Optimal. Leaf size=111 \[ -\frac{x^4 \sqrt{1-a^2 x^2}}{5 a}-\frac{x^3 \sqrt{1-a^2 x^2}}{4 a^2}-\frac{4 x^2 \sqrt{1-a^2 x^2}}{15 a^3}-\frac{(45 a x+64) \sqrt{1-a^2 x^2}}{120 a^5}+\frac{3 \sin ^{-1}(a x)}{8 a^5} \]
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Rubi [A] time = 0.10464, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {6124, 833, 780, 216} \[ -\frac{x^4 \sqrt{1-a^2 x^2}}{5 a}-\frac{x^3 \sqrt{1-a^2 x^2}}{4 a^2}-\frac{4 x^2 \sqrt{1-a^2 x^2}}{15 a^3}-\frac{(45 a x+64) \sqrt{1-a^2 x^2}}{120 a^5}+\frac{3 \sin ^{-1}(a x)}{8 a^5} \]
Antiderivative was successfully verified.
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Rule 6124
Rule 833
Rule 780
Rule 216
Rubi steps
\begin{align*} \int e^{\tanh ^{-1}(a x)} x^4 \, dx &=\int \frac{x^4 (1+a x)}{\sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{x^4 \sqrt{1-a^2 x^2}}{5 a}-\frac{\int \frac{x^3 \left (-4 a-5 a^2 x\right )}{\sqrt{1-a^2 x^2}} \, dx}{5 a^2}\\ &=-\frac{x^3 \sqrt{1-a^2 x^2}}{4 a^2}-\frac{x^4 \sqrt{1-a^2 x^2}}{5 a}+\frac{\int \frac{x^2 \left (15 a^2+16 a^3 x\right )}{\sqrt{1-a^2 x^2}} \, dx}{20 a^4}\\ &=-\frac{4 x^2 \sqrt{1-a^2 x^2}}{15 a^3}-\frac{x^3 \sqrt{1-a^2 x^2}}{4 a^2}-\frac{x^4 \sqrt{1-a^2 x^2}}{5 a}-\frac{\int \frac{x \left (-32 a^3-45 a^4 x\right )}{\sqrt{1-a^2 x^2}} \, dx}{60 a^6}\\ &=-\frac{4 x^2 \sqrt{1-a^2 x^2}}{15 a^3}-\frac{x^3 \sqrt{1-a^2 x^2}}{4 a^2}-\frac{x^4 \sqrt{1-a^2 x^2}}{5 a}-\frac{(64+45 a x) \sqrt{1-a^2 x^2}}{120 a^5}+\frac{3 \int \frac{1}{\sqrt{1-a^2 x^2}} \, dx}{8 a^4}\\ &=-\frac{4 x^2 \sqrt{1-a^2 x^2}}{15 a^3}-\frac{x^3 \sqrt{1-a^2 x^2}}{4 a^2}-\frac{x^4 \sqrt{1-a^2 x^2}}{5 a}-\frac{(64+45 a x) \sqrt{1-a^2 x^2}}{120 a^5}+\frac{3 \sin ^{-1}(a x)}{8 a^5}\\ \end{align*}
Mathematica [A] time = 0.0521778, size = 60, normalized size = 0.54 \[ \frac{45 \sin ^{-1}(a x)-\sqrt{1-a^2 x^2} \left (24 a^4 x^4+30 a^3 x^3+32 a^2 x^2+45 a x+64\right )}{120 a^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.039, size = 127, normalized size = 1.1 \begin{align*} -{\frac{{x}^{4}}{5\,a}\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{4\,{x}^{2}}{15\,{a}^{3}}\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{8}{15\,{a}^{5}}\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{{x}^{3}}{4\,{a}^{2}}\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{3\,x}{8\,{a}^{4}}\sqrt{-{a}^{2}{x}^{2}+1}}+{\frac{3}{8\,{a}^{4}}\arctan \left ({x\sqrt{{a}^{2}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ){\frac{1}{\sqrt{{a}^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.45737, size = 158, normalized size = 1.42 \begin{align*} -\frac{\sqrt{-a^{2} x^{2} + 1} x^{4}}{5 \, a} - \frac{\sqrt{-a^{2} x^{2} + 1} x^{3}}{4 \, a^{2}} - \frac{4 \, \sqrt{-a^{2} x^{2} + 1} x^{2}}{15 \, a^{3}} - \frac{3 \, \sqrt{-a^{2} x^{2} + 1} x}{8 \, a^{4}} + \frac{3 \, \arcsin \left (\frac{a^{2} x}{\sqrt{a^{2}}}\right )}{8 \, \sqrt{a^{2}} a^{4}} - \frac{8 \, \sqrt{-a^{2} x^{2} + 1}}{15 \, a^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.3653, size = 176, normalized size = 1.59 \begin{align*} -\frac{{\left (24 \, a^{4} x^{4} + 30 \, a^{3} x^{3} + 32 \, a^{2} x^{2} + 45 \, a x + 64\right )} \sqrt{-a^{2} x^{2} + 1} + 90 \, \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right )}{120 \, a^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 5.94124, size = 221, normalized size = 1.99 \begin{align*} a \left (\begin{cases} - \frac{x^{4} \sqrt{- a^{2} x^{2} + 1}}{5 a^{2}} - \frac{4 x^{2} \sqrt{- a^{2} x^{2} + 1}}{15 a^{4}} - \frac{8 \sqrt{- a^{2} x^{2} + 1}}{15 a^{6}} & \text{for}\: a \neq 0 \\\frac{x^{6}}{6} & \text{otherwise} \end{cases}\right ) + \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.26046, size = 92, normalized size = 0.83 \begin{align*} -\frac{1}{120} \, \sqrt{-a^{2} x^{2} + 1}{\left ({\left (2 \,{\left (3 \, x{\left (\frac{4 \, x}{a} + \frac{5}{a^{2}}\right )} + \frac{16}{a^{3}}\right )} x + \frac{45}{a^{4}}\right )} x + \frac{64}{a^{5}}\right )} + \frac{3 \, \arcsin \left (a x\right ) \mathrm{sgn}\left (a\right )}{8 \, a^{4}{\left | a \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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