Optimal. Leaf size=128 \[ -\frac{8 a^2 \sqrt{1-a^2 x^2}}{3 c x}-\frac{3 a \sqrt{1-a^2 x^2}}{2 c x^2}-\frac{4 \sqrt{1-a^2 x^2}}{3 c x^3}+\frac{a x+1}{c x^3 \sqrt{1-a^2 x^2}}-\frac{3 a^3 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )}{2 c} \]
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Rubi [A] time = 0.166154, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.304, Rules used = {6148, 823, 835, 807, 266, 63, 208} \[ -\frac{8 a^2 \sqrt{1-a^2 x^2}}{3 c x}-\frac{3 a \sqrt{1-a^2 x^2}}{2 c x^2}-\frac{4 \sqrt{1-a^2 x^2}}{3 c x^3}+\frac{a x+1}{c x^3 \sqrt{1-a^2 x^2}}-\frac{3 a^3 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )}{2 c} \]
Antiderivative was successfully verified.
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Rule 6148
Rule 823
Rule 835
Rule 807
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{e^{\tanh ^{-1}(a x)}}{x^4 \left (c-a^2 c x^2\right )} \, dx &=\frac{\int \frac{1+a x}{x^4 \left (1-a^2 x^2\right )^{3/2}} \, dx}{c}\\ &=\frac{1+a x}{c x^3 \sqrt{1-a^2 x^2}}+\frac{\int \frac{4 a^2+3 a^3 x}{x^4 \sqrt{1-a^2 x^2}} \, dx}{a^2 c}\\ &=\frac{1+a x}{c x^3 \sqrt{1-a^2 x^2}}-\frac{4 \sqrt{1-a^2 x^2}}{3 c x^3}-\frac{\int \frac{-9 a^3-8 a^4 x}{x^3 \sqrt{1-a^2 x^2}} \, dx}{3 a^2 c}\\ &=\frac{1+a x}{c x^3 \sqrt{1-a^2 x^2}}-\frac{4 \sqrt{1-a^2 x^2}}{3 c x^3}-\frac{3 a \sqrt{1-a^2 x^2}}{2 c x^2}+\frac{\int \frac{16 a^4+9 a^5 x}{x^2 \sqrt{1-a^2 x^2}} \, dx}{6 a^2 c}\\ &=\frac{1+a x}{c x^3 \sqrt{1-a^2 x^2}}-\frac{4 \sqrt{1-a^2 x^2}}{3 c x^3}-\frac{3 a \sqrt{1-a^2 x^2}}{2 c x^2}-\frac{8 a^2 \sqrt{1-a^2 x^2}}{3 c x}+\frac{\left (3 a^3\right ) \int \frac{1}{x \sqrt{1-a^2 x^2}} \, dx}{2 c}\\ &=\frac{1+a x}{c x^3 \sqrt{1-a^2 x^2}}-\frac{4 \sqrt{1-a^2 x^2}}{3 c x^3}-\frac{3 a \sqrt{1-a^2 x^2}}{2 c x^2}-\frac{8 a^2 \sqrt{1-a^2 x^2}}{3 c x}+\frac{\left (3 a^3\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-a^2 x}} \, dx,x,x^2\right )}{4 c}\\ &=\frac{1+a x}{c x^3 \sqrt{1-a^2 x^2}}-\frac{4 \sqrt{1-a^2 x^2}}{3 c x^3}-\frac{3 a \sqrt{1-a^2 x^2}}{2 c x^2}-\frac{8 a^2 \sqrt{1-a^2 x^2}}{3 c x}-\frac{(3 a) \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 c}\\ &=\frac{1+a x}{c x^3 \sqrt{1-a^2 x^2}}-\frac{4 \sqrt{1-a^2 x^2}}{3 c x^3}-\frac{3 a \sqrt{1-a^2 x^2}}{2 c x^2}-\frac{8 a^2 \sqrt{1-a^2 x^2}}{3 c x}-\frac{3 a^3 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )}{2 c}\\ \end{align*}
Mathematica [A] time = 0.0366877, size = 91, normalized size = 0.71 \[ -\frac{-16 a^4 x^4-9 a^3 x^3+8 a^2 x^2+9 a^3 x^3 \sqrt{1-a^2 x^2} \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )+3 a x+2}{6 c x^3 \sqrt{1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 140, normalized size = 1.1 \begin{align*} -{\frac{1}{c} \left ({\frac{5\,{a}^{2}}{3\,x}\sqrt{-{a}^{2}{x}^{2}+1}}+{a}^{3}{\it Artanh} \left ({\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) +{{a}^{2}\sqrt{-{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,a \left ( x-{a}^{-1} \right ) } \left ( x-{a}^{-1} \right ) ^{-1}}-a \left ( -{\frac{1}{2\,{x}^{2}}\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{{a}^{2}}{2}{\it Artanh} \left ({\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) } \right ) +{\frac{1}{3\,{x}^{3}}\sqrt{-{a}^{2}{x}^{2}+1}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01802, size = 200, normalized size = 1.56 \begin{align*} -\frac{\frac{3 \, a^{4} \log \left (\sqrt{-a^{2} x^{2} + 1} + 1\right )}{c} - \frac{3 \, a^{4} \log \left (\sqrt{-a^{2} x^{2} + 1} - 1\right )}{c} + \frac{2 \,{\left (3 \,{\left (a^{2} x^{2} - 1\right )} a^{4} + 2 \, a^{4}\right )}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} c - \sqrt{-a^{2} x^{2} + 1} c}}{4 \, a} + \frac{8 \, a^{4} x^{4} - 4 \, a^{2} x^{2} - 1}{3 \, \sqrt{a x + 1} \sqrt{-a x + 1} c x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.54098, size = 215, normalized size = 1.68 \begin{align*} \frac{6 \, a^{4} x^{4} - 6 \, a^{3} x^{3} + 9 \,{\left (a^{4} x^{4} - a^{3} x^{3}\right )} \log \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right ) -{\left (16 \, a^{3} x^{3} - 7 \, a^{2} x^{2} - a x - 2\right )} \sqrt{-a^{2} x^{2} + 1}}{6 \,{\left (a c x^{4} - c x^{3}\right )}} \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{\int \frac{a}{- a^{2} x^{5} \sqrt{- a^{2} x^{2} + 1} + x^{3} \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{1}{- a^{2} x^{6} \sqrt{- a^{2} x^{2} + 1} + x^{4} \sqrt{- a^{2} x^{2} + 1}}\, dx}{c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.18719, size = 382, normalized size = 2.98 \begin{align*} -\frac{{\left (a^{4} + \frac{2 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )} a^{2}}{x} + \frac{18 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{2}}{x^{2}} - \frac{69 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{3}}{a^{2} x^{3}}\right )} a^{6} x^{3}}{24 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{3} c{\left (\frac{\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a}{a^{2} x} - 1\right )}{\left | a \right |}} - \frac{3 \, a^{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 \, c{\left | a \right |}} - \frac{\frac{21 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )} a^{4} c^{2}}{x} + \frac{3 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{2} a^{2} c^{2}}{x^{2}} + \frac{{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{3} c^{2}}{x^{3}}}{24 \, a^{2} c^{3}{\left | a \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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