Optimal. Leaf size=110 \[ \frac {a^3 (c+3 d) \cos ^3(e+f x)}{3 f}-\frac {4 a^3 (c+d) \cos (e+f x)}{f}-\frac {3 a^3 (4 c+5 d) \sin (e+f x) \cos (e+f x)}{8 f}+\frac {5}{8} a^3 x (4 c+3 d)-\frac {a^3 d \sin ^3(e+f x) \cos (e+f x)}{4 f} \]
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Rubi [A] time = 0.10, antiderivative size = 117, normalized size of antiderivative = 1.06, number of steps used = 8, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {2751, 2645, 2638, 2635, 8, 2633} \[ \frac {a^3 (4 c+3 d) \cos ^3(e+f x)}{12 f}-\frac {a^3 (4 c+3 d) \cos (e+f x)}{f}-\frac {3 a^3 (4 c+3 d) \sin (e+f x) \cos (e+f x)}{8 f}+\frac {5}{8} a^3 x (4 c+3 d)-\frac {d \cos (e+f x) (a \sin (e+f x)+a)^3}{4 f} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2633
Rule 2635
Rule 2638
Rule 2645
Rule 2751
Rubi steps
\begin {align*} \int (a+a \sin (e+f x))^3 (c+d \sin (e+f x)) \, dx &=-\frac {d \cos (e+f x) (a+a \sin (e+f x))^3}{4 f}+\frac {1}{4} (4 c+3 d) \int (a+a \sin (e+f x))^3 \, dx\\ &=-\frac {d \cos (e+f x) (a+a \sin (e+f x))^3}{4 f}+\frac {1}{4} (4 c+3 d) \int \left (a^3+3 a^3 \sin (e+f x)+3 a^3 \sin ^2(e+f x)+a^3 \sin ^3(e+f x)\right ) \, dx\\ &=\frac {1}{4} a^3 (4 c+3 d) x-\frac {d \cos (e+f x) (a+a \sin (e+f x))^3}{4 f}+\frac {1}{4} \left (a^3 (4 c+3 d)\right ) \int \sin ^3(e+f x) \, dx+\frac {1}{4} \left (3 a^3 (4 c+3 d)\right ) \int \sin (e+f x) \, dx+\frac {1}{4} \left (3 a^3 (4 c+3 d)\right ) \int \sin ^2(e+f x) \, dx\\ &=\frac {1}{4} a^3 (4 c+3 d) x-\frac {3 a^3 (4 c+3 d) \cos (e+f x)}{4 f}-\frac {3 a^3 (4 c+3 d) \cos (e+f x) \sin (e+f x)}{8 f}-\frac {d \cos (e+f x) (a+a \sin (e+f x))^3}{4 f}+\frac {1}{8} \left (3 a^3 (4 c+3 d)\right ) \int 1 \, dx-\frac {\left (a^3 (4 c+3 d)\right ) \operatorname {Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cos (e+f x)\right )}{4 f}\\ &=\frac {5}{8} a^3 (4 c+3 d) x-\frac {a^3 (4 c+3 d) \cos (e+f x)}{f}+\frac {a^3 (4 c+3 d) \cos ^3(e+f x)}{12 f}-\frac {3 a^3 (4 c+3 d) \cos (e+f x) \sin (e+f x)}{8 f}-\frac {d \cos (e+f x) (a+a \sin (e+f x))^3}{4 f}\\ \end {align*}
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Mathematica [A] time = 0.52, size = 120, normalized size = 1.09 \[ -\frac {a^3 \cos (e+f x) \left (30 (4 c+3 d) \sin ^{-1}\left (\frac {\sqrt {1-\sin (e+f x)}}{\sqrt {2}}\right )+\sqrt {\cos ^2(e+f x)} \left (8 (c+3 d) \sin ^2(e+f x)+9 (4 c+5 d) \sin (e+f x)+88 c+6 d \sin ^3(e+f x)+72 d\right )\right )}{24 f \sqrt {\cos ^2(e+f x)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 108, normalized size = 0.98 \[ \frac {8 \, {\left (a^{3} c + 3 \, a^{3} d\right )} \cos \left (f x + e\right )^{3} + 15 \, {\left (4 \, a^{3} c + 3 \, a^{3} d\right )} f x - 96 \, {\left (a^{3} c + a^{3} d\right )} \cos \left (f x + e\right ) + 3 \, {\left (2 \, a^{3} d \cos \left (f x + e\right )^{3} - {\left (12 \, a^{3} c + 17 \, a^{3} d\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{24 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 138, normalized size = 1.25 \[ a^{3} c x - \frac {a^{3} d \cos \left (f x + e\right )}{f} + \frac {a^{3} d \sin \left (4 \, f x + 4 \, e\right )}{32 \, f} + \frac {3}{8} \, {\left (4 \, a^{3} c + 5 \, a^{3} d\right )} x + \frac {{\left (a^{3} c + 3 \, a^{3} d\right )} \cos \left (3 \, f x + 3 \, e\right )}{12 \, f} - \frac {3 \, {\left (5 \, a^{3} c + 3 \, a^{3} d\right )} \cos \left (f x + e\right )}{4 \, f} - \frac {{\left (3 \, a^{3} c + 4 \, a^{3} d\right )} \sin \left (2 \, f x + 2 \, e\right )}{4 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.25, size = 178, normalized size = 1.62 \[ \frac {-\frac {a^{3} c \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}+a^{3} d \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )+3 a^{3} c \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-a^{3} d \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )-3 a^{3} c \cos \left (f x +e \right )+3 a^{3} d \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )+a^{3} c \left (f x +e \right )-a^{3} d \cos \left (f x +e \right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 171, normalized size = 1.55 \[ \frac {32 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} a^{3} c + 72 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} a^{3} c + 96 \, {\left (f x + e\right )} a^{3} c + 96 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} a^{3} d + 3 \, {\left (12 \, f x + 12 \, e + \sin \left (4 \, f x + 4 \, e\right ) - 8 \, \sin \left (2 \, f x + 2 \, e\right )\right )} a^{3} d + 72 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} a^{3} d - 288 \, a^{3} c \cos \left (f x + e\right ) - 96 \, a^{3} d \cos \left (f x + e\right )}{96 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.07, size = 330, normalized size = 3.00 \[ \frac {5\,a^3\,\mathrm {atan}\left (\frac {5\,a^3\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (4\,c+3\,d\right )}{4\,\left (5\,a^3\,c+\frac {15\,a^3\,d}{4}\right )}\right )\,\left (4\,c+3\,d\right )}{4\,f}-\frac {5\,a^3\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )-\frac {f\,x}{2}\right )\,\left (4\,c+3\,d\right )}{4\,f}-\frac {\frac {22\,a^3\,c}{3}+6\,a^3\,d+\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (3\,a^3\,c+\frac {15\,a^3\,d}{4}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,\left (6\,a^3\,c+2\,a^3\,d\right )-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7\,\left (3\,a^3\,c+\frac {15\,a^3\,d}{4}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (3\,a^3\,c+\frac {23\,a^3\,d}{4}\right )-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,\left (3\,a^3\,c+\frac {23\,a^3\,d}{4}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (22\,a^3\,c+18\,a^3\,d\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (\frac {70\,a^3\,c}{3}+22\,a^3\,d\right )}{f\,\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8+4\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+6\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+4\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.00, size = 371, normalized size = 3.37 \[ \begin {cases} \frac {3 a^{3} c x \sin ^{2}{\left (e + f x \right )}}{2} + \frac {3 a^{3} c x \cos ^{2}{\left (e + f x \right )}}{2} + a^{3} c x - \frac {a^{3} c \sin ^{2}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} - \frac {3 a^{3} c \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{2 f} - \frac {2 a^{3} c \cos ^{3}{\left (e + f x \right )}}{3 f} - \frac {3 a^{3} c \cos {\left (e + f x \right )}}{f} + \frac {3 a^{3} d x \sin ^{4}{\left (e + f x \right )}}{8} + \frac {3 a^{3} d x \sin ^{2}{\left (e + f x \right )} \cos ^{2}{\left (e + f x \right )}}{4} + \frac {3 a^{3} d x \sin ^{2}{\left (e + f x \right )}}{2} + \frac {3 a^{3} d x \cos ^{4}{\left (e + f x \right )}}{8} + \frac {3 a^{3} d x \cos ^{2}{\left (e + f x \right )}}{2} - \frac {5 a^{3} d \sin ^{3}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{8 f} - \frac {3 a^{3} d \sin ^{2}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} - \frac {3 a^{3} d \sin {\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{8 f} - \frac {3 a^{3} d \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{2 f} - \frac {2 a^{3} d \cos ^{3}{\left (e + f x \right )}}{f} - \frac {a^{3} d \cos {\left (e + f x \right )}}{f} & \text {for}\: f \neq 0 \\x \left (c + d \sin {\relax (e )}\right ) \left (a \sin {\relax (e )} + a\right )^{3} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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