Optimal. Leaf size=105 \[ -\frac{2 \left (a+\frac{1}{x}\right )}{3 a^2 c^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}-\frac{6 a+\frac{7}{x}}{3 a^2 c^3 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{x \sqrt{1-\frac{1}{a^2 x^2}}}{c^3}+\frac{2 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a c^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.29333, antiderivative size = 105, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318, Rules used = {6177, 852, 1805, 807, 266, 63, 208} \[ -\frac{2 \left (a+\frac{1}{x}\right )}{3 a^2 c^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}-\frac{6 a+\frac{7}{x}}{3 a^2 c^3 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{x \sqrt{1-\frac{1}{a^2 x^2}}}{c^3}+\frac{2 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a c^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6177
Rule 852
Rule 1805
Rule 807
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{e^{-\coth ^{-1}(a x)}}{\left (c-\frac{c}{a x}\right )^3} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{1}{x^2 \left (c-\frac{c x}{a}\right )^2 \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{c}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{\left (c+\frac{c x}{a}\right )^2}{x^2 \left (1-\frac{x^2}{a^2}\right )^{5/2}} \, dx,x,\frac{1}{x}\right )}{c^5}\\ &=-\frac{2 \left (a+\frac{1}{x}\right )}{3 a^2 c^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}+\frac{\operatorname{Subst}\left (\int \frac{-3 c^2-\frac{6 c^2 x}{a}-\frac{4 c^2 x^2}{a^2}}{x^2 \left (1-\frac{x^2}{a^2}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{3 c^5}\\ &=-\frac{2 \left (a+\frac{1}{x}\right )}{3 a^2 c^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}-\frac{6 a+\frac{7}{x}}{3 a^2 c^3 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{\operatorname{Subst}\left (\int \frac{3 c^2+\frac{6 c^2 x}{a}}{x^2 \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{3 c^5}\\ &=-\frac{2 \left (a+\frac{1}{x}\right )}{3 a^2 c^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}-\frac{6 a+\frac{7}{x}}{3 a^2 c^3 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{\sqrt{1-\frac{1}{a^2 x^2}} x}{c^3}-\frac{2 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{a c^3}\\ &=-\frac{2 \left (a+\frac{1}{x}\right )}{3 a^2 c^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}-\frac{6 a+\frac{7}{x}}{3 a^2 c^3 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{\sqrt{1-\frac{1}{a^2 x^2}} x}{c^3}-\frac{\operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x}{a^2}}} \, dx,x,\frac{1}{x^2}\right )}{a c^3}\\ &=-\frac{2 \left (a+\frac{1}{x}\right )}{3 a^2 c^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}-\frac{6 a+\frac{7}{x}}{3 a^2 c^3 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{\sqrt{1-\frac{1}{a^2 x^2}} x}{c^3}+\frac{(2 a) \operatorname{Subst}\left (\int \frac{1}{a^2-a^2 x^2} \, dx,x,\sqrt{1-\frac{1}{a^2 x^2}}\right )}{c^3}\\ &=-\frac{2 \left (a+\frac{1}{x}\right )}{3 a^2 c^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}-\frac{6 a+\frac{7}{x}}{3 a^2 c^3 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{\sqrt{1-\frac{1}{a^2 x^2}} x}{c^3}+\frac{2 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a c^3}\\ \end{align*}
Mathematica [A] time = 0.0636371, size = 94, normalized size = 0.9 \[ \frac{3 a^3 x^3-11 a^2 x^2+6 a x \sqrt{1-\frac{1}{a^2 x^2}} (a x-1) \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )-4 a x+10}{3 a^2 c^3 x \sqrt{1-\frac{1}{a^2 x^2}} (a x-1)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.138, size = 344, normalized size = 3.3 \begin{align*}{\frac{ax+1}{12\,a{c}^{3} \left ( ax-1 \right ) ^{3}}\sqrt{{\frac{ax-1}{ax+1}}} \left ( 24\,\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}{\sqrt{{a}^{2}}}} \right ){x}^{3}{a}^{4}+27\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }{x}^{3}{a}^{3}-72\,\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}-15\,\sqrt{{a}^{2}} \left ( \left ( ax-1 \right ) \left ( ax+1 \right ) \right ) ^{3/2}xa-81\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }{x}^{2}{a}^{2}+72\,\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}+13\, \left ( \left ( ax-1 \right ) \left ( ax+1 \right ) \right ) ^{3/2}\sqrt{{a}^{2}}+81\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }xa-24\,a\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}{\sqrt{{a}^{2}}}} \right ) -27\,\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 ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.993697, size = 185, normalized size = 1.76 \begin{align*} \frac{1}{6} \, a{\left (\frac{\frac{14 \,{\left (a x - 1\right )}}{a x + 1} - \frac{27 \,{\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + 1}{a^{2} c^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{5}{2}} - a^{2} c^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}} + \frac{12 \, \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2} c^{3}} - \frac{12 \, \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right )}{a^{2} c^{3}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.89322, size = 309, normalized size = 2.94 \begin{align*} \frac{6 \,{\left (a^{2} x^{2} - 2 \, a x + 1\right )} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right ) - 6 \,{\left (a^{2} x^{2} - 2 \, a x + 1\right )} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right ) +{\left (3 \, a^{3} x^{3} - 11 \, a^{2} x^{2} - 4 \, a x + 10\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{3 \,{\left (a^{3} c^{3} x^{2} - 2 \, a^{2} c^{3} x + a c^{3}\right )}} \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*} \frac{a^{3} \int \frac{x^{3} \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}{a^{3} x^{3} - 3 a^{2} x^{2} + 3 a x - 1}\, dx}{c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.19961, size = 80, normalized size = 0.76 \begin{align*} -\frac{2 \, \log \left ({\left | -x{\left | a \right |} + \sqrt{a^{2} x^{2} - 1} \right |}\right ) \mathrm{sgn}\left (a x + 1\right )}{c^{3}{\left | a \right |}} + \frac{\sqrt{a^{2} x^{2} - 1} \mathrm{sgn}\left (a x + 1\right )}{a c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]