Optimal. Leaf size=18 \[ -\frac{2 d}{b \sqrt{d \tan (a+b x)}} \]
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Rubi [A] time = 0.0414481, antiderivative size = 18, 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} \[ -\frac{2 d}{b \sqrt{d \tan (a+b x)}} \]
Antiderivative was successfully verified.
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Rule 2591
Rule 30
Rubi steps
\begin{align*} \int \csc ^2(a+b x) \sqrt{d \tan (a+b x)} \, dx &=\frac{d \operatorname{Subst}\left (\int \frac{1}{x^{3/2}} \, dx,x,d \tan (a+b x)\right )}{b}\\ &=-\frac{2 d}{b \sqrt{d \tan (a+b x)}}\\ \end{align*}
Mathematica [A] time = 0.0727914, size = 18, normalized size = 1. \[ -\frac{2 d}{b \sqrt{d \tan (a+b x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.136, size = 38, normalized size = 2.1 \begin{align*} -2\,{\frac{\cos \left ( bx+a \right ) }{b\sin \left ( bx+a \right ) }\sqrt{{\frac{d\sin \left ( bx+a \right ) }{\cos \left ( bx+a \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.26284, size = 31, normalized size = 1.72 \begin{align*} -\frac{2 \, \sqrt{d \tan \left (b x + a\right )}}{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 = 1.63919, size = 92, normalized size = 5.11 \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 )}{b \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 \sqrt{d \tan{\left (a + b x \right )}} \csc ^{2}{\left (a + b x \right )}\, dx \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{d \tan \left (b x + a\right )} \csc \left (b x + a\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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