Optimal. Leaf size=125 \[ \frac{2 a^3 (a x+1)}{c \sqrt{1-a^2 x^2}}-\frac{8 a^2 \sqrt{1-a^2 x^2}}{3 c x}-\frac{a \sqrt{1-a^2 x^2}}{c x^2}-\frac{\sqrt{1-a^2 x^2}}{3 c x^3}-\frac{3 a^3 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )}{c} \]
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Rubi [A] time = 0.329187, antiderivative size = 125, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.421, Rules used = {6128, 852, 1805, 1807, 807, 266, 63, 208} \[ \frac{2 a^3 (a x+1)}{c \sqrt{1-a^2 x^2}}-\frac{8 a^2 \sqrt{1-a^2 x^2}}{3 c x}-\frac{a \sqrt{1-a^2 x^2}}{c x^2}-\frac{\sqrt{1-a^2 x^2}}{3 c x^3}-\frac{3 a^3 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )}{c} \]
Antiderivative was successfully verified.
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Rule 6128
Rule 852
Rule 1805
Rule 1807
Rule 807
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{e^{\tanh ^{-1}(a x)}}{x^4 (c-a c x)} \, dx &=c \int \frac{\sqrt{1-a^2 x^2}}{x^4 (c-a c x)^2} \, dx\\ &=\frac{\int \frac{(c+a c x)^2}{x^4 \left (1-a^2 x^2\right )^{3/2}} \, dx}{c^3}\\ &=\frac{2 a^3 (1+a x)}{c \sqrt{1-a^2 x^2}}-\frac{\int \frac{-c^2-2 a c^2 x-2 a^2 c^2 x^2-2 a^3 c^2 x^3}{x^4 \sqrt{1-a^2 x^2}} \, dx}{c^3}\\ &=\frac{2 a^3 (1+a x)}{c \sqrt{1-a^2 x^2}}-\frac{\sqrt{1-a^2 x^2}}{3 c x^3}+\frac{\int \frac{6 a c^2+8 a^2 c^2 x+6 a^3 c^2 x^2}{x^3 \sqrt{1-a^2 x^2}} \, dx}{3 c^3}\\ &=\frac{2 a^3 (1+a x)}{c \sqrt{1-a^2 x^2}}-\frac{\sqrt{1-a^2 x^2}}{3 c x^3}-\frac{a \sqrt{1-a^2 x^2}}{c x^2}-\frac{\int \frac{-16 a^2 c^2-18 a^3 c^2 x}{x^2 \sqrt{1-a^2 x^2}} \, dx}{6 c^3}\\ &=\frac{2 a^3 (1+a x)}{c \sqrt{1-a^2 x^2}}-\frac{\sqrt{1-a^2 x^2}}{3 c x^3}-\frac{a \sqrt{1-a^2 x^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}{c}\\ &=\frac{2 a^3 (1+a x)}{c \sqrt{1-a^2 x^2}}-\frac{\sqrt{1-a^2 x^2}}{3 c x^3}-\frac{a \sqrt{1-a^2 x^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 )}{2 c}\\ &=\frac{2 a^3 (1+a x)}{c \sqrt{1-a^2 x^2}}-\frac{\sqrt{1-a^2 x^2}}{3 c x^3}-\frac{a \sqrt{1-a^2 x^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 )}{c}\\ &=\frac{2 a^3 (1+a x)}{c \sqrt{1-a^2 x^2}}-\frac{\sqrt{1-a^2 x^2}}{3 c x^3}-\frac{a \sqrt{1-a^2 x^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 )}{c}\\ \end{align*}
Mathematica [A] time = 0.0410326, size = 91, normalized size = 0.73 \[ -\frac{-14 a^4 x^4-9 a^3 x^3+7 a^2 x^2+9 a^3 x^3 \sqrt{1-a^2 x^2} \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )+3 a x+1}{3 c x^3 \sqrt{1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 142, normalized size = 1.1 \begin{align*} -{\frac{1}{c} \left ({\frac{8\,{a}^{2}}{3\,x}\sqrt{-{a}^{2}{x}^{2}+1}}+2\,{a}^{3}{\it Artanh} \left ({\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) +2\,{{a}^{2}\sqrt{-{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,a \left ( x-{a}^{-1} \right ) } \left ( x-{a}^{-1} \right ) ^{-1}}-2\,a \left ( -1/2\,{\frac{\sqrt{-{a}^{2}{x}^{2}+1}}{{x}^{2}}}-1/2\,{a}^{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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{a x + 1}{\sqrt{-a^{2} x^{2} + 1}{\left (a c x - c\right )} x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.57952, size = 217, normalized size = 1.74 \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 (14 \, a^{3} x^{3} - 5 \, a^{2} x^{2} - 2 \, a x - 1\right )} \sqrt{-a^{2} x^{2} + 1}}{3 \,{\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 x}{a x^{5} \sqrt{- a^{2} x^{2} + 1} - x^{4} \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{1}{a x^{5} \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 = 1.26543, size = 382, normalized size = 3.06 \begin{align*} -\frac{{\left (a^{4} + \frac{5 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )} a^{2}}{x} + \frac{27 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{2}}{x^{2}} - \frac{129 \,{\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 )}{c{\left | a \right |}} - \frac{\frac{33 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )} a^{4} c^{2}}{x} + \frac{6 \,{\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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