Optimal. Leaf size=135 \[ c^4 x \sqrt{1-\frac{1}{a^2 x^2}}-\frac{32 c^4 \sqrt{1-\frac{1}{a^2 x^2}}}{3 a}+\frac{5 c^4 \sqrt{1-\frac{1}{a^2 x^2}}}{2 a^2 x}-\frac{c^4 \sqrt{1-\frac{1}{a^2 x^2}}}{3 a^3 x^2}-\frac{5 c^4 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a}-\frac{25 c^4 \csc ^{-1}(a x)}{2 a} \]
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Rubi [A] time = 0.437769, antiderivative size = 135, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {6177, 1807, 1809, 844, 216, 266, 63, 208} \[ c^4 x \sqrt{1-\frac{1}{a^2 x^2}}-\frac{32 c^4 \sqrt{1-\frac{1}{a^2 x^2}}}{3 a}+\frac{5 c^4 \sqrt{1-\frac{1}{a^2 x^2}}}{2 a^2 x}-\frac{c^4 \sqrt{1-\frac{1}{a^2 x^2}}}{3 a^3 x^2}-\frac{5 c^4 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a}-\frac{25 c^4 \csc ^{-1}(a x)}{2 a} \]
Antiderivative was successfully verified.
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Rule 6177
Rule 1807
Rule 1809
Rule 844
Rule 216
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int e^{-\coth ^{-1}(a x)} \left (c-\frac{c}{a x}\right )^4 \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{\left (c-\frac{c x}{a}\right )^5}{x^2 \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{c}\\ &=c^4 \sqrt{1-\frac{1}{a^2 x^2}} x+\frac{\operatorname{Subst}\left (\int \frac{\frac{5 c^5}{a}-\frac{10 c^5 x}{a^2}+\frac{10 c^5 x^2}{a^3}-\frac{5 c^5 x^3}{a^4}+\frac{c^5 x^4}{a^5}}{x \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{c}\\ &=-\frac{c^4 \sqrt{1-\frac{1}{a^2 x^2}}}{3 a^3 x^2}+c^4 \sqrt{1-\frac{1}{a^2 x^2}} x-\frac{a^2 \operatorname{Subst}\left (\int \frac{-\frac{15 c^5}{a^3}+\frac{30 c^5 x}{a^4}-\frac{32 c^5 x^2}{a^5}+\frac{15 c^5 x^3}{a^6}}{x \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{3 c}\\ &=-\frac{c^4 \sqrt{1-\frac{1}{a^2 x^2}}}{3 a^3 x^2}+\frac{5 c^4 \sqrt{1-\frac{1}{a^2 x^2}}}{2 a^2 x}+c^4 \sqrt{1-\frac{1}{a^2 x^2}} x+\frac{a^4 \operatorname{Subst}\left (\int \frac{\frac{30 c^5}{a^5}-\frac{75 c^5 x}{a^6}+\frac{64 c^5 x^2}{a^7}}{x \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{6 c}\\ &=-\frac{32 c^4 \sqrt{1-\frac{1}{a^2 x^2}}}{3 a}-\frac{c^4 \sqrt{1-\frac{1}{a^2 x^2}}}{3 a^3 x^2}+\frac{5 c^4 \sqrt{1-\frac{1}{a^2 x^2}}}{2 a^2 x}+c^4 \sqrt{1-\frac{1}{a^2 x^2}} x-\frac{a^6 \operatorname{Subst}\left (\int \frac{-\frac{30 c^5}{a^7}+\frac{75 c^5 x}{a^8}}{x \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{6 c}\\ &=-\frac{32 c^4 \sqrt{1-\frac{1}{a^2 x^2}}}{3 a}-\frac{c^4 \sqrt{1-\frac{1}{a^2 x^2}}}{3 a^3 x^2}+\frac{5 c^4 \sqrt{1-\frac{1}{a^2 x^2}}}{2 a^2 x}+c^4 \sqrt{1-\frac{1}{a^2 x^2}} x-\frac{\left (25 c^4\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{2 a^2}+\frac{\left (5 c^4\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{a}\\ &=-\frac{32 c^4 \sqrt{1-\frac{1}{a^2 x^2}}}{3 a}-\frac{c^4 \sqrt{1-\frac{1}{a^2 x^2}}}{3 a^3 x^2}+\frac{5 c^4 \sqrt{1-\frac{1}{a^2 x^2}}}{2 a^2 x}+c^4 \sqrt{1-\frac{1}{a^2 x^2}} x-\frac{25 c^4 \csc ^{-1}(a x)}{2 a}+\frac{\left (5 c^4\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x}{a^2}}} \, dx,x,\frac{1}{x^2}\right )}{2 a}\\ &=-\frac{32 c^4 \sqrt{1-\frac{1}{a^2 x^2}}}{3 a}-\frac{c^4 \sqrt{1-\frac{1}{a^2 x^2}}}{3 a^3 x^2}+\frac{5 c^4 \sqrt{1-\frac{1}{a^2 x^2}}}{2 a^2 x}+c^4 \sqrt{1-\frac{1}{a^2 x^2}} x-\frac{25 c^4 \csc ^{-1}(a x)}{2 a}-\left (5 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{32 c^4 \sqrt{1-\frac{1}{a^2 x^2}}}{3 a}-\frac{c^4 \sqrt{1-\frac{1}{a^2 x^2}}}{3 a^3 x^2}+\frac{5 c^4 \sqrt{1-\frac{1}{a^2 x^2}}}{2 a^2 x}+c^4 \sqrt{1-\frac{1}{a^2 x^2}} x-\frac{25 c^4 \csc ^{-1}(a x)}{2 a}-\frac{5 c^4 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a}\\ \end{align*}
Mathematica [A] time = 0.145488, size = 175, normalized size = 1.3 \[ \frac{c^4 \left (6 a^5 x^5-64 a^4 x^4+9 a^3 x^3+62 a^2 x^2+90 a^4 x^4 \sqrt{1-\frac{1}{a^2 x^2}} \sin ^{-1}\left (\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{2}}\right )-30 a^4 x^4 \sqrt{1-\frac{1}{a^2 x^2}} \sin ^{-1}\left (\frac{1}{a x}\right )-30 a^4 x^4 \sqrt{1-\frac{1}{a^2 x^2}} \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )-15 a x+2\right )}{6 a^5 x^4 \sqrt{1-\frac{1}{a^2 x^2}}} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.136, size = 290, normalized size = 2.2 \begin{align*}{\frac{ \left ( ax+1 \right ){c}^{4}}{6\,{a}^{4}{x}^{3}}\sqrt{{\frac{ax-1}{ax+1}}} \left ( -66\,\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}{x}^{4}{a}^{4}+66\, \left ({a}^{2}{x}^{2}-1 \right ) ^{3/2}\sqrt{{a}^{2}}{x}^{2}{a}^{2}-75\,\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}{x}^{3}{a}^{3}+66\,\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}}{\sqrt{{a}^{2}}}} \right ){x}^{3}{a}^{4}-75\,{a}^{3}{x}^{3}\sqrt{{a}^{2}}\arctan \left ({\frac{1}{\sqrt{{a}^{2}{x}^{2}-1}}} \right ) +96\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }{x}^{3}{a}^{3}-96\,\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}{\sqrt{{a}^{2}}}} \right ){x}^{3}{a}^{4}-15\,\sqrt{{a}^{2}} \left ({a}^{2}{x}^{2}-1 \right ) ^{3/2}xa+2\, \left ({a}^{2}{x}^{2}-1 \right ) ^{3/2}\sqrt{{a}^{2}} \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.
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Maxima [A] time = 1.58953, size = 301, normalized size = 2.23 \begin{align*} \frac{1}{3} \,{\left (\frac{75 \, c^{4} \arctan \left (\sqrt{\frac{a x - 1}{a x + 1}}\right )}{a^{2}} - \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{87 \, c^{4} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{7}{2}} + 61 \, c^{4} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{5}{2}} - 55 \, c^{4} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}} - 45 \, c^{4} \sqrt{\frac{a x - 1}{a x + 1}}}{\frac{2 \,{\left (a x - 1\right )} a^{2}}{a x + 1} - \frac{2 \,{\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 = 1.82614, size = 365, normalized size = 2.7 \begin{align*} \frac{150 \, a^{3} c^{4} x^{3} \arctan \left (\sqrt{\frac{a x - 1}{a x + 1}}\right ) - 30 \, a^{3} c^{4} x^{3} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right ) + 30 \, a^{3} c^{4} x^{3} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right ) +{\left (6 \, a^{4} c^{4} x^{4} - 58 \, a^{3} c^{4} x^{3} - 49 \, a^{2} c^{4} x^{2} + 13 \, a c^{4} x - 2 \, c^{4}\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{6 \, a^{4} x^{3}} \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*} \frac{c^{4} \left (\int a^{4} \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}\, dx + \int \frac{\sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}{x^{4}}\, dx + \int - \frac{4 a \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}{x^{3}}\, dx + \int \frac{6 a^{2} \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}{x^{2}}\, dx + \int - \frac{4 a^{3} \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}{x}\, dx\right )}{a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.21072, size = 358, normalized size = 2.65 \begin{align*} \frac{25 \, c^{4} \arctan \left (-x{\left | a \right |} + \sqrt{a^{2} x^{2} - 1}\right ) \mathrm{sgn}\left (a x + 1\right )}{a} + \frac{5 \, c^{4} \log \left ({\left | -x{\left | a \right |} + \sqrt{a^{2} x^{2} - 1} \right |}\right ) \mathrm{sgn}\left (a x + 1\right )}{{\left | a \right |}} + \frac{\sqrt{a^{2} x^{2} - 1} c^{4} \mathrm{sgn}\left (a x + 1\right )}{a} - \frac{15 \,{\left (x{\left | a \right |} - \sqrt{a^{2} x^{2} - 1}\right )}^{5} c^{4}{\left | a \right |} \mathrm{sgn}\left (a x + 1\right ) + 60 \,{\left (x{\left | a \right |} - \sqrt{a^{2} x^{2} - 1}\right )}^{4} a c^{4} \mathrm{sgn}\left (a x + 1\right ) + 132 \,{\left (x{\left | a \right |} - \sqrt{a^{2} x^{2} - 1}\right )}^{2} a c^{4} \mathrm{sgn}\left (a x + 1\right ) - 15 \,{\left (x{\left | a \right |} - \sqrt{a^{2} x^{2} - 1}\right )} c^{4}{\left | a \right |} \mathrm{sgn}\left (a x + 1\right ) + 64 \, a c^{4} \mathrm{sgn}\left (a x + 1\right )}{3 \,{\left ({\left (x{\left | a \right |} - \sqrt{a^{2} x^{2} - 1}\right )}^{2} + 1\right )}^{3} a{\left | a \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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