Optimal. Leaf size=92 \[ -\frac{c^3 \left (1-a^2 x^2\right )^{3/2}}{2 x^2}+\frac{a c^3 (a x+4) \sqrt{1-a^2 x^2}}{2 x}-\frac{1}{2} a^2 c^3 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )+2 a^2 c^3 \sin ^{-1}(a x) \]
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Rubi [A] time = 0.179135, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.421, Rules used = {6128, 1807, 813, 844, 216, 266, 63, 208} \[ -\frac{c^3 \left (1-a^2 x^2\right )^{3/2}}{2 x^2}+\frac{a c^3 (a x+4) \sqrt{1-a^2 x^2}}{2 x}-\frac{1}{2} a^2 c^3 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )+2 a^2 c^3 \sin ^{-1}(a x) \]
Antiderivative was successfully verified.
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Rule 6128
Rule 1807
Rule 813
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)^3}{x^3} \, dx &=c \int \frac{(c-a c x)^2 \sqrt{1-a^2 x^2}}{x^3} \, dx\\ &=-\frac{c^3 \left (1-a^2 x^2\right )^{3/2}}{2 x^2}-\frac{1}{2} c \int \frac{\left (4 a c^2-a^2 c^2 x\right ) \sqrt{1-a^2 x^2}}{x^2} \, dx\\ &=\frac{a c^3 (4+a x) \sqrt{1-a^2 x^2}}{2 x}-\frac{c^3 \left (1-a^2 x^2\right )^{3/2}}{2 x^2}+\frac{1}{4} c \int \frac{2 a^2 c^2+8 a^3 c^2 x}{x \sqrt{1-a^2 x^2}} \, dx\\ &=\frac{a c^3 (4+a x) \sqrt{1-a^2 x^2}}{2 x}-\frac{c^3 \left (1-a^2 x^2\right )^{3/2}}{2 x^2}+\frac{1}{2} \left (a^2 c^3\right ) \int \frac{1}{x \sqrt{1-a^2 x^2}} \, dx+\left (2 a^3 c^3\right ) \int \frac{1}{\sqrt{1-a^2 x^2}} \, dx\\ &=\frac{a c^3 (4+a x) \sqrt{1-a^2 x^2}}{2 x}-\frac{c^3 \left (1-a^2 x^2\right )^{3/2}}{2 x^2}+2 a^2 c^3 \sin ^{-1}(a x)+\frac{1}{4} \left (a^2 c^3\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-a^2 x}} \, dx,x,x^2\right )\\ &=\frac{a c^3 (4+a x) \sqrt{1-a^2 x^2}}{2 x}-\frac{c^3 \left (1-a^2 x^2\right )^{3/2}}{2 x^2}+2 a^2 c^3 \sin ^{-1}(a x)-\frac{1}{2} c^3 \operatorname{Subst}\left (\int \frac{1}{\frac{1}{a^2}-\frac{x^2}{a^2}} \, dx,x,\sqrt{1-a^2 x^2}\right )\\ &=\frac{a c^3 (4+a x) \sqrt{1-a^2 x^2}}{2 x}-\frac{c^3 \left (1-a^2 x^2\right )^{3/2}}{2 x^2}+2 a^2 c^3 \sin ^{-1}(a x)-\frac{1}{2} a^2 c^3 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )\\ \end{align*}
Mathematica [A] time = 0.123942, size = 155, normalized size = 1.68 \[ \frac{c^3 \left (-4 a^4 x^4-8 a^3 x^3+6 a^2 x^2+a^2 x^2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)-14 a^2 x^2 \sqrt{1-a^2 x^2} \sin ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{2}}\right )-2 a^2 x^2 \sqrt{1-a^2 x^2} \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )+8 a x-2\right )}{4 x^2 \sqrt{1-a^2 x^2}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.04, size = 116, normalized size = 1.3 \begin{align*}{c}^{3}{a}^{2}\sqrt{-{a}^{2}{x}^{2}+1}+2\,{\frac{{c}^{3}{a}^{3}}{\sqrt{{a}^{2}}}\arctan \left ({\frac{\sqrt{{a}^{2}}x}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) }+2\,{\frac{{c}^{3}a\sqrt{-{a}^{2}{x}^{2}+1}}{x}}-{\frac{{c}^{3}}{2\,{x}^{2}}\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{{c}^{3}{a}^{2}}{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 = 1.43181, size = 161, normalized size = 1.75 \begin{align*} \frac{2 \, a^{3} c^{3} \arcsin \left (\frac{a^{2} x}{\sqrt{a^{2}}}\right )}{\sqrt{a^{2}}} - \frac{1}{2} \, a^{2} c^{3} \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} a^{2} c^{3} + \frac{2 \, \sqrt{-a^{2} x^{2} + 1} a c^{3}}{x} - \frac{\sqrt{-a^{2} x^{2} + 1} c^{3}}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.61909, size = 246, normalized size = 2.67 \begin{align*} -\frac{8 \, a^{2} c^{3} x^{2} \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right ) - a^{2} c^{3} x^{2} \log \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right ) - 2 \, a^{2} c^{3} x^{2} -{\left (2 \, a^{2} c^{3} x^{2} + 4 \, a c^{3} x - c^{3}\right )} \sqrt{-a^{2} x^{2} + 1}}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 5.46274, size = 228, normalized size = 2.48 \begin{align*} - a^{4} c^{3} \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^{3} c^{3} \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 ) - 2 a c^{3} \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 ) + c^{3} \left (\begin{cases} - \frac{a^{2} \operatorname{acosh}{\left (\frac{1}{a x} \right )}}{2} - \frac{a \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{2 x} & \text{for}\: \frac{1}{\left |{a^{2} x^{2}}\right |} > 1 \\\frac{i a^{2} \operatorname{asin}{\left (\frac{1}{a x} \right )}}{2} - \frac{i a}{2 x \sqrt{1 - \frac{1}{a^{2} x^{2}}}} + \frac{i}{2 a x^{3} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} & \text{otherwise} \end{cases}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.30847, size = 286, normalized size = 3.11 \begin{align*} \frac{2 \, a^{3} c^{3} \arcsin \left (a x\right ) \mathrm{sgn}\left (a\right )}{{\left | a \right |}} - \frac{a^{3} c^{3} \log \left (\frac{{\left | -2 \, \sqrt{-a^{2} x^{2} + 1}{\left | a \right |} - 2 \, a \right |}}{2 \, a^{2}{\left | x \right |}}\right )}{2 \,{\left | a \right |}} + \sqrt{-a^{2} x^{2} + 1} a^{2} c^{3} + \frac{{\left (a^{3} c^{3} - \frac{8 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )} a c^{3}}{x}\right )} a^{4} x^{2}}{8 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{2}{\left | a \right |}} + \frac{\frac{8 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )} a c^{3}{\left | a \right |}}{x} - \frac{{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{2} c^{3}{\left | a \right |}}{a x^{2}}}{8 \, a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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