Optimal. Leaf size=65 \[ -\frac {12 E\left (\left .\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right )\right |2\right )}{25 b^2}+\frac {4 \sin ^{\frac {3}{2}}(a+b x) \cos (a+b x)}{25 b^2}+\frac {2 x \sin ^{\frac {5}{2}}(a+b x)}{5 b} \]
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Rubi [A] time = 0.03, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3443, 2635, 2639} \[ -\frac {12 E\left (\left .\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right )\right |2\right )}{25 b^2}+\frac {4 \sin ^{\frac {3}{2}}(a+b x) \cos (a+b x)}{25 b^2}+\frac {2 x \sin ^{\frac {5}{2}}(a+b x)}{5 b} \]
Antiderivative was successfully verified.
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Rule 2635
Rule 2639
Rule 3443
Rubi steps
\begin {align*} \int x \cos (a+b x) \sin ^{\frac {3}{2}}(a+b x) \, dx &=\frac {2 x \sin ^{\frac {5}{2}}(a+b x)}{5 b}-\frac {2 \int \sin ^{\frac {5}{2}}(a+b x) \, dx}{5 b}\\ &=\frac {4 \cos (a+b x) \sin ^{\frac {3}{2}}(a+b x)}{25 b^2}+\frac {2 x \sin ^{\frac {5}{2}}(a+b x)}{5 b}-\frac {6 \int \sqrt {\sin (a+b x)} \, dx}{25 b}\\ &=-\frac {12 E\left (\left .\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right )\right |2\right )}{25 b^2}+\frac {4 \cos (a+b x) \sin ^{\frac {3}{2}}(a+b x)}{25 b^2}+\frac {2 x \sin ^{\frac {5}{2}}(a+b x)}{5 b}\\ \end {align*}
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Mathematica [C] time = 0.90, size = 108, normalized size = 1.66 \[ \frac {\sqrt {\sin (a+b x)} \left (4 \tan \left (\frac {1}{2} (a+b x)\right ) \sqrt {\sec ^2\left (\frac {1}{2} (a+b x)\right )} \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};-\tan ^2\left (\frac {1}{2} (a+b x)\right )\right )+2 \sin (2 (a+b x))-5 b x \cos (2 (a+b x))-12 \tan \left (\frac {1}{2} (a+b x)\right )+5 b x\right )}{25 b^2} \]
Antiderivative was successfully verified.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x \cos \left (b x + a\right ) \sin \left (b x + a\right )^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.05, size = 0, normalized size = 0.00 \[ \int x \cos \left (b x +a \right ) \left (\sin ^{\frac {3}{2}}\left (b x +a \right )\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x \cos \left (b x + a\right ) \sin \left (b x + a\right )^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int x\,\cos \left (a+b\,x\right )\,{\sin \left (a+b\,x\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x \sin ^{\frac {3}{2}}{\left (a + b x \right )} \cos {\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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