3.40.48 \(\int \frac {-4-4 x+2 x^2+5 x^3+2 x^4+(7 x^2+6 x^3) \log (\frac {3 e^{x+x^2}}{4})+(-2+3 x+6 x^2) \log ^2(\frac {3 e^{x+x^2}}{4})+(1+2 x) \log ^3(\frac {3 e^{x+x^2}}{4})}{x^3+3 x^2 \log (\frac {3 e^{x+x^2}}{4})+3 x \log ^2(\frac {3 e^{x+x^2}}{4})+\log ^3(\frac {3 e^{x+x^2}}{4})} \, dx\)

Optimal. Leaf size=25 \[ 1+x+\left (-x+\frac {1}{x+\log \left (\frac {3 e^{x+x^2}}{4}\right )}\right )^2 \]

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Rubi [F]  time = 0.48, 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 {-4-4 x+2 x^2+5 x^3+2 x^4+\left (7 x^2+6 x^3\right ) \log \left (\frac {3 e^{x+x^2}}{4}\right )+\left (-2+3 x+6 x^2\right ) \log ^2\left (\frac {3 e^{x+x^2}}{4}\right )+(1+2 x) \log ^3\left (\frac {3 e^{x+x^2}}{4}\right )}{x^3+3 x^2 \log \left (\frac {3 e^{x+x^2}}{4}\right )+3 x \log ^2\left (\frac {3 e^{x+x^2}}{4}\right )+\log ^3\left (\frac {3 e^{x+x^2}}{4}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-4 - 4*x + 2*x^2 + 5*x^3 + 2*x^4 + (7*x^2 + 6*x^3)*Log[(3*E^(x + x^2))/4] + (-2 + 3*x + 6*x^2)*Log[(3*E^(
x + x^2))/4]^2 + (1 + 2*x)*Log[(3*E^(x + x^2))/4]^3)/(x^3 + 3*x^2*Log[(3*E^(x + x^2))/4] + 3*x*Log[(3*E^(x + x
^2))/4]^2 + Log[(3*E^(x + x^2))/4]^3),x]

[Out]

x + x^2 + (x + Log[(3*E^(x + x^2))/4])^(-2) + 4*Defer[Int][x/(x + Log[(3*E^(x + x^2))/4])^2, x] + 4*Defer[Int]
[x^2/(x + Log[(3*E^(x + x^2))/4])^2, x] - 2*Defer[Int][(x + Log[(3*E^(x + x^2))/4])^(-1), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-4-4 x+2 x^2+5 x^3+2 x^4+x^2 (7+6 x) \log \left (\frac {3 e^{x+x^2}}{4}\right )+\left (-2+3 x+6 x^2\right ) \log ^2\left (\frac {3 e^{x+x^2}}{4}\right )+(1+2 x) \log ^3\left (\frac {3 e^{x+x^2}}{4}\right )}{\left (x+\log \left (\frac {3 e^{x+x^2}}{4}\right )\right )^3} \, dx\\ &=\int \left (1+2 x-\frac {4 (1+x)}{\left (x+\log \left (\frac {3 e^{x+x^2}}{4}\right )\right )^3}+\frac {4 x (1+x)}{\left (x+\log \left (\frac {3 e^{x+x^2}}{4}\right )\right )^2}-\frac {2}{x+\log \left (\frac {3 e^{x+x^2}}{4}\right )}\right ) \, dx\\ &=x+x^2-2 \int \frac {1}{x+\log \left (\frac {3 e^{x+x^2}}{4}\right )} \, dx-4 \int \frac {1+x}{\left (x+\log \left (\frac {3 e^{x+x^2}}{4}\right )\right )^3} \, dx+4 \int \frac {x (1+x)}{\left (x+\log \left (\frac {3 e^{x+x^2}}{4}\right )\right )^2} \, dx\\ &=x+x^2+\frac {1}{\left (x+\log \left (\frac {3 e^{x+x^2}}{4}\right )\right )^2}-2 \int \frac {1}{x+\log \left (\frac {3 e^{x+x^2}}{4}\right )} \, dx+4 \int \left (\frac {x}{\left (x+\log \left (\frac {3 e^{x+x^2}}{4}\right )\right )^2}+\frac {x^2}{\left (x+\log \left (\frac {3 e^{x+x^2}}{4}\right )\right )^2}\right ) \, dx\\ &=x+x^2+\frac {1}{\left (x+\log \left (\frac {3 e^{x+x^2}}{4}\right )\right )^2}-2 \int \frac {1}{x+\log \left (\frac {3 e^{x+x^2}}{4}\right )} \, dx+4 \int \frac {x}{\left (x+\log \left (\frac {3 e^{x+x^2}}{4}\right )\right )^2} \, dx+4 \int \frac {x^2}{\left (x+\log \left (\frac {3 e^{x+x^2}}{4}\right )\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.04, size = 40, normalized size = 1.60 \begin {gather*} x+x^2+\frac {1}{\left (x+\log \left (\frac {3 e^{x+x^2}}{4}\right )\right )^2}-\frac {2 x}{x+\log \left (\frac {3 e^{x+x^2}}{4}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-4 - 4*x + 2*x^2 + 5*x^3 + 2*x^4 + (7*x^2 + 6*x^3)*Log[(3*E^(x + x^2))/4] + (-2 + 3*x + 6*x^2)*Log[
(3*E^(x + x^2))/4]^2 + (1 + 2*x)*Log[(3*E^(x + x^2))/4]^3)/(x^3 + 3*x^2*Log[(3*E^(x + x^2))/4] + 3*x*Log[(3*E^
(x + x^2))/4]^2 + Log[(3*E^(x + x^2))/4]^3),x]

[Out]

x + x^2 + (x + Log[(3*E^(x + x^2))/4])^(-2) - (2*x)/(x + Log[(3*E^(x + x^2))/4])

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fricas [B]  time = 0.82, size = 88, normalized size = 3.52 \begin {gather*} \frac {x^{6} + 5 \, x^{5} + 8 \, x^{4} + 2 \, x^{3} + {\left (x^{2} + x\right )} \log \left (\frac {3}{4}\right )^{2} - 4 \, x^{2} + 2 \, {\left (x^{4} + 3 \, x^{3} + 2 \, x^{2} - x\right )} \log \left (\frac {3}{4}\right ) + 1}{x^{4} + 4 \, x^{3} + 4 \, x^{2} + 2 \, {\left (x^{2} + 2 \, x\right )} \log \left (\frac {3}{4}\right ) + \log \left (\frac {3}{4}\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x+1)*log(3/4*exp(x)*exp(x^2))^3+(6*x^2+3*x-2)*log(3/4*exp(x)*exp(x^2))^2+(6*x^3+7*x^2)*log(3/4*e
xp(x)*exp(x^2))+2*x^4+5*x^3+2*x^2-4*x-4)/(log(3/4*exp(x)*exp(x^2))^3+3*x*log(3/4*exp(x)*exp(x^2))^2+3*x^2*log(
3/4*exp(x)*exp(x^2))+x^3),x, algorithm="fricas")

[Out]

(x^6 + 5*x^5 + 8*x^4 + 2*x^3 + (x^2 + x)*log(3/4)^2 - 4*x^2 + 2*(x^4 + 3*x^3 + 2*x^2 - x)*log(3/4) + 1)/(x^4 +
4*x^3 + 4*x^2 + 2*(x^2 + 2*x)*log(3/4) + log(3/4)^2)

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giac [A]  time = 0.14, size = 44, normalized size = 1.76 \begin {gather*} x^{2} + x - \frac {2 \, x^{3} + 4 \, x^{2} + 2 \, x \log \relax (3) - 4 \, x \log \relax (2) - 1}{{\left (x^{2} + 2 \, x + \log \relax (3) - 2 \, \log \relax (2)\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x+1)*log(3/4*exp(x)*exp(x^2))^3+(6*x^2+3*x-2)*log(3/4*exp(x)*exp(x^2))^2+(6*x^3+7*x^2)*log(3/4*e
xp(x)*exp(x^2))+2*x^4+5*x^3+2*x^2-4*x-4)/(log(3/4*exp(x)*exp(x^2))^3+3*x*log(3/4*exp(x)*exp(x^2))^2+3*x^2*log(
3/4*exp(x)*exp(x^2))+x^3),x, algorithm="giac")

[Out]

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

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maple [B]  time = 1.60, size = 54, normalized size = 2.16

method	result	size
default	\(x^{2}+x +\frac {-2 x^{3}-4 x^{2}-2 x \left (\ln \left (\frac {3 \,{\mathrm e}^{x} {\mathrm e}^{x^{2}}}{4}\right )-x^{2}-x \right )+1}{\left (\ln \left (\frac {3 \,{\mathrm e}^{x} {\mathrm e}^{x^{2}}}{4}\right )+x \right )^{2}}\)	\(54\)
risch	\(x^{2}+x -\frac {4 i \left (-i+\pi x \,\mathrm {csgn}\left (i {\mathrm e}^{x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{x^{2}}\right ) \mathrm {csgn}\left (i {\mathrm e}^{\left (x +1\right ) x}\right )-\pi x \,\mathrm {csgn}\left (i {\mathrm e}^{x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{\left (x +1\right ) x}\right )^{2}-\pi x \,\mathrm {csgn}\left (i {\mathrm e}^{x^{2}}\right ) \mathrm {csgn}\left (i {\mathrm e}^{\left (x +1\right ) x}\right )^{2}+\pi x \mathrm {csgn}\left (i {\mathrm e}^{\left (x +1\right ) x}\right )^{3}+2 i \ln \relax (3) x -4 i x \ln \relax (2)+2 i x^{2}+2 i x \ln \left ({\mathrm e}^{x}\right )+2 i x \ln \left ({\mathrm e}^{x^{2}}\right )\right )}{\left (\pi \,\mathrm {csgn}\left (i {\mathrm e}^{x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{x^{2}}\right ) \mathrm {csgn}\left (i {\mathrm e}^{\left (x +1\right ) x}\right )-\pi \,\mathrm {csgn}\left (i {\mathrm e}^{x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{\left (x +1\right ) x}\right )^{2}-\pi \,\mathrm {csgn}\left (i {\mathrm e}^{x^{2}}\right ) \mathrm {csgn}\left (i {\mathrm e}^{\left (x +1\right ) x}\right )^{2}+\pi \mathrm {csgn}\left (i {\mathrm e}^{\left (x +1\right ) x}\right )^{3}+2 i \ln \relax (3)-4 i \ln \relax (2)+2 i x +2 i \ln \left ({\mathrm e}^{x}\right )+2 i \ln \left ({\mathrm e}^{x^{2}}\right )\right )^{2}}\)	\(249\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x+1)*ln(3/4*exp(x)*exp(x^2))^3+(6*x^2+3*x-2)*ln(3/4*exp(x)*exp(x^2))^2+(6*x^3+7*x^2)*ln(3/4*exp(x)*exp
(x^2))+2*x^4+5*x^3+2*x^2-4*x-4)/(ln(3/4*exp(x)*exp(x^2))^3+3*x*ln(3/4*exp(x)*exp(x^2))^2+3*x^2*ln(3/4*exp(x)*e
xp(x^2))+x^3),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [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(((2*x+1)*log(3/4*exp(x)*exp(x^2))^3+(6*x^2+3*x-2)*log(3/4*exp(x)*exp(x^2))^2+(6*x^3+7*x^2)*log(3/4*e
xp(x)*exp(x^2))+2*x^4+5*x^3+2*x^2-4*x-4)/(log(3/4*exp(x)*exp(x^2))^3+3*x*log(3/4*exp(x)*exp(x^2))^2+3*x^2*log(
3/4*exp(x)*exp(x^2))+x^3),x, algorithm="maxima")

[Out]

Timed out

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mupad [B]  time = 0.25, size = 35, normalized size = 1.40 \begin {gather*} x-\frac {2\,x^3+4\,x^2+2\,\ln \left (\frac {3}{4}\right )\,x-1}{{\left (x^2+2\,x+\ln \left (\frac {3}{4}\right )\right )}^2}+x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log((3*exp(x^2)*exp(x))/4)^2*(3*x + 6*x^2 - 2) - 4*x + log((3*exp(x^2)*exp(x))/4)*(7*x^2 + 6*x^3) + 2*x^2
+ 5*x^3 + 2*x^4 + log((3*exp(x^2)*exp(x))/4)^3*(2*x + 1) - 4)/(3*x*log((3*exp(x^2)*exp(x))/4)^2 + 3*x^2*log((
3*exp(x^2)*exp(x))/4) + log((3*exp(x^2)*exp(x))/4)^3 + x^3),x)

[Out]

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

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sympy [B]  time = 3.24, size = 83, normalized size = 3.32 \begin {gather*} x^{2} + x + \frac {- 2 x^{3} - 4 x^{2} + x \left (- 2 \log {\relax (3 )} + 4 \log {\relax (2 )}\right ) + 1}{x^{4} + 4 x^{3} + x^{2} \left (- 4 \log {\relax (2 )} + 2 \log {\relax (3 )} + 4\right ) + x \left (- 8 \log {\relax (2 )} + 4 \log {\relax (3 )}\right ) - 4 \log {\relax (2 )} \log {\relax (3 )} + \log {\relax (3 )}^{2} + 4 \log {\relax (2 )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x+1)*ln(3/4*exp(x)*exp(x**2))**3+(6*x**2+3*x-2)*ln(3/4*exp(x)*exp(x**2))**2+(6*x**3+7*x**2)*ln(3
/4*exp(x)*exp(x**2))+2*x**4+5*x**3+2*x**2-4*x-4)/(ln(3/4*exp(x)*exp(x**2))**3+3*x*ln(3/4*exp(x)*exp(x**2))**2+
3*x**2*ln(3/4*exp(x)*exp(x**2))+x**3),x)

[Out]

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

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