3.12.47 \(\int \frac {(-x+2 x \log (x)+(1+e^x+2 x) \log ^2(x)) \sqrt {\frac {x^4+(2 x^2+2 e^x x^2+2 x^3+2 x^4) \log (x)+(1+e^{2 x}+2 x+3 x^2+2 x^3+x^4+e^x (2+2 x+2 x^2)) \log ^2(x)}{\log ^2(x)}}}{x^2 \log (x)+(1+e^x+x+x^2) \log ^2(x)} \, dx\)

Optimal. Leaf size=30 \[ \sqrt {\left (-1-e^x-x-x^2-\frac {x^2}{\log (x)}\right )^2} \]

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Rubi [B]  time = 2.24, antiderivative size = 203, normalized size of antiderivative = 6.77, number of steps used = 12, number of rules used = 7, integrand size = 128, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.055, Rules used = {6688, 6719, 6742, 2194, 2306, 2309, 2178} \begin {gather*} \frac {x^2 \left (x^2+\left (x^2+x+e^x+1\right ) \log (x)\right )}{\log (x) \sqrt {\frac {\left (x^2+\left (x^2+x+e^x+1\right ) \log (x)\right )^2}{\log ^2(x)}}}+\frac {x^2 \left (x^2+\left (x^2+x+e^x+1\right ) \log (x)\right )}{\log ^2(x) \sqrt {\frac {\left (x^2+\left (x^2+x+e^x+1\right ) \log (x)\right )^2}{\log ^2(x)}}}+\frac {x \left (x^2+\left (x^2+x+e^x+1\right ) \log (x)\right )}{\log (x) \sqrt {\frac {\left (x^2+\left (x^2+x+e^x+1\right ) \log (x)\right )^2}{\log ^2(x)}}}+\frac {e^x \left (x^2+\left (x^2+x+e^x+1\right ) \log (x)\right )}{\log (x) \sqrt {\frac {\left (x^2+\left (x^2+x+e^x+1\right ) \log (x)\right )^2}{\log ^2(x)}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((-x + 2*x*Log[x] + (1 + E^x + 2*x)*Log[x]^2)*Sqrt[(x^4 + (2*x^2 + 2*E^x*x^2 + 2*x^3 + 2*x^4)*Log[x] + (1
+ E^(2*x) + 2*x + 3*x^2 + 2*x^3 + x^4 + E^x*(2 + 2*x + 2*x^2))*Log[x]^2)/Log[x]^2])/(x^2*Log[x] + (1 + E^x + x
+ x^2)*Log[x]^2),x]

[Out]

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

Rule 2178

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - (c*f)/d))*ExpIntegral
Ei[(f*g*(c + d*x)*Log[F])/d])/d, x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !$UseGamma === True

Rule 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rule 2306

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Log
[c*x^n])^(p + 1))/(b*d*n*(p + 1)), x] - Dist[(m + 1)/(b*n*(p + 1)), Int[(d*x)^m*(a + b*Log[c*x^n])^(p + 1), x]
, x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1] && LtQ[p, -1]

Rule 2309

Int[((a_.) + Log[(c_.)*(x_)]*(b_.))^(p_)*(x_)^(m_.), x_Symbol] :> Dist[1/c^(m + 1), Subst[Int[E^((m + 1)*x)*(a
+ b*x)^p, x], x, Log[c*x]], x] /; FreeQ[{a, b, c, p}, x] && IntegerQ[m]

Rule 6688

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

Rule 6719

Int[(u_.)*((a_.)*(v_)^(m_.)*(w_)^(n_.))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a*v^m*w^n)^FracPart[p])/(v^(m*F
racPart[p])*w^(n*FracPart[p])), Int[u*v^(m*p)*w^(n*p), x], x] /; FreeQ[{a, m, n, p}, x] &&  !IntegerQ[p] &&  !
FreeQ[v, x] &&  !FreeQ[w, 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 \frac {\left (x^2+\left (1+e^x+x+x^2\right ) \log (x)\right ) \left (-x+2 x \log (x)+\left (1+e^x+2 x\right ) \log ^2(x)\right )}{\log ^3(x) \sqrt {\frac {\left (x^2+\left (1+e^x+x+x^2\right ) \log (x)\right )^2}{\log ^2(x)}}} \, dx\\ &=\frac {\left (x^2+\left (1+e^x+x+x^2\right ) \log (x)\right ) \int \frac {-x+2 x \log (x)+\left (1+e^x+2 x\right ) \log ^2(x)}{\log ^2(x)} \, dx}{\log (x) \sqrt {\frac {\left (x^2+\left (1+e^x+x+x^2\right ) \log (x)\right )^2}{\log ^2(x)}}}\\ &=\frac {\left (x^2+\left (1+e^x+x+x^2\right ) \log (x)\right ) \int \left (e^x+\frac {-x+2 x \log (x)+\log ^2(x)+2 x \log ^2(x)}{\log ^2(x)}\right ) \, dx}{\log (x) \sqrt {\frac {\left (x^2+\left (1+e^x+x+x^2\right ) \log (x)\right )^2}{\log ^2(x)}}}\\ &=\frac {\left (x^2+\left (1+e^x+x+x^2\right ) \log (x)\right ) \int e^x \, dx}{\log (x) \sqrt {\frac {\left (x^2+\left (1+e^x+x+x^2\right ) \log (x)\right )^2}{\log ^2(x)}}}+\frac {\left (x^2+\left (1+e^x+x+x^2\right ) \log (x)\right ) \int \frac {-x+2 x \log (x)+\log ^2(x)+2 x \log ^2(x)}{\log ^2(x)} \, dx}{\log (x) \sqrt {\frac {\left (x^2+\left (1+e^x+x+x^2\right ) \log (x)\right )^2}{\log ^2(x)}}}\\ &=\frac {e^x \left (x^2+\left (1+e^x+x+x^2\right ) \log (x)\right )}{\log (x) \sqrt {\frac {\left (x^2+\left (1+e^x+x+x^2\right ) \log (x)\right )^2}{\log ^2(x)}}}+\frac {\left (x^2+\left (1+e^x+x+x^2\right ) \log (x)\right ) \int \left (1+2 x-\frac {x}{\log ^2(x)}+\frac {2 x}{\log (x)}\right ) \, dx}{\log (x) \sqrt {\frac {\left (x^2+\left (1+e^x+x+x^2\right ) \log (x)\right )^2}{\log ^2(x)}}}\\ &=\frac {e^x \left (x^2+\left (1+e^x+x+x^2\right ) \log (x)\right )}{\log (x) \sqrt {\frac {\left (x^2+\left (1+e^x+x+x^2\right ) \log (x)\right )^2}{\log ^2(x)}}}+\frac {x \left (x^2+\left (1+e^x+x+x^2\right ) \log (x)\right )}{\log (x) \sqrt {\frac {\left (x^2+\left (1+e^x+x+x^2\right ) \log (x)\right )^2}{\log ^2(x)}}}+\frac {x^2 \left (x^2+\left (1+e^x+x+x^2\right ) \log (x)\right )}{\log (x) \sqrt {\frac {\left (x^2+\left (1+e^x+x+x^2\right ) \log (x)\right )^2}{\log ^2(x)}}}-\frac {\left (x^2+\left (1+e^x+x+x^2\right ) \log (x)\right ) \int \frac {x}{\log ^2(x)} \, dx}{\log (x) \sqrt {\frac {\left (x^2+\left (1+e^x+x+x^2\right ) \log (x)\right )^2}{\log ^2(x)}}}+\frac {\left (2 \left (x^2+\left (1+e^x+x+x^2\right ) \log (x)\right )\right ) \int \frac {x}{\log (x)} \, dx}{\log (x) \sqrt {\frac {\left (x^2+\left (1+e^x+x+x^2\right ) \log (x)\right )^2}{\log ^2(x)}}}\\ &=\frac {x^2 \left (x^2+\left (1+e^x+x+x^2\right ) \log (x)\right )}{\log ^2(x) \sqrt {\frac {\left (x^2+\left (1+e^x+x+x^2\right ) \log (x)\right )^2}{\log ^2(x)}}}+\frac {e^x \left (x^2+\left (1+e^x+x+x^2\right ) \log (x)\right )}{\log (x) \sqrt {\frac {\left (x^2+\left (1+e^x+x+x^2\right ) \log (x)\right )^2}{\log ^2(x)}}}+\frac {x \left (x^2+\left (1+e^x+x+x^2\right ) \log (x)\right )}{\log (x) \sqrt {\frac {\left (x^2+\left (1+e^x+x+x^2\right ) \log (x)\right )^2}{\log ^2(x)}}}+\frac {x^2 \left (x^2+\left (1+e^x+x+x^2\right ) \log (x)\right )}{\log (x) \sqrt {\frac {\left (x^2+\left (1+e^x+x+x^2\right ) \log (x)\right )^2}{\log ^2(x)}}}-\frac {\left (2 \left (x^2+\left (1+e^x+x+x^2\right ) \log (x)\right )\right ) \int \frac {x}{\log (x)} \, dx}{\log (x) \sqrt {\frac {\left (x^2+\left (1+e^x+x+x^2\right ) \log (x)\right )^2}{\log ^2(x)}}}+\frac {\left (2 \left (x^2+\left (1+e^x+x+x^2\right ) \log (x)\right )\right ) \operatorname {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,\log (x)\right )}{\log (x) \sqrt {\frac {\left (x^2+\left (1+e^x+x+x^2\right ) \log (x)\right )^2}{\log ^2(x)}}}\\ &=\frac {x^2 \left (x^2+\left (1+e^x+x+x^2\right ) \log (x)\right )}{\log ^2(x) \sqrt {\frac {\left (x^2+\left (1+e^x+x+x^2\right ) \log (x)\right )^2}{\log ^2(x)}}}+\frac {e^x \left (x^2+\left (1+e^x+x+x^2\right ) \log (x)\right )}{\log (x) \sqrt {\frac {\left (x^2+\left (1+e^x+x+x^2\right ) \log (x)\right )^2}{\log ^2(x)}}}+\frac {x \left (x^2+\left (1+e^x+x+x^2\right ) \log (x)\right )}{\log (x) \sqrt {\frac {\left (x^2+\left (1+e^x+x+x^2\right ) \log (x)\right )^2}{\log ^2(x)}}}+\frac {x^2 \left (x^2+\left (1+e^x+x+x^2\right ) \log (x)\right )}{\log (x) \sqrt {\frac {\left (x^2+\left (1+e^x+x+x^2\right ) \log (x)\right )^2}{\log ^2(x)}}}+\frac {2 \text {Ei}(2 \log (x)) \left (x^2+\left (1+e^x+x+x^2\right ) \log (x)\right )}{\log (x) \sqrt {\frac {\left (x^2+\left (1+e^x+x+x^2\right ) \log (x)\right )^2}{\log ^2(x)}}}-\frac {\left (2 \left (x^2+\left (1+e^x+x+x^2\right ) \log (x)\right )\right ) \operatorname {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,\log (x)\right )}{\log (x) \sqrt {\frac {\left (x^2+\left (1+e^x+x+x^2\right ) \log (x)\right )^2}{\log ^2(x)}}}\\ &=\frac {x^2 \left (x^2+\left (1+e^x+x+x^2\right ) \log (x)\right )}{\log ^2(x) \sqrt {\frac {\left (x^2+\left (1+e^x+x+x^2\right ) \log (x)\right )^2}{\log ^2(x)}}}+\frac {e^x \left (x^2+\left (1+e^x+x+x^2\right ) \log (x)\right )}{\log (x) \sqrt {\frac {\left (x^2+\left (1+e^x+x+x^2\right ) \log (x)\right )^2}{\log ^2(x)}}}+\frac {x \left (x^2+\left (1+e^x+x+x^2\right ) \log (x)\right )}{\log (x) \sqrt {\frac {\left (x^2+\left (1+e^x+x+x^2\right ) \log (x)\right )^2}{\log ^2(x)}}}+\frac {x^2 \left (x^2+\left (1+e^x+x+x^2\right ) \log (x)\right )}{\log (x) \sqrt {\frac {\left (x^2+\left (1+e^x+x+x^2\right ) \log (x)\right )^2}{\log ^2(x)}}}\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.29, size = 61, normalized size = 2.03 \begin {gather*} \frac {\left (x^2+\left (e^x+x+x^2\right ) \log (x)\right ) \sqrt {\frac {\left (x^2+\left (1+e^x+x+x^2\right ) \log (x)\right )^2}{\log ^2(x)}}}{x^2+\left (1+e^x+x+x^2\right ) \log (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((-x + 2*x*Log[x] + (1 + E^x + 2*x)*Log[x]^2)*Sqrt[(x^4 + (2*x^2 + 2*E^x*x^2 + 2*x^3 + 2*x^4)*Log[x]
+ (1 + E^(2*x) + 2*x + 3*x^2 + 2*x^3 + x^4 + E^x*(2 + 2*x + 2*x^2))*Log[x]^2)/Log[x]^2])/(x^2*Log[x] + (1 + E
^x + x + x^2)*Log[x]^2),x]

[Out]

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

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fricas [A]  time = 0.81, size = 19, normalized size = 0.63 \begin {gather*} \frac {x^{2} + {\left (x^{2} + x + e^{x}\right )} \log \relax (x)}{\log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((exp(x)+2*x+1)*log(x)^2+2*x*log(x)-x)*(((exp(x)^2+(2*x^2+2*x+2)*exp(x)+x^4+2*x^3+3*x^2+2*x+1)*log(x
)^2+(2*exp(x)*x^2+2*x^4+2*x^3+2*x^2)*log(x)+x^4)/log(x)^2)^(1/2)/((exp(x)+x^2+x+1)*log(x)^2+x^2*log(x)),x, alg
orithm="fricas")

[Out]

(x^2 + (x^2 + x + e^x)*log(x))/log(x)

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giac [A]  time = 0.61, size = 31, normalized size = 1.03 \begin {gather*} -\frac {1}{3} \, \log \relax (x)^{3} + \frac {3}{2} \, x^{2} - \frac {1}{2} \, \log \relax (x)^{2} + 2 \, x + \frac {x^{2}}{\log \relax (x)} + e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((exp(x)+2*x+1)*log(x)^2+2*x*log(x)-x)*(((exp(x)^2+(2*x^2+2*x+2)*exp(x)+x^4+2*x^3+3*x^2+2*x+1)*log(x
)^2+(2*exp(x)*x^2+2*x^4+2*x^3+2*x^2)*log(x)+x^4)/log(x)^2)^(1/2)/((exp(x)+x^2+x+1)*log(x)^2+x^2*log(x)),x, alg
orithm="giac")

[Out]

-1/3*log(x)^3 + 3/2*x^2 - 1/2*log(x)^2 + 2*x + x^2/log(x) + e^x

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maple [B]  time = 0.07, size = 233, normalized size = 7.77

method	result	size
risch	\(\frac {\sqrt {\frac {\left (x^{2} \ln \relax (x )+{\mathrm e}^{x} \ln \relax (x )+x \ln \relax (x )+x^{2}+\ln \relax (x )\right )^{2}}{\ln \relax (x )^{2}}}\, \ln \relax (x ) x^{2}}{x^{2} \ln \relax (x )+{\mathrm e}^{x} \ln \relax (x )+x \ln \relax (x )+x^{2}+\ln \relax (x )}+\frac {\sqrt {\frac {\left (x^{2} \ln \relax (x )+{\mathrm e}^{x} \ln \relax (x )+x \ln \relax (x )+x^{2}+\ln \relax (x )\right )^{2}}{\ln \relax (x )^{2}}}\, \ln \relax (x ) x}{x^{2} \ln \relax (x )+{\mathrm e}^{x} \ln \relax (x )+x \ln \relax (x )+x^{2}+\ln \relax (x )}+\frac {\sqrt {\frac {\left (x^{2} \ln \relax (x )+{\mathrm e}^{x} \ln \relax (x )+x \ln \relax (x )+x^{2}+\ln \relax (x )\right )^{2}}{\ln \relax (x )^{2}}}\, \ln \relax (x ) {\mathrm e}^{x}}{x^{2} \ln \relax (x )+{\mathrm e}^{x} \ln \relax (x )+x \ln \relax (x )+x^{2}+\ln \relax (x )}+\frac {\sqrt {\frac {\left (x^{2} \ln \relax (x )+{\mathrm e}^{x} \ln \relax (x )+x \ln \relax (x )+x^{2}+\ln \relax (x )\right )^{2}}{\ln \relax (x )^{2}}}\, x^{2}}{x^{2} \ln \relax (x )+{\mathrm e}^{x} \ln \relax (x )+x \ln \relax (x )+x^{2}+\ln \relax (x )}\)	\(233\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((exp(x)+2*x+1)*ln(x)^2+2*x*ln(x)-x)*(((exp(x)^2+(2*x^2+2*x+2)*exp(x)+x^4+2*x^3+3*x^2+2*x+1)*ln(x)^2+(2*ex
p(x)*x^2+2*x^4+2*x^3+2*x^2)*ln(x)+x^4)/ln(x)^2)^(1/2)/((exp(x)+x^2+x+1)*ln(x)^2+x^2*ln(x)),x,method=_RETURNVER
BOSE)

[Out]

((x^2*ln(x)+exp(x)*ln(x)+x*ln(x)+x^2+ln(x))^2/ln(x)^2)^(1/2)/(x^2*ln(x)+exp(x)*ln(x)+x*ln(x)+x^2+ln(x))*ln(x)*
x^2+((x^2*ln(x)+exp(x)*ln(x)+x*ln(x)+x^2+ln(x))^2/ln(x)^2)^(1/2)/(x^2*ln(x)+exp(x)*ln(x)+x*ln(x)+x^2+ln(x))*ln
(x)*x+((x^2*ln(x)+exp(x)*ln(x)+x*ln(x)+x^2+ln(x))^2/ln(x)^2)^(1/2)/(x^2*ln(x)+exp(x)*ln(x)+x*ln(x)+x^2+ln(x))*
ln(x)*exp(x)+((x^2*ln(x)+exp(x)*ln(x)+x*ln(x)+x^2+ln(x))^2/ln(x)^2)^(1/2)/(x^2*ln(x)+exp(x)*ln(x)+x*ln(x)+x^2+
ln(x))*x^2

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maxima [A]  time = 0.51, size = 22, normalized size = 0.73 \begin {gather*} \frac {x^{2} + {\left (x^{2} + x\right )} \log \relax (x) + e^{x} \log \relax (x)}{\log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((exp(x)+2*x+1)*log(x)^2+2*x*log(x)-x)*(((exp(x)^2+(2*x^2+2*x+2)*exp(x)+x^4+2*x^3+3*x^2+2*x+1)*log(x
)^2+(2*exp(x)*x^2+2*x^4+2*x^3+2*x^2)*log(x)+x^4)/log(x)^2)^(1/2)/((exp(x)+x^2+x+1)*log(x)^2+x^2*log(x)),x, alg
orithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {\sqrt {\frac {{\ln \relax (x)}^2\,\left (2\,x+{\mathrm {e}}^{2\,x}+{\mathrm {e}}^x\,\left (2\,x^2+2\,x+2\right )+3\,x^2+2\,x^3+x^4+1\right )+\ln \relax (x)\,\left (2\,x^2\,{\mathrm {e}}^x+2\,x^2+2\,x^3+2\,x^4\right )+x^4}{{\ln \relax (x)}^2}}\,\left (\left (2\,x+{\mathrm {e}}^x+1\right )\,{\ln \relax (x)}^2+2\,x\,\ln \relax (x)-x\right )}{x^2\,\ln \relax (x)+{\ln \relax (x)}^2\,\left (x+{\mathrm {e}}^x+x^2+1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((log(x)^2*(2*x + exp(2*x) + exp(x)*(2*x + 2*x^2 + 2) + 3*x^2 + 2*x^3 + x^4 + 1) + log(x)*(2*x^2*exp(x) +
2*x^2 + 2*x^3 + 2*x^4) + x^4)/log(x)^2)^(1/2)*(log(x)^2*(2*x + exp(x) + 1) - x + 2*x*log(x)))/(x^2*log(x) + l
og(x)^2*(x + exp(x) + x^2 + 1)),x)

[Out]

int((((log(x)^2*(2*x + exp(2*x) + exp(x)*(2*x + 2*x^2 + 2) + 3*x^2 + 2*x^3 + x^4 + 1) + log(x)*(2*x^2*exp(x) +
2*x^2 + 2*x^3 + 2*x^4) + x^4)/log(x)^2)^(1/2)*(log(x)^2*(2*x + exp(x) + 1) - x + 2*x*log(x)))/(x^2*log(x) + l
og(x)^2*(x + exp(x) + x^2 + 1)), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((exp(x)+2*x+1)*ln(x)**2+2*x*ln(x)-x)*(((exp(x)**2+(2*x**2+2*x+2)*exp(x)+x**4+2*x**3+3*x**2+2*x+1)*l
n(x)**2+(2*exp(x)*x**2+2*x**4+2*x**3+2*x**2)*ln(x)+x**4)/ln(x)**2)**(1/2)/((exp(x)+x**2+x+1)*ln(x)**2+x**2*ln(
x)),x)

[Out]

Timed out

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