Optimal. Leaf size=59 \[ \frac{1}{2} c^2 (2-a x) \sqrt{1-a^2 x^2}-c^2 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )-\frac{1}{2} c^2 \sin ^{-1}(a x) \]
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Rubi [A] time = 0.101381, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.368, Rules used = {6128, 815, 844, 216, 266, 63, 208} \[ \frac{1}{2} c^2 (2-a x) \sqrt{1-a^2 x^2}-c^2 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )-\frac{1}{2} c^2 \sin ^{-1}(a x) \]
Antiderivative was successfully verified.
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Rule 6128
Rule 815
Rule 844
Rule 216
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{e^{\tanh ^{-1}(a x)} (c-a c x)^2}{x} \, dx &=c \int \frac{(c-a c x) \sqrt{1-a^2 x^2}}{x} \, dx\\ &=\frac{1}{2} c^2 (2-a x) \sqrt{1-a^2 x^2}-\frac{c \int \frac{-2 a^2 c+a^3 c x}{x \sqrt{1-a^2 x^2}} \, dx}{2 a^2}\\ &=\frac{1}{2} c^2 (2-a x) \sqrt{1-a^2 x^2}+c^2 \int \frac{1}{x \sqrt{1-a^2 x^2}} \, dx-\frac{1}{2} \left (a c^2\right ) \int \frac{1}{\sqrt{1-a^2 x^2}} \, dx\\ &=\frac{1}{2} c^2 (2-a x) \sqrt{1-a^2 x^2}-\frac{1}{2} c^2 \sin ^{-1}(a x)+\frac{1}{2} c^2 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-a^2 x}} \, dx,x,x^2\right )\\ &=\frac{1}{2} c^2 (2-a x) \sqrt{1-a^2 x^2}-\frac{1}{2} c^2 \sin ^{-1}(a x)-\frac{c^2 \operatorname{Subst}\left (\int \frac{1}{\frac{1}{a^2}-\frac{x^2}{a^2}} \, dx,x,\sqrt{1-a^2 x^2}\right )}{a^2}\\ &=\frac{1}{2} c^2 (2-a x) \sqrt{1-a^2 x^2}-\frac{1}{2} c^2 \sin ^{-1}(a x)-c^2 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )\\ \end{align*}
Mathematica [B] time = 0.0777885, size = 125, normalized size = 2.12 \[ \frac{c^2 \left (a^3 x^3-2 a^2 x^2+\sqrt{1-a^2 x^2} \sin ^{-1}(a x)+4 \sqrt{1-a^2 x^2} \sin ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{2}}\right )-2 \sqrt{1-a^2 x^2} \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )-a x+2\right )}{2 \sqrt{1-a^2 x^2}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.037, size = 86, normalized size = 1.5 \begin{align*} -{\frac{a{c}^{2}x}{2}\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{a{c}^{2}}{2}\arctan \left ({x\sqrt{{a}^{2}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ){\frac{1}{\sqrt{{a}^{2}}}}}+{c}^{2}\sqrt{-{a}^{2}{x}^{2}+1}-{c}^{2}{\it Artanh} \left ({\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.42381, size = 120, normalized size = 2.03 \begin{align*} -\frac{1}{2} \, \sqrt{-a^{2} x^{2} + 1} a c^{2} x - \frac{a c^{2} \arcsin \left (\frac{a^{2} x}{\sqrt{a^{2}}}\right )}{2 \, \sqrt{a^{2}}} - c^{2} \log \left (\frac{2 \, \sqrt{-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac{2}{{\left | x \right |}}\right ) + \sqrt{-a^{2} x^{2} + 1} c^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.70968, size = 169, normalized size = 2.86 \begin{align*} c^{2} \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right ) + c^{2} \log \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right ) - \frac{1}{2} \,{\left (a c^{2} x - 2 \, c^{2}\right )} \sqrt{-a^{2} x^{2} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 12.0665, size = 201, normalized size = 3.41 \begin{align*} a^{3} c^{2} \left (\begin{cases} - \frac{i x \sqrt{a^{2} x^{2} - 1}}{2 a^{2}} - \frac{i \operatorname{acosh}{\left (a x \right )}}{2 a^{3}} & \text{for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac{x^{3}}{2 \sqrt{- a^{2} x^{2} + 1}} - \frac{x}{2 a^{2} \sqrt{- a^{2} x^{2} + 1}} + \frac{\operatorname{asin}{\left (a x \right )}}{2 a^{3}} & \text{otherwise} \end{cases}\right ) - a^{2} c^{2} \left (\begin{cases} \frac{x^{2}}{2} & \text{for}\: a^{2} = 0 \\- \frac{\sqrt{- a^{2} x^{2} + 1}}{a^{2}} & \text{otherwise} \end{cases}\right ) - a c^{2} \left (\begin{cases} \sqrt{\frac{1}{a^{2}}} \operatorname{asin}{\left (x \sqrt{a^{2}} \right )} & \text{for}\: a^{2} > 0 \\\sqrt{- \frac{1}{a^{2}}} \operatorname{asinh}{\left (x \sqrt{- a^{2}} \right )} & \text{for}\: a^{2} < 0 \end{cases}\right ) + c^{2} \left (\begin{cases} - \operatorname{acosh}{\left (\frac{1}{a x} \right )} & \text{for}\: \frac{1}{\left |{a^{2} x^{2}}\right |} > 1 \\i \operatorname{asin}{\left (\frac{1}{a x} \right )} & \text{otherwise} \end{cases}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.44031, size = 113, normalized size = 1.92 \begin{align*} -\frac{a c^{2} \arcsin \left (a x\right ) \mathrm{sgn}\left (a\right )}{2 \,{\left | a \right |}} - \frac{a c^{2} \log \left (\frac{{\left | -2 \, \sqrt{-a^{2} x^{2} + 1}{\left | a \right |} - 2 \, a \right |}}{2 \, a^{2}{\left | x \right |}}\right )}{{\left | a \right |}} - \frac{1}{2} \,{\left (a c^{2} x - 2 \, c^{2}\right )} \sqrt{-a^{2} x^{2} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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