Optimal. Leaf size=51 \[ -\frac{2 \left (a+\frac{1}{x}\right )^2}{a \sqrt{1-\frac{1}{a^2 x^2}}}-3 a \sqrt{1-\frac{1}{a^2 x^2}}+3 a \csc ^{-1}(a x) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0740333, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {6169, 853, 669, 641, 216} \[ -\frac{2 \left (a+\frac{1}{x}\right )^2}{a \sqrt{1-\frac{1}{a^2 x^2}}}-3 a \sqrt{1-\frac{1}{a^2 x^2}}+3 a \csc ^{-1}(a x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6169
Rule 853
Rule 669
Rule 641
Rule 216
Rubi steps
\begin{align*} \int \frac{e^{3 \coth ^{-1}(a x)}}{x^2} \, dx &=-\operatorname{Subst}\left (\int \frac{\left (1+\frac{x}{a}\right )^2}{\left (1-\frac{x}{a}\right ) \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )\\ &=-\operatorname{Subst}\left (\int \frac{\left (1+\frac{x}{a}\right )^3}{\left (1-\frac{x^2}{a^2}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{2 \left (a+\frac{1}{x}\right )^2}{a \sqrt{1-\frac{1}{a^2 x^2}}}+3 \operatorname{Subst}\left (\int \frac{1+\frac{x}{a}}{\sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )\\ &=-3 a \sqrt{1-\frac{1}{a^2 x^2}}-\frac{2 \left (a+\frac{1}{x}\right )^2}{a \sqrt{1-\frac{1}{a^2 x^2}}}+3 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )\\ &=-3 a \sqrt{1-\frac{1}{a^2 x^2}}-\frac{2 \left (a+\frac{1}{x}\right )^2}{a \sqrt{1-\frac{1}{a^2 x^2}}}+3 a \csc ^{-1}(a x)\\ \end{align*}
Mathematica [A] time = 0.0817021, size = 41, normalized size = 0.8 \[ \frac{a \sqrt{1-\frac{1}{a^2 x^2}} (1-5 a x)}{a x-1}+3 a \sin ^{-1}\left (\frac{1}{a x}\right ) \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.169, size = 593, normalized size = 11.6 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.47312, size = 97, normalized size = 1.9 \begin{align*} -2 \, a{\left (\frac{\frac{3 \,{\left (a x - 1\right )}}{a x + 1} + 2}{\left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}} + \sqrt{\frac{a x - 1}{a x + 1}}} + 3 \, \arctan \left (\sqrt{\frac{a x - 1}{a x + 1}}\right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.89969, size = 162, normalized size = 3.18 \begin{align*} -\frac{6 \,{\left (a^{2} x^{2} - a x\right )} \arctan \left (\sqrt{\frac{a x - 1}{a x + 1}}\right ) +{\left (5 \, a^{2} x^{2} + 4 \, a x - 1\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{a x^{2} - x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.18608, size = 115, normalized size = 2.25 \begin{align*} -2 \, a{\left (\frac{\frac{3 \,{\left (a x - 1\right )}}{a x + 1} + 2}{\frac{{\left (a x - 1\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{a x + 1} + \sqrt{\frac{a x - 1}{a x + 1}}} + 3 \, \arctan \left (\sqrt{\frac{a x - 1}{a x + 1}}\right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]