3.39.57 \(\int \frac {-4 x^5-4 x^4 \log (3)+e^x (50 x+25 \log (3))+e^x (25 x^2+25 x \log (3)) \log (x^2+x \log (3))}{-x^6-x^5 \log (3)+e^x (25 x^2+25 x \log (3)) \log (x^2+x \log (3))} \, dx\)

Optimal. Leaf size=23 \[ x+\log \left (-\frac {1}{25} e^{-x} x^4+\log (x (x+\log (3)))\right ) \]

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Rubi [F]  time = 11.45, 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 {-4 x^5-4 x^4 \log (3)+e^x (50 x+25 \log (3))+e^x \left (25 x^2+25 x \log (3)\right ) \log \left (x^2+x \log (3)\right )}{-x^6-x^5 \log (3)+e^x \left (25 x^2+25 x \log (3)\right ) \log \left (x^2+x \log (3)\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-4*x^5 - 4*x^4*Log[3] + E^x*(50*x + 25*Log[3]) + E^x*(25*x^2 + 25*x*Log[3])*Log[x^2 + x*Log[3]])/(-x^6 -
x^5*Log[3] + E^x*(25*x^2 + 25*x*Log[3])*Log[x^2 + x*Log[3]]),x]

[Out]

x + Defer[Int][(2*x + Log[3])/(x*(x + Log[3])*Log[x*(x + Log[3])]), x] - (4 - Log[3])*Log[3]^3*Defer[Int][(x^4
- 25*E^x*Log[x*(x + Log[3])])^(-1), x] - Log[3]^4*Defer[Int][(x^4 - 25*E^x*Log[x*(x + Log[3])])^(-1), x] + Lo
g[3]^2*Log[81]*Defer[Int][(x^4 - 25*E^x*Log[x*(x + Log[3])])^(-1), x] - 4*Log[3]^2*Defer[Int][x/(x^4 - 25*E^x*
Log[x*(x + Log[3])]), x] + (4 - Log[3])*Log[3]^2*Defer[Int][x/(x^4 - 25*E^x*Log[x*(x + Log[3])]), x] + Log[3]^
3*Defer[Int][x/(x^4 - 25*E^x*Log[x*(x + Log[3])]), x] - (4 - Log[3])*Log[3]*Defer[Int][x^2/(x^4 - 25*E^x*Log[x
*(x + Log[3])]), x] - Log[3]^2*Defer[Int][x^2/(x^4 - 25*E^x*Log[x*(x + Log[3])]), x] + Log[81]*Defer[Int][x^2/
(x^4 - 25*E^x*Log[x*(x + Log[3])]), x] + (4 - Log[3])*Defer[Int][x^3/(x^4 - 25*E^x*Log[x*(x + Log[3])]), x] +
Log[3]*Defer[Int][x^3/(x^4 - 25*E^x*Log[x*(x + Log[3])]), x] - Defer[Int][x^4/(x^4 - 25*E^x*Log[x*(x + Log[3])
]), x] - 4*Log[3]^4*Defer[Int][1/((x + Log[3])*(x^4 - 25*E^x*Log[x*(x + Log[3])])), x] + (4 - Log[3])*Log[3]^4
*Defer[Int][1/((x + Log[3])*(x^4 - 25*E^x*Log[x*(x + Log[3])])), x] + Log[3]^5*Defer[Int][1/((x + Log[3])*(x^4
- 25*E^x*Log[x*(x + Log[3])])), x] - Log[3]^2*Defer[Int][x/(Log[x*(x + Log[3])]*(x^4 - 25*E^x*Log[x*(x + Log[
3])])), x] + Log[3]*Defer[Int][x^2/(Log[x*(x + Log[3])]*(x^4 - 25*E^x*Log[x*(x + Log[3])])), x] - 2*Defer[Int]
[x^3/(Log[x*(x + Log[3])]*(x^4 - 25*E^x*Log[x*(x + Log[3])])), x] - Log[3]^4*Defer[Int][1/((x + Log[3])*Log[x*
(x + Log[3])]*(x^4 - 25*E^x*Log[x*(x + Log[3])])), x] - Log[3]^3*Defer[Int][1/(Log[x*(x + Log[3])]*(-x^4 + 25*
E^x*Log[x*(x + Log[3])])), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {4 x^5+4 x^4 \log (3)-e^x (50 x+25 \log (3))-e^x \left (25 x^2+25 x \log (3)\right ) \log \left (x^2+x \log (3)\right )}{x (x+\log (3)) \left (x^4-25 e^x \log (x (x+\log (3)))\right )} \, dx\\ &=\int \left (\frac {x^3 \left (-2 x-\log (3)-x^2 \log (x (x+\log (3)))+4 x \left (1-\frac {\log (3)}{4}\right ) \log (x (x+\log (3)))+4 \log (3) \log (x (x+\log (3)))\right )}{(x+\log (3)) \log (x (x+\log (3))) \left (x^4-25 e^x \log (x (x+\log (3)))\right )}+\frac {2 x+\log (3)+x^2 \log (x (x+\log (3)))+x \log (3) \log (x (x+\log (3)))}{x (x+\log (3)) \log (x (x+\log (3)))}\right ) \, dx\\ &=\int \frac {x^3 \left (-2 x-\log (3)-x^2 \log (x (x+\log (3)))+4 x \left (1-\frac {\log (3)}{4}\right ) \log (x (x+\log (3)))+4 \log (3) \log (x (x+\log (3)))\right )}{(x+\log (3)) \log (x (x+\log (3))) \left (x^4-25 e^x \log (x (x+\log (3)))\right )} \, dx+\int \frac {2 x+\log (3)+x^2 \log (x (x+\log (3)))+x \log (3) \log (x (x+\log (3)))}{x (x+\log (3)) \log (x (x+\log (3)))} \, dx\\ &=\int \frac {x^3 (-2 x-\log (3)-(-4+x) (x+\log (3)) \log (x (x+\log (3))))}{(x+\log (3)) \log (x (x+\log (3))) \left (x^4-25 e^x \log (x (x+\log (3)))\right )} \, dx+\int \frac {2 x+\log (3)+x (x+\log (3)) \log (x (x+\log (3)))}{x (x+\log (3)) \log (x (x+\log (3)))} \, dx\\ &=\int \left (1+\frac {2 x+\log (3)}{x (x+\log (3)) \log (x (x+\log (3)))}\right ) \, dx+\int \left (\frac {x \log (3) \left (2 x+\log (3)+x^2 \log (x (x+\log (3)))-4 x \left (1-\frac {\log (3)}{4}\right ) \log (x (x+\log (3)))-4 \log (3) \log (x (x+\log (3)))\right )}{\log (x (x+\log (3))) \left (x^4-25 e^x \log (x (x+\log (3)))\right )}+\frac {\log ^3(3) \left (2 x+\log (3)+x^2 \log (x (x+\log (3)))-4 x \left (1-\frac {\log (3)}{4}\right ) \log (x (x+\log (3)))-4 \log (3) \log (x (x+\log (3)))\right )}{(x+\log (3)) \log (x (x+\log (3))) \left (x^4-25 e^x \log (x (x+\log (3)))\right )}+\frac {x^2 \left (-2 x-\log (3)-x^2 \log (x (x+\log (3)))+4 x \left (1-\frac {\log (3)}{4}\right ) \log (x (x+\log (3)))+4 \log (3) \log (x (x+\log (3)))\right )}{\log (x (x+\log (3))) \left (x^4-25 e^x \log (x (x+\log (3)))\right )}+\frac {\log ^2(3) \left (-2 x-\log (3)-x^2 \log (x (x+\log (3)))+4 x \left (1-\frac {\log (3)}{4}\right ) \log (x (x+\log (3)))+4 \log (3) \log (x (x+\log (3)))\right )}{\log (x (x+\log (3))) \left (x^4-25 e^x \log (x (x+\log (3)))\right )}\right ) \, dx\\ &=x+\log (3) \int \frac {x \left (2 x+\log (3)+x^2 \log (x (x+\log (3)))-4 x \left (1-\frac {\log (3)}{4}\right ) \log (x (x+\log (3)))-4 \log (3) \log (x (x+\log (3)))\right )}{\log (x (x+\log (3))) \left (x^4-25 e^x \log (x (x+\log (3)))\right )} \, dx+\log ^2(3) \int \frac {-2 x-\log (3)-x^2 \log (x (x+\log (3)))+4 x \left (1-\frac {\log (3)}{4}\right ) \log (x (x+\log (3)))+4 \log (3) \log (x (x+\log (3)))}{\log (x (x+\log (3))) \left (x^4-25 e^x \log (x (x+\log (3)))\right )} \, dx+\log ^3(3) \int \frac {2 x+\log (3)+x^2 \log (x (x+\log (3)))-4 x \left (1-\frac {\log (3)}{4}\right ) \log (x (x+\log (3)))-4 \log (3) \log (x (x+\log (3)))}{(x+\log (3)) \log (x (x+\log (3))) \left (x^4-25 e^x \log (x (x+\log (3)))\right )} \, dx+\int \frac {2 x+\log (3)}{x (x+\log (3)) \log (x (x+\log (3)))} \, dx+\int \frac {x^2 \left (-2 x-\log (3)-x^2 \log (x (x+\log (3)))+4 x \left (1-\frac {\log (3)}{4}\right ) \log (x (x+\log (3)))+4 \log (3) \log (x (x+\log (3)))\right )}{\log (x (x+\log (3))) \left (x^4-25 e^x \log (x (x+\log (3)))\right )} \, dx\\ &=x+\log (3) \int \frac {x (2 x+\log (3)+(-4+x) (x+\log (3)) \log (x (x+\log (3))))}{\log (x (x+\log (3))) \left (x^4-25 e^x \log (x (x+\log (3)))\right )} \, dx+\log ^2(3) \int \frac {-2 x-\log (3)-(-4+x) (x+\log (3)) \log (x (x+\log (3)))}{\log (x (x+\log (3))) \left (x^4-25 e^x \log (x (x+\log (3)))\right )} \, dx+\log ^3(3) \int \frac {2 x+\log (3)+(-4+x) (x+\log (3)) \log (x (x+\log (3)))}{(x+\log (3)) \log (x (x+\log (3))) \left (x^4-25 e^x \log (x (x+\log (3)))\right )} \, dx+\int \frac {2 x+\log (3)}{x (x+\log (3)) \log (x (x+\log (3)))} \, dx+\int \frac {x^2 (-2 x-\log (3)-(-4+x) (x+\log (3)) \log (x (x+\log (3))))}{\log (x (x+\log (3))) \left (x^4-25 e^x \log (x (x+\log (3)))\right )} \, dx\\ &=x+\log (3) \int \left (\frac {x^3}{x^4-25 e^x \log (x (x+\log (3)))}-\frac {4 x^2 \left (1-\frac {\log (3)}{4}\right )}{x^4-25 e^x \log (x (x+\log (3)))}-\frac {4 x \log (3)}{x^4-25 e^x \log (x (x+\log (3)))}+\frac {2 x^2}{\log (x (x+\log (3))) \left (x^4-25 e^x \log (x (x+\log (3)))\right )}+\frac {x \log (3)}{\log (x (x+\log (3))) \left (x^4-25 e^x \log (x (x+\log (3)))\right )}\right ) \, dx+\log ^2(3) \int \left (-\frac {x^2}{x^4-25 e^x \log (x (x+\log (3)))}+\frac {4 x \left (1-\frac {\log (3)}{4}\right )}{x^4-25 e^x \log (x (x+\log (3)))}+\frac {\log (81)}{x^4-25 e^x \log (x (x+\log (3)))}-\frac {2 x}{\log (x (x+\log (3))) \left (x^4-25 e^x \log (x (x+\log (3)))\right )}+\frac {\log (3)}{\log (x (x+\log (3))) \left (-x^4+25 e^x \log (x (x+\log (3)))\right )}\right ) \, dx+\log ^3(3) \int \left (\frac {x^2}{(x+\log (3)) \left (x^4-25 e^x \log (x (x+\log (3)))\right )}-\frac {4 x \left (1-\frac {\log (3)}{4}\right )}{(x+\log (3)) \left (x^4-25 e^x \log (x (x+\log (3)))\right )}-\frac {4 \log (3)}{(x+\log (3)) \left (x^4-25 e^x \log (x (x+\log (3)))\right )}+\frac {2 x}{(x+\log (3)) \log (x (x+\log (3))) \left (x^4-25 e^x \log (x (x+\log (3)))\right )}+\frac {\log (3)}{(x+\log (3)) \log (x (x+\log (3))) \left (x^4-25 e^x \log (x (x+\log (3)))\right )}\right ) \, dx+\int \frac {2 x+\log (3)}{x (x+\log (3)) \log (x (x+\log (3)))} \, dx+\int \left (-\frac {x^4}{x^4-25 e^x \log (x (x+\log (3)))}+\frac {4 x^3 \left (1-\frac {\log (3)}{4}\right )}{x^4-25 e^x \log (x (x+\log (3)))}+\frac {x^2 \log (81)}{x^4-25 e^x \log (x (x+\log (3)))}-\frac {2 x^3}{\log (x (x+\log (3))) \left (x^4-25 e^x \log (x (x+\log (3)))\right )}-\frac {x^2 \log (3)}{\log (x (x+\log (3))) \left (x^4-25 e^x \log (x (x+\log (3)))\right )}\right ) \, dx\\ &=x-2 \int \frac {x^3}{\log (x (x+\log (3))) \left (x^4-25 e^x \log (x (x+\log (3)))\right )} \, dx+(4-\log (3)) \int \frac {x^3}{x^4-25 e^x \log (x (x+\log (3)))} \, dx+\log (3) \int \frac {x^3}{x^4-25 e^x \log (x (x+\log (3)))} \, dx-\log (3) \int \frac {x^2}{\log (x (x+\log (3))) \left (x^4-25 e^x \log (x (x+\log (3)))\right )} \, dx+(2 \log (3)) \int \frac {x^2}{\log (x (x+\log (3))) \left (x^4-25 e^x \log (x (x+\log (3)))\right )} \, dx-((4-\log (3)) \log (3)) \int \frac {x^2}{x^4-25 e^x \log (x (x+\log (3)))} \, dx-\log ^2(3) \int \frac {x^2}{x^4-25 e^x \log (x (x+\log (3)))} \, dx+\log ^2(3) \int \frac {x}{\log (x (x+\log (3))) \left (x^4-25 e^x \log (x (x+\log (3)))\right )} \, dx-\left (2 \log ^2(3)\right ) \int \frac {x}{\log (x (x+\log (3))) \left (x^4-25 e^x \log (x (x+\log (3)))\right )} \, dx-\left (4 \log ^2(3)\right ) \int \frac {x}{x^4-25 e^x \log (x (x+\log (3)))} \, dx+\left ((4-\log (3)) \log ^2(3)\right ) \int \frac {x}{x^4-25 e^x \log (x (x+\log (3)))} \, dx+\log ^3(3) \int \frac {x^2}{(x+\log (3)) \left (x^4-25 e^x \log (x (x+\log (3)))\right )} \, dx+\log ^3(3) \int \frac {1}{\log (x (x+\log (3))) \left (-x^4+25 e^x \log (x (x+\log (3)))\right )} \, dx+\left (2 \log ^3(3)\right ) \int \frac {x}{(x+\log (3)) \log (x (x+\log (3))) \left (x^4-25 e^x \log (x (x+\log (3)))\right )} \, dx-\left ((4-\log (3)) \log ^3(3)\right ) \int \frac {x}{(x+\log (3)) \left (x^4-25 e^x \log (x (x+\log (3)))\right )} \, dx+\log ^4(3) \int \frac {1}{(x+\log (3)) \log (x (x+\log (3))) \left (x^4-25 e^x \log (x (x+\log (3)))\right )} \, dx-\left (4 \log ^4(3)\right ) \int \frac {1}{(x+\log (3)) \left (x^4-25 e^x \log (x (x+\log (3)))\right )} \, dx+\log (81) \int \frac {x^2}{x^4-25 e^x \log (x (x+\log (3)))} \, dx+\left (\log ^2(3) \log (81)\right ) \int \frac {1}{x^4-25 e^x \log (x (x+\log (3)))} \, dx+\int \frac {2 x+\log (3)}{x (x+\log (3)) \log (x (x+\log (3)))} \, dx-\int \frac {x^4}{x^4-25 e^x \log (x (x+\log (3)))} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [F]  time = 2.32, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-4 x^5-4 x^4 \log (3)+e^x (50 x+25 \log (3))+e^x \left (25 x^2+25 x \log (3)\right ) \log \left (x^2+x \log (3)\right )}{-x^6-x^5 \log (3)+e^x \left (25 x^2+25 x \log (3)\right ) \log \left (x^2+x \log (3)\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(-4*x^5 - 4*x^4*Log[3] + E^x*(50*x + 25*Log[3]) + E^x*(25*x^2 + 25*x*Log[3])*Log[x^2 + x*Log[3]])/(-
x^6 - x^5*Log[3] + E^x*(25*x^2 + 25*x*Log[3])*Log[x^2 + x*Log[3]]),x]

[Out]

Integrate[(-4*x^5 - 4*x^4*Log[3] + E^x*(50*x + 25*Log[3]) + E^x*(25*x^2 + 25*x*Log[3])*Log[x^2 + x*Log[3]])/(-
x^6 - x^5*Log[3] + E^x*(25*x^2 + 25*x*Log[3])*Log[x^2 + x*Log[3]]), x]

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fricas [A]  time = 0.55, size = 26, normalized size = 1.13 \begin {gather*} x + \log \left (-{\left (x^{4} - 25 \, e^{x} \log \left (x^{2} + x \log \relax (3)\right )\right )} e^{\left (-x\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((25*x*log(3)+25*x^2)*exp(x)*log(x*log(3)+x^2)+(25*log(3)+50*x)*exp(x)-4*x^4*log(3)-4*x^5)/((25*x*lo
g(3)+25*x^2)*exp(x)*log(x*log(3)+x^2)-x^5*log(3)-x^6),x, algorithm="fricas")

[Out]

x + log(-(x^4 - 25*e^x*log(x^2 + x*log(3)))*e^(-x))

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giac [A]  time = 0.18, size = 20, normalized size = 0.87 \begin {gather*} \log \left (-x^{4} + 25 \, e^{x} \log \left (x^{2} + x \log \relax (3)\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((25*x*log(3)+25*x^2)*exp(x)*log(x*log(3)+x^2)+(25*log(3)+50*x)*exp(x)-4*x^4*log(3)-4*x^5)/((25*x*lo
g(3)+25*x^2)*exp(x)*log(x*log(3)+x^2)-x^5*log(3)-x^6),x, algorithm="giac")

[Out]

log(-x^4 + 25*e^x*log(x^2 + x*log(3)))

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maple [C]  time = 0.08, size = 119, normalized size = 5.17

method	result	size
risch	\(x +\ln \left (\ln \left (\ln \relax (3)+x \right )+\frac {i \left (2 i x^{4}-25 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \left (\ln \relax (3)+x \right )\right ) \mathrm {csgn}\left (i x \left (\ln \relax (3)+x \right )\right ) {\mathrm e}^{x}+25 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \left (\ln \relax (3)+x \right )\right )^{2} {\mathrm e}^{x}+25 \pi \,\mathrm {csgn}\left (i \left (\ln \relax (3)+x \right )\right ) \mathrm {csgn}\left (i x \left (\ln \relax (3)+x \right )\right )^{2} {\mathrm e}^{x}-25 \pi \mathrm {csgn}\left (i x \left (\ln \relax (3)+x \right )\right )^{3} {\mathrm e}^{x}-50 i {\mathrm e}^{x} \ln \relax (x )\right ) {\mathrm e}^{-x}}{50}\right )\)	\(119\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((25*x*ln(3)+25*x^2)*exp(x)*ln(x*ln(3)+x^2)+(25*ln(3)+50*x)*exp(x)-4*x^4*ln(3)-4*x^5)/((25*x*ln(3)+25*x^2)
*exp(x)*ln(x*ln(3)+x^2)-x^5*ln(3)-x^6),x,method=_RETURNVERBOSE)

[Out]

x+ln(ln(ln(3)+x)+1/50*I*(2*I*x^4-25*Pi*csgn(I*x)*csgn(I*(ln(3)+x))*csgn(I*x*(ln(3)+x))*exp(x)+25*Pi*csgn(I*x)*
csgn(I*x*(ln(3)+x))^2*exp(x)+25*Pi*csgn(I*(ln(3)+x))*csgn(I*x*(ln(3)+x))^2*exp(x)-25*Pi*csgn(I*x*(ln(3)+x))^3*
exp(x)-50*I*exp(x)*ln(x))*exp(-x))

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maxima [A]  time = 0.50, size = 28, normalized size = 1.22 \begin {gather*} x + \log \left (-\frac {1}{25} \, {\left (x^{4} - 25 \, e^{x} \log \left (x + \log \relax (3)\right ) - 25 \, e^{x} \log \relax (x)\right )} e^{\left (-x\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((25*x*log(3)+25*x^2)*exp(x)*log(x*log(3)+x^2)+(25*log(3)+50*x)*exp(x)-4*x^4*log(3)-4*x^5)/((25*x*lo
g(3)+25*x^2)*exp(x)*log(x*log(3)+x^2)-x^5*log(3)-x^6),x, algorithm="maxima")

[Out]

x + log(-1/25*(x^4 - 25*e^x*log(x + log(3)) - 25*e^x*log(x))*e^(-x))

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mupad [B]  time = 2.82, size = 22, normalized size = 0.96 \begin {gather*} x+\ln \left (\ln \left (x^2+\ln \relax (3)\,x\right )-\frac {x^4\,{\mathrm {e}}^{-x}}{25}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(x)*(50*x + 25*log(3)) - 4*x^4*log(3) - 4*x^5 + exp(x)*log(x*log(3) + x^2)*(25*x*log(3) + 25*x^2))/(x
^5*log(3) + x^6 - exp(x)*log(x*log(3) + x^2)*(25*x*log(3) + 25*x^2)),x)

[Out]

x + log(log(x*log(3) + x^2) - (x^4*exp(-x))/25)

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sympy [A]  time = 0.85, size = 31, normalized size = 1.35 \begin {gather*} \log {\left (- \frac {x^{4}}{25 \log {\left (x^{2} + x \log {\relax (3 )} \right )}} + e^{x} \right )} + \log {\left (\log {\left (x^{2} + x \log {\relax (3 )} \right )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((25*x*ln(3)+25*x**2)*exp(x)*ln(x*ln(3)+x**2)+(25*ln(3)+50*x)*exp(x)-4*x**4*ln(3)-4*x**5)/((25*x*ln(
3)+25*x**2)*exp(x)*ln(x*ln(3)+x**2)-x**5*ln(3)-x**6),x)

[Out]

log(-x**4/(25*log(x**2 + x*log(3))) + exp(x)) + log(log(x**2 + x*log(3)))

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