3.85.4 \(\int \frac {50 x+25 x^2+(50 x+25 x^2) \log (x)+(90+60 x+10 x^2+(-30-10 x) \log (3+x)) \log (1+\log (x))+(10 x+5 x^2+(10 x+5 x^2) \log (x)) \log ^2(1+\log (x))}{6 x+2 x^2+(6 x+2 x^2) \log (x)} \, dx\)

Optimal. Leaf size=22 \[ \frac {5}{2} (3+x-\log (3+x)) \left (5+\log ^2(1+\log (x))\right ) \]

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Rubi [F]  time = 0.91, 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 {50 x+25 x^2+\left (50 x+25 x^2\right ) \log (x)+\left (90+60 x+10 x^2+(-30-10 x) \log (3+x)\right ) \log (1+\log (x))+\left (10 x+5 x^2+\left (10 x+5 x^2\right ) \log (x)\right ) \log ^2(1+\log (x))}{6 x+2 x^2+\left (6 x+2 x^2\right ) \log (x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(50*x + 25*x^2 + (50*x + 25*x^2)*Log[x] + (90 + 60*x + 10*x^2 + (-30 - 10*x)*Log[3 + x])*Log[1 + Log[x]] +
(10*x + 5*x^2 + (10*x + 5*x^2)*Log[x])*Log[1 + Log[x]]^2)/(6*x + 2*x^2 + (6*x + 2*x^2)*Log[x]),x]

[Out]

(25*x)/2 - (25*Log[3 + x])/2 + (15*Log[1 + Log[x]]^2)/2 + 5*Defer[Int][Log[1 + Log[x]]/(1 + Log[x]), x] - 5*De
fer[Int][(Log[3 + x]*Log[1 + Log[x]])/(x*(1 + Log[x])), x] + (5*Defer[Int][Log[1 + Log[x]]^2, x])/2 - (5*Defer
[Int][Log[1 + Log[x]]^2/(3 + x), x])/2

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {5 \left (5 x (2+x)+2 (3+x) (3+x-\log (3+x)) \log (1+\log (x))+x (2+x) \log ^2(1+\log (x))+x (2+x) \log (x) \left (5+\log ^2(1+\log (x))\right )\right )}{2 x (3+x) (1+\log (x))} \, dx\\ &=\frac {5}{2} \int \frac {5 x (2+x)+2 (3+x) (3+x-\log (3+x)) \log (1+\log (x))+x (2+x) \log ^2(1+\log (x))+x (2+x) \log (x) \left (5+\log ^2(1+\log (x))\right )}{x (3+x) (1+\log (x))} \, dx\\ &=\frac {5}{2} \int \left (\frac {5 (2+x)}{3+x}+\frac {2 (3+x-\log (3+x)) \log (1+\log (x))}{x (1+\log (x))}+\frac {(2+x) \log ^2(1+\log (x))}{3+x}\right ) \, dx\\ &=\frac {5}{2} \int \frac {(2+x) \log ^2(1+\log (x))}{3+x} \, dx+5 \int \frac {(3+x-\log (3+x)) \log (1+\log (x))}{x (1+\log (x))} \, dx+\frac {25}{2} \int \frac {2+x}{3+x} \, dx\\ &=\frac {5}{2} \int \left (\log ^2(1+\log (x))-\frac {\log ^2(1+\log (x))}{3+x}\right ) \, dx+5 \int \left (\frac {\log (1+\log (x))}{1+\log (x)}+\frac {3 \log (1+\log (x))}{x (1+\log (x))}-\frac {\log (3+x) \log (1+\log (x))}{x (1+\log (x))}\right ) \, dx+\frac {25}{2} \int \left (1+\frac {1}{-3-x}\right ) \, dx\\ &=\frac {25 x}{2}-\frac {25}{2} \log (3+x)+\frac {5}{2} \int \log ^2(1+\log (x)) \, dx-\frac {5}{2} \int \frac {\log ^2(1+\log (x))}{3+x} \, dx+5 \int \frac {\log (1+\log (x))}{1+\log (x)} \, dx-5 \int \frac {\log (3+x) \log (1+\log (x))}{x (1+\log (x))} \, dx+15 \int \frac {\log (1+\log (x))}{x (1+\log (x))} \, dx\\ &=\frac {25 x}{2}-\frac {25}{2} \log (3+x)+\frac {5}{2} \int \log ^2(1+\log (x)) \, dx-\frac {5}{2} \int \frac {\log ^2(1+\log (x))}{3+x} \, dx+5 \int \frac {\log (1+\log (x))}{1+\log (x)} \, dx-5 \int \frac {\log (3+x) \log (1+\log (x))}{x (1+\log (x))} \, dx+15 \operatorname {Subst}\left (\int \frac {\log (1+x)}{1+x} \, dx,x,\log (x)\right )\\ &=\frac {25 x}{2}-\frac {25}{2} \log (3+x)+\frac {5}{2} \int \log ^2(1+\log (x)) \, dx-\frac {5}{2} \int \frac {\log ^2(1+\log (x))}{3+x} \, dx+5 \int \frac {\log (1+\log (x))}{1+\log (x)} \, dx-5 \int \frac {\log (3+x) \log (1+\log (x))}{x (1+\log (x))} \, dx+15 \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,1+\log (x)\right )\\ &=\frac {25 x}{2}-\frac {25}{2} \log (3+x)+\frac {15}{2} \log ^2(1+\log (x))+\frac {5}{2} \int \log ^2(1+\log (x)) \, dx-\frac {5}{2} \int \frac {\log ^2(1+\log (x))}{3+x} \, dx+5 \int \frac {\log (1+\log (x))}{1+\log (x)} \, dx-5 \int \frac {\log (3+x) \log (1+\log (x))}{x (1+\log (x))} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.23, size = 31, normalized size = 1.41 \begin {gather*} \frac {5}{2} \left (5 x-5 \log (3+x)+(3+x-\log (3+x)) \log ^2(1+\log (x))\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(50*x + 25*x^2 + (50*x + 25*x^2)*Log[x] + (90 + 60*x + 10*x^2 + (-30 - 10*x)*Log[3 + x])*Log[1 + Log
[x]] + (10*x + 5*x^2 + (10*x + 5*x^2)*Log[x])*Log[1 + Log[x]]^2)/(6*x + 2*x^2 + (6*x + 2*x^2)*Log[x]),x]

[Out]

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

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fricas [A]  time = 0.95, size = 28, normalized size = 1.27 \begin {gather*} \frac {5}{2} \, {\left (x - \log \left (x + 3\right ) + 3\right )} \log \left (\log \relax (x) + 1\right )^{2} + \frac {25}{2} \, x - \frac {25}{2} \, \log \left (x + 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((5*x^2+10*x)*log(x)+5*x^2+10*x)*log(log(x)+1)^2+((-10*x-30)*log(3+x)+10*x^2+60*x+90)*log(log(x)+1)
+(25*x^2+50*x)*log(x)+25*x^2+50*x)/((2*x^2+6*x)*log(x)+2*x^2+6*x),x, algorithm="fricas")

[Out]

5/2*(x - log(x + 3) + 3)*log(log(x) + 1)^2 + 25/2*x - 25/2*log(x + 3)

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giac [A]  time = 0.26, size = 36, normalized size = 1.64 \begin {gather*} \frac {5}{2} \, {\left (x - \log \left (x + 3\right )\right )} \log \left (\log \relax (x) + 1\right )^{2} + \frac {15}{2} \, \log \left (\log \relax (x) + 1\right )^{2} + \frac {25}{2} \, x - \frac {25}{2} \, \log \left (x + 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((5*x^2+10*x)*log(x)+5*x^2+10*x)*log(log(x)+1)^2+((-10*x-30)*log(3+x)+10*x^2+60*x+90)*log(log(x)+1)
+(25*x^2+50*x)*log(x)+25*x^2+50*x)/((2*x^2+6*x)*log(x)+2*x^2+6*x),x, algorithm="giac")

[Out]

5/2*(x - log(x + 3))*log(log(x) + 1)^2 + 15/2*log(log(x) + 1)^2 + 25/2*x - 25/2*log(x + 3)

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maple [A]  time = 0.06, size = 30, normalized size = 1.36

method	result	size
risch	\(\left (\frac {5 x}{2}-\frac {5 \ln \left (3+x \right )}{2}+\frac {15}{2}\right ) \ln \left (\ln \relax (x )+1\right )^{2}+\frac {25 x}{2}-\frac {25 \ln \left (3+x \right )}{2}\)	\(30\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((5*x^2+10*x)*ln(x)+5*x^2+10*x)*ln(ln(x)+1)^2+((-10*x-30)*ln(3+x)+10*x^2+60*x+90)*ln(ln(x)+1)+(25*x^2+50*
x)*ln(x)+25*x^2+50*x)/((2*x^2+6*x)*ln(x)+2*x^2+6*x),x,method=_RETURNVERBOSE)

[Out]

(5/2*x-5/2*ln(3+x)+15/2)*ln(ln(x)+1)^2+25/2*x-25/2*ln(3+x)

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maxima [A]  time = 0.40, size = 28, normalized size = 1.27 \begin {gather*} \frac {5}{2} \, {\left (x - \log \left (x + 3\right ) + 3\right )} \log \left (\log \relax (x) + 1\right )^{2} + \frac {25}{2} \, x - \frac {25}{2} \, \log \left (x + 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((5*x^2+10*x)*log(x)+5*x^2+10*x)*log(log(x)+1)^2+((-10*x-30)*log(3+x)+10*x^2+60*x+90)*log(log(x)+1)
+(25*x^2+50*x)*log(x)+25*x^2+50*x)/((2*x^2+6*x)*log(x)+2*x^2+6*x),x, algorithm="maxima")

[Out]

5/2*(x - log(x + 3) + 3)*log(log(x) + 1)^2 + 25/2*x - 25/2*log(x + 3)

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mupad [B]  time = 5.47, size = 47, normalized size = 2.14 \begin {gather*} \left (\frac {5\,x^3+15\,x^2}{2\,x\,\left (x+3\right )}-\frac {5\,\ln \left (x+3\right )}{2}+\frac {15}{2}\right )\,{\ln \left (\ln \relax (x)+1\right )}^2+\frac {25\,x}{2}-\frac {25\,\ln \left (x+3\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((50*x + log(log(x) + 1)*(60*x + 10*x^2 - log(x + 3)*(10*x + 30) + 90) + log(log(x) + 1)^2*(10*x + log(x)*(
10*x + 5*x^2) + 5*x^2) + log(x)*(50*x + 25*x^2) + 25*x^2)/(6*x + log(x)*(6*x + 2*x^2) + 2*x^2),x)

[Out]

(25*x)/2 - (25*log(x + 3))/2 + log(log(x) + 1)^2*((15*x^2 + 5*x^3)/(2*x*(x + 3)) - (5*log(x + 3))/2 + 15/2)

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sympy [A]  time = 0.81, size = 37, normalized size = 1.68 \begin {gather*} \frac {25 x}{2} + \left (\frac {5 x}{2} - \frac {5 \log {\left (x + 3 \right )}}{2} + \frac {15}{2}\right ) \log {\left (\log {\relax (x )} + 1 \right )}^{2} - \frac {25 \log {\left (x + 3 \right )}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((5*x**2+10*x)*ln(x)+5*x**2+10*x)*ln(ln(x)+1)**2+((-10*x-30)*ln(3+x)+10*x**2+60*x+90)*ln(ln(x)+1)+(
25*x**2+50*x)*ln(x)+25*x**2+50*x)/((2*x**2+6*x)*ln(x)+2*x**2+6*x),x)

[Out]

25*x/2 + (5*x/2 - 5*log(x + 3)/2 + 15/2)*log(log(x) + 1)**2 - 25*log(x + 3)/2

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