Optimal. Leaf size=70 \[ \frac{(a x+1)^3}{3 a^2 c \left (1-a^2 x^2\right )^{3/2}}-\frac{2 (a x+1)}{a^2 c \sqrt{1-a^2 x^2}}+\frac{\sin ^{-1}(a x)}{a^2 c} \]
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Rubi [A] time = 0.0860757, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {6148, 789, 653, 216} \[ \frac{(a x+1)^3}{3 a^2 c \left (1-a^2 x^2\right )^{3/2}}-\frac{2 (a x+1)}{a^2 c \sqrt{1-a^2 x^2}}+\frac{\sin ^{-1}(a x)}{a^2 c} \]
Antiderivative was successfully verified.
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Rule 6148
Rule 789
Rule 653
Rule 216
Rubi steps
\begin{align*} \int \frac{e^{3 \tanh ^{-1}(a x)} x}{c-a^2 c x^2} \, dx &=\frac{\int \frac{x (1+a x)^3}{\left (1-a^2 x^2\right )^{5/2}} \, dx}{c}\\ &=\frac{(1+a x)^3}{3 a^2 c \left (1-a^2 x^2\right )^{3/2}}-\frac{\int \frac{(1+a x)^2}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{a c}\\ &=\frac{(1+a x)^3}{3 a^2 c \left (1-a^2 x^2\right )^{3/2}}-\frac{2 (1+a x)}{a^2 c \sqrt{1-a^2 x^2}}+\frac{\int \frac{1}{\sqrt{1-a^2 x^2}} \, dx}{a c}\\ &=\frac{(1+a x)^3}{3 a^2 c \left (1-a^2 x^2\right )^{3/2}}-\frac{2 (1+a x)}{a^2 c \sqrt{1-a^2 x^2}}+\frac{\sin ^{-1}(a x)}{a^2 c}\\ \end{align*}
Mathematica [A] time = 0.0617958, size = 70, normalized size = 1. \[ \frac{-7 a^2 x^2+3 (a x-1) \sqrt{1-a^2 x^2} \sin ^{-1}(a x)-2 a x+5}{3 a^2 c (a x-1) \sqrt{1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.038, size = 155, normalized size = 2.2 \begin{align*} -5\,{\frac{x}{ac\sqrt{-{a}^{2}{x}^{2}+1}}}+{\frac{1}{ac}\arctan \left ({x\sqrt{{a}^{2}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ){\frac{1}{\sqrt{{a}^{2}}}}}-3\,{\frac{1}{{a}^{2}c\sqrt{-{a}^{2}{x}^{2}+1}}}-{\frac{4}{3\,{a}^{3}c} \left ( x-{a}^{-1} \right ) ^{-1}{\frac{1}{\sqrt{-{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,a \left ( x-{a}^{-1} \right ) }}}}+{\frac{8\,x}{3\,ac}{\frac{1}{\sqrt{-{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,a \left ( x-{a}^{-1} \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.75188, size = 572, normalized size = 8.17 \begin{align*} -\frac{a^{2} c{\left (\frac{2 \, c}{\sqrt{a^{2} c^{2}} \sqrt{-a^{2} x^{2} + 1} a^{3} c x + \sqrt{-a^{2} x^{2} + 1} a^{3} c^{2}} + \frac{2 \, c}{\sqrt{a^{2} c^{2}} \sqrt{-a^{2} x^{2} + 1} a^{3} c x - \sqrt{-a^{2} x^{2} + 1} a^{3} c^{2}} - \frac{2 \, \sqrt{a^{2} c^{2}}}{\sqrt{a^{2} c^{2}} \sqrt{-a^{2} x^{2} + 1} a^{4} c x + \sqrt{-a^{2} x^{2} + 1} a^{4} c^{2}} + \frac{2 \, \sqrt{a^{2} c^{2}}}{\sqrt{a^{2} c^{2}} \sqrt{-a^{2} x^{2} + 1} a^{4} c x - \sqrt{-a^{2} x^{2} + 1} a^{4} c^{2}} - \frac{8 \, x}{\sqrt{a^{2} c^{2}} \sqrt{-a^{2} x^{2} + 1} a} + \frac{8}{\sqrt{a^{2} c^{2}} \sqrt{-a^{2} x^{2} + 1} a^{2}} + \frac{12 \, \sqrt{a^{2} c^{2}} x}{\sqrt{-a^{2} x^{2} + 1} a^{3} c^{2}} + \frac{\sqrt{a^{2} c^{2}}}{\sqrt{-a^{2} x^{2} + 1} a^{4} c^{2}} + \frac{3 \, \left (a^{2} c^{2}\right )^{\frac{3}{2}} x}{\sqrt{-a^{2} x^{2} + 1} a^{5} c^{4}} - \frac{3 \, \sqrt{a^{2} c^{2}} \arcsin \left (\frac{x}{c \sqrt{\frac{1}{a^{2} c^{2}}}}\right )}{a^{5} c^{3} \sqrt{\frac{1}{a^{2} c^{2}}}}\right )}}{3 \, \sqrt{a^{2} c^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.62036, size = 217, normalized size = 3.1 \begin{align*} -\frac{5 \, a^{2} x^{2} - 10 \, a x + 6 \,{\left (a^{2} x^{2} - 2 \, a x + 1\right )} \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right ) - \sqrt{-a^{2} x^{2} + 1}{\left (7 \, a x - 5\right )} + 5}{3 \,{\left (a^{4} c x^{2} - 2 \, a^{3} c x + a^{2} c\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^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{3 a x^{2}}{a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{3 a^{2} x^{3}}{a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{a^{3} x^{4}}{a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx}{c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19037, size = 151, normalized size = 2.16 \begin{align*} \frac{\arcsin \left (a x\right ) \mathrm{sgn}\left (a\right )}{a c{\left | a \right |}} + \frac{2 \,{\left (\frac{12 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}}{a^{2} x} - \frac{3 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{2}}{a^{4} x^{2}} - 5\right )}}{3 \, a c{\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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