Optimal. Leaf size=62 \[ \frac{2 \sin (a+b x)}{5 b \sec ^{\frac{3}{2}}(a+b x)}+\frac{6 \sqrt{\cos (a+b x)} \sqrt{\sec (a+b x)} E\left (\left .\frac{1}{2} (a+b x)\right |2\right )}{5 b} \]
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Rubi [A] time = 0.0286673, antiderivative size = 62, 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 = {3769, 3771, 2639} \[ \frac{2 \sin (a+b x)}{5 b \sec ^{\frac{3}{2}}(a+b x)}+\frac{6 \sqrt{\cos (a+b x)} \sqrt{\sec (a+b x)} E\left (\left .\frac{1}{2} (a+b x)\right |2\right )}{5 b} \]
Antiderivative was successfully verified.
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Rule 3769
Rule 3771
Rule 2639
Rubi steps
\begin{align*} \int \frac{1}{\sec ^{\frac{5}{2}}(a+b x)} \, dx &=\frac{2 \sin (a+b x)}{5 b \sec ^{\frac{3}{2}}(a+b x)}+\frac{3}{5} \int \frac{1}{\sqrt{\sec (a+b x)}} \, dx\\ &=\frac{2 \sin (a+b x)}{5 b \sec ^{\frac{3}{2}}(a+b x)}+\frac{1}{5} \left (3 \sqrt{\cos (a+b x)} \sqrt{\sec (a+b x)}\right ) \int \sqrt{\cos (a+b x)} \, dx\\ &=\frac{6 \sqrt{\cos (a+b x)} E\left (\left .\frac{1}{2} (a+b x)\right |2\right ) \sqrt{\sec (a+b x)}}{5 b}+\frac{2 \sin (a+b x)}{5 b \sec ^{\frac{3}{2}}(a+b x)}\\ \end{align*}
Mathematica [A] time = 0.0693045, size = 55, normalized size = 0.89 \[ \frac{\sqrt{\sec (a+b x)} \left (\sin (a+b x)+\sin (3 (a+b x))+12 \sqrt{\cos (a+b x)} E\left (\left .\frac{1}{2} (a+b x)\right |2\right )\right )}{10 b} \]
Antiderivative was successfully verified.
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Maple [B] time = 1.212, size = 202, normalized size = 3.3 \begin{align*} -{\frac{2}{5\,b}\sqrt{ \left ( 2\, \left ( \cos \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}-1 \right ) \left ( \sin \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) ^{2}} \left ( -8\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{6}\cos \left ( 1/2\,bx+a/2 \right ) +8\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{4}\cos \left ( 1/2\,bx+a/2 \right ) -3\,\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 ) \right ){\frac{1}{\sqrt{-2\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{4}+ \left ( \sin \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) ^{2}}}} \left ( \sin \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) ^{-1}{\frac{1}{\sqrt{2\, \left ( \cos \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}-1}}}} \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{1}{\sec \left (b x + a\right )^{\frac{5}{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 (\frac{1}{\sec \left (b x + a\right )^{\frac{5}{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 \frac{1}{\sec ^{\frac{5}{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 \frac{1}{\sec \left (b x + a\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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