Optimal. Leaf size=85 \[ \frac{4 \cos (a+b x)}{25 b^2 \csc ^{\frac{3}{2}}(a+b x)}-\frac{12 \sqrt{\sin (a+b x)} \sqrt{\csc (a+b x)} E\left (\left .\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right )\right |2\right )}{25 b^2}+\frac{2 x}{5 b \csc ^{\frac{5}{2}}(a+b x)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0437454, antiderivative size = 85, normalized size of antiderivative = 1., 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.
[In]
[Out]
Rule 4213
Rule 3769
Rule 3771
Rule 2639
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*}
Mathematica [C] time = 0.997851, size = 114, normalized size = 1.34 \[ \frac{\tan \left (\frac{1}{2} (a+b x)\right ) \left (4 \sqrt{2} \sqrt{\frac{1}{\cos (a+b x)+1}} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{3}{4},\frac{7}{4},-\tan ^2\left (\frac{1}{2} (a+b x)\right )\right )+10 b x \sin (a+b x)+5 b x \sin (2 (a+b x))+4 \cos (a+b x)+2 \cos (2 (a+b x))-10\right )}{25 b^2 \sqrt{\csc (a+b x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.103, size = 0, normalized size = 0. \begin{align*} \int{x\cos \left ( bx+a \right ) \left ( \csc \left ( bx+a \right ) \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x \cos \left (b x + a\right )}{\csc \left (b x + a\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x \cos{\left (a + b x \right )}}{\csc ^{\frac{3}{2}}{\left (a + b x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x \cos \left (b x + a\right )}{\csc \left (b x + a\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]