Optimal. Leaf size=120 \[ \frac{3 a c^4 \left (1-a^2 x^2\right )^{3/2}}{2 x^2}-\frac{c^4 \left (1-a^2 x^2\right )^{3/2}}{3 x^3}-\frac{a^2 c^4 (6-a x) \sqrt{1-a^2 x^2}}{2 x}-\frac{1}{2} a^3 c^4 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )-3 a^3 c^4 \sin ^{-1}(a x) \]
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Rubi [A] time = 0.249044, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 9, 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} \[ \frac{3 a c^4 \left (1-a^2 x^2\right )^{3/2}}{2 x^2}-\frac{c^4 \left (1-a^2 x^2\right )^{3/2}}{3 x^3}-\frac{a^2 c^4 (6-a x) \sqrt{1-a^2 x^2}}{2 x}-\frac{1}{2} a^3 c^4 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )-3 a^3 c^4 \sin ^{-1}(a x) \]
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)^4}{x^4} \, dx &=c \int \frac{(c-a c x)^3 \sqrt{1-a^2 x^2}}{x^4} \, dx\\ &=-\frac{c^4 \left (1-a^2 x^2\right )^{3/2}}{3 x^3}-\frac{1}{3} c \int \frac{\sqrt{1-a^2 x^2} \left (9 a c^3-9 a^2 c^3 x+3 a^3 c^3 x^2\right )}{x^3} \, dx\\ &=-\frac{c^4 \left (1-a^2 x^2\right )^{3/2}}{3 x^3}+\frac{3 a c^4 \left (1-a^2 x^2\right )^{3/2}}{2 x^2}+\frac{1}{6} c \int \frac{\left (18 a^2 c^3+3 a^3 c^3 x\right ) \sqrt{1-a^2 x^2}}{x^2} \, dx\\ &=-\frac{a^2 c^4 (6-a x) \sqrt{1-a^2 x^2}}{2 x}-\frac{c^4 \left (1-a^2 x^2\right )^{3/2}}{3 x^3}+\frac{3 a c^4 \left (1-a^2 x^2\right )^{3/2}}{2 x^2}-\frac{1}{12} c \int \frac{-6 a^3 c^3+36 a^4 c^3 x}{x \sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{a^2 c^4 (6-a x) \sqrt{1-a^2 x^2}}{2 x}-\frac{c^4 \left (1-a^2 x^2\right )^{3/2}}{3 x^3}+\frac{3 a c^4 \left (1-a^2 x^2\right )^{3/2}}{2 x^2}+\frac{1}{2} \left (a^3 c^4\right ) \int \frac{1}{x \sqrt{1-a^2 x^2}} \, dx-\left (3 a^4 c^4\right ) \int \frac{1}{\sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{a^2 c^4 (6-a x) \sqrt{1-a^2 x^2}}{2 x}-\frac{c^4 \left (1-a^2 x^2\right )^{3/2}}{3 x^3}+\frac{3 a c^4 \left (1-a^2 x^2\right )^{3/2}}{2 x^2}-3 a^3 c^4 \sin ^{-1}(a x)+\frac{1}{4} \left (a^3 c^4\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-a^2 x}} \, dx,x,x^2\right )\\ &=-\frac{a^2 c^4 (6-a x) \sqrt{1-a^2 x^2}}{2 x}-\frac{c^4 \left (1-a^2 x^2\right )^{3/2}}{3 x^3}+\frac{3 a c^4 \left (1-a^2 x^2\right )^{3/2}}{2 x^2}-3 a^3 c^4 \sin ^{-1}(a x)-\frac{1}{2} \left (a 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 )\\ &=-\frac{a^2 c^4 (6-a x) \sqrt{1-a^2 x^2}}{2 x}-\frac{c^4 \left (1-a^2 x^2\right )^{3/2}}{3 x^3}+\frac{3 a c^4 \left (1-a^2 x^2\right )^{3/2}}{2 x^2}-3 a^3 c^4 \sin ^{-1}(a x)-\frac{1}{2} a^3 c^4 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )\\ \end{align*}
Mathematica [A] time = 0.135312, size = 164, normalized size = 1.37 \[ \frac{c^4 \left (12 a^5 x^5+32 a^4 x^4-30 a^3 x^3-28 a^2 x^2+3 a^3 x^3 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)+78 a^3 x^3 \sqrt{1-a^2 x^2} \sin ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{2}}\right )-6 a^3 x^3 \sqrt{1-a^2 x^2} \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )+18 a x-4\right )}{12 x^3 \sqrt{1-a^2 x^2}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.042, size = 140, normalized size = 1.2 \begin{align*} -{c}^{4}{a}^{3}\sqrt{-{a}^{2}{x}^{2}+1}-3\,{\frac{{c}^{4}{a}^{4}}{\sqrt{{a}^{2}}}\arctan \left ({\frac{\sqrt{{a}^{2}}x}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) }-{\frac{8\,{c}^{4}{a}^{2}}{3\,x}\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{{c}^{4}{a}^{3}}{2}{\it Artanh} \left ({\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) }+{\frac{3\,{c}^{4}a}{2\,{x}^{2}}\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{{c}^{4}}{3\,{x}^{3}}\sqrt{-{a}^{2}{x}^{2}+1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.43947, size = 193, normalized size = 1.61 \begin{align*} -\frac{3 \, a^{4} c^{4} \arcsin \left (\frac{a^{2} x}{\sqrt{a^{2}}}\right )}{\sqrt{a^{2}}} - \frac{1}{2} \, a^{3} c^{4} \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^{3} c^{4} - \frac{8 \, \sqrt{-a^{2} x^{2} + 1} a^{2} c^{4}}{3 \, x} + \frac{3 \, \sqrt{-a^{2} x^{2} + 1} a c^{4}}{2 \, x^{2}} - \frac{\sqrt{-a^{2} x^{2} + 1} c^{4}}{3 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.64987, size = 274, normalized size = 2.28 \begin{align*} \frac{36 \, a^{3} c^{4} x^{3} \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right ) + 3 \, a^{3} c^{4} x^{3} \log \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right ) - 6 \, a^{3} c^{4} x^{3} -{\left (6 \, a^{3} c^{4} x^{3} + 16 \, a^{2} c^{4} x^{2} - 9 \, a c^{4} x + 2 \, c^{4}\right )} \sqrt{-a^{2} x^{2} + 1}}{6 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 7.75275, size = 359, normalized size = 2.99 \begin{align*} a^{5} 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 ) - 3 a^{4} 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 ) + 2 a^{3} 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 ) + 2 a^{2} 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 ) - 3 a c^{4} \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 ) + c^{4} \left (\begin{cases} - \frac{2 i a^{2} \sqrt{a^{2} x^{2} - 1}}{3 x} - \frac{i \sqrt{a^{2} x^{2} - 1}}{3 x^{3}} & \text{for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac{2 a^{2} \sqrt{- a^{2} x^{2} + 1}}{3 x} - \frac{\sqrt{- a^{2} x^{2} + 1}}{3 x^{3}} & \text{otherwise} \end{cases}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.24267, size = 365, normalized size = 3.04 \begin{align*} -\frac{3 \, a^{4} c^{4} \arcsin \left (a x\right ) \mathrm{sgn}\left (a\right )}{{\left | a \right |}} - \frac{a^{4} 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 )}{2 \,{\left | a \right |}} - \sqrt{-a^{2} x^{2} + 1} a^{3} c^{4} + \frac{{\left (a^{4} c^{4} - \frac{9 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )} a^{2} c^{4}}{x} + \frac{33 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{2} c^{4}}{x^{2}}\right )} a^{6} x^{3}}{24 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{3}{\left | a \right |}} - \frac{\frac{33 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )} a^{4} c^{4}}{x} - \frac{9 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{2} a^{2} c^{4}}{x^{2}} + \frac{{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{3} c^{4}}{x^{3}}}{24 \, a^{2}{\left | a \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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