3.91.25 \(\int \frac {x+(2-2 x-4 x^2) \log (\frac {4}{5} e^{-x-x^2} x \log (4))}{x} \, dx\)

Optimal. Leaf size=23 \[ x+\log ^2\left (\frac {4}{5} e^{-x-x^2} x \log (4)\right ) \]

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Rubi [A]  time = 0.09, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {14, 76, 6686} \begin {gather*} \log ^2\left (\frac {4}{5} e^{-x^2-x} x \log (4)\right )+x \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

x + Log[(4*E^(-x - x^2)*x*Log[4])/5]^2

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 76

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && (NeQ[n, -1] || EqQ[p, 1]) && N
eQ[b*e + a*f, 0] && ( !IntegerQ[n] || LtQ[9*p + 5*n, 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && Rational
Q[a, b, d, e, f])) && (NeQ[n + p + 3, 0] || EqQ[p, 1])

Rule 6686

Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[(q*y^(m + 1))/(m + 1), x] /;  !F
alseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

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

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Mathematica [A]  time = 0.02, size = 20, normalized size = 0.87 \begin {gather*} x+\log ^2\left (\frac {4}{5} e^{-x (1+x)} x \log (4)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

x + Log[(4*x*Log[4])/(5*E^(x*(1 + x)))]^2

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fricas [A]  time = 0.53, size = 20, normalized size = 0.87 \begin {gather*} \log \left (\frac {8}{5} \, x e^{\left (-x^{2} - x\right )} \log \relax (2)\right )^{2} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x^2-2*x+2)*log(8/5*x*log(2)/exp(x)/exp(x^2))+x)/x,x, algorithm="fricas")

[Out]

log(8/5*x*e^(-x^2 - x)*log(2))^2 + x

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giac [B]  time = 0.15, size = 74, normalized size = 3.22 \begin {gather*} x^{4} + 2 \, x^{3} + x^{2} {\left (2 \, \log \relax (5) - 6 \, \log \relax (2) - 2 \, \log \left (\log \relax (2)\right ) + 1\right )} + x {\left (2 \, \log \relax (5) - 6 \, \log \relax (2) - 2 \, \log \left (\log \relax (2)\right ) + 1\right )} - 2 \, {\left (x^{2} + x\right )} \log \relax (x) - 2 \, {\left (\log \relax (5) - 3 \, \log \relax (2) - \log \left (\log \relax (2)\right )\right )} \log \relax (x) + \log \relax (x)^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x^2-2*x+2)*log(8/5*x*log(2)/exp(x)/exp(x^2))+x)/x,x, algorithm="giac")

[Out]

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

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maple [B]  time = 0.27, size = 135, normalized size = 5.87

method	result	size
default	\(x -2 \ln \left (x \,{\mathrm e}^{-\left (x +1\right ) x}\right ) x^{2}+2 \ln \left (x \,{\mathrm e}^{-\left (x +1\right ) x}\right ) \ln \relax (x )-2 \ln \left (x \,{\mathrm e}^{-\left (x +1\right ) x}\right ) x +2 x^{2} \ln \relax (x )-x^{2}+2 x \ln \relax (x )-\ln \relax (x )^{2}-x^{4}-2 x^{3}+2 x^{2} \ln \relax (5)-6 x^{2} \ln \relax (2)-2 x^{2} \ln \left (\ln \relax (2)\right )-2 \ln \relax (5) \ln \relax (x )+2 x \ln \relax (5)+6 \ln \relax (2) \ln \relax (x )-6 x \ln \relax (2)+2 \ln \relax (x ) \ln \left (\ln \relax (2)\right )-2 x \ln \left (\ln \relax (2)\right )\)	\(135\)
risch	\(x -2 x^{2} \ln \left (\ln \relax (2)\right )+2 \ln \relax (x ) \ln \left (\ln \relax (2)\right )+2 x^{2} \ln \relax (5)+\ln \relax (x )^{2}-x^{4}-2 x^{3}-x^{2}+6 \ln \relax (2) \ln \relax (x )+2 x \ln \relax (5)-6 x \ln \relax (2)-6 x^{2} \ln \relax (2)-i \ln \relax (x ) \pi \mathrm {csgn}\left (i x \,{\mathrm e}^{-\left (x +1\right ) x}\right )^{3}+i \pi \,x^{2} \mathrm {csgn}\left (i x \,{\mathrm e}^{-\left (x +1\right ) x}\right )^{3}+i \pi x \mathrm {csgn}\left (i {\mathrm e}^{-\left (x +1\right ) x}\right )^{3}-i \ln \relax (x ) \pi \mathrm {csgn}\left (i {\mathrm e}^{-\left (x +1\right ) x}\right )^{3}+i \pi \,x^{2} \mathrm {csgn}\left (i {\mathrm e}^{-\left (x +1\right ) x}\right )^{3}+i \pi x \mathrm {csgn}\left (i x \,{\mathrm e}^{-\left (x +1\right ) x}\right )^{3}+2 x \ln \left ({\mathrm e}^{x}\right )-2 \ln \relax (5) \ln \relax (x )-2 x \ln \left (\ln \relax (2)\right )+2 x^{2} \ln \left ({\mathrm e}^{x}\right )+i \ln \relax (x ) \pi \,\mathrm {csgn}\left (i {\mathrm e}^{-\left (x +1\right ) x}\right ) \mathrm {csgn}\left (i x \,{\mathrm e}^{-\left (x +1\right ) x}\right )^{2}-i \pi \,x^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \,{\mathrm e}^{-\left (x +1\right ) x}\right )^{2}-i \pi \,x^{2} \mathrm {csgn}\left (i {\mathrm e}^{-x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{-\left (x +1\right ) x}\right )^{2}-i \pi x \mathrm {csgn}\left (i {\mathrm e}^{-\left (x +1\right ) x}\right )^{2} \mathrm {csgn}\left (i {\mathrm e}^{-x^{2}}\right )-i \pi x \,\mathrm {csgn}\left (i {\mathrm e}^{-\left (x +1\right ) x}\right ) \mathrm {csgn}\left (i x \,{\mathrm e}^{-\left (x +1\right ) x}\right )^{2}-i \pi x \,\mathrm {csgn}\left (i {\mathrm e}^{-x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{-\left (x +1\right ) x}\right )^{2}-i \pi \,x^{2} \mathrm {csgn}\left (i {\mathrm e}^{-\left (x +1\right ) x}\right )^{2} \mathrm {csgn}\left (i {\mathrm e}^{-x^{2}}\right )+\left (2 x^{2}+2 x -2 \ln \relax (x )\right ) \ln \left ({\mathrm e}^{x^{2}}\right )-i \pi \,x^{2} \mathrm {csgn}\left (i {\mathrm e}^{-\left (x +1\right ) x}\right ) \mathrm {csgn}\left (i x \,{\mathrm e}^{-\left (x +1\right ) x}\right )^{2}-i \pi x \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \,{\mathrm e}^{-\left (x +1\right ) x}\right )^{2}+i \ln \relax (x ) \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \,{\mathrm e}^{-\left (x +1\right ) x}\right )^{2}+i \ln \relax (x ) \pi \,\mathrm {csgn}\left (i {\mathrm e}^{-x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{-\left (x +1\right ) x}\right )^{2}+i \ln \relax (x ) \pi \mathrm {csgn}\left (i {\mathrm e}^{-\left (x +1\right ) x}\right )^{2} \mathrm {csgn}\left (i {\mathrm e}^{-x^{2}}\right )-i \ln \relax (x ) \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i {\mathrm e}^{-\left (x +1\right ) x}\right ) \mathrm {csgn}\left (i x \,{\mathrm e}^{-\left (x +1\right ) x}\right )-i \ln \relax (x ) \pi \,\mathrm {csgn}\left (i {\mathrm e}^{-x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{-\left (x +1\right ) x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{-x^{2}}\right )+i \pi \,x^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i {\mathrm e}^{-\left (x +1\right ) x}\right ) \mathrm {csgn}\left (i x \,{\mathrm e}^{-\left (x +1\right ) x}\right )+i \pi \,x^{2} \mathrm {csgn}\left (i {\mathrm e}^{-x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{-\left (x +1\right ) x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{-x^{2}}\right )+i \pi x \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i {\mathrm e}^{-\left (x +1\right ) x}\right ) \mathrm {csgn}\left (i x \,{\mathrm e}^{-\left (x +1\right ) x}\right )+i \pi x \,\mathrm {csgn}\left (i {\mathrm e}^{-x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{-\left (x +1\right ) x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{-x^{2}}\right )-2 \ln \relax (x ) \ln \left ({\mathrm e}^{x}\right )\)	\(779\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-4*x^2-2*x+2)*ln(8/5*x*ln(2)/exp(x)/exp(x^2))+x)/x,x,method=_RETURNVERBOSE)

[Out]

x-2*ln(x*exp(-(x+1)*x))*x^2+2*ln(x*exp(-(x+1)*x))*ln(x)-2*ln(x*exp(-(x+1)*x))*x+2*x^2*ln(x)-x^2+2*x*ln(x)-ln(x
)^2-x^4-2*x^3+2*x^2*ln(5)-6*x^2*ln(2)-2*x^2*ln(ln(2))-2*ln(5)*ln(x)+2*x*ln(5)+6*ln(2)*ln(x)-6*x*ln(2)+2*ln(x)*
ln(ln(2))-2*x*ln(ln(2))

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maxima [B]  time = 0.43, size = 167, normalized size = 7.26 \begin {gather*} -2 \, x^{2} \log \left (\frac {8}{5} \, x e^{\left (-x^{2} - x\right )} \log \relax (2)\right ) - 2 \, x \log \left (\frac {8}{5} \, x e^{\left (-x^{2} - x\right )} \log \relax (2)\right ) + 2 \, \log \left (\frac {8}{5} \, x e^{\left (-x^{2} - x\right )} \log \relax (2)\right ) \log \relax (x) + x + \frac {2 \, {\left (x^{2} \log \relax (2) + x \log \relax (2) - \log \relax (2) \log \relax (x)\right )} \log \relax (x)}{\log \relax (2)} - \frac {3 \, x^{4} \log \relax (2) + 2 \, x^{3} \log \relax (2) - 3 \, x^{2} \log \relax (2)}{3 \, \log \relax (2)} - \frac {4 \, x^{3} \log \relax (2) + 3 \, x^{2} \log \relax (2) - 6 \, x \log \relax (2)}{3 \, \log \relax (2)} - \frac {x^{2} \log \relax (2) - \log \relax (2) \log \relax (x)^{2} + 2 \, x \log \relax (2)}{\log \relax (2)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x^2-2*x+2)*log(8/5*x*log(2)/exp(x)/exp(x^2))+x)/x,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 7.35, size = 24, normalized size = 1.04 \begin {gather*} x+{\left (x+\ln \relax (5)-\ln \relax (8)-\ln \left (\ln \relax (2)\right )-\ln \relax (x)+x^2\right )}^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x - log((8*x*exp(-x)*exp(-x^2)*log(2))/5)*(2*x + 4*x^2 - 2))/x,x)

[Out]

x + (x + log(5) - log(8) - log(log(2)) - log(x) + x^2)^2

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sympy [A]  time = 0.24, size = 20, normalized size = 0.87 \begin {gather*} x + \log {\left (\frac {8 x e^{- x} e^{- x^{2}} \log {\relax (2 )}}{5} \right )}^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x**2-2*x+2)*ln(8/5*x*ln(2)/exp(x)/exp(x**2))+x)/x,x)

[Out]

x + log(8*x*exp(-x)*exp(-x**2)*log(2)/5)**2

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