3.24.39 \(\int \frac {4-10 x+(5600-12400 x+7150 x^2-1600 x^3+125 x^4) \log (3)+(-750+800 x-150 x^2) \log (3) \log (4 x)+25 \log (3) \log ^2(4 x)}{4 x-5 x^2+(6400 x-6400 x^2+2400 x^3-400 x^4+25 x^5) \log (3)+(-800 x+400 x^2-50 x^3) \log (3) \log (4 x)+25 x \log (3) \log ^2(4 x)} \, dx\)

Optimal. Leaf size=29 \[ \log \left (x \left (4-5 \left (x-5 \log (3) \left (-(4-x)^2+\log (4 x)\right )^2\right )\right )\right ) \]

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Rubi [B]  time = 0.84, antiderivative size = 81, normalized size of antiderivative = 2.79, number of steps used = 3, number of rules used = 2, integrand size = 126, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.016, Rules used = {6742, 6684} \begin {gather*} \log \left (25 x^4 \log (3)-400 x^3 \log (3)-50 x^2 \log (3) \log (4 x)+2400 x^2 \log (3)+25 \log (3) \log ^2(4 x)+400 x \log (3) \log (4 x)-5 x (1+1280 \log (3))-800 \log (3) \log (4 x)+4 (1+1600 \log (3))\right )+\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(4 - 10*x + (5600 - 12400*x + 7150*x^2 - 1600*x^3 + 125*x^4)*Log[3] + (-750 + 800*x - 150*x^2)*Log[3]*Log[
4*x] + 25*Log[3]*Log[4*x]^2)/(4*x - 5*x^2 + (6400*x - 6400*x^2 + 2400*x^3 - 400*x^4 + 25*x^5)*Log[3] + (-800*x
+ 400*x^2 - 50*x^3)*Log[3]*Log[4*x] + 25*x*Log[3]*Log[4*x]^2),x]

[Out]

Log[x] + Log[2400*x^2*Log[3] - 400*x^3*Log[3] + 25*x^4*Log[3] - 5*x*(1 + 1280*Log[3]) + 4*(1 + 1600*Log[3]) -
800*Log[3]*Log[4*x] + 400*x*Log[3]*Log[4*x] - 50*x^2*Log[3]*Log[4*x] + 25*Log[3]*Log[4*x]^2]

Rule 6684

Int[(u_)/(y_), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*Log[RemoveContent[y, x]], x] /;  !Fa
lseQ[q]]

Rule 6742

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {1}{x}+\frac {5 \left (-160 \log (3)+950 x^2 \log (3)-240 x^3 \log (3)+20 x^4 \log (3)-x (1+1200 \log (3))+10 \log (3) \log (4 x)+80 x \log (3) \log (4 x)-20 x^2 \log (3) \log (4 x)\right )}{x \left (2400 x^2 \log (3)-400 x^3 \log (3)+25 x^4 \log (3)-5 x (1+1280 \log (3))+4 (1+1600 \log (3))-800 \log (3) \log (4 x)+400 x \log (3) \log (4 x)-50 x^2 \log (3) \log (4 x)+25 \log (3) \log ^2(4 x)\right )}\right ) \, dx\\ &=\log (x)+5 \int \frac {-160 \log (3)+950 x^2 \log (3)-240 x^3 \log (3)+20 x^4 \log (3)-x (1+1200 \log (3))+10 \log (3) \log (4 x)+80 x \log (3) \log (4 x)-20 x^2 \log (3) \log (4 x)}{x \left (2400 x^2 \log (3)-400 x^3 \log (3)+25 x^4 \log (3)-5 x (1+1280 \log (3))+4 (1+1600 \log (3))-800 \log (3) \log (4 x)+400 x \log (3) \log (4 x)-50 x^2 \log (3) \log (4 x)+25 \log (3) \log ^2(4 x)\right )} \, dx\\ &=\log (x)+\log \left (2400 x^2 \log (3)-400 x^3 \log (3)+25 x^4 \log (3)-5 x (1+1280 \log (3))+4 (1+1600 \log (3))-800 \log (3) \log (4 x)+400 x \log (3) \log (4 x)-50 x^2 \log (3) \log (4 x)+25 \log (3) \log ^2(4 x)\right )\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.07, size = 77, normalized size = 2.66 \begin {gather*} \log (x)+\log \left (4-5 x+6400 \log (3)-6400 x \log (3)+2400 x^2 \log (3)-400 x^3 \log (3)+25 x^4 \log (3)-800 \log (3) \log (4 x)+400 x \log (3) \log (4 x)-50 x^2 \log (3) \log (4 x)+25 \log (3) \log ^2(4 x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(4 - 10*x + (5600 - 12400*x + 7150*x^2 - 1600*x^3 + 125*x^4)*Log[3] + (-750 + 800*x - 150*x^2)*Log[3
]*Log[4*x] + 25*Log[3]*Log[4*x]^2)/(4*x - 5*x^2 + (6400*x - 6400*x^2 + 2400*x^3 - 400*x^4 + 25*x^5)*Log[3] + (
-800*x + 400*x^2 - 50*x^3)*Log[3]*Log[4*x] + 25*x*Log[3]*Log[4*x]^2),x]

[Out]

Log[x] + Log[4 - 5*x + 6400*Log[3] - 6400*x*Log[3] + 2400*x^2*Log[3] - 400*x^3*Log[3] + 25*x^4*Log[3] - 800*Lo
g[3]*Log[4*x] + 400*x*Log[3]*Log[4*x] - 50*x^2*Log[3]*Log[4*x] + 25*Log[3]*Log[4*x]^2]

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fricas [B]  time = 0.58, size = 59, normalized size = 2.03 \begin {gather*} \log \left (-50 \, {\left (x^{2} - 8 \, x + 16\right )} \log \relax (3) \log \left (4 \, x\right ) + 25 \, \log \relax (3) \log \left (4 \, x\right )^{2} + 25 \, {\left (x^{4} - 16 \, x^{3} + 96 \, x^{2} - 256 \, x + 256\right )} \log \relax (3) - 5 \, x + 4\right ) + \log \left (4 \, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((25*log(3)*log(4*x)^2+(-150*x^2+800*x-750)*log(3)*log(4*x)+(125*x^4-1600*x^3+7150*x^2-12400*x+5600)*
log(3)-10*x+4)/(25*x*log(3)*log(4*x)^2+(-50*x^3+400*x^2-800*x)*log(3)*log(4*x)+(25*x^5-400*x^4+2400*x^3-6400*x
^2+6400*x)*log(3)-5*x^2+4*x),x, algorithm="fricas")

[Out]

log(-50*(x^2 - 8*x + 16)*log(3)*log(4*x) + 25*log(3)*log(4*x)^2 + 25*(x^4 - 16*x^3 + 96*x^2 - 256*x + 256)*log
(3) - 5*x + 4) + log(4*x)

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giac [B]  time = 0.28, size = 77, normalized size = 2.66 \begin {gather*} \log \left (-25 \, x^{4} \log \relax (3) + 400 \, x^{3} \log \relax (3) + 50 \, x^{2} \log \relax (3) \log \left (4 \, x\right ) - 2400 \, x^{2} \log \relax (3) - 400 \, x \log \relax (3) \log \left (4 \, x\right ) - 25 \, \log \relax (3) \log \left (4 \, x\right )^{2} + 6400 \, x \log \relax (3) + 800 \, \log \relax (3) \log \left (4 \, x\right ) + 5 \, x - 6400 \, \log \relax (3) - 4\right ) + \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((25*log(3)*log(4*x)^2+(-150*x^2+800*x-750)*log(3)*log(4*x)+(125*x^4-1600*x^3+7150*x^2-12400*x+5600)*
log(3)-10*x+4)/(25*x*log(3)*log(4*x)^2+(-50*x^3+400*x^2-800*x)*log(3)*log(4*x)+(25*x^5-400*x^4+2400*x^3-6400*x
^2+6400*x)*log(3)-5*x^2+4*x),x, algorithm="giac")

[Out]

log(-25*x^4*log(3) + 400*x^3*log(3) + 50*x^2*log(3)*log(4*x) - 2400*x^2*log(3) - 400*x*log(3)*log(4*x) - 25*lo
g(3)*log(4*x)^2 + 6400*x*log(3) + 800*log(3)*log(4*x) + 5*x - 6400*log(3) - 4) + log(x)

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maple [B]  time = 0.19, size = 68, normalized size = 2.34

method	result	size
risch	\(\ln \relax (x )+\ln \left (\ln \left (4 x \right )^{2}+\left (-2 x^{2}+16 x -32\right ) \ln \left (4 x \right )+\frac {25 x^{4} \ln \relax (3)-400 x^{3} \ln \relax (3)+2400 x^{2} \ln \relax (3)-6400 x \ln \relax (3)+6400 \ln \relax (3)-5 x +4}{25 \ln \relax (3)}\right )\)	\(68\)
norman	\(\ln \left (4 x \right )+\ln \left (25 x^{4} \ln \relax (3)-50 \ln \relax (3) \ln \left (4 x \right ) x^{2}-400 x^{3} \ln \relax (3)+25 \ln \relax (3) \ln \left (4 x \right )^{2}+400 \ln \relax (3) \ln \left (4 x \right ) x +2400 x^{2} \ln \relax (3)-800 \ln \relax (3) \ln \left (4 x \right )-6400 x \ln \relax (3)+6400 \ln \relax (3)-5 x +4\right )\)	\(80\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((25*ln(3)*ln(4*x)^2+(-150*x^2+800*x-750)*ln(3)*ln(4*x)+(125*x^4-1600*x^3+7150*x^2-12400*x+5600)*ln(3)-10*x
+4)/(25*x*ln(3)*ln(4*x)^2+(-50*x^3+400*x^2-800*x)*ln(3)*ln(4*x)+(25*x^5-400*x^4+2400*x^3-6400*x^2+6400*x)*ln(3
)-5*x^2+4*x),x,method=_RETURNVERBOSE)

[Out]

ln(x)+ln(ln(4*x)^2+(-2*x^2+16*x-32)*ln(4*x)+1/25*(25*x^4*ln(3)-400*x^3*ln(3)+2400*x^2*ln(3)-6400*x*ln(3)+6400*
ln(3)-5*x+4)/ln(3))

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maxima [B]  time = 1.12, size = 108, normalized size = 3.72 \begin {gather*} \log \relax (x) + \log \left (\frac {25 \, x^{4} \log \relax (3) - 400 \, x^{3} \log \relax (3) - 100 \, {\left (\log \relax (3) \log \relax (2) - 24 \, \log \relax (3)\right )} x^{2} + 100 \, \log \relax (3) \log \relax (2)^{2} + 25 \, \log \relax (3) \log \relax (x)^{2} + 5 \, {\left (160 \, \log \relax (3) \log \relax (2) - 1280 \, \log \relax (3) - 1\right )} x - 1600 \, \log \relax (3) \log \relax (2) - 50 \, {\left (x^{2} \log \relax (3) - 8 \, x \log \relax (3) - 2 \, \log \relax (3) \log \relax (2) + 16 \, \log \relax (3)\right )} \log \relax (x) + 6400 \, \log \relax (3) + 4}{25 \, \log \relax (3)}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((25*log(3)*log(4*x)^2+(-150*x^2+800*x-750)*log(3)*log(4*x)+(125*x^4-1600*x^3+7150*x^2-12400*x+5600)*
log(3)-10*x+4)/(25*x*log(3)*log(4*x)^2+(-50*x^3+400*x^2-800*x)*log(3)*log(4*x)+(25*x^5-400*x^4+2400*x^3-6400*x
^2+6400*x)*log(3)-5*x^2+4*x),x, algorithm="maxima")

[Out]

log(x) + log(1/25*(25*x^4*log(3) - 400*x^3*log(3) - 100*(log(3)*log(2) - 24*log(3))*x^2 + 100*log(3)*log(2)^2
+ 25*log(3)*log(x)^2 + 5*(160*log(3)*log(2) - 1280*log(3) - 1)*x - 1600*log(3)*log(2) - 50*(x^2*log(3) - 8*x*l
og(3) - 2*log(3)*log(2) + 16*log(3))*log(x) + 6400*log(3) + 4)/log(3))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {25\,\ln \relax (3)\,{\ln \left (4\,x\right )}^2-\ln \relax (3)\,\left (150\,x^2-800\,x+750\right )\,\ln \left (4\,x\right )-10\,x+\ln \relax (3)\,\left (125\,x^4-1600\,x^3+7150\,x^2-12400\,x+5600\right )+4}{4\,x-5\,x^2+\ln \relax (3)\,\left (25\,x^5-400\,x^4+2400\,x^3-6400\,x^2+6400\,x\right )-\ln \left (4\,x\right )\,\ln \relax (3)\,\left (50\,x^3-400\,x^2+800\,x\right )+25\,x\,{\ln \left (4\,x\right )}^2\,\ln \relax (3)} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((25*log(4*x)^2*log(3) - 10*x + log(3)*(7150*x^2 - 12400*x - 1600*x^3 + 125*x^4 + 5600) - log(4*x)*log(3)*(
150*x^2 - 800*x + 750) + 4)/(4*x - 5*x^2 + log(3)*(6400*x - 6400*x^2 + 2400*x^3 - 400*x^4 + 25*x^5) - log(4*x)
*log(3)*(800*x - 400*x^2 + 50*x^3) + 25*x*log(4*x)^2*log(3)),x)

[Out]

int((25*log(4*x)^2*log(3) - 10*x + log(3)*(7150*x^2 - 12400*x - 1600*x^3 + 125*x^4 + 5600) - log(4*x)*log(3)*(
150*x^2 - 800*x + 750) + 4)/(4*x - 5*x^2 + log(3)*(6400*x - 6400*x^2 + 2400*x^3 - 400*x^4 + 25*x^5) - log(4*x)
*log(3)*(800*x - 400*x^2 + 50*x^3) + 25*x*log(4*x)^2*log(3)), x)

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sympy [B]  time = 0.46, size = 73, normalized size = 2.52 \begin {gather*} \log {\relax (x )} + \log {\left (\left (- 2 x^{2} + 16 x - 32\right ) \log {\left (4 x \right )} + \frac {25 x^{4} \log {\relax (3 )} - 400 x^{3} \log {\relax (3 )} + 2400 x^{2} \log {\relax (3 )} - 6400 x \log {\relax (3 )} - 5 x + 4 + 6400 \log {\relax (3 )}}{25 \log {\relax (3 )}} + \log {\left (4 x \right )}^{2} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((25*ln(3)*ln(4*x)**2+(-150*x**2+800*x-750)*ln(3)*ln(4*x)+(125*x**4-1600*x**3+7150*x**2-12400*x+5600)
*ln(3)-10*x+4)/(25*x*ln(3)*ln(4*x)**2+(-50*x**3+400*x**2-800*x)*ln(3)*ln(4*x)+(25*x**5-400*x**4+2400*x**3-6400
*x**2+6400*x)*ln(3)-5*x**2+4*x),x)

[Out]

log(x) + log((-2*x**2 + 16*x - 32)*log(4*x) + (25*x**4*log(3) - 400*x**3*log(3) + 2400*x**2*log(3) - 6400*x*lo
g(3) - 5*x + 4 + 6400*log(3))/(25*log(3)) + log(4*x)**2)

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