3.61.44 \(\int \frac {6+4 x-x^2-4 \log ^4(2)}{4-4 x+x^2} \, dx\)

Optimal. Leaf size=20 \[ -x+\frac {10-4 \log ^4(2)}{2-x} \]

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 21, normalized size of antiderivative = 1.05, number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {27, 683} \begin {gather*} \frac {2 \left (5-2 \log ^4(2)\right )}{2-x}-x \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-x + (2*(5 - 2*Log[2]^4))/(2 - x)

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 683

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[2*c*d - b*e,
0] && IGtQ[p, 0] &&  !(EqQ[m, 3] && NeQ[p, 1])

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 19, normalized size = 0.95 \begin {gather*} -x+\frac {2 \left (-5+2 \log ^4(2)\right )}{-2+x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-x + (2*(-5 + 2*Log[2]^4))/(-2 + x)

________________________________________________________________________________________

fricas [A]  time = 0.73, size = 22, normalized size = 1.10 \begin {gather*} \frac {4 \, \log \relax (2)^{4} - x^{2} + 2 \, x - 10}{x - 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*log(2)^4-x^2+4*x+6)/(x^2-4*x+4),x, algorithm="fricas")

[Out]

(4*log(2)^4 - x^2 + 2*x - 10)/(x - 2)

________________________________________________________________________________________

giac [A]  time = 0.15, size = 19, normalized size = 0.95 \begin {gather*} -x + \frac {2 \, {\left (2 \, \log \relax (2)^{4} - 5\right )}}{x - 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*log(2)^4-x^2+4*x+6)/(x^2-4*x+4),x, algorithm="giac")

[Out]

-x + 2*(2*log(2)^4 - 5)/(x - 2)

________________________________________________________________________________________

maple [A]  time = 0.38, size = 20, normalized size = 1.00

method	result	size
gosper	\(\frac {4 \ln \relax (2)^{4}-x^{2}-6}{x -2}\)	\(20\)
default	\(-x -\frac {10-4 \ln \relax (2)^{4}}{x -2}\)	\(20\)
risch	\(-x +\frac {4 \ln \relax (2)^{4}}{x -2}-\frac {10}{x -2}\)	\(23\)
meijerg	\(\frac {7 x}{2 \left (1-\frac {x}{2}\right )}-\frac {\ln \relax (2)^{4} x}{1-\frac {x}{2}}-\frac {x \left (-\frac {3 x}{2}+6\right )}{3 \left (1-\frac {x}{2}\right )}\)	\(41\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-4*ln(2)^4-x^2+4*x+6)/(x^2-4*x+4),x,method=_RETURNVERBOSE)

[Out]

(4*ln(2)^4-x^2-6)/(x-2)

________________________________________________________________________________________

maxima [A]  time = 0.34, size = 19, normalized size = 0.95 \begin {gather*} -x + \frac {2 \, {\left (2 \, \log \relax (2)^{4} - 5\right )}}{x - 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*log(2)^4-x^2+4*x+6)/(x^2-4*x+4),x, algorithm="maxima")

[Out]

-x + 2*(2*log(2)^4 - 5)/(x - 2)

________________________________________________________________________________________

mupad [B]  time = 4.22, size = 18, normalized size = 0.90 \begin {gather*} \frac {4\,{\ln \relax (2)}^4-10}{x-2}-x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x - 4*log(2)^4 - x^2 + 6)/(x^2 - 4*x + 4),x)

[Out]

(4*log(2)^4 - 10)/(x - 2) - x

________________________________________________________________________________________

sympy [A]  time = 0.11, size = 14, normalized size = 0.70 \begin {gather*} - x - \frac {10 - 4 \log {\relax (2 )}^{4}}{x - 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x - (10 - 4*log(2)**4)/(x - 2)

________________________________________________________________________________________