Optimal. Leaf size=108 \[ -\frac {4 \cos (a+b x) \csc ^{\frac {5}{2}}(a+b x)}{35 b^2}-\frac {12 \cos (a+b x) \sqrt {\csc (a+b x)}}{35 b^2}-\frac {12 \sqrt {\sin (a+b x)} \sqrt {\csc (a+b x)} E\left (\left .\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right )\right |2\right )}{35 b^2}-\frac {2 x \csc ^{\frac {7}{2}}(a+b x)}{7 b} \]
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Rubi [A] time = 0.06, antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {4213, 3768, 3771, 2639} \[ -\frac {4 \cos (a+b x) \csc ^{\frac {5}{2}}(a+b x)}{35 b^2}-\frac {12 \cos (a+b x) \sqrt {\csc (a+b x)}}{35 b^2}-\frac {12 \sqrt {\sin (a+b x)} \sqrt {\csc (a+b x)} E\left (\left .\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right )\right |2\right )}{35 b^2}-\frac {2 x \csc ^{\frac {7}{2}}(a+b x)}{7 b} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 3768
Rule 3771
Rule 4213
Rubi steps
\begin {align*} \int x \cos (a+b x) \csc ^{\frac {9}{2}}(a+b x) \, dx &=-\frac {2 x \csc ^{\frac {7}{2}}(a+b x)}{7 b}+\frac {2 \int \csc ^{\frac {7}{2}}(a+b x) \, dx}{7 b}\\ &=-\frac {4 \cos (a+b x) \csc ^{\frac {5}{2}}(a+b x)}{35 b^2}-\frac {2 x \csc ^{\frac {7}{2}}(a+b x)}{7 b}+\frac {6 \int \csc ^{\frac {3}{2}}(a+b x) \, dx}{35 b}\\ &=-\frac {12 \cos (a+b x) \sqrt {\csc (a+b x)}}{35 b^2}-\frac {4 \cos (a+b x) \csc ^{\frac {5}{2}}(a+b x)}{35 b^2}-\frac {2 x \csc ^{\frac {7}{2}}(a+b x)}{7 b}-\frac {6 \int \frac {1}{\sqrt {\csc (a+b x)}} \, dx}{35 b}\\ &=-\frac {12 \cos (a+b x) \sqrt {\csc (a+b x)}}{35 b^2}-\frac {4 \cos (a+b x) \csc ^{\frac {5}{2}}(a+b x)}{35 b^2}-\frac {2 x \csc ^{\frac {7}{2}}(a+b x)}{7 b}-\frac {\left (6 \sqrt {\csc (a+b x)} \sqrt {\sin (a+b x)}\right ) \int \sqrt {\sin (a+b x)} \, dx}{35 b}\\ &=-\frac {12 \cos (a+b x) \sqrt {\csc (a+b x)}}{35 b^2}-\frac {4 \cos (a+b x) \csc ^{\frac {5}{2}}(a+b x)}{35 b^2}-\frac {2 x \csc ^{\frac {7}{2}}(a+b x)}{7 b}-\frac {12 \sqrt {\csc (a+b x)} E\left (\left .\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right )\right |2\right ) \sqrt {\sin (a+b x)}}{35 b^2}\\ \end {align*}
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Mathematica [A] time = 0.27, size = 73, normalized size = 0.68 \[ -\frac {2 \csc ^{\frac {7}{2}}(a+b x) \left (\sin (2 (a+b x))+6 \sin ^3(a+b x) \cos (a+b x)-6 \sin ^{\frac {7}{2}}(a+b x) E\left (\left .\frac {1}{4} (-2 a-2 b x+\pi )\right |2\right )+5 b x\right )}{35 b^2} \]
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 \[ \int x \cos \left (b x + a\right ) \csc \left (b x + a\right )^{\frac {9}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.08, size = 0, normalized size = 0.00 \[ \int x \cos \left (b x +a \right ) \left (\csc ^{\frac {9}{2}}\left (b x +a \right )\right )\, dx \]
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 \[ \int x \cos \left (b x + a\right ) \csc \left (b x + a\right )^{\frac {9}{2}}\,{d x} \]
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 \[ \int x\,\cos \left (a+b\,x\right )\,{\left (\frac {1}{\sin \left (a+b\,x\right )}\right )}^{9/2} \,d x \]
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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