Optimal. Leaf size=103 \[ -\frac{4 \sin (a+b x) \sec ^{\frac{5}{2}}(a+b x)}{35 b^2}-\frac{12 \sin (a+b x) \sqrt{\sec (a+b x)}}{35 b^2}+\frac{12 \sqrt{\cos (a+b x)} \sqrt{\sec (a+b x)} E\left (\left .\frac{1}{2} (a+b x)\right |2\right )}{35 b^2}+\frac{2 x \sec ^{\frac{7}{2}}(a+b x)}{7 b} \]
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Rubi [A] time = 0.0620826, antiderivative size = 103, normalized size of antiderivative = 1., 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 = {4212, 3768, 3771, 2639} \[ -\frac{4 \sin (a+b x) \sec ^{\frac{5}{2}}(a+b x)}{35 b^2}-\frac{12 \sin (a+b x) \sqrt{\sec (a+b x)}}{35 b^2}+\frac{12 \sqrt{\cos (a+b x)} \sqrt{\sec (a+b x)} E\left (\left .\frac{1}{2} (a+b x)\right |2\right )}{35 b^2}+\frac{2 x \sec ^{\frac{7}{2}}(a+b x)}{7 b} \]
Antiderivative was successfully verified.
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Rule 4212
Rule 3768
Rule 3771
Rule 2639
Rubi steps
\begin{align*} \int x \sec ^{\frac{9}{2}}(a+b x) \sin (a+b x) \, dx &=\frac{2 x \sec ^{\frac{7}{2}}(a+b x)}{7 b}-\frac{2 \int \sec ^{\frac{7}{2}}(a+b x) \, dx}{7 b}\\ &=\frac{2 x \sec ^{\frac{7}{2}}(a+b x)}{7 b}-\frac{4 \sec ^{\frac{5}{2}}(a+b x) \sin (a+b x)}{35 b^2}-\frac{6 \int \sec ^{\frac{3}{2}}(a+b x) \, dx}{35 b}\\ &=\frac{2 x \sec ^{\frac{7}{2}}(a+b x)}{7 b}-\frac{12 \sqrt{\sec (a+b x)} \sin (a+b x)}{35 b^2}-\frac{4 \sec ^{\frac{5}{2}}(a+b x) \sin (a+b x)}{35 b^2}+\frac{6 \int \frac{1}{\sqrt{\sec (a+b x)}} \, dx}{35 b}\\ &=\frac{2 x \sec ^{\frac{7}{2}}(a+b x)}{7 b}-\frac{12 \sqrt{\sec (a+b x)} \sin (a+b x)}{35 b^2}-\frac{4 \sec ^{\frac{5}{2}}(a+b x) \sin (a+b x)}{35 b^2}+\frac{\left (6 \sqrt{\cos (a+b x)} \sqrt{\sec (a+b x)}\right ) \int \sqrt{\cos (a+b x)} \, dx}{35 b}\\ &=\frac{12 \sqrt{\cos (a+b x)} E\left (\left .\frac{1}{2} (a+b x)\right |2\right ) \sqrt{\sec (a+b x)}}{35 b^2}+\frac{2 x \sec ^{\frac{7}{2}}(a+b x)}{7 b}-\frac{12 \sqrt{\sec (a+b x)} \sin (a+b x)}{35 b^2}-\frac{4 \sec ^{\frac{5}{2}}(a+b x) \sin (a+b x)}{35 b^2}\\ \end{align*}
Mathematica [A] time = 0.286994, size = 65, normalized size = 0.63 \[ \frac{\sec ^{\frac{7}{2}}(a+b x) \left (-10 \sin (2 (a+b x))-3 \sin (4 (a+b x))+24 \cos ^{\frac{7}{2}}(a+b x) E\left (\left .\frac{1}{2} (a+b x)\right |2\right )+20 b x\right )}{70 b^2} \]
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{9}{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{9}{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{9}{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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