Optimal. Leaf size=39 \[ -\frac {\sqrt {1-a^2 x^2} (2-a x)}{2 a^2}-\frac {\sin ^{-1}(a x)}{2 a^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {6124, 780, 216} \[ -\frac {\sqrt {1-a^2 x^2} (2-a x)}{2 a^2}-\frac {\sin ^{-1}(a x)}{2 a^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 216
Rule 780
Rule 6124
Rubi steps
\begin {align*} \int e^{-\tanh ^{-1}(a x)} x \, dx &=\int \frac {x (1-a x)}{\sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {(2-a x) \sqrt {1-a^2 x^2}}{2 a^2}-\frac {\int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{2 a}\\ &=-\frac {(2-a x) \sqrt {1-a^2 x^2}}{2 a^2}-\frac {\sin ^{-1}(a x)}{2 a^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.03, size = 34, normalized size = 0.87 \[ \frac {(a x-2) \sqrt {1-a^2 x^2}-\sin ^{-1}(a x)}{2 a^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.78, size = 48, normalized size = 1.23 \[ \frac {\sqrt {-a^{2} x^{2} + 1} {\left (a x - 2\right )} + 2 \, \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right )}{2 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.17, size = 41, normalized size = 1.05 \[ \frac {1}{2} \, \sqrt {-a^{2} x^{2} + 1} {\left (\frac {x}{a} - \frac {2}{a^{2}}\right )} - \frac {\arcsin \left (a x\right ) \mathrm {sgn}\relax (a)}{2 \, a {\left | a \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.04, size = 119, normalized size = 3.05 \[ \frac {x \sqrt {-a^{2} x^{2}+1}}{2 a}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 a \sqrt {a^{2}}}-\frac {\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{a^{2}}-\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}\right )}{a \sqrt {a^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.41, size = 45, normalized size = 1.15 \[ \frac {\sqrt {-a^{2} x^{2} + 1} x}{2 \, a} - \frac {\arcsin \left (a x\right )}{2 \, a^{2}} - \frac {\sqrt {-a^{2} x^{2} + 1}}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.78, size = 58, normalized size = 1.49 \[ \frac {\sqrt {1-a^2\,x^2}\,\left (\frac {1}{\sqrt {-a^2}}+\frac {x\,\sqrt {-a^2}}{2\,a}\right )-\frac {\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{2\,a}}{\sqrt {-a^2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}{a x + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________