Optimal. Leaf size=269 \[ c^3 x \left (1-\frac{1}{a x}\right )^{3/2} \left (\frac{1}{a x}+1\right )^{9/2}+\frac{6 c^3 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{9/2}}{5 a}-\frac{21 c^3 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{7/2}}{20 a}-\frac{29 c^3 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{5/2}}{20 a}-\frac{17 c^3 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{3/2}}{8 a}-\frac{27 c^3 \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}{8 a}+\frac{3 c^3 \csc ^{-1}(a x)}{8 a}+\frac{3 c^3 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right )}{a} \]
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Rubi [A] time = 0.191917, antiderivative size = 269, normalized size of antiderivative = 1., number of steps used = 12, 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} \[ c^3 x \left (1-\frac{1}{a x}\right )^{3/2} \left (\frac{1}{a x}+1\right )^{9/2}+\frac{6 c^3 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{9/2}}{5 a}-\frac{21 c^3 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{7/2}}{20 a}-\frac{29 c^3 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{5/2}}{20 a}-\frac{17 c^3 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{3/2}}{8 a}-\frac{27 c^3 \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}{8 a}+\frac{3 c^3 \csc ^{-1}(a x)}{8 a}+\frac{3 c^3 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right )}{a} \]
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 )^3 \, dx &=-\left (c^3 \operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a}\right )^{3/2} \left (1+\frac{x}{a}\right )^{9/2}}{x^2} \, dx,x,\frac{1}{x}\right )\right )\\ &=c^3 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{9/2} x-c^3 \operatorname{Subst}\left (\int \frac{\left (\frac{3}{a}-\frac{6 x}{a^2}\right ) \sqrt{1-\frac{x}{a}} \left (1+\frac{x}{a}\right )^{7/2}}{x} \, dx,x,\frac{1}{x}\right )\\ &=\frac{6 c^3 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{9/2}}{5 a}+c^3 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{9/2} x-\frac{1}{5} \left (a c^3\right ) \operatorname{Subst}\left (\int \frac{\left (\frac{15}{a^2}-\frac{21 x}{a^3}\right ) \left (1+\frac{x}{a}\right )^{7/2}}{x \sqrt{1-\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{21 c^3 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{7/2}}{20 a}+\frac{6 c^3 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{9/2}}{5 a}+c^3 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{9/2} x+\frac{1}{20} \left (a^2 c^3\right ) \operatorname{Subst}\left (\int \frac{\left (-\frac{60}{a^3}+\frac{87 x}{a^4}\right ) \left (1+\frac{x}{a}\right )^{5/2}}{x \sqrt{1-\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{29 c^3 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{5/2}}{20 a}-\frac{21 c^3 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{7/2}}{20 a}+\frac{6 c^3 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{9/2}}{5 a}+c^3 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{9/2} x-\frac{1}{60} \left (a^3 c^3\right ) \operatorname{Subst}\left (\int \frac{\left (\frac{180}{a^4}-\frac{255 x}{a^5}\right ) \left (1+\frac{x}{a}\right )^{3/2}}{x \sqrt{1-\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{17 c^3 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/2}}{8 a}-\frac{29 c^3 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{5/2}}{20 a}-\frac{21 c^3 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{7/2}}{20 a}+\frac{6 c^3 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{9/2}}{5 a}+c^3 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{9/2} x+\frac{1}{120} \left (a^4 c^3\right ) \operatorname{Subst}\left (\int \frac{\left (-\frac{360}{a^5}+\frac{405 x}{a^6}\right ) \sqrt{1+\frac{x}{a}}}{x \sqrt{1-\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{27 c^3 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}{8 a}-\frac{17 c^3 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/2}}{8 a}-\frac{29 c^3 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{5/2}}{20 a}-\frac{21 c^3 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{7/2}}{20 a}+\frac{6 c^3 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{9/2}}{5 a}+c^3 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{9/2} x-\frac{1}{120} \left (a^5 c^3\right ) \operatorname{Subst}\left (\int \frac{\frac{360}{a^6}-\frac{45 x}{a^7}}{x \sqrt{1-\frac{x}{a}} \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{27 c^3 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}{8 a}-\frac{17 c^3 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/2}}{8 a}-\frac{29 c^3 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{5/2}}{20 a}-\frac{21 c^3 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{7/2}}{20 a}+\frac{6 c^3 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{9/2}}{5 a}+c^3 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{9/2} x+\frac{\left (3 c^3\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x}{a}} \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{8 a^2}-\frac{\left (3 c^3\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{27 c^3 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}{8 a}-\frac{17 c^3 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/2}}{8 a}-\frac{29 c^3 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{5/2}}{20 a}-\frac{21 c^3 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{7/2}}{20 a}+\frac{6 c^3 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{9/2}}{5 a}+c^3 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{9/2} x+\frac{\left (3 c^3\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{8 a^2}+\frac{\left (3 c^3\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{27 c^3 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}{8 a}-\frac{17 c^3 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/2}}{8 a}-\frac{29 c^3 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{5/2}}{20 a}-\frac{21 c^3 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{7/2}}{20 a}+\frac{6 c^3 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{9/2}}{5 a}+c^3 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{9/2} x+\frac{3 c^3 \csc ^{-1}(a x)}{8 a}+\frac{3 c^3 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}\right )}{a}\\ \end{align*}
Mathematica [A] time = 0.187653, size = 110, normalized size = 0.41 \[ \frac{c^3 \left (\sqrt{1-\frac{1}{a^2 x^2}} \left (40 a^5 x^5-152 a^4 x^4-55 a^3 x^3+24 a^2 x^2+30 a x+8\right )+120 a^4 x^4 \log \left (x \left (\sqrt{1-\frac{1}{a^2 x^2}}+1\right )\right )+15 a^4 x^4 \sin ^{-1}\left (\frac{1}{a x}\right )\right )}{40 a^5 x^4} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.189, size = 281, normalized size = 1. \begin{align*}{\frac{ \left ( ax-1 \right ) ^{2}{c}^{3}}{ \left ( 40\,ax+40 \right ){a}^{6}{x}^{5}} \left ( -120\,\sqrt{{a}^{2}}\sqrt{{a}^{2}{x}^{2}-1}{x}^{6}{a}^{6}+120\,\sqrt{{a}^{2}} \left ({a}^{2}{x}^{2}-1 \right ) ^{3/2}{x}^{4}{a}^{4}+15\,\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}{x}^{5}{a}^{5}+15\,\sqrt{{a}^{2}}\arctan \left ({\frac{1}{\sqrt{{a}^{2}{x}^{2}-1}}} \right ){x}^{5}{a}^{5}+120\,\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}}{\sqrt{{a}^{2}}}} \right ){x}^{5}{a}^{6}+25\, \left ({a}^{2}{x}^{2}-1 \right ) ^{3/2}\sqrt{{a}^{2}}{x}^{3}{a}^{3}-32\, \left ({a}^{2}{x}^{2}-1 \right ) ^{3/2}\sqrt{{a}^{2}}{x}^{2}{a}^{2}-30\,\sqrt{{a}^{2}} \left ({a}^{2}{x}^{2}-1 \right ) ^{3/2}xa-8\, \left ({a}^{2}{x}^{2}-1 \right ) ^{3/2}\sqrt{{a}^{2}} \right ) \left ({\frac{ax-1}{ax+1}} \right ) ^{-{\frac{3}{2}}}{\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.
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Maxima [A] time = 1.53196, size = 408, normalized size = 1.52 \begin{align*} -\frac{1}{20} \,{\left (\frac{15 \, c^{3} \arctan \left (\sqrt{\frac{a x - 1}{a x + 1}}\right )}{a^{2}} - \frac{60 \, c^{3} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2}} + \frac{60 \, c^{3} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right )}{a^{2}} - \frac{135 \, c^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{11}{2}} + 575 \, c^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{9}{2}} + 842 \, c^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{7}{2}} + 298 \, c^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{5}{2}} - 465 \, c^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}} - 105 \, c^{3} \sqrt{\frac{a x - 1}{a x + 1}}}{\frac{4 \,{\left (a x - 1\right )} a^{2}}{a x + 1} + \frac{5 \,{\left (a x - 1\right )}^{2} a^{2}}{{\left (a x + 1\right )}^{2}} - \frac{5 \,{\left (a x - 1\right )}^{4} a^{2}}{{\left (a x + 1\right )}^{4}} - \frac{4 \,{\left (a x - 1\right )}^{5} a^{2}}{{\left (a x + 1\right )}^{5}} - \frac{{\left (a x - 1\right )}^{6} a^{2}}{{\left (a x + 1\right )}^{6}} + a^{2}}\right )} a \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.71341, size = 419, normalized size = 1.56 \begin{align*} -\frac{30 \, a^{5} c^{3} x^{5} \arctan \left (\sqrt{\frac{a x - 1}{a x + 1}}\right ) - 120 \, a^{5} c^{3} x^{5} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right ) + 120 \, a^{5} c^{3} x^{5} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right ) -{\left (40 \, a^{6} c^{3} x^{6} - 112 \, a^{5} c^{3} x^{5} - 207 \, a^{4} c^{3} x^{4} - 31 \, a^{3} c^{3} x^{3} + 54 \, a^{2} c^{3} x^{2} + 38 \, a c^{3} x + 8 \, c^{3}\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{40 \, a^{6} x^{5}} \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 = 1.2177, size = 401, normalized size = 1.49 \begin{align*} -\frac{1}{20} \,{\left (\frac{15 \, c^{3} \arctan \left (\sqrt{\frac{a x - 1}{a x + 1}}\right )}{a^{2}} - \frac{60 \, c^{3} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2}} + \frac{60 \, c^{3} \log \left ({\left | \sqrt{\frac{a x - 1}{a x + 1}} - 1 \right |}\right )}{a^{2}} + \frac{40 \, c^{3} \sqrt{\frac{a x - 1}{a x + 1}}}{a^{2}{\left (\frac{a x - 1}{a x + 1} - 1\right )}} + \frac{\frac{810 \,{\left (a x - 1\right )} c^{3} \sqrt{\frac{a x - 1}{a x + 1}}}{a x + 1} + \frac{912 \,{\left (a x - 1\right )}^{2} c^{3} \sqrt{\frac{a x - 1}{a x + 1}}}{{\left (a x + 1\right )}^{2}} + \frac{470 \,{\left (a x - 1\right )}^{3} c^{3} \sqrt{\frac{a x - 1}{a x + 1}}}{{\left (a x + 1\right )}^{3}} + \frac{95 \,{\left (a x - 1\right )}^{4} c^{3} \sqrt{\frac{a x - 1}{a x + 1}}}{{\left (a x + 1\right )}^{4}} + 145 \, c^{3} \sqrt{\frac{a x - 1}{a x + 1}}}{a^{2}{\left (\frac{a x - 1}{a x + 1} + 1\right )}^{5}}\right )} a \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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