Optimal. Leaf size=85 \[ \frac {4 \cos (a+b x)}{25 b^2 \csc ^{\frac {3}{2}}(a+b x)}-\frac {12 \sqrt {\sin (a+b x)} \sqrt {\csc (a+b x)} E\left (\left .\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right )\right |2\right )}{25 b^2}+\frac {2 x}{5 b \csc ^{\frac {5}{2}}(a+b x)} \]
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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, 3769, 3771, 2639} \[ \frac {4 \cos (a+b x)}{25 b^2 \csc ^{\frac {3}{2}}(a+b x)}-\frac {12 \sqrt {\sin (a+b x)} \sqrt {\csc (a+b x)} E\left (\left .\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right )\right |2\right )}{25 b^2}+\frac {2 x}{5 b \csc ^{\frac {5}{2}}(a+b x)} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 3769
Rule 3771
Rule 4213
Rubi steps
\begin {align*} \int \frac {x \cos (a+b x)}{\csc ^{\frac {3}{2}}(a+b x)} \, dx &=\frac {2 x}{5 b \csc ^{\frac {5}{2}}(a+b x)}-\frac {2 \int \frac {1}{\csc ^{\frac {5}{2}}(a+b x)} \, dx}{5 b}\\ &=\frac {2 x}{5 b \csc ^{\frac {5}{2}}(a+b x)}+\frac {4 \cos (a+b x)}{25 b^2 \csc ^{\frac {3}{2}}(a+b x)}-\frac {6 \int \frac {1}{\sqrt {\csc (a+b x)}} \, dx}{25 b}\\ &=\frac {2 x}{5 b \csc ^{\frac {5}{2}}(a+b x)}+\frac {4 \cos (a+b x)}{25 b^2 \csc ^{\frac {3}{2}}(a+b x)}-\frac {\left (6 \sqrt {\csc (a+b x)} \sqrt {\sin (a+b x)}\right ) \int \sqrt {\sin (a+b x)} \, dx}{25 b}\\ &=\frac {2 x}{5 b \csc ^{\frac {5}{2}}(a+b x)}+\frac {4 \cos (a+b x)}{25 b^2 \csc ^{\frac {3}{2}}(a+b x)}-\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)}}{25 b^2}\\ \end {align*}
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Mathematica [C] time = 0.98, size = 114, normalized size = 1.34 \[ \frac {\tan \left (\frac {1}{2} (a+b x)\right ) \left (4 \sqrt {2} \sqrt {\frac {1}{\cos (a+b x)+1}} \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};-\tan ^2\left (\frac {1}{2} (a+b x)\right )\right )+10 b x \sin (a+b x)+5 b x \sin (2 (a+b x))+4 \cos (a+b x)+2 \cos (2 (a+b x))-10\right )}{25 b^2 \sqrt {\csc (a+b x)}} \]
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 \[ \int \frac {x \cos \left (b x + a\right )}{\csc \left (b x + a\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.05, size = 0, normalized size = 0.00 \[ \int \frac {x \cos \left (b x +a \right )}{\csc \left (b x +a \right )^{\frac {3}{2}}}\, dx \]
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 \[ \int \frac {x \cos \left (b x + a\right )}{\csc \left (b x + a\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x\,\cos \left (a+b\,x\right )}{{\left (\frac {1}{\sin \left (a+b\,x\right )}\right )}^{3/2}} \,d x \]
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 \[ \int \frac {x \cos {\left (a + b x \right )}}{\csc ^{\frac {3}{2}}{\left (a + b x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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