Integrand size = 8, antiderivative size = 53 \[ \int e^{\text {sech}^{-1}(a x)} x \, dx=\frac {x}{2 a}+\frac {1}{2} e^{\text {sech}^{-1}(a x)} x^2+\frac {\sqrt {\frac {1}{1+a x}} \sqrt {1+a x} \arcsin (a x)}{2 a^2} \]
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Time = 0.01 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6470, 8, 41, 222} \[ \int e^{\text {sech}^{-1}(a x)} x \, dx=\frac {\sqrt {\frac {1}{a x+1}} \sqrt {a x+1} \arcsin (a x)}{2 a^2}+\frac {1}{2} x^2 e^{\text {sech}^{-1}(a x)}+\frac {x}{2 a} \]
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Rule 8
Rule 41
Rule 222
Rule 6470
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} e^{\text {sech}^{-1}(a x)} x^2+\frac {\int 1 \, dx}{2 a}+\frac {\left (\sqrt {\frac {1}{1+a x}} \sqrt {1+a x}\right ) \int \frac {1}{\sqrt {1-a x} \sqrt {1+a x}} \, dx}{2 a} \\ & = \frac {x}{2 a}+\frac {1}{2} e^{\text {sech}^{-1}(a x)} x^2+\frac {\left (\sqrt {\frac {1}{1+a x}} \sqrt {1+a x}\right ) \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{2 a} \\ & = \frac {x}{2 a}+\frac {1}{2} e^{\text {sech}^{-1}(a x)} x^2+\frac {\sqrt {\frac {1}{1+a x}} \sqrt {1+a x} \arcsin (a x)}{2 a^2} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.07 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.42 \[ \int e^{\text {sech}^{-1}(a x)} x \, dx=\frac {2 a x+a x \sqrt {\frac {1-a x}{1+a x}} (1+a x)+i \log \left (-2 i a x+2 \sqrt {\frac {1-a x}{1+a x}} (1+a x)\right )}{2 a^2} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.05 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.74
method | result | size |
default | \(\frac {\sqrt {\frac {a x +1}{a x}}\, x \sqrt {-\frac {a x -1}{a x}}\, \left (\sqrt {-a^{2} x^{2}+1}\, x \,\operatorname {csgn}\left (a \right ) a +\arctan \left (\frac {\operatorname {csgn}\left (a \right ) a x}{\sqrt {-a^{2} x^{2}+1}}\right )\right ) \operatorname {csgn}\left (a \right )}{2 \sqrt {-a^{2} x^{2}+1}\, a}+\frac {x}{a}\) | \(92\) |
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Time = 0.25 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.49 \[ \int e^{\text {sech}^{-1}(a x)} x \, dx=\frac {a^{2} x^{2} \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}} + 2 \, a x - \arctan \left (\sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}}\right )}{2 \, a^{2}} \]
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\[ \int e^{\text {sech}^{-1}(a x)} x \, dx=\frac {\int 1\, dx + \int a x \sqrt {-1 + \frac {1}{a x}} \sqrt {1 + \frac {1}{a x}}\, dx}{a} \]
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\[ \int e^{\text {sech}^{-1}(a x)} x \, dx=\int { x {\left (\sqrt {\frac {1}{a x} + 1} \sqrt {\frac {1}{a x} - 1} + \frac {1}{a x}\right )} \,d x } \]
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\[ \int e^{\text {sech}^{-1}(a x)} x \, dx=\int { x {\left (\sqrt {\frac {1}{a x} + 1} \sqrt {\frac {1}{a x} - 1} + \frac {1}{a x}\right )} \,d x } \]
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Time = 10.97 (sec) , antiderivative size = 303, normalized size of antiderivative = 5.72 \[ \int e^{\text {sech}^{-1}(a x)} x \, dx=\frac {\ln \left (\frac {{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^2}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^2}+1\right )\,1{}\mathrm {i}}{2\,a^2}-\frac {\ln \left (\frac {\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}}{\sqrt {\frac {1}{a\,x}+1}-1}\right )\,1{}\mathrm {i}}{2\,a^2}+\frac {\frac {1{}\mathrm {i}}{32\,a^2}+\frac {{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^2\,1{}\mathrm {i}}{16\,a^2\,{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^2}-\frac {{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^4\,15{}\mathrm {i}}{32\,a^2\,{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^4}}{\frac {{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^2}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^2}+\frac {2\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^4}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^4}+\frac {{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^6}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^6}}+\frac {x}{a}+\frac {{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^2\,1{}\mathrm {i}}{32\,a^2\,{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^2} \]
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