3.79.76 \(\int \frac {x^2+e^{-x+\frac {-x-e^4 x+\log ^2(4)}{x}} x (-x+x^2+\log ^2(4))+e^{\frac {-x-e^4 x+\log ^2(4)}{x}} (x^2-x \log ^2(4))}{x^2} \, dx\)

Optimal. Leaf size=33 \[ x+e^{-e^4+\frac {-x+\log ^2(4)}{x}} \left (x-e^{-x} x\right ) \]

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Rubi [A]  time = 1.01, antiderivative size = 66, normalized size of antiderivative = 2.00, number of steps used = 6, number of rules used = 3, integrand size = 77, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.039, Rules used = {14, 6742, 2288} \begin {gather*} -\frac {e^{-x+\frac {\log ^2(4)}{x}-e^4-1} \left (x^2+\log ^2(4)\right )}{x \left (\frac {\log ^2(4)}{x^2}+1\right )}+x+x e^{\frac {\log ^2(4)}{x}-e^4-1} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^2 + E^(-x + (-x - E^4*x + Log[4]^2)/x)*x*(-x + x^2 + Log[4]^2) + E^((-x - E^4*x + Log[4]^2)/x)*(x^2 - x
*Log[4]^2))/x^2,x]

[Out]

x + E^(-1 - E^4 + Log[4]^2/x)*x - (E^(-1 - E^4 - x + Log[4]^2/x)*(x^2 + Log[4]^2))/(x*(1 + Log[4]^2/x^2))

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 2288

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = (v*y)/(Log[F]*D[u, x])}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (1+\frac {e^{-1-e^4-x+\frac {\log ^2(4)}{x}} \left (-x+e^x x+x^2+\log ^2(4)-e^x \log ^2(4)\right )}{x}\right ) \, dx\\ &=x+\int \frac {e^{-1-e^4-x+\frac {\log ^2(4)}{x}} \left (-x+e^x x+x^2+\log ^2(4)-e^x \log ^2(4)\right )}{x} \, dx\\ &=x+\int \left (\frac {e^{-1-e^4+\frac {\log ^2(4)}{x}} \left (x-\log ^2(4)\right )}{x}+\frac {e^{-1-e^4-x+\frac {\log ^2(4)}{x}} \left (-x+x^2+\log ^2(4)\right )}{x}\right ) \, dx\\ &=x+\int \frac {e^{-1-e^4+\frac {\log ^2(4)}{x}} \left (x-\log ^2(4)\right )}{x} \, dx+\int \frac {e^{-1-e^4-x+\frac {\log ^2(4)}{x}} \left (-x+x^2+\log ^2(4)\right )}{x} \, dx\\ &=x+e^{-1-e^4+\frac {\log ^2(4)}{x}} x-\frac {e^{-1-e^4-x+\frac {\log ^2(4)}{x}} \left (x^2+\log ^2(4)\right )}{x \left (1+\frac {\log ^2(4)}{x^2}\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.13, size = 43, normalized size = 1.30 \begin {gather*} \left (1+e^{-1-e^4+\frac {\log ^2(4)}{x}}-e^{-1-e^4-x+\frac {\log ^2(4)}{x}}\right ) x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^2 + E^(-x + (-x - E^4*x + Log[4]^2)/x)*x*(-x + x^2 + Log[4]^2) + E^((-x - E^4*x + Log[4]^2)/x)*(x
^2 - x*Log[4]^2))/x^2,x]

[Out]

(1 + E^(-1 - E^4 + Log[4]^2/x) - E^(-1 - E^4 - x + Log[4]^2/x))*x

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fricas [A]  time = 0.84, size = 50, normalized size = 1.52 \begin {gather*} x e^{\left (-\frac {x e^{4} - 4 \, \log \relax (2)^{2} + x}{x}\right )} + x - e^{\left (-\frac {x^{2} + x e^{4} - 4 \, \log \relax (2)^{2} - x \log \relax (x) + x}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*log(2)^2+x^2-x)*exp((4*log(2)^2-x*exp(4)-x)/x)*exp(log(x)-x)+(-4*x*log(2)^2+x^2)*exp((4*log(2)^2
-x*exp(4)-x)/x)+x^2)/x^2,x, algorithm="fricas")

[Out]

x*e^(-(x*e^4 - 4*log(2)^2 + x)/x) + x - e^(-(x^2 + x*e^4 - 4*log(2)^2 - x*log(x) + x)/x)

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giac [A]  time = 0.16, size = 46, normalized size = 1.39 \begin {gather*} -x e^{\left (-\frac {x^{2} + x e^{4} - 4 \, \log \relax (2)^{2} + x}{x}\right )} + x e^{\left (-\frac {x e^{4} - 4 \, \log \relax (2)^{2} + x}{x}\right )} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*log(2)^2+x^2-x)*exp((4*log(2)^2-x*exp(4)-x)/x)*exp(log(x)-x)+(-4*x*log(2)^2+x^2)*exp((4*log(2)^2
-x*exp(4)-x)/x)+x^2)/x^2,x, algorithm="giac")

[Out]

-x*e^(-(x^2 + x*e^4 - 4*log(2)^2 + x)/x) + x*e^(-(x*e^4 - 4*log(2)^2 + x)/x) + x

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maple [A]  time = 0.19, size = 47, normalized size = 1.42

method	result	size
risch	\(x \,{\mathrm e}^{-\frac {-4 \ln \relax (2)^{2}+x \,{\mathrm e}^{4}+x}{x}}-x \,{\mathrm e}^{-\frac {-4 \ln \relax (2)^{2}+x \,{\mathrm e}^{4}+x^{2}+x}{x}}+x\)	\(47\)
default	\(x +\frac {{\mathrm e}^{\frac {4 \ln \relax (2)^{2}-x \,{\mathrm e}^{4}-x}{x}} x^{2}-{\mathrm e}^{\frac {4 \ln \relax (2)^{2}-x \,{\mathrm e}^{4}-x}{x}} {\mathrm e}^{\ln \relax (x )-x} x}{x}\)	\(62\)
norman	\(\frac {x^{2}+{\mathrm e}^{\frac {4 \ln \relax (2)^{2}-x \,{\mathrm e}^{4}-x}{x}} x^{2}-{\mathrm e}^{\frac {4 \ln \relax (2)^{2}-x \,{\mathrm e}^{4}-x}{x}} {\mathrm e}^{\ln \relax (x )-x} x}{x}\)	\(63\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((4*ln(2)^2+x^2-x)*exp((4*ln(2)^2-x*exp(4)-x)/x)*exp(ln(x)-x)+(-4*x*ln(2)^2+x^2)*exp((4*ln(2)^2-x*exp(4)-x
)/x)+x^2)/x^2,x,method=_RETURNVERBOSE)

[Out]

x*exp(-(-4*ln(2)^2+x*exp(4)+x)/x)-x*exp(-(-4*ln(2)^2+x*exp(4)+x^2+x)/x)+x

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maxima [C]  time = 0.54, size = 71, normalized size = 2.15 \begin {gather*} 4 \, {\rm Ei}\left (\frac {4 \, \log \relax (2)^{2}}{x}\right ) e^{\left (-e^{4} - 1\right )} \log \relax (2)^{2} - 4 \, e^{\left (-e^{4} - 1\right )} \Gamma \left (-1, -\frac {4 \, \log \relax (2)^{2}}{x}\right ) \log \relax (2)^{2} - x e^{\left (-x + \frac {4 \, \log \relax (2)^{2}}{x} - e^{4} - 1\right )} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*log(2)^2+x^2-x)*exp((4*log(2)^2-x*exp(4)-x)/x)*exp(log(x)-x)+(-4*x*log(2)^2+x^2)*exp((4*log(2)^2
-x*exp(4)-x)/x)+x^2)/x^2,x, algorithm="maxima")

[Out]

4*Ei(4*log(2)^2/x)*e^(-e^4 - 1)*log(2)^2 - 4*e^(-e^4 - 1)*gamma(-1, -4*log(2)^2/x)*log(2)^2 - x*e^(-x + 4*log(
2)^2/x - e^4 - 1) + x

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mupad [B]  time = 5.56, size = 45, normalized size = 1.36 \begin {gather*} x+x\,{\mathrm {e}}^{-{\mathrm {e}}^4}\,{\mathrm {e}}^{-1}\,{\mathrm {e}}^{\frac {4\,{\ln \relax (2)}^2}{x}}-x\,{\mathrm {e}}^{-{\mathrm {e}}^4}\,{\mathrm {e}}^{-x}\,{\mathrm {e}}^{-1}\,{\mathrm {e}}^{\frac {4\,{\ln \relax (2)}^2}{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2 - exp(-(x + x*exp(4) - 4*log(2)^2)/x)*(4*x*log(2)^2 - x^2) + exp(log(x) - x)*exp(-(x + x*exp(4) - 4*l
og(2)^2)/x)*(4*log(2)^2 - x + x^2))/x^2,x)

[Out]

x + x*exp(-exp(4))*exp(-1)*exp((4*log(2)^2)/x) - x*exp(-exp(4))*exp(-x)*exp(-1)*exp((4*log(2)^2)/x)

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sympy [A]  time = 14.62, size = 24, normalized size = 0.73 \begin {gather*} x + \left (x - x e^{- x}\right ) e^{\frac {- x e^{4} - x + 4 \log {\relax (2 )}^{2}}{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*ln(2)**2+x**2-x)*exp((4*ln(2)**2-x*exp(4)-x)/x)*exp(ln(x)-x)+(-4*x*ln(2)**2+x**2)*exp((4*ln(2)**
2-x*exp(4)-x)/x)+x**2)/x**2,x)

[Out]

x + (x - x*exp(-x))*exp((-x*exp(4) - x + 4*log(2)**2)/x)

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