3.42.14 \(\int \frac {-2 x^4-2 x^4 \log (x)+(5 x^2+x^4+(5 x^2+x^4) \log (x)) \log (5+x^2)+(2 x^2+2 x^2 \log (x)+(-5-x^2+(-5-x^2) \log (x)) \log (5+x^2)) \log (1+\log (x))+(-5+9 x^2+2 x^4+(10 x^2+2 x^4) \log (x)) \log (5+x^2) \log (\frac {x}{2 \log (5+x^2)})}{(5 x+x^3+(5 x+x^3) \log (x)) \log (5+x^2)} \, dx\)

Optimal. Leaf size=26 \[ \left (x^2-\log (1+\log (x))\right ) \log \left (\frac {x}{2 \log \left (5+x^2\right )}\right ) \]

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Rubi [F]  time = 3.52, 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 {-2 x^4-2 x^4 \log (x)+\left (5 x^2+x^4+\left (5 x^2+x^4\right ) \log (x)\right ) \log \left (5+x^2\right )+\left (2 x^2+2 x^2 \log (x)+\left (-5-x^2+\left (-5-x^2\right ) \log (x)\right ) \log \left (5+x^2\right )\right ) \log (1+\log (x))+\left (-5+9 x^2+2 x^4+\left (10 x^2+2 x^4\right ) \log (x)\right ) \log \left (5+x^2\right ) \log \left (\frac {x}{2 \log \left (5+x^2\right )}\right )}{\left (5 x+x^3+\left (5 x+x^3\right ) \log (x)\right ) \log \left (5+x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-2*x^4 - 2*x^4*Log[x] + (5*x^2 + x^4 + (5*x^2 + x^4)*Log[x])*Log[5 + x^2] + (2*x^2 + 2*x^2*Log[x] + (-5 -
x^2 + (-5 - x^2)*Log[x])*Log[5 + x^2])*Log[1 + Log[x]] + (-5 + 9*x^2 + 2*x^4 + (10*x^2 + 2*x^4)*Log[x])*Log[5
+ x^2]*Log[x/(2*Log[5 + x^2])])/((5*x + x^3 + (5*x + x^3)*Log[x])*Log[5 + x^2]),x]

[Out]

x^2/2 + Log[x] - (1 + Log[x])*Log[1 + Log[x]] + 5*Log[Log[5 + x^2]] - LogIntegral[5 + x^2] - Defer[Int][Log[1
+ Log[x]]/((I*Sqrt[5] - x)*Log[5 + x^2]), x] + Defer[Int][Log[1 + Log[x]]/((I*Sqrt[5] + x)*Log[5 + x^2]), x] -
Defer[Int][Log[x/(2*Log[5 + x^2])]/(x*(1 + Log[x])), x] + 2*Defer[Int][(x*Log[x/(2*Log[5 + x^2])])/(1 + Log[x
]), x] + 2*Defer[Int][(x*Log[x]*Log[x/(2*Log[5 + x^2])])/(1 + Log[x]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-2 x^4-2 x^4 \log (x)+\left (5 x^2+x^4+\left (5 x^2+x^4\right ) \log (x)\right ) \log \left (5+x^2\right )+\left (2 x^2+2 x^2 \log (x)+\left (-5-x^2+\left (-5-x^2\right ) \log (x)\right ) \log \left (5+x^2\right )\right ) \log (1+\log (x))+\left (-5+9 x^2+2 x^4+\left (10 x^2+2 x^4\right ) \log (x)\right ) \log \left (5+x^2\right ) \log \left (\frac {x}{2 \log \left (5+x^2\right )}\right )}{x \left (5+x^2\right ) (1+\log (x)) \log \left (5+x^2\right )} \, dx\\ &=\int \left (\frac {\left (-2 x^2+5 \log \left (5+x^2\right )+x^2 \log \left (5+x^2\right )\right ) \left (x^2-\log (1+\log (x))\right )}{x \left (5+x^2\right ) \log \left (5+x^2\right )}+\frac {\left (-1+2 x^2+2 x^2 \log (x)\right ) \log \left (\frac {x}{2 \log \left (5+x^2\right )}\right )}{x (1+\log (x))}\right ) \, dx\\ &=\int \frac {\left (-2 x^2+5 \log \left (5+x^2\right )+x^2 \log \left (5+x^2\right )\right ) \left (x^2-\log (1+\log (x))\right )}{x \left (5+x^2\right ) \log \left (5+x^2\right )} \, dx+\int \frac {\left (-1+2 x^2+2 x^2 \log (x)\right ) \log \left (\frac {x}{2 \log \left (5+x^2\right )}\right )}{x (1+\log (x))} \, dx\\ &=\int \left (\frac {-2 x^3+5 x \log \left (5+x^2\right )+x^3 \log \left (5+x^2\right )}{\left (5+x^2\right ) \log \left (5+x^2\right )}-\frac {\left (-2 x^2+5 \log \left (5+x^2\right )+x^2 \log \left (5+x^2\right )\right ) \log (1+\log (x))}{x \left (5+x^2\right ) \log \left (5+x^2\right )}\right ) \, dx+\int \left (-\frac {\log \left (\frac {x}{2 \log \left (5+x^2\right )}\right )}{x (1+\log (x))}+\frac {2 x \log \left (\frac {x}{2 \log \left (5+x^2\right )}\right )}{1+\log (x)}+\frac {2 x \log (x) \log \left (\frac {x}{2 \log \left (5+x^2\right )}\right )}{1+\log (x)}\right ) \, dx\\ &=2 \int \frac {x \log \left (\frac {x}{2 \log \left (5+x^2\right )}\right )}{1+\log (x)} \, dx+2 \int \frac {x \log (x) \log \left (\frac {x}{2 \log \left (5+x^2\right )}\right )}{1+\log (x)} \, dx+\int \frac {-2 x^3+5 x \log \left (5+x^2\right )+x^3 \log \left (5+x^2\right )}{\left (5+x^2\right ) \log \left (5+x^2\right )} \, dx-\int \frac {\left (-2 x^2+5 \log \left (5+x^2\right )+x^2 \log \left (5+x^2\right )\right ) \log (1+\log (x))}{x \left (5+x^2\right ) \log \left (5+x^2\right )} \, dx-\int \frac {\log \left (\frac {x}{2 \log \left (5+x^2\right )}\right )}{x (1+\log (x))} \, dx\\ &=2 \int \frac {x \log \left (\frac {x}{2 \log \left (5+x^2\right )}\right )}{1+\log (x)} \, dx+2 \int \frac {x \log (x) \log \left (\frac {x}{2 \log \left (5+x^2\right )}\right )}{1+\log (x)} \, dx+\int \left (x-\frac {2 x^3}{\left (5+x^2\right ) \log \left (5+x^2\right )}\right ) \, dx-\int \frac {\left (1-\frac {2 x^2}{\left (5+x^2\right ) \log \left (5+x^2\right )}\right ) \log (1+\log (x))}{x} \, dx-\int \frac {\log \left (\frac {x}{2 \log \left (5+x^2\right )}\right )}{x (1+\log (x))} \, dx\\ &=\frac {x^2}{2}-2 \int \frac {x^3}{\left (5+x^2\right ) \log \left (5+x^2\right )} \, dx+2 \int \frac {x \log \left (\frac {x}{2 \log \left (5+x^2\right )}\right )}{1+\log (x)} \, dx+2 \int \frac {x \log (x) \log \left (\frac {x}{2 \log \left (5+x^2\right )}\right )}{1+\log (x)} \, dx-\int \left (\frac {\log (1+\log (x))}{x}-\frac {2 x \log (1+\log (x))}{\left (5+x^2\right ) \log \left (5+x^2\right )}\right ) \, dx-\int \frac {\log \left (\frac {x}{2 \log \left (5+x^2\right )}\right )}{x (1+\log (x))} \, dx\\ &=\frac {x^2}{2}+2 \int \frac {x \log (1+\log (x))}{\left (5+x^2\right ) \log \left (5+x^2\right )} \, dx+2 \int \frac {x \log \left (\frac {x}{2 \log \left (5+x^2\right )}\right )}{1+\log (x)} \, dx+2 \int \frac {x \log (x) \log \left (\frac {x}{2 \log \left (5+x^2\right )}\right )}{1+\log (x)} \, dx-\int \frac {\log (1+\log (x))}{x} \, dx-\int \frac {\log \left (\frac {x}{2 \log \left (5+x^2\right )}\right )}{x (1+\log (x))} \, dx-\operatorname {Subst}\left (\int \frac {x}{(5+x) \log (5+x)} \, dx,x,x^2\right )\\ &=\frac {x^2}{2}+2 \int \left (-\frac {\log (1+\log (x))}{2 \left (i \sqrt {5}-x\right ) \log \left (5+x^2\right )}+\frac {\log (1+\log (x))}{2 \left (i \sqrt {5}+x\right ) \log \left (5+x^2\right )}\right ) \, dx+2 \int \frac {x \log \left (\frac {x}{2 \log \left (5+x^2\right )}\right )}{1+\log (x)} \, dx+2 \int \frac {x \log (x) \log \left (\frac {x}{2 \log \left (5+x^2\right )}\right )}{1+\log (x)} \, dx-\int \frac {\log \left (\frac {x}{2 \log \left (5+x^2\right )}\right )}{x (1+\log (x))} \, dx-\operatorname {Subst}\left (\int \frac {-5+x}{x \log (x)} \, dx,x,5+x^2\right )-\operatorname {Subst}(\int \log (1+x) \, dx,x,\log (x))\\ &=\frac {x^2}{2}+2 \int \frac {x \log \left (\frac {x}{2 \log \left (5+x^2\right )}\right )}{1+\log (x)} \, dx+2 \int \frac {x \log (x) \log \left (\frac {x}{2 \log \left (5+x^2\right )}\right )}{1+\log (x)} \, dx-\int \frac {\log (1+\log (x))}{\left (i \sqrt {5}-x\right ) \log \left (5+x^2\right )} \, dx+\int \frac {\log (1+\log (x))}{\left (i \sqrt {5}+x\right ) \log \left (5+x^2\right )} \, dx-\int \frac {\log \left (\frac {x}{2 \log \left (5+x^2\right )}\right )}{x (1+\log (x))} \, dx-\operatorname {Subst}\left (\int \left (\frac {1}{\log (x)}-\frac {5}{x \log (x)}\right ) \, dx,x,5+x^2\right )-\operatorname {Subst}(\int \log (x) \, dx,x,1+\log (x))\\ &=\frac {x^2}{2}+\log (x)-(1+\log (x)) \log (1+\log (x))+2 \int \frac {x \log \left (\frac {x}{2 \log \left (5+x^2\right )}\right )}{1+\log (x)} \, dx+2 \int \frac {x \log (x) \log \left (\frac {x}{2 \log \left (5+x^2\right )}\right )}{1+\log (x)} \, dx+5 \operatorname {Subst}\left (\int \frac {1}{x \log (x)} \, dx,x,5+x^2\right )-\int \frac {\log (1+\log (x))}{\left (i \sqrt {5}-x\right ) \log \left (5+x^2\right )} \, dx+\int \frac {\log (1+\log (x))}{\left (i \sqrt {5}+x\right ) \log \left (5+x^2\right )} \, dx-\int \frac {\log \left (\frac {x}{2 \log \left (5+x^2\right )}\right )}{x (1+\log (x))} \, dx-\operatorname {Subst}\left (\int \frac {1}{\log (x)} \, dx,x,5+x^2\right )\\ &=\frac {x^2}{2}+\log (x)-(1+\log (x)) \log (1+\log (x))-\text {li}\left (5+x^2\right )+2 \int \frac {x \log \left (\frac {x}{2 \log \left (5+x^2\right )}\right )}{1+\log (x)} \, dx+2 \int \frac {x \log (x) \log \left (\frac {x}{2 \log \left (5+x^2\right )}\right )}{1+\log (x)} \, dx+5 \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,\log \left (5+x^2\right )\right )-\int \frac {\log (1+\log (x))}{\left (i \sqrt {5}-x\right ) \log \left (5+x^2\right )} \, dx+\int \frac {\log (1+\log (x))}{\left (i \sqrt {5}+x\right ) \log \left (5+x^2\right )} \, dx-\int \frac {\log \left (\frac {x}{2 \log \left (5+x^2\right )}\right )}{x (1+\log (x))} \, dx\\ &=\frac {x^2}{2}+\log (x)-(1+\log (x)) \log (1+\log (x))+5 \log \left (\log \left (5+x^2\right )\right )-\text {li}\left (5+x^2\right )+2 \int \frac {x \log \left (\frac {x}{2 \log \left (5+x^2\right )}\right )}{1+\log (x)} \, dx+2 \int \frac {x \log (x) \log \left (\frac {x}{2 \log \left (5+x^2\right )}\right )}{1+\log (x)} \, dx-\int \frac {\log (1+\log (x))}{\left (i \sqrt {5}-x\right ) \log \left (5+x^2\right )} \, dx+\int \frac {\log (1+\log (x))}{\left (i \sqrt {5}+x\right ) \log \left (5+x^2\right )} \, dx-\int \frac {\log \left (\frac {x}{2 \log \left (5+x^2\right )}\right )}{x (1+\log (x))} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.22, size = 26, normalized size = 1.00 \begin {gather*} \left (x^2-\log (1+\log (x))\right ) \log \left (\frac {x}{2 \log \left (5+x^2\right )}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-2*x^4 - 2*x^4*Log[x] + (5*x^2 + x^4 + (5*x^2 + x^4)*Log[x])*Log[5 + x^2] + (2*x^2 + 2*x^2*Log[x] +
(-5 - x^2 + (-5 - x^2)*Log[x])*Log[5 + x^2])*Log[1 + Log[x]] + (-5 + 9*x^2 + 2*x^4 + (10*x^2 + 2*x^4)*Log[x])
*Log[5 + x^2]*Log[x/(2*Log[5 + x^2])])/((5*x + x^3 + (5*x + x^3)*Log[x])*Log[5 + x^2]),x]

[Out]

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

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fricas [A]  time = 0.90, size = 36, normalized size = 1.38 \begin {gather*} x^{2} \log \left (\frac {x}{2 \, \log \left (x^{2} + 5\right )}\right ) - \log \left (\frac {x}{2 \, \log \left (x^{2} + 5\right )}\right ) \log \left (\log \relax (x) + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((-x^2-5)*log(x)-x^2-5)*log(x^2+5)+2*x^2*log(x)+2*x^2)*log(log(x)+1)+((2*x^4+10*x^2)*log(x)+2*x^4+
9*x^2-5)*log(x^2+5)*log(1/2*x/log(x^2+5))+((x^4+5*x^2)*log(x)+x^4+5*x^2)*log(x^2+5)-2*x^4*log(x)-2*x^4)/((x^3+
5*x)*log(x)+x^3+5*x)/log(x^2+5),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.36, size = 38, normalized size = 1.46 \begin {gather*} x^{2} \log \relax (x) - {\left (x^{2} - \log \left (\log \relax (x) + 1\right )\right )} \log \left (2 \, \log \left (x^{2} + 5\right )\right ) - \log \relax (x) \log \left (\log \relax (x) + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((-x^2-5)*log(x)-x^2-5)*log(x^2+5)+2*x^2*log(x)+2*x^2)*log(log(x)+1)+((2*x^4+10*x^2)*log(x)+2*x^4+
9*x^2-5)*log(x^2+5)*log(1/2*x/log(x^2+5))+((x^4+5*x^2)*log(x)+x^4+5*x^2)*log(x^2+5)-2*x^4*log(x)-2*x^4)/((x^3+
5*x)*log(x)+x^3+5*x)/log(x^2+5),x, algorithm="giac")

[Out]

x^2*log(x) - (x^2 - log(log(x) + 1))*log(2*log(x^2 + 5)) - log(x)*log(log(x) + 1)

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maple [C]  time = 0.22, size = 299, normalized size = 11.50

method	result	size
risch	\(\left (-x^{2}+\ln \left (\ln \relax (x )+1\right )\right ) \ln \left (\ln \left (x^{2}+5\right )\right )-\ln \relax (x ) \ln \left (\ln \relax (x )+1\right )+x^{2} \ln \relax (x )+\frac {i \pi \ln \left (\ln \relax (x )+1\right ) \mathrm {csgn}\left (\frac {i x}{\ln \left (x^{2}+5\right )}\right )^{3}}{2}+\frac {i \pi \ln \left (\ln \relax (x )+1\right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (\frac {i}{\ln \left (x^{2}+5\right )}\right ) \mathrm {csgn}\left (\frac {i x}{\ln \left (x^{2}+5\right )}\right )}{2}-\frac {i \pi \,x^{2} \mathrm {csgn}\left (\frac {i x}{\ln \left (x^{2}+5\right )}\right )^{3}}{2}-\frac {i \pi \ln \left (\ln \relax (x )+1\right ) \mathrm {csgn}\left (\frac {i}{\ln \left (x^{2}+5\right )}\right ) \mathrm {csgn}\left (\frac {i x}{\ln \left (x^{2}+5\right )}\right )^{2}}{2}-\frac {i \pi \ln \left (\ln \relax (x )+1\right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (\frac {i x}{\ln \left (x^{2}+5\right )}\right )^{2}}{2}+\frac {i \pi \,x^{2} \mathrm {csgn}\left (\frac {i}{\ln \left (x^{2}+5\right )}\right ) \mathrm {csgn}\left (\frac {i x}{\ln \left (x^{2}+5\right )}\right )^{2}}{2}-\frac {i \pi \,x^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (\frac {i}{\ln \left (x^{2}+5\right )}\right ) \mathrm {csgn}\left (\frac {i x}{\ln \left (x^{2}+5\right )}\right )}{2}+\frac {i \pi \,x^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (\frac {i x}{\ln \left (x^{2}+5\right )}\right )^{2}}{2}-x^{2} \ln \relax (2)+\ln \relax (2) \ln \left (\ln \relax (x )+1\right )\)	\(299\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((((-x^2-5)*ln(x)-x^2-5)*ln(x^2+5)+2*x^2*ln(x)+2*x^2)*ln(ln(x)+1)+((2*x^4+10*x^2)*ln(x)+2*x^4+9*x^2-5)*ln(
x^2+5)*ln(1/2*x/ln(x^2+5))+((x^4+5*x^2)*ln(x)+x^4+5*x^2)*ln(x^2+5)-2*x^4*ln(x)-2*x^4)/((x^3+5*x)*ln(x)+x^3+5*x
)/ln(x^2+5),x,method=_RETURNVERBOSE)

[Out]

(-x^2+ln(ln(x)+1))*ln(ln(x^2+5))-ln(x)*ln(ln(x)+1)+x^2*ln(x)+1/2*I*Pi*ln(ln(x)+1)*csgn(I*x/ln(x^2+5))^3+1/2*I*
Pi*ln(ln(x)+1)*csgn(I*x)*csgn(I/ln(x^2+5))*csgn(I*x/ln(x^2+5))-1/2*I*Pi*x^2*csgn(I*x/ln(x^2+5))^3-1/2*I*Pi*ln(
ln(x)+1)*csgn(I/ln(x^2+5))*csgn(I*x/ln(x^2+5))^2-1/2*I*Pi*ln(ln(x)+1)*csgn(I*x)*csgn(I*x/ln(x^2+5))^2+1/2*I*Pi
*x^2*csgn(I/ln(x^2+5))*csgn(I*x/ln(x^2+5))^2-1/2*I*Pi*x^2*csgn(I*x)*csgn(I/ln(x^2+5))*csgn(I*x/ln(x^2+5))+1/2*
I*Pi*x^2*csgn(I*x)*csgn(I*x/ln(x^2+5))^2-x^2*ln(2)+ln(2)*ln(ln(x)+1)

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maxima [A]  time = 0.50, size = 47, normalized size = 1.81 \begin {gather*} -x^{2} \log \relax (2) + x^{2} \log \relax (x) + {\left (\log \relax (2) - \log \relax (x)\right )} \log \left (\log \relax (x) + 1\right ) - {\left (x^{2} - \log \left (\log \relax (x) + 1\right )\right )} \log \left (\log \left (x^{2} + 5\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((-x^2-5)*log(x)-x^2-5)*log(x^2+5)+2*x^2*log(x)+2*x^2)*log(log(x)+1)+((2*x^4+10*x^2)*log(x)+2*x^4+
9*x^2-5)*log(x^2+5)*log(1/2*x/log(x^2+5))+((x^4+5*x^2)*log(x)+x^4+5*x^2)*log(x^2+5)-2*x^4*log(x)-2*x^4)/((x^3+
5*x)*log(x)+x^3+5*x)/log(x^2+5),x, algorithm="maxima")

[Out]

-x^2*log(2) + x^2*log(x) + (log(2) - log(x))*log(log(x) + 1) - (x^2 - log(log(x) + 1))*log(log(x^2 + 5))

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mupad [B]  time = 4.51, size = 26, normalized size = 1.00 \begin {gather*} \left (\ln \left (\ln \relax (x)+1\right )-x^2\right )\,\left (\ln \left (\ln \left (x^2+5\right )\right )+\ln \relax (2)-\ln \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x^2 + 5)*(log(x)*(5*x^2 + x^4) + 5*x^2 + x^4) - 2*x^4*log(x) + log(log(x) + 1)*(2*x^2*log(x) - log(x^
2 + 5)*(log(x)*(x^2 + 5) + x^2 + 5) + 2*x^2) - 2*x^4 + log(x/(2*log(x^2 + 5)))*log(x^2 + 5)*(log(x)*(10*x^2 +
2*x^4) + 9*x^2 + 2*x^4 - 5))/(log(x^2 + 5)*(5*x + log(x)*(5*x + x^3) + x^3)),x)

[Out]

(log(log(x) + 1) - x^2)*(log(log(x^2 + 5)) + log(2) - log(x))

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sympy [A]  time = 2.26, size = 20, normalized size = 0.77 \begin {gather*} \left (x^{2} - \log {\left (\log {\relax (x )} + 1 \right )}\right ) \log {\left (\frac {x}{2 \log {\left (x^{2} + 5 \right )}} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((-x**2-5)*ln(x)-x**2-5)*ln(x**2+5)+2*x**2*ln(x)+2*x**2)*ln(ln(x)+1)+((2*x**4+10*x**2)*ln(x)+2*x**
4+9*x**2-5)*ln(x**2+5)*ln(1/2*x/ln(x**2+5))+((x**4+5*x**2)*ln(x)+x**4+5*x**2)*ln(x**2+5)-2*x**4*ln(x)-2*x**4)/
((x**3+5*x)*ln(x)+x**3+5*x)/ln(x**2+5),x)

[Out]

(x**2 - log(log(x) + 1))*log(x/(2*log(x**2 + 5)))

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