Optimal. Leaf size=108 \[ \frac {4 \cos (a+b x)}{49 b^2 \csc ^{\frac {5}{2}}(a+b x)}+\frac {20 \cos (a+b x)}{147 b^2 \sqrt {\csc (a+b x)}}-\frac {20 \sqrt {\sin (a+b x)} \sqrt {\csc (a+b x)} F\left (\left .\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right )\right |2\right )}{147 b^2}+\frac {2 x}{7 b \csc ^{\frac {7}{2}}(a+b x)} \]
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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, 3769, 3771, 2641} \[ \frac {4 \cos (a+b x)}{49 b^2 \csc ^{\frac {5}{2}}(a+b x)}+\frac {20 \cos (a+b x)}{147 b^2 \sqrt {\csc (a+b x)}}-\frac {20 \sqrt {\sin (a+b x)} \sqrt {\csc (a+b x)} F\left (\left .\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right )\right |2\right )}{147 b^2}+\frac {2 x}{7 b \csc ^{\frac {7}{2}}(a+b x)} \]
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)}{\csc ^{\frac {5}{2}}(a+b x)} \, dx &=\frac {2 x}{7 b \csc ^{\frac {7}{2}}(a+b x)}-\frac {2 \int \frac {1}{\csc ^{\frac {7}{2}}(a+b x)} \, dx}{7 b}\\ &=\frac {2 x}{7 b \csc ^{\frac {7}{2}}(a+b x)}+\frac {4 \cos (a+b x)}{49 b^2 \csc ^{\frac {5}{2}}(a+b x)}-\frac {10 \int \frac {1}{\csc ^{\frac {3}{2}}(a+b x)} \, dx}{49 b}\\ &=\frac {2 x}{7 b \csc ^{\frac {7}{2}}(a+b x)}+\frac {4 \cos (a+b x)}{49 b^2 \csc ^{\frac {5}{2}}(a+b x)}+\frac {20 \cos (a+b x)}{147 b^2 \sqrt {\csc (a+b x)}}-\frac {10 \int \sqrt {\csc (a+b x)} \, dx}{147 b}\\ &=\frac {2 x}{7 b \csc ^{\frac {7}{2}}(a+b x)}+\frac {4 \cos (a+b x)}{49 b^2 \csc ^{\frac {5}{2}}(a+b x)}+\frac {20 \cos (a+b x)}{147 b^2 \sqrt {\csc (a+b x)}}-\frac {\left (10 \sqrt {\csc (a+b x)} \sqrt {\sin (a+b x)}\right ) \int \frac {1}{\sqrt {\sin (a+b x)}} \, dx}{147 b}\\ &=\frac {2 x}{7 b \csc ^{\frac {7}{2}}(a+b x)}+\frac {4 \cos (a+b x)}{49 b^2 \csc ^{\frac {5}{2}}(a+b x)}+\frac {20 \cos (a+b x)}{147 b^2 \sqrt {\csc (a+b x)}}-\frac {20 \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)}}{147 b^2}\\ \end {align*}
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Mathematica [A] time = 0.39, size = 93, normalized size = 0.86 \[ \frac {\sqrt {\csc (a+b x)} \left (52 \sin (2 (a+b x))-6 \sin (4 (a+b x))-84 b x \cos (2 (a+b x))+21 b x \cos (4 (a+b x))+80 \sqrt {\sin (a+b x)} F\left (\left .\frac {1}{4} (-2 a-2 b x+\pi )\right |2\right )+63 b x\right )}{588 b^2} \]
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 \[ \int \frac {x \cos \left (b x + a\right )}{\csc \left (b x + a\right )^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {x \cos \left (b x +a \right )}{\csc \left (b x +a \right )^{\frac {5}{2}}}\, dx \]
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 \[ \int \frac {x \cos \left (b x + a\right )}{\csc \left (b x + a\right )^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {x\,\cos \left (a+b\,x\right )}{{\left (\frac {1}{\sin \left (a+b\,x\right )}\right )}^{5/2}} \,d x \]
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 \[ \int \frac {x \cos {\left (a + b x \right )}}{\csc ^{\frac {5}{2}}{\left (a + b x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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