Optimal. Leaf size=252 \[ \frac{\sqrt{1-a^2 x^2}}{2 c^2 (1-a x) \sqrt{c-a^2 c x^2}}+\frac{\sqrt{1-a^2 x^2}}{8 c^2 (a x+1) \sqrt{c-a^2 c x^2}}+\frac{\sqrt{1-a^2 x^2}}{8 c^2 (1-a x)^2 \sqrt{c-a^2 c x^2}}+\frac{\sqrt{1-a^2 x^2} \log (x)}{c^2 \sqrt{c-a^2 c x^2}}-\frac{11 \sqrt{1-a^2 x^2} \log (1-a x)}{16 c^2 \sqrt{c-a^2 c x^2}}-\frac{5 \sqrt{1-a^2 x^2} \log (a x+1)}{16 c^2 \sqrt{c-a^2 c x^2}} \]
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Rubi [A] time = 0.243718, antiderivative size = 252, 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, 88} \[ \frac{\sqrt{1-a^2 x^2}}{2 c^2 (1-a x) \sqrt{c-a^2 c x^2}}+\frac{\sqrt{1-a^2 x^2}}{8 c^2 (a x+1) \sqrt{c-a^2 c x^2}}+\frac{\sqrt{1-a^2 x^2}}{8 c^2 (1-a x)^2 \sqrt{c-a^2 c x^2}}+\frac{\sqrt{1-a^2 x^2} \log (x)}{c^2 \sqrt{c-a^2 c x^2}}-\frac{11 \sqrt{1-a^2 x^2} \log (1-a x)}{16 c^2 \sqrt{c-a^2 c x^2}}-\frac{5 \sqrt{1-a^2 x^2} \log (a x+1)}{16 c^2 \sqrt{c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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Rule 6153
Rule 6150
Rule 88
Rubi steps
\begin{align*} \int \frac{e^{\tanh ^{-1}(a x)}}{x \left (c-a^2 c x^2\right )^{5/2}} \, dx &=\frac{\sqrt{1-a^2 x^2} \int \frac{e^{\tanh ^{-1}(a x)}}{x \left (1-a^2 x^2\right )^{5/2}} \, dx}{c^2 \sqrt{c-a^2 c x^2}}\\ &=\frac{\sqrt{1-a^2 x^2} \int \frac{1}{x (1-a x)^3 (1+a x)^2} \, dx}{c^2 \sqrt{c-a^2 c x^2}}\\ &=\frac{\sqrt{1-a^2 x^2} \int \left (\frac{1}{x}-\frac{a}{4 (-1+a x)^3}+\frac{a}{2 (-1+a x)^2}-\frac{11 a}{16 (-1+a x)}-\frac{a}{8 (1+a x)^2}-\frac{5 a}{16 (1+a x)}\right ) \, dx}{c^2 \sqrt{c-a^2 c x^2}}\\ &=\frac{\sqrt{1-a^2 x^2}}{8 c^2 (1-a x)^2 \sqrt{c-a^2 c x^2}}+\frac{\sqrt{1-a^2 x^2}}{2 c^2 (1-a x) \sqrt{c-a^2 c x^2}}+\frac{\sqrt{1-a^2 x^2}}{8 c^2 (1+a x) \sqrt{c-a^2 c x^2}}+\frac{\sqrt{1-a^2 x^2} \log (x)}{c^2 \sqrt{c-a^2 c x^2}}-\frac{11 \sqrt{1-a^2 x^2} \log (1-a x)}{16 c^2 \sqrt{c-a^2 c x^2}}-\frac{5 \sqrt{1-a^2 x^2} \log (1+a x)}{16 c^2 \sqrt{c-a^2 c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0725986, size = 86, normalized size = 0.34 \[ \frac{\sqrt{1-a^2 x^2} \left (\frac{8}{1-a x}+\frac{2}{a x+1}+\frac{2}{(a x-1)^2}-11 \log (1-a x)-5 \log (a x+1)+16 \log (x)\right )}{16 c^2 \sqrt{c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.093, size = 193, normalized size = 0.8 \begin{align*} -{\frac{16\,{a}^{3}\ln \left ( x \right ){x}^{3}-5\,{a}^{3}{x}^{3}\ln \left ( ax+1 \right ) -11\,\ln \left ( ax-1 \right ){x}^{3}{a}^{3}-16\,{a}^{2}\ln \left ( x \right ){x}^{2}+5\,\ln \left ( ax+1 \right ){a}^{2}{x}^{2}+11\,\ln \left ( ax-1 \right ){a}^{2}{x}^{2}-6\,{a}^{2}{x}^{2}-16\,a\ln \left ( x \right ) x+5\,ax\ln \left ( ax+1 \right ) +11\,\ln \left ( ax-1 \right ) xa-2\,ax+16\,\ln \left ( x \right ) -5\,\ln \left ( ax+1 \right ) -11\,\ln \left ( ax-1 \right ) +12}{ \left ( 16\,{a}^{2}{x}^{2}-16 \right ){c}^{3} \left ( ax-1 \right ) ^{2} \left ( ax+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{5}{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^{7} c^{3} x^{8} - a^{6} c^{3} x^{7} - 3 \, a^{5} c^{3} x^{6} + 3 \, a^{4} c^{3} x^{5} + 3 \, a^{3} c^{3} x^{4} - 3 \, a^{2} c^{3} x^{3} - a c^{3} x^{2} + c^{3} 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{5}{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{5}{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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