Optimal. Leaf size=65 \[ -\frac {4 E\left (\left .\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right )\right |2\right )}{3 b^2}-\frac {4 \cos (a+b x)}{3 b^2 \sqrt {\sin (a+b x)}}-\frac {2 x}{3 b \sin ^{\frac {3}{2}}(a+b x)} \]
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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, 2636, 2639} \[ -\frac {4 E\left (\left .\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right )\right |2\right )}{3 b^2}-\frac {4 \cos (a+b x)}{3 b^2 \sqrt {\sin (a+b x)}}-\frac {2 x}{3 b \sin ^{\frac {3}{2}}(a+b x)} \]
Antiderivative was successfully verified.
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Rule 2636
Rule 2639
Rule 3443
Rubi steps
\begin {align*} \int \frac {x \cos (a+b x)}{\sin ^{\frac {5}{2}}(a+b x)} \, dx &=-\frac {2 x}{3 b \sin ^{\frac {3}{2}}(a+b x)}+\frac {2 \int \frac {1}{\sin ^{\frac {3}{2}}(a+b x)} \, dx}{3 b}\\ &=-\frac {2 x}{3 b \sin ^{\frac {3}{2}}(a+b x)}-\frac {4 \cos (a+b x)}{3 b^2 \sqrt {\sin (a+b x)}}-\frac {2 \int \sqrt {\sin (a+b x)} \, dx}{3 b}\\ &=-\frac {4 E\left (\left .\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right )\right |2\right )}{3 b^2}-\frac {2 x}{3 b \sin ^{\frac {3}{2}}(a+b x)}-\frac {4 \cos (a+b x)}{3 b^2 \sqrt {\sin (a+b x)}}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 56, normalized size = 0.86 \[ -\frac {2 \left (\sin (2 (a+b x))-2 \sin ^{\frac {3}{2}}(a+b x) E\left (\left .\frac {1}{4} (-2 a-2 b x+\pi )\right |2\right )+b x\right )}{3 b^2 \sin ^{\frac {3}{2}}(a+b x)} \]
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 \[ \int \frac {x \cos \left (b x + a\right )}{\sin \left (b x + a\right )^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {x \cos \left (b x +a \right )}{\sin \left (b x +a \right )^{\frac {5}{2}}}\, dx \]
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 \[ \int \frac {x \cos \left (b x + a\right )}{\sin \left (b x + a\right )^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {x\,\cos \left (a+b\,x\right )}{{\sin \left (a+b\,x\right )}^{5/2}} \,d x \]
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 \[ \int \frac {x \cos {\left (a + b x \right )}}{\sin ^{\frac {5}{2}}{\left (a + b x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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