Optimal. Leaf size=38 \[ \frac {4 F\left (\left .\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right )\right |2\right )}{b^2}-\frac {2 x}{b \sqrt {\sin (a+b x)}} \]
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Rubi [A] time = 0.02, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {3443, 2641} \[ \frac {4 F\left (\left .\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right )\right |2\right )}{b^2}-\frac {2 x}{b \sqrt {\sin (a+b x)}} \]
Antiderivative was successfully verified.
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Rule 2641
Rule 3443
Rubi steps
\begin {align*} \int \frac {x \cos (a+b x)}{\sin ^{\frac {3}{2}}(a+b x)} \, dx &=-\frac {2 x}{b \sqrt {\sin (a+b x)}}+\frac {2 \int \frac {1}{\sqrt {\sin (a+b x)}} \, dx}{b}\\ &=\frac {4 F\left (\left .\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right )\right |2\right )}{b^2}-\frac {2 x}{b \sqrt {\sin (a+b x)}}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 37, normalized size = 0.97 \[ \frac {2 \left (-\frac {b x}{\sqrt {\sin (a+b x)}}-2 F\left (\left .\frac {1}{4} (-2 a-2 b x+\pi )\right |2\right )\right )}{b^2} \]
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 \[ \int \frac {x \cos \left (b x + a\right )}{\sin \left (b x + a\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.04, size = 0, normalized size = 0.00 \[ \int \frac {x \cos \left (b x +a \right )}{\sin \left (b x +a \right )^{\frac {3}{2}}}\, dx \]
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 \[ \int \frac {x \cos \left (b x + a\right )}{\sin \left (b x + a\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {x\,\cos \left (a+b\,x\right )}{{\sin \left (a+b\,x\right )}^{3/2}} \,d x \]
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 \[ \int \frac {x \cos {\left (a + b x \right )}}{\sin ^{\frac {3}{2}}{\left (a + b x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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