Optimal. Leaf size=20 \[ -\frac{2 d}{5 b (d \tan (a+b x))^{5/2}} \]
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Rubi [A] time = 0.0429692, 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} \[ -\frac{2 d}{5 b (d \tan (a+b x))^{5/2}} \]
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))^{3/2}} \, dx &=\frac{d \operatorname{Subst}\left (\int \frac{1}{x^{7/2}} \, dx,x,d \tan (a+b x)\right )}{b}\\ &=-\frac{2 d}{5 b (d \tan (a+b x))^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.120176, size = 20, normalized size = 1. \[ -\frac{2 d}{5 b (d \tan (a+b x))^{5/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.129, size = 38, normalized size = 1.9 \begin{align*} -{\frac{2\,\cos \left ( bx+a \right ) }{5\,b\sin \left ( bx+a \right ) } \left ({\frac{d\sin \left ( bx+a \right ) }{\cos \left ( bx+a \right ) }} \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.10128, size = 31, normalized size = 1.55 \begin{align*} -\frac{2}{5 \, \left (d \tan \left (b x + a\right )\right )^{\frac{3}{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 = 2.1955, size = 135, normalized size = 6.75 \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 )^{3}}{5 \,{\left (b d^{2} \cos \left (b x + a\right )^{2} - b d^{2}\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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\csc ^{2}{\left (a + b x \right )}}{\left (d \tan{\left (a + b x \right )}\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15788, size = 35, normalized size = 1.75 \begin{align*} -\frac{2}{5 \, \sqrt{d \tan \left (b x + a\right )} b d \tan \left (b x + a\right )^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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