3.60.21 \(\int \frac {-84 x^2-32 x^4+(-80 x+2 x^2-32 x^3) \log (2)+(-20-8 x^2) \log ^2(2)}{4 x^4+4 x^3 \log (2)+x^2 \log ^2(2)} \, dx\)

Optimal. Leaf size=33 \[ 4-4 \left (e^4-\frac {5}{x}+2 x\right )-\frac {2+2 x}{-2 x-\log (2)} \]

________________________________________________________________________________________

Rubi [A]  time = 0.09, antiderivative size = 27, normalized size of antiderivative = 0.82, number of steps used = 4, number of rules used = 3, integrand size = 64, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.047, Rules used = {1594, 27, 1620} \begin {gather*} -8 x+\frac {20}{x}+\frac {4-\log (4)}{2 (2 x+\log (2))} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-84*x^2 - 32*x^4 + (-80*x + 2*x^2 - 32*x^3)*Log[2] + (-20 - 8*x^2)*Log[2]^2)/(4*x^4 + 4*x^3*Log[2] + x^2*
Log[2]^2),x]

[Out]

20/x - 8*x + (4 - Log[4])/(2*(2*x + Log[2]))

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 1594

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^
(q - p) + c*x^(r - p))^n, x] /; FreeQ[{a, b, c, p, q, r}, x] && IntegerQ[n] && PosQ[q - p] && PosQ[r - p]

Rule 1620

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)
^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2]) &&
GtQ[Expon[Px, x], 2]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-84 x^2-32 x^4+\left (-80 x+2 x^2-32 x^3\right ) \log (2)+\left (-20-8 x^2\right ) \log ^2(2)}{x^2 \left (4 x^2+4 x \log (2)+\log ^2(2)\right )} \, dx\\ &=\int \frac {-84 x^2-32 x^4+\left (-80 x+2 x^2-32 x^3\right ) \log (2)+\left (-20-8 x^2\right ) \log ^2(2)}{x^2 (2 x+\log (2))^2} \, dx\\ &=\int \left (-8-\frac {20}{x^2}+\frac {-4+\log (4)}{(2 x+\log (2))^2}\right ) \, dx\\ &=\frac {20}{x}-8 x+\frac {4-\log (4)}{2 (2 x+\log (2))}\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.02, size = 24, normalized size = 0.73 \begin {gather*} -2 \left (-\frac {10}{x}+4 x+\frac {-2+\log (2)}{4 x+\log (4)}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-84*x^2 - 32*x^4 + (-80*x + 2*x^2 - 32*x^3)*Log[2] + (-20 - 8*x^2)*Log[2]^2)/(4*x^4 + 4*x^3*Log[2]
+ x^2*Log[2]^2),x]

[Out]

-2*(-10/x + 4*x + (-2 + Log[2])/(4*x + Log[4]))

________________________________________________________________________________________

fricas [A]  time = 0.83, size = 34, normalized size = 1.03 \begin {gather*} -\frac {16 \, x^{3} + {\left (8 \, x^{2} + x - 20\right )} \log \relax (2) - 42 \, x}{2 \, x^{2} + x \log \relax (2)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-8*x^2-20)*log(2)^2+(-32*x^3+2*x^2-80*x)*log(2)-32*x^4-84*x^2)/(x^2*log(2)^2+4*x^3*log(2)+4*x^4),x
, algorithm="fricas")

[Out]

-(16*x^3 + (8*x^2 + x - 20)*log(2) - 42*x)/(2*x^2 + x*log(2))

________________________________________________________________________________________

giac [A]  time = 0.16, size = 30, normalized size = 0.91 \begin {gather*} -8 \, x - \frac {x \log \relax (2) - 42 \, x - 20 \, \log \relax (2)}{2 \, x^{2} + x \log \relax (2)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-8*x^2-20)*log(2)^2+(-32*x^3+2*x^2-80*x)*log(2)-32*x^4-84*x^2)/(x^2*log(2)^2+4*x^3*log(2)+4*x^4),x
, algorithm="giac")

[Out]

-8*x - (x*log(2) - 42*x - 20*log(2))/(2*x^2 + x*log(2))

________________________________________________________________________________________

maple [A]  time = 0.08, size = 26, normalized size = 0.79

method	result	size
default	\(-8 x -\frac {2 \left (\frac {\ln \relax (2)}{2}-1\right )}{\ln \relax (2)+2 x}+\frac {20}{x}\)	\(26\)
risch	\(-8 x +\frac {\left (-\ln \relax (2)+42\right ) x +20 \ln \relax (2)}{x \left (\ln \relax (2)+2 x \right )}\)	\(30\)
norman	\(\frac {\left (4 \ln \relax (2)^{2}-\ln \relax (2)+42\right ) x -16 x^{3}+20 \ln \relax (2)}{x \left (\ln \relax (2)+2 x \right )}\)	\(37\)
gosper	\(\frac {4 x \ln \relax (2)^{2}-16 x^{3}-x \ln \relax (2)+20 \ln \relax (2)+42 x}{x \left (\ln \relax (2)+2 x \right )}\)	\(38\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-8*x^2-20)*ln(2)^2+(-32*x^3+2*x^2-80*x)*ln(2)-32*x^4-84*x^2)/(x^2*ln(2)^2+4*x^3*ln(2)+4*x^4),x,method=_R
ETURNVERBOSE)

[Out]

-8*x-2*(1/2*ln(2)-1)/(ln(2)+2*x)+20/x

________________________________________________________________________________________

maxima [A]  time = 0.36, size = 29, normalized size = 0.88 \begin {gather*} -8 \, x - \frac {x {\left (\log \relax (2) - 42\right )} - 20 \, \log \relax (2)}{2 \, x^{2} + x \log \relax (2)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-8*x^2-20)*log(2)^2+(-32*x^3+2*x^2-80*x)*log(2)-32*x^4-84*x^2)/(x^2*log(2)^2+4*x^3*log(2)+4*x^4),x
, algorithm="maxima")

[Out]

-8*x - (x*(log(2) - 42) - 20*log(2))/(2*x^2 + x*log(2))

________________________________________________________________________________________

mupad [B]  time = 0.13, size = 28, normalized size = 0.85 \begin {gather*} \frac {20\,\ln \relax (2)-x\,\left (\ln \relax (2)-42\right )}{x\,\left (2\,x+\ln \relax (2)\right )}-8\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(2)*(80*x - 2*x^2 + 32*x^3) + log(2)^2*(8*x^2 + 20) + 84*x^2 + 32*x^4)/(x^2*log(2)^2 + 4*x^3*log(2) +
4*x^4),x)

[Out]

(20*log(2) - x*(log(2) - 42))/(x*(2*x + log(2))) - 8*x

________________________________________________________________________________________

sympy [A]  time = 0.37, size = 26, normalized size = 0.79 \begin {gather*} - 8 x - \frac {x \left (-42 + \log {\relax (2 )}\right ) - 20 \log {\relax (2 )}}{2 x^{2} + x \log {\relax (2 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-8*x**2-20)*ln(2)**2+(-32*x**3+2*x**2-80*x)*ln(2)-32*x**4-84*x**2)/(x**2*ln(2)**2+4*x**3*ln(2)+4*x
**4),x)

[Out]

-8*x - (x*(-42 + log(2)) - 20*log(2))/(2*x**2 + x*log(2))

________________________________________________________________________________________