\(\int \frac {188-62 \log (x^2)+(-3+\log (x^2)) \log (3-\log (x^2))}{-45+15 \log (x^2)} \, dx\) [6229]

Optimal result

Integrand size = 35, antiderivative size = 21 \[ \int \frac {188-62 \log \left (x^2\right )+\left (-3+\log \left (x^2\right )\right ) \log \left (3-\log \left (x^2\right )\right )}{-45+15 \log \left (x^2\right )} \, dx=-1-4 x+\frac {1}{15} x \left (-2+\log \left (3-\log \left (x^2\right )\right )\right ) \]

[Out]

1/15*x*(ln(3-ln(x^2))-2)-1-4*x

Rubi [F]

\[ \int \frac {188-62 \log \left (x^2\right )+\left (-3+\log \left (x^2\right )\right ) \log \left (3-\log \left (x^2\right )\right )}{-45+15 \log \left (x^2\right )} \, dx=\int \frac {188-62 \log \left (x^2\right )+\left (-3+\log \left (x^2\right )\right ) \log \left (3-\log \left (x^2\right )\right )}{-45+15 \log \left (x^2\right )} \, dx \]

[In]

Int[(188 - 62*Log[x^2] + (-3 + Log[x^2])*Log[3 - Log[x^2]])/(-45 + 15*Log[x^2]),x]

[Out]

(-62*x)/15 + (E^(3/2)*x*ExpIntegralEi[(-3 + Log[x^2])/2])/(15*Sqrt[x^2]) + Defer[Int][Log[3 - Log[x^2]], x]/15

Rubi steps \begin{align*} \text {integral}& = \int \frac {-188+62 \log \left (x^2\right )-\left (-3+\log \left (x^2\right )\right ) \log \left (3-\log \left (x^2\right )\right )}{15 \left (3-\log \left (x^2\right )\right )} \, dx \\ & = \frac {1}{15} \int \frac {-188+62 \log \left (x^2\right )-\left (-3+\log \left (x^2\right )\right ) \log \left (3-\log \left (x^2\right )\right )}{3-\log \left (x^2\right )} \, dx \\ & = \frac {1}{15} \int \left (-\frac {2 \left (-94+31 \log \left (x^2\right )\right )}{-3+\log \left (x^2\right )}+\log \left (3-\log \left (x^2\right )\right )\right ) \, dx \\ & = \frac {1}{15} \int \log \left (3-\log \left (x^2\right )\right ) \, dx-\frac {2}{15} \int \frac {-94+31 \log \left (x^2\right )}{-3+\log \left (x^2\right )} \, dx \\ & = \frac {1}{15} \int \log \left (3-\log \left (x^2\right )\right ) \, dx-\frac {2}{15} \int \left (31-\frac {1}{-3+\log \left (x^2\right )}\right ) \, dx \\ & = -\frac {62 x}{15}+\frac {1}{15} \int \log \left (3-\log \left (x^2\right )\right ) \, dx+\frac {2}{15} \int \frac {1}{-3+\log \left (x^2\right )} \, dx \\ & = -\frac {62 x}{15}+\frac {1}{15} \int \log \left (3-\log \left (x^2\right )\right ) \, dx+\frac {x \text {Subst}\left (\int \frac {e^{x/2}}{-3+x} \, dx,x,\log \left (x^2\right )\right )}{15 \sqrt {x^2}} \\ & = -\frac {62 x}{15}+\frac {e^{3/2} x \text {Ei}\left (\frac {1}{2} \left (-3+\log \left (x^2\right )\right )\right )}{15 \sqrt {x^2}}+\frac {1}{15} \int \log \left (3-\log \left (x^2\right )\right ) \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90 \[ \int \frac {188-62 \log \left (x^2\right )+\left (-3+\log \left (x^2\right )\right ) \log \left (3-\log \left (x^2\right )\right )}{-45+15 \log \left (x^2\right )} \, dx=\frac {1}{15} \left (-62 x+x \log \left (3-\log \left (x^2\right )\right )\right ) \]

[In]

Integrate[(188 - 62*Log[x^2] + (-3 + Log[x^2])*Log[3 - Log[x^2]])/(-45 + 15*Log[x^2]),x]

[Out]

(-62*x + x*Log[3 - Log[x^2]])/15

Maple [A] (verified)

Time = 0.68 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.81

method	result	size
norman	\(-\frac {62 x}{15}+\frac {x \ln \left (3-\ln \left (x^{2}\right )\right )}{15}\)	\(17\)
parallelrisch	\(-\frac {62 x}{15}+\frac {x \ln \left (3-\ln \left (x^{2}\right )\right )}{15}\)	\(17\)

[In]

int(((ln(x^2)-3)*ln(3-ln(x^2))-62*ln(x^2)+188)/(15*ln(x^2)-45),x,method=_RETURNVERBOSE)

[Out]

-62/15*x+1/15*x*ln(3-ln(x^2))

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.76 \[ \int \frac {188-62 \log \left (x^2\right )+\left (-3+\log \left (x^2\right )\right ) \log \left (3-\log \left (x^2\right )\right )}{-45+15 \log \left (x^2\right )} \, dx=\frac {1}{15} \, x \log \left (-\log \left (x^{2}\right ) + 3\right ) - \frac {62}{15} \, x \]

[In]

integrate(((log(x^2)-3)*log(3-log(x^2))-62*log(x^2)+188)/(15*log(x^2)-45),x, algorithm="fricas")

[Out]

1/15*x*log(-log(x^2) + 3) - 62/15*x

Sympy [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.71 \[ \int \frac {188-62 \log \left (x^2\right )+\left (-3+\log \left (x^2\right )\right ) \log \left (3-\log \left (x^2\right )\right )}{-45+15 \log \left (x^2\right )} \, dx=\frac {x \log {\left (3 - \log {\left (x^{2} \right )} \right )}}{15} - \frac {62 x}{15} \]

[In]

integrate(((ln(x**2)-3)*ln(3-ln(x**2))-62*ln(x**2)+188)/(15*ln(x**2)-45),x)

[Out]

x*log(3 - log(x**2))/15 - 62*x/15

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.67 \[ \int \frac {188-62 \log \left (x^2\right )+\left (-3+\log \left (x^2\right )\right ) \log \left (3-\log \left (x^2\right )\right )}{-45+15 \log \left (x^2\right )} \, dx=\frac {1}{15} \, x \log \left (-2 \, \log \left (x\right ) + 3\right ) - \frac {62}{15} \, x \]

[In]

integrate(((log(x^2)-3)*log(3-log(x^2))-62*log(x^2)+188)/(15*log(x^2)-45),x, algorithm="maxima")

[Out]

1/15*x*log(-2*log(x) + 3) - 62/15*x

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.76 \[ \int \frac {188-62 \log \left (x^2\right )+\left (-3+\log \left (x^2\right )\right ) \log \left (3-\log \left (x^2\right )\right )}{-45+15 \log \left (x^2\right )} \, dx=\frac {1}{15} \, x \log \left (-\log \left (x^{2}\right ) + 3\right ) - \frac {62}{15} \, x \]

[In]

integrate(((log(x^2)-3)*log(3-log(x^2))-62*log(x^2)+188)/(15*log(x^2)-45),x, algorithm="giac")

[Out]

1/15*x*log(-log(x^2) + 3) - 62/15*x

Mupad [B] (verification not implemented)

Time = 12.49 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.67 \[ \int \frac {188-62 \log \left (x^2\right )+\left (-3+\log \left (x^2\right )\right ) \log \left (3-\log \left (x^2\right )\right )}{-45+15 \log \left (x^2\right )} \, dx=\frac {x\,\left (\ln \left (3-\ln \left (x^2\right )\right )-62\right )}{15} \]

[In]

int((log(3 - log(x^2))*(log(x^2) - 3) - 62*log(x^2) + 188)/(15*log(x^2) - 45),x)

[Out]

(x*(log(3 - log(x^2)) - 62))/15