Optimal. Leaf size=315 \[ -\frac {\left (-6 a^3 d^2+60 a^2 b c d+a b^2 \left (100 c^2+71 d^2\right )+90 b^3 c d\right ) \sin (e+f x) \cos (e+f x)}{120 f}+\frac {1}{8} x \left (4 a^3 \left (2 c^2+d^2\right )+24 a^2 b c d+3 a b^2 \left (4 c^2+3 d^2\right )+6 b^3 c d\right )-\frac {\left (-3 a^4 d^2+30 a^3 b c d+4 a^2 b^2 \left (20 c^2+13 d^2\right )+120 a b^3 c d+4 b^4 \left (5 c^2+4 d^2\right )\right ) \cos (e+f x)}{30 b f}-\frac {\left (3 a d (10 b c-a d)+4 b^2 \left (5 c^2+4 d^2\right )\right ) \cos (e+f x) (a+b \sin (e+f x))^2}{60 b f}-\frac {d (10 b c-a d) \cos (e+f x) (a+b \sin (e+f x))^3}{20 b f}-\frac {d^2 \cos (e+f x) (a+b \sin (e+f x))^4}{5 b f} \]
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Rubi [A] time = 0.46, antiderivative size = 315, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2791, 2753, 2734} \[ -\frac {\left (4 a^2 b^2 \left (20 c^2+13 d^2\right )+30 a^3 b c d-3 a^4 d^2+120 a b^3 c d+4 b^4 \left (5 c^2+4 d^2\right )\right ) \cos (e+f x)}{30 b f}-\frac {\left (60 a^2 b c d-6 a^3 d^2+a b^2 \left (100 c^2+71 d^2\right )+90 b^3 c d\right ) \sin (e+f x) \cos (e+f x)}{120 f}+\frac {1}{8} x \left (24 a^2 b c d+4 a^3 \left (2 c^2+d^2\right )+3 a b^2 \left (4 c^2+3 d^2\right )+6 b^3 c d\right )-\frac {\left (3 a d (10 b c-a d)+4 b^2 \left (5 c^2+4 d^2\right )\right ) \cos (e+f x) (a+b \sin (e+f x))^2}{60 b f}-\frac {d (10 b c-a d) \cos (e+f x) (a+b \sin (e+f x))^3}{20 b f}-\frac {d^2 \cos (e+f x) (a+b \sin (e+f x))^4}{5 b f} \]
Antiderivative was successfully verified.
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Rule 2734
Rule 2753
Rule 2791
Rubi steps
\begin {align*} \int (a+b \sin (e+f x))^3 (c+d \sin (e+f x))^2 \, dx &=-\frac {d^2 \cos (e+f x) (a+b \sin (e+f x))^4}{5 b f}+\frac {\int (a+b \sin (e+f x))^3 \left (b \left (5 c^2+4 d^2\right )+d (10 b c-a d) \sin (e+f x)\right ) \, dx}{5 b}\\ &=-\frac {d (10 b c-a d) \cos (e+f x) (a+b \sin (e+f x))^3}{20 b f}-\frac {d^2 \cos (e+f x) (a+b \sin (e+f x))^4}{5 b f}+\frac {\int (a+b \sin (e+f x))^2 \left (b \left (20 a c^2+30 b c d+13 a d^2\right )+\left (3 a d (10 b c-a d)+4 b^2 \left (5 c^2+4 d^2\right )\right ) \sin (e+f x)\right ) \, dx}{20 b}\\ &=-\frac {\left (3 a d (10 b c-a d)+4 b^2 \left (5 c^2+4 d^2\right )\right ) \cos (e+f x) (a+b \sin (e+f x))^2}{60 b f}-\frac {d (10 b c-a d) \cos (e+f x) (a+b \sin (e+f x))^3}{20 b f}-\frac {d^2 \cos (e+f x) (a+b \sin (e+f x))^4}{5 b f}+\frac {\int (a+b \sin (e+f x)) \left (b \left (150 a b c d+8 b^2 \left (5 c^2+4 d^2\right )+a^2 \left (60 c^2+33 d^2\right )\right )+\left (60 a^2 b c d+90 b^3 c d-6 a^3 d^2+a b^2 \left (100 c^2+71 d^2\right )\right ) \sin (e+f x)\right ) \, dx}{60 b}\\ &=\frac {1}{8} \left (24 a^2 b c d+6 b^3 c d+4 a^3 \left (2 c^2+d^2\right )+3 a b^2 \left (4 c^2+3 d^2\right )\right ) x-\frac {\left (30 a^3 b c d+120 a b^3 c d-3 a^4 d^2+4 b^4 \left (5 c^2+4 d^2\right )+4 a^2 b^2 \left (20 c^2+13 d^2\right )\right ) \cos (e+f x)}{30 b f}-\frac {\left (60 a^2 b c d+90 b^3 c d-6 a^3 d^2+a b^2 \left (100 c^2+71 d^2\right )\right ) \cos (e+f x) \sin (e+f x)}{120 f}-\frac {\left (3 a d (10 b c-a d)+4 b^2 \left (5 c^2+4 d^2\right )\right ) \cos (e+f x) (a+b \sin (e+f x))^2}{60 b f}-\frac {d (10 b c-a d) \cos (e+f x) (a+b \sin (e+f x))^3}{20 b f}-\frac {d^2 \cos (e+f x) (a+b \sin (e+f x))^4}{5 b f}\\ \end {align*}
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Mathematica [A] time = 1.59, size = 246, normalized size = 0.78 \[ \frac {10 b \left (12 a^2 d^2+24 a b c d+b^2 \left (4 c^2+5 d^2\right )\right ) \cos (3 (e+f x))+15 \left (4 (e+f x) \left (4 a^3 \left (2 c^2+d^2\right )+24 a^2 b c d+3 a b^2 \left (4 c^2+3 d^2\right )+6 b^3 c d\right )-8 \left (a^3 d^2+6 a^2 b c d+3 a b^2 \left (c^2+d^2\right )+2 b^3 c d\right ) \sin (2 (e+f x))+b^2 d (3 a d+2 b c) \sin (4 (e+f x))\right )-60 \left (16 a^3 c d+6 a^2 b \left (4 c^2+3 d^2\right )+36 a b^2 c d+b^3 \left (6 c^2+5 d^2\right )\right ) \cos (e+f x)-6 b^3 d^2 \cos (5 (e+f x))}{480 f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 253, normalized size = 0.80 \[ -\frac {24 \, b^{3} d^{2} \cos \left (f x + e\right )^{5} - 40 \, {\left (b^{3} c^{2} + 6 \, a b^{2} c d + {\left (3 \, a^{2} b + 2 \, b^{3}\right )} d^{2}\right )} \cos \left (f x + e\right )^{3} - 15 \, {\left (4 \, {\left (2 \, a^{3} + 3 \, a b^{2}\right )} c^{2} + 6 \, {\left (4 \, a^{2} b + b^{3}\right )} c d + {\left (4 \, a^{3} + 9 \, a b^{2}\right )} d^{2}\right )} f x + 120 \, {\left ({\left (3 \, a^{2} b + b^{3}\right )} c^{2} + 2 \, {\left (a^{3} + 3 \, a b^{2}\right )} c d + {\left (3 \, a^{2} b + b^{3}\right )} d^{2}\right )} \cos \left (f x + e\right ) - 15 \, {\left (2 \, {\left (2 \, b^{3} c d + 3 \, a b^{2} d^{2}\right )} \cos \left (f x + e\right )^{3} - {\left (12 \, a b^{2} c^{2} + 2 \, {\left (12 \, a^{2} b + 5 \, b^{3}\right )} c d + {\left (4 \, a^{3} + 15 \, a b^{2}\right )} d^{2}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{120 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.14, size = 274, normalized size = 0.87 \[ -\frac {b^{3} d^{2} \cos \left (5 \, f x + 5 \, e\right )}{80 \, f} + \frac {1}{8} \, {\left (8 \, a^{3} c^{2} + 12 \, a b^{2} c^{2} + 24 \, a^{2} b c d + 6 \, b^{3} c d + 4 \, a^{3} d^{2} + 9 \, a b^{2} d^{2}\right )} x + \frac {{\left (4 \, b^{3} c^{2} + 24 \, a b^{2} c d + 12 \, a^{2} b d^{2} + 5 \, b^{3} d^{2}\right )} \cos \left (3 \, f x + 3 \, e\right )}{48 \, f} - \frac {{\left (24 \, a^{2} b c^{2} + 6 \, b^{3} c^{2} + 16 \, a^{3} c d + 36 \, a b^{2} c d + 18 \, a^{2} b d^{2} + 5 \, b^{3} d^{2}\right )} \cos \left (f x + e\right )}{8 \, f} + \frac {{\left (2 \, b^{3} c d + 3 \, a b^{2} d^{2}\right )} \sin \left (4 \, f x + 4 \, e\right )}{32 \, f} - \frac {{\left (3 \, a b^{2} c^{2} + 6 \, a^{2} b c d + 2 \, b^{3} c d + a^{3} d^{2} + 3 \, a b^{2} d^{2}\right )} \sin \left (2 \, f x + 2 \, e\right )}{4 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.32, size = 325, normalized size = 1.03 \[ \frac {a^{3} c^{2} \left (f x +e \right )-2 a^{3} c d \cos \left (f x +e \right )+a^{3} d^{2} \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-3 a^{2} b \,c^{2} \cos \left (f x +e \right )+6 a^{2} b c d \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-a^{2} b \,d^{2} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )+3 a \,b^{2} c^{2} \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-2 a \,b^{2} c d \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )+3 a \,b^{2} d^{2} \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )-\frac {b^{3} c^{2} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}+2 b^{3} c d \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )-\frac {b^{3} d^{2} \left (\frac {8}{3}+\sin ^{4}\left (f x +e \right )+\frac {4 \left (\sin ^{2}\left (f x +e \right )\right )}{3}\right ) \cos \left (f x +e \right )}{5}}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.52, size = 314, normalized size = 1.00 \[ \frac {480 \, {\left (f x + e\right )} a^{3} c^{2} + 360 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} a b^{2} c^{2} + 160 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} b^{3} c^{2} + 720 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} a^{2} b c d + 960 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} a b^{2} c d + 30 \, {\left (12 \, f x + 12 \, e + \sin \left (4 \, f x + 4 \, e\right ) - 8 \, \sin \left (2 \, f x + 2 \, e\right )\right )} b^{3} c d + 120 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} a^{3} d^{2} + 480 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} a^{2} b d^{2} + 45 \, {\left (12 \, f x + 12 \, e + \sin \left (4 \, f x + 4 \, e\right ) - 8 \, \sin \left (2 \, f x + 2 \, e\right )\right )} a b^{2} d^{2} - 32 \, {\left (3 \, \cos \left (f x + e\right )^{5} - 10 \, \cos \left (f x + e\right )^{3} + 15 \, \cos \left (f x + e\right )\right )} b^{3} d^{2} - 1440 \, a^{2} b c^{2} \cos \left (f x + e\right ) - 960 \, a^{3} c d \cos \left (f x + e\right )}{480 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.52, size = 358, normalized size = 1.14 \[ -\frac {90\,b^3\,c^2\,\cos \left (e+f\,x\right )+75\,b^3\,d^2\,\cos \left (e+f\,x\right )-10\,b^3\,c^2\,\cos \left (3\,e+3\,f\,x\right )-\frac {25\,b^3\,d^2\,\cos \left (3\,e+3\,f\,x\right )}{2}+\frac {3\,b^3\,d^2\,\cos \left (5\,e+5\,f\,x\right )}{2}+30\,a^3\,d^2\,\sin \left (2\,e+2\,f\,x\right )-30\,a^2\,b\,d^2\,\cos \left (3\,e+3\,f\,x\right )+90\,a\,b^2\,c^2\,\sin \left (2\,e+2\,f\,x\right )+90\,a\,b^2\,d^2\,\sin \left (2\,e+2\,f\,x\right )-\frac {45\,a\,b^2\,d^2\,\sin \left (4\,e+4\,f\,x\right )}{4}+240\,a^3\,c\,d\,\cos \left (e+f\,x\right )+360\,a^2\,b\,c^2\,\cos \left (e+f\,x\right )+270\,a^2\,b\,d^2\,\cos \left (e+f\,x\right )+60\,b^3\,c\,d\,\sin \left (2\,e+2\,f\,x\right )-\frac {15\,b^3\,c\,d\,\sin \left (4\,e+4\,f\,x\right )}{2}-120\,a^3\,c^2\,f\,x-60\,a^3\,d^2\,f\,x-60\,a\,b^2\,c\,d\,\cos \left (3\,e+3\,f\,x\right )+180\,a^2\,b\,c\,d\,\sin \left (2\,e+2\,f\,x\right )-180\,a\,b^2\,c^2\,f\,x-135\,a\,b^2\,d^2\,f\,x+540\,a\,b^2\,c\,d\,\cos \left (e+f\,x\right )-90\,b^3\,c\,d\,f\,x-360\,a^2\,b\,c\,d\,f\,x}{120\,f} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.29, size = 729, normalized size = 2.31 \[ \begin {cases} a^{3} c^{2} x - \frac {2 a^{3} c d \cos {\left (e + f x \right )}}{f} + \frac {a^{3} d^{2} x \sin ^{2}{\left (e + f x \right )}}{2} + \frac {a^{3} d^{2} x \cos ^{2}{\left (e + f x \right )}}{2} - \frac {a^{3} d^{2} \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{2 f} - \frac {3 a^{2} b c^{2} \cos {\left (e + f x \right )}}{f} + 3 a^{2} b c d x \sin ^{2}{\left (e + f x \right )} + 3 a^{2} b c d x \cos ^{2}{\left (e + f x \right )} - \frac {3 a^{2} b c d \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} - \frac {3 a^{2} b d^{2} \sin ^{2}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} - \frac {2 a^{2} b d^{2} \cos ^{3}{\left (e + f x \right )}}{f} + \frac {3 a b^{2} c^{2} x \sin ^{2}{\left (e + f x \right )}}{2} + \frac {3 a b^{2} c^{2} x \cos ^{2}{\left (e + f x \right )}}{2} - \frac {3 a b^{2} c^{2} \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{2 f} - \frac {6 a b^{2} c d \sin ^{2}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} - \frac {4 a b^{2} c d \cos ^{3}{\left (e + f x \right )}}{f} + \frac {9 a b^{2} d^{2} x \sin ^{4}{\left (e + f x \right )}}{8} + \frac {9 a b^{2} d^{2} x \sin ^{2}{\left (e + f x \right )} \cos ^{2}{\left (e + f x \right )}}{4} + \frac {9 a b^{2} d^{2} x \cos ^{4}{\left (e + f x \right )}}{8} - \frac {15 a b^{2} d^{2} \sin ^{3}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{8 f} - \frac {9 a b^{2} d^{2} \sin {\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{8 f} - \frac {b^{3} c^{2} \sin ^{2}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} - \frac {2 b^{3} c^{2} \cos ^{3}{\left (e + f x \right )}}{3 f} + \frac {3 b^{3} c d x \sin ^{4}{\left (e + f x \right )}}{4} + \frac {3 b^{3} c d x \sin ^{2}{\left (e + f x \right )} \cos ^{2}{\left (e + f x \right )}}{2} + \frac {3 b^{3} c d x \cos ^{4}{\left (e + f x \right )}}{4} - \frac {5 b^{3} c d \sin ^{3}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{4 f} - \frac {3 b^{3} c d \sin {\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{4 f} - \frac {b^{3} d^{2} \sin ^{4}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} - \frac {4 b^{3} d^{2} \sin ^{2}{\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{3 f} - \frac {8 b^{3} d^{2} \cos ^{5}{\left (e + f x \right )}}{15 f} & \text {for}\: f \neq 0 \\x \left (a + b \sin {\relax (e )}\right )^{3} \left (c + d \sin {\relax (e )}\right )^{2} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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