Optimal. Leaf size=80 \[ -\frac{2 x}{15 a c^3 \sqrt{1-a^2 x^2}}-\frac{x}{15 a c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac{a x+1}{5 a^2 c^3 \left (1-a^2 x^2\right )^{5/2}} \]
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Rubi [A] time = 0.0690854, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {6148, 778, 192, 191} \[ -\frac{2 x}{15 a c^3 \sqrt{1-a^2 x^2}}-\frac{x}{15 a c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac{a x+1}{5 a^2 c^3 \left (1-a^2 x^2\right )^{5/2}} \]
Antiderivative was successfully verified.
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Rule 6148
Rule 778
Rule 192
Rule 191
Rubi steps
\begin{align*} \int \frac{e^{\tanh ^{-1}(a x)} x}{\left (c-a^2 c x^2\right )^3} \, dx &=\frac{\int \frac{x (1+a x)}{\left (1-a^2 x^2\right )^{7/2}} \, dx}{c^3}\\ &=\frac{1+a x}{5 a^2 c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac{\int \frac{1}{\left (1-a^2 x^2\right )^{5/2}} \, dx}{5 a c^3}\\ &=\frac{1+a x}{5 a^2 c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac{x}{15 a c^3 \left (1-a^2 x^2\right )^{3/2}}-\frac{2 \int \frac{1}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{15 a c^3}\\ &=\frac{1+a x}{5 a^2 c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac{x}{15 a c^3 \left (1-a^2 x^2\right )^{3/2}}-\frac{2 x}{15 a c^3 \sqrt{1-a^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.0311734, size = 68, normalized size = 0.85 \[ \frac{-2 a^4 x^4+2 a^3 x^3+3 a^2 x^2-3 a x+3}{15 a^2 c^3 (a x-1)^2 (a x+1) \sqrt{1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.032, size = 58, normalized size = 0.7 \begin{align*}{\frac{2\,{x}^{4}{a}^{4}-2\,{x}^{3}{a}^{3}-3\,{a}^{2}{x}^{2}+3\,ax-3}{ \left ( 15\,ax-15 \right ){c}^{3}{a}^{2}} \left ( -{a}^{2}{x}^{2}+1 \right ) ^{-{\frac{3}{2}}}} \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^{2}}{{\left (a^{6} c^{3} x^{6} - 3 \, a^{4} c^{3} x^{4} + 3 \, a^{2} c^{3} x^{2} - c^{3}\right )} \sqrt{a x + 1} \sqrt{-a x + 1}}\,{d x} + \frac{1}{5 \,{\left (-a^{2} x^{2} + 1\right )}^{\frac{5}{2}} a^{2} c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.58584, size = 293, normalized size = 3.66 \begin{align*} \frac{3 \, a^{5} x^{5} - 3 \, a^{4} x^{4} - 6 \, a^{3} x^{3} + 6 \, a^{2} x^{2} + 3 \, a x +{\left (2 \, a^{4} x^{4} - 2 \, a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 3\right )} \sqrt{-a^{2} x^{2} + 1} - 3}{15 \,{\left (a^{7} c^{3} x^{5} - a^{6} c^{3} x^{4} - 2 \, a^{5} c^{3} x^{3} + 2 \, a^{4} c^{3} x^{2} + a^{3} c^{3} x - a^{2} c^{3}\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{x}{- a^{6} x^{6} \sqrt{- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 3 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{a x^{2}}{- a^{6} x^{6} \sqrt{- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 3 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx}{c^{3}} \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}{{\left (a^{2} c x^{2} - c\right )}^{3} \sqrt{-a^{2} x^{2} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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