Optimal. Leaf size=104 \[ \frac{\left (1-a^2 x^2\right )^{3/2}}{a c^2 (1-a x)^2}+\frac{\left (1-a^2 x^2\right )^{3/2}}{3 a c^2 (1-a x)^3}-\frac{6 \sqrt{1-a^2 x^2}}{a c^2 (1-a x)}+\frac{3 \sin ^{-1}(a x)}{a c^2} \]
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Rubi [A] time = 0.185402, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {6131, 6128, 1639, 793, 663, 216} \[ \frac{\left (1-a^2 x^2\right )^{3/2}}{a c^2 (1-a x)^2}+\frac{\left (1-a^2 x^2\right )^{3/2}}{3 a c^2 (1-a x)^3}-\frac{6 \sqrt{1-a^2 x^2}}{a c^2 (1-a x)}+\frac{3 \sin ^{-1}(a x)}{a c^2} \]
Antiderivative was successfully verified.
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Rule 6131
Rule 6128
Rule 1639
Rule 793
Rule 663
Rule 216
Rubi steps
\begin{align*} \int \frac{e^{\tanh ^{-1}(a x)}}{\left (c-\frac{c}{a x}\right )^2} \, dx &=\frac{a^2 \int \frac{e^{\tanh ^{-1}(a x)} x^2}{(1-a x)^2} \, dx}{c^2}\\ &=\frac{a^2 \int \frac{x^2 \sqrt{1-a^2 x^2}}{(1-a x)^3} \, dx}{c^2}\\ &=\frac{\left (1-a^2 x^2\right )^{3/2}}{a c^2 (1-a x)^2}-\frac{\int \frac{\left (2 a^2-3 a^3 x\right ) \sqrt{1-a^2 x^2}}{(1-a x)^3} \, dx}{a^2 c^2}\\ &=\frac{\left (1-a^2 x^2\right )^{3/2}}{3 a c^2 (1-a x)^3}+\frac{\left (1-a^2 x^2\right )^{3/2}}{a c^2 (1-a x)^2}-\frac{3 \int \frac{\sqrt{1-a^2 x^2}}{(1-a x)^2} \, dx}{c^2}\\ &=-\frac{6 \sqrt{1-a^2 x^2}}{a c^2 (1-a x)}+\frac{\left (1-a^2 x^2\right )^{3/2}}{3 a c^2 (1-a x)^3}+\frac{\left (1-a^2 x^2\right )^{3/2}}{a c^2 (1-a x)^2}+\frac{3 \int \frac{1}{\sqrt{1-a^2 x^2}} \, dx}{c^2}\\ &=-\frac{6 \sqrt{1-a^2 x^2}}{a c^2 (1-a x)}+\frac{\left (1-a^2 x^2\right )^{3/2}}{3 a c^2 (1-a x)^3}+\frac{\left (1-a^2 x^2\right )^{3/2}}{a c^2 (1-a x)^2}+\frac{3 \sin ^{-1}(a x)}{a c^2}\\ \end{align*}
Mathematica [A] time = 0.133426, size = 53, normalized size = 0.51 \[ \frac{\frac{\sqrt{1-a^2 x^2} \left (-3 a^2 x^2+19 a x-14\right )}{(a x-1)^2}+9 \sin ^{-1}(a x)}{3 a c^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.047, size = 140, normalized size = 1.4 \begin{align*} -{\frac{1}{a{c}^{2}}\sqrt{-{a}^{2}{x}^{2}+1}}+3\,{\frac{1}{{c}^{2}\sqrt{{a}^{2}}}\arctan \left ({\frac{\sqrt{{a}^{2}}x}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) }+{\frac{2}{3\,{a}^{3}{c}^{2}}\sqrt{-{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,a \left ( x-{a}^{-1} \right ) } \left ( x-{a}^{-1} \right ) ^{-2}}+{\frac{13}{3\,{a}^{2}{c}^{2}}\sqrt{-{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,a \left ( x-{a}^{-1} \right ) } \left ( x-{a}^{-1} \right ) ^{-1}} \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 (c - \frac{c}{a x}\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.11676, size = 246, normalized size = 2.37 \begin{align*} -\frac{14 \, a^{2} x^{2} - 28 \, a x + 18 \,{\left (a^{2} x^{2} - 2 \, a x + 1\right )} \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right ) +{\left (3 \, a^{2} x^{2} - 19 \, a x + 14\right )} \sqrt{-a^{2} x^{2} + 1} + 14}{3 \,{\left (a^{3} c^{2} x^{2} - 2 \, a^{2} c^{2} x + a c^{2}\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{a^{2} \left (\int \frac{x^{2}}{a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} - 2 a x \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{a x^{3}}{a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} - 2 a x \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx\right )}{c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18989, size = 171, normalized size = 1.64 \begin{align*} \frac{3 \, \arcsin \left (a x\right ) \mathrm{sgn}\left (a\right )}{c^{2}{\left | a \right |}} - \frac{\sqrt{-a^{2} x^{2} + 1}}{a c^{2}} + \frac{2 \,{\left (\frac{24 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}}{a^{2} x} - \frac{9 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{2}}{a^{4} x^{2}} - 11\right )}}{3 \, c^{2}{\left (\frac{\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a}{a^{2} x} - 1\right )}^{3}{\left | a \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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