Optimal. Leaf size=49 \[ c x \sqrt{1-\frac{1}{a^2 x^2}}-\frac{2 c \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a}-\frac{c \csc ^{-1}(a x)}{a} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.145407, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35, Rules used = {6177, 1807, 844, 216, 266, 63, 208} \[ c x \sqrt{1-\frac{1}{a^2 x^2}}-\frac{2 c \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a}-\frac{c \csc ^{-1}(a x)}{a} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6177
Rule 1807
Rule 844
Rule 216
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int e^{-\coth ^{-1}(a x)} \left (c-\frac{c}{a x}\right ) \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{\left (c-\frac{c x}{a}\right )^2}{x^2 \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{c}\\ &=c \sqrt{1-\frac{1}{a^2 x^2}} x+\frac{\operatorname{Subst}\left (\int \frac{\frac{2 c^2}{a}-\frac{c^2 x}{a^2}}{x \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{c}\\ &=c \sqrt{1-\frac{1}{a^2 x^2}} x-\frac{c \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{a^2}+\frac{(2 c) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{a}\\ &=c \sqrt{1-\frac{1}{a^2 x^2}} x-\frac{c \csc ^{-1}(a x)}{a}+\frac{c \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x}{a^2}}} \, dx,x,\frac{1}{x^2}\right )}{a}\\ &=c \sqrt{1-\frac{1}{a^2 x^2}} x-\frac{c \csc ^{-1}(a x)}{a}-(2 a c) \operatorname{Subst}\left (\int \frac{1}{a^2-a^2 x^2} \, dx,x,\sqrt{1-\frac{1}{a^2 x^2}}\right )\\ &=c \sqrt{1-\frac{1}{a^2 x^2}} x-\frac{c \csc ^{-1}(a x)}{a}-\frac{2 c \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a}\\ \end{align*}
Mathematica [A] time = 0.101654, size = 73, normalized size = 1.49 \[ \frac{c \left (a x \sqrt{1-\frac{1}{a^2 x^2}}-2 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )-2 \sin ^{-1}\left (\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{2}}\right )-2 \sin ^{-1}\left (\frac{1}{a x}\right )\right )}{a} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.128, size = 136, normalized size = 2.8 \begin{align*} -{\frac{c \left ( ax+1 \right ) }{a}\sqrt{{\frac{ax-1}{ax+1}}} \left ( \sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}+\arctan \left ({\frac{1}{\sqrt{{a}^{2}{x}^{2}-1}}} \right ) \sqrt{{a}^{2}}+2\,a\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}{\sqrt{{a}^{2}}}} \right ) -2\,\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 [B] time = 1.59328, size = 154, normalized size = 3.14 \begin{align*} -2 \, a{\left (\frac{c \sqrt{\frac{a x - 1}{a x + 1}}}{\frac{{\left (a x - 1\right )} a^{2}}{a x + 1} - a^{2}} - \frac{c \arctan \left (\sqrt{\frac{a x - 1}{a x + 1}}\right )}{a^{2}} + \frac{c \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac{c \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 = 2.00251, size = 223, normalized size = 4.55 \begin{align*} \frac{2 \, c \arctan \left (\sqrt{\frac{a x - 1}{a x + 1}}\right ) - 2 \, c \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right ) + 2 \, c \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right ) +{\left (a c x + c\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{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*} \frac{c \left (\int a \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}\, dx + \int - \frac{\sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}{x}\, dx\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.18115, size = 115, normalized size = 2.35 \begin{align*} \frac{2 \, c \arctan \left (-x{\left | a \right |} + \sqrt{a^{2} x^{2} - 1}\right ) \mathrm{sgn}\left (a x + 1\right )}{a} + \frac{2 \, c \log \left ({\left | -x{\left | a \right |} + \sqrt{a^{2} x^{2} - 1} \right |}\right ) \mathrm{sgn}\left (a x + 1\right )}{{\left | a \right |}} + \frac{\sqrt{a^{2} x^{2} - 1} c \mathrm{sgn}\left (a x + 1\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]