Optimal. Leaf size=79 \[ \frac{b c x \tan ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{c^2 x^2-1}}\right )}{\sqrt{d} e \sqrt{c^2 x^2}}-\frac{a+b \csc ^{-1}(c x)}{e \sqrt{d+e x^2}} \]
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Rubi [A] time = 0.117238, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {5237, 446, 93, 204} \[ \frac{b c x \tan ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{c^2 x^2-1}}\right )}{\sqrt{d} e \sqrt{c^2 x^2}}-\frac{a+b \csc ^{-1}(c x)}{e \sqrt{d+e x^2}} \]
Antiderivative was successfully verified.
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Rule 5237
Rule 446
Rule 93
Rule 204
Rubi steps
\begin{align*} \int \frac{x \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}} \, dx &=-\frac{a+b \csc ^{-1}(c x)}{e \sqrt{d+e x^2}}-\frac{(b c x) \int \frac{1}{x \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}} \, dx}{e \sqrt{c^2 x^2}}\\ &=-\frac{a+b \csc ^{-1}(c x)}{e \sqrt{d+e x^2}}-\frac{(b c x) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{-1+c^2 x} \sqrt{d+e x}} \, dx,x,x^2\right )}{2 e \sqrt{c^2 x^2}}\\ &=-\frac{a+b \csc ^{-1}(c x)}{e \sqrt{d+e x^2}}-\frac{(b c x) \operatorname{Subst}\left (\int \frac{1}{-d-x^2} \, dx,x,\frac{\sqrt{d+e x^2}}{\sqrt{-1+c^2 x^2}}\right )}{e \sqrt{c^2 x^2}}\\ &=-\frac{a+b \csc ^{-1}(c x)}{e \sqrt{d+e x^2}}+\frac{b c x \tan ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{-1+c^2 x^2}}\right )}{\sqrt{d} e \sqrt{c^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.145208, size = 96, normalized size = 1.22 \[ -\frac{a+b \csc ^{-1}(c x)}{e \sqrt{d+e x^2}}-\frac{b c x \sqrt{1-\frac{1}{c^2 x^2}} \tan ^{-1}\left (\frac{\sqrt{d} \sqrt{c^2 x^2-1}}{\sqrt{d+e x^2}}\right )}{\sqrt{d} e \sqrt{c^2 x^2-1}} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.764, size = 0, normalized size = 0. \begin{align*} \int{x \left ( a+b{\rm arccsc} \left (cx\right ) \right ) \left ( e{x}^{2}+d \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [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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Fricas [A] time = 3.67538, size = 625, normalized size = 7.91 \begin{align*} \left [-\frac{{\left (b e x^{2} + b d\right )} \sqrt{-d} \log \left (\frac{{\left (c^{4} d^{2} - 6 \, c^{2} d e + e^{2}\right )} x^{4} - 8 \,{\left (c^{2} d^{2} - d e\right )} x^{2} + 4 \, \sqrt{c^{2} x^{2} - 1}{\left ({\left (c^{2} d - e\right )} x^{2} - 2 \, d\right )} \sqrt{e x^{2} + d} \sqrt{-d} + 8 \, d^{2}}{x^{4}}\right ) + 4 \, \sqrt{e x^{2} + d}{\left (b d \operatorname{arccsc}\left (c x\right ) + a d\right )}}{4 \,{\left (d e^{2} x^{2} + d^{2} e\right )}}, \frac{{\left (b e x^{2} + b d\right )} \sqrt{d} \arctan \left (-\frac{\sqrt{c^{2} x^{2} - 1}{\left ({\left (c^{2} d - e\right )} x^{2} - 2 \, d\right )} \sqrt{e x^{2} + d} \sqrt{d}}{2 \,{\left (c^{2} d e x^{4} +{\left (c^{2} d^{2} - d e\right )} x^{2} - d^{2}\right )}}\right ) - 2 \, \sqrt{e x^{2} + d}{\left (b d \operatorname{arccsc}\left (c x\right ) + a d\right )}}{2 \,{\left (d e^{2} x^{2} + d^{2} e\right )}}\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 (b \operatorname{arccsc}\left (c x\right ) + a\right )} x}{{\left (e x^{2} + d\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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