3.48.41 \(\int \frac {-2 x-2 x^2+e^x (2+2 x)+e^x (-2-2 x) \log (\frac {1+x}{5})+\log (\frac {x^2}{3}) (-2 x-2 x^2+e^x (1+x+x^2)+e^x (-1-2 x-x^2) \log (\frac {1+x}{5})) \log (\log (\frac {x^2}{3}))}{(1+x) \log (\frac {x^2}{3})} \, dx\)

Optimal. Leaf size=31 \[ x \left (e^x-x-e^x \log \left (\frac {1+x}{5}\right )\right ) \log \left (\log \left (\frac {x^2}{3}\right )\right ) \]

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Rubi [F]  time = 22.33, 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 x^2+e^x (2+2 x)+e^x (-2-2 x) \log \left (\frac {1+x}{5}\right )+\log \left (\frac {x^2}{3}\right ) \left (-2 x-2 x^2+e^x \left (1+x+x^2\right )+e^x \left (-1-2 x-x^2\right ) \log \left (\frac {1+x}{5}\right )\right ) \log \left (\log \left (\frac {x^2}{3}\right )\right )}{(1+x) \log \left (\frac {x^2}{3}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

-(x^2*Log[Log[x^2/3]]) - 2*Defer[Int][(E^x*Log[1/5 + x/5])/Log[x^2/3], x] - 2*Defer[Int][E^x/(Log[3] - Log[x^2
]), x] + Defer[Int][E^x*x*Log[Log[x^2/3]], x] + Defer[Int][(E^x*Log[Log[x^2/3]])/(1 + x), x] - Defer[Int][E^x*
Log[1/5 + x/5]*Log[Log[x^2/3]], x] - 2*Defer[Int][(E^x*Log[1/5 + x/5]*Log[Log[x^2/3]])/(-1 - x), x] - Defer[In
t][E^x*x*Log[1/5 + x/5]*Log[Log[x^2/3]], x] - 2*Defer[Int][(E^x*Log[1/5 + x/5]*Log[Log[x^2/3]])/(1 + x), x]

Rubi steps

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

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Mathematica [A]  time = 0.12, size = 31, normalized size = 1.00 \begin {gather*} -x \left (-e^x+x+e^x \log \left (\frac {1+x}{5}\right )\right ) \log \left (\log \left (\frac {x^2}{3}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.75, size = 28, normalized size = 0.90 \begin {gather*} -{\left (x e^{x} \log \left (\frac {1}{5} \, x + \frac {1}{5}\right ) + x^{2} - x e^{x}\right )} \log \left (\log \left (\frac {1}{3} \, x^{2}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 1.17, size = 53, normalized size = 1.71 \begin {gather*} x e^{x} \log \relax (5) \log \left (\log \left (\frac {1}{3} \, x^{2}\right )\right ) - x e^{x} \log \left (x + 1\right ) \log \left (\log \left (\frac {1}{3} \, x^{2}\right )\right ) - x^{2} \log \left (\log \left (\frac {1}{3} \, x^{2}\right )\right ) + x e^{x} \log \left (\log \left (\frac {1}{3} \, x^{2}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 0.90, size = 61, normalized size = 1.97

method	result	size
risch	\(\left (-{\mathrm e}^{x} \ln \left (\frac {x}{5}+\frac {1}{5}\right ) x -x^{2}+{\mathrm e}^{x} x \right ) \ln \left (-\ln \relax (3)+2 \ln \relax (x )-\frac {i \pi \,\mathrm {csgn}\left (i x^{2}\right ) \left (-\mathrm {csgn}\left (i x^{2}\right )+\mathrm {csgn}\left (i x \right )\right )^{2}}{2}\right )\)	\(61\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.48, size = 34, normalized size = 1.10 \begin {gather*} {\left (x {\left (\log \relax (5) + 1\right )} e^{x} - x e^{x} \log \left (x + 1\right ) - x^{2}\right )} \log \left (-\log \relax (3) + 2 \, \log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 3.66, size = 85, normalized size = 2.74 \begin {gather*} -\ln \left (\ln \left (\frac {x^2}{3}\right )\right )\,\left (\frac {x^4+x^3}{x\,\left (x+1\right )}+\frac {{\mathrm {e}}^x\,\left (x-x^3\right )}{x\,\left (x+1\right )}-\frac {{\mathrm {e}}^x\,\left (x^2+x\right )}{x\,\left (x+1\right )}+\frac {{\mathrm {e}}^x\,\ln \left (\frac {x}{5}+\frac {1}{5}\right )\,\left (x^3+x^2\right )}{x\,\left (x+1\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(2*x - exp(x)*(2*x + 2) + 2*x^2 + exp(x)*log(x/5 + 1/5)*(2*x + 2) + log(log(x^2/3))*log(x^2/3)*(2*x + 2*x
^2 - exp(x)*(x + x^2 + 1) + exp(x)*log(x/5 + 1/5)*(2*x + x^2 + 1)))/(log(x^2/3)*(x + 1)),x)

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x**2-2*x-1)*exp(x)*ln(1/5*x+1/5)+(x**2+x+1)*exp(x)-2*x**2-2*x)*ln(1/3*x**2)*ln(ln(1/3*x**2))+(-2
*x-2)*exp(x)*ln(1/5*x+1/5)+(2*x+2)*exp(x)-2*x**2-2*x)/(x+1)/ln(1/3*x**2),x)

[Out]

Timed out

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