3.2.11 \(\int \frac {e^{\frac {4+x^2+e^4 (3 x-2 x^2)}{e^4 (4+x^2)}} (40-20 x-14 x^2-3 x^3+x^4)}{16+8 x^2+x^4} \, dx\) [111]

Optimal. Leaf size=26 \[ e^{\frac {1}{e^4}-\frac {-3+2 x}{\frac {4}{x}+x}} (2+x) \]

[Out]

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

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Rubi [F]
time = 2.84, 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 {\exp \left (\frac {4+x^2+e^4 \left (3 x-2 x^2\right )}{e^4 \left (4+x^2\right )}\right ) \left (40-20 x-14 x^2-3 x^3+x^4\right )}{16+8 x^2+x^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((4 + x^2 + E^4*(3*x - 2*x^2))/(E^4*(4 + x^2)))*(40 - 20*x - 14*x^2 - 3*x^3 + x^4))/(16 + 8*x^2 + x^4),
x]

[Out]

Defer[Int][E^((4 + 3*E^4*x + (1 - 2*E^4)*x^2)/(E^4*(4 + x^2))), x] - 7*Defer[Int][E^((4 + 3*E^4*x + (1 - 2*E^4
)*x^2)/(E^4*(4 + x^2)))/(2*I - x)^2, x] + (3/2 - 2*I)*Defer[Int][E^((4 + 3*E^4*x + (1 - 2*E^4)*x^2)/(E^4*(4 +
x^2)))/(2*I - x), x] - 7*Defer[Int][E^((4 + 3*E^4*x + (1 - 2*E^4)*x^2)/(E^4*(4 + x^2)))/(2*I + x)^2, x] - (3/2
+ 2*I)*Defer[Int][E^((4 + 3*E^4*x + (1 - 2*E^4)*x^2)/(E^4*(4 + x^2)))/(2*I + x), x] - 8*Defer[Int][(E^((4 + 3
*E^4*x + (1 - 2*E^4)*x^2)/(E^4*(4 + x^2)))*x)/(4 + x^2)^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\exp \left (\frac {4+x^2+e^4 \left (3 x-2 x^2\right )}{e^4 \left (4+x^2\right )}\right ) \left (40-20 x-14 x^2-3 x^3+x^4\right )}{\left (4+x^2\right )^2} \, dx\\ &=\int \frac {\exp \left (\frac {4+3 e^4 x+\left (1-2 e^4\right ) x^2}{e^4 \left (4+x^2\right )}\right ) \left (40-20 x-14 x^2-3 x^3+x^4\right )}{\left (4+x^2\right )^2} \, dx\\ &=\int \left (\exp \left (\frac {4+3 e^4 x+\left (1-2 e^4\right ) x^2}{e^4 \left (4+x^2\right )}\right )-\frac {8 \exp \left (\frac {4+3 e^4 x+\left (1-2 e^4\right ) x^2}{e^4 \left (4+x^2\right )}\right ) (-14+x)}{\left (4+x^2\right )^2}+\frac {\exp \left (\frac {4+3 e^4 x+\left (1-2 e^4\right ) x^2}{e^4 \left (4+x^2\right )}\right ) (-22-3 x)}{4+x^2}\right ) \, dx\\ &=-\left (8 \int \frac {\exp \left (\frac {4+3 e^4 x+\left (1-2 e^4\right ) x^2}{e^4 \left (4+x^2\right )}\right ) (-14+x)}{\left (4+x^2\right )^2} \, dx\right )+\int \exp \left (\frac {4+3 e^4 x+\left (1-2 e^4\right ) x^2}{e^4 \left (4+x^2\right )}\right ) \, dx+\int \frac {\exp \left (\frac {4+3 e^4 x+\left (1-2 e^4\right ) x^2}{e^4 \left (4+x^2\right )}\right ) (-22-3 x)}{4+x^2} \, dx\\ &=-\left (8 \int \left (-\frac {14 \exp \left (\frac {4+3 e^4 x+\left (1-2 e^4\right ) x^2}{e^4 \left (4+x^2\right )}\right )}{\left (4+x^2\right )^2}+\frac {\exp \left (\frac {4+3 e^4 x+\left (1-2 e^4\right ) x^2}{e^4 \left (4+x^2\right )}\right ) x}{\left (4+x^2\right )^2}\right ) \, dx\right )+\int \exp \left (\frac {4+3 e^4 x+\left (1-2 e^4\right ) x^2}{e^4 \left (4+x^2\right )}\right ) \, dx+\int \left (\frac {\left (\frac {3}{2}-\frac {11 i}{2}\right ) \exp \left (\frac {4+3 e^4 x+\left (1-2 e^4\right ) x^2}{e^4 \left (4+x^2\right )}\right )}{2 i-x}-\frac {\left (\frac {3}{2}+\frac {11 i}{2}\right ) \exp \left (\frac {4+3 e^4 x+\left (1-2 e^4\right ) x^2}{e^4 \left (4+x^2\right )}\right )}{2 i+x}\right ) \, dx\\ &=\left (-\frac {3}{2}-\frac {11 i}{2}\right ) \int \frac {\exp \left (\frac {4+3 e^4 x+\left (1-2 e^4\right ) x^2}{e^4 \left (4+x^2\right )}\right )}{2 i+x} \, dx+\left (\frac {3}{2}-\frac {11 i}{2}\right ) \int \frac {\exp \left (\frac {4+3 e^4 x+\left (1-2 e^4\right ) x^2}{e^4 \left (4+x^2\right )}\right )}{2 i-x} \, dx-8 \int \frac {\exp \left (\frac {4+3 e^4 x+\left (1-2 e^4\right ) x^2}{e^4 \left (4+x^2\right )}\right ) x}{\left (4+x^2\right )^2} \, dx+112 \int \frac {\exp \left (\frac {4+3 e^4 x+\left (1-2 e^4\right ) x^2}{e^4 \left (4+x^2\right )}\right )}{\left (4+x^2\right )^2} \, dx+\int \exp \left (\frac {4+3 e^4 x+\left (1-2 e^4\right ) x^2}{e^4 \left (4+x^2\right )}\right ) \, dx\\ &=\left (-\frac {3}{2}-\frac {11 i}{2}\right ) \int \frac {\exp \left (\frac {4+3 e^4 x+\left (1-2 e^4\right ) x^2}{e^4 \left (4+x^2\right )}\right )}{2 i+x} \, dx+\left (\frac {3}{2}-\frac {11 i}{2}\right ) \int \frac {\exp \left (\frac {4+3 e^4 x+\left (1-2 e^4\right ) x^2}{e^4 \left (4+x^2\right )}\right )}{2 i-x} \, dx-8 \int \frac {\exp \left (\frac {4+3 e^4 x+\left (1-2 e^4\right ) x^2}{e^4 \left (4+x^2\right )}\right ) x}{\left (4+x^2\right )^2} \, dx+112 \int \left (-\frac {\exp \left (\frac {4+3 e^4 x+\left (1-2 e^4\right ) x^2}{e^4 \left (4+x^2\right )}\right )}{16 (2 i-x)^2}-\frac {\exp \left (\frac {4+3 e^4 x+\left (1-2 e^4\right ) x^2}{e^4 \left (4+x^2\right )}\right )}{16 (2 i+x)^2}-\frac {\exp \left (\frac {4+3 e^4 x+\left (1-2 e^4\right ) x^2}{e^4 \left (4+x^2\right )}\right )}{8 \left (-4-x^2\right )}\right ) \, dx+\int \exp \left (\frac {4+3 e^4 x+\left (1-2 e^4\right ) x^2}{e^4 \left (4+x^2\right )}\right ) \, dx\\ &=\left (-\frac {3}{2}-\frac {11 i}{2}\right ) \int \frac {\exp \left (\frac {4+3 e^4 x+\left (1-2 e^4\right ) x^2}{e^4 \left (4+x^2\right )}\right )}{2 i+x} \, dx+\left (\frac {3}{2}-\frac {11 i}{2}\right ) \int \frac {\exp \left (\frac {4+3 e^4 x+\left (1-2 e^4\right ) x^2}{e^4 \left (4+x^2\right )}\right )}{2 i-x} \, dx-7 \int \frac {\exp \left (\frac {4+3 e^4 x+\left (1-2 e^4\right ) x^2}{e^4 \left (4+x^2\right )}\right )}{(2 i-x)^2} \, dx-7 \int \frac {\exp \left (\frac {4+3 e^4 x+\left (1-2 e^4\right ) x^2}{e^4 \left (4+x^2\right )}\right )}{(2 i+x)^2} \, dx-8 \int \frac {\exp \left (\frac {4+3 e^4 x+\left (1-2 e^4\right ) x^2}{e^4 \left (4+x^2\right )}\right ) x}{\left (4+x^2\right )^2} \, dx-14 \int \frac {\exp \left (\frac {4+3 e^4 x+\left (1-2 e^4\right ) x^2}{e^4 \left (4+x^2\right )}\right )}{-4-x^2} \, dx+\int \exp \left (\frac {4+3 e^4 x+\left (1-2 e^4\right ) x^2}{e^4 \left (4+x^2\right )}\right ) \, dx\\ &=\left (-\frac {3}{2}-\frac {11 i}{2}\right ) \int \frac {\exp \left (\frac {4+3 e^4 x+\left (1-2 e^4\right ) x^2}{e^4 \left (4+x^2\right )}\right )}{2 i+x} \, dx+\left (\frac {3}{2}-\frac {11 i}{2}\right ) \int \frac {\exp \left (\frac {4+3 e^4 x+\left (1-2 e^4\right ) x^2}{e^4 \left (4+x^2\right )}\right )}{2 i-x} \, dx-7 \int \frac {\exp \left (\frac {4+3 e^4 x+\left (1-2 e^4\right ) x^2}{e^4 \left (4+x^2\right )}\right )}{(2 i-x)^2} \, dx-7 \int \frac {\exp \left (\frac {4+3 e^4 x+\left (1-2 e^4\right ) x^2}{e^4 \left (4+x^2\right )}\right )}{(2 i+x)^2} \, dx-8 \int \frac {\exp \left (\frac {4+3 e^4 x+\left (1-2 e^4\right ) x^2}{e^4 \left (4+x^2\right )}\right ) x}{\left (4+x^2\right )^2} \, dx-14 \int \left (-\frac {i \exp \left (\frac {4+3 e^4 x+\left (1-2 e^4\right ) x^2}{e^4 \left (4+x^2\right )}\right )}{4 (2 i-x)}-\frac {i \exp \left (\frac {4+3 e^4 x+\left (1-2 e^4\right ) x^2}{e^4 \left (4+x^2\right )}\right )}{4 (2 i+x)}\right ) \, dx+\int \exp \left (\frac {4+3 e^4 x+\left (1-2 e^4\right ) x^2}{e^4 \left (4+x^2\right )}\right ) \, dx\\ &=\left (-\frac {3}{2}-\frac {11 i}{2}\right ) \int \frac {\exp \left (\frac {4+3 e^4 x+\left (1-2 e^4\right ) x^2}{e^4 \left (4+x^2\right )}\right )}{2 i+x} \, dx+\frac {7}{2} i \int \frac {\exp \left (\frac {4+3 e^4 x+\left (1-2 e^4\right ) x^2}{e^4 \left (4+x^2\right )}\right )}{2 i-x} \, dx+\frac {7}{2} i \int \frac {\exp \left (\frac {4+3 e^4 x+\left (1-2 e^4\right ) x^2}{e^4 \left (4+x^2\right )}\right )}{2 i+x} \, dx+\left (\frac {3}{2}-\frac {11 i}{2}\right ) \int \frac {\exp \left (\frac {4+3 e^4 x+\left (1-2 e^4\right ) x^2}{e^4 \left (4+x^2\right )}\right )}{2 i-x} \, dx-7 \int \frac {\exp \left (\frac {4+3 e^4 x+\left (1-2 e^4\right ) x^2}{e^4 \left (4+x^2\right )}\right )}{(2 i-x)^2} \, dx-7 \int \frac {\exp \left (\frac {4+3 e^4 x+\left (1-2 e^4\right ) x^2}{e^4 \left (4+x^2\right )}\right )}{(2 i+x)^2} \, dx-8 \int \frac {\exp \left (\frac {4+3 e^4 x+\left (1-2 e^4\right ) x^2}{e^4 \left (4+x^2\right )}\right ) x}{\left (4+x^2\right )^2} \, dx+\int \exp \left (\frac {4+3 e^4 x+\left (1-2 e^4\right ) x^2}{e^4 \left (4+x^2\right )}\right ) \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.13, size = 24, normalized size = 0.92 \begin {gather*} e^{\frac {1}{e^4}+\frac {(3-2 x) x}{4+x^2}} (2+x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((4 + x^2 + E^4*(3*x - 2*x^2))/(E^4*(4 + x^2)))*(40 - 20*x - 14*x^2 - 3*x^3 + x^4))/(16 + 8*x^2 +
x^4),x]

[Out]

E^(E^(-4) + ((3 - 2*x)*x)/(4 + x^2))*(2 + x)

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Maple [A]
time = 0.18, size = 36, normalized size = 1.38

method	result	size
risch	\(\left (2+x \right ) {\mathrm e}^{-\frac {\left (2 x^{2} {\mathrm e}^{4}-3 x \,{\mathrm e}^{4}-x^{2}-4\right ) {\mathrm e}^{-4}}{x^{2}+4}}\)	\(36\)
gosper	\(\left (2+x \right ) {\mathrm e}^{-\frac {\left (2 x^{2} {\mathrm e}^{4}-3 x \,{\mathrm e}^{4}-x^{2}-4\right ) {\mathrm e}^{-4}}{x^{2}+4}}\)	\(38\)
norman	\(\frac {x^{3} {\mathrm e}^{\frac {\left (\left (-2 x^{2}+3 x \right ) {\mathrm e}^{4}+x^{2}+4\right ) {\mathrm e}^{-4}}{x^{2}+4}}+4 x \,{\mathrm e}^{\frac {\left (\left (-2 x^{2}+3 x \right ) {\mathrm e}^{4}+x^{2}+4\right ) {\mathrm e}^{-4}}{x^{2}+4}}+2 x^{2} {\mathrm e}^{\frac {\left (\left (-2 x^{2}+3 x \right ) {\mathrm e}^{4}+x^{2}+4\right ) {\mathrm e}^{-4}}{x^{2}+4}}+8 \,{\mathrm e}^{\frac {\left (\left (-2 x^{2}+3 x \right ) {\mathrm e}^{4}+x^{2}+4\right ) {\mathrm e}^{-4}}{x^{2}+4}}}{x^{2}+4}\)	\(144\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4-3*x^3-14*x^2-20*x+40)*exp(((-2*x^2+3*x)*exp(4)+x^2+4)/(x^2+4)/exp(4))/(x^4+8*x^2+16),x,method=_RETURN
VERBOSE)

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4-3*x^3-14*x^2-20*x+40)*exp(((-2*x^2+3*x)*exp(4)+x^2+4)/(x^2+4)/exp(4))/(x^4+8*x^2+16),x, algorit
hm="maxima")

[Out]

integrate((x^4 - 3*x^3 - 14*x^2 - 20*x + 40)*e^((x^2 - (2*x^2 - 3*x)*e^4 + 4)*e^(-4)/(x^2 + 4))/(x^4 + 8*x^2 +
16), x)

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Fricas [A]
time = 0.30, size = 33, normalized size = 1.27 \begin {gather*} {\left (x + 2\right )} e^{\left (\frac {{\left (x^{2} - {\left (2 \, x^{2} - 3 \, x\right )} e^{4} + 4\right )} e^{\left (-4\right )}}{x^{2} + 4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4-3*x^3-14*x^2-20*x+40)*exp(((-2*x^2+3*x)*exp(4)+x^2+4)/(x^2+4)/exp(4))/(x^4+8*x^2+16),x, algorit
hm="fricas")

[Out]

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

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Sympy [A]
time = 0.11, size = 29, normalized size = 1.12 \begin {gather*} \left (x + 2\right ) e^{\frac {x^{2} + \left (- 2 x^{2} + 3 x\right ) e^{4} + 4}{\left (x^{2} + 4\right ) e^{4}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**4-3*x**3-14*x**2-20*x+40)*exp(((-2*x**2+3*x)*exp(4)+x**2+4)/(x**2+4)/exp(4))/(x**4+8*x**2+16),x)

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (24) = 48\).
time = 0.45, size = 49, normalized size = 1.88 \begin {gather*} x e^{\left (-\frac {2 \, x^{2} - 3 \, x}{x^{2} + 4} + e^{\left (-4\right )}\right )} + 2 \, e^{\left (-\frac {2 \, x^{2} - 3 \, x}{x^{2} + 4} + e^{\left (-4\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4-3*x^3-14*x^2-20*x+40)*exp(((-2*x^2+3*x)*exp(4)+x^2+4)/(x^2+4)/exp(4))/(x^4+8*x^2+16),x, algorit
hm="giac")

[Out]

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

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Mupad [B]
time = 0.40, size = 54, normalized size = 2.08 \begin {gather*} {\mathrm {e}}^{-\frac {2\,x^2}{x^2+4}}\,{\mathrm {e}}^{\frac {x^2\,{\mathrm {e}}^{-4}}{x^2+4}}\,{\mathrm {e}}^{\frac {4\,{\mathrm {e}}^{-4}}{x^2+4}}\,{\mathrm {e}}^{\frac {3\,x}{x^2+4}}\,\left (x+2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp((exp(-4)*(exp(4)*(3*x - 2*x^2) + x^2 + 4))/(x^2 + 4))*(20*x + 14*x^2 + 3*x^3 - x^4 - 40))/(8*x^2 + x
^4 + 16),x)

[Out]

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

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