Optimal. Leaf size=69 \[ -\frac{8 c d^3 (d \csc (a+b x))^{3/2}}{21 b (c \sec (a+b x))^{3/2}}-\frac{2 c d (d \csc (a+b x))^{7/2}}{7 b (c \sec (a+b x))^{3/2}} \]
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Rubi [A] time = 0.0996172, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {2625, 2619} \[ -\frac{8 c d^3 (d \csc (a+b x))^{3/2}}{21 b (c \sec (a+b x))^{3/2}}-\frac{2 c d (d \csc (a+b x))^{7/2}}{7 b (c \sec (a+b x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 2625
Rule 2619
Rubi steps
\begin{align*} \int \frac{(d \csc (a+b x))^{9/2}}{\sqrt{c \sec (a+b x)}} \, dx &=-\frac{2 c d (d \csc (a+b x))^{7/2}}{7 b (c \sec (a+b x))^{3/2}}+\frac{1}{7} \left (4 d^2\right ) \int \frac{(d \csc (a+b x))^{5/2}}{\sqrt{c \sec (a+b x)}} \, dx\\ &=-\frac{8 c d^3 (d \csc (a+b x))^{3/2}}{21 b (c \sec (a+b x))^{3/2}}-\frac{2 c d (d \csc (a+b x))^{7/2}}{7 b (c \sec (a+b x))^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.25782, size = 45, normalized size = 0.65 \[ \frac{2 c d (2 \cos (2 (a+b x))-5) (d \csc (a+b x))^{7/2}}{21 b (c \sec (a+b x))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.164, size = 54, normalized size = 0.8 \begin{align*}{\frac{ \left ( 8\, \left ( \cos \left ( bx+a \right ) \right ) ^{2}-14 \right ) \cos \left ( bx+a \right ) \sin \left ( bx+a \right ) }{21\,b} \left ({\frac{d}{\sin \left ( bx+a \right ) }} \right ) ^{{\frac{9}{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{9}{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 = 2.32804, size = 185, normalized size = 2.68 \begin{align*} -\frac{2 \,{\left (4 \, d^{4} \cos \left (b x + a\right )^{4} - 7 \, d^{4} \cos \left (b x + a\right )^{2}\right )} \sqrt{\frac{c}{\cos \left (b x + a\right )}} \sqrt{\frac{d}{\sin \left (b x + a\right )}}}{21 \,{\left (b c \cos \left (b x + a\right )^{2} - b c\right )} \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{9}{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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