3.81.98 \(\int \frac {4+5 e^{12+2 x^2-4 e^3 x^2+2 e^6 x^2}+5 x^2+e^{6+x^2-2 e^3 x^2+e^6 x^2} (18 x-16 e^3 x+8 e^6 x)}{5 e^{12+2 x^2-4 e^3 x^2+2 e^6 x^2}+10 e^{6+x^2-2 e^3 x^2+e^6 x^2} x+5 x^2} \, dx\)

Optimal. Leaf size=26 \[ 8+e+x-\frac {4}{5 \left (e^{6+\left (x-e^3 x\right )^2}+x\right )} \]

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Rubi [F]  time = 2.73, 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 {4+5 e^{12+2 x^2-4 e^3 x^2+2 e^6 x^2}+5 x^2+e^{6+x^2-2 e^3 x^2+e^6 x^2} \left (18 x-16 e^3 x+8 e^6 x\right )}{5 e^{12+2 x^2-4 e^3 x^2+2 e^6 x^2}+10 e^{6+x^2-2 e^3 x^2+e^6 x^2} x+5 x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(4 + 5*E^(12 + 2*x^2 - 4*E^3*x^2 + 2*E^6*x^2) + 5*x^2 + E^(6 + x^2 - 2*E^3*x^2 + E^6*x^2)*(18*x - 16*E^3*x
+ 8*E^6*x))/(5*E^(12 + 2*x^2 - 4*E^3*x^2 + 2*E^6*x^2) + 10*E^(6 + x^2 - 2*E^3*x^2 + E^6*x^2)*x + 5*x^2),x]

[Out]

x + (4*Defer[Int][E^(4*E^3*x^2)/(E^(6 + (1 + E^6)*x^2) + E^(2*E^3*x^2)*x)^2, x])/5 + (2*(9 - 8*E^3 + 4*E^6)*De
fer[Int][(E^(6 + (1 + E^3)^2*x^2)*x)/(E^(6 + (1 + E^6)*x^2) + E^(2*E^3*x^2)*x)^2, x])/5 + 2*Defer[Int][(E^(4*E
^3*x^2)*x^2)/(E^(6 + (1 + E^6)*x^2) + E^(2*E^3*x^2)*x)^2, x] - 2*Defer[Int][(E^(2*E^3*x^2)*x)/(E^(6 + (1 + E^6
)*x^2) + E^(2*E^3*x^2)*x), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {5 e^{12+2 \left (1+e^6\right ) x^2}+18 e^{6+\left (1+e^3\right )^2 x^2} \left (1+\frac {4}{9} e^3 \left (-2+e^3\right )\right ) x+e^{4 e^3 x^2} \left (4+5 x^2\right )}{5 \left (e^{6+\left (1+e^6\right ) x^2}+e^{2 e^3 x^2} x\right )^2} \, dx\\ &=\frac {1}{5} \int \frac {5 e^{12+2 \left (1+e^6\right ) x^2}+18 e^{6+\left (1+e^3\right )^2 x^2} \left (1+\frac {4}{9} e^3 \left (-2+e^3\right )\right ) x+e^{4 e^3 x^2} \left (4+5 x^2\right )}{\left (e^{6+\left (1+e^6\right ) x^2}+e^{2 e^3 x^2} x\right )^2} \, dx\\ &=\frac {1}{5} \int \left (5-\frac {10 e^{2 e^3 x^2} x}{e^{6+\left (1+e^6\right ) x^2}+e^{2 e^3 x^2} x}+\frac {2 \left (2 e^{4 e^3 x^2}+9 e^{6+\left (1+e^3\right )^2 x^2} \left (1+\frac {4}{9} e^3 \left (-2+e^3\right )\right ) x+5 e^{4 e^3 x^2} x^2\right )}{\left (e^{6+\left (1+e^6\right ) x^2}+e^{2 e^3 x^2} x\right )^2}\right ) \, dx\\ &=x+\frac {2}{5} \int \frac {2 e^{4 e^3 x^2}+9 e^{6+\left (1+e^3\right )^2 x^2} \left (1+\frac {4}{9} e^3 \left (-2+e^3\right )\right ) x+5 e^{4 e^3 x^2} x^2}{\left (e^{6+\left (1+e^6\right ) x^2}+e^{2 e^3 x^2} x\right )^2} \, dx-2 \int \frac {e^{2 e^3 x^2} x}{e^{6+\left (1+e^6\right ) x^2}+e^{2 e^3 x^2} x} \, dx\\ &=x+\frac {2}{5} \int \left (\frac {2 e^{4 e^3 x^2}}{\left (e^{6+\left (1+e^6\right ) x^2}+e^{2 e^3 x^2} x\right )^2}+\frac {e^{6+\left (1+e^3\right )^2 x^2} \left (9-8 e^3+4 e^6\right ) x}{\left (e^{6+\left (1+e^6\right ) x^2}+e^{2 e^3 x^2} x\right )^2}+\frac {5 e^{4 e^3 x^2} x^2}{\left (e^{6+\left (1+e^6\right ) x^2}+e^{2 e^3 x^2} x\right )^2}\right ) \, dx-2 \int \frac {e^{2 e^3 x^2} x}{e^{6+\left (1+e^6\right ) x^2}+e^{2 e^3 x^2} x} \, dx\\ &=x+\frac {4}{5} \int \frac {e^{4 e^3 x^2}}{\left (e^{6+\left (1+e^6\right ) x^2}+e^{2 e^3 x^2} x\right )^2} \, dx+2 \int \frac {e^{4 e^3 x^2} x^2}{\left (e^{6+\left (1+e^6\right ) x^2}+e^{2 e^3 x^2} x\right )^2} \, dx-2 \int \frac {e^{2 e^3 x^2} x}{e^{6+\left (1+e^6\right ) x^2}+e^{2 e^3 x^2} x} \, dx+\frac {1}{5} \left (2 \left (9-8 e^3+4 e^6\right )\right ) \int \frac {e^{6+\left (1+e^3\right )^2 x^2} x}{\left (e^{6+\left (1+e^6\right ) x^2}+e^{2 e^3 x^2} x\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [F]  time = 10.04, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {4+5 e^{12+2 x^2-4 e^3 x^2+2 e^6 x^2}+5 x^2+e^{6+x^2-2 e^3 x^2+e^6 x^2} \left (18 x-16 e^3 x+8 e^6 x\right )}{5 e^{12+2 x^2-4 e^3 x^2+2 e^6 x^2}+10 e^{6+x^2-2 e^3 x^2+e^6 x^2} x+5 x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(4 + 5*E^(12 + 2*x^2 - 4*E^3*x^2 + 2*E^6*x^2) + 5*x^2 + E^(6 + x^2 - 2*E^3*x^2 + E^6*x^2)*(18*x - 16
*E^3*x + 8*E^6*x))/(5*E^(12 + 2*x^2 - 4*E^3*x^2 + 2*E^6*x^2) + 10*E^(6 + x^2 - 2*E^3*x^2 + E^6*x^2)*x + 5*x^2)
,x]

[Out]

Integrate[(4 + 5*E^(12 + 2*x^2 - 4*E^3*x^2 + 2*E^6*x^2) + 5*x^2 + E^(6 + x^2 - 2*E^3*x^2 + E^6*x^2)*(18*x - 16
*E^3*x + 8*E^6*x))/(5*E^(12 + 2*x^2 - 4*E^3*x^2 + 2*E^6*x^2) + 10*E^(6 + x^2 - 2*E^3*x^2 + E^6*x^2)*x + 5*x^2)
, x]

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fricas [B]  time = 0.93, size = 54, normalized size = 2.08 \begin {gather*} \frac {5 \, x^{2} + 5 \, x e^{\left (x^{2} e^{6} - 2 \, x^{2} e^{3} + x^{2} + 6\right )} - 4}{5 \, {\left (x + e^{\left (x^{2} e^{6} - 2 \, x^{2} e^{3} + x^{2} + 6\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*exp(x^2*exp(3)^2-2*x^2*exp(3)+x^2+6)^2+(8*x*exp(3)^2-16*x*exp(3)+18*x)*exp(x^2*exp(3)^2-2*x^2*exp
(3)+x^2+6)+5*x^2+4)/(5*exp(x^2*exp(3)^2-2*x^2*exp(3)+x^2+6)^2+10*x*exp(x^2*exp(3)^2-2*x^2*exp(3)+x^2+6)+5*x^2)
,x, algorithm="fricas")

[Out]

1/5*(5*x^2 + 5*x*e^(x^2*e^6 - 2*x^2*e^3 + x^2 + 6) - 4)/(x + e^(x^2*e^6 - 2*x^2*e^3 + x^2 + 6))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*exp(x^2*exp(3)^2-2*x^2*exp(3)+x^2+6)^2+(8*x*exp(3)^2-16*x*exp(3)+18*x)*exp(x^2*exp(3)^2-2*x^2*exp
(3)+x^2+6)+5*x^2+4)/(5*exp(x^2*exp(3)^2-2*x^2*exp(3)+x^2+6)^2+10*x*exp(x^2*exp(3)^2-2*x^2*exp(3)+x^2+6)+5*x^2)
,x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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maple [A]  time = 0.11, size = 28, normalized size = 1.08

method	result	size
risch	\(x -\frac {4}{5 \left ({\mathrm e}^{x^{2} {\mathrm e}^{6}-2 x^{2} {\mathrm e}^{3}+x^{2}+6}+x \right )}\)	\(28\)
norman	\(\frac {-\frac {4}{5}+x^{2}+x \,{\mathrm e}^{x^{2} {\mathrm e}^{6}-2 x^{2} {\mathrm e}^{3}+x^{2}+6}}{{\mathrm e}^{x^{2} {\mathrm e}^{6}-2 x^{2} {\mathrm e}^{3}+x^{2}+6}+x}\)	\(55\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5*exp(x^2*exp(3)^2-2*x^2*exp(3)+x^2+6)^2+(8*x*exp(3)^2-16*x*exp(3)+18*x)*exp(x^2*exp(3)^2-2*x^2*exp(3)+x^
2+6)+5*x^2+4)/(5*exp(x^2*exp(3)^2-2*x^2*exp(3)+x^2+6)^2+10*x*exp(x^2*exp(3)^2-2*x^2*exp(3)+x^2+6)+5*x^2),x,met
hod=_RETURNVERBOSE)

[Out]

x-4/5/(exp(x^2*exp(6)-2*x^2*exp(3)+x^2+6)+x)

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maxima [B]  time = 0.49, size = 59, normalized size = 2.27 \begin {gather*} \frac {5 \, x e^{\left (x^{2} e^{6} + x^{2} + 6\right )} + {\left (5 \, x^{2} - 4\right )} e^{\left (2 \, x^{2} e^{3}\right )}}{5 \, {\left (x e^{\left (2 \, x^{2} e^{3}\right )} + e^{\left (x^{2} e^{6} + x^{2} + 6\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*exp(x^2*exp(3)^2-2*x^2*exp(3)+x^2+6)^2+(8*x*exp(3)^2-16*x*exp(3)+18*x)*exp(x^2*exp(3)^2-2*x^2*exp
(3)+x^2+6)+5*x^2+4)/(5*exp(x^2*exp(3)^2-2*x^2*exp(3)+x^2+6)^2+10*x*exp(x^2*exp(3)^2-2*x^2*exp(3)+x^2+6)+5*x^2)
,x, algorithm="maxima")

[Out]

1/5*(5*x*e^(x^2*e^6 + x^2 + 6) + (5*x^2 - 4)*e^(2*x^2*e^3))/(x*e^(2*x^2*e^3) + e^(x^2*e^6 + x^2 + 6))

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mupad [B]  time = 5.74, size = 33, normalized size = 1.27 \begin {gather*} x-\frac {4}{5\,x+5\,{\mathrm {e}}^{-2\,x^2\,{\mathrm {e}}^3}\,{\mathrm {e}}^{x^2\,{\mathrm {e}}^6}\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5*exp(2*x^2*exp(6) - 4*x^2*exp(3) + 2*x^2 + 12) + 5*x^2 + exp(x^2*exp(6) - 2*x^2*exp(3) + x^2 + 6)*(18*x
- 16*x*exp(3) + 8*x*exp(6)) + 4)/(5*exp(2*x^2*exp(6) - 4*x^2*exp(3) + 2*x^2 + 12) + 10*x*exp(x^2*exp(6) - 2*x^
2*exp(3) + x^2 + 6) + 5*x^2),x)

[Out]

x - 4/(5*x + 5*exp(-2*x^2*exp(3))*exp(x^2*exp(6))*exp(x^2)*exp(6))

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sympy [A]  time = 0.18, size = 29, normalized size = 1.12 \begin {gather*} x - \frac {4}{5 x + 5 e^{- 2 x^{2} e^{3} + x^{2} + x^{2} e^{6} + 6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*exp(x**2*exp(3)**2-2*x**2*exp(3)+x**2+6)**2+(8*x*exp(3)**2-16*x*exp(3)+18*x)*exp(x**2*exp(3)**2-2
*x**2*exp(3)+x**2+6)+5*x**2+4)/(5*exp(x**2*exp(3)**2-2*x**2*exp(3)+x**2+6)**2+10*x*exp(x**2*exp(3)**2-2*x**2*e
xp(3)+x**2+6)+5*x**2),x)

[Out]

x - 4/(5*x + 5*exp(-2*x**2*exp(3) + x**2 + x**2*exp(6) + 6))

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