3.10.54 \(\int \frac {-6 e+18 x+(-2 e+30 x) \log (x)+14 x \log ^2(x)+2 x \log ^3(x)+(4 e+9 x+(e+6 x) \log (x)+x \log ^2(x)) \log (x^2)}{9 x^2+6 x^2 \log (x)+x^2 \log ^2(x)} \, dx\)

Optimal. Leaf size=21 \[ \left (1+\log (x)-\frac {e}{x (3+\log (x))}\right ) \log \left (x^2\right ) \]

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Rubi [F]  time = 2.55, 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 {-6 e+18 x+(-2 e+30 x) \log (x)+14 x \log ^2(x)+2 x \log ^3(x)+\left (4 e+9 x+(e+6 x) \log (x)+x \log ^2(x)\right ) \log \left (x^2\right )}{9 x^2+6 x^2 \log (x)+x^2 \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-6*E + 18*x + (-2*E + 30*x)*Log[x] + 14*x*Log[x]^2 + 2*x*Log[x]^3 + (4*E + 9*x + (E + 6*x)*Log[x] + x*Log
[x]^2)*Log[x^2])/(9*x^2 + 6*x^2*Log[x] + x^2*Log[x]^2),x]

[Out]

(8*E)/x + 6*E^4*ExpIntegralEi[-3 - Log[x]] + 2*Log[x] + Log[x]^2 + 8*E^4*ExpIntegralEi[-3 - Log[x]]*(3 + Log[x
]) - 4*E^4*ExpIntegralEi[-3 - Log[x]]*Log[x^2] - (9*Log[x^2])/(3 + Log[x]) - (4*E*Log[x^2])/(x*(3 + Log[x])) +
18*Log[3 + Log[x]] + E*Defer[Int][(Log[x]*Log[x^2])/(x^2*(3 + Log[x])^2), x] + 6*Defer[Int][(Log[x]*Log[x^2])
/(x*(3 + Log[x])^2), x] + Defer[Int][(Log[x]^2*Log[x^2])/(x*(3 + Log[x])^2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-6 e+18 x+(-2 e+30 x) \log (x)+14 x \log ^2(x)+2 x \log ^3(x)+\left (4 e+9 x+(e+6 x) \log (x)+x \log ^2(x)\right ) \log \left (x^2\right )}{x^2 (3+\log (x))^2} \, dx\\ &=\int \left (-\frac {6 e}{x^2 (3+\log (x))^2}+\frac {18}{x (3+\log (x))^2}-\frac {2 (e-15 x) \log (x)}{x^2 (3+\log (x))^2}+\frac {14 \log ^2(x)}{x (3+\log (x))^2}+\frac {2 \log ^3(x)}{x (3+\log (x))^2}+\frac {\left (4 e+9 x+e \log (x)+6 x \log (x)+x \log ^2(x)\right ) \log \left (x^2\right )}{x^2 (3+\log (x))^2}\right ) \, dx\\ &=-\left (2 \int \frac {(e-15 x) \log (x)}{x^2 (3+\log (x))^2} \, dx\right )+2 \int \frac {\log ^3(x)}{x (3+\log (x))^2} \, dx+14 \int \frac {\log ^2(x)}{x (3+\log (x))^2} \, dx+18 \int \frac {1}{x (3+\log (x))^2} \, dx-(6 e) \int \frac {1}{x^2 (3+\log (x))^2} \, dx+\int \frac {\left (4 e+9 x+e \log (x)+6 x \log (x)+x \log ^2(x)\right ) \log \left (x^2\right )}{x^2 (3+\log (x))^2} \, dx\\ &=\frac {6 e}{x (3+\log (x))}-2 \int \left (-\frac {3 (e-15 x)}{x^2 (3+\log (x))^2}+\frac {e-15 x}{x^2 (3+\log (x))}\right ) \, dx+2 \operatorname {Subst}\left (\int \frac {x^3}{(3+x)^2} \, dx,x,\log (x)\right )+14 \operatorname {Subst}\left (\int \frac {x^2}{(3+x)^2} \, dx,x,\log (x)\right )+18 \operatorname {Subst}\left (\int \frac {1}{x^2} \, dx,x,3+\log (x)\right )+(6 e) \int \frac {1}{x^2 (3+\log (x))} \, dx+\int \left (\frac {4 e \log \left (x^2\right )}{x^2 (3+\log (x))^2}+\frac {9 \log \left (x^2\right )}{x (3+\log (x))^2}+\frac {e \log (x) \log \left (x^2\right )}{x^2 (3+\log (x))^2}+\frac {6 \log (x) \log \left (x^2\right )}{x (3+\log (x))^2}+\frac {\log ^2(x) \log \left (x^2\right )}{x (3+\log (x))^2}\right ) \, dx\\ &=-\frac {18}{3+\log (x)}+\frac {6 e}{x (3+\log (x))}-2 \int \frac {e-15 x}{x^2 (3+\log (x))} \, dx+2 \operatorname {Subst}\left (\int \left (-6+x-\frac {27}{(3+x)^2}+\frac {27}{3+x}\right ) \, dx,x,\log (x)\right )+6 \int \frac {e-15 x}{x^2 (3+\log (x))^2} \, dx+6 \int \frac {\log (x) \log \left (x^2\right )}{x (3+\log (x))^2} \, dx+9 \int \frac {\log \left (x^2\right )}{x (3+\log (x))^2} \, dx+14 \operatorname {Subst}\left (\int \left (1+\frac {9}{(3+x)^2}-\frac {6}{3+x}\right ) \, dx,x,\log (x)\right )+e \int \frac {\log (x) \log \left (x^2\right )}{x^2 (3+\log (x))^2} \, dx+(4 e) \int \frac {\log \left (x^2\right )}{x^2 (3+\log (x))^2} \, dx+(6 e) \operatorname {Subst}\left (\int \frac {e^{-x}}{3+x} \, dx,x,\log (x)\right )+\int \frac {\log ^2(x) \log \left (x^2\right )}{x (3+\log (x))^2} \, dx\\ &=6 e^4 \text {Ei}(-3-\log (x))+2 \log (x)+\log ^2(x)-\frac {90}{3+\log (x)}+\frac {6 e}{x (3+\log (x))}-4 e^4 \text {Ei}(-3-\log (x)) \log \left (x^2\right )-\frac {9 \log \left (x^2\right )}{3+\log (x)}-\frac {4 e \log \left (x^2\right )}{x (3+\log (x))}-30 \log (3+\log (x))-2 \int \left (\frac {e}{x^2 (3+\log (x))}-\frac {15}{x (3+\log (x))}\right ) \, dx+6 \int \left (\frac {e}{x^2 (3+\log (x))^2}-\frac {15}{x (3+\log (x))^2}\right ) \, dx+6 \int \frac {\log (x) \log \left (x^2\right )}{x (3+\log (x))^2} \, dx+18 \int \frac {1}{x (3+\log (x))} \, dx+e \int \frac {\log (x) \log \left (x^2\right )}{x^2 (3+\log (x))^2} \, dx-(8 e) \int \frac {-e^3 x \text {Ei}(-3-\log (x))-\frac {1}{3+\log (x)}}{x^2} \, dx+\int \frac {\log ^2(x) \log \left (x^2\right )}{x (3+\log (x))^2} \, dx\\ &=6 e^4 \text {Ei}(-3-\log (x))+2 \log (x)+\log ^2(x)-\frac {90}{3+\log (x)}+\frac {6 e}{x (3+\log (x))}-4 e^4 \text {Ei}(-3-\log (x)) \log \left (x^2\right )-\frac {9 \log \left (x^2\right )}{3+\log (x)}-\frac {4 e \log \left (x^2\right )}{x (3+\log (x))}-30 \log (3+\log (x))+6 \int \frac {\log (x) \log \left (x^2\right )}{x (3+\log (x))^2} \, dx+18 \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,3+\log (x)\right )+30 \int \frac {1}{x (3+\log (x))} \, dx-90 \int \frac {1}{x (3+\log (x))^2} \, dx+e \int \frac {\log (x) \log \left (x^2\right )}{x^2 (3+\log (x))^2} \, dx-(2 e) \int \frac {1}{x^2 (3+\log (x))} \, dx+(6 e) \int \frac {1}{x^2 (3+\log (x))^2} \, dx-(8 e) \int \left (-\frac {e^3 \text {Ei}(-3-\log (x))}{x}-\frac {1}{x^2 (3+\log (x))}\right ) \, dx+\int \frac {\log ^2(x) \log \left (x^2\right )}{x (3+\log (x))^2} \, dx\\ &=6 e^4 \text {Ei}(-3-\log (x))+2 \log (x)+\log ^2(x)-\frac {90}{3+\log (x)}-4 e^4 \text {Ei}(-3-\log (x)) \log \left (x^2\right )-\frac {9 \log \left (x^2\right )}{3+\log (x)}-\frac {4 e \log \left (x^2\right )}{x (3+\log (x))}-12 \log (3+\log (x))+6 \int \frac {\log (x) \log \left (x^2\right )}{x (3+\log (x))^2} \, dx+30 \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,3+\log (x)\right )-90 \operatorname {Subst}\left (\int \frac {1}{x^2} \, dx,x,3+\log (x)\right )+e \int \frac {\log (x) \log \left (x^2\right )}{x^2 (3+\log (x))^2} \, dx-(2 e) \operatorname {Subst}\left (\int \frac {e^{-x}}{3+x} \, dx,x,\log (x)\right )-(6 e) \int \frac {1}{x^2 (3+\log (x))} \, dx+(8 e) \int \frac {1}{x^2 (3+\log (x))} \, dx+\left (8 e^4\right ) \int \frac {\text {Ei}(-3-\log (x))}{x} \, dx+\int \frac {\log ^2(x) \log \left (x^2\right )}{x (3+\log (x))^2} \, dx\\ &=4 e^4 \text {Ei}(-3-\log (x))+2 \log (x)+\log ^2(x)-4 e^4 \text {Ei}(-3-\log (x)) \log \left (x^2\right )-\frac {9 \log \left (x^2\right )}{3+\log (x)}-\frac {4 e \log \left (x^2\right )}{x (3+\log (x))}+18 \log (3+\log (x))+6 \int \frac {\log (x) \log \left (x^2\right )}{x (3+\log (x))^2} \, dx+e \int \frac {\log (x) \log \left (x^2\right )}{x^2 (3+\log (x))^2} \, dx-(6 e) \operatorname {Subst}\left (\int \frac {e^{-x}}{3+x} \, dx,x,\log (x)\right )+(8 e) \operatorname {Subst}\left (\int \frac {e^{-x}}{3+x} \, dx,x,\log (x)\right )+\left (8 e^4\right ) \operatorname {Subst}(\int \text {Ei}(-3-x) \, dx,x,\log (x))+\int \frac {\log ^2(x) \log \left (x^2\right )}{x (3+\log (x))^2} \, dx\\ &=\frac {8 e}{x}+6 e^4 \text {Ei}(-3-\log (x))+2 \log (x)+\log ^2(x)+8 e^4 \text {Ei}(-3-\log (x)) (3+\log (x))-4 e^4 \text {Ei}(-3-\log (x)) \log \left (x^2\right )-\frac {9 \log \left (x^2\right )}{3+\log (x)}-\frac {4 e \log \left (x^2\right )}{x (3+\log (x))}+18 \log (3+\log (x))+6 \int \frac {\log (x) \log \left (x^2\right )}{x (3+\log (x))^2} \, dx+e \int \frac {\log (x) \log \left (x^2\right )}{x^2 (3+\log (x))^2} \, dx+\int \frac {\log ^2(x) \log \left (x^2\right )}{x (3+\log (x))^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.16, size = 41, normalized size = 1.95 \begin {gather*} \frac {-e \log \left (x^2\right )+3 x \log (x) \left (2+\log \left (x^2\right )\right )+x \log ^2(x) \left (2+\log \left (x^2\right )\right )}{x (3+\log (x))} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-6*E + 18*x + (-2*E + 30*x)*Log[x] + 14*x*Log[x]^2 + 2*x*Log[x]^3 + (4*E + 9*x + (E + 6*x)*Log[x] +
x*Log[x]^2)*Log[x^2])/(9*x^2 + 6*x^2*Log[x] + x^2*Log[x]^2),x]

[Out]

(-(E*Log[x^2]) + 3*x*Log[x]*(2 + Log[x^2]) + x*Log[x]^2*(2 + Log[x^2]))/(x*(3 + Log[x]))

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fricas [A]  time = 0.86, size = 37, normalized size = 1.76 \begin {gather*} \frac {2 \, {\left (x \log \relax (x)^{3} + 4 \, x \log \relax (x)^{2} + {\left (3 \, x - e\right )} \log \relax (x)\right )}}{x \log \relax (x) + 3 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x*log(x)^2+(exp(1)+6*x)*log(x)+4*exp(1)+9*x)*log(x^2)+2*x*log(x)^3+14*x*log(x)^2+(-2*exp(1)+30*x)*
log(x)-6*exp(1)+18*x)/(x^2*log(x)^2+6*x^2*log(x)+9*x^2),x, algorithm="fricas")

[Out]

2*(x*log(x)^3 + 4*x*log(x)^2 + (3*x - e)*log(x))/(x*log(x) + 3*x)

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giac [A]  time = 0.49, size = 37, normalized size = 1.76 \begin {gather*} \frac {2 \, {\left (x \log \relax (x)^{3} + 4 \, x \log \relax (x)^{2} + 3 \, x \log \relax (x) - e \log \relax (x)\right )}}{x \log \relax (x) + 3 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x*log(x)^2+(exp(1)+6*x)*log(x)+4*exp(1)+9*x)*log(x^2)+2*x*log(x)^3+14*x*log(x)^2+(-2*exp(1)+30*x)*
log(x)-6*exp(1)+18*x)/(x^2*log(x)^2+6*x^2*log(x)+9*x^2),x, algorithm="giac")

[Out]

2*(x*log(x)^3 + 4*x*log(x)^2 + 3*x*log(x) - e*log(x))/(x*log(x) + 3*x)

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maple [C]  time = 0.12, size = 142, normalized size = 6.76

method	result	size
risch	\(2 \ln \relax (x )^{2}-\frac {i \pi \ln \relax (x ) \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right ) x -2 i \pi \ln \relax (x ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2} x +i \pi \ln \relax (x ) \mathrm {csgn}\left (i x^{2}\right )^{3} x -4 x \ln \relax (x )+4 \,{\mathrm e}}{2 x}+\frac {i {\mathrm e} \left (\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}-12 i\right )}{2 x \left (3+\ln \relax (x )\right )}\)	\(142\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x*ln(x)^2+(exp(1)+6*x)*ln(x)+4*exp(1)+9*x)*ln(x^2)+2*x*ln(x)^3+14*x*ln(x)^2+(-2*exp(1)+30*x)*ln(x)-6*exp
(1)+18*x)/(x^2*ln(x)^2+6*x^2*ln(x)+9*x^2),x,method=_RETURNVERBOSE)

[Out]

2*ln(x)^2-1/2*(I*Pi*ln(x)*csgn(I*x)^2*csgn(I*x^2)*x-2*I*Pi*ln(x)*csgn(I*x)*csgn(I*x^2)^2*x+I*Pi*ln(x)*csgn(I*x
^2)^3*x-4*x*ln(x)+4*exp(1))/x+1/2*I/x*exp(1)*(Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2+Pi*csgn(
I*x^2)^3-12*I)/(3+ln(x))

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maxima [A]  time = 0.82, size = 37, normalized size = 1.76 \begin {gather*} \frac {2 \, {\left (x \log \relax (x)^{3} + 4 \, x \log \relax (x)^{2} + {\left (3 \, x - e\right )} \log \relax (x)\right )}}{x \log \relax (x) + 3 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x*log(x)^2+(exp(1)+6*x)*log(x)+4*exp(1)+9*x)*log(x^2)+2*x*log(x)^3+14*x*log(x)^2+(-2*exp(1)+30*x)*
log(x)-6*exp(1)+18*x)/(x^2*log(x)^2+6*x^2*log(x)+9*x^2),x, algorithm="maxima")

[Out]

2*(x*log(x)^3 + 4*x*log(x)^2 + (3*x - e)*log(x))/(x*log(x) + 3*x)

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mupad [B]  time = 0.83, size = 34, normalized size = 1.62 \begin {gather*} \frac {\ln \relax (x)\,\left (2\,x+x\,\ln \left (x^2\right )\right )}{x}-\frac {\ln \left (x^2\right )\,\mathrm {e}}{x\,\left (\ln \relax (x)+3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((18*x - 6*exp(1) + 14*x*log(x)^2 + 2*x*log(x)^3 + log(x)*(30*x - 2*exp(1)) + log(x^2)*(9*x + 4*exp(1) + x*
log(x)^2 + log(x)*(6*x + exp(1))))/(6*x^2*log(x) + x^2*log(x)^2 + 9*x^2),x)

[Out]

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

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sympy [A]  time = 0.28, size = 31, normalized size = 1.48 \begin {gather*} 2 \log {\relax (x )}^{2} + 2 \log {\relax (x )} + \frac {6 e}{x \log {\relax (x )} + 3 x} - \frac {2 e}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x*ln(x)**2+(exp(1)+6*x)*ln(x)+4*exp(1)+9*x)*ln(x**2)+2*x*ln(x)**3+14*x*ln(x)**2+(-2*exp(1)+30*x)*l
n(x)-6*exp(1)+18*x)/(x**2*ln(x)**2+6*x**2*ln(x)+9*x**2),x)

[Out]

2*log(x)**2 + 2*log(x) + 6*E/(x*log(x) + 3*x) - 2*E/x

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