Optimal. Leaf size=176 \[ \frac{x \sqrt{1-a^2 x^2}}{a^3 c \sqrt{c-a^2 c x^2}}+\frac{\sqrt{1-a^2 x^2}}{2 a^4 c (1-a x) \sqrt{c-a^2 c x^2}}+\frac{5 \sqrt{1-a^2 x^2} \log (1-a x)}{4 a^4 c \sqrt{c-a^2 c x^2}}-\frac{\sqrt{1-a^2 x^2} \log (a x+1)}{4 a^4 c \sqrt{c-a^2 c x^2}} \]
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Rubi [A] time = 0.231571, antiderivative size = 176, 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} \[ \frac{x \sqrt{1-a^2 x^2}}{a^3 c \sqrt{c-a^2 c x^2}}+\frac{\sqrt{1-a^2 x^2}}{2 a^4 c (1-a x) \sqrt{c-a^2 c x^2}}+\frac{5 \sqrt{1-a^2 x^2} \log (1-a x)}{4 a^4 c \sqrt{c-a^2 c x^2}}-\frac{\sqrt{1-a^2 x^2} \log (a x+1)}{4 a^4 c \sqrt{c-a^2 c x^2}} \]
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^3}{\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^3}{\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{x^3}{(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}{a^3}+\frac{1}{2 a^3 (-1+a x)^2}+\frac{5}{4 a^3 (-1+a x)}-\frac{1}{4 a^3 (1+a x)}\right ) \, dx}{c \sqrt{c-a^2 c x^2}}\\ &=\frac{x \sqrt{1-a^2 x^2}}{a^3 c \sqrt{c-a^2 c x^2}}+\frac{\sqrt{1-a^2 x^2}}{2 a^4 c (1-a x) \sqrt{c-a^2 c x^2}}+\frac{5 \sqrt{1-a^2 x^2} \log (1-a x)}{4 a^4 c \sqrt{c-a^2 c x^2}}-\frac{\sqrt{1-a^2 x^2} \log (1+a x)}{4 a^4 c \sqrt{c-a^2 c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0524373, size = 71, normalized size = 0.4 \[ \frac{\sqrt{1-a^2 x^2} \left (4 a x+\frac{2}{1-a x}+5 \log (1-a x)-\log (a x+1)\right )}{4 a^4 c \sqrt{c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.092, size = 102, normalized size = 0.6 \begin{align*}{\frac{-4\,{a}^{2}{x}^{2}+ax\ln \left ( ax+1 \right ) -5\,\ln \left ( ax-1 \right ) xa+4\,ax-\ln \left ( ax+1 \right ) +5\,\ln \left ( ax-1 \right ) +2}{ \left ( 4\,{a}^{2}{x}^{2}-4 \right ){c}^{2}{a}^{4} \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*} -a \int -\frac{x^{4}}{{\left (a^{2} c^{\frac{3}{2}} x^{2} - c^{\frac{3}{2}}\right )}{\left (a x + 1\right )}{\left (a x - 1\right )}}\,{d x} - \frac{1}{2 \,{\left (a^{6} c^{\frac{3}{2}} x^{2} - a^{4} c^{\frac{3}{2}}\right )}} + \frac{\log \left (-a^{2} c x^{2} + c\right )}{2 \, a^{4} c^{\frac{3}{2}}} \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} x^{3}}{a^{5} c^{2} x^{5} - a^{4} c^{2} x^{4} - 2 \, a^{3} c^{2} x^{3} + 2 \, a^{2} c^{2} x^{2} + a c^{2} x - 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*} \int \frac{x^{3} \left (a x + 1\right )}{\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{{\left (a x + 1\right )} x^{3}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac{3}{2}} \sqrt{-a^{2} x^{2} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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