Optimal. Leaf size=75 \[ \frac {d \sin ^3(a+b x)}{9 b^2}+\frac {2 d \sin (a+b x)}{3 b^2}-\frac {2 (c+d x) \cos (a+b x)}{3 b}-\frac {(c+d x) \sin ^2(a+b x) \cos (a+b x)}{3 b} \]
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Rubi [A] time = 0.04, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {3310, 3296, 2637} \[ \frac {d \sin ^3(a+b x)}{9 b^2}+\frac {2 d \sin (a+b x)}{3 b^2}-\frac {2 (c+d x) \cos (a+b x)}{3 b}-\frac {(c+d x) \sin ^2(a+b x) \cos (a+b x)}{3 b} \]
Antiderivative was successfully verified.
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Rule 2637
Rule 3296
Rule 3310
Rubi steps
\begin {align*} \int (c+d x) \sin ^3(a+b x) \, dx &=-\frac {(c+d x) \cos (a+b x) \sin ^2(a+b x)}{3 b}+\frac {d \sin ^3(a+b x)}{9 b^2}+\frac {2}{3} \int (c+d x) \sin (a+b x) \, dx\\ &=-\frac {2 (c+d x) \cos (a+b x)}{3 b}-\frac {(c+d x) \cos (a+b x) \sin ^2(a+b x)}{3 b}+\frac {d \sin ^3(a+b x)}{9 b^2}+\frac {(2 d) \int \cos (a+b x) \, dx}{3 b}\\ &=-\frac {2 (c+d x) \cos (a+b x)}{3 b}+\frac {2 d \sin (a+b x)}{3 b^2}-\frac {(c+d x) \cos (a+b x) \sin ^2(a+b x)}{3 b}+\frac {d \sin ^3(a+b x)}{9 b^2}\\ \end {align*}
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Mathematica [A] time = 0.19, size = 59, normalized size = 0.79 \[ \frac {-27 b (c+d x) \cos (a+b x)+3 b (c+d x) \cos (3 (a+b x))+d (27 \sin (a+b x)-\sin (3 (a+b x)))}{36 b^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.59, size = 62, normalized size = 0.83 \[ \frac {3 \, {\left (b d x + b c\right )} \cos \left (b x + a\right )^{3} - 9 \, {\left (b d x + b c\right )} \cos \left (b x + a\right ) - {\left (d \cos \left (b x + a\right )^{2} - 7 \, d\right )} \sin \left (b x + a\right )}{9 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.21, size = 69, normalized size = 0.92 \[ \frac {{\left (b d x + b c\right )} \cos \left (3 \, b x + 3 \, a\right )}{12 \, b^{2}} - \frac {3 \, {\left (b d x + b c\right )} \cos \left (b x + a\right )}{4 \, b^{2}} - \frac {d \sin \left (3 \, b x + 3 \, a\right )}{36 \, b^{2}} + \frac {3 \, d \sin \left (b x + a\right )}{4 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 95, normalized size = 1.27 \[ \frac {\frac {d \left (-\frac {\left (b x +a \right ) \left (2+\sin ^{2}\left (b x +a \right )\right ) \cos \left (b x +a \right )}{3}+\frac {\left (\sin ^{3}\left (b x +a \right )\right )}{9}+\frac {2 \sin \left (b x +a \right )}{3}\right )}{b}+\frac {d a \left (2+\sin ^{2}\left (b x +a \right )\right ) \cos \left (b x +a \right )}{3 b}-\frac {c \left (2+\sin ^{2}\left (b x +a \right )\right ) \cos \left (b x +a \right )}{3}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 104, normalized size = 1.39 \[ \frac {12 \, {\left (\cos \left (b x + a\right )^{3} - 3 \, \cos \left (b x + a\right )\right )} c - \frac {12 \, {\left (\cos \left (b x + a\right )^{3} - 3 \, \cos \left (b x + a\right )\right )} a d}{b} + \frac {{\left (3 \, {\left (b x + a\right )} \cos \left (3 \, b x + 3 \, a\right ) - 27 \, {\left (b x + a\right )} \cos \left (b x + a\right ) - \sin \left (3 \, b x + 3 \, a\right ) + 27 \, \sin \left (b x + a\right )\right )} d}{b}}{36 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.63, size = 79, normalized size = 1.05 \[ \frac {7\,d\,\sin \left (a+b\,x\right )}{9\,b^2}-\frac {c\,\cos \left (a+b\,x\right )-\frac {c\,{\cos \left (a+b\,x\right )}^3}{3}+d\,x\,\cos \left (a+b\,x\right )-\frac {d\,x\,{\cos \left (a+b\,x\right )}^3}{3}}{b}-\frac {d\,{\cos \left (a+b\,x\right )}^2\,\sin \left (a+b\,x\right )}{9\,b^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.25, size = 126, normalized size = 1.68 \[ \begin {cases} - \frac {c \sin ^{2}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{b} - \frac {2 c \cos ^{3}{\left (a + b x \right )}}{3 b} - \frac {d x \sin ^{2}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{b} - \frac {2 d x \cos ^{3}{\left (a + b x \right )}}{3 b} + \frac {7 d \sin ^{3}{\left (a + b x \right )}}{9 b^{2}} + \frac {2 d \sin {\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{3 b^{2}} & \text {for}\: b \neq 0 \\\left (c x + \frac {d x^{2}}{2}\right ) \sin ^{3}{\relax (a )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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