Optimal. Leaf size=62 \[ x \sqrt{1-\frac{1}{a^2 x^2}}-\frac{4 \sqrt{1-\frac{1}{a^2 x^2}}}{a-\frac{1}{x}}+\frac{3 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.810646, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.875, Rules used = {6168, 6742, 651, 264, 266, 63, 208} \[ x \sqrt{1-\frac{1}{a^2 x^2}}-\frac{4 \sqrt{1-\frac{1}{a^2 x^2}}}{a-\frac{1}{x}}+\frac{3 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6168
Rule 6742
Rule 651
Rule 264
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int e^{3 \coth ^{-1}(a x)} \, dx &=-\operatorname{Subst}\left (\int \frac{\left (1+\frac{x}{a}\right )^2}{x^2 \left (1-\frac{x}{a}\right ) \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )\\ &=-\operatorname{Subst}\left (\int \left (\frac{4}{a (a-x) \sqrt{1-\frac{x^2}{a^2}}}+\frac{1}{x^2 \sqrt{1-\frac{x^2}{a^2}}}+\frac{3}{a x \sqrt{1-\frac{x^2}{a^2}}}\right ) \, dx,x,\frac{1}{x}\right )\\ &=-\frac{3 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{a}-\frac{4 \operatorname{Subst}\left (\int \frac{1}{(a-x) \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{a}-\operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{4 \sqrt{1-\frac{1}{a^2 x^2}}}{a-\frac{1}{x}}+\sqrt{1-\frac{1}{a^2 x^2}} x-\frac{3 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x}{a^2}}} \, dx,x,\frac{1}{x^2}\right )}{2 a}\\ &=-\frac{4 \sqrt{1-\frac{1}{a^2 x^2}}}{a-\frac{1}{x}}+\sqrt{1-\frac{1}{a^2 x^2}} x+(3 a) \operatorname{Subst}\left (\int \frac{1}{a^2-a^2 x^2} \, dx,x,\sqrt{1-\frac{1}{a^2 x^2}}\right )\\ &=-\frac{4 \sqrt{1-\frac{1}{a^2 x^2}}}{a-\frac{1}{x}}+\sqrt{1-\frac{1}{a^2 x^2}} x+\frac{3 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a}\\ \end{align*}
Mathematica [A] time = 0.0471783, size = 54, normalized size = 0.87 \[ \frac{x \sqrt{1-\frac{1}{a^2 x^2}} (a x-5)}{a x-1}+\frac{3 \log \left (a x \left (\sqrt{1-\frac{1}{a^2 x^2}}+1\right )\right )}{a} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.136, size = 247, normalized size = 4. \begin{align*}{\frac{1}{a \left ( ax+1 \right ) } \left ( 3\,\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}{\sqrt{{a}^{2}}}} \right ){x}^{2}{a}^{3}+3\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }{x}^{2}{a}^{2}-6\,\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}{\sqrt{{a}^{2}}}} \right ) x{a}^{2}-2\, \left ( \left ( ax-1 \right ) \left ( ax+1 \right ) \right ) ^{3/2}\sqrt{{a}^{2}}-6\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }xa+3\,a\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}{\sqrt{{a}^{2}}}} \right ) +3\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) } \right ){\frac{1}{\sqrt{{a}^{2}}}}{\frac{1}{\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}} \left ({\frac{ax-1}{ax+1}} \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.00855, size = 149, normalized size = 2.4 \begin{align*} -a{\left (\frac{2 \,{\left (\frac{3 \,{\left (a x - 1\right )}}{a x + 1} - 2\right )}}{a^{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}} - a^{2} \sqrt{\frac{a x - 1}{a x + 1}}} - \frac{3 \, \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2}} + \frac{3 \, \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right )}{a^{2}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.84083, size = 217, normalized size = 3.5 \begin{align*} \frac{3 \,{\left (a x - 1\right )} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right ) - 3 \,{\left (a x - 1\right )} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right ) +{\left (a^{2} x^{2} - 4 \, a x - 5\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{a^{2} x - a} \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}{\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 [B] time = 1.16671, size = 161, normalized size = 2.6 \begin{align*} a{\left (\frac{3 \, \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac{3 \, \log \left ({\left | \sqrt{\frac{a x - 1}{a x + 1}} - 1 \right |}\right )}{a^{2}} - \frac{2 \,{\left (\frac{3 \,{\left (a x - 1\right )}}{a x + 1} - 2\right )}}{a^{2}{\left (\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}}\right )}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]