3.55.44 \(\int \frac {-23 x-86 x^2-238 x^3-40 x^4+100 x^5+(1+3 x-40 x^2-113 x^3-20 x^4+50 x^5) \log (x)+(x+4 x^2+3 x^3) \log ^2(x)+(2-50 x+40 x^2+(1-25 x+20 x^2) \log (x)) \log (2+\log (x))}{2 x+x \log (x)} \, dx\)

Optimal. Leaf size=30 \[ (5 (-x+x (-4+2 x))+\log (x)) \left (x (1+x)^2+\log (2+\log (x))\right ) \]

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Rubi [F]  time = 1.14, 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 {-23 x-86 x^2-238 x^3-40 x^4+100 x^5+\left (1+3 x-40 x^2-113 x^3-20 x^4+50 x^5\right ) \log (x)+\left (x+4 x^2+3 x^3\right ) \log ^2(x)+\left (2-50 x+40 x^2+\left (1-25 x+20 x^2\right ) \log (x)\right ) \log (2+\log (x))}{2 x+x \log (x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-23*x - 86*x^2 - 238*x^3 - 40*x^4 + 100*x^5 + (1 + 3*x - 40*x^2 - 113*x^3 - 20*x^4 + 50*x^5)*Log[x] + (x
+ 4*x^2 + 3*x^3)*Log[x]^2 + (2 - 50*x + 40*x^2 + (1 - 25*x + 20*x^2)*Log[x])*Log[2 + Log[x]])/(2*x + x*Log[x])
,x]

[Out]

2*x - 21*x^2 - 38*x^3 - 5*x^4 + 10*x^5 - 2*x*(1 + x)^2 - (19*ExpIntegralEi[2 + Log[x]])/E^2 - (70*ExpIntegralE
i[2*(2 + Log[x])])/E^4 - (226*ExpIntegralEi[3*(2 + Log[x])])/E^6 - (40*ExpIntegralEi[4*(2 + Log[x])])/E^8 + (1
00*ExpIntegralEi[5*(2 + Log[x])])/E^10 + x*Log[x] + 2*x^2*Log[x] + x^3*Log[x] + (2 + Log[x])*Log[2 + Log[x]] -
2*Defer[Int][(1 + 3*x - 40*x^2 - 113*x^3 - 20*x^4 + 50*x^5)/(x*(2 + Log[x])), x] - 25*Defer[Int][Log[2 + Log[
x]], x] + 20*Defer[Int][x*Log[2 + Log[x]], x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-23 x-86 x^2-238 x^3-40 x^4+100 x^5+\left (1+3 x-40 x^2-113 x^3-20 x^4+50 x^5\right ) \log (x)+\left (x+4 x^2+3 x^3\right ) \log ^2(x)+\left (2-50 x+40 x^2+\left (1-25 x+20 x^2\right ) \log (x)\right ) \log (2+\log (x))}{x (2+\log (x))} \, dx\\ &=\int \left (-\frac {23}{2+\log (x)}-\frac {86 x}{2+\log (x)}-\frac {238 x^2}{2+\log (x)}-\frac {40 x^3}{2+\log (x)}+\frac {100 x^4}{2+\log (x)}+\frac {\left (1+3 x-40 x^2-113 x^3-20 x^4+50 x^5\right ) \log (x)}{x (2+\log (x))}+\frac {(1+x) (1+3 x) \log ^2(x)}{2+\log (x)}+\frac {\left (1-25 x+20 x^2\right ) \log (2+\log (x))}{x}\right ) \, dx\\ &=-\left (23 \int \frac {1}{2+\log (x)} \, dx\right )-40 \int \frac {x^3}{2+\log (x)} \, dx-86 \int \frac {x}{2+\log (x)} \, dx+100 \int \frac {x^4}{2+\log (x)} \, dx-238 \int \frac {x^2}{2+\log (x)} \, dx+\int \frac {\left (1+3 x-40 x^2-113 x^3-20 x^4+50 x^5\right ) \log (x)}{x (2+\log (x))} \, dx+\int \frac {(1+x) (1+3 x) \log ^2(x)}{2+\log (x)} \, dx+\int \frac {\left (1-25 x+20 x^2\right ) \log (2+\log (x))}{x} \, dx\\ &=-\left (23 \operatorname {Subst}\left (\int \frac {e^x}{2+x} \, dx,x,\log (x)\right )\right )-40 \operatorname {Subst}\left (\int \frac {e^{4 x}}{2+x} \, dx,x,\log (x)\right )-86 \operatorname {Subst}\left (\int \frac {e^{2 x}}{2+x} \, dx,x,\log (x)\right )+100 \operatorname {Subst}\left (\int \frac {e^{5 x}}{2+x} \, dx,x,\log (x)\right )-238 \operatorname {Subst}\left (\int \frac {e^{3 x}}{2+x} \, dx,x,\log (x)\right )+\int \left (-2 (1+x) (1+3 x)+(1+x) (1+3 x) \log (x)+\frac {4 (1+x) (1+3 x)}{2+\log (x)}\right ) \, dx+\int \left (\frac {1+3 x-40 x^2-113 x^3-20 x^4+50 x^5}{x}-\frac {2 \left (1+3 x-40 x^2-113 x^3-20 x^4+50 x^5\right )}{x (2+\log (x))}\right ) \, dx+\int \left (-25 \log (2+\log (x))+\frac {\log (2+\log (x))}{x}+20 x \log (2+\log (x))\right ) \, dx\\ &=-\frac {23 \text {Ei}(2+\log (x))}{e^2}-\frac {86 \text {Ei}(2 (2+\log (x)))}{e^4}-\frac {238 \text {Ei}(3 (2+\log (x)))}{e^6}-\frac {40 \text {Ei}(4 (2+\log (x)))}{e^8}+\frac {100 \text {Ei}(5 (2+\log (x)))}{e^{10}}-2 \int (1+x) (1+3 x) \, dx-2 \int \frac {1+3 x-40 x^2-113 x^3-20 x^4+50 x^5}{x (2+\log (x))} \, dx+4 \int \frac {(1+x) (1+3 x)}{2+\log (x)} \, dx+20 \int x \log (2+\log (x)) \, dx-25 \int \log (2+\log (x)) \, dx+\int \frac {1+3 x-40 x^2-113 x^3-20 x^4+50 x^5}{x} \, dx+\int (1+x) (1+3 x) \log (x) \, dx+\int \frac {\log (2+\log (x))}{x} \, dx\\ &=-2 x (1+x)^2-\frac {23 \text {Ei}(2+\log (x))}{e^2}-\frac {86 \text {Ei}(2 (2+\log (x)))}{e^4}-\frac {238 \text {Ei}(3 (2+\log (x)))}{e^6}-\frac {40 \text {Ei}(4 (2+\log (x)))}{e^8}+\frac {100 \text {Ei}(5 (2+\log (x)))}{e^{10}}-2 \int \frac {1+3 x-40 x^2-113 x^3-20 x^4+50 x^5}{x (2+\log (x))} \, dx+4 \int \left (\frac {1}{2+\log (x)}+\frac {4 x}{2+\log (x)}+\frac {3 x^2}{2+\log (x)}\right ) \, dx+20 \int x \log (2+\log (x)) \, dx-25 \int \log (2+\log (x)) \, dx+\int \left (3+\frac {1}{x}-40 x-113 x^2-20 x^3+50 x^4\right ) \, dx+\int \left (\log (x)+4 x \log (x)+3 x^2 \log (x)\right ) \, dx+\operatorname {Subst}(\int \log (2+x) \, dx,x,\log (x))\\ &=3 x-20 x^2-\frac {113 x^3}{3}-5 x^4+10 x^5-2 x (1+x)^2-\frac {23 \text {Ei}(2+\log (x))}{e^2}-\frac {86 \text {Ei}(2 (2+\log (x)))}{e^4}-\frac {238 \text {Ei}(3 (2+\log (x)))}{e^6}-\frac {40 \text {Ei}(4 (2+\log (x)))}{e^8}+\frac {100 \text {Ei}(5 (2+\log (x)))}{e^{10}}+\log (x)-2 \int \frac {1+3 x-40 x^2-113 x^3-20 x^4+50 x^5}{x (2+\log (x))} \, dx+3 \int x^2 \log (x) \, dx+4 \int x \log (x) \, dx+4 \int \frac {1}{2+\log (x)} \, dx+12 \int \frac {x^2}{2+\log (x)} \, dx+16 \int \frac {x}{2+\log (x)} \, dx+20 \int x \log (2+\log (x)) \, dx-25 \int \log (2+\log (x)) \, dx+\int \log (x) \, dx+\operatorname {Subst}(\int \log (x) \, dx,x,2+\log (x))\\ &=2 x-21 x^2-38 x^3-5 x^4+10 x^5-2 x (1+x)^2-\frac {23 \text {Ei}(2+\log (x))}{e^2}-\frac {86 \text {Ei}(2 (2+\log (x)))}{e^4}-\frac {238 \text {Ei}(3 (2+\log (x)))}{e^6}-\frac {40 \text {Ei}(4 (2+\log (x)))}{e^8}+\frac {100 \text {Ei}(5 (2+\log (x)))}{e^{10}}+x \log (x)+2 x^2 \log (x)+x^3 \log (x)+(2+\log (x)) \log (2+\log (x))-2 \int \frac {1+3 x-40 x^2-113 x^3-20 x^4+50 x^5}{x (2+\log (x))} \, dx+4 \operatorname {Subst}\left (\int \frac {e^x}{2+x} \, dx,x,\log (x)\right )+12 \operatorname {Subst}\left (\int \frac {e^{3 x}}{2+x} \, dx,x,\log (x)\right )+16 \operatorname {Subst}\left (\int \frac {e^{2 x}}{2+x} \, dx,x,\log (x)\right )+20 \int x \log (2+\log (x)) \, dx-25 \int \log (2+\log (x)) \, dx\\ &=2 x-21 x^2-38 x^3-5 x^4+10 x^5-2 x (1+x)^2-\frac {19 \text {Ei}(2+\log (x))}{e^2}-\frac {70 \text {Ei}(2 (2+\log (x)))}{e^4}-\frac {226 \text {Ei}(3 (2+\log (x)))}{e^6}-\frac {40 \text {Ei}(4 (2+\log (x)))}{e^8}+\frac {100 \text {Ei}(5 (2+\log (x)))}{e^{10}}+x \log (x)+2 x^2 \log (x)+x^3 \log (x)+(2+\log (x)) \log (2+\log (x))-2 \int \frac {1+3 x-40 x^2-113 x^3-20 x^4+50 x^5}{x (2+\log (x))} \, dx+20 \int x \log (2+\log (x)) \, dx-25 \int \log (2+\log (x)) \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.06, size = 25, normalized size = 0.83 \begin {gather*} (5 x (-5+2 x)+\log (x)) \left (x (1+x)^2+\log (2+\log (x))\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-23*x - 86*x^2 - 238*x^3 - 40*x^4 + 100*x^5 + (1 + 3*x - 40*x^2 - 113*x^3 - 20*x^4 + 50*x^5)*Log[x]
+ (x + 4*x^2 + 3*x^3)*Log[x]^2 + (2 - 50*x + 40*x^2 + (1 - 25*x + 20*x^2)*Log[x])*Log[2 + Log[x]])/(2*x + x*L
og[x]),x]

[Out]

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

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fricas [A]  time = 0.54, size = 51, normalized size = 1.70 \begin {gather*} 10 \, x^{5} - 5 \, x^{4} - 40 \, x^{3} - 25 \, x^{2} + {\left (x^{3} + 2 \, x^{2} + x\right )} \log \relax (x) + {\left (10 \, x^{2} - 25 \, x + \log \relax (x)\right )} \log \left (\log \relax (x) + 2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((20*x^2-25*x+1)*log(x)+40*x^2-50*x+2)*log(log(x)+2)+(3*x^3+4*x^2+x)*log(x)^2+(50*x^5-20*x^4-113*x^
3-40*x^2+3*x+1)*log(x)+100*x^5-40*x^4-238*x^3-86*x^2-23*x)/(x*log(x)+2*x),x, algorithm="fricas")

[Out]

10*x^5 - 5*x^4 - 40*x^3 - 25*x^2 + (x^3 + 2*x^2 + x)*log(x) + (10*x^2 - 25*x + log(x))*log(log(x) + 2)

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giac [A]  time = 0.16, size = 51, normalized size = 1.70 \begin {gather*} 10 \, x^{5} - 5 \, x^{4} - 40 \, x^{3} - 25 \, x^{2} + {\left (x^{3} + 2 \, x^{2} + x\right )} \log \relax (x) + {\left (10 \, x^{2} - 25 \, x + \log \relax (x)\right )} \log \left (\log \relax (x) + 2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((20*x^2-25*x+1)*log(x)+40*x^2-50*x+2)*log(log(x)+2)+(3*x^3+4*x^2+x)*log(x)^2+(50*x^5-20*x^4-113*x^
3-40*x^2+3*x+1)*log(x)+100*x^5-40*x^4-238*x^3-86*x^2-23*x)/(x*log(x)+2*x),x, algorithm="giac")

[Out]

10*x^5 - 5*x^4 - 40*x^3 - 25*x^2 + (x^3 + 2*x^2 + x)*log(x) + (10*x^2 - 25*x + log(x))*log(log(x) + 2)

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maple [A]  time = 0.04, size = 56, normalized size = 1.87

method	result	size
risch	\(\left (10 x^{2}-25 x +\ln \relax (x )\right ) \ln \left (\ln \relax (x )+2\right )+10 x^{5}-5 x^{4}+x^{3} \ln \relax (x )-40 x^{3}+2 x^{2} \ln \relax (x )-25 x^{2}+x \ln \relax (x )\)	\(56\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((20*x^2-25*x+1)*ln(x)+40*x^2-50*x+2)*ln(ln(x)+2)+(3*x^3+4*x^2+x)*ln(x)^2+(50*x^5-20*x^4-113*x^3-40*x^2+3
*x+1)*ln(x)+100*x^5-40*x^4-238*x^3-86*x^2-23*x)/(x*ln(x)+2*x),x,method=_RETURNVERBOSE)

[Out]

(10*x^2-25*x+ln(x))*ln(ln(x)+2)+10*x^5-5*x^4+x^3*ln(x)-40*x^3+2*x^2*ln(x)-25*x^2+x*ln(x)

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maxima [A]  time = 0.40, size = 51, normalized size = 1.70 \begin {gather*} 10 \, x^{5} - 5 \, x^{4} - 40 \, x^{3} - 25 \, x^{2} + {\left (x^{3} + 2 \, x^{2} + x\right )} \log \relax (x) + {\left (10 \, x^{2} - 25 \, x + \log \relax (x)\right )} \log \left (\log \relax (x) + 2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((20*x^2-25*x+1)*log(x)+40*x^2-50*x+2)*log(log(x)+2)+(3*x^3+4*x^2+x)*log(x)^2+(50*x^5-20*x^4-113*x^
3-40*x^2+3*x+1)*log(x)+100*x^5-40*x^4-238*x^3-86*x^2-23*x)/(x*log(x)+2*x),x, algorithm="maxima")

[Out]

10*x^5 - 5*x^4 - 40*x^3 - 25*x^2 + (x^3 + 2*x^2 + x)*log(x) + (10*x^2 - 25*x + log(x))*log(log(x) + 2)

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mupad [B]  time = 3.62, size = 51, normalized size = 1.70 \begin {gather*} \ln \relax (x)\,\left (x^3+2\,x^2+x\right )+\ln \left (\ln \relax (x)+2\right )\,\left (\ln \relax (x)-25\,x+10\,x^2\right )-25\,x^2-40\,x^3-5\,x^4+10\,x^5 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(23*x - log(log(x) + 2)*(log(x)*(20*x^2 - 25*x + 1) - 50*x + 40*x^2 + 2) - log(x)*(3*x - 40*x^2 - 113*x^3
- 20*x^4 + 50*x^5 + 1) + 86*x^2 + 238*x^3 + 40*x^4 - 100*x^5 - log(x)^2*(x + 4*x^2 + 3*x^3))/(2*x + x*log(x))
,x)

[Out]

log(x)*(x + 2*x^2 + x^3) + log(log(x) + 2)*(log(x) - 25*x + 10*x^2) - 25*x^2 - 40*x^3 - 5*x^4 + 10*x^5

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sympy [A]  time = 0.37, size = 51, normalized size = 1.70 \begin {gather*} 10 x^{5} - 5 x^{4} - 40 x^{3} - 25 x^{2} + \left (10 x^{2} - 25 x + \log {\relax (x )}\right ) \log {\left (\log {\relax (x )} + 2 \right )} + \left (x^{3} + 2 x^{2} + x\right ) \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((20*x**2-25*x+1)*ln(x)+40*x**2-50*x+2)*ln(ln(x)+2)+(3*x**3+4*x**2+x)*ln(x)**2+(50*x**5-20*x**4-113
*x**3-40*x**2+3*x+1)*ln(x)+100*x**5-40*x**4-238*x**3-86*x**2-23*x)/(x*ln(x)+2*x),x)

[Out]

10*x**5 - 5*x**4 - 40*x**3 - 25*x**2 + (10*x**2 - 25*x + log(x))*log(log(x) + 2) + (x**3 + 2*x**2 + x)*log(x)

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