\(\int \frac {e^{\frac {9+123 x+30 x^2}{10 \log (\frac {2+x^2}{x})}} (18+246 x+51 x^2-123 x^3-30 x^4+(246 x+120 x^2+123 x^3+60 x^4) \log (\frac {2+x^2}{x})+(20+10 x^2) \log ^2(\frac {2+x^2}{x}))}{(20+10 x^2) \log ^2(\frac {2+x^2}{x})} \, dx\) [9541]

Optimal result

Integrand size = 120, antiderivative size = 27 \[ \int \frac {e^{\frac {9+123 x+30 x^2}{10 \log \left (\frac {2+x^2}{x}\right )}} \left (18+246 x+51 x^2-123 x^3-30 x^4+\left (246 x+120 x^2+123 x^3+60 x^4\right ) \log \left (\frac {2+x^2}{x}\right )+\left (20+10 x^2\right ) \log ^2\left (\frac {2+x^2}{x}\right )\right )}{\left (20+10 x^2\right ) \log ^2\left (\frac {2+x^2}{x}\right )} \, dx=e^{\frac {3 \left (x+\left (\frac {1}{10}+x\right ) (3+x)\right )}{\log \left (\frac {2}{x}+x\right )}} x \]

[Out]

x*exp(3/ln(x+2/x)*(x+(3+x)*(1/10+x)))

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(164\) vs. \(2(27)=54\).

Time = 0.80 (sec) , antiderivative size = 164, normalized size of antiderivative = 6.07, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.008, Rules used = {2326} \[ \int \frac {e^{\frac {9+123 x+30 x^2}{10 \log \left (\frac {2+x^2}{x}\right )}} \left (18+246 x+51 x^2-123 x^3-30 x^4+\left (246 x+120 x^2+123 x^3+60 x^4\right ) \log \left (\frac {2+x^2}{x}\right )+\left (20+10 x^2\right ) \log ^2\left (\frac {2+x^2}{x}\right )\right )}{\left (20+10 x^2\right ) \log ^2\left (\frac {2+x^2}{x}\right )} \, dx=-\frac {e^{\frac {3 \left (10 x^2+41 x+3\right )}{10 \log \left (\frac {x^2+2}{x}\right )}} \left (-10 x^4-41 x^3+17 x^2+\left (20 x^4+41 x^3+40 x^2+82 x\right ) \log \left (\frac {x^2+2}{x}\right )+82 x+6\right )}{\left (x^2+2\right ) \left (\frac {x \left (10 x^2+41 x+3\right ) \left (2-\frac {x^2+2}{x^2}\right )}{\left (x^2+2\right ) \log ^2\left (\frac {x^2+2}{x}\right )}-\frac {20 x+41}{\log \left (\frac {x^2+2}{x}\right )}\right ) \log ^2\left (\frac {x^2+2}{x}\right )} \]

[In]

Int[(E^((9 + 123*x + 30*x^2)/(10*Log[(2 + x^2)/x]))*(18 + 246*x + 51*x^2 - 123*x^3 - 30*x^4 + (246*x + 120*x^2
+ 123*x^3 + 60*x^4)*Log[(2 + x^2)/x] + (20 + 10*x^2)*Log[(2 + x^2)/x]^2))/((20 + 10*x^2)*Log[(2 + x^2)/x]^2),
x]

[Out]

-((E^((3*(3 + 41*x + 10*x^2))/(10*Log[(2 + x^2)/x]))*(6 + 82*x + 17*x^2 - 41*x^3 - 10*x^4 + (82*x + 40*x^2 + 4
1*x^3 + 20*x^4)*Log[(2 + x^2)/x]))/((2 + x^2)*((x*(3 + 41*x + 10*x^2)*(2 - (2 + x^2)/x^2))/((2 + x^2)*Log[(2 +
x^2)/x]^2) - (41 + 20*x)/Log[(2 + x^2)/x])*Log[(2 + x^2)/x]^2))

Rule 2326

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = v*(y/(Log[F]*D[u, x]))}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rubi steps \begin{align*} \text {integral}& = -\frac {e^{\frac {3 \left (3+41 x+10 x^2\right )}{10 \log \left (\frac {2+x^2}{x}\right )}} \left (6+82 x+17 x^2-41 x^3-10 x^4+\left (82 x+40 x^2+41 x^3+20 x^4\right ) \log \left (\frac {2+x^2}{x}\right )\right )}{\left (2+x^2\right ) \left (\frac {x \left (3+41 x+10 x^2\right ) \left (2-\frac {2+x^2}{x^2}\right )}{\left (2+x^2\right ) \log ^2\left (\frac {2+x^2}{x}\right )}-\frac {41+20 x}{\log \left (\frac {2+x^2}{x}\right )}\right ) \log ^2\left (\frac {2+x^2}{x}\right )} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.04 \[ \int \frac {e^{\frac {9+123 x+30 x^2}{10 \log \left (\frac {2+x^2}{x}\right )}} \left (18+246 x+51 x^2-123 x^3-30 x^4+\left (246 x+120 x^2+123 x^3+60 x^4\right ) \log \left (\frac {2+x^2}{x}\right )+\left (20+10 x^2\right ) \log ^2\left (\frac {2+x^2}{x}\right )\right )}{\left (20+10 x^2\right ) \log ^2\left (\frac {2+x^2}{x}\right )} \, dx=e^{\frac {3 \left (3+41 x+10 x^2\right )}{10 \log \left (\frac {2}{x}+x\right )}} x \]

[In]

Integrate[(E^((9 + 123*x + 30*x^2)/(10*Log[(2 + x^2)/x]))*(18 + 246*x + 51*x^2 - 123*x^3 - 30*x^4 + (246*x + 1
20*x^2 + 123*x^3 + 60*x^4)*Log[(2 + x^2)/x] + (20 + 10*x^2)*Log[(2 + x^2)/x]^2))/((20 + 10*x^2)*Log[(2 + x^2)/
x]^2),x]

[Out]

E^((3*(3 + 41*x + 10*x^2))/(10*Log[2/x + x]))*x

Maple [A] (verified)

Time = 1.27 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.04

method	result	size
risch	\({\mathrm e}^{\frac {3 x^{2}+\frac {123}{10} x +\frac {9}{10}}{\ln \left (\frac {x^{2}+2}{x}\right )}} x\)	\(28\)
parallelrisch	\({\mathrm e}^{\frac {3 x^{2}+\frac {123}{10} x +\frac {9}{10}}{\ln \left (\frac {x^{2}+2}{x}\right )}} x\)	\(28\)

[In]

int(((10*x^2+20)*ln((x^2+2)/x)^2+(60*x^4+123*x^3+120*x^2+246*x)*ln((x^2+2)/x)-30*x^4-123*x^3+51*x^2+246*x+18)*
exp(1/10*(30*x^2+123*x+9)/ln((x^2+2)/x))/(10*x^2+20)/ln((x^2+2)/x)^2,x,method=_RETURNVERBOSE)

[Out]

exp(3/10*(10*x^2+41*x+3)/ln((x^2+2)/x))*x

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {e^{\frac {9+123 x+30 x^2}{10 \log \left (\frac {2+x^2}{x}\right )}} \left (18+246 x+51 x^2-123 x^3-30 x^4+\left (246 x+120 x^2+123 x^3+60 x^4\right ) \log \left (\frac {2+x^2}{x}\right )+\left (20+10 x^2\right ) \log ^2\left (\frac {2+x^2}{x}\right )\right )}{\left (20+10 x^2\right ) \log ^2\left (\frac {2+x^2}{x}\right )} \, dx=x e^{\left (\frac {3 \, {\left (10 \, x^{2} + 41 \, x + 3\right )}}{10 \, \log \left (\frac {x^{2} + 2}{x}\right )}\right )} \]

[In]

integrate(((10*x^2+20)*log((x^2+2)/x)^2+(60*x^4+123*x^3+120*x^2+246*x)*log((x^2+2)/x)-30*x^4-123*x^3+51*x^2+24
6*x+18)*exp(1/10*(30*x^2+123*x+9)/log((x^2+2)/x))/(10*x^2+20)/log((x^2+2)/x)^2,x, algorithm="fricas")

[Out]

x*e^(3/10*(10*x^2 + 41*x + 3)/log((x^2 + 2)/x))

Sympy [F(-2)]

Exception generated. \[ \int \frac {e^{\frac {9+123 x+30 x^2}{10 \log \left (\frac {2+x^2}{x}\right )}} \left (18+246 x+51 x^2-123 x^3-30 x^4+\left (246 x+120 x^2+123 x^3+60 x^4\right ) \log \left (\frac {2+x^2}{x}\right )+\left (20+10 x^2\right ) \log ^2\left (\frac {2+x^2}{x}\right )\right )}{\left (20+10 x^2\right ) \log ^2\left (\frac {2+x^2}{x}\right )} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(((10*x**2+20)*ln((x**2+2)/x)**2+(60*x**4+123*x**3+120*x**2+246*x)*ln((x**2+2)/x)-30*x**4-123*x**3+51
*x**2+246*x+18)*exp(1/10*(30*x**2+123*x+9)/ln((x**2+2)/x))/(10*x**2+20)/ln((x**2+2)/x)**2,x)

[Out]

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {e^{\frac {9+123 x+30 x^2}{10 \log \left (\frac {2+x^2}{x}\right )}} \left (18+246 x+51 x^2-123 x^3-30 x^4+\left (246 x+120 x^2+123 x^3+60 x^4\right ) \log \left (\frac {2+x^2}{x}\right )+\left (20+10 x^2\right ) \log ^2\left (\frac {2+x^2}{x}\right )\right )}{\left (20+10 x^2\right ) \log ^2\left (\frac {2+x^2}{x}\right )} \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate(((10*x^2+20)*log((x^2+2)/x)^2+(60*x^4+123*x^3+120*x^2+246*x)*log((x^2+2)/x)-30*x^4-123*x^3+51*x^2+24
6*x+18)*exp(1/10*(30*x^2+123*x+9)/log((x^2+2)/x))/(10*x^2+20)/log((x^2+2)/x)^2,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is undefined.

Giac [F]

\[ \int \frac {e^{\frac {9+123 x+30 x^2}{10 \log \left (\frac {2+x^2}{x}\right )}} \left (18+246 x+51 x^2-123 x^3-30 x^4+\left (246 x+120 x^2+123 x^3+60 x^4\right ) \log \left (\frac {2+x^2}{x}\right )+\left (20+10 x^2\right ) \log ^2\left (\frac {2+x^2}{x}\right )\right )}{\left (20+10 x^2\right ) \log ^2\left (\frac {2+x^2}{x}\right )} \, dx=\int { -\frac {{\left (30 \, x^{4} + 123 \, x^{3} - 10 \, {\left (x^{2} + 2\right )} \log \left (\frac {x^{2} + 2}{x}\right )^{2} - 51 \, x^{2} - 3 \, {\left (20 \, x^{4} + 41 \, x^{3} + 40 \, x^{2} + 82 \, x\right )} \log \left (\frac {x^{2} + 2}{x}\right ) - 246 \, x - 18\right )} e^{\left (\frac {3 \, {\left (10 \, x^{2} + 41 \, x + 3\right )}}{10 \, \log \left (\frac {x^{2} + 2}{x}\right )}\right )}}{10 \, {\left (x^{2} + 2\right )} \log \left (\frac {x^{2} + 2}{x}\right )^{2}} \,d x } \]

[In]

integrate(((10*x^2+20)*log((x^2+2)/x)^2+(60*x^4+123*x^3+120*x^2+246*x)*log((x^2+2)/x)-30*x^4-123*x^3+51*x^2+24
6*x+18)*exp(1/10*(30*x^2+123*x+9)/log((x^2+2)/x))/(10*x^2+20)/log((x^2+2)/x)^2,x, algorithm="giac")

[Out]

integrate(-1/10*(30*x^4 + 123*x^3 - 10*(x^2 + 2)*log((x^2 + 2)/x)^2 - 51*x^2 - 3*(20*x^4 + 41*x^3 + 40*x^2 + 8
2*x)*log((x^2 + 2)/x) - 246*x - 18)*e^(3/10*(10*x^2 + 41*x + 3)/log((x^2 + 2)/x))/((x^2 + 2)*log((x^2 + 2)/x)^
2), x)

Mupad [B] (verification not implemented)

Time = 15.67 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.30 \[ \int \frac {e^{\frac {9+123 x+30 x^2}{10 \log \left (\frac {2+x^2}{x}\right )}} \left (18+246 x+51 x^2-123 x^3-30 x^4+\left (246 x+120 x^2+123 x^3+60 x^4\right ) \log \left (\frac {2+x^2}{x}\right )+\left (20+10 x^2\right ) \log ^2\left (\frac {2+x^2}{x}\right )\right )}{\left (20+10 x^2\right ) \log ^2\left (\frac {2+x^2}{x}\right )} \, dx=x\,{\mathrm {e}}^{\frac {3\,x^2}{\ln \left (\frac {1}{x}\right )+\ln \left (x^2+2\right )}}\,{\mathrm {e}}^{\frac {9}{10\,\left (\ln \left (\frac {1}{x}\right )+\ln \left (x^2+2\right )\right )}}\,{\mathrm {e}}^{\frac {123\,x}{10\,\left (\ln \left (\frac {1}{x}\right )+\ln \left (x^2+2\right )\right )}} \]

[In]

int((exp(((123*x)/10 + 3*x^2 + 9/10)/log((x^2 + 2)/x))*(246*x + log((x^2 + 2)/x)*(246*x + 120*x^2 + 123*x^3 +
60*x^4) + log((x^2 + 2)/x)^2*(10*x^2 + 20) + 51*x^2 - 123*x^3 - 30*x^4 + 18))/(log((x^2 + 2)/x)^2*(10*x^2 + 20
)),x)

[Out]

x*exp((3*x^2)/(log(1/x) + log(x^2 + 2)))*exp(9/(10*(log(1/x) + log(x^2 + 2))))*exp((123*x)/(10*(log(1/x) + log
(x^2 + 2))))