Optimal. Leaf size=66 \[ -a c \sqrt{1-a^2 x^2}-\frac{c \sqrt{1-a^2 x^2}}{x}-3 a c \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )+3 a c \sin ^{-1}(a x) \]
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Rubi [A] time = 0.191489, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.348, Rules used = {6148, 1807, 1809, 844, 216, 266, 63, 208} \[ -a c \sqrt{1-a^2 x^2}-\frac{c \sqrt{1-a^2 x^2}}{x}-3 a c \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )+3 a c \sin ^{-1}(a x) \]
Antiderivative was successfully verified.
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Rule 6148
Rule 1807
Rule 1809
Rule 844
Rule 216
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{e^{3 \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )}{x^2} \, dx &=c \int \frac{(1+a x)^3}{x^2 \sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{c \sqrt{1-a^2 x^2}}{x}-c \int \frac{-3 a-3 a^2 x-a^3 x^2}{x \sqrt{1-a^2 x^2}} \, dx\\ &=-a c \sqrt{1-a^2 x^2}-\frac{c \sqrt{1-a^2 x^2}}{x}+\frac{c \int \frac{3 a^3+3 a^4 x}{x \sqrt{1-a^2 x^2}} \, dx}{a^2}\\ &=-a c \sqrt{1-a^2 x^2}-\frac{c \sqrt{1-a^2 x^2}}{x}+(3 a c) \int \frac{1}{x \sqrt{1-a^2 x^2}} \, dx+\left (3 a^2 c\right ) \int \frac{1}{\sqrt{1-a^2 x^2}} \, dx\\ &=-a c \sqrt{1-a^2 x^2}-\frac{c \sqrt{1-a^2 x^2}}{x}+3 a c \sin ^{-1}(a x)+\frac{1}{2} (3 a c) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-a^2 x}} \, dx,x,x^2\right )\\ &=-a c \sqrt{1-a^2 x^2}-\frac{c \sqrt{1-a^2 x^2}}{x}+3 a c \sin ^{-1}(a x)-\frac{(3 c) \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}\\ &=-a c \sqrt{1-a^2 x^2}-\frac{c \sqrt{1-a^2 x^2}}{x}+3 a c \sin ^{-1}(a x)-3 a c \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )\\ \end{align*}
Mathematica [A] time = 0.0631801, size = 52, normalized size = 0.79 \[ c \left (-\frac{\sqrt{1-a^2 x^2} (a x+1)}{x}-3 a \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )+3 a \sin ^{-1}(a x)\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.043, size = 122, normalized size = 1.9 \begin{align*}{{a}^{3}c{x}^{2}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}-{ac{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}+{cx{a}^{2}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}+3\,{\frac{{a}^{2}c}{\sqrt{{a}^{2}}}\arctan \left ({\frac{\sqrt{{a}^{2}}x}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) }-{\frac{c}{x}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}-3\,ca{\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 [B] time = 1.45713, size = 169, normalized size = 2.56 \begin{align*} \frac{a^{3} c x^{2}}{\sqrt{-a^{2} x^{2} + 1}} + \frac{a^{2} c x}{\sqrt{-a^{2} x^{2} + 1}} + \frac{3 \, a^{2} c \arcsin \left (\frac{a^{2} x}{\sqrt{a^{2}}}\right )}{\sqrt{a^{2}}} - 3 \, a c \log \left (\frac{2 \, \sqrt{-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac{2}{{\left | x \right |}}\right ) - \frac{a c}{\sqrt{-a^{2} x^{2} + 1}} - \frac{c}{\sqrt{-a^{2} x^{2} + 1} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.61367, size = 184, normalized size = 2.79 \begin{align*} -\frac{6 \, a c x \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right ) - 3 \, a c x \log \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right ) + a c x + \sqrt{-a^{2} x^{2} + 1}{\left (a c x + c\right )}}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 9.7716, size = 150, normalized size = 2.27 \begin{align*} a^{3} c \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 ) + 3 a^{2} c \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 ) + 3 a c \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 ) + c \left (\begin{cases} - \frac{i \sqrt{a^{2} x^{2} - 1}}{x} & \text{for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac{\sqrt{- a^{2} x^{2} + 1}}{x} & \text{otherwise} \end{cases}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.15961, size = 177, normalized size = 2.68 \begin{align*} \frac{a^{4} c x}{2 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}{\left | a \right |}} + \frac{3 \, a^{2} c \arcsin \left (a x\right ) \mathrm{sgn}\left (a\right )}{{\left | a \right |}} - \frac{3 \, a^{2} c \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 |}} - \sqrt{-a^{2} x^{2} + 1} a c - \frac{{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )} c}{2 \, x{\left | a \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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