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

Optimal. Leaf size=27 \[ e^{\frac {3 \left (x+\left (\frac {1}{10}+x\right ) (3+x)\right )}{\log \left (\frac {2}{x}+x\right )}} x \]

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Rubi [B]  time = 1.18, antiderivative size = 164, normalized size of antiderivative = 6.07, number of steps used = 1, number of rules used = 1, integrand size = 120, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.008, Rules used = {2288} \begin {gather*} -\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 )} \end {gather*}

Antiderivative was successfully verified.

[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 2288

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 {gather*} \begin {aligned} \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 {aligned} \end {gather*}

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Mathematica [A]  time = 0.07, size = 28, normalized size = 1.04 \begin {gather*} e^{\frac {3 \left (3+41 x+10 x^2\right )}{10 \log \left (\frac {2}{x}+x\right )}} x \end {gather*}

Antiderivative was successfully verified.

[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

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fricas [A]  time = 0.55, size = 27, normalized size = 1.00 \begin {gather*} x e^{\left (\frac {3 \, {\left (10 \, x^{2} + 41 \, x + 3\right )}}{10 \, \log \left (\frac {x^{2} + 2}{x}\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [A]  time = 0.06, size = 28, normalized size = 1.04

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

Verification of antiderivative is not currently implemented for this CAS.

[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]

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

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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.

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mupad [B]  time = 8.82, size = 62, normalized size = 2.30 \begin {gather*} 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 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))))

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sympy [A]  time = 3.35, size = 24, normalized size = 0.89 \begin {gather*} x e^{\frac {3 x^{2} + \frac {123 x}{10} + \frac {9}{10}}{\log {\left (\frac {x^{2} + 2}{x} \right )}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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]

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

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