3.97.87 \(\int \frac {-15 x \log (3)+5 x \log (3) \log (x)+(5+5 x) \log (3) \log (1+x)}{(288 x+288 x^2+(-192 x-192 x^2) \log (x)+(32 x+32 x^2) \log ^2(x)) \log ^2(1+x)} \, dx\)

Optimal. Leaf size=18 \[ -\frac {5 \log (3)}{32 (-3+\log (x)) \log (1+x)} \]

________________________________________________________________________________________

Rubi [F]  time = 0.90, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {-15 x \log (3)+5 x \log (3) \log (x)+(5+5 x) \log (3) \log (1+x)}{\left (288 x+288 x^2+\left (-192 x-192 x^2\right ) \log (x)+\left (32 x+32 x^2\right ) \log ^2(x)\right ) \log ^2(1+x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-15*x*Log[3] + 5*x*Log[3]*Log[x] + (5 + 5*x)*Log[3]*Log[1 + x])/((288*x + 288*x^2 + (-192*x - 192*x^2)*Lo
g[x] + (32*x + 32*x^2)*Log[x]^2)*Log[1 + x]^2),x]

[Out]

(5*Log[3]*Defer[Int][1/((1 + x)*(-3 + Log[x])*Log[1 + x]^2), x])/32 + (5*Log[3]*Defer[Int][1/(x*(-3 + Log[x])^
2*Log[1 + x]), x])/32

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {5 \log (3) (-3 x+x \log (x)+(1+x) \log (1+x))}{32 x (1+x) (3-\log (x))^2 \log ^2(1+x)} \, dx\\ &=\frac {1}{32} (5 \log (3)) \int \frac {-3 x+x \log (x)+(1+x) \log (1+x)}{x (1+x) (3-\log (x))^2 \log ^2(1+x)} \, dx\\ &=\frac {1}{32} (5 \log (3)) \int \left (\frac {1}{(1+x) (-3+\log (x)) \log ^2(1+x)}+\frac {1}{x (-3+\log (x))^2 \log (1+x)}\right ) \, dx\\ &=\frac {1}{32} (5 \log (3)) \int \frac {1}{(1+x) (-3+\log (x)) \log ^2(1+x)} \, dx+\frac {1}{32} (5 \log (3)) \int \frac {1}{x (-3+\log (x))^2 \log (1+x)} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.09, size = 20, normalized size = 1.11 \begin {gather*} \frac {5 \log (3)}{32 (3-\log (x)) \log (1+x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-15*x*Log[3] + 5*x*Log[3]*Log[x] + (5 + 5*x)*Log[3]*Log[1 + x])/((288*x + 288*x^2 + (-192*x - 192*x
^2)*Log[x] + (32*x + 32*x^2)*Log[x]^2)*Log[1 + x]^2),x]

[Out]

(5*Log[3])/(32*(3 - Log[x])*Log[1 + x])

________________________________________________________________________________________

fricas [A]  time = 0.68, size = 16, normalized size = 0.89 \begin {gather*} -\frac {5 \, \log \relax (3)}{32 \, {\left (\log \relax (x) - 3\right )} \log \left (x + 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*x+5)*log(3)*log(x+1)+5*x*log(3)*log(x)-15*x*log(3))/((32*x^2+32*x)*log(x)^2+(-192*x^2-192*x)*log
(x)+288*x^2+288*x)/log(x+1)^2,x, algorithm="fricas")

[Out]

-5/32*log(3)/((log(x) - 3)*log(x + 1))

________________________________________________________________________________________

giac [A]  time = 0.13, size = 20, normalized size = 1.11 \begin {gather*} -\frac {5 \, \log \relax (3)}{32 \, {\left (\log \left (x + 1\right ) \log \relax (x) - 3 \, \log \left (x + 1\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*x+5)*log(3)*log(x+1)+5*x*log(3)*log(x)-15*x*log(3))/((32*x^2+32*x)*log(x)^2+(-192*x^2-192*x)*log
(x)+288*x^2+288*x)/log(x+1)^2,x, algorithm="giac")

[Out]

-5/32*log(3)/(log(x + 1)*log(x) - 3*log(x + 1))

________________________________________________________________________________________

maple [A]  time = 0.06, size = 17, normalized size = 0.94

method	result	size
risch	\(-\frac {5 \ln \relax (3)}{32 \left (\ln \relax (x )-3\right ) \ln \left (x +1\right )}\)	\(17\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((5*x+5)*ln(3)*ln(x+1)+5*x*ln(3)*ln(x)-15*x*ln(3))/((32*x^2+32*x)*ln(x)^2+(-192*x^2-192*x)*ln(x)+288*x^2+2
88*x)/ln(x+1)^2,x,method=_RETURNVERBOSE)

[Out]

-5/32*ln(3)/(ln(x)-3)/ln(x+1)

________________________________________________________________________________________

maxima [A]  time = 0.49, size = 16, normalized size = 0.89 \begin {gather*} -\frac {5 \, \log \relax (3)}{32 \, {\left (\log \relax (x) - 3\right )} \log \left (x + 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*x+5)*log(3)*log(x+1)+5*x*log(3)*log(x)-15*x*log(3))/((32*x^2+32*x)*log(x)^2+(-192*x^2-192*x)*log
(x)+288*x^2+288*x)/log(x+1)^2,x, algorithm="maxima")

[Out]

-5/32*log(3)/((log(x) - 3)*log(x + 1))

________________________________________________________________________________________

mupad [B]  time = 6.46, size = 110, normalized size = 6.11 \begin {gather*} \frac {\frac {5\,\ln \relax (3)}{32\,x}-\frac {5\,\ln \relax (3)\,\ln \relax (x)}{64\,x}}{\ln \relax (x)-3}-\frac {\frac {5\,\ln \relax (3)}{32\,\left (\ln \relax (x)-3\right )}+\frac {5\,\ln \left (x+1\right )\,\ln \relax (3)\,\left (x+1\right )}{32\,x\,{\left (\ln \relax (x)-3\right )}^2}}{\ln \left (x+1\right )}+\frac {5\,\ln \relax (3)}{64\,x}-\frac {\frac {5\,\left (\ln \relax (3)-2\,x\,\ln \relax (3)\right )}{64\,x}-\frac {5\,\ln \relax (3)\,\ln \relax (x)}{64\,x}}{{\ln \relax (x)}^2-6\,\ln \relax (x)+9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5*x*log(3)*log(x) - 15*x*log(3) + log(x + 1)*log(3)*(5*x + 5))/(log(x + 1)^2*(288*x + log(x)^2*(32*x + 32
*x^2) - log(x)*(192*x + 192*x^2) + 288*x^2)),x)

[Out]

((5*log(3))/(32*x) - (5*log(3)*log(x))/(64*x))/(log(x) - 3) - ((5*log(3))/(32*(log(x) - 3)) + (5*log(x + 1)*lo
g(3)*(x + 1))/(32*x*(log(x) - 3)^2))/log(x + 1) + (5*log(3))/(64*x) - ((5*(log(3) - 2*x*log(3)))/(64*x) - (5*l
og(3)*log(x))/(64*x))/(log(x)^2 - 6*log(x) + 9)

________________________________________________________________________________________

sympy [A]  time = 0.26, size = 17, normalized size = 0.94 \begin {gather*} - \frac {5 \log {\relax (3 )}}{\left (32 \log {\relax (x )} - 96\right ) \log {\left (x + 1 \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*x+5)*ln(3)*ln(x+1)+5*x*ln(3)*ln(x)-15*x*ln(3))/((32*x**2+32*x)*ln(x)**2+(-192*x**2-192*x)*ln(x)+
288*x**2+288*x)/ln(x+1)**2,x)

[Out]

-5*log(3)/((32*log(x) - 96)*log(x + 1))

________________________________________________________________________________________