Optimal. Leaf size=92 \[ \frac{\sec ^3(a+b x)}{3 b \sqrt{\csc (a+b x)}}+\frac{5 \sec (a+b x)}{6 b \sqrt{\csc (a+b x)}}+\frac{5 \sqrt{\sin (a+b x)} \sqrt{\csc (a+b x)} F\left (\left .\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right )\right |2\right )}{6 b} \]
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Rubi [A] time = 0.0813144, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {2626, 3771, 2641} \[ \frac{\sec ^3(a+b x)}{3 b \sqrt{\csc (a+b x)}}+\frac{5 \sec (a+b x)}{6 b \sqrt{\csc (a+b x)}}+\frac{5 \sqrt{\sin (a+b x)} \sqrt{\csc (a+b x)} F\left (\left .\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right )\right |2\right )}{6 b} \]
Antiderivative was successfully verified.
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Rule 2626
Rule 3771
Rule 2641
Rubi steps
\begin{align*} \int \sqrt{\csc (a+b x)} \sec ^4(a+b x) \, dx &=\frac{\sec ^3(a+b x)}{3 b \sqrt{\csc (a+b x)}}+\frac{5}{6} \int \sqrt{\csc (a+b x)} \sec ^2(a+b x) \, dx\\ &=\frac{5 \sec (a+b x)}{6 b \sqrt{\csc (a+b x)}}+\frac{\sec ^3(a+b x)}{3 b \sqrt{\csc (a+b x)}}+\frac{5}{12} \int \sqrt{\csc (a+b x)} \, dx\\ &=\frac{5 \sec (a+b x)}{6 b \sqrt{\csc (a+b x)}}+\frac{\sec ^3(a+b x)}{3 b \sqrt{\csc (a+b x)}}+\frac{1}{12} \left (5 \sqrt{\csc (a+b x)} \sqrt{\sin (a+b x)}\right ) \int \frac{1}{\sqrt{\sin (a+b x)}} \, dx\\ &=\frac{5 \sec (a+b x)}{6 b \sqrt{\csc (a+b x)}}+\frac{\sec ^3(a+b x)}{3 b \sqrt{\csc (a+b x)}}+\frac{5 \sqrt{\csc (a+b x)} F\left (\left .\frac{1}{2} \left (a-\frac{\pi }{2}+b x\right )\right |2\right ) \sqrt{\sin (a+b x)}}{6 b}\\ \end{align*}
Mathematica [A] time = 0.378396, size = 64, normalized size = 0.7 \[ \frac{\sqrt{\csc (a+b x)} \left (\tan (a+b x) \left (2 \sec ^2(a+b x)+5\right )-5 \sqrt{\sin (a+b x)} F\left (\left .\frac{1}{4} (-2 a-2 b x+\pi )\right |2\right )\right )}{6 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 1.954, size = 168, normalized size = 1.8 \begin{align*} -{\frac{1}{ \left ( 12\,\sin \left ( bx+a \right ) -12 \right ) \left ( \sin \left ( bx+a \right ) +1 \right ) \cos \left ( bx+a \right ) b}\sqrt{ \left ( \cos \left ( bx+a \right ) \right ) ^{2}\sin \left ( bx+a \right ) } \left ( 5\,\sqrt{\sin \left ( bx+a \right ) +1}\sqrt{-2\,\sin \left ( bx+a \right ) +2}\sqrt{-\sin \left ( bx+a \right ) }{\it EllipticF} \left ( \sqrt{\sin \left ( bx+a \right ) +1},1/2\,\sqrt{2} \right ) \left ( \cos \left ( bx+a \right ) \right ) ^{2}+10\, \left ( \cos \left ( bx+a \right ) \right ) ^{2}\sin \left ( bx+a \right ) +4\,\sin \left ( bx+a \right ) \right ){\frac{1}{\sqrt{-\sin \left ( bx+a \right ) \left ( \sin \left ( bx+a \right ) -1 \right ) \left ( \sin \left ( bx+a \right ) +1 \right ) }}}{\frac{1}{\sqrt{\sin \left ( bx+a \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [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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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sqrt{\csc \left (b x + a\right )} \sec \left (b x + a\right )^{4}, x\right ) \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 \sqrt{\csc \left (b x + a\right )} \sec \left (b x + a\right )^{4}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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