Optimal. Leaf size=140 \[ \frac{c^4 \sqrt{1-a^2 x^2}}{a}-\frac{32 c^4 \sqrt{1-a^2 x^2}}{3 a^2 x}+\frac{5 c^4 \sqrt{1-a^2 x^2}}{2 a^3 x^2}-\frac{c^4 \sqrt{1-a^2 x^2}}{3 a^4 x^3}+\frac{25 c^4 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )}{2 a}+\frac{5 c^4 \sin ^{-1}(a x)}{a} \]
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Rubi [A] time = 0.419824, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 9, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.409, Rules used = {6131, 6128, 1807, 1809, 844, 216, 266, 63, 208} \[ \frac{c^4 \sqrt{1-a^2 x^2}}{a}-\frac{32 c^4 \sqrt{1-a^2 x^2}}{3 a^2 x}+\frac{5 c^4 \sqrt{1-a^2 x^2}}{2 a^3 x^2}-\frac{c^4 \sqrt{1-a^2 x^2}}{3 a^4 x^3}+\frac{25 c^4 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )}{2 a}+\frac{5 c^4 \sin ^{-1}(a x)}{a} \]
Antiderivative was successfully verified.
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Rule 6131
Rule 6128
Rule 1807
Rule 1809
Rule 844
Rule 216
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int e^{-\tanh ^{-1}(a x)} \left (c-\frac{c}{a x}\right )^4 \, dx &=\frac{c^4 \int \frac{e^{-\tanh ^{-1}(a x)} (1-a x)^4}{x^4} \, dx}{a^4}\\ &=\frac{c^4 \int \frac{(1-a x)^5}{x^4 \sqrt{1-a^2 x^2}} \, dx}{a^4}\\ &=-\frac{c^4 \sqrt{1-a^2 x^2}}{3 a^4 x^3}-\frac{c^4 \int \frac{15 a-32 a^2 x+30 a^3 x^2-15 a^4 x^3+3 a^5 x^4}{x^3 \sqrt{1-a^2 x^2}} \, dx}{3 a^4}\\ &=-\frac{c^4 \sqrt{1-a^2 x^2}}{3 a^4 x^3}+\frac{5 c^4 \sqrt{1-a^2 x^2}}{2 a^3 x^2}+\frac{c^4 \int \frac{64 a^2-75 a^3 x+30 a^4 x^2-6 a^5 x^3}{x^2 \sqrt{1-a^2 x^2}} \, dx}{6 a^4}\\ &=-\frac{c^4 \sqrt{1-a^2 x^2}}{3 a^4 x^3}+\frac{5 c^4 \sqrt{1-a^2 x^2}}{2 a^3 x^2}-\frac{32 c^4 \sqrt{1-a^2 x^2}}{3 a^2 x}-\frac{c^4 \int \frac{75 a^3-30 a^4 x+6 a^5 x^2}{x \sqrt{1-a^2 x^2}} \, dx}{6 a^4}\\ &=\frac{c^4 \sqrt{1-a^2 x^2}}{a}-\frac{c^4 \sqrt{1-a^2 x^2}}{3 a^4 x^3}+\frac{5 c^4 \sqrt{1-a^2 x^2}}{2 a^3 x^2}-\frac{32 c^4 \sqrt{1-a^2 x^2}}{3 a^2 x}+\frac{c^4 \int \frac{-75 a^5+30 a^6 x}{x \sqrt{1-a^2 x^2}} \, dx}{6 a^6}\\ &=\frac{c^4 \sqrt{1-a^2 x^2}}{a}-\frac{c^4 \sqrt{1-a^2 x^2}}{3 a^4 x^3}+\frac{5 c^4 \sqrt{1-a^2 x^2}}{2 a^3 x^2}-\frac{32 c^4 \sqrt{1-a^2 x^2}}{3 a^2 x}+\left (5 c^4\right ) \int \frac{1}{\sqrt{1-a^2 x^2}} \, dx-\frac{\left (25 c^4\right ) \int \frac{1}{x \sqrt{1-a^2 x^2}} \, dx}{2 a}\\ &=\frac{c^4 \sqrt{1-a^2 x^2}}{a}-\frac{c^4 \sqrt{1-a^2 x^2}}{3 a^4 x^3}+\frac{5 c^4 \sqrt{1-a^2 x^2}}{2 a^3 x^2}-\frac{32 c^4 \sqrt{1-a^2 x^2}}{3 a^2 x}+\frac{5 c^4 \sin ^{-1}(a x)}{a}-\frac{\left (25 c^4\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-a^2 x}} \, dx,x,x^2\right )}{4 a}\\ &=\frac{c^4 \sqrt{1-a^2 x^2}}{a}-\frac{c^4 \sqrt{1-a^2 x^2}}{3 a^4 x^3}+\frac{5 c^4 \sqrt{1-a^2 x^2}}{2 a^3 x^2}-\frac{32 c^4 \sqrt{1-a^2 x^2}}{3 a^2 x}+\frac{5 c^4 \sin ^{-1}(a x)}{a}+\frac{\left (25 c^4\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{1}{a^2}-\frac{x^2}{a^2}} \, dx,x,\sqrt{1-a^2 x^2}\right )}{2 a^3}\\ &=\frac{c^4 \sqrt{1-a^2 x^2}}{a}-\frac{c^4 \sqrt{1-a^2 x^2}}{3 a^4 x^3}+\frac{5 c^4 \sqrt{1-a^2 x^2}}{2 a^3 x^2}-\frac{32 c^4 \sqrt{1-a^2 x^2}}{3 a^2 x}+\frac{5 c^4 \sin ^{-1}(a x)}{a}+\frac{25 c^4 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )}{2 a}\\ \end{align*}
Mathematica [A] time = 0.239494, size = 85, normalized size = 0.61 \[ \frac{c^4 \left (\frac{\sqrt{1-a^2 x^2} \left (6 a^3 x^3-64 a^2 x^2+15 a x-2\right )}{a^3 x^3}+75 \log \left (\sqrt{1-a^2 x^2}+1\right )-75 \log (a x)+30 \sin ^{-1}(a x)\right )}{6 a} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.052, size = 232, normalized size = 1.7 \begin{align*} -11\,{\frac{{c}^{4} \left ( -{a}^{2}{x}^{2}+1 \right ) ^{3/2}}{{a}^{2}x}}-11\,{c}^{4}x\sqrt{-{a}^{2}{x}^{2}+1}-11\,{\frac{{c}^{4}}{\sqrt{{a}^{2}}}\arctan \left ({\frac{\sqrt{{a}^{2}}x}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) }+{\frac{25\,{c}^{4}}{2\,a}{\it Artanh} \left ({\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) }-{\frac{25\,{c}^{4}}{2\,a}\sqrt{-{a}^{2}{x}^{2}+1}}+16\,{\frac{{c}^{4}\sqrt{-{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,a \left ( x+{a}^{-1} \right ) }}{a}}+16\,{\frac{{c}^{4}}{\sqrt{{a}^{2}}}\arctan \left ({\frac{\sqrt{{a}^{2}}x}{\sqrt{-{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,a \left ( x+{a}^{-1} \right ) }}} \right ) }+{\frac{5\,{c}^{4}}{2\,{x}^{2}{a}^{3}} \left ( -{a}^{2}{x}^{2}+1 \right ) ^{{\frac{3}{2}}}}-{\frac{{c}^{4}}{3\,{a}^{4}{x}^{3}} \left ( -{a}^{2}{x}^{2}+1 \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.13532, size = 286, normalized size = 2.04 \begin{align*} -\frac{60 \, a^{3} c^{4} x^{3} \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right ) + 75 \, a^{3} c^{4} x^{3} \log \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right ) - 6 \, a^{3} c^{4} x^{3} -{\left (6 \, a^{3} c^{4} x^{3} - 64 \, a^{2} c^{4} x^{2} + 15 \, a c^{4} x - 2 \, c^{4}\right )} \sqrt{-a^{2} x^{2} + 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 \frac{\sqrt{- a^{2} x^{2} + 1}}{a x^{5} + x^{4}}\, dx + \int - \frac{4 a x \sqrt{- a^{2} x^{2} + 1}}{a x^{5} + x^{4}}\, dx + \int \frac{6 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1}}{a x^{5} + x^{4}}\, dx + \int - \frac{4 a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1}}{a x^{5} + x^{4}}\, dx + \int \frac{a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1}}{a x^{5} + x^{4}}\, dx\right )}{a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.18686, size = 354, normalized size = 2.53 \begin{align*} \frac{{\left (c^{4} - \frac{15 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )} c^{4}}{a^{2} x} + \frac{129 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{2} c^{4}}{a^{4} x^{2}}\right )} a^{6} x^{3}}{24 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{3}{\left | a \right |}} + \frac{5 \, c^{4} \arcsin \left (a x\right ) \mathrm{sgn}\left (a\right )}{{\left | a \right |}} + \frac{25 \, c^{4} \log \left (\frac{{\left | -2 \, \sqrt{-a^{2} x^{2} + 1}{\left | a \right |} - 2 \, a \right |}}{2 \, a^{2}{\left | x \right |}}\right )}{2 \,{\left | a \right |}} + \frac{\sqrt{-a^{2} x^{2} + 1} c^{4}}{a} - \frac{\frac{129 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )} c^{4}}{x} - \frac{15 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{2} c^{4}}{a^{2} x^{2}} + \frac{{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{3} c^{4}}{a^{4} x^{3}}}{24 \, a^{2}{\left | a \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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