Optimal. Leaf size=99 \[ \frac{4 a (a x+1)}{3 c^2 \left (1-a^2 x^2\right )^{3/2}}-\frac{\sqrt{1-a^2 x^2}}{c^2 x}+\frac{a (11 a x+9)}{3 c^2 \sqrt{1-a^2 x^2}}-\frac{3 a \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )}{c^2} \]
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Rubi [A] time = 0.266828, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.368, Rules used = {6128, 852, 1805, 807, 266, 63, 208} \[ \frac{4 a (a x+1)}{3 c^2 \left (1-a^2 x^2\right )^{3/2}}-\frac{\sqrt{1-a^2 x^2}}{c^2 x}+\frac{a (11 a x+9)}{3 c^2 \sqrt{1-a^2 x^2}}-\frac{3 a \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )}{c^2} \]
Antiderivative was successfully verified.
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Rule 6128
Rule 852
Rule 1805
Rule 807
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{e^{\tanh ^{-1}(a x)}}{x^2 (c-a c x)^2} \, dx &=c \int \frac{\sqrt{1-a^2 x^2}}{x^2 (c-a c x)^3} \, dx\\ &=\frac{\int \frac{(c+a c x)^3}{x^2 \left (1-a^2 x^2\right )^{5/2}} \, dx}{c^5}\\ &=\frac{4 a (1+a x)}{3 c^2 \left (1-a^2 x^2\right )^{3/2}}-\frac{\int \frac{-3 c^3-9 a c^3 x-8 a^2 c^3 x^2}{x^2 \left (1-a^2 x^2\right )^{3/2}} \, dx}{3 c^5}\\ &=\frac{4 a (1+a x)}{3 c^2 \left (1-a^2 x^2\right )^{3/2}}+\frac{a (9+11 a x)}{3 c^2 \sqrt{1-a^2 x^2}}+\frac{\int \frac{3 c^3+9 a c^3 x}{x^2 \sqrt{1-a^2 x^2}} \, dx}{3 c^5}\\ &=\frac{4 a (1+a x)}{3 c^2 \left (1-a^2 x^2\right )^{3/2}}+\frac{a (9+11 a x)}{3 c^2 \sqrt{1-a^2 x^2}}-\frac{\sqrt{1-a^2 x^2}}{c^2 x}+\frac{(3 a) \int \frac{1}{x \sqrt{1-a^2 x^2}} \, dx}{c^2}\\ &=\frac{4 a (1+a x)}{3 c^2 \left (1-a^2 x^2\right )^{3/2}}+\frac{a (9+11 a x)}{3 c^2 \sqrt{1-a^2 x^2}}-\frac{\sqrt{1-a^2 x^2}}{c^2 x}+\frac{(3 a) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-a^2 x}} \, dx,x,x^2\right )}{2 c^2}\\ &=\frac{4 a (1+a x)}{3 c^2 \left (1-a^2 x^2\right )^{3/2}}+\frac{a (9+11 a x)}{3 c^2 \sqrt{1-a^2 x^2}}-\frac{\sqrt{1-a^2 x^2}}{c^2 x}-\frac{3 \operatorname{Subst}\left (\int \frac{1}{\frac{1}{a^2}-\frac{x^2}{a^2}} \, dx,x,\sqrt{1-a^2 x^2}\right )}{a c^2}\\ &=\frac{4 a (1+a x)}{3 c^2 \left (1-a^2 x^2\right )^{3/2}}+\frac{a (9+11 a x)}{3 c^2 \sqrt{1-a^2 x^2}}-\frac{\sqrt{1-a^2 x^2}}{c^2 x}-\frac{3 a \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )}{c^2}\\ \end{align*}
Mathematica [A] time = 0.0415937, size = 91, normalized size = 0.92 \[ \frac{14 a^3 x^3-5 a^2 x^2-9 a x (a x-1) \sqrt{1-a^2 x^2} \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )-16 a x+3}{3 c^2 x (a x-1) \sqrt{1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.045, size = 118, normalized size = 1.2 \begin{align*}{\frac{1}{{c}^{2}} \left ( -{\frac{1}{x}\sqrt{-{a}^{2}{x}^{2}+1}}-3\,a{\it Artanh} \left ({\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) +{\frac{2}{3\,a}\sqrt{-{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,a \left ( x-{a}^{-1} \right ) } \left ( x-{a}^{-1} \right ) ^{-2}}-{\frac{11}{3}\sqrt{-{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,a \left ( x-{a}^{-1} \right ) } \left ( x-{a}^{-1} \right ) ^{-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 )}^{2} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.60348, size = 252, normalized size = 2.55 \begin{align*} \frac{13 \, a^{3} x^{3} - 26 \, a^{2} x^{2} + 13 \, a x + 9 \,{\left (a^{3} x^{3} - 2 \, a^{2} x^{2} + a x\right )} \log \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right ) -{\left (14 \, a^{2} x^{2} - 19 \, a x + 3\right )} \sqrt{-a^{2} x^{2} + 1}}{3 \,{\left (a^{2} c^{2} x^{3} - 2 \, a c^{2} x^{2} + c^{2} x\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^{2} x^{4} \sqrt{- a^{2} x^{2} + 1} - 2 a x^{3} \sqrt{- a^{2} x^{2} + 1} + x^{2} \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{1}{a^{2} x^{4} \sqrt{- a^{2} x^{2} + 1} - 2 a x^{3} \sqrt{- a^{2} x^{2} + 1} + x^{2} \sqrt{- a^{2} x^{2} + 1}}\, dx}{c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [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 )}^{2} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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