3.73.70 \(\int \frac {30 x^4+2 x^6+e^{\frac {-1+x^2}{x^2}} (30+2 x^2)+(2 e^{\frac {-1+x^2}{x^2}} x^4+2 x^6) \log (5 e^{\frac {-1+x^2}{x^2}}+5 x^2)}{e^{\frac {-1+x^2}{x^2}} x^3+x^5} \, dx\)

Optimal. Leaf size=22 \[ \left (15+x^2\right ) \log \left (5 \left (e^{1-\frac {1}{x^2}}+x^2\right )\right ) \]

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Rubi [F]  time = 1.87, 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 {30 x^4+2 x^6+e^{\frac {-1+x^2}{x^2}} \left (30+2 x^2\right )+\left (2 e^{\frac {-1+x^2}{x^2}} x^4+2 x^6\right ) \log \left (5 e^{\frac {-1+x^2}{x^2}}+5 x^2\right )}{e^{\frac {-1+x^2}{x^2}} x^3+x^5} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(30*x^4 + 2*x^6 + E^((-1 + x^2)/x^2)*(30 + 2*x^2) + (2*E^((-1 + x^2)/x^2)*x^4 + 2*x^6)*Log[5*E^((-1 + x^2)
/x^2) + 5*x^2])/(E^((-1 + x^2)/x^2)*x^3 + x^5),x]

[Out]

30*Log[x] + x^2*Log[5*(E^(1 - x^(-2)) + x^2)] + 15*E*Defer[Subst][Defer[Int][1/(x^2*(E + E^x^(-1)*x)), x], x,
x^2] - 15*E*Defer[Subst][Defer[Int][1/(x*(E + E^x^(-1)*x)), x], x, x^2]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {2 \left (15+x^2\right ) \left (e+e^{\frac {1}{x^2}} x^4\right )}{x^3 \left (e+e^{\frac {1}{x^2}} x^2\right )}+2 x \log \left (5 \left (e^{1-\frac {1}{x^2}}+x^2\right )\right )\right ) \, dx\\ &=2 \int \frac {\left (15+x^2\right ) \left (e+e^{\frac {1}{x^2}} x^4\right )}{x^3 \left (e+e^{\frac {1}{x^2}} x^2\right )} \, dx+2 \int x \log \left (5 \left (e^{1-\frac {1}{x^2}}+x^2\right )\right ) \, dx\\ &=x^2 \log \left (5 \left (e^{1-\frac {1}{x^2}}+x^2\right )\right )-\int \frac {2 \left (e+e^{\frac {1}{x^2}} x^4\right )}{e x+e^{\frac {1}{x^2}} x^3} \, dx+\operatorname {Subst}\left (\int \frac {(15+x) \left (e+e^{\frac {1}{x}} x^2\right )}{x^2 \left (e+e^{\frac {1}{x}} x\right )} \, dx,x,x^2\right )\\ &=x^2 \log \left (5 \left (e^{1-\frac {1}{x^2}}+x^2\right )\right )-2 \int \frac {e+e^{\frac {1}{x^2}} x^4}{e x+e^{\frac {1}{x^2}} x^3} \, dx+\operatorname {Subst}\left (\int \left (\frac {15+x}{x}-\frac {e (-1+x) (15+x)}{x^2 \left (e+e^{\frac {1}{x}} x\right )}\right ) \, dx,x,x^2\right )\\ &=x^2 \log \left (5 \left (e^{1-\frac {1}{x^2}}+x^2\right )\right )-2 \int \left (x-\frac {e \left (-1+x^2\right )}{x \left (e+e^{\frac {1}{x^2}} x^2\right )}\right ) \, dx-e \operatorname {Subst}\left (\int \frac {(-1+x) (15+x)}{x^2 \left (e+e^{\frac {1}{x}} x\right )} \, dx,x,x^2\right )+\operatorname {Subst}\left (\int \frac {15+x}{x} \, dx,x,x^2\right )\\ &=-x^2+x^2 \log \left (5 \left (e^{1-\frac {1}{x^2}}+x^2\right )\right )-e \operatorname {Subst}\left (\int \left (\frac {1}{e+e^{\frac {1}{x}} x}-\frac {15}{x^2 \left (e+e^{\frac {1}{x}} x\right )}+\frac {14}{x \left (e+e^{\frac {1}{x}} x\right )}\right ) \, dx,x,x^2\right )+(2 e) \int \frac {-1+x^2}{x \left (e+e^{\frac {1}{x^2}} x^2\right )} \, dx+\operatorname {Subst}\left (\int \left (1+\frac {15}{x}\right ) \, dx,x,x^2\right )\\ &=30 \log (x)+x^2 \log \left (5 \left (e^{1-\frac {1}{x^2}}+x^2\right )\right )-e \operatorname {Subst}\left (\int \frac {1}{e+e^{\frac {1}{x}} x} \, dx,x,x^2\right )+e \operatorname {Subst}\left (\int \frac {-1+x}{x \left (e+e^{\frac {1}{x}} x\right )} \, dx,x,x^2\right )-(14 e) \operatorname {Subst}\left (\int \frac {1}{x \left (e+e^{\frac {1}{x}} x\right )} \, dx,x,x^2\right )+(15 e) \operatorname {Subst}\left (\int \frac {1}{x^2 \left (e+e^{\frac {1}{x}} x\right )} \, dx,x,x^2\right )\\ &=30 \log (x)+x^2 \log \left (5 \left (e^{1-\frac {1}{x^2}}+x^2\right )\right )-e \operatorname {Subst}\left (\int \frac {1}{e+e^{\frac {1}{x}} x} \, dx,x,x^2\right )+e \operatorname {Subst}\left (\int \left (\frac {1}{e+e^{\frac {1}{x}} x}-\frac {1}{x \left (e+e^{\frac {1}{x}} x\right )}\right ) \, dx,x,x^2\right )-(14 e) \operatorname {Subst}\left (\int \frac {1}{x \left (e+e^{\frac {1}{x}} x\right )} \, dx,x,x^2\right )+(15 e) \operatorname {Subst}\left (\int \frac {1}{x^2 \left (e+e^{\frac {1}{x}} x\right )} \, dx,x,x^2\right )\\ &=30 \log (x)+x^2 \log \left (5 \left (e^{1-\frac {1}{x^2}}+x^2\right )\right )-e \operatorname {Subst}\left (\int \frac {1}{x \left (e+e^{\frac {1}{x}} x\right )} \, dx,x,x^2\right )-(14 e) \operatorname {Subst}\left (\int \frac {1}{x \left (e+e^{\frac {1}{x}} x\right )} \, dx,x,x^2\right )+(15 e) \operatorname {Subst}\left (\int \frac {1}{x^2 \left (e+e^{\frac {1}{x}} x\right )} \, dx,x,x^2\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.29, size = 41, normalized size = 1.86 \begin {gather*} 1-\frac {15}{x^2}+x^2 \log \left (5 \left (e^{1-\frac {1}{x^2}}+x^2\right )\right )+15 \log \left (e+e^{\frac {1}{x^2}} x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(30*x^4 + 2*x^6 + E^((-1 + x^2)/x^2)*(30 + 2*x^2) + (2*E^((-1 + x^2)/x^2)*x^4 + 2*x^6)*Log[5*E^((-1
+ x^2)/x^2) + 5*x^2])/(E^((-1 + x^2)/x^2)*x^3 + x^5),x]

[Out]

1 - 15/x^2 + x^2*Log[5*(E^(1 - x^(-2)) + x^2)] + 15*Log[E + E^x^(-2)*x^2]

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fricas [A]  time = 0.72, size = 25, normalized size = 1.14 \begin {gather*} {\left (x^{2} + 15\right )} \log \left (5 \, x^{2} + 5 \, e^{\left (\frac {x^{2} - 1}{x^{2}}\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^4*exp((x^2-1)/x^2)+2*x^6)*log(5*exp((x^2-1)/x^2)+5*x^2)+(2*x^2+30)*exp((x^2-1)/x^2)+2*x^6+30*x
^4)/(x^3*exp((x^2-1)/x^2)+x^5),x, algorithm="fricas")

[Out]

(x^2 + 15)*log(5*x^2 + 5*e^((x^2 - 1)/x^2))

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giac [A]  time = 0.17, size = 41, normalized size = 1.86 \begin {gather*} x^{2} \log \left (5 \, x^{2} + 5 \, e^{\left (\frac {x^{2} - 1}{x^{2}}\right )}\right ) + 15 \, \log \left (x^{2} + e^{\left (\frac {x^{2} - 1}{x^{2}}\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^4*exp((x^2-1)/x^2)+2*x^6)*log(5*exp((x^2-1)/x^2)+5*x^2)+(2*x^2+30)*exp((x^2-1)/x^2)+2*x^6+30*x
^4)/(x^3*exp((x^2-1)/x^2)+x^5),x, algorithm="giac")

[Out]

x^2*log(5*x^2 + 5*e^((x^2 - 1)/x^2)) + 15*log(x^2 + e^((x^2 - 1)/x^2))

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maple [A]  time = 0.04, size = 45, normalized size = 2.05

method	result	size
risch	\(x^{2} \ln \left (5 \,{\mathrm e}^{\frac {\left (x -1\right ) \left (x +1\right )}{x^{2}}}+5 x^{2}\right )+15 \ln \left ({\mathrm e}^{\frac {\left (x -1\right ) \left (x +1\right )}{x^{2}}}+x^{2}\right )-15\)	\(45\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x^4*exp((x^2-1)/x^2)+2*x^6)*ln(5*exp((x^2-1)/x^2)+5*x^2)+(2*x^2+30)*exp((x^2-1)/x^2)+2*x^6+30*x^4)/(x^
3*exp((x^2-1)/x^2)+x^5),x,method=_RETURNVERBOSE)

[Out]

x^2*ln(5*exp((x-1)*(x+1)/x^2)+5*x^2)+15*ln(exp((x-1)*(x+1)/x^2)+x^2)-15

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maxima [B]  time = 0.48, size = 51, normalized size = 2.32 \begin {gather*} \frac {x^{4} \log \relax (5) + x^{4} \log \left (x^{2} e^{\left (\frac {1}{x^{2}}\right )} + e\right ) - 15}{x^{2}} + 30 \, \log \relax (x) + 15 \, \log \left (\frac {x^{2} e^{\left (\frac {1}{x^{2}}\right )} + e}{x^{2}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^4*exp((x^2-1)/x^2)+2*x^6)*log(5*exp((x^2-1)/x^2)+5*x^2)+(2*x^2+30)*exp((x^2-1)/x^2)+2*x^6+30*x
^4)/(x^3*exp((x^2-1)/x^2)+x^5),x, algorithm="maxima")

[Out]

(x^4*log(5) + x^4*log(x^2*e^(x^(-2)) + e) - 15)/x^2 + 30*log(x) + 15*log((x^2*e^(x^(-2)) + e)/x^2)

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mupad [B]  time = 4.46, size = 38, normalized size = 1.73 \begin {gather*} 15\,\ln \left (\mathrm {e}\,{\mathrm {e}}^{-\frac {1}{x^2}}+x^2\right )+x^2\,\ln \left (5\,\mathrm {e}\,{\mathrm {e}}^{-\frac {1}{x^2}}+5\,x^2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(5*exp((x^2 - 1)/x^2) + 5*x^2)*(2*x^4*exp((x^2 - 1)/x^2) + 2*x^6) + exp((x^2 - 1)/x^2)*(2*x^2 + 30) +
30*x^4 + 2*x^6)/(x^3*exp((x^2 - 1)/x^2) + x^5),x)

[Out]

15*log(exp(1)*exp(-1/x^2) + x^2) + x^2*log(5*exp(1)*exp(-1/x^2) + 5*x^2)

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sympy [A]  time = 0.50, size = 37, normalized size = 1.68 \begin {gather*} x^{2} \log {\left (5 x^{2} + 5 e^{\frac {x^{2} - 1}{x^{2}}} \right )} + 15 \log {\left (x^{2} + e^{\frac {x^{2} - 1}{x^{2}}} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x**4*exp((x**2-1)/x**2)+2*x**6)*ln(5*exp((x**2-1)/x**2)+5*x**2)+(2*x**2+30)*exp((x**2-1)/x**2)+2
*x**6+30*x**4)/(x**3*exp((x**2-1)/x**2)+x**5),x)

[Out]

x**2*log(5*x**2 + 5*exp((x**2 - 1)/x**2)) + 15*log(x**2 + exp((x**2 - 1)/x**2))

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