Optimal. Leaf size=101 \[ \frac{1}{4} a c^4 x \left (1-a^2 x^2\right )^{3/2}-c^4 \left (1-a^2 x^2\right )^{3/2}+\frac{1}{8} c^4 (8-13 a x) \sqrt{1-a^2 x^2}-c^4 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )-\frac{13}{8} c^4 \sin ^{-1}(a x) \]
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Rubi [A] time = 0.246412, antiderivative size = 101, 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, 1809, 815, 844, 216, 266, 63, 208} \[ \frac{1}{4} a c^4 x \left (1-a^2 x^2\right )^{3/2}-c^4 \left (1-a^2 x^2\right )^{3/2}+\frac{1}{8} c^4 (8-13 a x) \sqrt{1-a^2 x^2}-c^4 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )-\frac{13}{8} c^4 \sin ^{-1}(a x) \]
Antiderivative was successfully verified.
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Rule 6128
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} \, dx &=c \int \frac{(c-a c x)^3 \sqrt{1-a^2 x^2}}{x} \, dx\\ &=\frac{1}{4} a c^4 x \left (1-a^2 x^2\right )^{3/2}-\frac{c \int \frac{\sqrt{1-a^2 x^2} \left (-4 a^2 c^3+13 a^3 c^3 x-12 a^4 c^3 x^2\right )}{x} \, dx}{4 a^2}\\ &=-c^4 \left (1-a^2 x^2\right )^{3/2}+\frac{1}{4} a c^4 x \left (1-a^2 x^2\right )^{3/2}+\frac{c \int \frac{\left (12 a^4 c^3-39 a^5 c^3 x\right ) \sqrt{1-a^2 x^2}}{x} \, dx}{12 a^4}\\ &=\frac{1}{8} c^4 (8-13 a x) \sqrt{1-a^2 x^2}-c^4 \left (1-a^2 x^2\right )^{3/2}+\frac{1}{4} a c^4 x \left (1-a^2 x^2\right )^{3/2}-\frac{c \int \frac{-24 a^6 c^3+39 a^7 c^3 x}{x \sqrt{1-a^2 x^2}} \, dx}{24 a^6}\\ &=\frac{1}{8} c^4 (8-13 a x) \sqrt{1-a^2 x^2}-c^4 \left (1-a^2 x^2\right )^{3/2}+\frac{1}{4} a c^4 x \left (1-a^2 x^2\right )^{3/2}+c^4 \int \frac{1}{x \sqrt{1-a^2 x^2}} \, dx-\frac{1}{8} \left (13 a c^4\right ) \int \frac{1}{\sqrt{1-a^2 x^2}} \, dx\\ &=\frac{1}{8} c^4 (8-13 a x) \sqrt{1-a^2 x^2}-c^4 \left (1-a^2 x^2\right )^{3/2}+\frac{1}{4} a c^4 x \left (1-a^2 x^2\right )^{3/2}-\frac{13}{8} c^4 \sin ^{-1}(a x)+\frac{1}{2} c^4 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-a^2 x}} \, dx,x,x^2\right )\\ &=\frac{1}{8} c^4 (8-13 a x) \sqrt{1-a^2 x^2}-c^4 \left (1-a^2 x^2\right )^{3/2}+\frac{1}{4} a c^4 x \left (1-a^2 x^2\right )^{3/2}-\frac{13}{8} c^4 \sin ^{-1}(a x)-\frac{c^4 \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}{8} c^4 (8-13 a x) \sqrt{1-a^2 x^2}-c^4 \left (1-a^2 x^2\right )^{3/2}+\frac{1}{4} a c^4 x \left (1-a^2 x^2\right )^{3/2}-\frac{13}{8} c^4 \sin ^{-1}(a x)-c^4 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )\\ \end{align*}
Mathematica [A] time = 0.0961237, size = 142, normalized size = 1.41 \[ \frac{c^4 \left (2 a^5 x^5-8 a^4 x^4+9 a^3 x^3+8 a^2 x^2+4 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)+34 \sqrt{1-a^2 x^2} \sin ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{2}}\right )-8 \sqrt{1-a^2 x^2} \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )-11 a x\right )}{8 \sqrt{1-a^2 x^2}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.04, size = 115, normalized size = 1.1 \begin{align*} -{\frac{{c}^{4}{a}^{3}{x}^{3}}{4}\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{11\,{c}^{4}ax}{8}\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{13\,{c}^{4}a}{8}\arctan \left ({x\sqrt{{a}^{2}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ){\frac{1}{\sqrt{{a}^{2}}}}}+{c}^{4}{a}^{2}{x}^{2}\sqrt{-{a}^{2}{x}^{2}+1}-{c}^{4}{\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 = 1.44072, size = 159, normalized size = 1.57 \begin{align*} -\frac{1}{4} \, \sqrt{-a^{2} x^{2} + 1} a^{3} c^{4} x^{3} + \sqrt{-a^{2} x^{2} + 1} a^{2} c^{4} x^{2} - \frac{11}{8} \, \sqrt{-a^{2} x^{2} + 1} a c^{4} x - \frac{13 \, a c^{4} \arcsin \left (\frac{a^{2} x}{\sqrt{a^{2}}}\right )}{8 \, \sqrt{a^{2}}} - c^{4} \log \left (\frac{2 \, \sqrt{-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac{2}{{\left | x \right |}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.66374, size = 212, normalized size = 2.1 \begin{align*} \frac{13}{4} \, c^{4} \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right ) + c^{4} \log \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right ) - \frac{1}{8} \,{\left (2 \, a^{3} c^{4} x^{3} - 8 \, a^{2} c^{4} x^{2} + 11 \, a c^{4} x\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 = 20.6904, size = 420, normalized size = 4.16 \begin{align*} a^{5} c^{4} \left (\begin{cases} - \frac{i x^{5}}{4 \sqrt{a^{2} x^{2} - 1}} - \frac{i x^{3}}{8 a^{2} \sqrt{a^{2} x^{2} - 1}} + \frac{3 i x}{8 a^{4} \sqrt{a^{2} x^{2} - 1}} - \frac{3 i \operatorname{acosh}{\left (a x \right )}}{8 a^{5}} & \text{for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac{x^{5}}{4 \sqrt{- a^{2} x^{2} + 1}} + \frac{x^{3}}{8 a^{2} \sqrt{- a^{2} x^{2} + 1}} - \frac{3 x}{8 a^{4} \sqrt{- a^{2} x^{2} + 1}} + \frac{3 \operatorname{asin}{\left (a x \right )}}{8 a^{5}} & \text{otherwise} \end{cases}\right ) - 3 a^{4} 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 ) + 2 a^{3} 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^{2} 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 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 ) + 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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.30434, size = 135, normalized size = 1.34 \begin{align*} -\frac{13 \, a c^{4} \arcsin \left (a x\right ) \mathrm{sgn}\left (a\right )}{8 \,{\left | a \right |}} - \frac{a 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{1}{8} \,{\left (11 \, a c^{4} + 2 \,{\left (a^{3} c^{4} x - 4 \, a^{2} c^{4}\right )} x\right )} \sqrt{-a^{2} x^{2} + 1} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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