Optimal. Leaf size=85 \[ \frac {4 \cos (a+b x)}{9 b^2 \sqrt {\csc (a+b x)}}-\frac {4 \sqrt {\sin (a+b x)} \sqrt {\csc (a+b x)} F\left (\left .\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right )\right |2\right )}{9 b^2}+\frac {2 x}{3 b \csc ^{\frac {3}{2}}(a+b x)} \]
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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, 2641} \[ \frac {4 \cos (a+b x)}{9 b^2 \sqrt {\csc (a+b x)}}-\frac {4 \sqrt {\sin (a+b x)} \sqrt {\csc (a+b x)} F\left (\left .\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right )\right |2\right )}{9 b^2}+\frac {2 x}{3 b \csc ^{\frac {3}{2}}(a+b x)} \]
Antiderivative was successfully verified.
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Rule 2641
Rule 3769
Rule 3771
Rule 4213
Rubi steps
\begin {align*} \int \frac {x \cos (a+b x)}{\sqrt {\csc (a+b x)}} \, dx &=\frac {2 x}{3 b \csc ^{\frac {3}{2}}(a+b x)}-\frac {2 \int \frac {1}{\csc ^{\frac {3}{2}}(a+b x)} \, dx}{3 b}\\ &=\frac {2 x}{3 b \csc ^{\frac {3}{2}}(a+b x)}+\frac {4 \cos (a+b x)}{9 b^2 \sqrt {\csc (a+b x)}}-\frac {2 \int \sqrt {\csc (a+b x)} \, dx}{9 b}\\ &=\frac {2 x}{3 b \csc ^{\frac {3}{2}}(a+b x)}+\frac {4 \cos (a+b x)}{9 b^2 \sqrt {\csc (a+b x)}}-\frac {\left (2 \sqrt {\csc (a+b x)} \sqrt {\sin (a+b x)}\right ) \int \frac {1}{\sqrt {\sin (a+b x)}} \, dx}{9 b}\\ &=\frac {2 x}{3 b \csc ^{\frac {3}{2}}(a+b x)}+\frac {4 \cos (a+b x)}{9 b^2 \sqrt {\csc (a+b x)}}-\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)}}{9 b^2}\\ \end {align*}
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Mathematica [A] time = 0.23, size = 65, normalized size = 0.76 \[ \frac {2 \sqrt {\csc (a+b x)} \left (3 b x \sin ^2(a+b x)+\sin (2 (a+b x))+2 \sqrt {\sin (a+b x)} F\left (\left .\frac {1}{4} (-2 a-2 b x+\pi )\right |2\right )\right )}{9 b^2} \]
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 \[ \int \frac {x \cos \left (b x + a\right )}{\sqrt {\csc \left (b x + a\right )}}\,{d x} \]
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 \[ \int \frac {x \cos \left (b x +a \right )}{\sqrt {\csc \left (b x +a \right )}}\, dx \]
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 \[ \int \frac {x \cos \left (b x + a\right )}{\sqrt {\csc \left (b x + a\right )}}\,{d x} \]
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 \[ \int \frac {x\,\cos \left (a+b\,x\right )}{\sqrt {\frac {1}{\sin \left (a+b\,x\right )}}} \,d x \]
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 \[ \int \frac {x \cos {\left (a + b x \right )}}{\sqrt {\csc {\left (a + b x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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