3.2.61 \(\int \frac {e^{e^{2 x^2}} (16+8 x-7 x^2-2 x^3+x^4)^{e^{e^{2 x^2}}} (-2+4 x+e^{2 x^2} (-16 x-4 x^2+4 x^3) \log (16+8 x-7 x^2-2 x^3+x^4))}{-4-x+x^2} \, dx\)

Optimal. Leaf size=20 \[ \left (\left (4+x-x^2\right )^2\right )^{e^{e^{2 x^2}}} \]

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Rubi [F]  time = 2.70, 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 {e^{e^{2 x^2}} \left (16+8 x-7 x^2-2 x^3+x^4\right )^{e^{e^{2 x^2}}} \left (-2+4 x+e^{2 x^2} \left (-16 x-4 x^2+4 x^3\right ) \log \left (16+8 x-7 x^2-2 x^3+x^4\right )\right )}{-4-x+x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^E^(2*x^2)*(16 + 8*x - 7*x^2 - 2*x^3 + x^4)^E^E^(2*x^2)*(-2 + 4*x + E^(2*x^2)*(-16*x - 4*x^2 + 4*x^3)*Lo
g[16 + 8*x - 7*x^2 - 2*x^3 + x^4]))/(-4 - x + x^2),x]

[Out]

4*Log[(4 + x - x^2)^2]*Defer[Int][E^(E^(2*x^2) + 2*x^2)*x*((4 + x - x^2)^2)^E^E^(2*x^2), x] + 4*Defer[Int][(E^
E^(2*x^2)*((4 + x - x^2)^2)^E^E^(2*x^2))/(-1 - Sqrt[17] + 2*x), x] + 4*Defer[Int][(E^E^(2*x^2)*((4 + x - x^2)^
2)^E^E^(2*x^2))/(-1 + Sqrt[17] + 2*x), x] - (16*Defer[Int][Defer[Int][E^(E^(2*x^2) + 2*x^2)*x*((4 + x - x^2)^2
)^E^E^(2*x^2), x]/(1 + Sqrt[17] - 2*x), x])/Sqrt[17] - (16*(17 + Sqrt[17])*Defer[Int][Defer[Int][E^(E^(2*x^2)
+ 2*x^2)*x*((4 + x - x^2)^2)^E^E^(2*x^2), x]/(-1 - Sqrt[17] + 2*x), x])/17 - (16*Defer[Int][Defer[Int][E^(E^(2
*x^2) + 2*x^2)*x*((4 + x - x^2)^2)^E^E^(2*x^2), x]/(-1 + Sqrt[17] + 2*x), x])/Sqrt[17] - (16*(17 - Sqrt[17])*D
efer[Int][Defer[Int][E^(E^(2*x^2) + 2*x^2)*x*((4 + x - x^2)^2)^E^E^(2*x^2), x]/(-1 + Sqrt[17] + 2*x), x])/17

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {2 e^{e^{2 x^2}} (-1+2 x) \left (16+8 x-7 x^2-2 x^3+x^4\right )^{e^{e^{2 x^2}}}}{-4-x+x^2}+4 e^{e^{2 x^2}+2 x^2} x \left (16+8 x-7 x^2-2 x^3+x^4\right )^{e^{e^{2 x^2}}} \log \left (\left (4+x-x^2\right )^2\right )\right ) \, dx\\ &=2 \int \frac {e^{e^{2 x^2}} (-1+2 x) \left (16+8 x-7 x^2-2 x^3+x^4\right )^{e^{e^{2 x^2}}}}{-4-x+x^2} \, dx+4 \int e^{e^{2 x^2}+2 x^2} x \left (16+8 x-7 x^2-2 x^3+x^4\right )^{e^{e^{2 x^2}}} \log \left (\left (4+x-x^2\right )^2\right ) \, dx\\ &=2 \int \left (\frac {2 e^{e^{2 x^2}} \left (16+8 x-7 x^2-2 x^3+x^4\right )^{e^{e^{2 x^2}}}}{-1-\sqrt {17}+2 x}+\frac {2 e^{e^{2 x^2}} \left (16+8 x-7 x^2-2 x^3+x^4\right )^{e^{e^{2 x^2}}}}{-1+\sqrt {17}+2 x}\right ) \, dx-4 \int \frac {2 (1-2 x) \int e^{e^{2 x^2}+2 x^2} x \left (\left (4+x-x^2\right )^2\right )^{e^{e^{2 x^2}}} \, dx}{4+x-x^2} \, dx+\left (4 \log \left (\left (4+x-x^2\right )^2\right )\right ) \int e^{e^{2 x^2}+2 x^2} x \left (\left (4+x-x^2\right )^2\right )^{e^{e^{2 x^2}}} \, dx\\ &=4 \int \frac {e^{e^{2 x^2}} \left (16+8 x-7 x^2-2 x^3+x^4\right )^{e^{e^{2 x^2}}}}{-1-\sqrt {17}+2 x} \, dx+4 \int \frac {e^{e^{2 x^2}} \left (16+8 x-7 x^2-2 x^3+x^4\right )^{e^{e^{2 x^2}}}}{-1+\sqrt {17}+2 x} \, dx-8 \int \frac {(1-2 x) \int e^{e^{2 x^2}+2 x^2} x \left (\left (4+x-x^2\right )^2\right )^{e^{e^{2 x^2}}} \, dx}{4+x-x^2} \, dx+\left (4 \log \left (\left (4+x-x^2\right )^2\right )\right ) \int e^{e^{2 x^2}+2 x^2} x \left (\left (4+x-x^2\right )^2\right )^{e^{e^{2 x^2}}} \, dx\\ &=4 \int \frac {e^{e^{2 x^2}} \left (\left (4+x-x^2\right )^2\right )^{e^{e^{2 x^2}}}}{-1-\sqrt {17}+2 x} \, dx+4 \int \frac {e^{e^{2 x^2}} \left (\left (4+x-x^2\right )^2\right )^{e^{e^{2 x^2}}}}{-1+\sqrt {17}+2 x} \, dx-8 \int \left (-\frac {\int e^{e^{2 x^2}+2 x^2} x \left (\left (4+x-x^2\right )^2\right )^{e^{e^{2 x^2}}} \, dx}{-4-x+x^2}+\frac {2 x \int e^{e^{2 x^2}+2 x^2} x \left (\left (4+x-x^2\right )^2\right )^{e^{e^{2 x^2}}} \, dx}{-4-x+x^2}\right ) \, dx+\left (4 \log \left (\left (4+x-x^2\right )^2\right )\right ) \int e^{e^{2 x^2}+2 x^2} x \left (\left (4+x-x^2\right )^2\right )^{e^{e^{2 x^2}}} \, dx\\ &=4 \int \frac {e^{e^{2 x^2}} \left (\left (4+x-x^2\right )^2\right )^{e^{e^{2 x^2}}}}{-1-\sqrt {17}+2 x} \, dx+4 \int \frac {e^{e^{2 x^2}} \left (\left (4+x-x^2\right )^2\right )^{e^{e^{2 x^2}}}}{-1+\sqrt {17}+2 x} \, dx+8 \int \frac {\int e^{e^{2 x^2}+2 x^2} x \left (\left (4+x-x^2\right )^2\right )^{e^{e^{2 x^2}}} \, dx}{-4-x+x^2} \, dx-16 \int \frac {x \int e^{e^{2 x^2}+2 x^2} x \left (\left (4+x-x^2\right )^2\right )^{e^{e^{2 x^2}}} \, dx}{-4-x+x^2} \, dx+\left (4 \log \left (\left (4+x-x^2\right )^2\right )\right ) \int e^{e^{2 x^2}+2 x^2} x \left (\left (4+x-x^2\right )^2\right )^{e^{e^{2 x^2}}} \, dx\\ &=4 \int \frac {e^{e^{2 x^2}} \left (\left (4+x-x^2\right )^2\right )^{e^{e^{2 x^2}}}}{-1-\sqrt {17}+2 x} \, dx+4 \int \frac {e^{e^{2 x^2}} \left (\left (4+x-x^2\right )^2\right )^{e^{e^{2 x^2}}}}{-1+\sqrt {17}+2 x} \, dx+8 \int \left (-\frac {2 \int e^{e^{2 x^2}+2 x^2} x \left (\left (4+x-x^2\right )^2\right )^{e^{e^{2 x^2}}} \, dx}{\sqrt {17} \left (1+\sqrt {17}-2 x\right )}-\frac {2 \int e^{e^{2 x^2}+2 x^2} x \left (\left (4+x-x^2\right )^2\right )^{e^{e^{2 x^2}}} \, dx}{\sqrt {17} \left (-1+\sqrt {17}+2 x\right )}\right ) \, dx-16 \int \left (\frac {\left (1+\frac {1}{\sqrt {17}}\right ) \int e^{e^{2 x^2}+2 x^2} x \left (\left (4+x-x^2\right )^2\right )^{e^{e^{2 x^2}}} \, dx}{-1-\sqrt {17}+2 x}+\frac {\left (1-\frac {1}{\sqrt {17}}\right ) \int e^{e^{2 x^2}+2 x^2} x \left (\left (4+x-x^2\right )^2\right )^{e^{e^{2 x^2}}} \, dx}{-1+\sqrt {17}+2 x}\right ) \, dx+\left (4 \log \left (\left (4+x-x^2\right )^2\right )\right ) \int e^{e^{2 x^2}+2 x^2} x \left (\left (4+x-x^2\right )^2\right )^{e^{e^{2 x^2}}} \, dx\\ &=4 \int \frac {e^{e^{2 x^2}} \left (\left (4+x-x^2\right )^2\right )^{e^{e^{2 x^2}}}}{-1-\sqrt {17}+2 x} \, dx+4 \int \frac {e^{e^{2 x^2}} \left (\left (4+x-x^2\right )^2\right )^{e^{e^{2 x^2}}}}{-1+\sqrt {17}+2 x} \, dx-\frac {16 \int \frac {\int e^{e^{2 x^2}+2 x^2} x \left (\left (4+x-x^2\right )^2\right )^{e^{e^{2 x^2}}} \, dx}{1+\sqrt {17}-2 x} \, dx}{\sqrt {17}}-\frac {16 \int \frac {\int e^{e^{2 x^2}+2 x^2} x \left (\left (4+x-x^2\right )^2\right )^{e^{e^{2 x^2}}} \, dx}{-1+\sqrt {17}+2 x} \, dx}{\sqrt {17}}-\frac {1}{17} \left (16 \left (17-\sqrt {17}\right )\right ) \int \frac {\int e^{e^{2 x^2}+2 x^2} x \left (\left (4+x-x^2\right )^2\right )^{e^{e^{2 x^2}}} \, dx}{-1+\sqrt {17}+2 x} \, dx-\frac {1}{17} \left (16 \left (17+\sqrt {17}\right )\right ) \int \frac {\int e^{e^{2 x^2}+2 x^2} x \left (\left (4+x-x^2\right )^2\right )^{e^{e^{2 x^2}}} \, dx}{-1-\sqrt {17}+2 x} \, dx+\left (4 \log \left (\left (4+x-x^2\right )^2\right )\right ) \int e^{e^{2 x^2}+2 x^2} x \left (\left (4+x-x^2\right )^2\right )^{e^{e^{2 x^2}}} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.54, size = 20, normalized size = 1.00 \begin {gather*} \left (\left (4+x-x^2\right )^2\right )^{e^{e^{2 x^2}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^E^(2*x^2)*(16 + 8*x - 7*x^2 - 2*x^3 + x^4)^E^E^(2*x^2)*(-2 + 4*x + E^(2*x^2)*(-16*x - 4*x^2 + 4*x
^3)*Log[16 + 8*x - 7*x^2 - 2*x^3 + x^4]))/(-4 - x + x^2),x]

[Out]

((4 + x - x^2)^2)^E^E^(2*x^2)

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fricas [A]  time = 0.77, size = 26, normalized size = 1.30 \begin {gather*} {\left (x^{4} - 2 \, x^{3} - 7 \, x^{2} + 8 \, x + 16\right )}^{e^{\left (e^{\left (2 \, x^{2}\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^3-4*x^2-16*x)*exp(x^2)^2*log(x^4-2*x^3-7*x^2+8*x+16)+4*x-2)*exp(exp(x^2)^2)*exp(log(x^4-2*x^3-
7*x^2+8*x+16)*exp(exp(x^2)^2))/(x^2-x-4),x, algorithm="fricas")

[Out]

(x^4 - 2*x^3 - 7*x^2 + 8*x + 16)^e^(e^(2*x^2))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 \, {\left (2 \, {\left (x^{3} - x^{2} - 4 \, x\right )} e^{\left (2 \, x^{2}\right )} \log \left (x^{4} - 2 \, x^{3} - 7 \, x^{2} + 8 \, x + 16\right ) + 2 \, x - 1\right )} {\left (x^{4} - 2 \, x^{3} - 7 \, x^{2} + 8 \, x + 16\right )}^{e^{\left (e^{\left (2 \, x^{2}\right )}\right )}} e^{\left (e^{\left (2 \, x^{2}\right )}\right )}}{x^{2} - x - 4}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^3-4*x^2-16*x)*exp(x^2)^2*log(x^4-2*x^3-7*x^2+8*x+16)+4*x-2)*exp(exp(x^2)^2)*exp(log(x^4-2*x^3-
7*x^2+8*x+16)*exp(exp(x^2)^2))/(x^2-x-4),x, algorithm="giac")

[Out]

integrate(2*(2*(x^3 - x^2 - 4*x)*e^(2*x^2)*log(x^4 - 2*x^3 - 7*x^2 + 8*x + 16) + 2*x - 1)*(x^4 - 2*x^3 - 7*x^2
+ 8*x + 16)^e^(e^(2*x^2))*e^(e^(2*x^2))/(x^2 - x - 4), x)

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maple [C]  time = 0.24, size = 107, normalized size = 5.35

method	result	size
risch	\({\mathrm e}^{\frac {\left (-i \pi \mathrm {csgn}\left (i \left (x^{2}-x -4\right )^{2}\right )^{3}+2 i \pi \mathrm {csgn}\left (i \left (x^{2}-x -4\right )^{2}\right )^{2} \mathrm {csgn}\left (i \left (x^{2}-x -4\right )\right )-i \pi \,\mathrm {csgn}\left (i \left (x^{2}-x -4\right )^{2}\right ) \mathrm {csgn}\left (i \left (x^{2}-x -4\right )\right )^{2}+4 \ln \left (x^{2}-x -4\right )\right ) {\mathrm e}^{{\mathrm e}^{2 x^{2}}}}{2}}\)	\(107\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((4*x^3-4*x^2-16*x)*exp(x^2)^2*ln(x^4-2*x^3-7*x^2+8*x+16)+4*x-2)*exp(exp(x^2)^2)*exp(ln(x^4-2*x^3-7*x^2+8*
x+16)*exp(exp(x^2)^2))/(x^2-x-4),x,method=_RETURNVERBOSE)

[Out]

exp(1/2*(-I*Pi*csgn(I*(x^2-x-4)^2)^3+2*I*Pi*csgn(I*(x^2-x-4)^2)^2*csgn(I*(x^2-x-4))-I*Pi*csgn(I*(x^2-x-4)^2)*c
sgn(I*(x^2-x-4))^2+4*ln(x^2-x-4))*exp(exp(2*x^2)))

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maxima [A]  time = 0.57, size = 18, normalized size = 0.90 \begin {gather*} {\left (x^{2} - x - 4\right )}^{2 \, e^{\left (e^{\left (2 \, x^{2}\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^3-4*x^2-16*x)*exp(x^2)^2*log(x^4-2*x^3-7*x^2+8*x+16)+4*x-2)*exp(exp(x^2)^2)*exp(log(x^4-2*x^3-
7*x^2+8*x+16)*exp(exp(x^2)^2))/(x^2-x-4),x, algorithm="maxima")

[Out]

(x^2 - x - 4)^(2*e^(e^(2*x^2)))

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mupad [B]  time = 0.51, size = 26, normalized size = 1.30 \begin {gather*} {\left (x^4-2\,x^3-7\,x^2+8\,x+16\right )}^{{\mathrm {e}}^{{\mathrm {e}}^{2\,x^2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(exp(2*x^2))*exp(exp(exp(2*x^2))*log(8*x - 7*x^2 - 2*x^3 + x^4 + 16))*(exp(2*x^2)*log(8*x - 7*x^2 - 2*
x^3 + x^4 + 16)*(16*x + 4*x^2 - 4*x^3) - 4*x + 2))/(x - x^2 + 4),x)

[Out]

(8*x - 7*x^2 - 2*x^3 + x^4 + 16)^exp(exp(2*x^2))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x**3-4*x**2-16*x)*exp(x**2)**2*ln(x**4-2*x**3-7*x**2+8*x+16)+4*x-2)*exp(exp(x**2)**2)*exp(ln(x**
4-2*x**3-7*x**2+8*x+16)*exp(exp(x**2)**2))/(x**2-x-4),x)

[Out]

Timed out

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