3.101.83 \(\int \frac {x^5+e^{\frac {e^2-4 e^{26} x+4 e^{50} x^2+(2 e x^2-4 e^{25} x^3) \log (x^2)+x^4 \log ^2(x^2)}{x^4}} (4 e^2-4 e x^2+8 e^{50} x^2+e^{25} (-12 e x+8 x^3)+(4 e x^2-4 e^{25} x^3-4 x^4) \log (x^2))}{x^5} \, dx\)

Optimal. Leaf size=30 \[ 3-e^{\left (-\frac {e}{x^2}+\frac {2 e^{25}}{x}-\log \left (x^2\right )\right )^2}+x \]

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Rubi [F]  time = 9.32, 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 {x^5+\exp \left (\frac {e^2-4 e^{26} x+4 e^{50} x^2+\left (2 e x^2-4 e^{25} x^3\right ) \log \left (x^2\right )+x^4 \log ^2\left (x^2\right )}{x^4}\right ) \left (4 e^2-4 e x^2+8 e^{50} x^2+e^{25} \left (-12 e x+8 x^3\right )+\left (4 e x^2-4 e^{25} x^3-4 x^4\right ) \log \left (x^2\right )\right )}{x^5} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(x^5 + E^((E^2 - 4*E^26*x + 4*E^50*x^2 + (2*E*x^2 - 4*E^25*x^3)*Log[x^2] + x^4*Log[x^2]^2)/x^4)*(4*E^2 - 4
*E*x^2 + 8*E^50*x^2 + E^25*(-12*E*x + 8*x^3) + (4*E*x^2 - 4*E^25*x^3 - 4*x^4)*Log[x^2]))/x^5,x]

[Out]

x + 4*Defer[Int][(E^(2 + (E^2*(1 - 2*E^24*x)^2)/x^4 + Log[x^2]^2)*(x^2)^((2*E - 4*E^25*x)/x^2))/x^5, x] - 12*D
efer[Int][(E^(26 + (E^2*(1 - 2*E^24*x)^2)/x^4 + Log[x^2]^2)*(x^2)^((2*E - 4*E^25*x)/x^2))/x^4, x] - 4*(1 - 2*E
^49)*Defer[Int][(E^(1 + (E^2*(1 - 2*E^24*x)^2)/x^4 + Log[x^2]^2)*(x^2)^((2*E - 4*E^25*x)/x^2))/x^3, x] + 8*Def
er[Int][E^(25 + (E^2*(1 - 2*E^24*x)^2)/x^4 + Log[x^2]^2)*(x^2)^(-1 + (2*E - 4*E^25*x)/x^2), x] + 4*Defer[Int][
(E^(1 + (E^2*(1 - 2*E^24*x)^2)/x^4 + Log[x^2]^2)*(x^2)^((2*E - 4*E^25*x)/x^2)*Log[x^2])/x^3, x] - 4*Defer[Int]
[(E^((E^2*(1 - 2*E^24*x)^2)/x^4 + Log[x^2]^2)*(x^2)^((2*E - 4*E^25*x)/x^2)*Log[x^2])/x, x] - 4*Defer[Int][E^(2
5 + (E^2*(1 - 2*E^24*x)^2)/x^4 + Log[x^2]^2)*(x^2)^(-1 + (2*E - 4*E^25*x)/x^2)*Log[x^2], x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (1+\frac {4 e^{\frac {e^2 \left (1-2 e^{24} x\right )^2}{x^4}+\log ^2\left (x^2\right )} \left (x^2\right )^{\frac {2 e-4 e^{25} x}{x^2}} \left (e-e^{25} x-x^2\right ) \left (e-2 e^{25} x+x^2 \log \left (x^2\right )\right )}{x^5}\right ) \, dx\\ &=x+4 \int \frac {e^{\frac {e^2 \left (1-2 e^{24} x\right )^2}{x^4}+\log ^2\left (x^2\right )} \left (x^2\right )^{\frac {2 e-4 e^{25} x}{x^2}} \left (e-e^{25} x-x^2\right ) \left (e-2 e^{25} x+x^2 \log \left (x^2\right )\right )}{x^5} \, dx\\ &=x+4 \int \left (\frac {e^{1+\frac {e^2 \left (1-2 e^{24} x\right )^2}{x^4}+\log ^2\left (x^2\right )} \left (x^2\right )^{\frac {2 e-4 e^{25} x}{x^2}} \left (-1+2 e^{24} x\right ) \left (-e+e^{25} x+x^2\right )}{x^5}-\frac {e^{\frac {e^2 \left (1-2 e^{24} x\right )^2}{x^4}+\log ^2\left (x^2\right )} \left (x^2\right )^{\frac {2 e-4 e^{25} x}{x^2}} \left (-e+e^{25} x+x^2\right ) \log \left (x^2\right )}{x^3}\right ) \, dx\\ &=x+4 \int \frac {e^{1+\frac {e^2 \left (1-2 e^{24} x\right )^2}{x^4}+\log ^2\left (x^2\right )} \left (x^2\right )^{\frac {2 e-4 e^{25} x}{x^2}} \left (-1+2 e^{24} x\right ) \left (-e+e^{25} x+x^2\right )}{x^5} \, dx-4 \int \frac {e^{\frac {e^2 \left (1-2 e^{24} x\right )^2}{x^4}+\log ^2\left (x^2\right )} \left (x^2\right )^{\frac {2 e-4 e^{25} x}{x^2}} \left (-e+e^{25} x+x^2\right ) \log \left (x^2\right )}{x^3} \, dx\\ &=x+4 \int \left (\frac {e^{2+\frac {e^2 \left (1-2 e^{24} x\right )^2}{x^4}+\log ^2\left (x^2\right )} \left (x^2\right )^{\frac {2 e-4 e^{25} x}{x^2}}}{x^5}-\frac {3 e^{26+\frac {e^2 \left (1-2 e^{24} x\right )^2}{x^4}+\log ^2\left (x^2\right )} \left (x^2\right )^{\frac {2 e-4 e^{25} x}{x^2}}}{x^4}+\frac {e^{1+\frac {e^2 \left (1-2 e^{24} x\right )^2}{x^4}+\log ^2\left (x^2\right )} \left (-1+2 e^{49}\right ) \left (x^2\right )^{\frac {2 e-4 e^{25} x}{x^2}}}{x^3}+2 e^{25+\frac {e^2 \left (1-2 e^{24} x\right )^2}{x^4}+\log ^2\left (x^2\right )} \left (x^2\right )^{-1+\frac {2 e-4 e^{25} x}{x^2}}\right ) \, dx-4 \int \left (-\frac {e^{1+\frac {e^2 \left (1-2 e^{24} x\right )^2}{x^4}+\log ^2\left (x^2\right )} \left (x^2\right )^{\frac {2 e-4 e^{25} x}{x^2}} \log \left (x^2\right )}{x^3}+\frac {e^{\frac {e^2 \left (1-2 e^{24} x\right )^2}{x^4}+\log ^2\left (x^2\right )} \left (x^2\right )^{\frac {2 e-4 e^{25} x}{x^2}} \log \left (x^2\right )}{x}+e^{25+\frac {e^2 \left (1-2 e^{24} x\right )^2}{x^4}+\log ^2\left (x^2\right )} \left (x^2\right )^{-1+\frac {2 e-4 e^{25} x}{x^2}} \log \left (x^2\right )\right ) \, dx\\ &=x+4 \int \frac {e^{2+\frac {e^2 \left (1-2 e^{24} x\right )^2}{x^4}+\log ^2\left (x^2\right )} \left (x^2\right )^{\frac {2 e-4 e^{25} x}{x^2}}}{x^5} \, dx+4 \int \frac {e^{1+\frac {e^2 \left (1-2 e^{24} x\right )^2}{x^4}+\log ^2\left (x^2\right )} \left (x^2\right )^{\frac {2 e-4 e^{25} x}{x^2}} \log \left (x^2\right )}{x^3} \, dx-4 \int \frac {e^{\frac {e^2 \left (1-2 e^{24} x\right )^2}{x^4}+\log ^2\left (x^2\right )} \left (x^2\right )^{\frac {2 e-4 e^{25} x}{x^2}} \log \left (x^2\right )}{x} \, dx-4 \int e^{25+\frac {e^2 \left (1-2 e^{24} x\right )^2}{x^4}+\log ^2\left (x^2\right )} \left (x^2\right )^{-1+\frac {2 e-4 e^{25} x}{x^2}} \log \left (x^2\right ) \, dx+8 \int e^{25+\frac {e^2 \left (1-2 e^{24} x\right )^2}{x^4}+\log ^2\left (x^2\right )} \left (x^2\right )^{-1+\frac {2 e-4 e^{25} x}{x^2}} \, dx-12 \int \frac {e^{26+\frac {e^2 \left (1-2 e^{24} x\right )^2}{x^4}+\log ^2\left (x^2\right )} \left (x^2\right )^{\frac {2 e-4 e^{25} x}{x^2}}}{x^4} \, dx-\left (4 \left (1-2 e^{49}\right )\right ) \int \frac {e^{1+\frac {e^2 \left (1-2 e^{24} x\right )^2}{x^4}+\log ^2\left (x^2\right )} \left (x^2\right )^{\frac {2 e-4 e^{25} x}{x^2}}}{x^3} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 1.84, size = 54, normalized size = 1.80 \begin {gather*} x-e^{\frac {e^2}{x^4}-\frac {4 e^{26}}{x^3}+\frac {4 e^{50}}{x^2}+\log ^2\left (x^2\right )} \left (x^2\right )^{-\frac {2 e \left (-1+2 e^{24} x\right )}{x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^5 + E^((E^2 - 4*E^26*x + 4*E^50*x^2 + (2*E*x^2 - 4*E^25*x^3)*Log[x^2] + x^4*Log[x^2]^2)/x^4)*(4*E
^2 - 4*E*x^2 + 8*E^50*x^2 + E^25*(-12*E*x + 8*x^3) + (4*E*x^2 - 4*E^25*x^3 - 4*x^4)*Log[x^2]))/x^5,x]

[Out]

x - E^(E^2/x^4 - (4*E^26)/x^3 + (4*E^50)/x^2 + Log[x^2]^2)/(x^2)^((2*E*(-1 + 2*E^24*x))/x^2)

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fricas [A]  time = 0.69, size = 55, normalized size = 1.83 \begin {gather*} x - e^{\left (\frac {x^{4} \log \left (x^{2}\right )^{2} + 4 \, x^{2} e^{50} - 4 \, x e^{26} - 2 \, {\left (2 \, x^{3} e^{25} - x^{2} e\right )} \log \left (x^{2}\right ) + e^{2}}{x^{4}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-4*x^3*exp(25)+4*x^2*exp(1)-4*x^4)*log(x^2)+8*x^2*exp(25)^2+(-12*x*exp(1)+8*x^3)*exp(25)+4*exp(1)
^2-4*x^2*exp(1))*exp((x^4*log(x^2)^2+(-4*x^3*exp(25)+2*x^2*exp(1))*log(x^2)+4*x^2*exp(25)^2-4*x*exp(1)*exp(25)
+exp(1)^2)/x^4)+x^5)/x^5,x, algorithm="fricas")

[Out]

x - e^((x^4*log(x^2)^2 + 4*x^2*e^50 - 4*x*e^26 - 2*(2*x^3*e^25 - x^2*e)*log(x^2) + e^2)/x^4)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{5} + 4 \, {\left (2 \, x^{2} e^{50} - x^{2} e + {\left (2 \, x^{3} - 3 \, x e\right )} e^{25} - {\left (x^{4} + x^{3} e^{25} - x^{2} e\right )} \log \left (x^{2}\right ) + e^{2}\right )} e^{\left (\frac {x^{4} \log \left (x^{2}\right )^{2} + 4 \, x^{2} e^{50} - 4 \, x e^{26} - 2 \, {\left (2 \, x^{3} e^{25} - x^{2} e\right )} \log \left (x^{2}\right ) + e^{2}}{x^{4}}\right )}}{x^{5}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-4*x^3*exp(25)+4*x^2*exp(1)-4*x^4)*log(x^2)+8*x^2*exp(25)^2+(-12*x*exp(1)+8*x^3)*exp(25)+4*exp(1)
^2-4*x^2*exp(1))*exp((x^4*log(x^2)^2+(-4*x^3*exp(25)+2*x^2*exp(1))*log(x^2)+4*x^2*exp(25)^2-4*x*exp(1)*exp(25)
+exp(1)^2)/x^4)+x^5)/x^5,x, algorithm="giac")

[Out]

integrate((x^5 + 4*(2*x^2*e^50 - x^2*e + (2*x^3 - 3*x*e)*e^25 - (x^4 + x^3*e^25 - x^2*e)*log(x^2) + e^2)*e^((x
^4*log(x^2)^2 + 4*x^2*e^50 - 4*x*e^26 - 2*(2*x^3*e^25 - x^2*e)*log(x^2) + e^2)/x^4))/x^5, x)

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maple [A]  time = 0.84, size = 57, normalized size = 1.90

method	result	size
risch	\(x -\left (x^{2}\right )^{-\frac {4 \,{\mathrm e}^{25}}{x}} \left (x^{2}\right )^{\frac {2 \,{\mathrm e}}{x^{2}}} {\mathrm e}^{\frac {x^{4} \ln \left (x^{2}\right )^{2}+4 x^{2} {\mathrm e}^{50}-4 x \,{\mathrm e}^{26}+{\mathrm e}^{2}}{x^{4}}}\)	\(57\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-4*x^3*exp(25)+4*x^2*exp(1)-4*x^4)*ln(x^2)+8*x^2*exp(25)^2+(-12*x*exp(1)+8*x^3)*exp(25)+4*exp(1)^2-4*x^
2*exp(1))*exp((x^4*ln(x^2)^2+(-4*x^3*exp(25)+2*x^2*exp(1))*ln(x^2)+4*x^2*exp(25)^2-4*x*exp(1)*exp(25)+exp(1)^2
)/x^4)+x^5)/x^5,x,method=_RETURNVERBOSE)

[Out]

x-(x^2)^(-4*exp(25)/x)*(x^2)^(2*exp(1)/x^2)*exp((x^4*ln(x^2)^2+4*x^2*exp(50)-4*x*exp(26)+exp(2))/x^4)

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maxima [A]  time = 0.64, size = 50, normalized size = 1.67 \begin {gather*} x - e^{\left (4 \, \log \relax (x)^{2} - \frac {8 \, e^{25} \log \relax (x)}{x} + \frac {4 \, e \log \relax (x)}{x^{2}} + \frac {4 \, e^{50}}{x^{2}} - \frac {4 \, e^{26}}{x^{3}} + \frac {e^{2}}{x^{4}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-4*x^3*exp(25)+4*x^2*exp(1)-4*x^4)*log(x^2)+8*x^2*exp(25)^2+(-12*x*exp(1)+8*x^3)*exp(25)+4*exp(1)
^2-4*x^2*exp(1))*exp((x^4*log(x^2)^2+(-4*x^3*exp(25)+2*x^2*exp(1))*log(x^2)+4*x^2*exp(25)^2-4*x*exp(1)*exp(25)
+exp(1)^2)/x^4)+x^5)/x^5,x, algorithm="maxima")

[Out]

x - e^(4*log(x)^2 - 8*e^25*log(x)/x + 4*e*log(x)/x^2 + 4*e^50/x^2 - 4*e^26/x^3 + e^2/x^4)

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mupad [B]  time = 7.48, size = 58, normalized size = 1.93 \begin {gather*} x-\frac {{\mathrm {e}}^{\frac {{\mathrm {e}}^2}{x^4}}\,{\mathrm {e}}^{-\frac {4\,{\mathrm {e}}^{26}}{x^3}}\,{\mathrm {e}}^{\frac {4\,{\mathrm {e}}^{50}}{x^2}}\,{\mathrm {e}}^{{\ln \left (x^2\right )}^2}\,{\left (x^2\right )}^{\frac {2\,\mathrm {e}}{x^2}}}{{\left (x^2\right )}^{\frac {4\,{\mathrm {e}}^{25}}{x}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp((exp(2) + log(x^2)*(2*x^2*exp(1) - 4*x^3*exp(25)) - 4*x*exp(26) + 4*x^2*exp(50) + x^4*log(x^2)^2)/x^
4)*(log(x^2)*(4*x^3*exp(25) - 4*x^2*exp(1) + 4*x^4) - 4*exp(2) + exp(25)*(12*x*exp(1) - 8*x^3) + 4*x^2*exp(1)
- 8*x^2*exp(50)) - x^5)/x^5,x)

[Out]

x - (exp(exp(2)/x^4)*exp(-(4*exp(26))/x^3)*exp((4*exp(50))/x^2)*exp(log(x^2)^2)*(x^2)^((2*exp(1))/x^2))/(x^2)^
((4*exp(25))/x)

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sympy [B]  time = 0.69, size = 56, normalized size = 1.87 \begin {gather*} x - e^{\frac {x^{4} \log {\left (x^{2} \right )}^{2} + 4 x^{2} e^{50} - 4 x e^{26} + \left (- 4 x^{3} e^{25} + 2 e x^{2}\right ) \log {\left (x^{2} \right )} + e^{2}}{x^{4}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-4*x**3*exp(25)+4*x**2*exp(1)-4*x**4)*ln(x**2)+8*x**2*exp(25)**2+(-12*x*exp(1)+8*x**3)*exp(25)+4*
exp(1)**2-4*x**2*exp(1))*exp((x**4*ln(x**2)**2+(-4*x**3*exp(25)+2*x**2*exp(1))*ln(x**2)+4*x**2*exp(25)**2-4*x*
exp(1)*exp(25)+exp(1)**2)/x**4)+x**5)/x**5,x)

[Out]

x - exp((x**4*log(x**2)**2 + 4*x**2*exp(50) - 4*x*exp(26) + (-4*x**3*exp(25) + 2*E*x**2)*log(x**2) + exp(2))/x
**4)

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