3.13.26 \(\int \frac {-21-15 e^{2 x}-6 x+e^x (36+3 x-3 x^2)+(36-30 e^x+3 x) \log (x)-15 \log ^2(x)+(e^x (-3-3 x)-3 \log (x)) \log (e^x x^2)}{x^2-2 e^x x^2+e^{2 x} x^2+(-2 x^2+2 e^x x^2) \log (x)+x^2 \log ^2(x)} \, dx\)

Optimal. Leaf size=32 \[ \frac {3 \left (5-\frac {x+\log \left (e^x x^2\right )}{1-e^x-\log (x)}\right )}{x} \]

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Rubi [F]  time = 4.63, 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 {-21-15 e^{2 x}-6 x+e^x \left (36+3 x-3 x^2\right )+\left (36-30 e^x+3 x\right ) \log (x)-15 \log ^2(x)+\left (e^x (-3-3 x)-3 \log (x)\right ) \log \left (e^x x^2\right )}{x^2-2 e^x x^2+e^{2 x} x^2+\left (-2 x^2+2 e^x x^2\right ) \log (x)+x^2 \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-21 - 15*E^(2*x) - 6*x + E^x*(36 + 3*x - 3*x^2) + (36 - 30*E^x + 3*x)*Log[x] - 15*Log[x]^2 + (E^x*(-3 - 3
*x) - 3*Log[x])*Log[E^x*x^2])/(x^2 - 2*E^x*x^2 + E^(2*x)*x^2 + (-2*x^2 + 2*E^x*x^2)*Log[x] + x^2*Log[x]^2),x]

[Out]

15/x - 3*Defer[Int][(-1 + E^x + Log[x])^(-2), x] - 3*Defer[Int][1/(x*(-1 + E^x + Log[x])^2), x] + 3*Defer[Int]
[Log[x]/(-1 + E^x + Log[x])^2, x] - 3*Defer[Int][(-1 + E^x + Log[x])^(-1), x] + 6*Defer[Int][1/(x^2*(-1 + E^x
+ Log[x])), x] + 3*Defer[Int][1/(x*(-1 + E^x + Log[x])), x] - 3*Defer[Int][Log[E^x*x^2]/(x^2*(-1 + E^x + Log[x
])^2), x] - 3*Defer[Int][Log[E^x*x^2]/(x*(-1 + E^x + Log[x])^2), x] + 3*Defer[Int][(Log[x]*Log[E^x*x^2])/(x*(-
1 + E^x + Log[x])^2), x] - 3*Defer[Int][Log[E^x*x^2]/(x^2*(-1 + E^x + Log[x])), x] - 3*Defer[Int][Log[E^x*x^2]
/(x*(-1 + E^x + Log[x])), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-21-15 e^{2 x}-6 x+e^x \left (36+3 x-3 x^2\right )+\left (36-30 e^x+3 x\right ) \log (x)-15 \log ^2(x)+\left (e^x (-3-3 x)-3 \log (x)\right ) \log \left (e^x x^2\right )}{x^2 \left (1-e^x-\log (x)\right )^2} \, dx\\ &=\int \left (-\frac {15}{x^2}-\frac {3 (1+x) \left (-2+x+\log \left (e^x x^2\right )\right )}{x^2 \left (-1+e^x+\log (x)\right )}+\frac {3 (-1-x+x \log (x)) \left (x+\log \left (e^x x^2\right )\right )}{x^2 \left (-1+e^x+\log (x)\right )^2}\right ) \, dx\\ &=\frac {15}{x}-3 \int \frac {(1+x) \left (-2+x+\log \left (e^x x^2\right )\right )}{x^2 \left (-1+e^x+\log (x)\right )} \, dx+3 \int \frac {(-1-x+x \log (x)) \left (x+\log \left (e^x x^2\right )\right )}{x^2 \left (-1+e^x+\log (x)\right )^2} \, dx\\ &=\frac {15}{x}+3 \int \left (-\frac {1}{\left (-1+e^x+\log (x)\right )^2}-\frac {1}{x \left (-1+e^x+\log (x)\right )^2}+\frac {\log (x)}{\left (-1+e^x+\log (x)\right )^2}-\frac {\log \left (e^x x^2\right )}{x^2 \left (-1+e^x+\log (x)\right )^2}-\frac {\log \left (e^x x^2\right )}{x \left (-1+e^x+\log (x)\right )^2}+\frac {\log (x) \log \left (e^x x^2\right )}{x \left (-1+e^x+\log (x)\right )^2}\right ) \, dx-3 \int \left (\frac {-2+x+\log \left (e^x x^2\right )}{x^2 \left (-1+e^x+\log (x)\right )}+\frac {-2+x+\log \left (e^x x^2\right )}{x \left (-1+e^x+\log (x)\right )}\right ) \, dx\\ &=\frac {15}{x}-3 \int \frac {1}{\left (-1+e^x+\log (x)\right )^2} \, dx-3 \int \frac {1}{x \left (-1+e^x+\log (x)\right )^2} \, dx+3 \int \frac {\log (x)}{\left (-1+e^x+\log (x)\right )^2} \, dx-3 \int \frac {\log \left (e^x x^2\right )}{x^2 \left (-1+e^x+\log (x)\right )^2} \, dx-3 \int \frac {\log \left (e^x x^2\right )}{x \left (-1+e^x+\log (x)\right )^2} \, dx+3 \int \frac {\log (x) \log \left (e^x x^2\right )}{x \left (-1+e^x+\log (x)\right )^2} \, dx-3 \int \frac {-2+x+\log \left (e^x x^2\right )}{x^2 \left (-1+e^x+\log (x)\right )} \, dx-3 \int \frac {-2+x+\log \left (e^x x^2\right )}{x \left (-1+e^x+\log (x)\right )} \, dx\\ &=\frac {15}{x}-3 \int \frac {1}{\left (-1+e^x+\log (x)\right )^2} \, dx-3 \int \frac {1}{x \left (-1+e^x+\log (x)\right )^2} \, dx+3 \int \frac {\log (x)}{\left (-1+e^x+\log (x)\right )^2} \, dx-3 \int \frac {\log \left (e^x x^2\right )}{x^2 \left (-1+e^x+\log (x)\right )^2} \, dx-3 \int \frac {\log \left (e^x x^2\right )}{x \left (-1+e^x+\log (x)\right )^2} \, dx+3 \int \frac {\log (x) \log \left (e^x x^2\right )}{x \left (-1+e^x+\log (x)\right )^2} \, dx-3 \int \left (-\frac {2}{x^2 \left (-1+e^x+\log (x)\right )}+\frac {1}{x \left (-1+e^x+\log (x)\right )}+\frac {\log \left (e^x x^2\right )}{x^2 \left (-1+e^x+\log (x)\right )}\right ) \, dx-3 \int \left (\frac {1}{-1+e^x+\log (x)}-\frac {2}{x \left (-1+e^x+\log (x)\right )}+\frac {\log \left (e^x x^2\right )}{x \left (-1+e^x+\log (x)\right )}\right ) \, dx\\ &=\frac {15}{x}-3 \int \frac {1}{\left (-1+e^x+\log (x)\right )^2} \, dx-3 \int \frac {1}{x \left (-1+e^x+\log (x)\right )^2} \, dx+3 \int \frac {\log (x)}{\left (-1+e^x+\log (x)\right )^2} \, dx-3 \int \frac {1}{-1+e^x+\log (x)} \, dx-3 \int \frac {1}{x \left (-1+e^x+\log (x)\right )} \, dx-3 \int \frac {\log \left (e^x x^2\right )}{x^2 \left (-1+e^x+\log (x)\right )^2} \, dx-3 \int \frac {\log \left (e^x x^2\right )}{x \left (-1+e^x+\log (x)\right )^2} \, dx+3 \int \frac {\log (x) \log \left (e^x x^2\right )}{x \left (-1+e^x+\log (x)\right )^2} \, dx-3 \int \frac {\log \left (e^x x^2\right )}{x^2 \left (-1+e^x+\log (x)\right )} \, dx-3 \int \frac {\log \left (e^x x^2\right )}{x \left (-1+e^x+\log (x)\right )} \, dx+6 \int \frac {1}{x^2 \left (-1+e^x+\log (x)\right )} \, dx+6 \int \frac {1}{x \left (-1+e^x+\log (x)\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.09, size = 34, normalized size = 1.06 \begin {gather*} \frac {3 \left (-5+5 e^x+x+5 \log (x)+\log \left (e^x x^2\right )\right )}{x \left (-1+e^x+\log (x)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-21 - 15*E^(2*x) - 6*x + E^x*(36 + 3*x - 3*x^2) + (36 - 30*E^x + 3*x)*Log[x] - 15*Log[x]^2 + (E^x*(
-3 - 3*x) - 3*Log[x])*Log[E^x*x^2])/(x^2 - 2*E^x*x^2 + E^(2*x)*x^2 + (-2*x^2 + 2*E^x*x^2)*Log[x] + x^2*Log[x]^
2),x]

[Out]

(3*(-5 + 5*E^x + x + 5*Log[x] + Log[E^x*x^2]))/(x*(-1 + E^x + Log[x]))

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fricas [A]  time = 1.21, size = 29, normalized size = 0.91 \begin {gather*} \frac {3 \, {\left (2 \, x + 5 \, e^{x} + 7 \, \log \relax (x) - 5\right )}}{x e^{x} + x \log \relax (x) - x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*log(x)+(-3*x-3)*exp(x))*log(exp(x)*x^2)-15*log(x)^2+(-30*exp(x)+3*x+36)*log(x)-15*exp(x)^2+(-3*
x^2+3*x+36)*exp(x)-6*x-21)/(x^2*log(x)^2+(2*exp(x)*x^2-2*x^2)*log(x)+exp(x)^2*x^2-2*exp(x)*x^2+x^2),x, algorit
hm="fricas")

[Out]

3*(2*x + 5*e^x + 7*log(x) - 5)/(x*e^x + x*log(x) - x)

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giac [A]  time = 0.77, size = 29, normalized size = 0.91 \begin {gather*} \frac {3 \, {\left (2 \, x + 5 \, e^{x} + 7 \, \log \relax (x) - 5\right )}}{x e^{x} + x \log \relax (x) - x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*log(x)+(-3*x-3)*exp(x))*log(exp(x)*x^2)-15*log(x)^2+(-30*exp(x)+3*x+36)*log(x)-15*exp(x)^2+(-3*
x^2+3*x+36)*exp(x)-6*x-21)/(x^2*log(x)^2+(2*exp(x)*x^2-2*x^2)*log(x)+exp(x)^2*x^2-2*exp(x)*x^2+x^2),x, algorit
hm="giac")

[Out]

3*(2*x + 5*e^x + 7*log(x) - 5)/(x*e^x + x*log(x) - x)

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maple [C]  time = 0.12, size = 177, normalized size = 5.53

method	result	size
risch	\(\frac {3 \ln \left ({\mathrm e}^{x}\right )}{x \left (-1+\ln \relax (x )+{\mathrm e}^{x}\right )}+\frac {-\frac {3 i \pi \,\mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x \right )^{2}}{2}+3 i \pi \mathrm {csgn}\left (i x^{2}\right )^{2} \mathrm {csgn}\left (i x \right )-\frac {3 i \pi \mathrm {csgn}\left (i x^{2}\right )^{3}}{2}-\frac {3 i \pi \,\mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i {\mathrm e}^{x}\right ) \mathrm {csgn}\left (i x^{2} {\mathrm e}^{x}\right )}{2}+\frac {3 i \pi \,\mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x^{2} {\mathrm e}^{x}\right )^{2}}{2}+\frac {3 i \pi \,\mathrm {csgn}\left (i {\mathrm e}^{x}\right ) \mathrm {csgn}\left (i x^{2} {\mathrm e}^{x}\right )^{2}}{2}-\frac {3 i \pi \mathrm {csgn}\left (i x^{2} {\mathrm e}^{x}\right )^{3}}{2}+3 x +15 \,{\mathrm e}^{x}+21 \ln \relax (x )-15}{x \left (-1+\ln \relax (x )+{\mathrm e}^{x}\right )}\)	\(177\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-3*ln(x)+(-3*x-3)*exp(x))*ln(exp(x)*x^2)-15*ln(x)^2+(-30*exp(x)+3*x+36)*ln(x)-15*exp(x)^2+(-3*x^2+3*x+36
)*exp(x)-6*x-21)/(x^2*ln(x)^2+(2*exp(x)*x^2-2*x^2)*ln(x)+exp(x)^2*x^2-2*exp(x)*x^2+x^2),x,method=_RETURNVERBOS
E)

[Out]

3/x/(-1+ln(x)+exp(x))*ln(exp(x))+3/2*(-I*Pi*csgn(I*x^2)*csgn(I*x)^2+2*I*Pi*csgn(I*x^2)^2*csgn(I*x)-I*Pi*csgn(I
*x^2)^3-I*Pi*csgn(I*x^2)*csgn(I*exp(x))*csgn(I*x^2*exp(x))+I*Pi*csgn(I*x^2)*csgn(I*x^2*exp(x))^2+I*Pi*csgn(I*e
xp(x))*csgn(I*x^2*exp(x))^2-I*Pi*csgn(I*x^2*exp(x))^3+2*x+10*exp(x)+14*ln(x)-10)/x/(-1+ln(x)+exp(x))

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maxima [A]  time = 0.46, size = 29, normalized size = 0.91 \begin {gather*} \frac {3 \, {\left (2 \, x + 5 \, e^{x} + 7 \, \log \relax (x) - 5\right )}}{x e^{x} + x \log \relax (x) - x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*log(x)+(-3*x-3)*exp(x))*log(exp(x)*x^2)-15*log(x)^2+(-30*exp(x)+3*x+36)*log(x)-15*exp(x)^2+(-3*
x^2+3*x+36)*exp(x)-6*x-21)/(x^2*log(x)^2+(2*exp(x)*x^2-2*x^2)*log(x)+exp(x)^2*x^2-2*exp(x)*x^2+x^2),x, algorit
hm="maxima")

[Out]

3*(2*x + 5*e^x + 7*log(x) - 5)/(x*e^x + x*log(x) - x)

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mupad [B]  time = 1.25, size = 30, normalized size = 0.94 \begin {gather*} \frac {3\,\left (2\,x+\ln \left (x^2\right )+5\,{\mathrm {e}}^x+5\,\ln \relax (x)-5\right )}{x\,\left ({\mathrm {e}}^x+\ln \relax (x)-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(6*x + 15*exp(2*x) + 15*log(x)^2 - exp(x)*(3*x - 3*x^2 + 36) - log(x)*(3*x - 30*exp(x) + 36) + log(x^2*ex
p(x))*(3*log(x) + exp(x)*(3*x + 3)) + 21)/(x^2*exp(2*x) - 2*x^2*exp(x) + x^2*log(x)^2 + x^2 + log(x)*(2*x^2*ex
p(x) - 2*x^2)),x)

[Out]

(3*(2*x + log(x^2) + 5*exp(x) + 5*log(x) - 5))/(x*(exp(x) + log(x) - 1))

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sympy [A]  time = 0.32, size = 22, normalized size = 0.69 \begin {gather*} \frac {6 x + 6 \log {\relax (x )}}{x e^{x} + x \log {\relax (x )} - x} + \frac {15}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*ln(x)+(-3*x-3)*exp(x))*ln(exp(x)*x**2)-15*ln(x)**2+(-30*exp(x)+3*x+36)*ln(x)-15*exp(x)**2+(-3*x
**2+3*x+36)*exp(x)-6*x-21)/(x**2*ln(x)**2+(2*exp(x)*x**2-2*x**2)*ln(x)+exp(x)**2*x**2-2*exp(x)*x**2+x**2),x)

[Out]

(6*x + 6*log(x))/(x*exp(x) + x*log(x) - x) + 15/x

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