Optimal. Leaf size=163 \[ -\frac{\sqrt{\frac{1-a x}{a x+1}} (a x+1)^4}{4 a^4}+\frac{\left (9 \sqrt{\frac{1-a x}{a x+1}}+4\right ) (a x+1)^3}{12 a^4}-\frac{\left (5 \sqrt{\frac{1-a x}{a x+1}}+8\right ) (a x+1)^2}{8 a^4}+\frac{\left (\sqrt{\frac{1-a x}{a x+1}}+8\right ) (a x+1)}{8 a^4}+\frac{\tan ^{-1}\left (\sqrt{\frac{1-a x}{a x+1}}\right )}{4 a^4} \]
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Rubi [A] time = 0.56946, antiderivative size = 163, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {6337, 1804, 1814, 639, 203} \[ -\frac{\sqrt{\frac{1-a x}{a x+1}} (a x+1)^4}{4 a^4}+\frac{\left (9 \sqrt{\frac{1-a x}{a x+1}}+4\right ) (a x+1)^3}{12 a^4}-\frac{\left (5 \sqrt{\frac{1-a x}{a x+1}}+8\right ) (a x+1)^2}{8 a^4}+\frac{\left (\sqrt{\frac{1-a x}{a x+1}}+8\right ) (a x+1)}{8 a^4}+\frac{\tan ^{-1}\left (\sqrt{\frac{1-a x}{a x+1}}\right )}{4 a^4} \]
Antiderivative was successfully verified.
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Rule 6337
Rule 1804
Rule 1814
Rule 639
Rule 203
Rubi steps
\begin{align*} \int e^{-\text{sech}^{-1}(a x)} x^3 \, dx &=\int \frac{x^3}{\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)^4 x (1+x)^2}{\left (1+x^2\right )^5} \, dx,x,\sqrt{\frac{1-a x}{1+a x}}\right )}{a^4}\\ &=-\frac{\sqrt{\frac{1-a x}{1+a x}} (1+a x)^4}{4 a^4}+\frac{\operatorname{Subst}\left (\int \frac{8-8 x-48 x^2+16 x^3+16 x^4-8 x^5}{\left (1+x^2\right )^4} \, dx,x,\sqrt{\frac{1-a x}{1+a x}}\right )}{2 a^4}\\ &=-\frac{\sqrt{\frac{1-a x}{1+a x}} (1+a x)^4}{4 a^4}+\frac{(1+a x)^3 \left (4+9 \sqrt{\frac{1-a x}{1+a x}}\right )}{12 a^4}-\frac{\operatorname{Subst}\left (\int \frac{24-144 x-96 x^2+48 x^3}{\left (1+x^2\right )^3} \, dx,x,\sqrt{\frac{1-a x}{1+a x}}\right )}{12 a^4}\\ &=-\frac{\sqrt{\frac{1-a x}{1+a x}} (1+a x)^4}{4 a^4}-\frac{(1+a x)^2 \left (8+5 \sqrt{\frac{1-a x}{1+a x}}\right )}{8 a^4}+\frac{(1+a x)^3 \left (4+9 \sqrt{\frac{1-a x}{1+a x}}\right )}{12 a^4}+\frac{\operatorname{Subst}\left (\int \frac{24-192 x}{\left (1+x^2\right )^2} \, dx,x,\sqrt{\frac{1-a x}{1+a x}}\right )}{48 a^4}\\ &=-\frac{\sqrt{\frac{1-a x}{1+a x}} (1+a x)^4}{4 a^4}+\frac{(1+a x) \left (8+\sqrt{\frac{1-a x}{1+a x}}\right )}{8 a^4}-\frac{(1+a x)^2 \left (8+5 \sqrt{\frac{1-a x}{1+a x}}\right )}{8 a^4}+\frac{(1+a x)^3 \left (4+9 \sqrt{\frac{1-a x}{1+a x}}\right )}{12 a^4}+\frac{\operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\sqrt{\frac{1-a x}{1+a x}}\right )}{4 a^4}\\ &=-\frac{\sqrt{\frac{1-a x}{1+a x}} (1+a x)^4}{4 a^4}+\frac{(1+a x) \left (8+\sqrt{\frac{1-a x}{1+a x}}\right )}{8 a^4}-\frac{(1+a x)^2 \left (8+5 \sqrt{\frac{1-a x}{1+a x}}\right )}{8 a^4}+\frac{(1+a x)^3 \left (4+9 \sqrt{\frac{1-a x}{1+a x}}\right )}{12 a^4}+\frac{\tan ^{-1}\left (\sqrt{\frac{1-a x}{1+a x}}\right )}{4 a^4}\\ \end{align*}
Mathematica [C] time = 0.113146, size = 97, normalized size = 0.6 \[ \frac{8 a^3 x^3+3 a \sqrt{\frac{1-a x}{a x+1}} \left (-2 a^3 x^4-2 a^2 x^3+a x^2+x\right )-3 i \log \left (2 \sqrt{\frac{1-a x}{a x+1}} (a x+1)-2 i a x\right )}{24 a^4} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 180., size = 0, normalized size = 0. \begin{align*} \int{{x}^{3} \left ({\frac{1}{ax}}+\sqrt{{\frac{1}{ax}}-1}\sqrt{1+{\frac{1}{ax}}} \right ) ^{-1}}\, dx \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^{3}}{\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 = 2.06981, size = 203, normalized size = 1.25 \begin{align*} \frac{8 \, a^{3} x^{3} - 3 \,{\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}} + 3 \, \arctan \left (\sqrt{\frac{a x + 1}{a x}} \sqrt{-\frac{a x - 1}{a x}}\right )}{24 \, 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^{4}}{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^{3}}{\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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