3.86.82 \(\int \frac {-391+16 x+(92-4 x) \log (23-x)+(-92+4 x) \log (x)}{-92 x+4 x^2+(23 x-x^2) \log (23-x)+(-23 x+x^2) \log (x)} \, dx\)

Optimal. Leaf size=19 \[ -2+\log \left (x^4 (4-\log (23-x)+\log (x))\right ) \]

________________________________________________________________________________________

Rubi [A]  time = 0.53, antiderivative size = 18, normalized size of antiderivative = 0.95, number of steps used = 4, number of rules used = 3, integrand size = 63, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {6741, 6742, 6684} \begin {gather*} 4 \log (x)+\log (-\log (23-x)+\log (x)+4) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-391 + 16*x + (92 - 4*x)*Log[23 - x] + (-92 + 4*x)*Log[x])/(-92*x + 4*x^2 + (23*x - x^2)*Log[23 - x] + (-
23*x + x^2)*Log[x]),x]

[Out]

4*Log[x] + Log[4 - Log[23 - x] + Log[x]]

Rule 6684

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

Rule 6741

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

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 \frac {391-16 x-(92-4 x) \log (23-x)-(-92+4 x) \log (x)}{(23-x) x (4-\log (23-x)+\log (x))} \, dx\\ &=\int \left (\frac {4}{x}+\frac {23}{(-23+x) x (-4+\log (23-x)-\log (x))}\right ) \, dx\\ &=4 \log (x)+23 \int \frac {1}{(-23+x) x (-4+\log (23-x)-\log (x))} \, dx\\ &=4 \log (x)+\log (4-\log (23-x)+\log (x))\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.13, size = 18, normalized size = 0.95 \begin {gather*} 4 \log (x)+\log (4-\log (23-x)+\log (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-391 + 16*x + (92 - 4*x)*Log[23 - x] + (-92 + 4*x)*Log[x])/(-92*x + 4*x^2 + (23*x - x^2)*Log[23 - x
] + (-23*x + x^2)*Log[x]),x]

[Out]

4*Log[x] + Log[4 - Log[23 - x] + Log[x]]

________________________________________________________________________________________

fricas [A]  time = 0.62, size = 18, normalized size = 0.95 \begin {gather*} 4 \, \log \relax (x) + \log \left (\log \relax (x) - \log \left (-x + 23\right ) + 4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x-92)*log(x)+(-4*x+92)*log(23-x)+16*x-391)/((x^2-23*x)*log(x)+(-x^2+23*x)*log(23-x)+4*x^2-92*x),
x, algorithm="fricas")

[Out]

4*log(x) + log(log(x) - log(-x + 23) + 4)

________________________________________________________________________________________

giac [A]  time = 0.15, size = 18, normalized size = 0.95 \begin {gather*} 4 \, \log \relax (x) + \log \left (-\log \relax (x) + \log \left (-x + 23\right ) - 4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x-92)*log(x)+(-4*x+92)*log(23-x)+16*x-391)/((x^2-23*x)*log(x)+(-x^2+23*x)*log(23-x)+4*x^2-92*x),
x, algorithm="giac")

[Out]

4*log(x) + log(-log(x) + log(-x + 23) - 4)

________________________________________________________________________________________

maple [A]  time = 0.06, size = 19, normalized size = 1.00

method	result	size
norman	\(4 \ln \relax (x )+\ln \left (4+\ln \relax (x )-\ln \left (23-x \right )\right )\)	\(19\)
risch	\(4 \ln \relax (x )+\ln \left (-\ln \relax (x )+\ln \left (23-x \right )-4\right )\)	\(19\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((4*x-92)*ln(x)+(-4*x+92)*ln(23-x)+16*x-391)/((x^2-23*x)*ln(x)+(-x^2+23*x)*ln(23-x)+4*x^2-92*x),x,method=_
RETURNVERBOSE)

[Out]

4*ln(x)+ln(4+ln(x)-ln(23-x))

________________________________________________________________________________________

maxima [A]  time = 0.43, size = 18, normalized size = 0.95 \begin {gather*} 4 \, \log \relax (x) + \log \left (-\log \relax (x) + \log \left (-x + 23\right ) - 4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x-92)*log(x)+(-4*x+92)*log(23-x)+16*x-391)/((x^2-23*x)*log(x)+(-x^2+23*x)*log(23-x)+4*x^2-92*x),
x, algorithm="maxima")

[Out]

4*log(x) + log(-log(x) + log(-x + 23) - 4)

________________________________________________________________________________________

mupad [B]  time = 5.66, size = 18, normalized size = 0.95 \begin {gather*} \ln \left (\ln \left (23-x\right )-\ln \relax (x)-4\right )+4\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(16*x - log(23 - x)*(4*x - 92) + log(x)*(4*x - 92) - 391)/(92*x - log(23 - x)*(23*x - x^2) + log(x)*(23*x
- x^2) - 4*x^2),x)

[Out]

log(log(23 - x) - log(x) - 4) + 4*log(x)

________________________________________________________________________________________

sympy [A]  time = 0.35, size = 15, normalized size = 0.79 \begin {gather*} 4 \log {\relax (x )} + \log {\left (- \log {\relax (x )} + \log {\left (23 - x \right )} - 4 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x-92)*ln(x)+(-4*x+92)*ln(23-x)+16*x-391)/((x**2-23*x)*ln(x)+(-x**2+23*x)*ln(23-x)+4*x**2-92*x),x
)

[Out]

4*log(x) + log(-log(x) + log(23 - x) - 4)

________________________________________________________________________________________