3.56.77 \(\int \frac {e^{x^2} (x+2 x^3) \log (x)+e^{x^2} (-6-12 x^2+(1+2 x^2) \log (5)) \log ^2(x)+e^{\log ^2(\frac {x+(-6+\log (5)) \log (x)}{\log (x)})} (-10+10 \log (x)) \log (\frac {x+(-6+\log (5)) \log (x)}{\log (x)})}{5 x \log (x)+(-30+5 \log (5)) \log ^2(x)} \, dx\)

Optimal. Leaf size=26 \[ e^{\log ^2\left (-6+\log (5)+\frac {x}{\log (x)}\right )}+\frac {e^{x^2} x}{5} \]

________________________________________________________________________________________

Rubi [A]  time = 1.59, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, integrand size = 104, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {6741, 12, 6742, 2226, 2204, 2212, 6688, 6706} \begin {gather*} \frac {e^{x^2} x}{5}+e^{\log ^2\left (\frac {x}{\log (x)}-6+\log (5)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^x^2*(x + 2*x^3)*Log[x] + E^x^2*(-6 - 12*x^2 + (1 + 2*x^2)*Log[5])*Log[x]^2 + E^Log[(x + (-6 + Log[5])*L
og[x])/Log[x]]^2*(-10 + 10*Log[x])*Log[(x + (-6 + Log[5])*Log[x])/Log[x]])/(5*x*Log[x] + (-30 + 5*Log[5])*Log[
x]^2),x]

[Out]

E^Log[-6 + Log[5] + x/Log[x]]^2 + (E^x^2*x)/5

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2204

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[(F^a*Sqrt[Pi]*Erfi[(c + d*x)*Rt[b*Log[F], 2
]])/(2*d*Rt[b*Log[F], 2]), x] /; FreeQ[{F, a, b, c, d}, x] && PosQ[b]

Rule 2212

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^(m
- n + 1)*F^(a + b*(c + d*x)^n))/(b*d*n*Log[F]), x] - Dist[(m - n + 1)/(b*n*Log[F]), Int[(c + d*x)^(m - n)*F^(
a + b*(c + d*x)^n), x], x] /; FreeQ[{F, a, b, c, d}, x] && IntegerQ[(2*(m + 1))/n] && LtQ[0, (m + 1)/n, 5] &&
IntegerQ[n] && (LtQ[0, n, m + 1] || LtQ[m, n, 0])

Rule 2226

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*(u_), x_Symbol] :> Int[ExpandLinearProduct[F^(a + b*(c + d*
x)^n), u, c, d, x], x] /; FreeQ[{F, a, b, c, d, n}, x] && PolynomialQ[u, x]

Rule 6688

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

Rule 6706

Int[(F_)^(v_)*(u_), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Simp[(q*F^v)/Log[F], x] /;  !FalseQ[q]
] /; FreeQ[F, x]

Rule 6741

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

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 {e^{x^2} \left (x+2 x^3\right ) \log (x)+e^{x^2} \left (-6-12 x^2+\left (1+2 x^2\right ) \log (5)\right ) \log ^2(x)+e^{\log ^2\left (\frac {x+(-6+\log (5)) \log (x)}{\log (x)}\right )} (-10+10 \log (x)) \log \left (\frac {x+(-6+\log (5)) \log (x)}{\log (x)}\right )}{5 \log (x) \left (x-6 \left (1-\frac {\log (5)}{6}\right ) \log (x)\right )} \, dx\\ &=\frac {1}{5} \int \frac {e^{x^2} \left (x+2 x^3\right ) \log (x)+e^{x^2} \left (-6-12 x^2+\left (1+2 x^2\right ) \log (5)\right ) \log ^2(x)+e^{\log ^2\left (\frac {x+(-6+\log (5)) \log (x)}{\log (x)}\right )} (-10+10 \log (x)) \log \left (\frac {x+(-6+\log (5)) \log (x)}{\log (x)}\right )}{\log (x) \left (x-6 \left (1-\frac {\log (5)}{6}\right ) \log (x)\right )} \, dx\\ &=\frac {1}{5} \int \left (e^{x^2} \left (1+2 x^2\right )+\frac {10 e^{\log ^2\left (-6 \left (1-\frac {\log (5)}{6}\right )+\frac {x}{\log (x)}\right )} (-1+\log (x)) \log \left (-6 \left (1-\frac {\log (5)}{6}\right )+\frac {x}{\log (x)}\right )}{\log (x) \left (x-6 \left (1-\frac {\log (5)}{6}\right ) \log (x)\right )}\right ) \, dx\\ &=\frac {1}{5} \int e^{x^2} \left (1+2 x^2\right ) \, dx+2 \int \frac {e^{\log ^2\left (-6 \left (1-\frac {\log (5)}{6}\right )+\frac {x}{\log (x)}\right )} (-1+\log (x)) \log \left (-6 \left (1-\frac {\log (5)}{6}\right )+\frac {x}{\log (x)}\right )}{\log (x) \left (x-6 \left (1-\frac {\log (5)}{6}\right ) \log (x)\right )} \, dx\\ &=\frac {1}{5} \int \left (e^{x^2}+2 e^{x^2} x^2\right ) \, dx+2 \int \frac {e^{\log ^2\left (-6 \left (1-\frac {\log (5)}{6}\right )+\frac {x}{\log (x)}\right )} (-1+\log (x)) \log \left (-6 \left (1-\frac {\log (5)}{6}\right )+\frac {x}{\log (x)}\right )}{\log (x) (x+(-6+\log (5)) \log (x))} \, dx\\ &=e^{\log ^2\left (-6+\log (5)+\frac {x}{\log (x)}\right )}+\frac {1}{5} \int e^{x^2} \, dx+\frac {2}{5} \int e^{x^2} x^2 \, dx\\ &=e^{\log ^2\left (-6+\log (5)+\frac {x}{\log (x)}\right )}+\frac {e^{x^2} x}{5}+\frac {1}{10} \sqrt {\pi } \text {erfi}(x)-\frac {1}{5} \int e^{x^2} \, dx\\ &=e^{\log ^2\left (-6+\log (5)+\frac {x}{\log (x)}\right )}+\frac {e^{x^2} x}{5}\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.14, size = 29, normalized size = 1.12 \begin {gather*} \frac {1}{5} \left (5 e^{\log ^2\left (-6+\log (5)+\frac {x}{\log (x)}\right )}+e^{x^2} x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^x^2*(x + 2*x^3)*Log[x] + E^x^2*(-6 - 12*x^2 + (1 + 2*x^2)*Log[5])*Log[x]^2 + E^Log[(x + (-6 + Log
[5])*Log[x])/Log[x]]^2*(-10 + 10*Log[x])*Log[(x + (-6 + Log[5])*Log[x])/Log[x]])/(5*x*Log[x] + (-30 + 5*Log[5]
)*Log[x]^2),x]

[Out]

(5*E^Log[-6 + Log[5] + x/Log[x]]^2 + E^x^2*x)/5

________________________________________________________________________________________

fricas [A]  time = 0.83, size = 26, normalized size = 1.00 \begin {gather*} \frac {1}{5} \, x e^{\left (x^{2}\right )} + e^{\left (\log \left (\frac {{\left (\log \relax (5) - 6\right )} \log \relax (x) + x}{\log \relax (x)}\right )^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((10*log(x)-10)*log(((log(5)-6)*log(x)+x)/log(x))*exp(log(((log(5)-6)*log(x)+x)/log(x))^2)+((2*x^2+1
)*log(5)-12*x^2-6)*exp(x^2)*log(x)^2+(2*x^3+x)*exp(x^2)*log(x))/((5*log(5)-30)*log(x)^2+5*x*log(x)),x, algorit
hm="fricas")

[Out]

1/5*x*e^(x^2) + e^(log(((log(5) - 6)*log(x) + x)/log(x))^2)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \mathit {undef} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((10*log(x)-10)*log(((log(5)-6)*log(x)+x)/log(x))*exp(log(((log(5)-6)*log(x)+x)/log(x))^2)+((2*x^2+1
)*log(5)-12*x^2-6)*exp(x^2)*log(x)^2+(2*x^3+x)*exp(x^2)*log(x))/((5*log(5)-30)*log(x)^2+5*x*log(x)),x, algorit
hm="giac")

[Out]

undef

________________________________________________________________________________________

maple [C]  time = 0.24, size = 178, normalized size = 6.85

method	result	size
risch	\(\frac {{\mathrm e}^{x^{2}} x}{5}+{\mathrm e}^{\frac {\left (i \pi \mathrm {csgn}\left (\frac {i \left (\ln \relax (5) \ln \relax (x )-6 \ln \relax (x )+x \right )}{\ln \relax (x )}\right )^{3}-i \pi \mathrm {csgn}\left (\frac {i \left (\ln \relax (5) \ln \relax (x )-6 \ln \relax (x )+x \right )}{\ln \relax (x )}\right )^{2} \mathrm {csgn}\left (\frac {i}{\ln \relax (x )}\right )-i \pi \mathrm {csgn}\left (\frac {i \left (\ln \relax (5) \ln \relax (x )-6 \ln \relax (x )+x \right )}{\ln \relax (x )}\right )^{2} \mathrm {csgn}\left (i \left (\ln \relax (5) \ln \relax (x )-6 \ln \relax (x )+x \right )\right )+i \pi \,\mathrm {csgn}\left (\frac {i \left (\ln \relax (5) \ln \relax (x )-6 \ln \relax (x )+x \right )}{\ln \relax (x )}\right ) \mathrm {csgn}\left (\frac {i}{\ln \relax (x )}\right ) \mathrm {csgn}\left (i \left (\ln \relax (5) \ln \relax (x )-6 \ln \relax (x )+x \right )\right )+2 \ln \left (\ln \relax (x )\right )-2 \ln \left (\ln \relax (5) \ln \relax (x )-6 \ln \relax (x )+x \right )\right )^{2}}{4}}\)	\(178\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((10*ln(x)-10)*ln(((ln(5)-6)*ln(x)+x)/ln(x))*exp(ln(((ln(5)-6)*ln(x)+x)/ln(x))^2)+((2*x^2+1)*ln(5)-12*x^2-
6)*exp(x^2)*ln(x)^2+(2*x^3+x)*exp(x^2)*ln(x))/((5*ln(5)-30)*ln(x)^2+5*x*ln(x)),x,method=_RETURNVERBOSE)

[Out]

1/5*exp(x^2)*x+exp(1/4*(I*Pi*csgn(I/ln(x)*(ln(5)*ln(x)-6*ln(x)+x))^3-I*Pi*csgn(I/ln(x)*(ln(5)*ln(x)-6*ln(x)+x)
)^2*csgn(I/ln(x))-I*Pi*csgn(I/ln(x)*(ln(5)*ln(x)-6*ln(x)+x))^2*csgn(I*(ln(5)*ln(x)-6*ln(x)+x))+I*Pi*csgn(I/ln(
x)*(ln(5)*ln(x)-6*ln(x)+x))*csgn(I/ln(x))*csgn(I*(ln(5)*ln(x)-6*ln(x)+x))+2*ln(ln(x))-2*ln(ln(5)*ln(x)-6*ln(x)
+x))^2)

________________________________________________________________________________________

maxima [A]  time = 0.54, size = 42, normalized size = 1.62 \begin {gather*} \frac {1}{5} \, x e^{\left (x^{2}\right )} + e^{\left (\log \left ({\left (\log \relax (5) - 6\right )} \log \relax (x) + x\right )^{2} - 2 \, \log \left ({\left (\log \relax (5) - 6\right )} \log \relax (x) + x\right ) \log \left (\log \relax (x)\right ) + \log \left (\log \relax (x)\right )^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((10*log(x)-10)*log(((log(5)-6)*log(x)+x)/log(x))*exp(log(((log(5)-6)*log(x)+x)/log(x))^2)+((2*x^2+1
)*log(5)-12*x^2-6)*exp(x^2)*log(x)^2+(2*x^3+x)*exp(x^2)*log(x))/((5*log(5)-30)*log(x)^2+5*x*log(x)),x, algorit
hm="maxima")

[Out]

1/5*x*e^(x^2) + e^(log((log(5) - 6)*log(x) + x)^2 - 2*log((log(5) - 6)*log(x) + x)*log(log(x)) + log(log(x))^2
)

________________________________________________________________________________________

mupad [B]  time = 3.82, size = 28, normalized size = 1.08 \begin {gather*} {\mathrm {e}}^{{\ln \left (\frac {x-6\,\ln \relax (x)+\ln \relax (5)\,\ln \relax (x)}{\ln \relax (x)}\right )}^2}+\frac {x\,{\mathrm {e}}^{x^2}}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x^2)*log(x)*(x + 2*x^3) + exp(log((x + log(x)*(log(5) - 6))/log(x))^2)*log((x + log(x)*(log(5) - 6))/
log(x))*(10*log(x) - 10) - exp(x^2)*log(x)^2*(12*x^2 - log(5)*(2*x^2 + 1) + 6))/(5*x*log(x) + log(x)^2*(5*log(
5) - 30)),x)

[Out]

exp(log((x - 6*log(x) + log(5)*log(x))/log(x))^2) + (x*exp(x^2))/5

________________________________________________________________________________________

sympy [A]  time = 12.91, size = 26, normalized size = 1.00 \begin {gather*} \frac {x e^{x^{2}}}{5} + e^{\log {\left (\frac {x + \left (-6 + \log {\relax (5 )}\right ) \log {\relax (x )}}{\log {\relax (x )}} \right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((10*ln(x)-10)*ln(((ln(5)-6)*ln(x)+x)/ln(x))*exp(ln(((ln(5)-6)*ln(x)+x)/ln(x))**2)+((2*x**2+1)*ln(5)
-12*x**2-6)*exp(x**2)*ln(x)**2+(2*x**3+x)*exp(x**2)*ln(x))/((5*ln(5)-30)*ln(x)**2+5*x*ln(x)),x)

[Out]

x*exp(x**2)/5 + exp(log((x + (-6 + log(5))*log(x))/log(x))**2)

________________________________________________________________________________________