Optimal. Leaf size=195 \[ c^2 x \sqrt{\frac{1}{a x}+1} \left (1-\frac{1}{a x}\right )^{7/2}+\frac{4 c^2 \sqrt{\frac{1}{a x}+1} \left (1-\frac{1}{a x}\right )^{5/2}}{3 a}+\frac{11 c^2 \sqrt{\frac{1}{a x}+1} \left (1-\frac{1}{a x}\right )^{3/2}}{6 a}+\frac{5 c^2 \sqrt{\frac{1}{a x}+1} \sqrt{1-\frac{1}{a x}}}{2 a}-\frac{c^2 \csc ^{-1}(a x)}{2 a}-\frac{3 c^2 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right )}{a} \]
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Rubi [A] time = 0.131351, antiderivative size = 195, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {6194, 97, 154, 157, 41, 216, 92, 208} \[ c^2 x \sqrt{\frac{1}{a x}+1} \left (1-\frac{1}{a x}\right )^{7/2}+\frac{4 c^2 \sqrt{\frac{1}{a x}+1} \left (1-\frac{1}{a x}\right )^{5/2}}{3 a}+\frac{11 c^2 \sqrt{\frac{1}{a x}+1} \left (1-\frac{1}{a x}\right )^{3/2}}{6 a}+\frac{5 c^2 \sqrt{\frac{1}{a x}+1} \sqrt{1-\frac{1}{a x}}}{2 a}-\frac{c^2 \csc ^{-1}(a x)}{2 a}-\frac{3 c^2 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right )}{a} \]
Antiderivative was successfully verified.
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Rule 6194
Rule 97
Rule 154
Rule 157
Rule 41
Rule 216
Rule 92
Rule 208
Rubi steps
\begin{align*} \int e^{-3 \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 )^{7/2} \sqrt{1+\frac{x}{a}}}{x^2} \, dx,x,\frac{1}{x}\right )\right )\\ &=c^2 \left (1-\frac{1}{a x}\right )^{7/2} \sqrt{1+\frac{1}{a x}} x-c^2 \operatorname{Subst}\left (\int \frac{\left (-\frac{3}{a}-\frac{4 x}{a^2}\right ) \left (1-\frac{x}{a}\right )^{5/2}}{x \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{4 c^2 \left (1-\frac{1}{a x}\right )^{5/2} \sqrt{1+\frac{1}{a x}}}{3 a}+c^2 \left (1-\frac{1}{a x}\right )^{7/2} \sqrt{1+\frac{1}{a x}} x-\frac{1}{3} \left (a c^2\right ) \operatorname{Subst}\left (\int \frac{\left (-\frac{9}{a^2}-\frac{11 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{11 c^2 \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{1+\frac{1}{a x}}}{6 a}+\frac{4 c^2 \left (1-\frac{1}{a x}\right )^{5/2} \sqrt{1+\frac{1}{a x}}}{3 a}+c^2 \left (1-\frac{1}{a x}\right )^{7/2} \sqrt{1+\frac{1}{a x}} x-\frac{1}{6} \left (a^2 c^2\right ) \operatorname{Subst}\left (\int \frac{\left (-\frac{18}{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{11 c^2 \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{1+\frac{1}{a x}}}{6 a}+\frac{4 c^2 \left (1-\frac{1}{a x}\right )^{5/2} \sqrt{1+\frac{1}{a x}}}{3 a}+c^2 \left (1-\frac{1}{a x}\right )^{7/2} \sqrt{1+\frac{1}{a x}} x-\frac{1}{6} \left (a^3 c^2\right ) \operatorname{Subst}\left (\int \frac{-\frac{18}{a^4}+\frac{3 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{11 c^2 \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{1+\frac{1}{a x}}}{6 a}+\frac{4 c^2 \left (1-\frac{1}{a x}\right )^{5/2} \sqrt{1+\frac{1}{a x}}}{3 a}+c^2 \left (1-\frac{1}{a x}\right )^{7/2} \sqrt{1+\frac{1}{a x}} x-\frac{c^2 \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{\left (3 c^2\right ) \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{11 c^2 \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{1+\frac{1}{a x}}}{6 a}+\frac{4 c^2 \left (1-\frac{1}{a x}\right )^{5/2} \sqrt{1+\frac{1}{a x}}}{3 a}+c^2 \left (1-\frac{1}{a x}\right )^{7/2} \sqrt{1+\frac{1}{a x}} x-\frac{c^2 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{2 a^2}-\frac{\left (3 c^2\right ) \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{5 c^2 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}{2 a}+\frac{11 c^2 \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{1+\frac{1}{a x}}}{6 a}+\frac{4 c^2 \left (1-\frac{1}{a x}\right )^{5/2} \sqrt{1+\frac{1}{a x}}}{3 a}+c^2 \left (1-\frac{1}{a x}\right )^{7/2} \sqrt{1+\frac{1}{a x}} x-\frac{c^2 \csc ^{-1}(a x)}{2 a}-\frac{3 c^2 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}\right )}{a}\\ \end{align*}
Mathematica [A] time = 0.133969, size = 94, normalized size = 0.48 \[ \frac{c^2 \left (\sqrt{1-\frac{1}{a^2 x^2}} \left (6 a^3 x^3+16 a^2 x^2-9 a x+2\right )-18 a^2 x^2 \log \left (x \left (\sqrt{1-\frac{1}{a^2 x^2}}+1\right )\right )-3 a^2 x^2 \sin ^{-1}\left (\frac{1}{a x}\right )\right )}{6 a^3 x^2} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.151, size = 233, normalized size = 1.2 \begin{align*} -{\frac{ \left ( ax+1 \right ) ^{2}{c}^{2}}{ \left ( 6\,ax-6 \right ){a}^{4}{x}^{3}} \left ({\frac{ax-1}{ax+1}} \right ) ^{{\frac{3}{2}}} \left ( -18\,\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}{x}^{4}{a}^{4}+18\, \left ({a}^{2}{x}^{2}-1 \right ) ^{3/2}\sqrt{{a}^{2}}{x}^{2}{a}^{2}+3\,\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}{x}^{3}{a}^{3}+18\,\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}}{\sqrt{{a}^{2}}}} \right ){x}^{3}{a}^{4}+3\,{a}^{3}{x}^{3}\sqrt{{a}^{2}}\arctan \left ({\frac{1}{\sqrt{{a}^{2}{x}^{2}-1}}} \right ) -9\,\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{{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.
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Maxima [A] time = 1.54839, size = 302, normalized size = 1.55 \begin{align*} \frac{1}{3} \, a{\left (\frac{3 \, c^{2} \arctan \left (\sqrt{\frac{a x - 1}{a x + 1}}\right )}{a^{2}} - \frac{9 \, c^{2} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2}} + \frac{9 \, c^{2} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right )}{a^{2}} - \frac{21 \, c^{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{7}{2}} - 17 \, c^{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{5}{2}} - 37 \, c^{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}} - 15 \, 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.
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Fricas [A] time = 1.32473, size = 359, normalized size = 1.84 \begin{align*} \frac{6 \, a^{3} c^{2} x^{3} \arctan \left (\sqrt{\frac{a x - 1}{a x + 1}}\right ) - 18 \, a^{3} c^{2} x^{3} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right ) + 18 \, 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} + 22 \, a^{3} c^{2} x^{3} + 7 \, a^{2} c^{2} x^{2} - 7 \, 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.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18631, size = 356, normalized size = 1.83 \begin{align*} \frac{c^{2} \arctan \left (-x{\left | a \right |} + \sqrt{a^{2} x^{2} - 1}\right ) \mathrm{sgn}\left (a x + 1\right )}{a} + \frac{3 \, c^{2} \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^{2} \mathrm{sgn}\left (a x + 1\right )}{a} + \frac{9 \,{\left (x{\left | a \right |} - \sqrt{a^{2} x^{2} - 1}\right )}^{5} c^{2}{\left | a \right |} \mathrm{sgn}\left (a x + 1\right ) + 12 \,{\left (x{\left | a \right |} - \sqrt{a^{2} x^{2} - 1}\right )}^{4} a c^{2} \mathrm{sgn}\left (a x + 1\right ) + 36 \,{\left (x{\left | a \right |} - \sqrt{a^{2} x^{2} - 1}\right )}^{2} a c^{2} \mathrm{sgn}\left (a x + 1\right ) - 9 \,{\left (x{\left | a \right |} - \sqrt{a^{2} x^{2} - 1}\right )} c^{2}{\left | a \right |} \mathrm{sgn}\left (a x + 1\right ) + 16 \, a c^{2} \mathrm{sgn}\left (a x + 1\right )}{3 \,{\left ({\left (x{\left | a \right |} - \sqrt{a^{2} x^{2} - 1}\right )}^{2} + 1\right )}^{3} a{\left | a \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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