Optimal. Leaf size=80 \[ -\frac{4 \sqrt{\cos (a+b x)} \sqrt{\sec (a+b x)} \text{EllipticF}\left (\frac{1}{2} (a+b x),2\right )}{15 b^2}-\frac{4 \sin (a+b x) \sec ^{\frac{3}{2}}(a+b x)}{15 b^2}+\frac{2 x \sec ^{\frac{5}{2}}(a+b x)}{5 b} \]
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Rubi [A] time = 0.0488525, antiderivative size = 80, 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 = {4212, 3768, 3771, 2641} \[ -\frac{4 \sin (a+b x) \sec ^{\frac{3}{2}}(a+b x)}{15 b^2}-\frac{4 \sqrt{\cos (a+b x)} \sqrt{\sec (a+b x)} F\left (\left .\frac{1}{2} (a+b x)\right |2\right )}{15 b^2}+\frac{2 x \sec ^{\frac{5}{2}}(a+b x)}{5 b} \]
Antiderivative was successfully verified.
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Rule 4212
Rule 3768
Rule 3771
Rule 2641
Rubi steps
\begin{align*} \int x \sec ^{\frac{7}{2}}(a+b x) \sin (a+b x) \, dx &=\frac{2 x \sec ^{\frac{5}{2}}(a+b x)}{5 b}-\frac{2 \int \sec ^{\frac{5}{2}}(a+b x) \, dx}{5 b}\\ &=\frac{2 x \sec ^{\frac{5}{2}}(a+b x)}{5 b}-\frac{4 \sec ^{\frac{3}{2}}(a+b x) \sin (a+b x)}{15 b^2}-\frac{2 \int \sqrt{\sec (a+b x)} \, dx}{15 b}\\ &=\frac{2 x \sec ^{\frac{5}{2}}(a+b x)}{5 b}-\frac{4 \sec ^{\frac{3}{2}}(a+b x) \sin (a+b x)}{15 b^2}-\frac{\left (2 \sqrt{\cos (a+b x)} \sqrt{\sec (a+b x)}\right ) \int \frac{1}{\sqrt{\cos (a+b x)}} \, dx}{15 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)}}{15 b^2}+\frac{2 x \sec ^{\frac{5}{2}}(a+b x)}{5 b}-\frac{4 \sec ^{\frac{3}{2}}(a+b x) \sin (a+b x)}{15 b^2}\\ \end{align*}
Mathematica [A] time = 0.242782, size = 61, normalized size = 0.76 \[ \frac{2 \sqrt{\sec (a+b x)} \left (-2 \sqrt{\cos (a+b x)} \text{EllipticF}\left (\frac{1}{2} (a+b x),2\right )-2 \tan (a+b x)+3 b x \sec ^2(a+b x)\right )}{15 b^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.095, size = 0, normalized size = 0. \begin{align*} \int x \left ( \sec \left ( bx+a \right ) \right ) ^{{\frac{7}{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{7}{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{7}{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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