Optimal. Leaf size=51 \[ \frac{2 \left (a+\frac{1}{x}\right )}{a^2 c \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{\tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a c} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.206206, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.438, Rules used = {6175, 6178, 852, 1805, 266, 63, 208} \[ \frac{2 \left (a+\frac{1}{x}\right )}{a^2 c \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{\tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a c} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6175
Rule 6178
Rule 852
Rule 1805
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{e^{\coth ^{-1}(a x)}}{c-a c x} \, dx &=-\frac{\int \frac{e^{\coth ^{-1}(a x)}}{\left (1-\frac{1}{a x}\right ) x} \, dx}{a c}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\sqrt{1-\frac{x^2}{a^2}}}{x \left (1-\frac{x}{a}\right )^2} \, dx,x,\frac{1}{x}\right )}{a c}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\left (1+\frac{x}{a}\right )^2}{x \left (1-\frac{x^2}{a^2}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{a c}\\ &=\frac{2 \left (a+\frac{1}{x}\right )}{a^2 c \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{\operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{a c}\\ &=\frac{2 \left (a+\frac{1}{x}\right )}{a^2 c \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{\operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x}{a^2}}} \, dx,x,\frac{1}{x^2}\right )}{2 a c}\\ &=\frac{2 \left (a+\frac{1}{x}\right )}{a^2 c \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{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}\\ &=\frac{2 \left (a+\frac{1}{x}\right )}{a^2 c \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{\tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a c}\\ \end{align*}
Mathematica [A] time = 0.0521113, size = 60, normalized size = 1.18 \[ \frac{2 a x \sqrt{1-\frac{1}{a^2 x^2}}+(1-a x) \log \left (a x \left (\sqrt{1-\frac{1}{a^2 x^2}}+1\right )\right )}{a c (a x-1)} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.132, size = 247, normalized size = 4.8 \begin{align*} -{\frac{1}{ac \left ( ax-1 \right ) } \left ( \ln \left ({ \left ({a}^{2}x+\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) } \right ){\frac{1}{\sqrt{{a}^{2}}}}} \right ){x}^{2}{a}^{3}+\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }{x}^{2}{a}^{2}-2\,\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}- \left ( \left ( ax-1 \right ) \left ( ax+1 \right ) \right ) ^{{\frac{3}{2}}}\sqrt{{a}^{2}}-2\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }xa+a\ln \left ({ \left ({a}^{2}x+\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) } \right ){\frac{1}{\sqrt{{a}^{2}}}}} \right ) +\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 ) }}}{\frac{1}{\sqrt{{\frac{ax-1}{ax+1}}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.03232, size = 105, normalized size = 2.06 \begin{align*} -a{\left (\frac{\log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2} c} - \frac{\log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right )}{a^{2} c} - \frac{2}{a^{2} c \sqrt{\frac{a x - 1}{a x + 1}}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.61373, size = 205, normalized size = 4.02 \begin{align*} -\frac{{\left (a x - 1\right )} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right ) -{\left (a x - 1\right )} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right ) - 2 \,{\left (a x + 1\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{a^{2} c x - a c} \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{\int \frac{1}{a x \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}} - \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}\, dx}{c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.18387, size = 107, normalized size = 2.1 \begin{align*} -a{\left (\frac{\log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2} c} - \frac{\log \left ({\left | \sqrt{\frac{a x - 1}{a x + 1}} - 1 \right |}\right )}{a^{2} c} - \frac{2}{a^{2} c \sqrt{\frac{a x - 1}{a x + 1}}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]