Optimal. Leaf size=78 \[ -\frac{1}{4} a^3 c^3 x^4 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}+\frac{3}{8} a c^3 x^2 \sqrt{1-\frac{1}{a^2 x^2}}-\frac{3 c^3 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{8 a} \]
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Rubi [A] time = 0.141355, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {6175, 6178, 266, 47, 63, 208} \[ -\frac{1}{4} a^3 c^3 x^4 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}+\frac{3}{8} a c^3 x^2 \sqrt{1-\frac{1}{a^2 x^2}}-\frac{3 c^3 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{8 a} \]
Antiderivative was successfully verified.
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Rule 6175
Rule 6178
Rule 266
Rule 47
Rule 63
Rule 208
Rubi steps
\begin{align*} \int e^{3 \coth ^{-1}(a x)} (c-a c x)^3 \, dx &=-\left (\left (a^3 c^3\right ) \int e^{3 \coth ^{-1}(a x)} \left (1-\frac{1}{a x}\right )^3 x^3 \, dx\right )\\ &=\left (a^3 c^3\right ) \operatorname{Subst}\left (\int \frac{\left (1-\frac{x^2}{a^2}\right )^{3/2}}{x^5} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{2} \left (a^3 c^3\right ) \operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a^2}\right )^{3/2}}{x^3} \, dx,x,\frac{1}{x^2}\right )\\ &=-\frac{1}{4} a^3 c^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} x^4-\frac{1}{8} \left (3 a c^3\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1-\frac{x}{a^2}}}{x^2} \, dx,x,\frac{1}{x^2}\right )\\ &=\frac{3}{8} a c^3 \sqrt{1-\frac{1}{a^2 x^2}} x^2-\frac{1}{4} a^3 c^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} x^4+\frac{\left (3 c^3\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x}{a^2}}} \, dx,x,\frac{1}{x^2}\right )}{16 a}\\ &=\frac{3}{8} a c^3 \sqrt{1-\frac{1}{a^2 x^2}} x^2-\frac{1}{4} a^3 c^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} x^4-\frac{1}{8} \left (3 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{3}{8} a c^3 \sqrt{1-\frac{1}{a^2 x^2}} x^2-\frac{1}{4} a^3 c^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} x^4-\frac{3 c^3 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{8 a}\\ \end{align*}
Mathematica [A] time = 0.136916, size = 64, normalized size = 0.82 \[ \frac{c^3 \left (a^2 x^2 \sqrt{1-\frac{1}{a^2 x^2}} \left (5-2 a^2 x^2\right )-3 \log \left (a x \left (\sqrt{1-\frac{1}{a^2 x^2}}+1\right )\right )\right )}{8 a} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.165, size = 124, normalized size = 1.6 \begin{align*} -{\frac{ \left ( ax-1 \right ) ^{2}{c}^{3}}{8\,ax+8} \left ( 2\,x \left ({a}^{2}{x}^{2}-1 \right ) ^{3/2}\sqrt{{a}^{2}}-3\,x\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}+3\,\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}}{\sqrt{{a}^{2}}}} \right ) \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 [B] time = 1.06614, size = 298, normalized size = 3.82 \begin{align*} -\frac{1}{8} \,{\left (\frac{3 \, c^{3} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac{3 \, c^{3} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right )}{a^{2}} + \frac{2 \,{\left (3 \, c^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{7}{2}} - 11 \, c^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{5}{2}} - 11 \, c^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}} + 3 \, c^{3} \sqrt{\frac{a x - 1}{a x + 1}}\right )}}{\frac{4 \,{\left (a x - 1\right )} a^{2}}{a x + 1} - \frac{6 \,{\left (a x - 1\right )}^{2} a^{2}}{{\left (a x + 1\right )}^{2}} + \frac{4 \,{\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 )} a \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.52967, size = 246, normalized size = 3.15 \begin{align*} -\frac{3 \, c^{3} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right ) - 3 \, c^{3} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right ) +{\left (2 \, a^{4} c^{3} x^{4} + 2 \, a^{3} c^{3} x^{3} - 5 \, a^{2} c^{3} x^{2} - 5 \, a c^{3} x\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{8 \, a} \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*} - c^{3} \left (\int \frac{3 a x}{\frac{a x \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}{a x + 1} - \frac{\sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}{a x + 1}}\, dx + \int - \frac{3 a^{2} x^{2}}{\frac{a x \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}{a x + 1} - \frac{\sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}{a x + 1}}\, dx + \int \frac{a^{3} x^{3}}{\frac{a x \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}{a x + 1} - \frac{\sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}{a x + 1}}\, dx + \int - \frac{1}{\frac{a x \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}{a x + 1} - \frac{\sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}{a x + 1}}\, dx\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.23131, size = 275, normalized size = 3.53 \begin{align*} -\frac{1}{16} \,{\left (\frac{3 \, c^{3} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + \frac{1}{\sqrt{\frac{a x - 1}{a x + 1}}} + 2\right )}{a^{2}} - \frac{3 \, c^{3} \log \left ({\left | \sqrt{\frac{a x - 1}{a x + 1}} + \frac{1}{\sqrt{\frac{a x - 1}{a x + 1}}} - 2 \right |}\right )}{a^{2}} - \frac{4 \,{\left (3 \, c^{3}{\left (\sqrt{\frac{a x - 1}{a x + 1}} + \frac{1}{\sqrt{\frac{a x - 1}{a x + 1}}}\right )}^{3} - 20 \, c^{3}{\left (\sqrt{\frac{a x - 1}{a x + 1}} + \frac{1}{\sqrt{\frac{a x - 1}{a x + 1}}}\right )}\right )}}{{\left ({\left (\sqrt{\frac{a x - 1}{a x + 1}} + \frac{1}{\sqrt{\frac{a x - 1}{a x + 1}}}\right )}^{2} - 4\right )}^{2} a^{2}}\right )} a \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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