Optimal. Leaf size=127 \[ -\frac{1}{4} a^3 c^3 x^4 \sqrt{1-\frac{1}{a^2 x^2}}+\frac{4}{3} a^2 c^3 x^3 \sqrt{1-\frac{1}{a^2 x^2}}-\frac{27}{8} a c^3 x^2 \sqrt{1-\frac{1}{a^2 x^2}}+\frac{20}{3} c^3 x \sqrt{1-\frac{1}{a^2 x^2}}-\frac{35 c^3 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{8 a} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.351761, antiderivative size = 127, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.389, Rules used = {6175, 6178, 1807, 807, 266, 63, 208} \[ -\frac{1}{4} a^3 c^3 x^4 \sqrt{1-\frac{1}{a^2 x^2}}+\frac{4}{3} a^2 c^3 x^3 \sqrt{1-\frac{1}{a^2 x^2}}-\frac{27}{8} a c^3 x^2 \sqrt{1-\frac{1}{a^2 x^2}}+\frac{20}{3} c^3 x \sqrt{1-\frac{1}{a^2 x^2}}-\frac{35 c^3 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{8 a} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6175
Rule 6178
Rule 1807
Rule 807
Rule 266
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 )^4}{x^5 \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )\\ &=-\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{16}{a}-\frac{27 x}{a^2}+\frac{16 x^2}{a^3}-\frac{4 x^3}{a^4}}{x^4 \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{4}{3} 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{81}{a^2}-\frac{80 x}{a^3}+\frac{12 x^2}{a^4}}{x^3 \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{27}{8} a c^3 \sqrt{1-\frac{1}{a^2 x^2}} x^2+\frac{4}{3} 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{160}{a^3}-\frac{105 x}{a^4}}{x^2 \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{20}{3} c^3 \sqrt{1-\frac{1}{a^2 x^2}} x-\frac{27}{8} a c^3 \sqrt{1-\frac{1}{a^2 x^2}} x^2+\frac{4}{3} 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 (35 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{20}{3} c^3 \sqrt{1-\frac{1}{a^2 x^2}} x-\frac{27}{8} a c^3 \sqrt{1-\frac{1}{a^2 x^2}} x^2+\frac{4}{3} 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 (35 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{20}{3} c^3 \sqrt{1-\frac{1}{a^2 x^2}} x-\frac{27}{8} a c^3 \sqrt{1-\frac{1}{a^2 x^2}} x^2+\frac{4}{3} 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 (35 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{20}{3} c^3 \sqrt{1-\frac{1}{a^2 x^2}} x-\frac{27}{8} a c^3 \sqrt{1-\frac{1}{a^2 x^2}} x^2+\frac{4}{3} 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{35 c^3 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{8 a}\\ \end{align*}
Mathematica [A] time = 0.188479, size = 72, normalized size = 0.57 \[ \frac{c^3 \left (a x \sqrt{1-\frac{1}{a^2 x^2}} \left (-6 a^3 x^3+32 a^2 x^2-81 a x+160\right )-105 \log \left (a x \left (\sqrt{1-\frac{1}{a^2 x^2}}+1\right )\right )\right )}{24 a} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.132, size = 196, normalized size = 1.5 \begin{align*} -{\frac{ \left ( ax+1 \right ){c}^{3}}{24\,a}\sqrt{{\frac{ax-1}{ax+1}}} \left ( 6\,\sqrt{{a}^{2}} \left ({a}^{2}{x}^{2}-1 \right ) ^{3/2}xa+87\,\sqrt{{a}^{2}}\sqrt{{a}^{2}{x}^{2}-1}xa-32\, \left ( \left ( ax-1 \right ) \left ( ax+1 \right ) \right ) ^{3/2}\sqrt{{a}^{2}}-87\,\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}}{\sqrt{{a}^{2}}}} \right ) a-192\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }+192\,a\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}{\sqrt{{a}^{2}}}} \right ) \right ){\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.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.05021, size = 298, normalized size = 2.35 \begin{align*} -\frac{1}{24} \,{\left (\frac{105 \, c^{3} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac{105 \, c^{3} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right )}{a^{2}} - \frac{2 \,{\left (279 \, c^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{7}{2}} - 511 \, c^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{5}{2}} + 385 \, c^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}} - 105 \, 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.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.59482, size = 270, normalized size = 2.13 \begin{align*} -\frac{105 \, c^{3} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right ) - 105 \, c^{3} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right ) +{\left (6 \, a^{4} c^{3} x^{4} - 26 \, a^{3} c^{3} x^{3} + 49 \, a^{2} c^{3} x^{2} - 79 \, a c^{3} x - 160 \, 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.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} - c^{3} \left (\int 3 a x \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}\, dx + \int - 3 a^{2} x^{2} \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}\, dx + \int a^{3} x^{3} \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}\, dx + \int - \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.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.14968, size = 147, normalized size = 1.16 \begin{align*} \frac{35 \, c^{3} \log \left ({\left | -x{\left | a \right |} + \sqrt{a^{2} x^{2} - 1} \right |}\right ) \mathrm{sgn}\left (a x + 1\right )}{8 \,{\left | a \right |}} + \frac{1}{24} \, \sqrt{a^{2} x^{2} - 1}{\left (\frac{160 \, c^{3} \mathrm{sgn}\left (a x + 1\right )}{a} -{\left (81 \, c^{3} \mathrm{sgn}\left (a x + 1\right ) + 2 \,{\left (3 \, a^{2} c^{3} x \mathrm{sgn}\left (a x + 1\right ) - 16 \, a c^{3} \mathrm{sgn}\left (a x + 1\right )\right )} x\right )} x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]