Optimal. Leaf size=105 \[ \frac{1}{5} a^4 c^4 x^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}-\frac{1}{4} a^3 c^4 x^4 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}+\frac{3}{8} a c^4 x^2 \sqrt{1-\frac{1}{a^2 x^2}}-\frac{3 c^4 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{8 a} \]
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Rubi [A] time = 0.165661, antiderivative size = 105, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.389, Rules used = {6175, 6178, 807, 266, 47, 63, 208} \[ \frac{1}{5} a^4 c^4 x^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}-\frac{1}{4} a^3 c^4 x^4 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}+\frac{3}{8} a c^4 x^2 \sqrt{1-\frac{1}{a^2 x^2}}-\frac{3 c^4 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{8 a} \]
Antiderivative was successfully verified.
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Rule 6175
Rule 6178
Rule 807
Rule 266
Rule 47
Rule 63
Rule 208
Rubi steps
\begin{align*} \int e^{3 \coth ^{-1}(a x)} (c-a c x)^4 \, dx &=\left (a^4 c^4\right ) \int e^{3 \coth ^{-1}(a x)} \left (1-\frac{1}{a x}\right )^4 x^4 \, dx\\ &=-\left (\left (a^4 c^4\right ) \operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a}\right ) \left (1-\frac{x^2}{a^2}\right )^{3/2}}{x^6} \, dx,x,\frac{1}{x}\right )\right )\\ &=\frac{1}{5} a^4 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} x^5+\left (a^3 c^4\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}{5} a^4 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} x^5+\frac{1}{2} \left (a^3 c^4\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^4 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} x^4+\frac{1}{5} a^4 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} x^5-\frac{1}{8} \left (3 a c^4\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^4 \sqrt{1-\frac{1}{a^2 x^2}} x^2-\frac{1}{4} a^3 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} x^4+\frac{1}{5} a^4 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} x^5+\frac{\left (3 c^4\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^4 \sqrt{1-\frac{1}{a^2 x^2}} x^2-\frac{1}{4} a^3 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} x^4+\frac{1}{5} a^4 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} x^5-\frac{1}{8} \left (3 a c^4\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^4 \sqrt{1-\frac{1}{a^2 x^2}} x^2-\frac{1}{4} a^3 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} x^4+\frac{1}{5} a^4 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} x^5-\frac{3 c^4 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{8 a}\\ \end{align*}
Mathematica [A] time = 0.18773, size = 80, normalized size = 0.76 \[ \frac{c^4 \left (a x \sqrt{1-\frac{1}{a^2 x^2}} \left (8 a^4 x^4-10 a^3 x^3-16 a^2 x^2+25 a x+8\right )-15 \log \left (a x \left (\sqrt{1-\frac{1}{a^2 x^2}}+1\right )\right )\right )}{40 a} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.18, size = 192, normalized size = 1.8 \begin{align*}{\frac{ \left ( ax-1 \right ) ^{2}{c}^{4}}{120\,a \left ( ax+1 \right ) } \left ( 24\, \left ({a}^{2}{x}^{2}-1 \right ) ^{3/2}\sqrt{{a}^{2}}{x}^{2}{a}^{2}-30\,\sqrt{{a}^{2}} \left ({a}^{2}{x}^{2}-1 \right ) ^{3/2}xa+16\, \left ({a}^{2}{x}^{2}-1 \right ) ^{3/2}\sqrt{{a}^{2}}+45\,\sqrt{{a}^{2}}\sqrt{{a}^{2}{x}^{2}-1}xa-40\, \left ( \left ( ax-1 \right ) \left ( ax+1 \right ) \right ) ^{3/2}\sqrt{{a}^{2}}-45\,\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}}{\sqrt{{a}^{2}}}} \right ) a \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 = 1.10712, size = 350, normalized size = 3.33 \begin{align*} -\frac{1}{40} \,{\left (\frac{15 \, c^{4} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac{15 \, c^{4} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right )}{a^{2}} - \frac{2 \,{\left (15 \, c^{4} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{9}{2}} - 70 \, c^{4} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{7}{2}} - 128 \, c^{4} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{5}{2}} + 70 \, c^{4} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}} - 15 \, c^{4} \sqrt{\frac{a x - 1}{a x + 1}}\right )}}{\frac{5 \,{\left (a x - 1\right )} a^{2}}{a x + 1} - \frac{10 \,{\left (a x - 1\right )}^{2} a^{2}}{{\left (a x + 1\right )}^{2}} + \frac{10 \,{\left (a x - 1\right )}^{3} a^{2}}{{\left (a x + 1\right )}^{3}} - \frac{5 \,{\left (a x - 1\right )}^{4} a^{2}}{{\left (a x + 1\right )}^{4}} + \frac{{\left (a x - 1\right )}^{5} a^{2}}{{\left (a x + 1\right )}^{5}} - a^{2}}\right )} a \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.61108, size = 285, normalized size = 2.71 \begin{align*} -\frac{15 \, c^{4} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right ) - 15 \, c^{4} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right ) -{\left (8 \, a^{5} c^{4} x^{5} - 2 \, a^{4} c^{4} x^{4} - 26 \, a^{3} c^{4} x^{3} + 9 \, a^{2} c^{4} x^{2} + 33 \, a c^{4} x + 8 \, c^{4}\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{40 \, 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^{4} \left (\int - \frac{4 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{6 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{4 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{a^{4} x^{4}}{\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 = 1.21687, size = 316, normalized size = 3.01 \begin{align*} -\frac{1}{40} \,{\left (\frac{15 \, c^{4} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac{15 \, c^{4} \log \left ({\left | \sqrt{\frac{a x - 1}{a x + 1}} - 1 \right |}\right )}{a^{2}} - \frac{2 \,{\left (\frac{70 \,{\left (a x - 1\right )} c^{4} \sqrt{\frac{a x - 1}{a x + 1}}}{a x + 1} - \frac{128 \,{\left (a x - 1\right )}^{2} c^{4} \sqrt{\frac{a x - 1}{a x + 1}}}{{\left (a x + 1\right )}^{2}} - \frac{70 \,{\left (a x - 1\right )}^{3} c^{4} \sqrt{\frac{a x - 1}{a x + 1}}}{{\left (a x + 1\right )}^{3}} + \frac{15 \,{\left (a x - 1\right )}^{4} c^{4} \sqrt{\frac{a x - 1}{a x + 1}}}{{\left (a x + 1\right )}^{4}} - 15 \, c^{4} \sqrt{\frac{a x - 1}{a x + 1}}\right )}}{a^{2}{\left (\frac{a x - 1}{a x + 1} - 1\right )}^{5}}\right )} a \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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