\(\int \frac {(1+9 x) \log (x)+(-22-9 x-\log (x)) \log (\frac {1}{2} (22+9 x+\log (x)))}{(22 x+9 x^2) \log ^2(x)+x \log ^3(x)} \, dx\) [8247]

Optimal result

Integrand size = 55, antiderivative size = 24 \[ \int \frac {(1+9 x) \log (x)+(-22-9 x-\log (x)) \log \left (\frac {1}{2} (22+9 x+\log (x))\right )}{\left (22 x+9 x^2\right ) \log ^2(x)+x \log ^3(x)} \, dx=4+\frac {\log \left (5 \left (2+x+\frac {1}{10} (2-x+\log (x))\right )\right )}{\log (x)} \]

[Out]

ln(1/2*ln(x)+9/2*x+11)/ln(x)+4

Rubi [F]

\[ \int \frac {(1+9 x) \log (x)+(-22-9 x-\log (x)) \log \left (\frac {1}{2} (22+9 x+\log (x))\right )}{\left (22 x+9 x^2\right ) \log ^2(x)+x \log ^3(x)} \, dx=\int \frac {(1+9 x) \log (x)+(-22-9 x-\log (x)) \log \left (\frac {1}{2} (22+9 x+\log (x))\right )}{\left (22 x+9 x^2\right ) \log ^2(x)+x \log ^3(x)} \, dx \]

[In]

Int[((1 + 9*x)*Log[x] + (-22 - 9*x - Log[x])*Log[(22 + 9*x + Log[x])/2])/((22*x + 9*x^2)*Log[x]^2 + x*Log[x]^3
),x]

[Out]

-(Log[2]/Log[x]) + Defer[Int][(1 + 9*x)/(x*(22 + 9*x)*Log[x]), x] - Defer[Int][1/(x*(22 + 9*x + Log[x])), x]/2
2 - (189*Defer[Int][1/((22 + 9*x)*(22 + 9*x + Log[x])), x])/22 - Defer[Int][Log[22 + 9*x + Log[x]]/(x*Log[x]^2
), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {\log (2)+\frac {(1+9 x) \log (x)}{22+9 x+\log (x)}-\log (22+9 x+\log (x))}{x \log ^2(x)} \, dx \\ & = \int \left (\frac {22 \log (2)+9 x \log (2)+9 x \log (x)+(1+\log (2)) \log (x)}{x \log ^2(x) (22+9 x+\log (x))}-\frac {\log (22+9 x+\log (x))}{x \log ^2(x)}\right ) \, dx \\ & = \int \frac {22 \log (2)+9 x \log (2)+9 x \log (x)+(1+\log (2)) \log (x)}{x \log ^2(x) (22+9 x+\log (x))} \, dx-\int \frac {\log (22+9 x+\log (x))}{x \log ^2(x)} \, dx \\ & = \int \left (\frac {\log (2)}{x \log ^2(x)}+\frac {1+9 x}{x (22+9 x) \log (x)}+\frac {-1-9 x}{x (22+9 x) (22+9 x+\log (x))}\right ) \, dx-\int \frac {\log (22+9 x+\log (x))}{x \log ^2(x)} \, dx \\ & = \log (2) \int \frac {1}{x \log ^2(x)} \, dx+\int \frac {1+9 x}{x (22+9 x) \log (x)} \, dx+\int \frac {-1-9 x}{x (22+9 x) (22+9 x+\log (x))} \, dx-\int \frac {\log (22+9 x+\log (x))}{x \log ^2(x)} \, dx \\ & = \log (2) \text {Subst}\left (\int \frac {1}{x^2} \, dx,x,\log (x)\right )+\int \frac {1+9 x}{x (22+9 x) \log (x)} \, dx+\int \left (-\frac {1}{22 x (22+9 x+\log (x))}-\frac {189}{22 (22+9 x) (22+9 x+\log (x))}\right ) \, dx-\int \frac {\log (22+9 x+\log (x))}{x \log ^2(x)} \, dx \\ & = -\frac {\log (2)}{\log (x)}-\frac {1}{22} \int \frac {1}{x (22+9 x+\log (x))} \, dx-\frac {189}{22} \int \frac {1}{(22+9 x) (22+9 x+\log (x))} \, dx+\int \frac {1+9 x}{x (22+9 x) \log (x)} \, dx-\int \frac {\log (22+9 x+\log (x))}{x \log ^2(x)} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {(1+9 x) \log (x)+(-22-9 x-\log (x)) \log \left (\frac {1}{2} (22+9 x+\log (x))\right )}{\left (22 x+9 x^2\right ) \log ^2(x)+x \log ^3(x)} \, dx=-\frac {\log (2)}{\log (x)}+\frac {\log (22+9 x+\log (x))}{\log (x)} \]

[In]

Integrate[((1 + 9*x)*Log[x] + (-22 - 9*x - Log[x])*Log[(22 + 9*x + Log[x])/2])/((22*x + 9*x^2)*Log[x]^2 + x*Lo
g[x]^3),x]

[Out]

-(Log[2]/Log[x]) + Log[22 + 9*x + Log[x]]/Log[x]

Maple [A] (verified)

Time = 0.51 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.67

method	result	size
risch	\(\frac {\ln \left (\frac {\ln \left (x \right )}{2}+\frac {9 x}{2}+11\right )}{\ln \left (x \right )}\)	\(16\)
parallelrisch	\(\frac {\ln \left (\frac {\ln \left (x \right )}{2}+\frac {9 x}{2}+11\right )}{\ln \left (x \right )}\)	\(16\)

[In]

int(((-ln(x)-9*x-22)*ln(1/2*ln(x)+9/2*x+11)+(9*x+1)*ln(x))/(x*ln(x)^3+(9*x^2+22*x)*ln(x)^2),x,method=_RETURNVE
RBOSE)

[Out]

ln(1/2*ln(x)+9/2*x+11)/ln(x)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.62 \[ \int \frac {(1+9 x) \log (x)+(-22-9 x-\log (x)) \log \left (\frac {1}{2} (22+9 x+\log (x))\right )}{\left (22 x+9 x^2\right ) \log ^2(x)+x \log ^3(x)} \, dx=\frac {\log \left (\frac {9}{2} \, x + \frac {1}{2} \, \log \left (x\right ) + 11\right )}{\log \left (x\right )} \]

[In]

integrate(((-log(x)-9*x-22)*log(1/2*log(x)+9/2*x+11)+(9*x+1)*log(x))/(x*log(x)^3+(9*x^2+22*x)*log(x)^2),x, alg
orithm="fricas")

[Out]

log(9/2*x + 1/2*log(x) + 11)/log(x)

Sympy [F(-2)]

Exception generated. \[ \int \frac {(1+9 x) \log (x)+(-22-9 x-\log (x)) \log \left (\frac {1}{2} (22+9 x+\log (x))\right )}{\left (22 x+9 x^2\right ) \log ^2(x)+x \log ^3(x)} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(((-ln(x)-9*x-22)*ln(1/2*ln(x)+9/2*x+11)+(9*x+1)*ln(x))/(x*ln(x)**3+(9*x**2+22*x)*ln(x)**2),x)

[Out]

Exception raised: TypeError >> '>' not supported between instances of 'Poly' and 'int'

Maxima [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.79 \[ \int \frac {(1+9 x) \log (x)+(-22-9 x-\log (x)) \log \left (\frac {1}{2} (22+9 x+\log (x))\right )}{\left (22 x+9 x^2\right ) \log ^2(x)+x \log ^3(x)} \, dx=-\frac {\log \left (2\right ) - \log \left (9 \, x + \log \left (x\right ) + 22\right )}{\log \left (x\right )} \]

[In]

integrate(((-log(x)-9*x-22)*log(1/2*log(x)+9/2*x+11)+(9*x+1)*log(x))/(x*log(x)^3+(9*x^2+22*x)*log(x)^2),x, alg
orithm="maxima")

[Out]

-(log(2) - log(9*x + log(x) + 22))/log(x)

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {(1+9 x) \log (x)+(-22-9 x-\log (x)) \log \left (\frac {1}{2} (22+9 x+\log (x))\right )}{\left (22 x+9 x^2\right ) \log ^2(x)+x \log ^3(x)} \, dx=-\frac {\log \left (2\right )}{\log \left (x\right )} + \frac {\log \left (9 \, x + \log \left (x\right ) + 22\right )}{\log \left (x\right )} \]

[In]

integrate(((-log(x)-9*x-22)*log(1/2*log(x)+9/2*x+11)+(9*x+1)*log(x))/(x*log(x)^3+(9*x^2+22*x)*log(x)^2),x, alg
orithm="giac")

[Out]

-log(2)/log(x) + log(9*x + log(x) + 22)/log(x)

Mupad [B] (verification not implemented)

Time = 14.25 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.79 \[ \int \frac {(1+9 x) \log (x)+(-22-9 x-\log (x)) \log \left (\frac {1}{2} (22+9 x+\log (x))\right )}{\left (22 x+9 x^2\right ) \log ^2(x)+x \log ^3(x)} \, dx=-\frac {\ln \left (2\right )-\ln \left (9\,x+\ln \left (x\right )+22\right )}{\ln \left (x\right )} \]

[In]

int(-(log((9*x)/2 + log(x)/2 + 11)*(9*x + log(x) + 22) - log(x)*(9*x + 1))/(log(x)^2*(22*x + 9*x^2) + x*log(x)
^3),x)

[Out]

-(log(2) - log(9*x + log(x) + 22))/log(x)