Optimal. Leaf size=165 \[ \frac{\sqrt{1-a^2 x^2}}{2 c (1-a x) \sqrt{c-a^2 c x^2}}+\frac{\sqrt{1-a^2 x^2} \log (x)}{c \sqrt{c-a^2 c x^2}}-\frac{3 \sqrt{1-a^2 x^2} \log (1-a x)}{4 c \sqrt{c-a^2 c x^2}}-\frac{\sqrt{1-a^2 x^2} \log (a x+1)}{4 c \sqrt{c-a^2 c x^2}} \]
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Rubi [A] time = 0.219334, antiderivative size = 165, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {6153, 6150, 72} \[ \frac{\sqrt{1-a^2 x^2}}{2 c (1-a x) \sqrt{c-a^2 c x^2}}+\frac{\sqrt{1-a^2 x^2} \log (x)}{c \sqrt{c-a^2 c x^2}}-\frac{3 \sqrt{1-a^2 x^2} \log (1-a x)}{4 c \sqrt{c-a^2 c x^2}}-\frac{\sqrt{1-a^2 x^2} \log (a x+1)}{4 c \sqrt{c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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Rule 6153
Rule 6150
Rule 72
Rubi steps
\begin{align*} \int \frac{e^{\tanh ^{-1}(a x)}}{x \left (c-a^2 c x^2\right )^{3/2}} \, dx &=\frac{\sqrt{1-a^2 x^2} \int \frac{e^{\tanh ^{-1}(a x)}}{x \left (1-a^2 x^2\right )^{3/2}} \, dx}{c \sqrt{c-a^2 c x^2}}\\ &=\frac{\sqrt{1-a^2 x^2} \int \frac{1}{x (1-a x)^2 (1+a x)} \, dx}{c \sqrt{c-a^2 c x^2}}\\ &=\frac{\sqrt{1-a^2 x^2} \int \left (\frac{1}{x}+\frac{a}{2 (-1+a x)^2}-\frac{3 a}{4 (-1+a x)}-\frac{a}{4 (1+a x)}\right ) \, dx}{c \sqrt{c-a^2 c x^2}}\\ &=\frac{\sqrt{1-a^2 x^2}}{2 c (1-a x) \sqrt{c-a^2 c x^2}}+\frac{\sqrt{1-a^2 x^2} \log (x)}{c \sqrt{c-a^2 c x^2}}-\frac{3 \sqrt{1-a^2 x^2} \log (1-a x)}{4 c \sqrt{c-a^2 c x^2}}-\frac{\sqrt{1-a^2 x^2} \log (1+a x)}{4 c \sqrt{c-a^2 c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0511237, size = 65, normalized size = 0.39 \[ \frac{\sqrt{1-a^2 x^2} \left (\frac{1}{2-2 a x}-\frac{3}{4} \log (1-a x)-\frac{1}{4} \log (a x+1)+\log (x)\right )}{c \sqrt{c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.095, size = 96, normalized size = 0.6 \begin{align*} -{\frac{4\,a\ln \left ( x \right ) x-ax\ln \left ( ax+1 \right ) -3\,\ln \left ( ax-1 \right ) xa-4\,\ln \left ( x \right ) +\ln \left ( ax+1 \right ) +3\,\ln \left ( ax-1 \right ) -2}{4\,{c}^{2} \left ( ax-1 \right ) \left ({a}^{2}{x}^{2}-1 \right ) }\sqrt{-{a}^{2}{x}^{2}+1}\sqrt{-c \left ({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}{{\left (-a^{2} c x^{2} + c\right )}^{\frac{3}{2}} \sqrt{-a^{2} x^{2} + 1} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1}}{a^{5} c^{2} x^{6} - a^{4} c^{2} x^{5} - 2 \, a^{3} c^{2} x^{4} + 2 \, a^{2} c^{2} x^{3} + a c^{2} x^{2} - c^{2} x}, 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*} \int \frac{a x + 1}{x \sqrt{- \left (a x - 1\right ) \left (a x + 1\right )} \left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac{3}{2}}}\, dx \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}{{\left (-a^{2} c x^{2} + c\right )}^{\frac{3}{2}} \sqrt{-a^{2} x^{2} + 1} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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