Optimal. Leaf size=132 \[ \frac{1}{5} a^4 c^4 x^5 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}-\frac{3}{4} a^3 c^4 x^4 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}+\frac{17}{15} a^2 c^4 x^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}-\frac{7}{8} a c^4 x^2 \sqrt{1-\frac{1}{a^2 x^2}}+\frac{7 c^4 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{8 a} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.30285, antiderivative size = 132, normalized size of antiderivative = 1., number of steps used = 9, 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} \[ \frac{1}{5} a^4 c^4 x^5 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}-\frac{3}{4} a^3 c^4 x^4 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}+\frac{17}{15} a^2 c^4 x^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}-\frac{7}{8} a c^4 x^2 \sqrt{1-\frac{1}{a^2 x^2}}+\frac{7 c^4 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{8 a} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6175
Rule 6178
Rule 1807
Rule 807
Rule 266
Rule 47
Rule 63
Rule 208
Rubi steps
\begin{align*} \int e^{\coth ^{-1}(a x)} (c-a c x)^4 \, dx &=\left (a^4 c^4\right ) \int e^{\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 )^3 \sqrt{1-\frac{x^2}{a^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 )^{3/2} x^5+\frac{1}{5} \left (a^4 c^4\right ) \operatorname{Subst}\left (\int \frac{\left (\frac{15}{a}-\frac{17 x}{a^2}+\frac{5 x^2}{a^3}\right ) \sqrt{1-\frac{x^2}{a^2}}}{x^5} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{3}{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 )^{3/2} x^5-\frac{1}{20} \left (a^4 c^4\right ) \operatorname{Subst}\left (\int \frac{\left (\frac{68}{a^2}-\frac{35 x}{a^3}\right ) \sqrt{1-\frac{x^2}{a^2}}}{x^4} \, dx,x,\frac{1}{x}\right )\\ &=\frac{17}{15} a^2 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} x^3-\frac{3}{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 )^{3/2} x^5+\frac{1}{4} \left (7 a c^4\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1-\frac{x^2}{a^2}}}{x^3} \, dx,x,\frac{1}{x}\right )\\ &=\frac{17}{15} a^2 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} x^3-\frac{3}{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 )^{3/2} x^5+\frac{1}{8} \left (7 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{7}{8} a c^4 \sqrt{1-\frac{1}{a^2 x^2}} x^2+\frac{17}{15} a^2 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} x^3-\frac{3}{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 )^{3/2} x^5-\frac{\left (7 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{7}{8} a c^4 \sqrt{1-\frac{1}{a^2 x^2}} x^2+\frac{17}{15} a^2 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} x^3-\frac{3}{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 )^{3/2} x^5+\frac{1}{8} \left (7 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{7}{8} a c^4 \sqrt{1-\frac{1}{a^2 x^2}} x^2+\frac{17}{15} a^2 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} x^3-\frac{3}{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 )^{3/2} x^5+\frac{7 c^4 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{8 a}\\ \end{align*}
Mathematica [A] time = 0.184801, size = 80, normalized size = 0.61 \[ \frac{c^4 \left (a x \sqrt{1-\frac{1}{a^2 x^2}} \left (24 a^4 x^4-90 a^3 x^3+112 a^2 x^2-15 a x-136\right )+105 \log \left (a x \left (\sqrt{1-\frac{1}{a^2 x^2}}+1\right )\right )\right )}{120 a} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.14, size = 183, normalized size = 1.4 \begin{align*}{\frac{ \left ( ax-1 \right ){c}^{4}}{120\,a} \left ( 24\, \left ({a}^{2}{x}^{2}-1 \right ) ^{3/2}\sqrt{{a}^{2}}{x}^{2}{a}^{2}-90\,\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}}-105\,\sqrt{{a}^{2}}\sqrt{{a}^{2}{x}^{2}-1}xa+120\, \left ( \left ( ax-1 \right ) \left ( ax+1 \right ) \right ) ^{3/2}\sqrt{{a}^{2}}+105\,\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.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 0.998874, size = 350, normalized size = 2.65 \begin{align*} \frac{1}{120} \,{\left (\frac{105 \, c^{4} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac{105 \, c^{4} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right )}{a^{2}} - \frac{2 \,{\left (105 \, c^{4} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{9}{2}} + 790 \, c^{4} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{7}{2}} - 896 \, c^{4} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{5}{2}} + 490 \, c^{4} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}} - 105 \, 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.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.64434, size = 296, normalized size = 2.24 \begin{align*} \frac{105 \, c^{4} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right ) - 105 \, c^{4} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right ) +{\left (24 \, a^{5} c^{4} x^{5} - 66 \, a^{4} c^{4} x^{4} + 22 \, a^{3} c^{4} x^{3} + 97 \, a^{2} c^{4} x^{2} - 151 \, a c^{4} x - 136 \, c^{4}\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{120 \, 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^{4} \left (\int - \frac{4 a x}{\sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}\, dx + \int \frac{6 a^{2} x^{2}}{\sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}\, dx + \int - \frac{4 a^{3} x^{3}}{\sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}\, dx + \int \frac{a^{4} x^{4}}{\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.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.22029, size = 316, normalized size = 2.39 \begin{align*} \frac{1}{120} \,{\left (\frac{105 \, c^{4} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac{105 \, c^{4} \log \left ({\left | \sqrt{\frac{a x - 1}{a x + 1}} - 1 \right |}\right )}{a^{2}} - \frac{2 \,{\left (\frac{490 \,{\left (a x - 1\right )} c^{4} \sqrt{\frac{a x - 1}{a x + 1}}}{a x + 1} - \frac{896 \,{\left (a x - 1\right )}^{2} c^{4} \sqrt{\frac{a x - 1}{a x + 1}}}{{\left (a x + 1\right )}^{2}} + \frac{790 \,{\left (a x - 1\right )}^{3} c^{4} \sqrt{\frac{a x - 1}{a x + 1}}}{{\left (a x + 1\right )}^{3}} + \frac{105 \,{\left (a x - 1\right )}^{4} c^{4} \sqrt{\frac{a x - 1}{a x + 1}}}{{\left (a x + 1\right )}^{4}} - 105 \, 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.
[In]
[Out]