Optimal. Leaf size=151 \[ -\frac{2 d^2 (c+d x) \sin ^3(a+b x)}{9 b^3}-\frac{4 d^2 (c+d x) \sin (a+b x)}{3 b^3}+\frac{2 d (c+d x)^2 \cos (a+b x)}{3 b^2}+\frac{d (c+d x)^2 \sin ^2(a+b x) \cos (a+b x)}{3 b^2}+\frac{2 d^3 \cos ^3(a+b x)}{27 b^4}-\frac{14 d^3 \cos (a+b x)}{9 b^4}+\frac{(c+d x)^3 \sin ^3(a+b x)}{3 b} \]
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Rubi [A] time = 0.134578, antiderivative size = 151, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {4404, 3311, 3296, 2638, 2633} \[ -\frac{2 d^2 (c+d x) \sin ^3(a+b x)}{9 b^3}-\frac{4 d^2 (c+d x) \sin (a+b x)}{3 b^3}+\frac{2 d (c+d x)^2 \cos (a+b x)}{3 b^2}+\frac{d (c+d x)^2 \sin ^2(a+b x) \cos (a+b x)}{3 b^2}+\frac{2 d^3 \cos ^3(a+b x)}{27 b^4}-\frac{14 d^3 \cos (a+b x)}{9 b^4}+\frac{(c+d x)^3 \sin ^3(a+b x)}{3 b} \]
Antiderivative was successfully verified.
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Rule 4404
Rule 3311
Rule 3296
Rule 2638
Rule 2633
Rubi steps
\begin{align*} \int (c+d x)^3 \cos (a+b x) \sin ^2(a+b x) \, dx &=\frac{(c+d x)^3 \sin ^3(a+b x)}{3 b}-\frac{d \int (c+d x)^2 \sin ^3(a+b x) \, dx}{b}\\ &=\frac{d (c+d x)^2 \cos (a+b x) \sin ^2(a+b x)}{3 b^2}-\frac{2 d^2 (c+d x) \sin ^3(a+b x)}{9 b^3}+\frac{(c+d x)^3 \sin ^3(a+b x)}{3 b}-\frac{(2 d) \int (c+d x)^2 \sin (a+b x) \, dx}{3 b}+\frac{\left (2 d^3\right ) \int \sin ^3(a+b x) \, dx}{9 b^3}\\ &=\frac{2 d (c+d x)^2 \cos (a+b x)}{3 b^2}+\frac{d (c+d x)^2 \cos (a+b x) \sin ^2(a+b x)}{3 b^2}-\frac{2 d^2 (c+d x) \sin ^3(a+b x)}{9 b^3}+\frac{(c+d x)^3 \sin ^3(a+b x)}{3 b}-\frac{\left (4 d^2\right ) \int (c+d x) \cos (a+b x) \, dx}{3 b^2}-\frac{\left (2 d^3\right ) \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cos (a+b x)\right )}{9 b^4}\\ &=-\frac{2 d^3 \cos (a+b x)}{9 b^4}+\frac{2 d (c+d x)^2 \cos (a+b x)}{3 b^2}+\frac{2 d^3 \cos ^3(a+b x)}{27 b^4}-\frac{4 d^2 (c+d x) \sin (a+b x)}{3 b^3}+\frac{d (c+d x)^2 \cos (a+b x) \sin ^2(a+b x)}{3 b^2}-\frac{2 d^2 (c+d x) \sin ^3(a+b x)}{9 b^3}+\frac{(c+d x)^3 \sin ^3(a+b x)}{3 b}+\frac{\left (4 d^3\right ) \int \sin (a+b x) \, dx}{3 b^3}\\ &=-\frac{14 d^3 \cos (a+b x)}{9 b^4}+\frac{2 d (c+d x)^2 \cos (a+b x)}{3 b^2}+\frac{2 d^3 \cos ^3(a+b x)}{27 b^4}-\frac{4 d^2 (c+d x) \sin (a+b x)}{3 b^3}+\frac{d (c+d x)^2 \cos (a+b x) \sin ^2(a+b x)}{3 b^2}-\frac{2 d^2 (c+d x) \sin ^3(a+b x)}{9 b^3}+\frac{(c+d x)^3 \sin ^3(a+b x)}{3 b}\\ \end{align*}
Mathematica [A] time = 0.991882, size = 121, normalized size = 0.8 \[ -\frac{-81 d \cos (a+b x) \left (b^2 (c+d x)^2-2 d^2\right )+d \cos (3 (a+b x)) \left (9 b^2 (c+d x)^2-2 d^2\right )+6 b (c+d x) \sin (a+b x) \left (\cos (2 (a+b x)) \left (3 b^2 (c+d x)^2-2 d^2\right )-3 b^2 (c+d x)^2+26 d^2\right )}{108 b^4} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.019, size = 447, normalized size = 3. \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.11837, size = 674, normalized size = 4.46 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.502087, size = 491, normalized size = 3.25 \begin{align*} -\frac{{\left (9 \, b^{2} d^{3} x^{2} + 18 \, b^{2} c d^{2} x + 9 \, b^{2} c^{2} d - 2 \, d^{3}\right )} \cos \left (b x + a\right )^{3} - 3 \,{\left (9 \, b^{2} d^{3} x^{2} + 18 \, b^{2} c d^{2} x + 9 \, b^{2} c^{2} d - 14 \, d^{3}\right )} \cos \left (b x + a\right ) - 3 \,{\left (3 \, b^{3} d^{3} x^{3} + 9 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{3} - 14 \, b c d^{2} -{\left (3 \, b^{3} d^{3} x^{3} + 9 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{3} - 2 \, b c d^{2} +{\left (9 \, b^{3} c^{2} d - 2 \, b d^{3}\right )} x\right )} \cos \left (b x + a\right )^{2} +{\left (9 \, b^{3} c^{2} d - 14 \, b d^{3}\right )} x\right )} \sin \left (b x + a\right )}{27 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 5.26756, size = 391, normalized size = 2.59 \begin{align*} \begin{cases} \frac{c^{3} \sin ^{3}{\left (a + b x \right )}}{3 b} + \frac{c^{2} d x \sin ^{3}{\left (a + b x \right )}}{b} + \frac{c d^{2} x^{2} \sin ^{3}{\left (a + b x \right )}}{b} + \frac{d^{3} x^{3} \sin ^{3}{\left (a + b x \right )}}{3 b} + \frac{c^{2} d \sin ^{2}{\left (a + b x \right )} \cos{\left (a + b x \right )}}{b^{2}} + \frac{2 c^{2} d \cos ^{3}{\left (a + b x \right )}}{3 b^{2}} + \frac{2 c d^{2} x \sin ^{2}{\left (a + b x \right )} \cos{\left (a + b x \right )}}{b^{2}} + \frac{4 c d^{2} x \cos ^{3}{\left (a + b x \right )}}{3 b^{2}} + \frac{d^{3} x^{2} \sin ^{2}{\left (a + b x \right )} \cos{\left (a + b x \right )}}{b^{2}} + \frac{2 d^{3} x^{2} \cos ^{3}{\left (a + b x \right )}}{3 b^{2}} - \frac{14 c d^{2} \sin ^{3}{\left (a + b x \right )}}{9 b^{3}} - \frac{4 c d^{2} \sin{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{3 b^{3}} - \frac{14 d^{3} x \sin ^{3}{\left (a + b x \right )}}{9 b^{3}} - \frac{4 d^{3} x \sin{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{3 b^{3}} - \frac{14 d^{3} \sin ^{2}{\left (a + b x \right )} \cos{\left (a + b x \right )}}{9 b^{4}} - \frac{40 d^{3} \cos ^{3}{\left (a + b x \right )}}{27 b^{4}} & \text{for}\: b \neq 0 \\\left (c^{3} x + \frac{3 c^{2} d x^{2}}{2} + c d^{2} x^{3} + \frac{d^{3} x^{4}}{4}\right ) \sin ^{2}{\left (a \right )} \cos{\left (a \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16585, size = 312, normalized size = 2.07 \begin{align*} -\frac{{\left (9 \, b^{2} d^{3} x^{2} + 18 \, b^{2} c d^{2} x + 9 \, b^{2} c^{2} d - 2 \, d^{3}\right )} \cos \left (3 \, b x + 3 \, a\right )}{108 \, b^{4}} + \frac{3 \,{\left (b^{2} d^{3} x^{2} + 2 \, b^{2} c d^{2} x + b^{2} c^{2} d - 2 \, d^{3}\right )} \cos \left (b x + a\right )}{4 \, b^{4}} - \frac{{\left (3 \, b^{3} d^{3} x^{3} + 9 \, b^{3} c d^{2} x^{2} + 9 \, b^{3} c^{2} d x + 3 \, b^{3} c^{3} - 2 \, b d^{3} x - 2 \, b c d^{2}\right )} \sin \left (3 \, b x + 3 \, a\right )}{36 \, b^{4}} + \frac{{\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x + b^{3} c^{3} - 6 \, b d^{3} x - 6 \, b c d^{2}\right )} \sin \left (b x + a\right )}{4 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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