Optimal. Leaf size=53 \[ \frac{2 x \sqrt{\sec (a+b x)}}{b}-\frac{4 \sqrt{\cos (a+b x)} \sqrt{\sec (a+b x)} \text{EllipticF}\left (\frac{1}{2} (a+b x),2\right )}{b^2} \]
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Rubi [A] time = 0.0358781, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {4212, 3771, 2641} \[ \frac{2 x \sqrt{\sec (a+b x)}}{b}-\frac{4 \sqrt{\cos (a+b x)} \sqrt{\sec (a+b x)} F\left (\left .\frac{1}{2} (a+b x)\right |2\right )}{b^2} \]
Antiderivative was successfully verified.
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Rule 4212
Rule 3771
Rule 2641
Rubi steps
\begin{align*} \int x \sec ^{\frac{3}{2}}(a+b x) \sin (a+b x) \, dx &=\frac{2 x \sqrt{\sec (a+b x)}}{b}-\frac{2 \int \sqrt{\sec (a+b x)} \, dx}{b}\\ &=\frac{2 x \sqrt{\sec (a+b x)}}{b}-\frac{\left (2 \sqrt{\cos (a+b x)} \sqrt{\sec (a+b x)}\right ) \int \frac{1}{\sqrt{\cos (a+b x)}} \, dx}{b}\\ &=\frac{2 x \sqrt{\sec (a+b x)}}{b}-\frac{4 \sqrt{\cos (a+b x)} F\left (\left .\frac{1}{2} (a+b x)\right |2\right ) \sqrt{\sec (a+b x)}}{b^2}\\ \end{align*}
Mathematica [A] time = 0.140606, size = 42, normalized size = 0.79 \[ \frac{2 \sqrt{\sec (a+b x)} \left (b x-2 \sqrt{\cos (a+b x)} \text{EllipticF}\left (\frac{1}{2} (a+b x),2\right )\right )}{b^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.101, size = 0, normalized size = 0. \begin{align*} \int x \left ( \sec \left ( bx+a \right ) \right ) ^{{\frac{3}{2}}}\sin \left ( bx+a \right ) \, 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 x \sec \left (b x + a\right )^{\frac{3}{2}} \sin \left (b x + a\right )\,{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 x \sec \left (b x + a\right )^{\frac{3}{2}} \sin \left (b x + a\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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