Optimal. Leaf size=88 \[ \frac{c^3 \sqrt{1-\frac{1}{a^2 x^2}} \left (4 a+\frac{1}{x}\right )}{2 a^2}+c^3 x \left (1-\frac{1}{a^2 x^2}\right )^{3/2}-\frac{2 c^3 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a}+\frac{c^3 \csc ^{-1}(a x)}{2 a} \]
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Rubi [A] time = 0.190626, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {6177, 1807, 815, 844, 216, 266, 63, 208} \[ \frac{c^3 \sqrt{1-\frac{1}{a^2 x^2}} \left (4 a+\frac{1}{x}\right )}{2 a^2}+c^3 x \left (1-\frac{1}{a^2 x^2}\right )^{3/2}-\frac{2 c^3 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a}+\frac{c^3 \csc ^{-1}(a x)}{2 a} \]
Antiderivative was successfully verified.
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Rule 6177
Rule 1807
Rule 815
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 )^3 \, dx &=-\left (c \operatorname{Subst}\left (\int \frac{\left (c-\frac{c x}{a}\right )^2 \sqrt{1-\frac{x^2}{a^2}}}{x^2} \, dx,x,\frac{1}{x}\right )\right )\\ &=c^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} x+c \operatorname{Subst}\left (\int \frac{\left (\frac{2 c^2}{a}+\frac{c^2 x}{a^2}\right ) \sqrt{1-\frac{x^2}{a^2}}}{x} \, dx,x,\frac{1}{x}\right )\\ &=\frac{c^3 \sqrt{1-\frac{1}{a^2 x^2}} \left (4 a+\frac{1}{x}\right )}{2 a^2}+c^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} x-\frac{1}{2} \left (a^2 c\right ) \operatorname{Subst}\left (\int \frac{-\frac{4 c^2}{a^3}-\frac{c^2 x}{a^4}}{x \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{c^3 \sqrt{1-\frac{1}{a^2 x^2}} \left (4 a+\frac{1}{x}\right )}{2 a^2}+c^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} x+\frac{c^3 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{2 a^2}+\frac{\left (2 c^3\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{a}\\ &=\frac{c^3 \sqrt{1-\frac{1}{a^2 x^2}} \left (4 a+\frac{1}{x}\right )}{2 a^2}+c^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} x+\frac{c^3 \csc ^{-1}(a x)}{2 a}+\frac{c^3 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x}{a^2}}} \, dx,x,\frac{1}{x^2}\right )}{a}\\ &=\frac{c^3 \sqrt{1-\frac{1}{a^2 x^2}} \left (4 a+\frac{1}{x}\right )}{2 a^2}+c^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} x+\frac{c^3 \csc ^{-1}(a x)}{2 a}-\left (2 a c^3\right ) \operatorname{Subst}\left (\int \frac{1}{a^2-a^2 x^2} \, dx,x,\sqrt{1-\frac{1}{a^2 x^2}}\right )\\ &=\frac{c^3 \sqrt{1-\frac{1}{a^2 x^2}} \left (4 a+\frac{1}{x}\right )}{2 a^2}+c^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} x+\frac{c^3 \csc ^{-1}(a x)}{2 a}-\frac{2 c^3 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a}\\ \end{align*}
Mathematica [A] time = 0.141962, size = 167, normalized size = 1.9 \[ \frac{c^3 \left (2 a^4 x^4+4 a^3 x^3-3 a^2 x^2+2 a^3 x^3 \sqrt{1-\frac{1}{a^2 x^2}} \sin ^{-1}\left (\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{2}}\right )+2 a^3 x^3 \sqrt{1-\frac{1}{a^2 x^2}} \sin ^{-1}\left (\frac{1}{a x}\right )-4 a^3 x^3 \sqrt{1-\frac{1}{a^2 x^2}} \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )-4 a x+1\right )}{2 a^4 x^3 \sqrt{1-\frac{1}{a^2 x^2}}} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.131, size = 200, normalized size = 2.3 \begin{align*} -{\frac{ \left ( ax-1 \right ){c}^{3}}{2\,{x}^{2}{a}^{3}} \left ( -4\,\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}{x}^{3}{a}^{3}+4\,\sqrt{{a}^{2}} \left ({a}^{2}{x}^{2}-1 \right ) ^{3/2}xa-\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}{x}^{2}{a}^{2}+4\,\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}}{\sqrt{{a}^{2}}}} \right ){x}^{2}{a}^{3}-{a}^{2}{x}^{2}\sqrt{{a}^{2}}\arctan \left ({\frac{1}{\sqrt{{a}^{2}{x}^{2}-1}}} \right ) - \left ({a}^{2}{x}^{2}-1 \right ) ^{{\frac{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.
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Maxima [B] time = 1.63146, size = 271, normalized size = 3.08 \begin{align*} -{\left (\frac{c^{3} \arctan \left (\sqrt{\frac{a x - 1}{a x + 1}}\right )}{a^{2}} + \frac{2 \, c^{3} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac{2 \, c^{3} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right )}{a^{2}} + \frac{3 \, c^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{5}{2}} - 6 \, c^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}} - 5 \, c^{3} \sqrt{\frac{a x - 1}{a x + 1}}}{\frac{{\left (a x - 1\right )} a^{2}}{a x + 1} - \frac{{\left (a x - 1\right )}^{2} a^{2}}{{\left (a x + 1\right )}^{2}} - \frac{{\left (a x - 1\right )}^{3} a^{2}}{{\left (a x + 1\right )}^{3}} + a^{2}}\right )} a \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.6649, size = 332, normalized size = 3.77 \begin{align*} -\frac{2 \, a^{2} c^{3} x^{2} \arctan \left (\sqrt{\frac{a x - 1}{a x + 1}}\right ) + 4 \, a^{2} c^{3} x^{2} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right ) - 4 \, a^{2} c^{3} x^{2} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right ) -{\left (2 \, a^{3} c^{3} x^{3} + 6 \, a^{2} c^{3} x^{2} + 3 \, a c^{3} x - c^{3}\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{2 \, a^{3} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{c^{3} \left (\int \frac{a^{3}}{\sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}\, dx + \int - \frac{1}{x^{3} \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}\, dx + \int \frac{3 a}{x^{2} \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}\, dx + \int - \frac{3 a^{2}}{x \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}\, dx\right )}{a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.21314, size = 263, normalized size = 2.99 \begin{align*} -{\left (\frac{c^{3} \arctan \left (\sqrt{\frac{a x - 1}{a x + 1}}\right )}{a^{2}} + \frac{2 \, c^{3} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac{2 \, c^{3} \log \left ({\left | \sqrt{\frac{a x - 1}{a x + 1}} - 1 \right |}\right )}{a^{2}} + \frac{2 \, c^{3} \sqrt{\frac{a x - 1}{a x + 1}}}{a^{2}{\left (\frac{a x - 1}{a x + 1} - 1\right )}} - \frac{\frac{5 \,{\left (a x - 1\right )} c^{3} \sqrt{\frac{a x - 1}{a x + 1}}}{a x + 1} + 3 \, c^{3} \sqrt{\frac{a x - 1}{a x + 1}}}{a^{2}{\left (\frac{a x - 1}{a x + 1} + 1\right )}^{2}}\right )} a \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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