Optimal. Leaf size=203 \[ \frac{(1-a x) (a x+1)^4}{5 a^5}+\frac{\sqrt{\frac{1-a x}{a x+1}} \left (5-6 \sqrt{\frac{1-a x}{a x+1}}\right ) (a x+1)^4}{10 a^5}-\frac{\left (45 \sqrt{\frac{1-a x}{a x+1}}+4\right ) (a x+1)^3}{30 a^5}+\frac{5 \sqrt{\frac{1-a x}{a x+1}} (a x+1)^2}{4 a^5}+\frac{\left (4-\sqrt{\frac{1-a x}{a x+1}}\right ) (a x+1)}{4 a^5}-\frac{\tan ^{-1}\left (\sqrt{\frac{1-a x}{a x+1}}\right )}{2 a^5} \]
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Rubi [A] time = 0.699971, antiderivative size = 203, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {6337, 1804, 1811, 1814, 639, 203} \[ \frac{(1-a x) (a x+1)^4}{5 a^5}+\frac{\sqrt{\frac{1-a x}{a x+1}} \left (5-6 \sqrt{\frac{1-a x}{a x+1}}\right ) (a x+1)^4}{10 a^5}-\frac{\left (45 \sqrt{\frac{1-a x}{a x+1}}+4\right ) (a x+1)^3}{30 a^5}+\frac{5 \sqrt{\frac{1-a x}{a x+1}} (a x+1)^2}{4 a^5}+\frac{\left (4-\sqrt{\frac{1-a x}{a x+1}}\right ) (a x+1)}{4 a^5}-\frac{\tan ^{-1}\left (\sqrt{\frac{1-a x}{a x+1}}\right )}{2 a^5} \]
Antiderivative was successfully verified.
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Rule 6337
Rule 1804
Rule 1811
Rule 1814
Rule 639
Rule 203
Rubi steps
\begin{align*} \int e^{2 \text{sech}^{-1}(a x)} x^4 \, dx &=\int x^4 \left (\frac{1}{a x}+\sqrt{\frac{1-a x}{1+a x}}+\frac{\sqrt{\frac{1-a x}{1+a x}}}{a x}\right )^2 \, dx\\ &=-\frac{4 \operatorname{Subst}\left (\int \frac{(-1+x)^2 x (1+x)^6}{\left (1+x^2\right )^6} \, dx,x,\sqrt{\frac{1-a x}{1+a x}}\right )}{a^5}\\ &=\frac{(1-a x) (1+a x)^4}{5 a^5}+\frac{2 \operatorname{Subst}\left (\int \frac{-42 x-40 x^2+130 x^3+80 x^4-30 x^5-40 x^6-10 x^7}{\left (1+x^2\right )^5} \, dx,x,\sqrt{\frac{1-a x}{1+a x}}\right )}{5 a^5}\\ &=\frac{(1-a x) (1+a x)^4}{5 a^5}+\frac{2 \operatorname{Subst}\left (\int \frac{x \left (-42-40 x+130 x^2+80 x^3-30 x^4-40 x^5-10 x^6\right )}{\left (1+x^2\right )^5} \, dx,x,\sqrt{\frac{1-a x}{1+a x}}\right )}{5 a^5}\\ &=\frac{(1-a x) (1+a x)^4}{5 a^5}+\frac{\sqrt{\frac{1-a x}{1+a x}} (1+a x)^4 \left (5-6 \sqrt{\frac{1-a x}{1+a x}}\right )}{10 a^5}-\frac{\operatorname{Subst}\left (\int \frac{160-48 x-960 x^2+160 x^3+320 x^4+80 x^5}{\left (1+x^2\right )^4} \, dx,x,\sqrt{\frac{1-a x}{1+a x}}\right )}{20 a^5}\\ &=\frac{(1-a x) (1+a x)^4}{5 a^5}+\frac{\sqrt{\frac{1-a x}{1+a x}} (1+a x)^4 \left (5-6 \sqrt{\frac{1-a x}{1+a x}}\right )}{10 a^5}-\frac{(1+a x)^3 \left (4+45 \sqrt{\frac{1-a x}{1+a x}}\right )}{30 a^5}+\frac{\operatorname{Subst}\left (\int \frac{480-480 x-1920 x^2-480 x^3}{\left (1+x^2\right )^3} \, dx,x,\sqrt{\frac{1-a x}{1+a x}}\right )}{120 a^5}\\ &=\frac{5 \sqrt{\frac{1-a x}{1+a x}} (1+a x)^2}{4 a^5}+\frac{(1-a x) (1+a x)^4}{5 a^5}+\frac{\sqrt{\frac{1-a x}{1+a x}} (1+a x)^4 \left (5-6 \sqrt{\frac{1-a x}{1+a x}}\right )}{10 a^5}-\frac{(1+a x)^3 \left (4+45 \sqrt{\frac{1-a x}{1+a x}}\right )}{30 a^5}-\frac{\operatorname{Subst}\left (\int \frac{480+1920 x}{\left (1+x^2\right )^2} \, dx,x,\sqrt{\frac{1-a x}{1+a x}}\right )}{480 a^5}\\ &=\frac{5 \sqrt{\frac{1-a x}{1+a x}} (1+a x)^2}{4 a^5}+\frac{(1-a x) (1+a x)^4}{5 a^5}+\frac{\sqrt{\frac{1-a x}{1+a x}} (1+a x)^4 \left (5-6 \sqrt{\frac{1-a x}{1+a x}}\right )}{10 a^5}+\frac{(1+a x) \left (4-\sqrt{\frac{1-a x}{1+a x}}\right )}{4 a^5}-\frac{(1+a x)^3 \left (4+45 \sqrt{\frac{1-a x}{1+a x}}\right )}{30 a^5}-\frac{\operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\sqrt{\frac{1-a x}{1+a x}}\right )}{2 a^5}\\ &=\frac{5 \sqrt{\frac{1-a x}{1+a x}} (1+a x)^2}{4 a^5}+\frac{(1-a x) (1+a x)^4}{5 a^5}+\frac{\sqrt{\frac{1-a x}{1+a x}} (1+a x)^4 \left (5-6 \sqrt{\frac{1-a x}{1+a x}}\right )}{10 a^5}+\frac{(1+a x) \left (4-\sqrt{\frac{1-a x}{1+a x}}\right )}{4 a^5}-\frac{(1+a x)^3 \left (4+45 \sqrt{\frac{1-a x}{1+a x}}\right )}{30 a^5}-\frac{\tan ^{-1}\left (\sqrt{\frac{1-a x}{1+a x}}\right )}{2 a^5}\\ \end{align*}
Mathematica [C] time = 0.141372, size = 105, normalized size = 0.52 \[ \frac{-12 a^5 x^5+40 a^3 x^3-15 a \sqrt{\frac{1-a x}{a x+1}} \left (-2 a^3 x^4-2 a^2 x^3+a x^2+x\right )+15 i \log \left (2 \sqrt{\frac{1-a x}{a x+1}} (a x+1)-2 i a x\right )}{60 a^5} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.206, size = 123, normalized size = 0.6 \begin{align*} -{\frac{{x}^{5}}{5}}+{\frac{2\,{x}^{3}}{3\,{a}^{2}}}+{\frac{x{\it csgn} \left ( a \right ) }{4\,{a}^{4}}\sqrt{-{\frac{ax-1}{ax}}}\sqrt{{\frac{ax+1}{ax}}} \left ( 2\,{\it csgn} \left ( a \right ){x}^{3}{a}^{3}\sqrt{-{a}^{2}{x}^{2}+1}-x\sqrt{-{a}^{2}{x}^{2}+1}{\it csgn} \left ( a \right ) a+\arctan \left ({x{\it csgn} \left ( a \right ) a{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ) \right ){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{2 \, x^{3}}{3 \, a^{2}} + \frac{2 \,{\left (-\frac{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} x}{4 \, a^{2}} + \frac{\sqrt{-a^{2} x^{2} + 1} x}{8 \, a^{2}} + \frac{\arcsin \left (\frac{a^{2} x}{\sqrt{a^{2}}}\right )}{8 \, \sqrt{a^{2}} a^{2}}\right )}}{a^{2}} - \int x^{4}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.09114, size = 225, normalized size = 1.11 \begin{align*} -\frac{12 \, a^{5} x^{5} - 40 \, a^{3} x^{3} - 15 \,{\left (2 \, a^{4} x^{4} - a^{2} x^{2}\right )} \sqrt{\frac{a x + 1}{a x}} \sqrt{-\frac{a x - 1}{a x}} + 15 \, \arctan \left (\sqrt{\frac{a x + 1}{a x}} \sqrt{-\frac{a x - 1}{a x}}\right )}{60 \, a^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 )}^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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