Optimal. Leaf size=88 \[ -\frac {12 E\left (\left .\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right )\right |2\right )}{35 b^2}-\frac {4 \cos (a+b x)}{35 b^2 \sin ^{\frac {5}{2}}(a+b x)}-\frac {12 \cos (a+b x)}{35 b^2 \sqrt {\sin (a+b x)}}-\frac {2 x}{7 b \sin ^{\frac {7}{2}}(a+b x)} \]
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Rubi [A] time = 0.04, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3443, 2636, 2639} \[ -\frac {12 E\left (\left .\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right )\right |2\right )}{35 b^2}-\frac {4 \cos (a+b x)}{35 b^2 \sin ^{\frac {5}{2}}(a+b x)}-\frac {12 \cos (a+b x)}{35 b^2 \sqrt {\sin (a+b x)}}-\frac {2 x}{7 b \sin ^{\frac {7}{2}}(a+b x)} \]
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 {9}{2}}(a+b x)} \, dx &=-\frac {2 x}{7 b \sin ^{\frac {7}{2}}(a+b x)}+\frac {2 \int \frac {1}{\sin ^{\frac {7}{2}}(a+b x)} \, dx}{7 b}\\ &=-\frac {2 x}{7 b \sin ^{\frac {7}{2}}(a+b x)}-\frac {4 \cos (a+b x)}{35 b^2 \sin ^{\frac {5}{2}}(a+b x)}+\frac {6 \int \frac {1}{\sin ^{\frac {3}{2}}(a+b x)} \, dx}{35 b}\\ &=-\frac {2 x}{7 b \sin ^{\frac {7}{2}}(a+b x)}-\frac {4 \cos (a+b x)}{35 b^2 \sin ^{\frac {5}{2}}(a+b x)}-\frac {12 \cos (a+b x)}{35 b^2 \sqrt {\sin (a+b x)}}-\frac {6 \int \sqrt {\sin (a+b x)} \, dx}{35 b}\\ &=-\frac {12 E\left (\left .\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right )\right |2\right )}{35 b^2}-\frac {2 x}{7 b \sin ^{\frac {7}{2}}(a+b x)}-\frac {4 \cos (a+b x)}{35 b^2 \sin ^{\frac {5}{2}}(a+b x)}-\frac {12 \cos (a+b x)}{35 b^2 \sqrt {\sin (a+b x)}}\\ \end {align*}
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Mathematica [A] time = 0.29, size = 73, normalized size = 0.83 \[ -\frac {2 \left (\sin (2 (a+b x))+6 \sin ^3(a+b x) \cos (a+b x)-6 \sin ^{\frac {7}{2}}(a+b x) E\left (\left .\frac {1}{4} (-2 a-2 b x+\pi )\right |2\right )+5 b x\right )}{35 b^2 \sin ^{\frac {7}{2}}(a+b x)} \]
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 \[ \int \frac {x \cos \left (b x + a\right )}{\sin \left (b x + a\right )^{\frac {9}{2}}}\,{d x} \]
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 \[ \int \frac {x \cos \left (b x +a \right )}{\sin \left (b x +a \right )^{\frac {9}{2}}}\, dx \]
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 \[ \int \frac {x \cos \left (b x + a\right )}{\sin \left (b x + a\right )^{\frac {9}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x\,\cos \left (a+b\,x\right )}{{\sin \left (a+b\,x\right )}^{9/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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