Optimal. Leaf size=152 \[ -\frac{1}{4} a^3 c^3 x^4 \sqrt{1-\frac{1}{a^2 x^2}}+2 a^2 c^3 x^3 \sqrt{1-\frac{1}{a^2 x^2}}-\frac{67}{8} a c^3 x^2 \sqrt{1-\frac{1}{a^2 x^2}}+30 c^3 x \sqrt{1-\frac{1}{a^2 x^2}}+\frac{32 c^3 \left (a-\frac{1}{x}\right )}{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{315 c^3 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{8 a} \]
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Rubi [A] time = 0.43646, antiderivative size = 152, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.444, Rules used = {6175, 6178, 1805, 1807, 807, 266, 63, 208} \[ -\frac{1}{4} a^3 c^3 x^4 \sqrt{1-\frac{1}{a^2 x^2}}+2 a^2 c^3 x^3 \sqrt{1-\frac{1}{a^2 x^2}}-\frac{67}{8} a c^3 x^2 \sqrt{1-\frac{1}{a^2 x^2}}+30 c^3 x \sqrt{1-\frac{1}{a^2 x^2}}+\frac{32 c^3 \left (a-\frac{1}{x}\right )}{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{315 c^3 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{8 a} \]
Antiderivative was successfully verified.
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Rule 6175
Rule 6178
Rule 1805
Rule 1807
Rule 807
Rule 266
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}{a}\right )^6}{x^5 \left (1-\frac{x^2}{a^2}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{32 c^3 \left (a-\frac{1}{x}\right )}{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}-\left (a^3 c^3\right ) \operatorname{Subst}\left (\int \frac{-1+\frac{6 x}{a}-\frac{16 x^2}{a^2}+\frac{26 x^3}{a^3}-\frac{31 x^4}{a^4}}{x^5 \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{32 c^3 \left (a-\frac{1}{x}\right )}{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{1}{4} a^3 c^3 \sqrt{1-\frac{1}{a^2 x^2}} x^4+\frac{1}{4} \left (a^3 c^3\right ) \operatorname{Subst}\left (\int \frac{-\frac{24}{a}+\frac{67 x}{a^2}-\frac{104 x^2}{a^3}+\frac{124 x^3}{a^4}}{x^4 \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{32 c^3 \left (a-\frac{1}{x}\right )}{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}+2 a^2 c^3 \sqrt{1-\frac{1}{a^2 x^2}} x^3-\frac{1}{4} a^3 c^3 \sqrt{1-\frac{1}{a^2 x^2}} x^4-\frac{1}{12} \left (a^3 c^3\right ) \operatorname{Subst}\left (\int \frac{-\frac{201}{a^2}+\frac{360 x}{a^3}-\frac{372 x^2}{a^4}}{x^3 \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{32 c^3 \left (a-\frac{1}{x}\right )}{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{67}{8} a c^3 \sqrt{1-\frac{1}{a^2 x^2}} x^2+2 a^2 c^3 \sqrt{1-\frac{1}{a^2 x^2}} x^3-\frac{1}{4} a^3 c^3 \sqrt{1-\frac{1}{a^2 x^2}} x^4+\frac{1}{24} \left (a^3 c^3\right ) \operatorname{Subst}\left (\int \frac{-\frac{720}{a^3}+\frac{945 x}{a^4}}{x^2 \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{32 c^3 \left (a-\frac{1}{x}\right )}{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}+30 c^3 \sqrt{1-\frac{1}{a^2 x^2}} x-\frac{67}{8} a c^3 \sqrt{1-\frac{1}{a^2 x^2}} x^2+2 a^2 c^3 \sqrt{1-\frac{1}{a^2 x^2}} x^3-\frac{1}{4} a^3 c^3 \sqrt{1-\frac{1}{a^2 x^2}} x^4+\frac{\left (315 c^3\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{8 a}\\ &=\frac{32 c^3 \left (a-\frac{1}{x}\right )}{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}+30 c^3 \sqrt{1-\frac{1}{a^2 x^2}} x-\frac{67}{8} a c^3 \sqrt{1-\frac{1}{a^2 x^2}} x^2+2 a^2 c^3 \sqrt{1-\frac{1}{a^2 x^2}} x^3-\frac{1}{4} a^3 c^3 \sqrt{1-\frac{1}{a^2 x^2}} x^4+\frac{\left (315 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{32 c^3 \left (a-\frac{1}{x}\right )}{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}+30 c^3 \sqrt{1-\frac{1}{a^2 x^2}} x-\frac{67}{8} a c^3 \sqrt{1-\frac{1}{a^2 x^2}} x^2+2 a^2 c^3 \sqrt{1-\frac{1}{a^2 x^2}} x^3-\frac{1}{4} a^3 c^3 \sqrt{1-\frac{1}{a^2 x^2}} x^4-\frac{1}{8} \left (315 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{32 c^3 \left (a-\frac{1}{x}\right )}{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}+30 c^3 \sqrt{1-\frac{1}{a^2 x^2}} x-\frac{67}{8} a c^3 \sqrt{1-\frac{1}{a^2 x^2}} x^2+2 a^2 c^3 \sqrt{1-\frac{1}{a^2 x^2}} x^3-\frac{1}{4} a^3 c^3 \sqrt{1-\frac{1}{a^2 x^2}} x^4-\frac{315 c^3 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{8 a}\\ \end{align*}
Mathematica [A] time = 0.218798, size = 86, normalized size = 0.57 \[ \frac{1}{8} c^3 \left (\frac{x \sqrt{1-\frac{1}{a^2 x^2}} \left (-2 a^4 x^4+14 a^3 x^3-51 a^2 x^2+173 a x+496\right )}{a x+1}-\frac{315 \log \left (a x \left (\sqrt{1-\frac{1}{a^2 x^2}}+1\right )\right )}{a}\right ) \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.138, size = 542, normalized size = 3.6 \begin{align*} -{\frac{{c}^{3}}{8\, \left ( ax-1 \right ) a} \left ( 2\, \left ({a}^{2}{x}^{2}-1 \right ) ^{3/2}\sqrt{{a}^{2}}{x}^{3}{a}^{3}+4\, \left ({a}^{2}{x}^{2}-1 \right ) ^{3/2}\sqrt{{a}^{2}}{x}^{2}{a}^{2}+69\,\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}{x}^{3}{a}^{3}-16\,\sqrt{{a}^{2}} \left ( \left ( ax-1 \right ) \left ( ax+1 \right ) \right ) ^{3/2}{x}^{2}{a}^{2}+2\,\sqrt{{a}^{2}} \left ({a}^{2}{x}^{2}-1 \right ) ^{3/2}xa+138\,\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}{x}^{2}{a}^{2}-69\,\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}}{\sqrt{{a}^{2}}}} \right ){x}^{2}{a}^{3}-32\,\sqrt{{a}^{2}} \left ( \left ( ax-1 \right ) \left ( ax+1 \right ) \right ) ^{3/2}xa-384\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }{x}^{2}{a}^{2}+384\,\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}{\sqrt{{a}^{2}}}} \right ){x}^{2}{a}^{3}+69\,\sqrt{{a}^{2}}\sqrt{{a}^{2}{x}^{2}-1}xa-138\,\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}}{\sqrt{{a}^{2}}}} \right ) x{a}^{2}+112\, \left ( \left ( ax-1 \right ) \left ( ax+1 \right ) \right ) ^{3/2}\sqrt{{a}^{2}}-768\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }xa+768\,\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}{\sqrt{{a}^{2}}}} \right ) x{a}^{2}-69\,\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}}{\sqrt{{a}^{2}}}} \right ) a-384\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }+384\,a\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}{\sqrt{{a}^{2}}}} \right ) \right ) \left ({\frac{ax-1}{ax+1}} \right ) ^{{\frac{3}{2}}}{\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 = 1.03389, size = 329, normalized size = 2.16 \begin{align*} -\frac{1}{8} \,{\left (\frac{315 \, c^{3} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac{315 \, c^{3} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right )}{a^{2}} - \frac{256 \, c^{3} \sqrt{\frac{a x - 1}{a x + 1}}}{a^{2}} - \frac{2 \,{\left (325 \, c^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{7}{2}} - 765 \, c^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{5}{2}} + 643 \, c^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}} - 187 \, 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 = 1.68883, size = 270, normalized size = 1.78 \begin{align*} -\frac{315 \, c^{3} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right ) - 315 \, c^{3} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right ) +{\left (2 \, a^{4} c^{3} x^{4} - 14 \, a^{3} c^{3} x^{3} + 51 \, a^{2} c^{3} x^{2} - 173 \, a c^{3} x - 496 \, c^{3}\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(-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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{undef} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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