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

Optimal. Leaf size=24 \[ 4+\frac {\log \left (5 \left (2+x+\frac {1}{10} (2-x+\log (x))\right )\right )}{\log (x)} \]

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Rubi [F]  time = 1.03, 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 {(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 \end {gather*}

Verification is not applicable to the result.

[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 {gather*} \begin {aligned} \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) \operatorname {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 {aligned} \end {gather*}

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Mathematica [A]  time = 0.24, size = 22, normalized size = 0.92 \begin {gather*} -\frac {\log (2)}{\log (x)}+\frac {\log (22+9 x+\log (x))}{\log (x)} \end {gather*}

Antiderivative was successfully verified.

[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]

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fricas [A]  time = 0.64, size = 15, normalized size = 0.62 \begin {gather*} \frac {\log \left (\frac {9}{2} \, x + \frac {1}{2} \, \log \relax (x) + 11\right )}{\log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [A]  time = 0.24, size = 22, normalized size = 0.92 \begin {gather*} -\frac {\log \relax (2)}{\log \relax (x)} + \frac {\log \left (9 \, x + \log \relax (x) + 22\right )}{\log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [A]  time = 0.03, size = 16, normalized size = 0.67

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maxima [A]  time = 0.50, size = 19, normalized size = 0.79 \begin {gather*} -\frac {\log \relax (2) - \log \left (9 \, x + \log \relax (x) + 22\right )}{\log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [B]  time = 5.42, size = 19, normalized size = 0.79 \begin {gather*} -\frac {\ln \relax (2)-\ln \left (9\,x+\ln \relax (x)+22\right )}{\ln \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [A]  time = 0.39, size = 15, normalized size = 0.62 \begin {gather*} \frac {\log {\left (\frac {9 x}{2} + \frac {\log {\relax (x )}}{2} + 11 \right )}}{\log {\relax (x )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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]

log(9*x/2 + log(x)/2 + 11)/log(x)

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