Optimal. Leaf size=85 \[ -\frac {4 \cos (a+b x) \csc ^{\frac {3}{2}}(a+b x)}{15 b^2}+\frac {4 \sqrt {\sin (a+b x)} \sqrt {\csc (a+b x)} F\left (\left .\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right )\right |2\right )}{15 b^2}-\frac {2 x \csc ^{\frac {5}{2}}(a+b x)}{5 b} \]
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Rubi [A] time = 0.04, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {4213, 3768, 3771, 2641} \[ -\frac {4 \cos (a+b x) \csc ^{\frac {3}{2}}(a+b x)}{15 b^2}+\frac {4 \sqrt {\sin (a+b x)} \sqrt {\csc (a+b x)} F\left (\left .\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right )\right |2\right )}{15 b^2}-\frac {2 x \csc ^{\frac {5}{2}}(a+b x)}{5 b} \]
Antiderivative was successfully verified.
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Rule 2641
Rule 3768
Rule 3771
Rule 4213
Rubi steps
\begin {align*} \int x \cos (a+b x) \csc ^{\frac {7}{2}}(a+b x) \, dx &=-\frac {2 x \csc ^{\frac {5}{2}}(a+b x)}{5 b}+\frac {2 \int \csc ^{\frac {5}{2}}(a+b x) \, dx}{5 b}\\ &=-\frac {4 \cos (a+b x) \csc ^{\frac {3}{2}}(a+b x)}{15 b^2}-\frac {2 x \csc ^{\frac {5}{2}}(a+b x)}{5 b}+\frac {2 \int \sqrt {\csc (a+b x)} \, dx}{15 b}\\ &=-\frac {4 \cos (a+b x) \csc ^{\frac {3}{2}}(a+b x)}{15 b^2}-\frac {2 x \csc ^{\frac {5}{2}}(a+b x)}{5 b}+\frac {\left (2 \sqrt {\csc (a+b x)} \sqrt {\sin (a+b x)}\right ) \int \frac {1}{\sqrt {\sin (a+b x)}} \, dx}{15 b}\\ &=-\frac {4 \cos (a+b x) \csc ^{\frac {3}{2}}(a+b x)}{15 b^2}-\frac {2 x \csc ^{\frac {5}{2}}(a+b x)}{5 b}+\frac {4 \sqrt {\csc (a+b x)} F\left (\left .\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right )\right |2\right ) \sqrt {\sin (a+b x)}}{15 b^2}\\ \end {align*}
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Mathematica [A] time = 0.29, size = 65, normalized size = 0.76 \[ -\frac {2 \sqrt {\csc (a+b x)} \left (2 \cot (a+b x)+3 b x \csc ^2(a+b x)+2 \sqrt {\sin (a+b x)} F\left (\left .\frac {1}{4} (-2 a-2 b x+\pi )\right |2\right )\right )}{15 b^2} \]
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 \[ \int x \cos \left (b x + a\right ) \csc \left (b x + a\right )^{\frac {7}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.06, size = 0, normalized size = 0.00 \[ \int x \cos \left (b x +a \right ) \left (\csc ^{\frac {7}{2}}\left (b x +a \right )\right )\, dx \]
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 \[ \int x \cos \left (b x + a\right ) \csc \left (b x + a\right )^{\frac {7}{2}}\,{d x} \]
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 \[ \int x\,\cos \left (a+b\,x\right )\,{\left (\frac {1}{\sin \left (a+b\,x\right )}\right )}^{7/2} \,d x \]
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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