Optimal. Leaf size=33 \[ -\frac{2 c d (d \csc (a+b x))^{3/2}}{3 b (c \sec (a+b x))^{3/2}} \]
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Rubi [A] time = 0.0480831, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04, Rules used = {2619} \[ -\frac{2 c d (d \csc (a+b x))^{3/2}}{3 b (c \sec (a+b x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 2619
Rubi steps
\begin{align*} \int \frac{(d \csc (a+b x))^{5/2}}{\sqrt{c \sec (a+b x)}} \, dx &=-\frac{2 c d (d \csc (a+b x))^{3/2}}{3 b (c \sec (a+b x))^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.111938, size = 33, normalized size = 1. \[ -\frac{2 c d (d \csc (a+b x))^{3/2}}{3 b (c \sec (a+b x))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.153, size = 42, normalized size = 1.3 \begin{align*} -{\frac{2\,\cos \left ( bx+a \right ) \sin \left ( bx+a \right ) }{3\,b} \left ({\frac{d}{\sin \left ( bx+a \right ) }} \right ) ^{{\frac{5}{2}}}{\frac{1}{\sqrt{{\frac{c}{\cos \left ( bx+a \right ) }}}}}} \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{\left (d \csc \left (b x + a\right )\right )^{\frac{5}{2}}}{\sqrt{c \sec \left (b x + a\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.06628, size = 116, normalized size = 3.52 \begin{align*} -\frac{2 \, d^{2} \sqrt{\frac{c}{\cos \left (b x + a\right )}} \sqrt{\frac{d}{\sin \left (b x + a\right )}} \cos \left (b x + a\right )^{2}}{3 \, b c \sin \left (b x + a\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 \frac{\left (d \csc \left (b x + a\right )\right )^{\frac{5}{2}}}{\sqrt{c \sec \left (b x + a\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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