Optimal. Leaf size=129 \[ \frac{1}{3} a^2 c^2 x^3 \sqrt{1-\frac{1}{a^2 x^2}}-\frac{5}{2} a c^2 x^2 \sqrt{1-\frac{1}{a^2 x^2}}+\frac{35}{3} c^2 x \sqrt{1-\frac{1}{a^2 x^2}}+\frac{16 c^2 \left (a-\frac{1}{x}\right )}{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{35 c^2 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{2 a} \]
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Rubi [A] time = 0.34967, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 9, 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} \[ \frac{1}{3} a^2 c^2 x^3 \sqrt{1-\frac{1}{a^2 x^2}}-\frac{5}{2} a c^2 x^2 \sqrt{1-\frac{1}{a^2 x^2}}+\frac{35}{3} c^2 x \sqrt{1-\frac{1}{a^2 x^2}}+\frac{16 c^2 \left (a-\frac{1}{x}\right )}{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{35 c^2 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{2 a} \]
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)^2 \, dx &=\left (a^2 c^2\right ) \int e^{-3 \coth ^{-1}(a x)} \left (1-\frac{1}{a x}\right )^2 x^2 \, dx\\ &=-\left (\left (a^2 c^2\right ) \operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a}\right )^5}{x^4 \left (1-\frac{x^2}{a^2}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )\right )\\ &=\frac{16 c^2 \left (a-\frac{1}{x}\right )}{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}+\left (a^2 c^2\right ) \operatorname{Subst}\left (\int \frac{-1+\frac{5 x}{a}-\frac{11 x^2}{a^2}+\frac{15 x^3}{a^3}}{x^4 \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{16 c^2 \left (a-\frac{1}{x}\right )}{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{1}{3} a^2 c^2 \sqrt{1-\frac{1}{a^2 x^2}} x^3-\frac{1}{3} \left (a^2 c^2\right ) \operatorname{Subst}\left (\int \frac{-\frac{15}{a}+\frac{35 x}{a^2}-\frac{45 x^2}{a^3}}{x^3 \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{16 c^2 \left (a-\frac{1}{x}\right )}{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{5}{2} a c^2 \sqrt{1-\frac{1}{a^2 x^2}} x^2+\frac{1}{3} a^2 c^2 \sqrt{1-\frac{1}{a^2 x^2}} x^3+\frac{1}{6} \left (a^2 c^2\right ) \operatorname{Subst}\left (\int \frac{-\frac{70}{a^2}+\frac{105 x}{a^3}}{x^2 \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{16 c^2 \left (a-\frac{1}{x}\right )}{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{35}{3} c^2 \sqrt{1-\frac{1}{a^2 x^2}} x-\frac{5}{2} a c^2 \sqrt{1-\frac{1}{a^2 x^2}} x^2+\frac{1}{3} a^2 c^2 \sqrt{1-\frac{1}{a^2 x^2}} x^3+\frac{\left (35 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{2 a}\\ &=\frac{16 c^2 \left (a-\frac{1}{x}\right )}{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{35}{3} c^2 \sqrt{1-\frac{1}{a^2 x^2}} x-\frac{5}{2} a c^2 \sqrt{1-\frac{1}{a^2 x^2}} x^2+\frac{1}{3} a^2 c^2 \sqrt{1-\frac{1}{a^2 x^2}} x^3+\frac{\left (35 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x}{a^2}}} \, dx,x,\frac{1}{x^2}\right )}{4 a}\\ &=\frac{16 c^2 \left (a-\frac{1}{x}\right )}{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{35}{3} c^2 \sqrt{1-\frac{1}{a^2 x^2}} x-\frac{5}{2} a c^2 \sqrt{1-\frac{1}{a^2 x^2}} x^2+\frac{1}{3} a^2 c^2 \sqrt{1-\frac{1}{a^2 x^2}} x^3-\frac{1}{2} \left (35 a c^2\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{16 c^2 \left (a-\frac{1}{x}\right )}{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{35}{3} c^2 \sqrt{1-\frac{1}{a^2 x^2}} x-\frac{5}{2} a c^2 \sqrt{1-\frac{1}{a^2 x^2}} x^2+\frac{1}{3} a^2 c^2 \sqrt{1-\frac{1}{a^2 x^2}} x^3-\frac{35 c^2 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{2 a}\\ \end{align*}
Mathematica [A] time = 0.150984, size = 78, normalized size = 0.6 \[ \frac{1}{6} c^2 \left (\frac{x \sqrt{1-\frac{1}{a^2 x^2}} \left (2 a^3 x^3-13 a^2 x^2+55 a x+166\right )}{a x+1}-\frac{105 \log \left (a x \left (\sqrt{1-\frac{1}{a^2 x^2}}+1\right )\right )}{a}\right ) \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.136, size = 474, normalized size = 3.7 \begin{align*} -{\frac{{c}^{2}}{6\, \left ( ax-1 \right ) a} \left ( 15\,\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}{x}^{3}{a}^{3}-2\,\sqrt{{a}^{2}} \left ( \left ( ax-1 \right ) \left ( ax+1 \right ) \right ) ^{3/2}{x}^{2}{a}^{2}+30\,\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}{x}^{2}{a}^{2}-15\,\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}}{\sqrt{{a}^{2}}}} \right ){x}^{2}{a}^{3}-4\,\sqrt{{a}^{2}} \left ( \left ( ax-1 \right ) \left ( ax+1 \right ) \right ) ^{3/2}xa-120\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }{x}^{2}{a}^{2}+120\,\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}+15\,\sqrt{{a}^{2}}\sqrt{{a}^{2}{x}^{2}-1}xa-30\,\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}}{\sqrt{{a}^{2}}}} \right ) x{a}^{2}+46\, \left ( \left ( ax-1 \right ) \left ( ax+1 \right ) \right ) ^{3/2}\sqrt{{a}^{2}}-240\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }xa+240\,\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}-15\,\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}}{\sqrt{{a}^{2}}}} \right ) a-120\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }+120\,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 = 1.11875, size = 275, normalized size = 2.13 \begin{align*} -\frac{1}{6} \, a{\left (\frac{105 \, c^{2} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac{105 \, c^{2} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right )}{a^{2}} - \frac{96 \, c^{2} \sqrt{\frac{a x - 1}{a x + 1}}}{a^{2}} + \frac{2 \,{\left (87 \, c^{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{5}{2}} - 136 \, c^{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}} + 57 \, c^{2} \sqrt{\frac{a x - 1}{a x + 1}}\right )}}{\frac{3 \,{\left (a x - 1\right )} a^{2}}{a x + 1} - \frac{3 \,{\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 )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.62261, size = 246, normalized size = 1.91 \begin{align*} -\frac{105 \, c^{2} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right ) - 105 \, c^{2} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right ) -{\left (2 \, a^{3} c^{2} x^{3} - 13 \, a^{2} c^{2} x^{2} + 55 \, a c^{2} x + 166 \, c^{2}\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{6 \, 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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