Optimal. Leaf size=105 \[ -\frac{1}{4} a^3 c^3 x^4 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}+\frac{2}{3} a^2 c^3 x^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}-\frac{5}{8} a c^3 x^2 \sqrt{1-\frac{1}{a^2 x^2}}+\frac{5 c^3 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{8 a} \]
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Rubi [A] time = 0.225517, antiderivative size = 105, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {6175, 6178, 1807, 807, 266, 47, 63, 208} \[ -\frac{1}{4} a^3 c^3 x^4 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}+\frac{2}{3} a^2 c^3 x^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}-\frac{5}{8} a c^3 x^2 \sqrt{1-\frac{1}{a^2 x^2}}+\frac{5 c^3 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{8 a} \]
Antiderivative was successfully verified.
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Rule 6175
Rule 6178
Rule 1807
Rule 807
Rule 266
Rule 47
Rule 63
Rule 208
Rubi steps
\begin{align*} \int e^{\coth ^{-1}(a x)} (c-a c x)^3 \, dx &=-\left (\left (a^3 c^3\right ) \int e^{\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}{a}\right )^2 \sqrt{1-\frac{x^2}{a^2}}}{x^5} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{1}{4} a^3 c^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} x^4-\frac{1}{4} \left (a^3 c^3\right ) \operatorname{Subst}\left (\int \frac{\left (\frac{8}{a}-\frac{5 x}{a^2}\right ) \sqrt{1-\frac{x^2}{a^2}}}{x^4} \, dx,x,\frac{1}{x}\right )\\ &=\frac{2}{3} a^2 c^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} x^3-\frac{1}{4} a^3 c^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} x^4+\frac{1}{4} \left (5 a c^3\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1-\frac{x^2}{a^2}}}{x^3} \, dx,x,\frac{1}{x}\right )\\ &=\frac{2}{3} a^2 c^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} x^3-\frac{1}{4} a^3 c^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} x^4+\frac{1}{8} \left (5 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{5}{8} a c^3 \sqrt{1-\frac{1}{a^2 x^2}} x^2+\frac{2}{3} a^2 c^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} x^3-\frac{1}{4} a^3 c^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} x^4-\frac{\left (5 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{5}{8} a c^3 \sqrt{1-\frac{1}{a^2 x^2}} x^2+\frac{2}{3} a^2 c^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} x^3-\frac{1}{4} a^3 c^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} x^4+\frac{1}{8} \left (5 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{5}{8} a c^3 \sqrt{1-\frac{1}{a^2 x^2}} x^2+\frac{2}{3} a^2 c^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} x^3-\frac{1}{4} a^3 c^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} x^4+\frac{5 c^3 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{8 a}\\ \end{align*}
Mathematica [A] time = 0.153045, size = 73, normalized size = 0.7 \[ \frac{c^3 \left (15 \log \left (a x \left (\sqrt{1-\frac{1}{a^2 x^2}}+1\right )\right )-a x \sqrt{1-\frac{1}{a^2 x^2}} \left (6 a^3 x^3-16 a^2 x^2+9 a x+16\right )\right )}{24 a} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.132, size = 141, normalized size = 1.3 \begin{align*} -{\frac{ \left ( ax-1 \right ){c}^{3}}{24\,a} \left ( 6\,\sqrt{{a}^{2}} \left ({a}^{2}{x}^{2}-1 \right ) ^{3/2}xa+15\,\sqrt{{a}^{2}}\sqrt{{a}^{2}{x}^{2}-1}xa-16\, \left ( \left ( ax-1 \right ) \left ( ax+1 \right ) \right ) ^{3/2}\sqrt{{a}^{2}}-15\,\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}}{\sqrt{{a}^{2}}}} \right ) a \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 = 0.991732, size = 298, normalized size = 2.84 \begin{align*} \frac{1}{24} \,{\left (\frac{15 \, c^{3} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac{15 \, c^{3} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right )}{a^{2}} + \frac{2 \,{\left (15 \, c^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{7}{2}} + 73 \, c^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{5}{2}} - 55 \, c^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}} + 15 \, 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 = 1.5496, size = 263, normalized size = 2.5 \begin{align*} \frac{15 \, c^{3} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right ) - 15 \, c^{3} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right ) -{\left (6 \, a^{4} c^{3} x^{4} - 10 \, a^{3} c^{3} x^{3} - 7 \, a^{2} c^{3} x^{2} + 25 \, a c^{3} x + 16 \, c^{3}\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{24 \, 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}{\sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}\, dx + \int - \frac{3 a^{2} x^{2}}{\sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}\, dx + \int \frac{a^{3} x^{3}}{\sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}\, dx + \int - \frac{1}{\sqrt{\frac{a x}{a x + 1} - \frac{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 = 1.21364, size = 270, normalized size = 2.57 \begin{align*} \frac{1}{24} \,{\left (\frac{15 \, c^{3} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac{15 \, c^{3} \log \left ({\left | \sqrt{\frac{a x - 1}{a x + 1}} - 1 \right |}\right )}{a^{2}} + \frac{2 \,{\left (\frac{55 \,{\left (a x - 1\right )} c^{3} \sqrt{\frac{a x - 1}{a x + 1}}}{a x + 1} - \frac{73 \,{\left (a x - 1\right )}^{2} c^{3} \sqrt{\frac{a x - 1}{a x + 1}}}{{\left (a x + 1\right )}^{2}} - \frac{15 \,{\left (a x - 1\right )}^{3} c^{3} \sqrt{\frac{a x - 1}{a x + 1}}}{{\left (a x + 1\right )}^{3}} - 15 \, c^{3} \sqrt{\frac{a x - 1}{a x + 1}}\right )}}{a^{2}{\left (\frac{a x - 1}{a x + 1} - 1\right )}^{4}}\right )} a \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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