Optimal. Leaf size=20 \[ -\frac{2 d}{7 b (d \tan (a+b x))^{7/2}} \]
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Rubi [A] time = 0.0428568, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2591, 30} \[ -\frac{2 d}{7 b (d \tan (a+b x))^{7/2}} \]
Antiderivative was successfully verified.
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Rule 2591
Rule 30
Rubi steps
\begin{align*} \int \frac{\csc ^2(a+b x)}{(d \tan (a+b x))^{5/2}} \, dx &=\frac{d \operatorname{Subst}\left (\int \frac{1}{x^{9/2}} \, dx,x,d \tan (a+b x)\right )}{b}\\ &=-\frac{2 d}{7 b (d \tan (a+b x))^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.159704, size = 20, normalized size = 1. \[ -\frac{2 d}{7 b (d \tan (a+b x))^{7/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.13, size = 38, normalized size = 1.9 \begin{align*} -{\frac{2\,\cos \left ( bx+a \right ) }{7\,b\sin \left ( bx+a \right ) } \left ({\frac{d\sin \left ( bx+a \right ) }{\cos \left ( bx+a \right ) }} \right ) ^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.08351, size = 31, normalized size = 1.55 \begin{align*} -\frac{2}{7 \, \left (d \tan \left (b x + a\right )\right )^{\frac{5}{2}} b \tan \left (b x + a\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.37278, size = 150, normalized size = 7.5 \begin{align*} -\frac{2 \, \sqrt{\frac{d \sin \left (b x + a\right )}{\cos \left (b x + a\right )}} \cos \left (b x + a\right )^{4}}{7 \,{\left (b d^{3} \cos \left (b x + a\right )^{4} - 2 \, b d^{3} \cos \left (b x + a\right )^{2} + b d^{3}\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 [A] time = 1.14885, size = 35, normalized size = 1.75 \begin{align*} -\frac{2}{7 \, \sqrt{d \tan \left (b x + a\right )} b d^{2} \tan \left (b x + a\right )^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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