Optimal. Leaf size=103 \[ \frac{c^4 \sqrt{1-\frac{1}{a^2 x^2}} \left (2 a+\frac{3}{x}\right )}{2 a^2}+\frac{c^4 x \left (1-\frac{1}{a^2 x^2}\right )^{3/2} \left (3 a+\frac{1}{x}\right )}{3 a}-\frac{c^4 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a}+\frac{3 c^4 \csc ^{-1}(a x)}{2 a} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.13326, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {6177, 813, 815, 844, 216, 266, 63, 208} \[ \frac{c^4 \sqrt{1-\frac{1}{a^2 x^2}} \left (2 a+\frac{3}{x}\right )}{2 a^2}+\frac{c^4 x \left (1-\frac{1}{a^2 x^2}\right )^{3/2} \left (3 a+\frac{1}{x}\right )}{3 a}-\frac{c^4 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a}+\frac{3 c^4 \csc ^{-1}(a x)}{2 a} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6177
Rule 813
Rule 815
Rule 844
Rule 216
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int e^{3 \coth ^{-1}(a x)} \left (c-\frac{c}{a x}\right )^4 \, dx &=-\left (c^3 \operatorname{Subst}\left (\int \frac{\left (c-\frac{c x}{a}\right ) \left (1-\frac{x^2}{a^2}\right )^{3/2}}{x^2} \, dx,x,\frac{1}{x}\right )\right )\\ &=\frac{c^4 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} \left (3 a+\frac{1}{x}\right ) x}{3 a}+\frac{1}{2} c^3 \operatorname{Subst}\left (\int \frac{\left (\frac{2 c}{a}+\frac{6 c x}{a^2}\right ) \sqrt{1-\frac{x^2}{a^2}}}{x} \, dx,x,\frac{1}{x}\right )\\ &=\frac{c^4 \sqrt{1-\frac{1}{a^2 x^2}} \left (2 a+\frac{3}{x}\right )}{2 a^2}+\frac{c^4 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} \left (3 a+\frac{1}{x}\right ) x}{3 a}-\frac{1}{4} \left (a^2 c^3\right ) \operatorname{Subst}\left (\int \frac{-\frac{4 c}{a^3}-\frac{6 c x}{a^4}}{x \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{c^4 \sqrt{1-\frac{1}{a^2 x^2}} \left (2 a+\frac{3}{x}\right )}{2 a^2}+\frac{c^4 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} \left (3 a+\frac{1}{x}\right ) x}{3 a}+\frac{\left (3 c^4\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{2 a^2}+\frac{c^4 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{a}\\ &=\frac{c^4 \sqrt{1-\frac{1}{a^2 x^2}} \left (2 a+\frac{3}{x}\right )}{2 a^2}+\frac{c^4 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} \left (3 a+\frac{1}{x}\right ) x}{3 a}+\frac{3 c^4 \csc ^{-1}(a x)}{2 a}+\frac{c^4 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x}{a^2}}} \, dx,x,\frac{1}{x^2}\right )}{2 a}\\ &=\frac{c^4 \sqrt{1-\frac{1}{a^2 x^2}} \left (2 a+\frac{3}{x}\right )}{2 a^2}+\frac{c^4 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} \left (3 a+\frac{1}{x}\right ) x}{3 a}+\frac{3 c^4 \csc ^{-1}(a x)}{2 a}-\left (a c^4\right ) \operatorname{Subst}\left (\int \frac{1}{a^2-a^2 x^2} \, dx,x,\sqrt{1-\frac{1}{a^2 x^2}}\right )\\ &=\frac{c^4 \sqrt{1-\frac{1}{a^2 x^2}} \left (2 a+\frac{3}{x}\right )}{2 a^2}+\frac{c^4 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} \left (3 a+\frac{1}{x}\right ) x}{3 a}+\frac{3 c^4 \csc ^{-1}(a x)}{2 a}-\frac{c^4 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a}\\ \end{align*}
Mathematica [A] time = 0.21715, size = 175, normalized size = 1.7 \[ -\frac{c^4 \left (-24 a^5 x^5-32 a^4 x^4+12 a^3 x^3+40 a^2 x^2+42 a^4 x^4 \sqrt{1-\frac{1}{a^2 x^2}} \sin ^{-1}\left (\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{2}}\right )-15 a^4 x^4 \sqrt{1-\frac{1}{a^2 x^2}} \sin ^{-1}\left (\frac{1}{a x}\right )+24 a^4 x^4 \sqrt{1-\frac{1}{a^2 x^2}} \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )+12 a x-8\right )}{24 a^5 x^4 \sqrt{1-\frac{1}{a^2 x^2}}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.168, size = 233, normalized size = 2.3 \begin{align*} -{\frac{ \left ( ax-1 \right ) ^{2}{c}^{4}}{ \left ( 6\,ax+6 \right ){a}^{4}{x}^{3}} \left ( -6\,\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}{x}^{4}{a}^{4}+6\, \left ({a}^{2}{x}^{2}-1 \right ) ^{3/2}\sqrt{{a}^{2}}{x}^{2}{a}^{2}-9\,\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}{x}^{3}{a}^{3}+6\,\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}}{\sqrt{{a}^{2}}}} \right ){x}^{3}{a}^{4}-9\,{a}^{3}{x}^{3}\sqrt{{a}^{2}}\arctan \left ({\frac{1}{\sqrt{{a}^{2}{x}^{2}-1}}} \right ) +3\,\sqrt{{a}^{2}} \left ({a}^{2}{x}^{2}-1 \right ) ^{3/2}xa-2\, \left ({a}^{2}{x}^{2}-1 \right ) ^{3/2}\sqrt{{a}^{2}} \right ) \left ({\frac{ax-1}{ax+1}} \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}}{\frac{1}{\sqrt{{a}^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.59112, size = 301, normalized size = 2.92 \begin{align*} -\frac{1}{3} \,{\left (\frac{9 \, c^{4} \arctan \left (\sqrt{\frac{a x - 1}{a x + 1}}\right )}{a^{2}} + \frac{3 \, c^{4} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac{3 \, c^{4} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right )}{a^{2}} - \frac{3 \, c^{4} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{7}{2}} + c^{4} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{5}{2}} + 29 \, c^{4} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}} + 15 \, c^{4} \sqrt{\frac{a x - 1}{a x + 1}}}{\frac{2 \,{\left (a x - 1\right )} a^{2}}{a x + 1} - \frac{2 \,{\left (a x - 1\right )}^{3} a^{2}}{{\left (a x + 1\right )}^{3}} - \frac{{\left (a x - 1\right )}^{4} a^{2}}{{\left (a x + 1\right )}^{4}} + a^{2}}\right )} a \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.72577, size = 358, normalized size = 3.48 \begin{align*} -\frac{18 \, a^{3} c^{4} x^{3} \arctan \left (\sqrt{\frac{a x - 1}{a x + 1}}\right ) + 6 \, a^{3} c^{4} x^{3} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right ) - 6 \, a^{3} c^{4} x^{3} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right ) -{\left (6 \, a^{4} c^{4} x^{4} + 14 \, a^{3} c^{4} x^{3} + 11 \, a^{2} c^{4} x^{2} + a c^{4} x - 2 \, c^{4}\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{6 \, a^{4} x^{3}} \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{c^{4} \left (\int - \frac{4 a}{\frac{a x^{4} \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}{a x + 1} - \frac{x^{3} \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}{a x + 1}}\, dx + \int \frac{6 a^{2}}{\frac{a x^{3} \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}{a x + 1} - \frac{x^{2} \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}{a x + 1}}\, dx + \int - \frac{4 a^{3}}{\frac{a x^{2} \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}{a x + 1} - \frac{x \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}{a x + 1}}\, dx + \int \frac{a^{4}}{\frac{a x \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}{a x + 1} - \frac{\sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}{a x + 1}}\, dx + \int \frac{1}{\frac{a x^{5} \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}{a x + 1} - \frac{x^{4} \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}{a x + 1}}\, dx\right )}{a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.20557, size = 311, normalized size = 3.02 \begin{align*} -\frac{1}{3} \,{\left (\frac{9 \, c^{4} \arctan \left (\sqrt{\frac{a x - 1}{a x + 1}}\right )}{a^{2}} + \frac{3 \, c^{4} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac{3 \, c^{4} \log \left ({\left | \sqrt{\frac{a x - 1}{a x + 1}} - 1 \right |}\right )}{a^{2}} + \frac{6 \, c^{4} \sqrt{\frac{a x - 1}{a x + 1}}}{a^{2}{\left (\frac{a x - 1}{a x + 1} - 1\right )}} - \frac{\frac{20 \,{\left (a x - 1\right )} c^{4} \sqrt{\frac{a x - 1}{a x + 1}}}{a x + 1} + \frac{3 \,{\left (a x - 1\right )}^{2} c^{4} \sqrt{\frac{a x - 1}{a x + 1}}}{{\left (a x + 1\right )}^{2}} + 9 \, c^{4} \sqrt{\frac{a x - 1}{a x + 1}}}{a^{2}{\left (\frac{a x - 1}{a x + 1} + 1\right )}^{3}}\right )} a \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]