3.27.34 \(\int \frac {18+108 x+9 x^2+e^{e^x} (-54 x+e^x (-90+63 x^2-9 x^4))}{4-4 x^2+x^4} \, dx\)

Optimal. Leaf size=26 \[ \frac {9 \left (x+\left (-2+e^{e^x}\right ) \left (-5+x^2\right )\right )}{2-x^2} \]

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Rubi [F]  time = 0.59, 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 {18+108 x+9 x^2+e^{e^x} \left (-54 x+e^x \left (-90+63 x^2-9 x^4\right )\right )}{4-4 x^2+x^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(18 + 108*x + 9*x^2 + E^E^x*(-54*x + E^x*(-90 + 63*x^2 - 9*x^4)))/(4 - 4*x^2 + x^4),x]

[Out]

-9*E^E^x + 54/(2 - x^2) + (9*x)/(2 - x^2) - (27*Defer[Int][E^(E^x + x)/(Sqrt[2] - x), x])/(2*Sqrt[2]) - (27*De
fer[Int][E^(E^x + x)/(Sqrt[2] + x), x])/(2*Sqrt[2]) - 54*Defer[Int][(E^E^x*x)/(-2 + x^2)^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {18+108 x+9 x^2+e^{e^x} \left (-54 x+e^x \left (-90+63 x^2-9 x^4\right )\right )}{\left (-2+x^2\right )^2} \, dx\\ &=\int \left (\frac {18}{\left (-2+x^2\right )^2}+\frac {108 x}{\left (-2+x^2\right )^2}-\frac {54 e^{e^x} x}{\left (-2+x^2\right )^2}+\frac {9 x^2}{\left (-2+x^2\right )^2}-\frac {9 e^{e^x+x} \left (-5+x^2\right )}{-2+x^2}\right ) \, dx\\ &=9 \int \frac {x^2}{\left (-2+x^2\right )^2} \, dx-9 \int \frac {e^{e^x+x} \left (-5+x^2\right )}{-2+x^2} \, dx+18 \int \frac {1}{\left (-2+x^2\right )^2} \, dx-54 \int \frac {e^{e^x} x}{\left (-2+x^2\right )^2} \, dx+108 \int \frac {x}{\left (-2+x^2\right )^2} \, dx\\ &=\frac {54}{2-x^2}+\frac {9 x}{2-x^2}-9 \int \left (e^{e^x+x}-\frac {3 e^{e^x+x}}{-2+x^2}\right ) \, dx-54 \int \frac {e^{e^x} x}{\left (-2+x^2\right )^2} \, dx\\ &=\frac {54}{2-x^2}+\frac {9 x}{2-x^2}-9 \int e^{e^x+x} \, dx+27 \int \frac {e^{e^x+x}}{-2+x^2} \, dx-54 \int \frac {e^{e^x} x}{\left (-2+x^2\right )^2} \, dx\\ &=\frac {54}{2-x^2}+\frac {9 x}{2-x^2}-9 \operatorname {Subst}\left (\int e^x \, dx,x,e^x\right )+27 \int \left (-\frac {e^{e^x+x}}{2 \sqrt {2} \left (\sqrt {2}-x\right )}-\frac {e^{e^x+x}}{2 \sqrt {2} \left (\sqrt {2}+x\right )}\right ) \, dx-54 \int \frac {e^{e^x} x}{\left (-2+x^2\right )^2} \, dx\\ &=-9 e^{e^x}+\frac {54}{2-x^2}+\frac {9 x}{2-x^2}-54 \int \frac {e^{e^x} x}{\left (-2+x^2\right )^2} \, dx-\frac {27 \int \frac {e^{e^x+x}}{\sqrt {2}-x} \, dx}{2 \sqrt {2}}-\frac {27 \int \frac {e^{e^x+x}}{\sqrt {2}+x} \, dx}{2 \sqrt {2}}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.16, size = 23, normalized size = 0.88 \begin {gather*} -\frac {9 \left (6+x+e^{e^x} \left (-5+x^2\right )\right )}{-2+x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(18 + 108*x + 9*x^2 + E^E^x*(-54*x + E^x*(-90 + 63*x^2 - 9*x^4)))/(4 - 4*x^2 + x^4),x]

[Out]

(-9*(6 + x + E^E^x*(-5 + x^2)))/(-2 + x^2)

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fricas [A]  time = 0.66, size = 21, normalized size = 0.81 \begin {gather*} -\frac {9 \, {\left ({\left (x^{2} - 5\right )} e^{\left (e^{x}\right )} + x + 6\right )}}{x^{2} - 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-9*x^4+63*x^2-90)*exp(x)-54*x)*exp(exp(x))+9*x^2+108*x+18)/(x^4-4*x^2+4),x, algorithm="fricas")

[Out]

-9*((x^2 - 5)*e^(e^x) + x + 6)/(x^2 - 2)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {9 \, {\left (x^{2} - {\left ({\left (x^{4} - 7 \, x^{2} + 10\right )} e^{x} + 6 \, x\right )} e^{\left (e^{x}\right )} + 12 \, x + 2\right )}}{x^{4} - 4 \, x^{2} + 4}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-9*x^4+63*x^2-90)*exp(x)-54*x)*exp(exp(x))+9*x^2+108*x+18)/(x^4-4*x^2+4),x, algorithm="giac")

[Out]

integrate(9*(x^2 - ((x^4 - 7*x^2 + 10)*e^x + 6*x)*e^(e^x) + 12*x + 2)/(x^4 - 4*x^2 + 4), x)

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maple [A]  time = 0.06, size = 27, normalized size = 1.04

method	result	size
norman	\(\frac {-9 x -9 \,{\mathrm e}^{{\mathrm e}^{x}} x^{2}+45 \,{\mathrm e}^{{\mathrm e}^{x}}-54}{x^{2}-2}\)	\(27\)
risch	\(\frac {-9 x -54}{x^{2}-2}-\frac {9 \left (x^{2}-5\right ) {\mathrm e}^{{\mathrm e}^{x}}}{x^{2}-2}\)	\(32\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-9*x^4+63*x^2-90)*exp(x)-54*x)*exp(exp(x))+9*x^2+108*x+18)/(x^4-4*x^2+4),x,method=_RETURNVERBOSE)

[Out]

(-9*x-9*exp(exp(x))*x^2+45*exp(exp(x))-54)/(x^2-2)

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maxima [A]  time = 0.56, size = 37, normalized size = 1.42 \begin {gather*} -\frac {9 \, {\left (x^{2} - 5\right )} e^{\left (e^{x}\right )}}{x^{2} - 2} - \frac {9 \, x}{x^{2} - 2} - \frac {54}{x^{2} - 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-9*x^4+63*x^2-90)*exp(x)-54*x)*exp(exp(x))+9*x^2+108*x+18)/(x^4-4*x^2+4),x, algorithm="maxima")

[Out]

-9*(x^2 - 5)*e^(e^x)/(x^2 - 2) - 9*x/(x^2 - 2) - 54/(x^2 - 2)

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mupad [B]  time = 0.17, size = 34, normalized size = 1.31 \begin {gather*} -\frac {9\,x+54}{x^2-2}-\frac {{\mathrm {e}}^{{\mathrm {e}}^x}\,\left (9\,x^2-45\right )}{x^2-2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((108*x + 9*x^2 - exp(exp(x))*(54*x + exp(x)*(9*x^4 - 63*x^2 + 90)) + 18)/(x^4 - 4*x^2 + 4),x)

[Out]

- (9*x + 54)/(x^2 - 2) - (exp(exp(x))*(9*x^2 - 45))/(x^2 - 2)

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sympy [A]  time = 0.18, size = 27, normalized size = 1.04 \begin {gather*} \frac {\left (45 - 9 x^{2}\right ) e^{e^{x}}}{x^{2} - 2} + \frac {- 9 x - 54}{x^{2} - 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-9*x**4+63*x**2-90)*exp(x)-54*x)*exp(exp(x))+9*x**2+108*x+18)/(x**4-4*x**2+4),x)

[Out]

(45 - 9*x**2)*exp(exp(x))/(x**2 - 2) + (-9*x - 54)/(x**2 - 2)

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