3.70.57 \(\int \frac {-2 x^2-4 x^3+x^4+(8 x+18 x^2-x^4) \log (4+x)+(-8 x-18 x^2+x^4) \log ^2(4+x)+((-8 x-2 x^2-4 x^3-x^4) \log (4+x)+(-8+6 x-2 x^2+3 x^3+x^4) \log ^2(4+x)) \log (\frac {2 x+(2-2 x) \log (4+x)}{\log (4+x)})}{(-4 x^3-x^4) \log (4+x)+(-4 x^2+3 x^3+x^4) \log ^2(4+x)} \, dx\)

Optimal. Leaf size=29 \[ -4+\frac {(-2+(-4+x) x) \log \left (2 \left (1-x+\frac {x}{\log (4+x)}\right )\right )}{x} \]

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Rubi [F]  time = 6.08, 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^2-4 x^3+x^4+\left (8 x+18 x^2-x^4\right ) \log (4+x)+\left (-8 x-18 x^2+x^4\right ) \log ^2(4+x)+\left (\left (-8 x-2 x^2-4 x^3-x^4\right ) \log (4+x)+\left (-8+6 x-2 x^2+3 x^3+x^4\right ) \log ^2(4+x)\right ) \log \left (\frac {2 x+(2-2 x) \log (4+x)}{\log (4+x)}\right )}{\left (-4 x^3-x^4\right ) \log (4+x)+\left (-4 x^2+3 x^3+x^4\right ) \log ^2(4+x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-2*x^2 - 4*x^3 + x^4 + (8*x + 18*x^2 - x^4)*Log[4 + x] + (-8*x - 18*x^2 + x^4)*Log[4 + x]^2 + ((-8*x - 2*
x^2 - 4*x^3 - x^4)*Log[4 + x] + (-8 + 6*x - 2*x^2 + 3*x^3 + x^4)*Log[4 + x]^2)*Log[(2*x + (2 - 2*x)*Log[4 + x]
)/Log[4 + x]])/((-4*x^3 - x^4)*Log[4 + x] + (-4*x^2 + 3*x^3 + x^4)*Log[4 + x]^2),x]

[Out]

x - 5*Log[1 - x] + 2*Log[x] + (15*Log[Log[4 + x]])/2 - LogIntegral[4 + x] + Defer[Int][1/(x*Log[4 + x]), x]/2
- 8*Defer[Int][(-x - Log[4 + x] + x*Log[4 + x])^(-1), x] - 5*Defer[Int][1/((-1 + x)*(-x - Log[4 + x] + x*Log[4
+ x])), x] + (5*Defer[Int][1/(x*(-x - Log[4 + x] + x*Log[4 + x])), x])/2 + Defer[Int][x/(-x - Log[4 + x] + x*
Log[4 + x]), x] + (75*Defer[Int][1/((4 + x)*(-x - Log[4 + x] + x*Log[4 + x])), x])/2 + Defer[Int][Log[2 - 2*x
+ (2*x)/Log[4 + x]], x] + 2*Defer[Int][Log[2 - 2*x + (2*x)/Log[4 + x]]/x^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 x^2+4 x^3-x^4-x \left (8+18 x-x^3\right ) \log (4+x)-x \left (-8-18 x+x^3\right ) \log ^2(4+x)-\left (8+2 x+4 x^2+x^3\right ) \log (4+x) (-x+(-1+x) \log (4+x)) \log \left (2-2 x+\frac {2 x}{\log (4+x)}\right )}{x^2 (4+x) \log (4+x) (x-(-1+x) \log (4+x))} \, dx\\ &=\int \left (\frac {2+4 x-x^2}{x (-x-\log (4+x)+x \log (4+x))}-\frac {2}{(4+x) \log (4+x) (-x-\log (4+x)+x \log (4+x))}-\frac {4 x}{(4+x) \log (4+x) (-x-\log (4+x)+x \log (4+x))}+\frac {x^2}{(4+x) \log (4+x) (-x-\log (4+x)+x \log (4+x))}+\frac {\left (-2-4 x+x^2\right ) \log (4+x)}{x (-x-\log (4+x)+x \log (4+x))}+\frac {\left (2+x^2\right ) \log \left (2-2 x+\frac {2 x}{\log (4+x)}\right )}{x^2}\right ) \, dx\\ &=-\left (2 \int \frac {1}{(4+x) \log (4+x) (-x-\log (4+x)+x \log (4+x))} \, dx\right )-4 \int \frac {x}{(4+x) \log (4+x) (-x-\log (4+x)+x \log (4+x))} \, dx+\int \frac {2+4 x-x^2}{x (-x-\log (4+x)+x \log (4+x))} \, dx+\int \frac {x^2}{(4+x) \log (4+x) (-x-\log (4+x)+x \log (4+x))} \, dx+\int \frac {\left (-2-4 x+x^2\right ) \log (4+x)}{x (-x-\log (4+x)+x \log (4+x))} \, dx+\int \frac {\left (2+x^2\right ) \log \left (2-2 x+\frac {2 x}{\log (4+x)}\right )}{x^2} \, dx\\ &=-\left (2 \int \left (-\frac {1}{x (4+x) \log (4+x)}+\frac {-1+x}{x (4+x) (-x-\log (4+x)+x \log (4+x))}\right ) \, dx\right )-4 \int \left (-\frac {1}{(4+x) \log (4+x)}+\frac {-1+x}{(4+x) (-x-\log (4+x)+x \log (4+x))}\right ) \, dx+\int \left (\frac {4}{-x-\log (4+x)+x \log (4+x)}+\frac {2}{x (-x-\log (4+x)+x \log (4+x))}-\frac {x}{-x-\log (4+x)+x \log (4+x)}\right ) \, dx+\int \left (-\frac {x}{(4+x) \log (4+x)}+\frac {(-1+x) x}{(4+x) (-x-\log (4+x)+x \log (4+x))}\right ) \, dx+\int \left (\frac {-2-4 x+x^2}{(-1+x) x}+\frac {-2-4 x+x^2}{(-1+x) (-x-\log (4+x)+x \log (4+x))}\right ) \, dx+\int \left (\log \left (2-2 x+\frac {2 x}{\log (4+x)}\right )+\frac {2 \log \left (2-2 x+\frac {2 x}{\log (4+x)}\right )}{x^2}\right ) \, dx\\ &=2 \int \frac {1}{x (4+x) \log (4+x)} \, dx+2 \int \frac {1}{x (-x-\log (4+x)+x \log (4+x))} \, dx-2 \int \frac {-1+x}{x (4+x) (-x-\log (4+x)+x \log (4+x))} \, dx+2 \int \frac {\log \left (2-2 x+\frac {2 x}{\log (4+x)}\right )}{x^2} \, dx+4 \int \frac {1}{(4+x) \log (4+x)} \, dx+4 \int \frac {1}{-x-\log (4+x)+x \log (4+x)} \, dx-4 \int \frac {-1+x}{(4+x) (-x-\log (4+x)+x \log (4+x))} \, dx+\int \frac {-2-4 x+x^2}{(-1+x) x} \, dx-\int \frac {x}{(4+x) \log (4+x)} \, dx-\int \frac {x}{-x-\log (4+x)+x \log (4+x)} \, dx+\int \frac {(-1+x) x}{(4+x) (-x-\log (4+x)+x \log (4+x))} \, dx+\int \frac {-2-4 x+x^2}{(-1+x) (-x-\log (4+x)+x \log (4+x))} \, dx+\int \log \left (2-2 x+\frac {2 x}{\log (4+x)}\right ) \, dx\\ &=2 \int \left (\frac {1}{4 x \log (4+x)}-\frac {1}{4 (4+x) \log (4+x)}\right ) \, dx+2 \int \frac {1}{x (-x-\log (4+x)+x \log (4+x))} \, dx-2 \int \left (-\frac {1}{4 x (-x-\log (4+x)+x \log (4+x))}+\frac {5}{4 (4+x) (-x-\log (4+x)+x \log (4+x))}\right ) \, dx+2 \int \frac {\log \left (2-2 x+\frac {2 x}{\log (4+x)}\right )}{x^2} \, dx+4 \int \frac {1}{-x-\log (4+x)+x \log (4+x)} \, dx-4 \int \left (\frac {1}{-x-\log (4+x)+x \log (4+x)}-\frac {5}{(4+x) (-x-\log (4+x)+x \log (4+x))}\right ) \, dx+4 \operatorname {Subst}\left (\int \frac {1}{x \log (x)} \, dx,x,4+x\right )+\int \left (1-\frac {5}{-1+x}+\frac {2}{x}\right ) \, dx-\int \frac {x}{-x-\log (4+x)+x \log (4+x)} \, dx+\int \left (-\frac {3}{-x-\log (4+x)+x \log (4+x)}-\frac {5}{(-1+x) (-x-\log (4+x)+x \log (4+x))}+\frac {x}{-x-\log (4+x)+x \log (4+x)}\right ) \, dx+\int \left (-\frac {5}{-x-\log (4+x)+x \log (4+x)}+\frac {x}{-x-\log (4+x)+x \log (4+x)}+\frac {20}{(4+x) (-x-\log (4+x)+x \log (4+x))}\right ) \, dx+\int \log \left (2-2 x+\frac {2 x}{\log (4+x)}\right ) \, dx-\operatorname {Subst}\left (\int \frac {-4+x}{x \log (x)} \, dx,x,4+x\right )\\ &=x-5 \log (1-x)+2 \log (x)+\frac {1}{2} \int \frac {1}{x \log (4+x)} \, dx-\frac {1}{2} \int \frac {1}{(4+x) \log (4+x)} \, dx+\frac {1}{2} \int \frac {1}{x (-x-\log (4+x)+x \log (4+x))} \, dx+2 \int \frac {1}{x (-x-\log (4+x)+x \log (4+x))} \, dx+2 \int \frac {\log \left (2-2 x+\frac {2 x}{\log (4+x)}\right )}{x^2} \, dx-\frac {5}{2} \int \frac {1}{(4+x) (-x-\log (4+x)+x \log (4+x))} \, dx-3 \int \frac {1}{-x-\log (4+x)+x \log (4+x)} \, dx+4 \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,\log (4+x)\right )-5 \int \frac {1}{-x-\log (4+x)+x \log (4+x)} \, dx-5 \int \frac {1}{(-1+x) (-x-\log (4+x)+x \log (4+x))} \, dx+2 \left (20 \int \frac {1}{(4+x) (-x-\log (4+x)+x \log (4+x))} \, dx\right )+\int \frac {x}{-x-\log (4+x)+x \log (4+x)} \, dx+\int \log \left (2-2 x+\frac {2 x}{\log (4+x)}\right ) \, dx-\operatorname {Subst}\left (\int \left (\frac {1}{\log (x)}-\frac {4}{x \log (x)}\right ) \, dx,x,4+x\right )\\ &=x-5 \log (1-x)+2 \log (x)+4 \log (\log (4+x))+\frac {1}{2} \int \frac {1}{x \log (4+x)} \, dx+\frac {1}{2} \int \frac {1}{x (-x-\log (4+x)+x \log (4+x))} \, dx-\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x \log (x)} \, dx,x,4+x\right )+2 \int \frac {1}{x (-x-\log (4+x)+x \log (4+x))} \, dx+2 \int \frac {\log \left (2-2 x+\frac {2 x}{\log (4+x)}\right )}{x^2} \, dx-\frac {5}{2} \int \frac {1}{(4+x) (-x-\log (4+x)+x \log (4+x))} \, dx-3 \int \frac {1}{-x-\log (4+x)+x \log (4+x)} \, dx+4 \operatorname {Subst}\left (\int \frac {1}{x \log (x)} \, dx,x,4+x\right )-5 \int \frac {1}{-x-\log (4+x)+x \log (4+x)} \, dx-5 \int \frac {1}{(-1+x) (-x-\log (4+x)+x \log (4+x))} \, dx+2 \left (20 \int \frac {1}{(4+x) (-x-\log (4+x)+x \log (4+x))} \, dx\right )+\int \frac {x}{-x-\log (4+x)+x \log (4+x)} \, dx+\int \log \left (2-2 x+\frac {2 x}{\log (4+x)}\right ) \, dx-\operatorname {Subst}\left (\int \frac {1}{\log (x)} \, dx,x,4+x\right )\\ &=x-5 \log (1-x)+2 \log (x)+4 \log (\log (4+x))-\text {li}(4+x)+\frac {1}{2} \int \frac {1}{x \log (4+x)} \, dx+\frac {1}{2} \int \frac {1}{x (-x-\log (4+x)+x \log (4+x))} \, dx-\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,\log (4+x)\right )+2 \int \frac {1}{x (-x-\log (4+x)+x \log (4+x))} \, dx+2 \int \frac {\log \left (2-2 x+\frac {2 x}{\log (4+x)}\right )}{x^2} \, dx-\frac {5}{2} \int \frac {1}{(4+x) (-x-\log (4+x)+x \log (4+x))} \, dx-3 \int \frac {1}{-x-\log (4+x)+x \log (4+x)} \, dx+4 \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,\log (4+x)\right )-5 \int \frac {1}{-x-\log (4+x)+x \log (4+x)} \, dx-5 \int \frac {1}{(-1+x) (-x-\log (4+x)+x \log (4+x))} \, dx+2 \left (20 \int \frac {1}{(4+x) (-x-\log (4+x)+x \log (4+x))} \, dx\right )+\int \frac {x}{-x-\log (4+x)+x \log (4+x)} \, dx+\int \log \left (2-2 x+\frac {2 x}{\log (4+x)}\right ) \, dx\\ &=x-5 \log (1-x)+2 \log (x)+\frac {15}{2} \log (\log (4+x))-\text {li}(4+x)+\frac {1}{2} \int \frac {1}{x \log (4+x)} \, dx+\frac {1}{2} \int \frac {1}{x (-x-\log (4+x)+x \log (4+x))} \, dx+2 \int \frac {1}{x (-x-\log (4+x)+x \log (4+x))} \, dx+2 \int \frac {\log \left (2-2 x+\frac {2 x}{\log (4+x)}\right )}{x^2} \, dx-\frac {5}{2} \int \frac {1}{(4+x) (-x-\log (4+x)+x \log (4+x))} \, dx-3 \int \frac {1}{-x-\log (4+x)+x \log (4+x)} \, dx-5 \int \frac {1}{-x-\log (4+x)+x \log (4+x)} \, dx-5 \int \frac {1}{(-1+x) (-x-\log (4+x)+x \log (4+x))} \, dx+2 \left (20 \int \frac {1}{(4+x) (-x-\log (4+x)+x \log (4+x))} \, dx\right )+\int \frac {x}{-x-\log (4+x)+x \log (4+x)} \, dx+\int \log \left (2-2 x+\frac {2 x}{\log (4+x)}\right ) \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.14, size = 47, normalized size = 1.62 \begin {gather*} \left (-\frac {2}{x}+x\right ) \log \left (2-2 x+\frac {2 x}{\log (4+x)}\right )+4 (\log (\log (4+x))-\log (-x+(-1+x) \log (4+x))) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-2*x^2 - 4*x^3 + x^4 + (8*x + 18*x^2 - x^4)*Log[4 + x] + (-8*x - 18*x^2 + x^4)*Log[4 + x]^2 + ((-8*
x - 2*x^2 - 4*x^3 - x^4)*Log[4 + x] + (-8 + 6*x - 2*x^2 + 3*x^3 + x^4)*Log[4 + x]^2)*Log[(2*x + (2 - 2*x)*Log[
4 + x])/Log[4 + x]])/((-4*x^3 - x^4)*Log[4 + x] + (-4*x^2 + 3*x^3 + x^4)*Log[4 + x]^2),x]

[Out]

(-2/x + x)*Log[2 - 2*x + (2*x)/Log[4 + x]] + 4*(Log[Log[4 + x]] - Log[-x + (-1 + x)*Log[4 + x]])

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fricas [A]  time = 0.79, size = 33, normalized size = 1.14 \begin {gather*} \frac {{\left (x^{2} - 4 \, x - 2\right )} \log \left (-\frac {2 \, {\left ({\left (x - 1\right )} \log \left (x + 4\right ) - x\right )}}{\log \left (x + 4\right )}\right )}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^4+3*x^3-2*x^2+6*x-8)*log(4+x)^2+(-x^4-4*x^3-2*x^2-8*x)*log(4+x))*log(((-2*x+2)*log(4+x)+2*x)/lo
g(4+x))+(x^4-18*x^2-8*x)*log(4+x)^2+(-x^4+18*x^2+8*x)*log(4+x)+x^4-4*x^3-2*x^2)/((x^4+3*x^3-4*x^2)*log(4+x)^2+
(-x^4-4*x^3)*log(4+x)),x, algorithm="fricas")

[Out]

(x^2 - 4*x - 2)*log(-2*((x - 1)*log(x + 4) - x)/log(x + 4))/x

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giac [B]  time = 0.33, size = 67, normalized size = 2.31 \begin {gather*} {\left (x - \frac {2}{x}\right )} \log \left (-2 \, x \log \left (x + 4\right ) + 2 \, x + 2 \, \log \left (x + 4\right )\right ) - {\left (x - \frac {2}{x}\right )} \log \left (\log \left (x + 4\right )\right ) - 4 \, \log \left (x \log \left (x + 4\right ) - x - \log \left (x + 4\right )\right ) + 4 \, \log \left (\log \left (x + 4\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^4+3*x^3-2*x^2+6*x-8)*log(4+x)^2+(-x^4-4*x^3-2*x^2-8*x)*log(4+x))*log(((-2*x+2)*log(4+x)+2*x)/lo
g(4+x))+(x^4-18*x^2-8*x)*log(4+x)^2+(-x^4+18*x^2+8*x)*log(4+x)+x^4-4*x^3-2*x^2)/((x^4+3*x^3-4*x^2)*log(4+x)^2+
(-x^4-4*x^3)*log(4+x)),x, algorithm="giac")

[Out]

(x - 2/x)*log(-2*x*log(x + 4) + 2*x + 2*log(x + 4)) - (x - 2/x)*log(log(x + 4)) - 4*log(x*log(x + 4) - x - log
(x + 4)) + 4*log(log(x + 4))

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maple [C]  time = 0.29, size = 541, normalized size = 18.66

method	result	size
risch	\(\frac {\left (x^{2}-2\right ) \ln \left (\left (\ln \left (4+x \right )-1\right ) x -\ln \left (4+x \right )\right )}{x}+\frac {-4 \ln \relax (2)-2 i \pi \mathrm {csgn}\left (\frac {i \left (\left (\ln \left (4+x \right )-1\right ) x -\ln \left (4+x \right )\right )}{\ln \left (4+x \right )}\right )^{3}+2 x^{2} \ln \relax (2)-2 i \pi \,x^{2} \mathrm {csgn}\left (\frac {i \left (\left (\ln \left (4+x \right )-1\right ) x -\ln \left (4+x \right )\right )}{\ln \left (4+x \right )}\right )^{2}-2 i \pi \,\mathrm {csgn}\left (\frac {i}{\ln \left (4+x \right )}\right ) \mathrm {csgn}\left (\frac {i \left (\left (\ln \left (4+x \right )-1\right ) x -\ln \left (4+x \right )\right )}{\ln \left (4+x \right )}\right )^{2}+2 i \pi \,x^{2}+2 i \pi \,\mathrm {csgn}\left (i \left (\left (\ln \left (4+x \right )-1\right ) x -\ln \left (4+x \right )\right )\right ) \mathrm {csgn}\left (\frac {i}{\ln \left (4+x \right )}\right ) \mathrm {csgn}\left (\frac {i \left (\left (\ln \left (4+x \right )-1\right ) x -\ln \left (4+x \right )\right )}{\ln \left (4+x \right )}\right )+i \pi \,x^{2} \mathrm {csgn}\left (\frac {i \left (\left (\ln \left (4+x \right )-1\right ) x -\ln \left (4+x \right )\right )}{\ln \left (4+x \right )}\right )^{3}-8 \ln \left (x -1\right ) x +4 \ln \left (\ln \left (4+x \right )\right )+8 \ln \left (\ln \left (4+x \right )\right ) x -8 \ln \left (\ln \left (4+x \right )-\frac {x}{x -1}\right ) x -2 x^{2} \ln \left (\ln \left (4+x \right )\right )-2 i \pi \,\mathrm {csgn}\left (i \left (\left (\ln \left (4+x \right )-1\right ) x -\ln \left (4+x \right )\right )\right ) \mathrm {csgn}\left (\frac {i \left (\left (\ln \left (4+x \right )-1\right ) x -\ln \left (4+x \right )\right )}{\ln \left (4+x \right )}\right )^{2}+4 i \pi \mathrm {csgn}\left (\frac {i \left (\left (\ln \left (4+x \right )-1\right ) x -\ln \left (4+x \right )\right )}{\ln \left (4+x \right )}\right )^{2}-i \pi \,x^{2} \mathrm {csgn}\left (i \left (\left (\ln \left (4+x \right )-1\right ) x -\ln \left (4+x \right )\right )\right ) \mathrm {csgn}\left (\frac {i}{\ln \left (4+x \right )}\right ) \mathrm {csgn}\left (\frac {i \left (\left (\ln \left (4+x \right )-1\right ) x -\ln \left (4+x \right )\right )}{\ln \left (4+x \right )}\right )+i \pi \,x^{2} \mathrm {csgn}\left (i \left (\left (\ln \left (4+x \right )-1\right ) x -\ln \left (4+x \right )\right )\right ) \mathrm {csgn}\left (\frac {i \left (\left (\ln \left (4+x \right )-1\right ) x -\ln \left (4+x \right )\right )}{\ln \left (4+x \right )}\right )^{2}+i \pi \,x^{2} \mathrm {csgn}\left (\frac {i}{\ln \left (4+x \right )}\right ) \mathrm {csgn}\left (\frac {i \left (\left (\ln \left (4+x \right )-1\right ) x -\ln \left (4+x \right )\right )}{\ln \left (4+x \right )}\right )^{2}-4 i \pi }{2 x}\)	\(541\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((x^4+3*x^3-2*x^2+6*x-8)*ln(4+x)^2+(-x^4-4*x^3-2*x^2-8*x)*ln(4+x))*ln(((-2*x+2)*ln(4+x)+2*x)/ln(4+x))+(x^
4-18*x^2-8*x)*ln(4+x)^2+(-x^4+18*x^2+8*x)*ln(4+x)+x^4-4*x^3-2*x^2)/((x^4+3*x^3-4*x^2)*ln(4+x)^2+(-x^4-4*x^3)*l
n(4+x)),x,method=_RETURNVERBOSE)

[Out]

(x^2-2)/x*ln((ln(4+x)-1)*x-ln(4+x))+1/2*(-4*ln(2)-2*I*Pi*csgn(I/ln(4+x)*((ln(4+x)-1)*x-ln(4+x)))^3+2*x^2*ln(2)
-2*I*Pi*x^2*csgn(I/ln(4+x)*((ln(4+x)-1)*x-ln(4+x)))^2-2*I*Pi*csgn(I/ln(4+x))*csgn(I/ln(4+x)*((ln(4+x)-1)*x-ln(
4+x)))^2+2*I*Pi*x^2+2*I*Pi*csgn(I*((ln(4+x)-1)*x-ln(4+x)))*csgn(I/ln(4+x))*csgn(I/ln(4+x)*((ln(4+x)-1)*x-ln(4+
x)))+I*Pi*x^2*csgn(I/ln(4+x)*((ln(4+x)-1)*x-ln(4+x)))^3-8*ln(x-1)*x+4*ln(ln(4+x))+8*ln(ln(4+x))*x-8*ln(ln(4+x)
-x/(x-1))*x-2*x^2*ln(ln(4+x))-2*I*Pi*csgn(I*((ln(4+x)-1)*x-ln(4+x)))*csgn(I/ln(4+x)*((ln(4+x)-1)*x-ln(4+x)))^2
+4*I*Pi*csgn(I/ln(4+x)*((ln(4+x)-1)*x-ln(4+x)))^2-I*Pi*x^2*csgn(I*((ln(4+x)-1)*x-ln(4+x)))*csgn(I/ln(4+x))*csg
n(I/ln(4+x)*((ln(4+x)-1)*x-ln(4+x)))+I*Pi*x^2*csgn(I*((ln(4+x)-1)*x-ln(4+x)))*csgn(I/ln(4+x)*((ln(4+x)-1)*x-ln
(4+x)))^2+I*Pi*x^2*csgn(I/ln(4+x))*csgn(I/ln(4+x)*((ln(4+x)-1)*x-ln(4+x)))^2-4*I*Pi)/x

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maxima [C]  time = 0.67, size = 87, normalized size = 3.00 \begin {gather*} \frac {-2 i \, \pi + {\left (i \, \pi + \log \relax (2)\right )} x^{2} + {\left (x^{2} - 2\right )} \log \left (x {\left (\log \left (x + 4\right ) - 1\right )} - \log \left (x + 4\right )\right ) - {\left (x^{2} - 4 \, x - 2\right )} \log \left (\log \left (x + 4\right )\right ) - 2 \, \log \relax (2)}{x} - 4 \, \log \left (x - 1\right ) - 4 \, \log \left (\frac {{\left (x - 1\right )} \log \left (x + 4\right ) - x}{x - 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^4+3*x^3-2*x^2+6*x-8)*log(4+x)^2+(-x^4-4*x^3-2*x^2-8*x)*log(4+x))*log(((-2*x+2)*log(4+x)+2*x)/lo
g(4+x))+(x^4-18*x^2-8*x)*log(4+x)^2+(-x^4+18*x^2+8*x)*log(4+x)+x^4-4*x^3-2*x^2)/((x^4+3*x^3-4*x^2)*log(4+x)^2+
(-x^4-4*x^3)*log(4+x)),x, algorithm="maxima")

[Out]

(-2*I*pi + (I*pi + log(2))*x^2 + (x^2 - 2)*log(x*(log(x + 4) - 1) - log(x + 4)) - (x^2 - 4*x - 2)*log(log(x +
4)) - 2*log(2))/x - 4*log(x - 1) - 4*log(((x - 1)*log(x + 4) - x)/(x - 1))

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mupad [B]  time = 4.73, size = 132, normalized size = 4.55 \begin {gather*} 4\,\ln \left (x-1\right )-4\,\ln \left (\frac {8\,x+8\,\ln \left (x+4\right )-8\,x\,\ln \left (x+4\right )}{x^2+3\,x-4}\right )-4\,\ln \left (x^2-x+5\right )+4\,\ln \left (\frac {\ln \left (x+4\right )\,\left (x^2-x+5\right )}{{\left (x-1\right )}^2\,\left (x+4\right )}\right )+x\,\ln \left (\frac {2\,x+2\,\ln \left (x+4\right )-2\,x\,\ln \left (x+4\right )}{\ln \left (x+4\right )}\right )-\frac {2\,\ln \left (\frac {2\,x+2\,\ln \left (x+4\right )-2\,x\,\ln \left (x+4\right )}{\ln \left (x+4\right )}\right )}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x + 4)^2*(8*x + 18*x^2 - x^4) - log(x + 4)*(8*x + 18*x^2 - x^4) + 2*x^2 + 4*x^3 - x^4 + log((2*x - lo
g(x + 4)*(2*x - 2))/log(x + 4))*(log(x + 4)*(8*x + 2*x^2 + 4*x^3 + x^4) - log(x + 4)^2*(6*x - 2*x^2 + 3*x^3 +
x^4 - 8)))/(log(x + 4)*(4*x^3 + x^4) - log(x + 4)^2*(3*x^3 - 4*x^2 + x^4)),x)

[Out]

4*log(x - 1) - 4*log((8*x + 8*log(x + 4) - 8*x*log(x + 4))/(3*x + x^2 - 4)) - 4*log(x^2 - x + 5) + 4*log((log(
x + 4)*(x^2 - x + 5))/((x - 1)^2*(x + 4))) + x*log((2*x + 2*log(x + 4) - 2*x*log(x + 4))/log(x + 4)) - (2*log(
(2*x + 2*log(x + 4) - 2*x*log(x + 4))/log(x + 4)))/x

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sympy [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: PolynomialError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x**4+3*x**3-2*x**2+6*x-8)*ln(4+x)**2+(-x**4-4*x**3-2*x**2-8*x)*ln(4+x))*ln(((-2*x+2)*ln(4+x)+2*x)
/ln(4+x))+(x**4-18*x**2-8*x)*ln(4+x)**2+(-x**4+18*x**2+8*x)*ln(4+x)+x**4-4*x**3-2*x**2)/((x**4+3*x**3-4*x**2)*
ln(4+x)**2+(-x**4-4*x**3)*ln(4+x)),x)

[Out]

Exception raised: PolynomialError

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