Optimal. Leaf size=194 \[ c^2 x \left (1-\frac{1}{a x}\right )^{3/2} \left (\frac{1}{a x}+1\right )^{5/2}+\frac{4 c^2 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{5/2}}{3 a}-\frac{7 c^2 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{3/2}}{6 a}-\frac{5 c^2 \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}{2 a}+\frac{3 c^2 \csc ^{-1}(a x)}{2 a}+\frac{c^2 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right )}{a} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.134112, antiderivative size = 194, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {6194, 97, 154, 157, 41, 216, 92, 208} \[ c^2 x \left (1-\frac{1}{a x}\right )^{3/2} \left (\frac{1}{a x}+1\right )^{5/2}+\frac{4 c^2 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{5/2}}{3 a}-\frac{7 c^2 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{3/2}}{6 a}-\frac{5 c^2 \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}{2 a}+\frac{3 c^2 \csc ^{-1}(a x)}{2 a}+\frac{c^2 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right )}{a} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6194
Rule 97
Rule 154
Rule 157
Rule 41
Rule 216
Rule 92
Rule 208
Rubi steps
\begin{align*} \int e^{\coth ^{-1}(a x)} \left (c-\frac{c}{a^2 x^2}\right )^2 \, dx &=-\left (c^2 \operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a}\right )^{3/2} \left (1+\frac{x}{a}\right )^{5/2}}{x^2} \, dx,x,\frac{1}{x}\right )\right )\\ &=c^2 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{5/2} x-c^2 \operatorname{Subst}\left (\int \frac{\left (\frac{1}{a}-\frac{4 x}{a^2}\right ) \sqrt{1-\frac{x}{a}} \left (1+\frac{x}{a}\right )^{3/2}}{x} \, dx,x,\frac{1}{x}\right )\\ &=\frac{4 c^2 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{5/2}}{3 a}+c^2 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{5/2} x-\frac{1}{3} \left (a c^2\right ) \operatorname{Subst}\left (\int \frac{\left (\frac{3}{a^2}-\frac{7 x}{a^3}\right ) \left (1+\frac{x}{a}\right )^{3/2}}{x \sqrt{1-\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{7 c^2 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/2}}{6 a}+\frac{4 c^2 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{5/2}}{3 a}+c^2 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{5/2} x+\frac{1}{6} \left (a^2 c^2\right ) \operatorname{Subst}\left (\int \frac{\left (-\frac{6}{a^3}+\frac{15 x}{a^4}\right ) \sqrt{1+\frac{x}{a}}}{x \sqrt{1-\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{5 c^2 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}{2 a}-\frac{7 c^2 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/2}}{6 a}+\frac{4 c^2 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{5/2}}{3 a}+c^2 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{5/2} x-\frac{1}{6} \left (a^3 c^2\right ) \operatorname{Subst}\left (\int \frac{\frac{6}{a^4}-\frac{9 x}{a^5}}{x \sqrt{1-\frac{x}{a}} \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{5 c^2 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}{2 a}-\frac{7 c^2 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/2}}{6 a}+\frac{4 c^2 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{5/2}}{3 a}+c^2 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{5/2} x+\frac{\left (3 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x}{a}} \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{2 a^2}-\frac{c^2 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x}{a}} \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{a}\\ &=-\frac{5 c^2 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}{2 a}-\frac{7 c^2 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/2}}{6 a}+\frac{4 c^2 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{5/2}}{3 a}+c^2 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{5/2} x+\frac{c^2 \operatorname{Subst}\left (\int \frac{1}{\frac{1}{a}-\frac{x^2}{a}} \, dx,x,\sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}\right )}{a^2}+\frac{\left (3 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{2 a^2}\\ &=-\frac{5 c^2 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}{2 a}-\frac{7 c^2 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/2}}{6 a}+\frac{4 c^2 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{5/2}}{3 a}+c^2 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{5/2} x+\frac{3 c^2 \csc ^{-1}(a x)}{2 a}+\frac{c^2 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}\right )}{a}\\ \end{align*}
Mathematica [A] time = 0.132348, size = 94, normalized size = 0.48 \[ \frac{c^2 \left (\sqrt{1-\frac{1}{a^2 x^2}} \left (6 a^3 x^3-8 a^2 x^2+3 a x+2\right )+6 a^2 x^2 \log \left (x \left (\sqrt{1-\frac{1}{a^2 x^2}}+1\right )\right )+9 a^2 x^2 \sin ^{-1}\left (\frac{1}{a x}\right )\right )}{6 a^3 x^2} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.135, size = 224, normalized size = 1.2 \begin{align*}{\frac{ \left ( ax-1 \right ){c}^{2}}{6\,{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 ){\frac{1}{\sqrt{{\frac{ax-1}{ax+1}}}}}{\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 [A] time = 1.52984, size = 301, normalized size = 1.55 \begin{align*} -\frac{1}{3} \, a{\left (\frac{9 \, c^{2} \arctan \left (\sqrt{\frac{a x - 1}{a x + 1}}\right )}{a^{2}} - \frac{3 \, c^{2} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2}} + \frac{3 \, c^{2} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right )}{a^{2}} - \frac{15 \, c^{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{7}{2}} + 29 \, c^{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{5}{2}} + c^{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}} + 3 \, c^{2} \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 )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.33358, size = 358, normalized size = 1.85 \begin{align*} -\frac{18 \, a^{3} c^{2} x^{3} \arctan \left (\sqrt{\frac{a x - 1}{a x + 1}}\right ) - 6 \, a^{3} c^{2} x^{3} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right ) + 6 \, a^{3} c^{2} x^{3} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right ) -{\left (6 \, a^{4} c^{2} x^{4} - 2 \, a^{3} c^{2} x^{3} - 5 \, a^{2} c^{2} x^{2} + 5 \, a c^{2} x + 2 \, c^{2}\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^{2} \left (\int \frac{a^{4}}{\sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}\, dx + \int \frac{1}{x^{4} \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}\, dx + \int - \frac{2 a^{2}}{x^{2} \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}\, dx\right )}{a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.20413, size = 309, normalized size = 1.59 \begin{align*} -\frac{1}{3} \, a{\left (\frac{9 \, c^{2} \arctan \left (\sqrt{\frac{a x - 1}{a x + 1}}\right )}{a^{2}} - \frac{3 \, c^{2} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2}} + \frac{3 \, c^{2} \log \left ({\left | \sqrt{\frac{a x - 1}{a x + 1}} - 1 \right |}\right )}{a^{2}} + \frac{6 \, c^{2} \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^{2} \sqrt{\frac{a x - 1}{a x + 1}}}{a x + 1} + \frac{9 \,{\left (a x - 1\right )}^{2} c^{2} \sqrt{\frac{a x - 1}{a x + 1}}}{{\left (a x + 1\right )}^{2}} + 3 \, c^{2} \sqrt{\frac{a x - 1}{a x + 1}}}{a^{2}{\left (\frac{a x - 1}{a x + 1} + 1\right )}^{3}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]