3.56.85 \(\int \frac {(58-9 x) \log ^2(x)+e^{\frac {1}{\log (x)}} (-20+40 \log ^2(x))}{100 e^{\frac {1}{\log (x)}} x \log ^2(x)+(145 x-15 x^2) \log ^2(x)} \, dx\)

Optimal. Leaf size=27 \[ \frac {1}{5} \log \left (x^2 \left (25+4 \left (1+5 e^{\frac {1}{\log (x)}}-x\right )+x\right )\right ) \]

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Rubi [F]  time = 1.48, 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 {(58-9 x) \log ^2(x)+e^{\frac {1}{\log (x)}} \left (-20+40 \log ^2(x)\right )}{100 e^{\frac {1}{\log (x)}} x \log ^2(x)+\left (145 x-15 x^2\right ) \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((58 - 9*x)*Log[x]^2 + E^Log[x]^(-1)*(-20 + 40*Log[x]^2))/(100*E^Log[x]^(-1)*x*Log[x]^2 + (145*x - 15*x^2)
*Log[x]^2),x]

[Out]

1/(5*Log[x]) + (2*Log[x])/5 - (3*Defer[Int][(29 + 20*E^Log[x]^(-1) - 3*x)^(-1), x])/5 - (3*Defer[Int][1/((29 +
20*E^Log[x]^(-1) - 3*x)*Log[x]^2), x])/5 + (29*Defer[Int][1/((29 + 20*E^Log[x]^(-1) - 3*x)*x*Log[x]^2), x])/5

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {(58-9 x) \log ^2(x)+e^{\frac {1}{\log (x)}} \left (-20+40 \log ^2(x)\right )}{5 \left (29+20 e^{\frac {1}{\log (x)}}-3 x\right ) x \log ^2(x)} \, dx\\ &=\frac {1}{5} \int \frac {(58-9 x) \log ^2(x)+e^{\frac {1}{\log (x)}} \left (-20+40 \log ^2(x)\right )}{\left (29+20 e^{\frac {1}{\log (x)}}-3 x\right ) x \log ^2(x)} \, dx\\ &=\frac {1}{5} \int \left (\frac {-1+2 \log ^2(x)}{x \log ^2(x)}-\frac {-29+3 x+3 x \log ^2(x)}{\left (29+20 e^{\frac {1}{\log (x)}}-3 x\right ) x \log ^2(x)}\right ) \, dx\\ &=\frac {1}{5} \int \frac {-1+2 \log ^2(x)}{x \log ^2(x)} \, dx-\frac {1}{5} \int \frac {-29+3 x+3 x \log ^2(x)}{\left (29+20 e^{\frac {1}{\log (x)}}-3 x\right ) x \log ^2(x)} \, dx\\ &=-\left (\frac {1}{5} \int \left (\frac {3}{29+20 e^{\frac {1}{\log (x)}}-3 x}+\frac {3}{\left (29+20 e^{\frac {1}{\log (x)}}-3 x\right ) \log ^2(x)}-\frac {29}{\left (29+20 e^{\frac {1}{\log (x)}}-3 x\right ) x \log ^2(x)}\right ) \, dx\right )+\frac {1}{5} \operatorname {Subst}\left (\int \frac {-1+2 x^2}{x^2} \, dx,x,\log (x)\right )\\ &=\frac {1}{5} \operatorname {Subst}\left (\int \left (2-\frac {1}{x^2}\right ) \, dx,x,\log (x)\right )-\frac {3}{5} \int \frac {1}{29+20 e^{\frac {1}{\log (x)}}-3 x} \, dx-\frac {3}{5} \int \frac {1}{\left (29+20 e^{\frac {1}{\log (x)}}-3 x\right ) \log ^2(x)} \, dx+\frac {29}{5} \int \frac {1}{\left (29+20 e^{\frac {1}{\log (x)}}-3 x\right ) x \log ^2(x)} \, dx\\ &=\frac {1}{5 \log (x)}+\frac {2 \log (x)}{5}-\frac {3}{5} \int \frac {1}{29+20 e^{\frac {1}{\log (x)}}-3 x} \, dx-\frac {3}{5} \int \frac {1}{\left (29+20 e^{\frac {1}{\log (x)}}-3 x\right ) \log ^2(x)} \, dx+\frac {29}{5} \int \frac {1}{\left (29+20 e^{\frac {1}{\log (x)}}-3 x\right ) x \log ^2(x)} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.50, size = 23, normalized size = 0.85 \begin {gather*} \frac {1}{5} \left (\log \left (29+20 e^{\frac {1}{\log (x)}}-3 x\right )+2 \log (x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((58 - 9*x)*Log[x]^2 + E^Log[x]^(-1)*(-20 + 40*Log[x]^2))/(100*E^Log[x]^(-1)*x*Log[x]^2 + (145*x - 1
5*x^2)*Log[x]^2),x]

[Out]

(Log[29 + 20*E^Log[x]^(-1) - 3*x] + 2*Log[x])/5

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fricas [A]  time = 1.40, size = 20, normalized size = 0.74 \begin {gather*} \frac {2}{5} \, \log \relax (x) + \frac {1}{5} \, \log \left (-3 \, x + 20 \, e^{\frac {1}{\log \relax (x)}} + 29\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((40*log(x)^2-20)*exp(1/log(x))+(-9*x+58)*log(x)^2)/(100*x*log(x)^2*exp(1/log(x))+(-15*x^2+145*x)*lo
g(x)^2),x, algorithm="fricas")

[Out]

2/5*log(x) + 1/5*log(-3*x + 20*e^(1/log(x)) + 29)

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giac [A]  time = 0.22, size = 20, normalized size = 0.74 \begin {gather*} \frac {2}{5} \, \log \relax (x) + \frac {1}{5} \, \log \left (-3 \, x + 20 \, e^{\frac {1}{\log \relax (x)}} + 29\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((40*log(x)^2-20)*exp(1/log(x))+(-9*x+58)*log(x)^2)/(100*x*log(x)^2*exp(1/log(x))+(-15*x^2+145*x)*lo
g(x)^2),x, algorithm="giac")

[Out]

2/5*log(x) + 1/5*log(-3*x + 20*e^(1/log(x)) + 29)

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maple [A]  time = 0.05, size = 19, normalized size = 0.70

method	result	size
risch	\(\frac {2 \ln \relax (x )}{5}+\frac {\ln \left (-\frac {3 x}{20}+{\mathrm e}^{\frac {1}{\ln \relax (x )}}+\frac {29}{20}\right )}{5}\)	\(19\)
norman	\(\frac {2 \ln \relax (x )}{5}+\frac {\ln \left (3 x -20 \,{\mathrm e}^{\frac {1}{\ln \relax (x )}}-29\right )}{5}\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((40*ln(x)^2-20)*exp(1/ln(x))+(-9*x+58)*ln(x)^2)/(100*x*ln(x)^2*exp(1/ln(x))+(-15*x^2+145*x)*ln(x)^2),x,me
thod=_RETURNVERBOSE)

[Out]

2/5*ln(x)+1/5*ln(-3/20*x+exp(1/ln(x))+29/20)

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maxima [A]  time = 0.45, size = 18, normalized size = 0.67 \begin {gather*} \frac {2}{5} \, \log \relax (x) + \frac {1}{5} \, \log \left (-\frac {3}{20} \, x + e^{\frac {1}{\log \relax (x)}} + \frac {29}{20}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((40*log(x)^2-20)*exp(1/log(x))+(-9*x+58)*log(x)^2)/(100*x*log(x)^2*exp(1/log(x))+(-15*x^2+145*x)*lo
g(x)^2),x, algorithm="maxima")

[Out]

2/5*log(x) + 1/5*log(-3/20*x + e^(1/log(x)) + 29/20)

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mupad [B]  time = 3.70, size = 18, normalized size = 0.67 \begin {gather*} \frac {\ln \left ({\mathrm {e}}^{\frac {1}{\ln \relax (x)}}-\frac {3\,x}{20}+\frac {29}{20}\right )}{5}+\frac {2\,\ln \relax (x)}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(1/log(x))*(40*log(x)^2 - 20) - log(x)^2*(9*x - 58))/(log(x)^2*(145*x - 15*x^2) + 100*x*exp(1/log(x))*
log(x)^2),x)

[Out]

log(exp(1/log(x)) - (3*x)/20 + 29/20)/5 + (2*log(x))/5

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sympy [A]  time = 0.36, size = 24, normalized size = 0.89 \begin {gather*} \frac {2 \log {\relax (x )}}{5} + \frac {\log {\left (- \frac {3 x}{20} + e^{\frac {1}{\log {\relax (x )}}} + \frac {29}{20} \right )}}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((40*ln(x)**2-20)*exp(1/ln(x))+(-9*x+58)*ln(x)**2)/(100*x*ln(x)**2*exp(1/ln(x))+(-15*x**2+145*x)*ln(
x)**2),x)

[Out]

2*log(x)/5 + log(-3*x/20 + exp(1/log(x)) + 29/20)/5

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