3.92.40 \(\int \frac {-837 x+e^x (-3-279 x-279 x^2) \log (3)+(-567 x+e^x (-189 x-189 x^2) \log (3)) \log (x)+(-96 x+e^x (-32 x-32 x^2) \log (3)) \log ^2(x)+(-279 e^x x \log (3)-189 e^x x \log (3) \log (x)-32 e^x x \log (3) \log ^2(x)) \log (\frac {93+32 \log (x)}{3+\log (x)})}{837 x+567 x \log (x)+96 x \log ^2(x)} \, dx\)

Optimal. Leaf size=27 \[ 3-x-\frac {1}{3} e^x \log (3) \left (x+\log \left (32-\frac {3}{3+\log (x)}\right )\right ) \]

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Rubi [B]  time = 4.05, antiderivative size = 109, normalized size of antiderivative = 4.04, number of steps used = 5, number of rules used = 4, integrand size = 131, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.031, Rules used = {6741, 12, 6742, 2288} \begin {gather*} -\frac {e^x \log (3) \left (279 x^2+32 x^2 \log ^2(x)+189 x^2 \log (x)+32 x \log ^2(x) \log \left (\frac {32 \log (x)+93}{\log (x)+3}\right )+189 x \log (x) \log \left (\frac {32 \log (x)+93}{\log (x)+3}\right )+279 x \log \left (\frac {32 \log (x)+93}{\log (x)+3}\right )\right )}{3 x (\log (x)+3) (32 \log (x)+93)}-x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-837*x + E^x*(-3 - 279*x - 279*x^2)*Log[3] + (-567*x + E^x*(-189*x - 189*x^2)*Log[3])*Log[x] + (-96*x + E
^x*(-32*x - 32*x^2)*Log[3])*Log[x]^2 + (-279*E^x*x*Log[3] - 189*E^x*x*Log[3]*Log[x] - 32*E^x*x*Log[3]*Log[x]^2
)*Log[(93 + 32*Log[x])/(3 + Log[x])])/(837*x + 567*x*Log[x] + 96*x*Log[x]^2),x]

[Out]

-x - (E^x*Log[3]*(279*x^2 + 189*x^2*Log[x] + 32*x^2*Log[x]^2 + 279*x*Log[(93 + 32*Log[x])/(3 + Log[x])] + 189*
x*Log[x]*Log[(93 + 32*Log[x])/(3 + Log[x])] + 32*x*Log[x]^2*Log[(93 + 32*Log[x])/(3 + Log[x])]))/(3*x*(3 + Log
[x])*(93 + 32*Log[x]))

Rule 12

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

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 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 {-837 x+e^x \left (-3-279 x-279 x^2\right ) \log (3)+\left (-567 x+e^x \left (-189 x-189 x^2\right ) \log (3)\right ) \log (x)+\left (-96 x+e^x \left (-32 x-32 x^2\right ) \log (3)\right ) \log ^2(x)+\left (-279 e^x x \log (3)-189 e^x x \log (3) \log (x)-32 e^x x \log (3) \log ^2(x)\right ) \log \left (\frac {93+32 \log (x)}{3+\log (x)}\right )}{3 x \left (279+189 \log (x)+32 \log ^2(x)\right )} \, dx\\ &=\frac {1}{3} \int \frac {-837 x+e^x \left (-3-279 x-279 x^2\right ) \log (3)+\left (-567 x+e^x \left (-189 x-189 x^2\right ) \log (3)\right ) \log (x)+\left (-96 x+e^x \left (-32 x-32 x^2\right ) \log (3)\right ) \log ^2(x)+\left (-279 e^x x \log (3)-189 e^x x \log (3) \log (x)-32 e^x x \log (3) \log ^2(x)\right ) \log \left (\frac {93+32 \log (x)}{3+\log (x)}\right )}{x \left (279+189 \log (x)+32 \log ^2(x)\right )} \, dx\\ &=\frac {1}{3} \int \left (-3-\frac {e^x \log (3) \left (3+279 x+279 x^2+189 x \log (x)+189 x^2 \log (x)+32 x \log ^2(x)+32 x^2 \log ^2(x)+279 x \log \left (\frac {93+32 \log (x)}{3+\log (x)}\right )+189 x \log (x) \log \left (\frac {93+32 \log (x)}{3+\log (x)}\right )+32 x \log ^2(x) \log \left (\frac {93+32 \log (x)}{3+\log (x)}\right )\right )}{x (3+\log (x)) (93+32 \log (x))}\right ) \, dx\\ &=-x-\frac {1}{3} \log (3) \int \frac {e^x \left (3+279 x+279 x^2+189 x \log (x)+189 x^2 \log (x)+32 x \log ^2(x)+32 x^2 \log ^2(x)+279 x \log \left (\frac {93+32 \log (x)}{3+\log (x)}\right )+189 x \log (x) \log \left (\frac {93+32 \log (x)}{3+\log (x)}\right )+32 x \log ^2(x) \log \left (\frac {93+32 \log (x)}{3+\log (x)}\right )\right )}{x (3+\log (x)) (93+32 \log (x))} \, dx\\ &=-x-\frac {e^x \log (3) \left (279 x^2+189 x^2 \log (x)+32 x^2 \log ^2(x)+279 x \log \left (\frac {93+32 \log (x)}{3+\log (x)}\right )+189 x \log (x) \log \left (\frac {93+32 \log (x)}{3+\log (x)}\right )+32 x \log ^2(x) \log \left (\frac {93+32 \log (x)}{3+\log (x)}\right )\right )}{3 x (3+\log (x)) (93+32 \log (x))}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.35, size = 37, normalized size = 1.37 \begin {gather*} \frac {1}{3} \left (-3 x-e^x x \log (3)-e^x \log (3) \log \left (\frac {93+32 \log (x)}{3+\log (x)}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-837*x + E^x*(-3 - 279*x - 279*x^2)*Log[3] + (-567*x + E^x*(-189*x - 189*x^2)*Log[3])*Log[x] + (-96
*x + E^x*(-32*x - 32*x^2)*Log[3])*Log[x]^2 + (-279*E^x*x*Log[3] - 189*E^x*x*Log[3]*Log[x] - 32*E^x*x*Log[3]*Lo
g[x]^2)*Log[(93 + 32*Log[x])/(3 + Log[x])])/(837*x + 567*x*Log[x] + 96*x*Log[x]^2),x]

[Out]

(-3*x - E^x*x*Log[3] - E^x*Log[3]*Log[(93 + 32*Log[x])/(3 + Log[x])])/3

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fricas [A]  time = 0.65, size = 31, normalized size = 1.15 \begin {gather*} -\frac {1}{3} \, x e^{x} \log \relax (3) - \frac {1}{3} \, e^{x} \log \relax (3) \log \left (\frac {32 \, \log \relax (x) + 93}{\log \relax (x) + 3}\right ) - x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-32*x*log(3)*exp(x)*log(x)^2-189*x*log(3)*exp(x)*log(x)-279*x*log(3)*exp(x))*log((32*log(x)+93)/(3
+log(x)))+((-32*x^2-32*x)*log(3)*exp(x)-96*x)*log(x)^2+((-189*x^2-189*x)*log(3)*exp(x)-567*x)*log(x)+(-279*x^2
-279*x-3)*log(3)*exp(x)-837*x)/(96*x*log(x)^2+567*x*log(x)+837*x),x, algorithm="fricas")

[Out]

-1/3*x*e^x*log(3) - 1/3*e^x*log(3)*log((32*log(x) + 93)/(log(x) + 3)) - x

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giac [A]  time = 0.25, size = 35, normalized size = 1.30 \begin {gather*} -\frac {1}{3} \, x e^{x} \log \relax (3) - \frac {1}{3} \, e^{x} \log \relax (3) \log \left (32 \, \log \relax (x) + 93\right ) + \frac {1}{3} \, e^{x} \log \relax (3) \log \left (\log \relax (x) + 3\right ) - x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-32*x*log(3)*exp(x)*log(x)^2-189*x*log(3)*exp(x)*log(x)-279*x*log(3)*exp(x))*log((32*log(x)+93)/(3
+log(x)))+((-32*x^2-32*x)*log(3)*exp(x)-96*x)*log(x)^2+((-189*x^2-189*x)*log(3)*exp(x)-567*x)*log(x)+(-279*x^2
-279*x-3)*log(3)*exp(x)-837*x)/(96*x*log(x)^2+567*x*log(x)+837*x),x, algorithm="giac")

[Out]

-1/3*x*e^x*log(3) - 1/3*e^x*log(3)*log(32*log(x) + 93) + 1/3*e^x*log(3)*log(log(x) + 3) - x

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maple [C]  time = 0.56, size = 172, normalized size = 6.37

method	result	size
risch	\(-\frac {\ln \relax (3) {\mathrm e}^{x} \ln \left (\ln \relax (x )+\frac {93}{32}\right )}{3}+\frac {\ln \relax (3) {\mathrm e}^{x} \ln \left (3+\ln \relax (x )\right )}{3}-x +\frac {i \pi \ln \relax (3) \mathrm {csgn}\left (\frac {i}{3+\ln \relax (x )}\right ) \mathrm {csgn}\left (i \left (\ln \relax (x )+\frac {93}{32}\right )\right ) \mathrm {csgn}\left (\frac {i \left (\ln \relax (x )+\frac {93}{32}\right )}{3+\ln \relax (x )}\right ) {\mathrm e}^{x}}{6}-\frac {i \pi \ln \relax (3) \mathrm {csgn}\left (\frac {i}{3+\ln \relax (x )}\right ) \mathrm {csgn}\left (\frac {i \left (\ln \relax (x )+\frac {93}{32}\right )}{3+\ln \relax (x )}\right )^{2} {\mathrm e}^{x}}{6}-\frac {i \pi \ln \relax (3) \mathrm {csgn}\left (i \left (\ln \relax (x )+\frac {93}{32}\right )\right ) \mathrm {csgn}\left (\frac {i \left (\ln \relax (x )+\frac {93}{32}\right )}{3+\ln \relax (x )}\right )^{2} {\mathrm e}^{x}}{6}+\frac {i \pi \ln \relax (3) \mathrm {csgn}\left (\frac {i \left (\ln \relax (x )+\frac {93}{32}\right )}{3+\ln \relax (x )}\right )^{3} {\mathrm e}^{x}}{6}-\frac {5 \,{\mathrm e}^{x} \ln \relax (3) \ln \relax (2)}{3}-\frac {x \ln \relax (3) {\mathrm e}^{x}}{3}\)	\(172\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-32*x*ln(3)*exp(x)*ln(x)^2-189*x*ln(3)*exp(x)*ln(x)-279*x*ln(3)*exp(x))*ln((32*ln(x)+93)/(3+ln(x)))+((-3
2*x^2-32*x)*ln(3)*exp(x)-96*x)*ln(x)^2+((-189*x^2-189*x)*ln(3)*exp(x)-567*x)*ln(x)+(-279*x^2-279*x-3)*ln(3)*ex
p(x)-837*x)/(96*x*ln(x)^2+567*x*ln(x)+837*x),x,method=_RETURNVERBOSE)

[Out]

-1/3*ln(3)*exp(x)*ln(ln(x)+93/32)+1/3*ln(3)*exp(x)*ln(3+ln(x))-x+1/6*I*Pi*ln(3)*csgn(I/(3+ln(x)))*csgn(I*(ln(x
)+93/32))*csgn(I/(3+ln(x))*(ln(x)+93/32))*exp(x)-1/6*I*Pi*ln(3)*csgn(I/(3+ln(x)))*csgn(I/(3+ln(x))*(ln(x)+93/3
2))^2*exp(x)-1/6*I*Pi*ln(3)*csgn(I*(ln(x)+93/32))*csgn(I/(3+ln(x))*(ln(x)+93/32))^2*exp(x)+1/6*I*Pi*ln(3)*csgn
(I/(3+ln(x))*(ln(x)+93/32))^3*exp(x)-5/3*exp(x)*ln(3)*ln(2)-1/3*x*ln(3)*exp(x)

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maxima [A]  time = 0.50, size = 35, normalized size = 1.30 \begin {gather*} -\frac {1}{3} \, x e^{x} \log \relax (3) - \frac {1}{3} \, e^{x} \log \relax (3) \log \left (32 \, \log \relax (x) + 93\right ) + \frac {1}{3} \, e^{x} \log \relax (3) \log \left (\log \relax (x) + 3\right ) - x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-32*x*log(3)*exp(x)*log(x)^2-189*x*log(3)*exp(x)*log(x)-279*x*log(3)*exp(x))*log((32*log(x)+93)/(3
+log(x)))+((-32*x^2-32*x)*log(3)*exp(x)-96*x)*log(x)^2+((-189*x^2-189*x)*log(3)*exp(x)-567*x)*log(x)+(-279*x^2
-279*x-3)*log(3)*exp(x)-837*x)/(96*x*log(x)^2+567*x*log(x)+837*x),x, algorithm="maxima")

[Out]

-1/3*x*e^x*log(3) - 1/3*e^x*log(3)*log(32*log(x) + 93) + 1/3*e^x*log(3)*log(log(x) + 3) - x

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mupad [B]  time = 7.39, size = 31, normalized size = 1.15 \begin {gather*} -x-\frac {x\,{\mathrm {e}}^x\,\ln \relax (3)}{3}-\frac {\ln \left (\frac {32\,\ln \relax (x)+93}{\ln \relax (x)+3}\right )\,{\mathrm {e}}^x\,\ln \relax (3)}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(837*x + log((32*log(x) + 93)/(log(x) + 3))*(279*x*exp(x)*log(3) + 189*x*exp(x)*log(3)*log(x) + 32*x*exp(
x)*log(3)*log(x)^2) + log(x)*(567*x + exp(x)*log(3)*(189*x + 189*x^2)) + log(x)^2*(96*x + exp(x)*log(3)*(32*x
+ 32*x^2)) + exp(x)*log(3)*(279*x + 279*x^2 + 3))/(837*x + 96*x*log(x)^2 + 567*x*log(x)),x)

[Out]

- x - (x*exp(x)*log(3))/3 - (log((32*log(x) + 93)/(log(x) + 3))*exp(x)*log(3))/3

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sympy [A]  time = 14.23, size = 29, normalized size = 1.07 \begin {gather*} - x + \frac {\left (- x \log {\relax (3 )} - \log {\relax (3 )} \log {\left (\frac {32 \log {\relax (x )} + 93}{\log {\relax (x )} + 3} \right )}\right ) e^{x}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-32*x*ln(3)*exp(x)*ln(x)**2-189*x*ln(3)*exp(x)*ln(x)-279*x*ln(3)*exp(x))*ln((32*ln(x)+93)/(3+ln(x)
))+((-32*x**2-32*x)*ln(3)*exp(x)-96*x)*ln(x)**2+((-189*x**2-189*x)*ln(3)*exp(x)-567*x)*ln(x)+(-279*x**2-279*x-
3)*ln(3)*exp(x)-837*x)/(96*x*ln(x)**2+567*x*ln(x)+837*x),x)

[Out]

-x + (-x*log(3) - log(3)*log((32*log(x) + 93)/(log(x) + 3)))*exp(x)/3

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