3.22.39 \(\int \frac {-12+300 x^3+(-4+100 x^3) \log (-\log (x)+\log (4+50 x^3))}{(-2 x-25 x^4) \log (x)+(2 x+25 x^4) \log (4+50 x^3)} \, dx\)

Optimal. Leaf size=18 \[ \left (3+\log \left (-\log (x)+\log \left (4+50 x^3\right )\right )\right )^2 \]

________________________________________________________________________________________

Rubi [A]  time = 0.23, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 63, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {6741, 12, 6686} \begin {gather*} \left (\log \left (\log \left (50 x^3+4\right )-\log (x)\right )+3\right )^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-12 + 300*x^3 + (-4 + 100*x^3)*Log[-Log[x] + Log[4 + 50*x^3]])/((-2*x - 25*x^4)*Log[x] + (2*x + 25*x^4)*L
og[4 + 50*x^3]),x]

[Out]

(3 + Log[-Log[x] + Log[4 + 50*x^3]])^2

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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]

Rule 6741

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =!= u]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {4 \left (1-25 x^3\right ) \left (3+\log \left (-\log (x)+\log \left (4+50 x^3\right )\right )\right )}{x \left (2+25 x^3\right ) \left (\log (x)-\log \left (4+50 x^3\right )\right )} \, dx\\ &=4 \int \frac {\left (1-25 x^3\right ) \left (3+\log \left (-\log (x)+\log \left (4+50 x^3\right )\right )\right )}{x \left (2+25 x^3\right ) \left (\log (x)-\log \left (4+50 x^3\right )\right )} \, dx\\ &=\left (3+\log \left (-\log (x)+\log \left (4+50 x^3\right )\right )\right )^2\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.03, size = 18, normalized size = 1.00 \begin {gather*} \left (3+\log \left (-\log (x)+\log \left (4+50 x^3\right )\right )\right )^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-12 + 300*x^3 + (-4 + 100*x^3)*Log[-Log[x] + Log[4 + 50*x^3]])/((-2*x - 25*x^4)*Log[x] + (2*x + 25*
x^4)*Log[4 + 50*x^3]),x]

[Out]

(3 + Log[-Log[x] + Log[4 + 50*x^3]])^2

________________________________________________________________________________________

fricas [A]  time = 1.02, size = 33, normalized size = 1.83 \begin {gather*} \log \left (\log \left (50 \, x^{3} + 4\right ) - \log \relax (x)\right )^{2} + 6 \, \log \left (\log \left (50 \, x^{3} + 4\right ) - \log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((100*x^3-4)*log(log(50*x^3+4)-log(x))+300*x^3-12)/((25*x^4+2*x)*log(50*x^3+4)+(-25*x^4-2*x)*log(x))
,x, algorithm="fricas")

[Out]

log(log(50*x^3 + 4) - log(x))^2 + 6*log(log(50*x^3 + 4) - log(x))

________________________________________________________________________________________

giac [A]  time = 0.32, size = 33, normalized size = 1.83 \begin {gather*} \log \left (\log \left (50 \, x^{3} + 4\right ) - \log \relax (x)\right )^{2} + 6 \, \log \left (\log \left (50 \, x^{3} + 4\right ) - \log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((100*x^3-4)*log(log(50*x^3+4)-log(x))+300*x^3-12)/((25*x^4+2*x)*log(50*x^3+4)+(-25*x^4-2*x)*log(x))
,x, algorithm="giac")

[Out]

log(log(50*x^3 + 4) - log(x))^2 + 6*log(log(50*x^3 + 4) - log(x))

________________________________________________________________________________________

maple [A]  time = 0.05, size = 34, normalized size = 1.89

method	result	size
risch	\(\ln \left (\ln \left (50 x^{3}+4\right )-\ln \relax (x )\right )^{2}+6 \ln \left (\ln \left (50 x^{3}+4\right )-\ln \relax (x )\right )\)	\(34\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((100*x^3-4)*ln(ln(50*x^3+4)-ln(x))+300*x^3-12)/((25*x^4+2*x)*ln(50*x^3+4)+(-25*x^4-2*x)*ln(x)),x,method=_
RETURNVERBOSE)

[Out]

ln(ln(50*x^3+4)-ln(x))^2+6*ln(ln(50*x^3+4)-ln(x))

________________________________________________________________________________________

maxima [B]  time = 0.94, size = 37, normalized size = 2.06 \begin {gather*} \log \left (\log \relax (2) + \log \left (25 \, x^{3} + 2\right ) - \log \relax (x)\right )^{2} + 6 \, \log \left (\log \relax (2) + \log \left (25 \, x^{3} + 2\right ) - \log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((100*x^3-4)*log(log(50*x^3+4)-log(x))+300*x^3-12)/((25*x^4+2*x)*log(50*x^3+4)+(-25*x^4-2*x)*log(x))
,x, algorithm="maxima")

[Out]

log(log(2) + log(25*x^3 + 2) - log(x))^2 + 6*log(log(2) + log(25*x^3 + 2) - log(x))

________________________________________________________________________________________

mupad [B]  time = 1.44, size = 31, normalized size = 1.72 \begin {gather*} \ln \left (\ln \left (50\,x^3+4\right )-\ln \relax (x)\right )\,\left (\ln \left (\ln \left (50\,x^3+4\right )-\ln \relax (x)\right )+6\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(log(50*x^3 + 4) - log(x))*(100*x^3 - 4) + 300*x^3 - 12)/(log(x)*(2*x + 25*x^4) - log(50*x^3 + 4)*(2*
x + 25*x^4)),x)

[Out]

log(log(50*x^3 + 4) - log(x))*(log(log(50*x^3 + 4) - log(x)) + 6)

________________________________________________________________________________________

sympy [A]  time = 0.99, size = 29, normalized size = 1.61 \begin {gather*} \log {\left (- \log {\relax (x )} + \log {\left (50 x^{3} + 4 \right )} \right )}^{2} + 6 \log {\left (- \log {\relax (x )} + \log {\left (50 x^{3} + 4 \right )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((100*x**3-4)*ln(ln(50*x**3+4)-ln(x))+300*x**3-12)/((25*x**4+2*x)*ln(50*x**3+4)+(-25*x**4-2*x)*ln(x)
),x)

[Out]

log(-log(x) + log(50*x**3 + 4))**2 + 6*log(-log(x) + log(50*x**3 + 4))

________________________________________________________________________________________