Optimal. Leaf size=26 \[ (2+x) \left (x+\frac {e^x+\frac {x^2}{\log \left (2+x^2\right )}}{x}\right ) \]
________________________________________________________________________________________
Rubi [F] time = 1.24, 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 = {} \begin {gather*} \int \frac {-4 x^4-2 x^5+\left (4 x^2+4 x^3+2 x^4+2 x^5\right ) \log \left (2+x^2\right )+\left (4 x^2+4 x^3+2 x^4+2 x^5+e^x \left (-4+4 x+2 x^3+x^4\right )\right ) \log ^2\left (2+x^2\right )}{\left (2 x^2+x^4\right ) \log ^2\left (2+x^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-4 x^4-2 x^5+\left (4 x^2+4 x^3+2 x^4+2 x^5\right ) \log \left (2+x^2\right )+\left (4 x^2+4 x^3+2 x^4+2 x^5+e^x \left (-4+4 x+2 x^3+x^4\right )\right ) \log ^2\left (2+x^2\right )}{x^2 \left (2+x^2\right ) \log ^2\left (2+x^2\right )} \, dx\\ &=\int \left (2 (1+x)+\frac {e^x \left (-2+2 x+x^2\right )}{x^2}-\frac {2 x^2 (2+x)}{\left (2+x^2\right ) \log ^2\left (2+x^2\right )}+\frac {2 (1+x)}{\log \left (2+x^2\right )}\right ) \, dx\\ &=(1+x)^2-2 \int \frac {x^2 (2+x)}{\left (2+x^2\right ) \log ^2\left (2+x^2\right )} \, dx+2 \int \frac {1+x}{\log \left (2+x^2\right )} \, dx+\int \frac {e^x \left (-2+2 x+x^2\right )}{x^2} \, dx\\ &=(1+x)^2-2 \int \left (\frac {2}{\log ^2\left (2+x^2\right )}+\frac {x}{\log ^2\left (2+x^2\right )}-\frac {2 (2+x)}{\left (2+x^2\right ) \log ^2\left (2+x^2\right )}\right ) \, dx+2 \int \frac {1+x}{\log \left (2+x^2\right )} \, dx+\int \left (e^x-\frac {2 e^x}{x^2}+\frac {2 e^x}{x}\right ) \, dx\\ &=(1+x)^2-2 \int \frac {e^x}{x^2} \, dx+2 \int \frac {e^x}{x} \, dx-2 \int \frac {x}{\log ^2\left (2+x^2\right )} \, dx+2 \int \frac {1+x}{\log \left (2+x^2\right )} \, dx-4 \int \frac {1}{\log ^2\left (2+x^2\right )} \, dx+4 \int \frac {2+x}{\left (2+x^2\right ) \log ^2\left (2+x^2\right )} \, dx+\int e^x \, dx\\ &=e^x+\frac {2 e^x}{x}+(1+x)^2+2 \text {Ei}(x)-2 \int \frac {e^x}{x} \, dx+2 \int \frac {1+x}{\log \left (2+x^2\right )} \, dx+4 \int \left (\frac {2}{\left (2+x^2\right ) \log ^2\left (2+x^2\right )}+\frac {x}{\left (2+x^2\right ) \log ^2\left (2+x^2\right )}\right ) \, dx-4 \int \frac {1}{\log ^2\left (2+x^2\right )} \, dx-\operatorname {Subst}\left (\int \frac {1}{\log ^2(2+x)} \, dx,x,x^2\right )\\ &=e^x+\frac {2 e^x}{x}+(1+x)^2+2 \int \frac {1+x}{\log \left (2+x^2\right )} \, dx-4 \int \frac {1}{\log ^2\left (2+x^2\right )} \, dx+4 \int \frac {x}{\left (2+x^2\right ) \log ^2\left (2+x^2\right )} \, dx+8 \int \frac {1}{\left (2+x^2\right ) \log ^2\left (2+x^2\right )} \, dx-\operatorname {Subst}\left (\int \frac {1}{\log ^2(x)} \, dx,x,2+x^2\right )\\ &=e^x+\frac {2 e^x}{x}+(1+x)^2+\frac {2+x^2}{\log \left (2+x^2\right )}+2 \int \frac {1+x}{\log \left (2+x^2\right )} \, dx+2 \operatorname {Subst}\left (\int \frac {1}{(2+x) \log ^2(2+x)} \, dx,x,x^2\right )-4 \int \frac {1}{\log ^2\left (2+x^2\right )} \, dx+8 \int \frac {1}{\left (2+x^2\right ) \log ^2\left (2+x^2\right )} \, dx-\operatorname {Subst}\left (\int \frac {1}{\log (x)} \, dx,x,2+x^2\right )\\ &=e^x+\frac {2 e^x}{x}+(1+x)^2+\frac {2+x^2}{\log \left (2+x^2\right )}-\text {li}\left (2+x^2\right )+2 \int \frac {1+x}{\log \left (2+x^2\right )} \, dx+2 \operatorname {Subst}\left (\int \frac {1}{x \log ^2(x)} \, dx,x,2+x^2\right )-4 \int \frac {1}{\log ^2\left (2+x^2\right )} \, dx+8 \int \frac {1}{\left (2+x^2\right ) \log ^2\left (2+x^2\right )} \, dx\\ &=e^x+\frac {2 e^x}{x}+(1+x)^2+\frac {2+x^2}{\log \left (2+x^2\right )}-\text {li}\left (2+x^2\right )+2 \int \frac {1+x}{\log \left (2+x^2\right )} \, dx+2 \operatorname {Subst}\left (\int \frac {1}{x^2} \, dx,x,\log \left (2+x^2\right )\right )-4 \int \frac {1}{\log ^2\left (2+x^2\right )} \, dx+8 \int \frac {1}{\left (2+x^2\right ) \log ^2\left (2+x^2\right )} \, dx\\ &=e^x+\frac {2 e^x}{x}+(1+x)^2-\frac {2}{\log \left (2+x^2\right )}+\frac {2+x^2}{\log \left (2+x^2\right )}-\text {li}\left (2+x^2\right )+2 \int \frac {1+x}{\log \left (2+x^2\right )} \, dx-4 \int \frac {1}{\log ^2\left (2+x^2\right )} \, dx+8 \int \frac {1}{\left (2+x^2\right ) \log ^2\left (2+x^2\right )} \, dx\\ \end {aligned} \end {gather*}
________________________________________________________________________________________
Mathematica [A] time = 1.09, size = 26, normalized size = 1.00 \begin {gather*} \frac {(2+x) \left (e^x+x^2+\frac {x^2}{\log \left (2+x^2\right )}\right )}{x} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.08, size = 43, normalized size = 1.65 \begin {gather*} \frac {x^{3} + 2 \, x^{2} + {\left (x^{3} + 2 \, x^{2} + {\left (x + 2\right )} e^{x}\right )} \log \left (x^{2} + 2\right )}{x \log \left (x^{2} + 2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.51, size = 62, normalized size = 2.38 \begin {gather*} \frac {x^{3} \log \left (x^{2} + 2\right ) + x^{3} + 2 \, x^{2} \log \left (x^{2} + 2\right ) + x e^{x} \log \left (x^{2} + 2\right ) + 2 \, x^{2} + 2 \, e^{x} \log \left (x^{2} + 2\right )}{x \log \left (x^{2} + 2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.19, size = 36, normalized size = 1.38
method | result | size |
risch | \(\frac {x^{3}+2 x^{2}+{\mathrm e}^{x} x +2 \,{\mathrm e}^{x}}{x}+\frac {x \left (2+x \right )}{\ln \left (x^{2}+2\right )}\) | \(36\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.41, size = 43, normalized size = 1.65 \begin {gather*} \frac {x^{3} + 2 \, x^{2} + {\left (x^{3} + 2 \, x^{2} + {\left (x + 2\right )} e^{x}\right )} \log \left (x^{2} + 2\right )}{x \log \left (x^{2} + 2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.15, size = 39, normalized size = 1.50 \begin {gather*} 2\,x+{\mathrm {e}}^x+\frac {2\,{\mathrm {e}}^x}{x}+\frac {x^2}{\ln \left (x^2+2\right )}+x^2+\frac {2\,x}{\ln \left (x^2+2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.34, size = 27, normalized size = 1.04 \begin {gather*} x^{2} + 2 x + \frac {x^{2} + 2 x}{\log {\left (x^{2} + 2 \right )}} + \frac {\left (x + 2\right ) e^{x}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________