Integrand size = 12, antiderivative size = 169 \[ \int e^{2 \text {sech}^{-1}(a x)} x^2 \, dx=\frac {(1+a x) \left (1-\sqrt {\frac {1-a x}{1+a x}}\right ) \left (1+\sqrt {\frac {1-a x}{1+a x}}\right )}{2 a^3}-\frac {\sqrt {\frac {1-a x}{1+a x}} (1+a x)^2 \left (1+\sqrt {\frac {1-a x}{1+a x}}\right )^3}{6 a^3}+\frac {(1+a x)^3 \left (1+\sqrt {\frac {1-a x}{1+a x}}\right )^4}{12 a^3}-\frac {2 \arctan \left (\sqrt {\frac {1-a x}{1+a x}}\right )}{a^3} \]
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Time = 0.34 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6472, 835, 12, 743, 737, 209} \[ \int e^{2 \text {sech}^{-1}(a x)} x^2 \, dx=-\frac {2 \arctan \left (\sqrt {\frac {1-a x}{a x+1}}\right )}{a^3}+\frac {(a x+1)^3 \left (\sqrt {\frac {1-a x}{a x+1}}+1\right )^4}{12 a^3}-\frac {\sqrt {\frac {1-a x}{a x+1}} (a x+1)^2 \left (\sqrt {\frac {1-a x}{a x+1}}+1\right )^3}{6 a^3}+\frac {(a x+1) \left (1-\sqrt {\frac {1-a x}{a x+1}}\right ) \left (\sqrt {\frac {1-a x}{a x+1}}+1\right )}{2 a^3} \]
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Rule 12
Rule 209
Rule 737
Rule 743
Rule 835
Rule 6472
Rubi steps \begin{align*} \text {integral}& = \int x^2 \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 \text {Subst}\left (\int \frac {x (1+x)^4}{\left (1+x^2\right )^4} \, dx,x,\sqrt {\frac {1-a x}{1+a x}}\right )}{a^3} \\ & = \frac {(1+a x)^3 \left (1+\sqrt {\frac {1-a x}{1+a x}}\right )^4}{12 a^3}-\frac {2 \text {Subst}\left (\int \frac {4 (1+x)^3}{\left (1+x^2\right )^3} \, dx,x,\sqrt {\frac {1-a x}{1+a x}}\right )}{3 a^3} \\ & = \frac {(1+a x)^3 \left (1+\sqrt {\frac {1-a x}{1+a x}}\right )^4}{12 a^3}-\frac {8 \text {Subst}\left (\int \frac {(1+x)^3}{\left (1+x^2\right )^3} \, dx,x,\sqrt {\frac {1-a x}{1+a x}}\right )}{3 a^3} \\ & = -\frac {\sqrt {\frac {1-a x}{1+a x}} (1+a x)^2 \left (1+\sqrt {\frac {1-a x}{1+a x}}\right )^3}{6 a^3}+\frac {(1+a x)^3 \left (1+\sqrt {\frac {1-a x}{1+a x}}\right )^4}{12 a^3}-\frac {2 \text {Subst}\left (\int \frac {(1+x)^2}{\left (1+x^2\right )^2} \, dx,x,\sqrt {\frac {1-a x}{1+a x}}\right )}{a^3} \\ & = \frac {(1+a x) \left (1-\sqrt {\frac {1-a x}{1+a x}}\right ) \left (1+\sqrt {\frac {1-a x}{1+a x}}\right )}{2 a^3}-\frac {\sqrt {\frac {1-a x}{1+a x}} (1+a x)^2 \left (1+\sqrt {\frac {1-a x}{1+a x}}\right )^3}{6 a^3}+\frac {(1+a x)^3 \left (1+\sqrt {\frac {1-a x}{1+a x}}\right )^4}{12 a^3}-\frac {2 \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {\frac {1-a x}{1+a x}}\right )}{a^3} \\ & = \frac {(1+a x) \left (1-\sqrt {\frac {1-a x}{1+a x}}\right ) \left (1+\sqrt {\frac {1-a x}{1+a x}}\right )}{2 a^3}-\frac {\sqrt {\frac {1-a x}{1+a x}} (1+a x)^2 \left (1+\sqrt {\frac {1-a x}{1+a x}}\right )^3}{6 a^3}+\frac {(1+a x)^3 \left (1+\sqrt {\frac {1-a x}{1+a x}}\right )^4}{12 a^3}-\frac {2 \arctan \left (\sqrt {\frac {1-a x}{1+a x}}\right )}{a^3} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.08 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.51 \[ \int e^{2 \text {sech}^{-1}(a x)} x^2 \, dx=\frac {2 x}{a^2}-\frac {x^3}{3}+\sqrt {\frac {1-a x}{1+a x}} \left (\frac {x}{a^2}+\frac {x^2}{a}\right )+\frac {i \log \left (-2 i a x+2 \sqrt {\frac {1-a x}{1+a x}} (1+a x)\right )}{a^3} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.05 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.62
method | result | size |
default | \(\frac {-\frac {1}{3} a^{2} x^{3}+x}{a^{2}}+\frac {\sqrt {\frac {a x +1}{a x}}\, x \sqrt {-\frac {a x -1}{a x}}\, \left (\sqrt {-a^{2} x^{2}+1}\, x \,\operatorname {csgn}\left (a \right ) a +\arctan \left (\frac {\operatorname {csgn}\left (a \right ) a x}{\sqrt {-a^{2} x^{2}+1}}\right )\right ) \operatorname {csgn}\left (a \right )}{a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {x}{a^{2}}\) | \(105\) |
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Time = 0.25 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.51 \[ \int e^{2 \text {sech}^{-1}(a x)} x^2 \, dx=-\frac {a^{3} x^{3} - 3 \, a^{2} x^{2} \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}} - 6 \, a x + 3 \, \arctan \left (\sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}}\right )}{3 \, a^{3}} \]
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\[ \int e^{2 \text {sech}^{-1}(a x)} x^2 \, dx=\frac {\int 2\, dx + \int \left (- a^{2} x^{2}\right )\, dx + \int 2 a x \sqrt {-1 + \frac {1}{a x}} \sqrt {1 + \frac {1}{a x}}\, dx}{a^{2}} \]
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\[ \int e^{2 \text {sech}^{-1}(a x)} x^2 \, dx=\int { x^{2} {\left (\sqrt {\frac {1}{a x} + 1} \sqrt {\frac {1}{a x} - 1} + \frac {1}{a x}\right )}^{2} \,d x } \]
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\[ \int e^{2 \text {sech}^{-1}(a x)} x^2 \, dx=\int { x^{2} {\left (\sqrt {\frac {1}{a x} + 1} \sqrt {\frac {1}{a x} - 1} + \frac {1}{a x}\right )}^{2} \,d x } \]
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Time = 12.78 (sec) , antiderivative size = 420, normalized size of antiderivative = 2.49 \[ \int e^{2 \text {sech}^{-1}(a x)} x^2 \, dx=\frac {\frac {1{}\mathrm {i}}{16\,a^3}+\frac {{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^2\,1{}\mathrm {i}}{8\,a^3\,{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^2}-\frac {{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^4\,15{}\mathrm {i}}{16\,a^3\,{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^4}}{\frac {{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^2}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^2}+\frac {2\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^4}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^4}+\frac {{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^6}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^6}}-\frac {x^3\,\left (\frac {a^2}{3}-\frac {2}{x^2}\right )}{a^2}+\frac {\left (\ln \left (\frac {{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^2}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^2}+1\right )-\ln \left (\frac {\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}}{\sqrt {\frac {1}{a\,x}+1}-1}\right )\right )\,2{}\mathrm {i}}{a^3}+\frac {\ln \left (\frac {\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}}{\sqrt {\frac {1}{a\,x}+1}-1}\right )\,1{}\mathrm {i}}{a^3}-\frac {\ln \left (\frac {2\,a\,\sqrt {\frac {a+\frac {1}{x}}{a}}-\frac {2}{x}+a\,\sqrt {-\frac {a-\frac {1}{x}}{a}}\,2{}\mathrm {i}}{2\,a+\frac {1}{x}-2\,a\,\sqrt {\frac {a+\frac {1}{x}}{a}}}\right )\,1{}\mathrm {i}}{a^3}+\frac {{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^2\,1{}\mathrm {i}}{16\,a^3\,{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^2} \]
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