Optimal. Leaf size=138 \[ -\frac{16 \left (a+\frac{1}{x}\right )}{5 a^2 c^2 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}-\frac{4 \left (5 a+\frac{11}{x}\right )}{15 a^2 c^2 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}-\frac{75 a+\frac{103}{x}}{15 a^2 c^2 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{x \sqrt{1-\frac{1}{a^2 x^2}}}{c^2}+\frac{5 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a c^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.401889, antiderivative size = 138, normalized size of antiderivative = 1., number of steps used = 9, 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} \[ -\frac{16 \left (a+\frac{1}{x}\right )}{5 a^2 c^2 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}-\frac{4 \left (5 a+\frac{11}{x}\right )}{15 a^2 c^2 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}-\frac{75 a+\frac{103}{x}}{15 a^2 c^2 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{x \sqrt{1-\frac{1}{a^2 x^2}}}{c^2}+\frac{5 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a c^2} \]
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^{3 \coth ^{-1}(a x)}}{\left (c-\frac{c}{a x}\right )^2} \, dx &=-\left (c^3 \operatorname{Subst}\left (\int \frac{\left (1-\frac{x^2}{a^2}\right )^{3/2}}{x^2 \left (c-\frac{c x}{a}\right )^5} \, dx,x,\frac{1}{x}\right )\right )\\ &=-\frac{\operatorname{Subst}\left (\int \frac{\left (c+\frac{c x}{a}\right )^5}{x^2 \left (1-\frac{x^2}{a^2}\right )^{7/2}} \, dx,x,\frac{1}{x}\right )}{c^7}\\ &=-\frac{16 \left (a+\frac{1}{x}\right )}{5 a^2 c^2 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}+\frac{\operatorname{Subst}\left (\int \frac{-5 c^5-\frac{25 c^5 x}{a}-\frac{39 c^5 x^2}{a^2}+\frac{5 c^5 x^3}{a^3}}{x^2 \left (1-\frac{x^2}{a^2}\right )^{5/2}} \, dx,x,\frac{1}{x}\right )}{5 c^7}\\ &=-\frac{16 \left (a+\frac{1}{x}\right )}{5 a^2 c^2 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}-\frac{4 \left (5 a+\frac{11}{x}\right )}{15 a^2 c^2 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}-\frac{\operatorname{Subst}\left (\int \frac{15 c^5+\frac{75 c^5 x}{a}+\frac{88 c^5 x^2}{a^2}}{x^2 \left (1-\frac{x^2}{a^2}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{15 c^7}\\ &=-\frac{16 \left (a+\frac{1}{x}\right )}{5 a^2 c^2 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}-\frac{4 \left (5 a+\frac{11}{x}\right )}{15 a^2 c^2 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}-\frac{75 a+\frac{103}{x}}{15 a^2 c^2 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{\operatorname{Subst}\left (\int \frac{-15 c^5-\frac{75 c^5 x}{a}}{x^2 \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{15 c^7}\\ &=-\frac{16 \left (a+\frac{1}{x}\right )}{5 a^2 c^2 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}-\frac{4 \left (5 a+\frac{11}{x}\right )}{15 a^2 c^2 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}-\frac{75 a+\frac{103}{x}}{15 a^2 c^2 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{\sqrt{1-\frac{1}{a^2 x^2}} x}{c^2}-\frac{5 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{a c^2}\\ &=-\frac{16 \left (a+\frac{1}{x}\right )}{5 a^2 c^2 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}-\frac{4 \left (5 a+\frac{11}{x}\right )}{15 a^2 c^2 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}-\frac{75 a+\frac{103}{x}}{15 a^2 c^2 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{\sqrt{1-\frac{1}{a^2 x^2}} x}{c^2}-\frac{5 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x}{a^2}}} \, dx,x,\frac{1}{x^2}\right )}{2 a c^2}\\ &=-\frac{16 \left (a+\frac{1}{x}\right )}{5 a^2 c^2 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}-\frac{4 \left (5 a+\frac{11}{x}\right )}{15 a^2 c^2 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}-\frac{75 a+\frac{103}{x}}{15 a^2 c^2 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{\sqrt{1-\frac{1}{a^2 x^2}} x}{c^2}+\frac{(5 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^2}\\ &=-\frac{16 \left (a+\frac{1}{x}\right )}{5 a^2 c^2 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}-\frac{4 \left (5 a+\frac{11}{x}\right )}{15 a^2 c^2 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}-\frac{75 a+\frac{103}{x}}{15 a^2 c^2 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{\sqrt{1-\frac{1}{a^2 x^2}} x}{c^2}+\frac{5 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a c^2}\\ \end{align*}
Mathematica [A] time = 0.075933, size = 104, normalized size = 0.75 \[ \frac{15 a^4 x^4-173 a^3 x^3+91 a^2 x^2+75 a x \sqrt{1-\frac{1}{a^2 x^2}} (a x-1)^2 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )+161 a x-118}{15 a^2 c^2 x \sqrt{1-\frac{1}{a^2 x^2}} (a x-1)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.172, size = 438, normalized size = 3.2 \begin{align*}{\frac{1}{15\,a \left ( ax-1 \right ) ^{2}{c}^{2} \left ( ax+1 \right ) } \left ( 75\,\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}{\sqrt{{a}^{2}}}} \right ){x}^{4}{a}^{5}+75\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }{x}^{4}{a}^{4}-300\,\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}-60\,\sqrt{{a}^{2}} \left ( \left ( ax-1 \right ) \left ( ax+1 \right ) \right ) ^{3/2}{x}^{2}{a}^{2}-300\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }{x}^{3}{a}^{3}+450\,\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}+97\,\sqrt{{a}^{2}} \left ( \left ( ax-1 \right ) \left ( ax+1 \right ) \right ) ^{3/2}xa+450\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }{x}^{2}{a}^{2}-300\,\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}-43\, \left ( \left ( ax-1 \right ) \left ( ax+1 \right ) \right ) ^{3/2}\sqrt{{a}^{2}}-300\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }xa+75\,a\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}{\sqrt{{a}^{2}}}} \right ) +75\,\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 = 1.08625, size = 207, normalized size = 1.5 \begin{align*} \frac{1}{15} \, a{\left (\frac{\frac{17 \,{\left (a x - 1\right )}}{a x + 1} + \frac{100 \,{\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} - \frac{150 \,{\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} + 3}{a^{2} c^{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{7}{2}} - a^{2} c^{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{5}{2}}} + \frac{75 \, \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2} c^{2}} - \frac{75 \, \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right )}{a^{2} c^{2}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.54351, size = 392, normalized size = 2.84 \begin{align*} \frac{75 \,{\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right ) - 75 \,{\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right ) +{\left (15 \, a^{4} x^{4} - 173 \, a^{3} x^{3} + 91 \, a^{2} x^{2} + 161 \, a x - 118\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{15 \,{\left (a^{4} c^{2} x^{3} - 3 \, a^{3} c^{2} x^{2} + 3 \, a^{2} c^{2} x - a c^{2}\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^{2} \int \frac{x^{2}}{\frac{a^{3} x^{3} \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}{a x + 1} - \frac{3 a^{2} x^{2} \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}{a x + 1} + \frac{3 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}{c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.23729, size = 224, normalized size = 1.62 \begin{align*} \frac{1}{15} \, a{\left (\frac{75 \, \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2} c^{2}} - \frac{75 \, \log \left ({\left | \sqrt{\frac{a x - 1}{a x + 1}} - 1 \right |}\right )}{a^{2} c^{2}} - \frac{{\left (a x + 1\right )}^{2}{\left (\frac{20 \,{\left (a x - 1\right )}}{a x + 1} + \frac{120 \,{\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + 3\right )}}{{\left (a x - 1\right )}^{2} a^{2} c^{2} \sqrt{\frac{a x - 1}{a x + 1}}} - \frac{30 \, \sqrt{\frac{a x - 1}{a x + 1}}}{a^{2} c^{2}{\left (\frac{a x - 1}{a x + 1} - 1\right )}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]