Optimal. Leaf size=33 \[ \frac{2 x}{b \sqrt{\cos (a+b x)}}-\frac{4 \text{EllipticF}\left (\frac{1}{2} (a+b x),2\right )}{b^2} \]
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Rubi [A] time = 0.0256239, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {3444, 2641} \[ \frac{2 x}{b \sqrt{\cos (a+b x)}}-\frac{4 F\left (\left .\frac{1}{2} (a+b x)\right |2\right )}{b^2} \]
Antiderivative was successfully verified.
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Rule 3444
Rule 2641
Rubi steps
\begin{align*} \int \frac{x \sin (a+b x)}{\cos ^{\frac{3}{2}}(a+b x)} \, dx &=\frac{2 x}{b \sqrt{\cos (a+b x)}}-\frac{2 \int \frac{1}{\sqrt{\cos (a+b x)}} \, dx}{b}\\ &=\frac{2 x}{b \sqrt{\cos (a+b x)}}-\frac{4 F\left (\left .\frac{1}{2} (a+b x)\right |2\right )}{b^2}\\ \end{align*}
Mathematica [A] time = 0.164701, size = 33, normalized size = 1. \[ \frac{2 x}{b \sqrt{\cos (a+b x)}}-\frac{4 \text{EllipticF}\left (\frac{1}{2} (a+b x),2\right )}{b^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.089, size = 0, normalized size = 0. \begin{align*} \int{x\sin \left ( bx+a \right ) \left ( \cos \left ( bx+a \right ) \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x \sin \left (b x + a\right )}{\cos \left (b x + a\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x \sin \left (b x + a\right )}{\cos \left (b x + a\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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