Optimal. Leaf size=106 \[ \frac{1}{3} a c^4 \left (1-a^2 x^2\right )^{3/2}-\frac{c^4 \left (1-a^2 x^2\right )^{3/2}}{x}-\frac{1}{2} a c^4 (6-a x) \sqrt{1-a^2 x^2}+3 a c^4 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )+\frac{1}{2} a c^4 \sin ^{-1}(a x) \]
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Rubi [A] time = 0.24186, antiderivative size = 106, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.474, Rules used = {6128, 1807, 1809, 815, 844, 216, 266, 63, 208} \[ \frac{1}{3} a c^4 \left (1-a^2 x^2\right )^{3/2}-\frac{c^4 \left (1-a^2 x^2\right )^{3/2}}{x}-\frac{1}{2} a c^4 (6-a x) \sqrt{1-a^2 x^2}+3 a c^4 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )+\frac{1}{2} a c^4 \sin ^{-1}(a x) \]
Antiderivative was successfully verified.
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Rule 6128
Rule 1807
Rule 1809
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)^4}{x^2} \, dx &=c \int \frac{(c-a c x)^3 \sqrt{1-a^2 x^2}}{x^2} \, dx\\ &=-\frac{c^4 \left (1-a^2 x^2\right )^{3/2}}{x}-c \int \frac{\sqrt{1-a^2 x^2} \left (3 a c^3-a^2 c^3 x+a^3 c^3 x^2\right )}{x} \, dx\\ &=\frac{1}{3} a c^4 \left (1-a^2 x^2\right )^{3/2}-\frac{c^4 \left (1-a^2 x^2\right )^{3/2}}{x}+\frac{c \int \frac{\left (-9 a^3 c^3+3 a^4 c^3 x\right ) \sqrt{1-a^2 x^2}}{x} \, dx}{3 a^2}\\ &=-\frac{1}{2} a c^4 (6-a x) \sqrt{1-a^2 x^2}+\frac{1}{3} a c^4 \left (1-a^2 x^2\right )^{3/2}-\frac{c^4 \left (1-a^2 x^2\right )^{3/2}}{x}-\frac{c \int \frac{18 a^5 c^3-3 a^6 c^3 x}{x \sqrt{1-a^2 x^2}} \, dx}{6 a^4}\\ &=-\frac{1}{2} a c^4 (6-a x) \sqrt{1-a^2 x^2}+\frac{1}{3} a c^4 \left (1-a^2 x^2\right )^{3/2}-\frac{c^4 \left (1-a^2 x^2\right )^{3/2}}{x}-\left (3 a c^4\right ) \int \frac{1}{x \sqrt{1-a^2 x^2}} \, dx+\frac{1}{2} \left (a^2 c^4\right ) \int \frac{1}{\sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{1}{2} a c^4 (6-a x) \sqrt{1-a^2 x^2}+\frac{1}{3} a c^4 \left (1-a^2 x^2\right )^{3/2}-\frac{c^4 \left (1-a^2 x^2\right )^{3/2}}{x}+\frac{1}{2} a c^4 \sin ^{-1}(a x)-\frac{1}{2} \left (3 a c^4\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-a^2 x}} \, dx,x,x^2\right )\\ &=-\frac{1}{2} a c^4 (6-a x) \sqrt{1-a^2 x^2}+\frac{1}{3} a c^4 \left (1-a^2 x^2\right )^{3/2}-\frac{c^4 \left (1-a^2 x^2\right )^{3/2}}{x}+\frac{1}{2} a c^4 \sin ^{-1}(a x)+\frac{\left (3 c^4\right ) \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}\\ &=-\frac{1}{2} a c^4 (6-a x) \sqrt{1-a^2 x^2}+\frac{1}{3} a c^4 \left (1-a^2 x^2\right )^{3/2}-\frac{c^4 \left (1-a^2 x^2\right )^{3/2}}{x}+\frac{1}{2} a c^4 \sin ^{-1}(a x)+3 a c^4 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )\\ \end{align*}
Mathematica [A] time = 0.127345, size = 152, normalized size = 1.43 \[ -\frac{c^4 \left (-2 a^5 x^5+9 a^4 x^4-14 a^3 x^3-15 a^2 x^2+9 a x \sqrt{1-a^2 x^2} \sin ^{-1}(a x)+24 a x \sqrt{1-a^2 x^2} \sin ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{2}}\right )-18 a x \sqrt{1-a^2 x^2} \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )+16 a x+6\right )}{6 x \sqrt{1-a^2 x^2}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.046, size = 136, normalized size = 1.3 \begin{align*} -{\frac{{c}^{4}{a}^{3}{x}^{2}}{3}\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{8\,{c}^{4}a}{3}\sqrt{-{a}^{2}{x}^{2}+1}}+{\frac{3\,{c}^{4}{a}^{2}x}{2}\sqrt{-{a}^{2}{x}^{2}+1}}+{\frac{{c}^{4}{a}^{2}}{2}\arctan \left ({x\sqrt{{a}^{2}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ){\frac{1}{\sqrt{{a}^{2}}}}}-{\frac{{c}^{4}}{x}\sqrt{-{a}^{2}{x}^{2}+1}}+3\,{c}^{4}a{\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 = 1.44635, size = 188, normalized size = 1.77 \begin{align*} -\frac{1}{3} \, \sqrt{-a^{2} x^{2} + 1} a^{3} c^{4} x^{2} + \frac{3}{2} \, \sqrt{-a^{2} x^{2} + 1} a^{2} c^{4} x + \frac{a^{2} c^{4} \arcsin \left (\frac{a^{2} x}{\sqrt{a^{2}}}\right )}{2 \, \sqrt{a^{2}}} + 3 \, a c^{4} \log \left (\frac{2 \, \sqrt{-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac{2}{{\left | x \right |}}\right ) - \frac{8}{3} \, \sqrt{-a^{2} x^{2} + 1} a c^{4} - \frac{\sqrt{-a^{2} x^{2} + 1} c^{4}}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.58546, size = 258, normalized size = 2.43 \begin{align*} -\frac{6 \, a c^{4} x \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right ) + 18 \, a c^{4} x \log \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right ) + 16 \, a c^{4} x +{\left (2 \, a^{3} c^{4} x^{3} - 9 \, a^{2} c^{4} x^{2} + 16 \, a c^{4} x + 6 \, c^{4}\right )} \sqrt{-a^{2} x^{2} + 1}}{6 \, x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 7.41335, size = 306, normalized size = 2.89 \begin{align*} a^{5} c^{4} \left (\begin{cases} - \frac{x^{2} \sqrt{- a^{2} x^{2} + 1}}{3 a^{2}} - \frac{2 \sqrt{- a^{2} x^{2} + 1}}{3 a^{4}} & \text{for}\: a \neq 0 \\\frac{x^{4}}{4} & \text{otherwise} \end{cases}\right ) - 3 a^{4} c^{4} \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 ) + 2 a^{3} c^{4} \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 ) + 2 a^{2} c^{4} \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^{4} \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^{4} \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 [A] time = 1.28515, size = 221, normalized size = 2.08 \begin{align*} \frac{a^{4} c^{4} x}{2 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}{\left | a \right |}} + \frac{a^{2} c^{4} \arcsin \left (a x\right ) \mathrm{sgn}\left (a\right )}{2 \,{\left | a \right |}} + \frac{3 \, a^{2} c^{4} \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{{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )} c^{4}}{2 \, x{\left | a \right |}} - \frac{1}{6} \,{\left (16 \, a c^{4} +{\left (2 \, a^{3} c^{4} x - 9 \, a^{2} c^{4}\right )} x\right )} \sqrt{-a^{2} x^{2} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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