3}\right )} x + \frac{1}{2} \,{\left (2 \, a^{2} c^{3} + 3 \, a^{2} c d^{2}\right )} x + \frac{{\left (12 \, a^{2} c^{2} d + 24 \, a^{2} c d^{2} + 5 \, a^{2} d^{3}\right )} \cos \left (3 \, f x + 3 \, e\right )}{48 \, f} - \frac{{\left (16 \, a^{2} c^{3} + 18 \, a^{2} c^{2} d + 36 \, a^{2} c d^{2} + 5 \, a^{2} d^{3}\right )} \cos \left (f x + e\right )}{8 \, f} - \frac{3 \,{\left (4 \, a^{2} c^{2} d + a^{2} d^{3}\right )} \cos \left (f x + e\right )}{4 \, f} + \frac{{\left (3 \, a^{2} c d^{2} + 2 \, a^{2} d^{3}\right )} \sin \left (4 \, f x + 4 \, e\right )}{32 \, f} - \frac{{\left (a^{2} c^{3} + 6 \, a^{2} c^{2} d + 3 \, a^{2} c d^{2} + 2 \, a^{2} d^{3}\right )} \sin \left (2 \, f x + 2 \, e\right )}{4 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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