Optimal. Leaf size=10 \[ \operatorname {ExpIntegralEi}\left (\frac {x}{x^2+2}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [F] time = 0.41, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {e^{\frac {x}{2+x^2}} \left (2-x^2\right )}{2 x+x^3} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
Rubi steps
\begin {align*} \int \frac {e^{\frac {x}{2+x^2}} \left (2-x^2\right )}{2 x+x^3} \, dx &=\int \frac {e^{\frac {x}{2+x^2}} \left (2-x^2\right )}{x \left (2+x^2\right )} \, dx\\ &=\int \left (\frac {e^{\frac {x}{2+x^2}}}{x}-\frac {2 e^{\frac {x}{2+x^2}} x}{2+x^2}\right ) \, dx\\ &=-\left (2 \int \frac {e^{\frac {x}{2+x^2}} x}{2+x^2} \, dx\right )+\int \frac {e^{\frac {x}{2+x^2}}}{x} \, dx\\ &=-\left (2 \int \left (-\frac {e^{\frac {x}{2+x^2}}}{2 \left (i \sqrt {2}-x\right )}+\frac {e^{\frac {x}{2+x^2}}}{2 \left (i \sqrt {2}+x\right )}\right ) \, dx\right )+\int \frac {e^{\frac {x}{2+x^2}}}{x} \, dx\\ &=\int \frac {e^{\frac {x}{2+x^2}}}{i \sqrt {2}-x} \, dx+\int \frac {e^{\frac {x}{2+x^2}}}{x} \, dx-\int \frac {e^{\frac {x}{2+x^2}}}{i \sqrt {2}+x} \, dx\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.11, size = 10, normalized size = 1.00 \[ \operatorname {ExpIntegralEi}\left (\frac {x}{x^2+2}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.43, size = 10, normalized size = 1.00 \[ {\rm Ei}\left (\frac {x}{x^{2} + 2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int -\frac {{\left (x^{2} - 2\right )} e^{\left (\frac {x}{x^{2} + 2}\right )}}{x^{3} + 2 \, x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.10, size = 0, normalized size = 0.00 \[ \int \frac {\left (-x^{2}+2\right ) {\mathrm e}^{\frac {x}{x^{2}+2}}}{x^{3}+2 x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\int \frac {{\left (x^{2} - 2\right )} e^{\left (\frac {x}{x^{2} + 2}\right )}}{x^{3} + 2 \, x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.24, size = 10, normalized size = 1.00 \[ \mathrm {ei}\left (\frac {x}{x^2+2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ - \int \left (- \frac {2 e^{\frac {x}{x^{2} + 2}}}{x^{3} + 2 x}\right )\, dx - \int \frac {x^{2} e^{\frac {x}{x^{2} + 2}}}{x^{3} + 2 x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________