3.40.45 \(\int \frac {-800-800 x+(1600-400 x-800 x^2) \log (x)+(-5 x+2 x^3) \log ^3(x)+((400+400 x) \log (x)+(-x-x^2) \log ^3(x)) \log (\frac {400-x \log ^2(x)}{\log ^2(x)})}{(-400-400 x) \log (x)+(x+x^2) \log ^3(x)} \, dx\)

Optimal. Leaf size=26 \[ 5-3 \log (1+x)+x \left (-1+x-\log \left (-x+\frac {400}{\log ^2(x)}\right )\right ) \]

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Rubi [F]  time = 2.34, 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 {-800-800 x+\left (1600-400 x-800 x^2\right ) \log (x)+\left (-5 x+2 x^3\right ) \log ^3(x)+\left ((400+400 x) \log (x)+\left (-x-x^2\right ) \log ^3(x)\right ) \log \left (\frac {400-x \log ^2(x)}{\log ^2(x)}\right )}{(-400-400 x) \log (x)+\left (x+x^2\right ) \log ^3(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-800 - 800*x + (1600 - 400*x - 800*x^2)*Log[x] + (-5*x + 2*x^3)*Log[x]^3 + ((400 + 400*x)*Log[x] + (-x -
x^2)*Log[x]^3)*Log[(400 - x*Log[x]^2)/Log[x]^2])/((-400 - 400*x)*Log[x] + (x + x^2)*Log[x]^3),x]

[Out]

-2*x + x^2 - 3*Log[1 + x] + 2*Defer[Int][1/((1 + x)*Log[x]), x] + 2*Defer[Int][x/((1 + x)*Log[x]), x] - 400*De
fer[Int][(400 - x*Log[x]^2)^(-1), x] - 800*Defer[Int][(-400 + x*Log[x]^2)^(-1), x] - 2*Defer[Int][(x*Log[x])/(
-400 + x*Log[x]^2), x] - Defer[Int][Log[-x + 400/Log[x]^2], x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {800+800 x-\left (1600-400 x-800 x^2\right ) \log (x)-\left (-5 x+2 x^3\right ) \log ^3(x)-\left ((400+400 x) \log (x)+\left (-x-x^2\right ) \log ^3(x)\right ) \log \left (\frac {400-x \log ^2(x)}{\log ^2(x)}\right )}{(1+x) \log (x) \left (400-x \log ^2(x)\right )} \, dx\\ &=\int \left (-\frac {400 \left (-4+x+2 x^2\right )}{(1+x) \left (-400+x \log ^2(x)\right )}-\frac {800}{(1+x) \log (x) \left (-400+x \log ^2(x)\right )}-\frac {800 x}{(1+x) \log (x) \left (-400+x \log ^2(x)\right )}+\frac {x \left (-5+2 x^2\right ) \log ^2(x)}{(1+x) \left (-400+x \log ^2(x)\right )}-\log \left (-x+\frac {400}{\log ^2(x)}\right )\right ) \, dx\\ &=-\left (400 \int \frac {-4+x+2 x^2}{(1+x) \left (-400+x \log ^2(x)\right )} \, dx\right )-800 \int \frac {1}{(1+x) \log (x) \left (-400+x \log ^2(x)\right )} \, dx-800 \int \frac {x}{(1+x) \log (x) \left (-400+x \log ^2(x)\right )} \, dx+\int \frac {x \left (-5+2 x^2\right ) \log ^2(x)}{(1+x) \left (-400+x \log ^2(x)\right )} \, dx-\int \log \left (-x+\frac {400}{\log ^2(x)}\right ) \, dx\\ &=-\left (400 \int \left (\frac {1}{400-x \log ^2(x)}+\frac {2 x}{-400+x \log ^2(x)}-\frac {3}{(1+x) \left (-400+x \log ^2(x)\right )}\right ) \, dx\right )-800 \int \left (-\frac {1}{400 (1+x) \log (x)}+\frac {x \log (x)}{400 (1+x) \left (-400+x \log ^2(x)\right )}\right ) \, dx-800 \int \left (-\frac {x}{400 (1+x) \log (x)}+\frac {x^2 \log (x)}{400 (1+x) \left (-400+x \log ^2(x)\right )}\right ) \, dx+\int \left (\frac {-5+2 x^2}{1+x}+\frac {400 \left (-5+2 x^2\right )}{(1+x) \left (-400+x \log ^2(x)\right )}\right ) \, dx-\int \log \left (-x+\frac {400}{\log ^2(x)}\right ) \, dx\\ &=2 \int \frac {1}{(1+x) \log (x)} \, dx+2 \int \frac {x}{(1+x) \log (x)} \, dx-2 \int \frac {x \log (x)}{(1+x) \left (-400+x \log ^2(x)\right )} \, dx-2 \int \frac {x^2 \log (x)}{(1+x) \left (-400+x \log ^2(x)\right )} \, dx-400 \int \frac {1}{400-x \log ^2(x)} \, dx+400 \int \frac {-5+2 x^2}{(1+x) \left (-400+x \log ^2(x)\right )} \, dx-800 \int \frac {x}{-400+x \log ^2(x)} \, dx+1200 \int \frac {1}{(1+x) \left (-400+x \log ^2(x)\right )} \, dx+\int \frac {-5+2 x^2}{1+x} \, dx-\int \log \left (-x+\frac {400}{\log ^2(x)}\right ) \, dx\\ &=2 \int \frac {1}{(1+x) \log (x)} \, dx+2 \int \frac {x}{(1+x) \log (x)} \, dx-2 \int \left (\frac {\log (x)}{-400+x \log ^2(x)}-\frac {\log (x)}{(1+x) \left (-400+x \log ^2(x)\right )}\right ) \, dx-2 \int \left (-\frac {\log (x)}{-400+x \log ^2(x)}+\frac {x \log (x)}{-400+x \log ^2(x)}+\frac {\log (x)}{(1+x) \left (-400+x \log ^2(x)\right )}\right ) \, dx-400 \int \frac {1}{400-x \log ^2(x)} \, dx+400 \int \left (-\frac {2}{-400+x \log ^2(x)}+\frac {2 x}{-400+x \log ^2(x)}-\frac {3}{(1+x) \left (-400+x \log ^2(x)\right )}\right ) \, dx-800 \int \frac {x}{-400+x \log ^2(x)} \, dx+1200 \int \frac {1}{(1+x) \left (-400+x \log ^2(x)\right )} \, dx+\int \left (-2+2 x-\frac {3}{1+x}\right ) \, dx-\int \log \left (-x+\frac {400}{\log ^2(x)}\right ) \, dx\\ &=-2 x+x^2-3 \log (1+x)+2 \int \frac {1}{(1+x) \log (x)} \, dx+2 \int \frac {x}{(1+x) \log (x)} \, dx-2 \int \frac {x \log (x)}{-400+x \log ^2(x)} \, dx-400 \int \frac {1}{400-x \log ^2(x)} \, dx-800 \int \frac {1}{-400+x \log ^2(x)} \, dx-\int \log \left (-x+\frac {400}{\log ^2(x)}\right ) \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.08, size = 27, normalized size = 1.04 \begin {gather*} -x+x^2-3 \log (1+x)-x \log \left (-x+\frac {400}{\log ^2(x)}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-800 - 800*x + (1600 - 400*x - 800*x^2)*Log[x] + (-5*x + 2*x^3)*Log[x]^3 + ((400 + 400*x)*Log[x] +
(-x - x^2)*Log[x]^3)*Log[(400 - x*Log[x]^2)/Log[x]^2])/((-400 - 400*x)*Log[x] + (x + x^2)*Log[x]^3),x]

[Out]

-x + x^2 - 3*Log[1 + x] - x*Log[-x + 400/Log[x]^2]

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fricas [A]  time = 0.65, size = 31, normalized size = 1.19 \begin {gather*} x^{2} - x \log \left (-\frac {x \log \relax (x)^{2} - 400}{\log \relax (x)^{2}}\right ) - x - 3 \, \log \left (x + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x^2-x)*log(x)^3+(400*x+400)*log(x))*log((-x*log(x)^2+400)/log(x)^2)+(2*x^3-5*x)*log(x)^3+(-800*x
^2-400*x+1600)*log(x)-800*x-800)/((x^2+x)*log(x)^3+(-400*x-400)*log(x)),x, algorithm="fricas")

[Out]

x^2 - x*log(-(x*log(x)^2 - 400)/log(x)^2) - x - 3*log(x + 1)

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giac [A]  time = 0.23, size = 33, normalized size = 1.27 \begin {gather*} x^{2} - x \log \left (-x \log \relax (x)^{2} + 400\right ) + x \log \left (\log \relax (x)^{2}\right ) - x - 3 \, \log \left (x + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x^2-x)*log(x)^3+(400*x+400)*log(x))*log((-x*log(x)^2+400)/log(x)^2)+(2*x^3-5*x)*log(x)^3+(-800*x
^2-400*x+1600)*log(x)-800*x-800)/((x^2+x)*log(x)^3+(-400*x-400)*log(x)),x, algorithm="giac")

[Out]

x^2 - x*log(-x*log(x)^2 + 400) + x*log(log(x)^2) - x - 3*log(x + 1)

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maple [C]  time = 0.15, size = 247, normalized size = 9.50

method	result	size
risch	\(-x \ln \left (x \ln \relax (x )^{2}-400\right )+2 x \ln \left (\ln \relax (x )\right )+i \pi x \mathrm {csgn}\left (\frac {i \left (x \ln \relax (x )^{2}-400\right )}{\ln \relax (x )^{2}}\right )^{2}-\frac {i \pi x \mathrm {csgn}\left (i \ln \relax (x )\right )^{2} \mathrm {csgn}\left (i \ln \relax (x )^{2}\right )}{2}-i \pi x +i \pi x \,\mathrm {csgn}\left (i \ln \relax (x )\right ) \mathrm {csgn}\left (i \ln \relax (x )^{2}\right )^{2}+\frac {i \pi x \,\mathrm {csgn}\left (i \left (x \ln \relax (x )^{2}-400\right )\right ) \mathrm {csgn}\left (\frac {i}{\ln \relax (x )^{2}}\right ) \mathrm {csgn}\left (\frac {i \left (x \ln \relax (x )^{2}-400\right )}{\ln \relax (x )^{2}}\right )}{2}-\frac {i \pi x \,\mathrm {csgn}\left (\frac {i}{\ln \relax (x )^{2}}\right ) \mathrm {csgn}\left (\frac {i \left (x \ln \relax (x )^{2}-400\right )}{\ln \relax (x )^{2}}\right )^{2}}{2}-\frac {i \pi x \mathrm {csgn}\left (\frac {i \left (x \ln \relax (x )^{2}-400\right )}{\ln \relax (x )^{2}}\right )^{3}}{2}-\frac {i \pi x \,\mathrm {csgn}\left (i \left (x \ln \relax (x )^{2}-400\right )\right ) \mathrm {csgn}\left (\frac {i \left (x \ln \relax (x )^{2}-400\right )}{\ln \relax (x )^{2}}\right )^{2}}{2}-\frac {i \pi x \mathrm {csgn}\left (i \ln \relax (x )^{2}\right )^{3}}{2}+x^{2}-x -3 \ln \left (x +1\right )\)	\(247\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-x^2-x)*ln(x)^3+(400*x+400)*ln(x))*ln((-x*ln(x)^2+400)/ln(x)^2)+(2*x^3-5*x)*ln(x)^3+(-800*x^2-400*x+160
0)*ln(x)-800*x-800)/((x^2+x)*ln(x)^3+(-400*x-400)*ln(x)),x,method=_RETURNVERBOSE)

[Out]

-x*ln(x*ln(x)^2-400)+2*x*ln(ln(x))+I*Pi*x*csgn(I/ln(x)^2*(x*ln(x)^2-400))^2-1/2*I*Pi*x*csgn(I*ln(x))^2*csgn(I*
ln(x)^2)-I*Pi*x+I*Pi*x*csgn(I*ln(x))*csgn(I*ln(x)^2)^2+1/2*I*Pi*x*csgn(I*(x*ln(x)^2-400))*csgn(I/ln(x)^2)*csgn
(I/ln(x)^2*(x*ln(x)^2-400))-1/2*I*Pi*x*csgn(I/ln(x)^2)*csgn(I/ln(x)^2*(x*ln(x)^2-400))^2-1/2*I*Pi*x*csgn(I/ln(
x)^2*(x*ln(x)^2-400))^3-1/2*I*Pi*x*csgn(I*(x*ln(x)^2-400))*csgn(I/ln(x)^2*(x*ln(x)^2-400))^2-1/2*I*Pi*x*csgn(I
*ln(x)^2)^3+x^2-x-3*ln(x+1)

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maxima [A]  time = 0.40, size = 32, normalized size = 1.23 \begin {gather*} x^{2} - x \log \left (-x \log \relax (x)^{2} + 400\right ) + 2 \, x \log \left (\log \relax (x)\right ) - x - 3 \, \log \left (x + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x^2-x)*log(x)^3+(400*x+400)*log(x))*log((-x*log(x)^2+400)/log(x)^2)+(2*x^3-5*x)*log(x)^3+(-800*x
^2-400*x+1600)*log(x)-800*x-800)/((x^2+x)*log(x)^3+(-400*x-400)*log(x)),x, algorithm="maxima")

[Out]

x^2 - x*log(-x*log(x)^2 + 400) + 2*x*log(log(x)) - x - 3*log(x + 1)

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mupad [B]  time = 2.35, size = 31, normalized size = 1.19 \begin {gather*} x^2-3\,\ln \left (x+1\right )-x\,\ln \left (-\frac {x\,{\ln \relax (x)}^2-400}{{\ln \relax (x)}^2}\right )-x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(800*x + log(x)^3*(5*x - 2*x^3) + log(-(x*log(x)^2 - 400)/log(x)^2)*(log(x)^3*(x + x^2) - log(x)*(400*x +
400)) + log(x)*(400*x + 800*x^2 - 1600) + 800)/(log(x)^3*(x + x^2) - log(x)*(400*x + 400)),x)

[Out]

x^2 - 3*log(x + 1) - x*log(-(x*log(x)^2 - 400)/log(x)^2) - x

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sympy [A]  time = 0.45, size = 27, normalized size = 1.04 \begin {gather*} x^{2} - x \log {\left (\frac {- x \log {\relax (x )}^{2} + 400}{\log {\relax (x )}^{2}} \right )} - x - 3 \log {\left (x + 1 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x**2-x)*ln(x)**3+(400*x+400)*ln(x))*ln((-x*ln(x)**2+400)/ln(x)**2)+(2*x**3-5*x)*ln(x)**3+(-800*x
**2-400*x+1600)*ln(x)-800*x-800)/((x**2+x)*ln(x)**3+(-400*x-400)*ln(x)),x)

[Out]

x**2 - x*log((-x*log(x)**2 + 400)/log(x)**2) - x - 3*log(x + 1)

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