Optimal. Leaf size=64 \[ \frac{x^2}{15 a^3}-\frac{x^3 e^{\text{sech}^{-1}(a x)}}{15 a^2}-\frac{2 x e^{\text{sech}^{-1}(a x)}}{15 a^4}+\frac{x^4}{20 a}+\frac{1}{5} x^5 e^{\text{sech}^{-1}(a x)} \]
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Rubi [A] time = 0.0384958, antiderivative size = 83, normalized size of antiderivative = 1.3, number of steps used = 5, number of rules used = 5, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {6335, 30, 100, 12, 74} \[ -\frac{x^2 \sqrt{1-a x}}{15 a^3 \sqrt{\frac{1}{a x+1}}}-\frac{2 \sqrt{1-a x}}{15 a^5 \sqrt{\frac{1}{a x+1}}}+\frac{x^4}{20 a}+\frac{1}{5} x^5 e^{\text{sech}^{-1}(a x)} \]
Warning: Unable to verify antiderivative.
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Rule 6335
Rule 30
Rule 100
Rule 12
Rule 74
Rubi steps
\begin{align*} \int e^{\text{sech}^{-1}(a x)} x^4 \, dx &=\frac{1}{5} e^{\text{sech}^{-1}(a x)} x^5+\frac{\int x^3 \, dx}{5 a}+\frac{\left (\sqrt{\frac{1}{1+a x}} \sqrt{1+a x}\right ) \int \frac{x^3}{\sqrt{1-a x} \sqrt{1+a x}} \, dx}{5 a}\\ &=\frac{x^4}{20 a}+\frac{1}{5} e^{\text{sech}^{-1}(a x)} x^5-\frac{x^2 \sqrt{1-a x}}{15 a^3 \sqrt{\frac{1}{1+a x}}}-\frac{\left (\sqrt{\frac{1}{1+a x}} \sqrt{1+a x}\right ) \int -\frac{2 x}{\sqrt{1-a x} \sqrt{1+a x}} \, dx}{15 a^3}\\ &=\frac{x^4}{20 a}+\frac{1}{5} e^{\text{sech}^{-1}(a x)} x^5-\frac{x^2 \sqrt{1-a x}}{15 a^3 \sqrt{\frac{1}{1+a x}}}+\frac{\left (2 \sqrt{\frac{1}{1+a x}} \sqrt{1+a x}\right ) \int \frac{x}{\sqrt{1-a x} \sqrt{1+a x}} \, dx}{15 a^3}\\ &=\frac{x^4}{20 a}+\frac{1}{5} e^{\text{sech}^{-1}(a x)} x^5-\frac{2 \sqrt{1-a x}}{15 a^5 \sqrt{\frac{1}{1+a x}}}-\frac{x^2 \sqrt{1-a x}}{15 a^3 \sqrt{\frac{1}{1+a x}}}\\ \end{align*}
Mathematica [A] time = 0.0899093, size = 65, normalized size = 1.02 \[ \frac{15 a^4 x^4+4 \sqrt{\frac{1-a x}{a x+1}} (a x+1)^2 \left (3 a^3 x^3-3 a^2 x^2+2 a x-2\right )}{60 a^5} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.189, size = 64, normalized size = 1. \begin{align*}{\frac{x \left ({a}^{2}{x}^{2}-1 \right ) \left ( 3\,{a}^{2}{x}^{2}+2 \right ) }{15\,{a}^{4}}\sqrt{-{\frac{ax-1}{ax}}}\sqrt{{\frac{ax+1}{ax}}}}+{\frac{{x}^{4}}{4\,a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.07833, size = 63, normalized size = 0.98 \begin{align*} \frac{x^{4}}{4 \, a} + \frac{{\left (3 \, a^{4} x^{4} - a^{2} x^{2} - 2\right )} \sqrt{a x + 1} \sqrt{-a x + 1}}{15 \, a^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.71408, size = 135, normalized size = 2.11 \begin{align*} \frac{15 \, a^{3} x^{4} + 4 \,{\left (3 \, a^{4} x^{5} - a^{2} x^{3} - 2 \, x\right )} \sqrt{\frac{a x + 1}{a x}} \sqrt{-\frac{a x - 1}{a x}}}{60 \, a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int x^{3}\, dx + \int a x^{4} \sqrt{-1 + \frac{1}{a x}} \sqrt{1 + \frac{1}{a x}}\, dx}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{4}{\left (\sqrt{\frac{1}{a x} + 1} \sqrt{\frac{1}{a x} - 1} + \frac{1}{a x}\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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