Optimal. Leaf size=162 \[ -\frac {a d \left (6 c^2+20 c d+9 d^2\right ) \sin (e+f x) \cos (e+f x)}{24 f}-\frac {a \left (3 c^3+16 c^2 d+12 c d^2+4 d^3\right ) \cos (e+f x)}{6 f}+\frac {1}{8} a x \left (8 c^3+12 c^2 d+12 c d^2+3 d^3\right )-\frac {a \cos (e+f x) (c+d \sin (e+f x))^3}{4 f}-\frac {a (3 c+4 d) \cos (e+f x) (c+d \sin (e+f x))^2}{12 f} \]
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Rubi [A] time = 0.19, antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {2753, 2734} \[ -\frac {a \left (16 c^2 d+3 c^3+12 c d^2+4 d^3\right ) \cos (e+f x)}{6 f}-\frac {a d \left (6 c^2+20 c d+9 d^2\right ) \sin (e+f x) \cos (e+f x)}{24 f}+\frac {1}{8} a x \left (12 c^2 d+8 c^3+12 c d^2+3 d^3\right )-\frac {a \cos (e+f x) (c+d \sin (e+f x))^3}{4 f}-\frac {a (3 c+4 d) \cos (e+f x) (c+d \sin (e+f x))^2}{12 f} \]
Antiderivative was successfully verified.
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Rule 2734
Rule 2753
Rubi steps
\begin {align*} \int (a+a \sin (e+f x)) (c+d \sin (e+f x))^3 \, dx &=-\frac {a \cos (e+f x) (c+d \sin (e+f x))^3}{4 f}+\frac {1}{4} \int (c+d \sin (e+f x))^2 (a (4 c+3 d)+a (3 c+4 d) \sin (e+f x)) \, dx\\ &=-\frac {a (3 c+4 d) \cos (e+f x) (c+d \sin (e+f x))^2}{12 f}-\frac {a \cos (e+f x) (c+d \sin (e+f x))^3}{4 f}+\frac {1}{12} \int (c+d \sin (e+f x)) \left (a \left (12 c^2+15 c d+8 d^2\right )+a \left (6 c^2+20 c d+9 d^2\right ) \sin (e+f x)\right ) \, dx\\ &=\frac {1}{8} a \left (8 c^3+12 c^2 d+12 c d^2+3 d^3\right ) x-\frac {a \left (3 c^3+16 c^2 d+12 c d^2+4 d^3\right ) \cos (e+f x)}{6 f}-\frac {a d \left (6 c^2+20 c d+9 d^2\right ) \cos (e+f x) \sin (e+f x)}{24 f}-\frac {a (3 c+4 d) \cos (e+f x) (c+d \sin (e+f x))^2}{12 f}-\frac {a \cos (e+f x) (c+d \sin (e+f x))^3}{4 f}\\ \end {align*}
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Mathematica [A] time = 0.80, size = 124, normalized size = 0.77 \[ \frac {a \left (3 \left (-8 d \left (3 c^2+3 c d+d^2\right ) \sin (2 (e+f x))+4 f x \left (8 c^3+12 c^2 d+12 c d^2+3 d^3\right )+d^3 \sin (4 (e+f x))\right )-24 \left (4 c^3+12 c^2 d+9 c d^2+3 d^3\right ) \cos (e+f x)+8 d^2 (3 c+d) \cos (3 (e+f x))\right )}{96 f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 145, normalized size = 0.90 \[ \frac {8 \, {\left (3 \, a c d^{2} + a d^{3}\right )} \cos \left (f x + e\right )^{3} + 3 \, {\left (8 \, a c^{3} + 12 \, a c^{2} d + 12 \, a c d^{2} + 3 \, a d^{3}\right )} f x - 24 \, {\left (a c^{3} + 3 \, a c^{2} d + 3 \, a c d^{2} + a d^{3}\right )} \cos \left (f x + e\right ) + 3 \, {\left (2 \, a d^{3} \cos \left (f x + e\right )^{3} - {\left (12 \, a c^{2} d + 12 \, a c d^{2} + 5 \, a d^{3}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{24 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.47, size = 191, normalized size = 1.18 \[ \frac {a c d^{2} \cos \left (3 \, f x + 3 \, e\right )}{4 \, f} + \frac {a d^{3} \cos \left (3 \, f x + 3 \, e\right )}{12 \, f} + \frac {a d^{3} \sin \left (4 \, f x + 4 \, e\right )}{32 \, f} - \frac {3 \, a c d^{2} \sin \left (2 \, f x + 2 \, e\right )}{4 \, f} + \frac {1}{2} \, {\left (2 \, a c^{3} + 3 \, a c d^{2}\right )} x + \frac {3}{8} \, {\left (4 \, a c^{2} d + a d^{3}\right )} x - \frac {{\left (4 \, a c^{3} + 9 \, a c d^{2}\right )} \cos \left (f x + e\right )}{4 \, f} - \frac {3 \, {\left (4 \, a c^{2} d + a d^{3}\right )} \cos \left (f x + e\right )}{4 \, f} - \frac {{\left (3 \, a c^{2} d + a d^{3}\right )} \sin \left (2 \, f x + 2 \, e\right )}{4 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.29, size = 182, normalized size = 1.12 \[ \frac {-a \,c^{3} \cos \left (f x +e \right )+3 a \,c^{2} d \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-a c \,d^{2} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )+a \,d^{3} \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )+a \,c^{3} \left (f x +e \right )-3 a \,c^{2} d \cos \left (f x +e \right )+3 a c \,d^{2} \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-\frac {a \,d^{3} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 175, normalized size = 1.08 \[ \frac {96 \, {\left (f x + e\right )} a c^{3} + 72 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} a c^{2} d + 96 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} a c d^{2} + 72 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} a c d^{2} + 32 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} a d^{3} + 3 \, {\left (12 \, f x + 12 \, e + \sin \left (4 \, f x + 4 \, e\right ) - 8 \, \sin \left (2 \, f x + 2 \, e\right )\right )} a d^{3} - 96 \, a c^{3} \cos \left (f x + e\right ) - 288 \, a c^{2} d \cos \left (f x + e\right )}{96 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.17, size = 460, normalized size = 2.84 \[ \frac {a\,\mathrm {atan}\left (\frac {a\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (8\,c^3+12\,c^2\,d+12\,c\,d^2+3\,d^3\right )}{4\,\left (2\,a\,c^3+3\,a\,c^2\,d+3\,a\,c\,d^2+\frac {3\,a\,d^3}{4}\right )}\right )\,\left (8\,c^3+12\,c^2\,d+12\,c\,d^2+3\,d^3\right )}{4\,f}-\frac {a\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )-\frac {f\,x}{2}\right )\,\left (8\,c^3+12\,c^2\,d+12\,c\,d^2+3\,d^3\right )}{4\,f}-\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (3\,a\,c^2\,d+3\,a\,c\,d^2+\frac {11\,a\,d^3}{4}\right )-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7\,\left (3\,a\,c^2\,d+3\,a\,c\,d^2+\frac {3\,a\,d^3}{4}\right )-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,\left (3\,a\,c^2\,d+3\,a\,c\,d^2+\frac {11\,a\,d^3}{4}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,\left (2\,a\,c^3+6\,a\,d\,c^2\right )+2\,a\,c^3+\frac {4\,a\,d^3}{3}+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (6\,a\,c^3+18\,a\,c^2\,d+12\,a\,c\,d^2+4\,a\,d^3\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (6\,a\,c^3+18\,a\,c^2\,d+16\,a\,c\,d^2+\frac {16\,a\,d^3}{3}\right )+\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (3\,a\,c^2\,d+3\,a\,c\,d^2+\frac {3\,a\,d^3}{4}\right )+4\,a\,c\,d^2+6\,a\,c^2\,d}{f\,\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8+4\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+6\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+4\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.68, size = 386, normalized size = 2.38 \[ \begin {cases} a c^{3} x - \frac {a c^{3} \cos {\left (e + f x \right )}}{f} + \frac {3 a c^{2} d x \sin ^{2}{\left (e + f x \right )}}{2} + \frac {3 a c^{2} d x \cos ^{2}{\left (e + f x \right )}}{2} - \frac {3 a c^{2} d \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{2 f} - \frac {3 a c^{2} d \cos {\left (e + f x \right )}}{f} + \frac {3 a c d^{2} x \sin ^{2}{\left (e + f x \right )}}{2} + \frac {3 a c d^{2} x \cos ^{2}{\left (e + f x \right )}}{2} - \frac {3 a c d^{2} \sin ^{2}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} - \frac {3 a c d^{2} \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{2 f} - \frac {2 a c d^{2} \cos ^{3}{\left (e + f x \right )}}{f} + \frac {3 a d^{3} x \sin ^{4}{\left (e + f x \right )}}{8} + \frac {3 a d^{3} x \sin ^{2}{\left (e + f x \right )} \cos ^{2}{\left (e + f x \right )}}{4} + \frac {3 a d^{3} x \cos ^{4}{\left (e + f x \right )}}{8} - \frac {5 a d^{3} \sin ^{3}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{8 f} - \frac {a d^{3} \sin ^{2}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} - \frac {3 a d^{3} \sin {\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{8 f} - \frac {2 a d^{3} \cos ^{3}{\left (e + f x \right )}}{3 f} & \text {for}\: f \neq 0 \\x \left (c + d \sin {\relax (e )}\right )^{3} \left (a \sin {\relax (e )} + a\right ) & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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