Optimal. Leaf size=53 \[ \frac {\sqrt {\frac {1}{a x+1}} \sqrt {a x+1} \sin ^{-1}(a x)}{2 a^2}+\frac {1}{2} x^2 e^{\text {sech}^{-1}(a x)}+\frac {x}{2 a} \]
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Rubi [A] time = 0.02, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6335, 8, 41, 216} \[ \frac {\sqrt {\frac {1}{a x+1}} \sqrt {a x+1} \sin ^{-1}(a x)}{2 a^2}+\frac {1}{2} x^2 e^{\text {sech}^{-1}(a x)}+\frac {x}{2 a} \]
Antiderivative was successfully verified.
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Rule 8
Rule 41
Rule 216
Rule 6335
Rubi steps
\begin {align*} \int e^{\text {sech}^{-1}(a x)} x \, dx &=\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} \sin ^{-1}(a x)}{2 a^2}\\ \end {align*}
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Mathematica [C] time = 0.08, size = 75, normalized size = 1.42 \[ \frac {2 a x+a x \sqrt {\frac {1-a x}{a x+1}} (a x+1)+i \log \left (2 \sqrt {\frac {1-a x}{a x+1}} (a x+1)-2 i a x\right )}{2 a^2} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.62, size = 79, normalized size = 1.49 \[ \frac {a^{2} x^{2} \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}} + 2 \, a x - \arctan \left (\sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}}\right )}{2 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x {\left (\sqrt {\frac {1}{a x} + 1} \sqrt {\frac {1}{a x} - 1} + \frac {1}{a x}\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.05, size = 92, normalized size = 1.74 \[ \frac {\sqrt {-\frac {a x -1}{a x}}\, x \sqrt {\frac {a x +1}{a x}}\, \left (x \sqrt {-a^{2} x^{2}+1}\, \mathrm {csgn}\relax (a ) a +\arctan \left (\frac {\mathrm {csgn}\relax (a ) a x}{\sqrt {-a^{2} x^{2}+1}}\right )\right ) \mathrm {csgn}\relax (a )}{2 \sqrt {-a^{2} x^{2}+1}\, a}+\frac {x}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {x}{a} + \frac {\frac {1}{2} \, \sqrt {-a^{2} x^{2} + 1} x + \frac {\arcsin \left (a x\right )}{2 \, a}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.96, size = 303, normalized size = 5.72 \[ \frac {\ln \left (\frac {{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^2}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^2}+1\right )\,1{}\mathrm {i}}{2\,a^2}-\frac {\ln \left (\frac {\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}}{\sqrt {\frac {1}{a\,x}+1}-1}\right )\,1{}\mathrm {i}}{2\,a^2}+\frac {\frac {1{}\mathrm {i}}{32\,a^2}+\frac {{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^2\,1{}\mathrm {i}}{16\,a^2\,{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^2}-\frac {{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^4\,15{}\mathrm {i}}{32\,a^2\,{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^4}}{\frac {{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^2}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^2}+\frac {2\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^4}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^4}+\frac {{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^6}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^6}}+\frac {x}{a}+\frac {{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^2\,1{}\mathrm {i}}{32\,a^2\,{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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