Optimal. Leaf size=58 \[ \frac{2 \sin (a+b x) \sqrt{\sec (a+b x)}}{b}-\frac{2 \sqrt{\cos (a+b x)} \sqrt{\sec (a+b x)} E\left (\left .\frac{1}{2} (a+b x)\right |2\right )}{b} \]
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Rubi [A] time = 0.0255371, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {3768, 3771, 2639} \[ \frac{2 \sin (a+b x) \sqrt{\sec (a+b x)}}{b}-\frac{2 \sqrt{\cos (a+b x)} \sqrt{\sec (a+b x)} E\left (\left .\frac{1}{2} (a+b x)\right |2\right )}{b} \]
Antiderivative was successfully verified.
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Rule 3768
Rule 3771
Rule 2639
Rubi steps
\begin{align*} \int \sec ^{\frac{3}{2}}(a+b x) \, dx &=\frac{2 \sqrt{\sec (a+b x)} \sin (a+b x)}{b}-\int \frac{1}{\sqrt{\sec (a+b x)}} \, dx\\ &=\frac{2 \sqrt{\sec (a+b x)} \sin (a+b x)}{b}-\left (\sqrt{\cos (a+b x)} \sqrt{\sec (a+b x)}\right ) \int \sqrt{\cos (a+b x)} \, dx\\ &=-\frac{2 \sqrt{\cos (a+b x)} E\left (\left .\frac{1}{2} (a+b x)\right |2\right ) \sqrt{\sec (a+b x)}}{b}+\frac{2 \sqrt{\sec (a+b x)} \sin (a+b x)}{b}\\ \end{align*}
Mathematica [A] time = 0.0475868, size = 45, normalized size = 0.78 \[ \frac{2 \sqrt{\sec (a+b x)} \left (\sin (a+b x)-\sqrt{\cos (a+b x)} E\left (\left .\frac{1}{2} (a+b x)\right |2\right )\right )}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 1.062, size = 101, normalized size = 1.7 \begin{align*} -2\,{\frac{\sqrt{2\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}-1}\sqrt{ \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}}{\it EllipticE} \left ( \cos \left ( 1/2\,bx+a/2 \right ) ,\sqrt{2} \right ) -2\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}\cos \left ( 1/2\,bx+a/2 \right ) }{\sin \left ( 1/2\,bx+a/2 \right ) \sqrt{2\, \left ( \cos \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}-1}b}} \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 \sec \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] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sec \left (b x + a\right )^{\frac{3}{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sec ^{\frac{3}{2}}{\left (a + b x \right )}\, dx \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 \sec \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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