Optimal. Leaf size=147 \[ -\frac{\sqrt{\frac{1-a x}{a x+1}} (a x+1)^5}{5 a^5}+\frac{\left (16 \sqrt{\frac{1-a x}{a x+1}}+5\right ) (a x+1)^4}{20 a^5}-\frac{\left (17 \sqrt{\frac{1-a x}{a x+1}}+15\right ) (a x+1)^3}{15 a^5}+\frac{\left (4 \sqrt{\frac{1-a x}{a x+1}}+9\right ) (a x+1)^2}{6 a^5}-\frac{x}{a^4} \]
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Rubi [A] time = 0.619208, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {6337, 1804, 1814, 12, 261} \[ -\frac{\sqrt{\frac{1-a x}{a x+1}} (a x+1)^5}{5 a^5}+\frac{\left (16 \sqrt{\frac{1-a x}{a x+1}}+5\right ) (a x+1)^4}{20 a^5}-\frac{\left (17 \sqrt{\frac{1-a x}{a x+1}}+15\right ) (a x+1)^3}{15 a^5}+\frac{\left (4 \sqrt{\frac{1-a x}{a x+1}}+9\right ) (a x+1)^2}{6 a^5}-\frac{x}{a^4} \]
Antiderivative was successfully verified.
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Rule 6337
Rule 1804
Rule 1814
Rule 12
Rule 261
Rubi steps
\begin{align*} \int e^{-\text{sech}^{-1}(a x)} x^4 \, dx &=\int \frac{x^4}{\frac{1}{a x}+\sqrt{\frac{1-a x}{1+a x}}+\frac{\sqrt{\frac{1-a x}{1+a x}}}{a x}} \, dx\\ &=\frac{4 \operatorname{Subst}\left (\int \frac{(-1+x)^5 x (1+x)^3}{\left (1+x^2\right )^6} \, dx,x,\sqrt{\frac{1-a x}{1+a x}}\right )}{a^5}\\ &=-\frac{\sqrt{\frac{1-a x}{1+a x}} (1+a x)^5}{5 a^5}-\frac{2 \operatorname{Subst}\left (\int \frac{-16+10 x+140 x^2-30 x^3-80 x^4+30 x^5+20 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{\sqrt{\frac{1-a x}{1+a x}} (1+a x)^5}{5 a^5}+\frac{(1+a x)^4 \left (5+16 \sqrt{\frac{1-a x}{1+a x}}\right )}{20 a^5}+\frac{\operatorname{Subst}\left (\int \frac{-128+560 x+800 x^2-320 x^3-160 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{\sqrt{\frac{1-a x}{1+a x}} (1+a x)^5}{5 a^5}+\frac{(1+a x)^4 \left (5+16 \sqrt{\frac{1-a x}{1+a x}}\right )}{20 a^5}-\frac{(1+a x)^3 \left (15+17 \sqrt{\frac{1-a x}{1+a x}}\right )}{15 a^5}-\frac{\operatorname{Subst}\left (\int \frac{-320+2400 x+960 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{\sqrt{\frac{1-a x}{1+a x}} (1+a x)^5}{5 a^5}+\frac{(1+a x)^2 \left (9+4 \sqrt{\frac{1-a x}{1+a x}}\right )}{6 a^5}+\frac{(1+a x)^4 \left (5+16 \sqrt{\frac{1-a x}{1+a x}}\right )}{20 a^5}-\frac{(1+a x)^3 \left (15+17 \sqrt{\frac{1-a x}{1+a x}}\right )}{15 a^5}+\frac{\operatorname{Subst}\left (\int \frac{1920 x}{\left (1+x^2\right )^2} \, dx,x,\sqrt{\frac{1-a x}{1+a x}}\right )}{480 a^5}\\ &=-\frac{\sqrt{\frac{1-a x}{1+a x}} (1+a x)^5}{5 a^5}+\frac{(1+a x)^2 \left (9+4 \sqrt{\frac{1-a x}{1+a x}}\right )}{6 a^5}+\frac{(1+a x)^4 \left (5+16 \sqrt{\frac{1-a x}{1+a x}}\right )}{20 a^5}-\frac{(1+a x)^3 \left (15+17 \sqrt{\frac{1-a x}{1+a x}}\right )}{15 a^5}+\frac{4 \operatorname{Subst}\left (\int \frac{x}{\left (1+x^2\right )^2} \, dx,x,\sqrt{\frac{1-a x}{1+a x}}\right )}{a^5}\\ &=-\frac{x}{a^4}-\frac{\sqrt{\frac{1-a x}{1+a x}} (1+a x)^5}{5 a^5}+\frac{(1+a x)^2 \left (9+4 \sqrt{\frac{1-a x}{1+a x}}\right )}{6 a^5}+\frac{(1+a x)^4 \left (5+16 \sqrt{\frac{1-a x}{1+a x}}\right )}{20 a^5}-\frac{(1+a x)^3 \left (15+17 \sqrt{\frac{1-a x}{1+a x}}\right )}{15 a^5}\\ \end{align*}
Mathematica [A] time = 0.0900951, size = 65, normalized size = 0.44 \[ \frac{15 a^4 x^4-4 \sqrt{\frac{1-a x}{a x+1}} (a x+1)^2 \left (3 a^3 x^3-3 a^2 x^2+2 a x-2\right )}{60 a^5} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.54, size = 531, normalized size = 3.6 \begin{align*} -{\frac{ax-1}{60\,{x}^{7}{a}^{12}} \left ( 15\,{a}^{10}{x}^{10} \left ({\frac{ax+1}{ax}} \right ) ^{7/2} \left ( -{\frac{ax-1}{ax}} \right ) ^{5/2}+30\,{a}^{8}{x}^{8} \left ({\frac{ax+1}{ax}} \right ) ^{7/2} \left ( -{\frac{ax-1}{ax}} \right ) ^{5/2}+30\, \left ({\frac{ax+1}{ax}} \right ) ^{7/2} \left ( -{\frac{ax-1}{ax}} \right ) ^{3/2}{x}^{8}{a}^{8}+30\,{x}^{6}\ln \left ({a}^{2}{x}^{2} \right ) \left ({\frac{ax+1}{ax}} \right ) ^{7/2} \left ( -{\frac{ax-1}{ax}} \right ) ^{5/2}{a}^{6}-30\,{a}^{7}{x}^{7} \left ({\frac{ax+1}{ax}} \right ) ^{7/2} \left ( -{\frac{ax-1}{ax}} \right ) ^{3/2}+60\, \left ({\frac{ax+1}{ax}} \right ) ^{7/2} \left ( -{\frac{ax-1}{ax}} \right ) ^{3/2}\ln \left ({a}^{2}{x}^{2} \right ){x}^{6}{a}^{6}+12\,{x}^{11}{a}^{11}-60\,{x}^{5}\ln \left ({a}^{2}{x}^{2} \right ) \left ({\frac{ax+1}{ax}} \right ) ^{7/2} \left ( -{\frac{ax-1}{ax}} \right ) ^{3/2}{a}^{5}+30\, \left ({\frac{ax+1}{ax}} \right ) ^{7/2}\sqrt{-{\frac{ax-1}{ax}}}\ln \left ({a}^{2}{x}^{2} \right ){x}^{6}{a}^{6}+12\,{x}^{10}{a}^{10}-60\, \left ({\frac{ax+1}{ax}} \right ) ^{7/2}\sqrt{-{\frac{ax-1}{ax}}}\ln \left ({a}^{2}{x}^{2} \right ){x}^{5}{a}^{5}-40\,{x}^{9}{a}^{9}+30\,{x}^{4}\ln \left ({a}^{2}{x}^{2} \right ) \left ({\frac{ax+1}{ax}} \right ) ^{7/2}\sqrt{-{\frac{ax-1}{ax}}}{a}^{4}-40\,{x}^{8}{a}^{8}+40\,{a}^{7}{x}^{7}+40\,{x}^{6}{a}^{6}-20\,{x}^{3}{a}^{3}-20\,{a}^{2}{x}^{2}+8\,ax+8 \right ) \left ({\frac{ax+1}{ax}} \right ) ^{-{\frac{7}{2}}} \left ( -{\frac{ax-1}{ax}} \right ) ^{-{\frac{7}{2}}}} \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*} \int \frac{x^{4}}{\sqrt{\frac{1}{a x} + 1} \sqrt{\frac{1}{a x} - 1} + \frac{1}{a x}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.03716, size = 135, normalized size = 0.92 \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*} a \int \frac{x^{5}}{a x \sqrt{-1 + \frac{1}{a x}} \sqrt{1 + \frac{1}{a x}} + 1}\, dx \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 \frac{x^{4}}{\sqrt{\frac{1}{a x} + 1} \sqrt{\frac{1}{a x} - 1} + \frac{1}{a x}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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