3.62.15 \(\int \frac {-104976 x+39366 x^2-1458 x^4+e^{2 x} (-5832 x+8019 x^2-1458 x^3-729 x^4+162 x^5)}{26244-5832 x^2+324 x^4+e^{4 x} (81-18 x^2+x^4)+e^{2 x} (2916-648 x^2+36 x^4)} \, dx\)

Optimal. Leaf size=33 \[ \frac {(4-x) x^2}{\left (2+\frac {e^{2 x}}{9}\right ) \left (-1+\frac {x^2}{9}\right )} \]

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Rubi [F]  time = 1.62, 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 {-104976 x+39366 x^2-1458 x^4+e^{2 x} \left (-5832 x+8019 x^2-1458 x^3-729 x^4+162 x^5\right )}{26244-5832 x^2+324 x^4+e^{4 x} \left (81-18 x^2+x^4\right )+e^{2 x} \left (2916-648 x^2+36 x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-104976*x + 39366*x^2 - 1458*x^4 + E^(2*x)*(-5832*x + 8019*x^2 - 1458*x^3 - 729*x^4 + 162*x^5))/(26244 -
5832*x^2 + 324*x^4 + E^(4*x)*(81 - 18*x^2 + x^4) + E^(2*x)*(2916 - 648*x^2 + 36*x^4)),x]

[Out]

324/(18 + E^(2*x)) - (81*x)/(18 + E^(2*x)) - (243*Defer[Int][1/((18 + E^(2*x))*(-3 + x)^2), x])/2 + 4374*Defer
[Int][1/((18 + E^(2*x))^2*(-3 + x)), x] - 243*Defer[Int][1/((18 + E^(2*x))*(-3 + x)), x] + (1701*Defer[Int][1/
((18 + E^(2*x))*(3 + x)^2), x])/2 - 30618*Defer[Int][1/((18 + E^(2*x))^2*(3 + x)), x] + 1701*Defer[Int][1/((18
+ E^(2*x))*(3 + x)), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {81 x \left (-18 \left (72-27 x+x^3\right )+e^{2 x} \left (-72+99 x-18 x^2-9 x^3+2 x^4\right )\right )}{\left (18+e^{2 x}\right )^2 \left (9-x^2\right )^2} \, dx\\ &=81 \int \frac {x \left (-18 \left (72-27 x+x^3\right )+e^{2 x} \left (-72+99 x-18 x^2-9 x^3+2 x^4\right )\right )}{\left (18+e^{2 x}\right )^2 \left (9-x^2\right )^2} \, dx\\ &=81 \int \left (-\frac {36 (-4+x) x^2}{\left (18+e^{2 x}\right )^2 \left (-9+x^2\right )}+\frac {x \left (-72+99 x-18 x^2-9 x^3+2 x^4\right )}{\left (18+e^{2 x}\right ) \left (-9+x^2\right )^2}\right ) \, dx\\ &=81 \int \frac {x \left (-72+99 x-18 x^2-9 x^3+2 x^4\right )}{\left (18+e^{2 x}\right ) \left (-9+x^2\right )^2} \, dx-2916 \int \frac {(-4+x) x^2}{\left (18+e^{2 x}\right )^2 \left (-9+x^2\right )} \, dx\\ &=81 \int \left (-\frac {9}{18+e^{2 x}}-\frac {3}{2 \left (18+e^{2 x}\right ) (-3+x)^2}-\frac {3}{\left (18+e^{2 x}\right ) (-3+x)}+\frac {2 x}{18+e^{2 x}}+\frac {21}{2 \left (18+e^{2 x}\right ) (3+x)^2}+\frac {21}{\left (18+e^{2 x}\right ) (3+x)}\right ) \, dx-2916 \int \left (-\frac {4}{\left (18+e^{2 x}\right )^2}-\frac {3}{2 \left (18+e^{2 x}\right )^2 (-3+x)}+\frac {x}{\left (18+e^{2 x}\right )^2}+\frac {21}{2 \left (18+e^{2 x}\right )^2 (3+x)}\right ) \, dx\\ &=-\left (\frac {243}{2} \int \frac {1}{\left (18+e^{2 x}\right ) (-3+x)^2} \, dx\right )+162 \int \frac {x}{18+e^{2 x}} \, dx-243 \int \frac {1}{\left (18+e^{2 x}\right ) (-3+x)} \, dx-729 \int \frac {1}{18+e^{2 x}} \, dx+\frac {1701}{2} \int \frac {1}{\left (18+e^{2 x}\right ) (3+x)^2} \, dx+1701 \int \frac {1}{\left (18+e^{2 x}\right ) (3+x)} \, dx-2916 \int \frac {x}{\left (18+e^{2 x}\right )^2} \, dx+4374 \int \frac {1}{\left (18+e^{2 x}\right )^2 (-3+x)} \, dx+11664 \int \frac {1}{\left (18+e^{2 x}\right )^2} \, dx-30618 \int \frac {1}{\left (18+e^{2 x}\right )^2 (3+x)} \, dx\\ &=\frac {9 x^2}{2}-9 \int \frac {e^{2 x} x}{18+e^{2 x}} \, dx-\frac {243}{2} \int \frac {1}{\left (18+e^{2 x}\right ) (-3+x)^2} \, dx+162 \int \frac {e^{2 x} x}{\left (18+e^{2 x}\right )^2} \, dx-162 \int \frac {x}{18+e^{2 x}} \, dx-243 \int \frac {1}{\left (18+e^{2 x}\right ) (-3+x)} \, dx-\frac {729}{2} \operatorname {Subst}\left (\int \frac {1}{x (18+x)} \, dx,x,e^{2 x}\right )+\frac {1701}{2} \int \frac {1}{\left (18+e^{2 x}\right ) (3+x)^2} \, dx+1701 \int \frac {1}{\left (18+e^{2 x}\right ) (3+x)} \, dx+4374 \int \frac {1}{\left (18+e^{2 x}\right )^2 (-3+x)} \, dx+5832 \operatorname {Subst}\left (\int \frac {1}{x (18+x)^2} \, dx,x,e^{2 x}\right )-30618 \int \frac {1}{\left (18+e^{2 x}\right )^2 (3+x)} \, dx\\ &=-\frac {81 x}{18+e^{2 x}}-\frac {9}{2} x \log \left (1+\frac {e^{2 x}}{18}\right )+\frac {9}{2} \int \log \left (1+\frac {e^{2 x}}{18}\right ) \, dx+9 \int \frac {e^{2 x} x}{18+e^{2 x}} \, dx-\frac {81}{4} \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,e^{2 x}\right )+\frac {81}{4} \operatorname {Subst}\left (\int \frac {1}{18+x} \, dx,x,e^{2 x}\right )+81 \int \frac {1}{18+e^{2 x}} \, dx-\frac {243}{2} \int \frac {1}{\left (18+e^{2 x}\right ) (-3+x)^2} \, dx-243 \int \frac {1}{\left (18+e^{2 x}\right ) (-3+x)} \, dx+\frac {1701}{2} \int \frac {1}{\left (18+e^{2 x}\right ) (3+x)^2} \, dx+1701 \int \frac {1}{\left (18+e^{2 x}\right ) (3+x)} \, dx+4374 \int \frac {1}{\left (18+e^{2 x}\right )^2 (-3+x)} \, dx+5832 \operatorname {Subst}\left (\int \left (\frac {1}{324 x}-\frac {1}{18 (18+x)^2}-\frac {1}{324 (18+x)}\right ) \, dx,x,e^{2 x}\right )-30618 \int \frac {1}{\left (18+e^{2 x}\right )^2 (3+x)} \, dx\\ &=\frac {324}{18+e^{2 x}}-\frac {9 x}{2}-\frac {81 x}{18+e^{2 x}}+\frac {9}{4} \log \left (18+e^{2 x}\right )+\frac {9}{4} \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {x}{18}\right )}{x} \, dx,x,e^{2 x}\right )-\frac {9}{2} \int \log \left (1+\frac {e^{2 x}}{18}\right ) \, dx+\frac {81}{2} \operatorname {Subst}\left (\int \frac {1}{x (18+x)} \, dx,x,e^{2 x}\right )-\frac {243}{2} \int \frac {1}{\left (18+e^{2 x}\right ) (-3+x)^2} \, dx-243 \int \frac {1}{\left (18+e^{2 x}\right ) (-3+x)} \, dx+\frac {1701}{2} \int \frac {1}{\left (18+e^{2 x}\right ) (3+x)^2} \, dx+1701 \int \frac {1}{\left (18+e^{2 x}\right ) (3+x)} \, dx+4374 \int \frac {1}{\left (18+e^{2 x}\right )^2 (-3+x)} \, dx-30618 \int \frac {1}{\left (18+e^{2 x}\right )^2 (3+x)} \, dx\\ &=\frac {324}{18+e^{2 x}}-\frac {9 x}{2}-\frac {81 x}{18+e^{2 x}}+\frac {9}{4} \log \left (18+e^{2 x}\right )-\frac {9}{4} \text {Li}_2\left (-\frac {e^{2 x}}{18}\right )+\frac {9}{4} \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,e^{2 x}\right )-\frac {9}{4} \operatorname {Subst}\left (\int \frac {1}{18+x} \, dx,x,e^{2 x}\right )-\frac {9}{4} \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {x}{18}\right )}{x} \, dx,x,e^{2 x}\right )-\frac {243}{2} \int \frac {1}{\left (18+e^{2 x}\right ) (-3+x)^2} \, dx-243 \int \frac {1}{\left (18+e^{2 x}\right ) (-3+x)} \, dx+\frac {1701}{2} \int \frac {1}{\left (18+e^{2 x}\right ) (3+x)^2} \, dx+1701 \int \frac {1}{\left (18+e^{2 x}\right ) (3+x)} \, dx+4374 \int \frac {1}{\left (18+e^{2 x}\right )^2 (-3+x)} \, dx-30618 \int \frac {1}{\left (18+e^{2 x}\right )^2 (3+x)} \, dx\\ &=\frac {324}{18+e^{2 x}}-\frac {81 x}{18+e^{2 x}}-\frac {243}{2} \int \frac {1}{\left (18+e^{2 x}\right ) (-3+x)^2} \, dx-243 \int \frac {1}{\left (18+e^{2 x}\right ) (-3+x)} \, dx+\frac {1701}{2} \int \frac {1}{\left (18+e^{2 x}\right ) (3+x)^2} \, dx+1701 \int \frac {1}{\left (18+e^{2 x}\right ) (3+x)} \, dx+4374 \int \frac {1}{\left (18+e^{2 x}\right )^2 (-3+x)} \, dx-30618 \int \frac {1}{\left (18+e^{2 x}\right )^2 (3+x)} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.28, size = 24, normalized size = 0.73 \begin {gather*} -\frac {81 (-4+x) x^2}{\left (18+e^{2 x}\right ) \left (-9+x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-104976*x + 39366*x^2 - 1458*x^4 + E^(2*x)*(-5832*x + 8019*x^2 - 1458*x^3 - 729*x^4 + 162*x^5))/(26
244 - 5832*x^2 + 324*x^4 + E^(4*x)*(81 - 18*x^2 + x^4) + E^(2*x)*(2916 - 648*x^2 + 36*x^4)),x]

[Out]

(-81*(-4 + x)*x^2)/((18 + E^(2*x))*(-9 + x^2))

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fricas [A]  time = 1.05, size = 30, normalized size = 0.91 \begin {gather*} -\frac {81 \, {\left (x^{3} - 4 \, x^{2}\right )}}{18 \, x^{2} + {\left (x^{2} - 9\right )} e^{\left (2 \, x\right )} - 162} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((162*x^5-729*x^4-1458*x^3+8019*x^2-5832*x)*exp(x)^2-1458*x^4+39366*x^2-104976*x)/((x^4-18*x^2+81)*e
xp(x)^4+(36*x^4-648*x^2+2916)*exp(x)^2+324*x^4-5832*x^2+26244),x, algorithm="fricas")

[Out]

-81*(x^3 - 4*x^2)/(18*x^2 + (x^2 - 9)*e^(2*x) - 162)

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giac [A]  time = 0.14, size = 34, normalized size = 1.03 \begin {gather*} -\frac {81 \, {\left (x^{3} - 4 \, x^{2}\right )}}{x^{2} e^{\left (2 \, x\right )} + 18 \, x^{2} - 9 \, e^{\left (2 \, x\right )} - 162} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((162*x^5-729*x^4-1458*x^3+8019*x^2-5832*x)*exp(x)^2-1458*x^4+39366*x^2-104976*x)/((x^4-18*x^2+81)*e
xp(x)^4+(36*x^4-648*x^2+2916)*exp(x)^2+324*x^4-5832*x^2+26244),x, algorithm="giac")

[Out]

-81*(x^3 - 4*x^2)/(x^2*e^(2*x) + 18*x^2 - 9*e^(2*x) - 162)

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maple [A]  time = 0.14, size = 24, normalized size = 0.73

method	result	size
risch	\(-\frac {81 \left (x -4\right ) x^{2}}{\left (x^{2}-9\right ) \left ({\mathrm e}^{2 x}+18\right )}\)	\(24\)
norman	\(\frac {-81 x^{3}+324 x^{2}}{\left ({\mathrm e}^{2 x}+18\right ) \left (x^{2}-9\right )}\)	\(28\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((162*x^5-729*x^4-1458*x^3+8019*x^2-5832*x)*exp(x)^2-1458*x^4+39366*x^2-104976*x)/((x^4-18*x^2+81)*exp(x)^
4+(36*x^4-648*x^2+2916)*exp(x)^2+324*x^4-5832*x^2+26244),x,method=_RETURNVERBOSE)

[Out]

-81*(x-4)*x^2/(x^2-9)/(exp(2*x)+18)

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maxima [A]  time = 0.39, size = 30, normalized size = 0.91 \begin {gather*} -\frac {81 \, {\left (x^{3} - 4 \, x^{2}\right )}}{18 \, x^{2} + {\left (x^{2} - 9\right )} e^{\left (2 \, x\right )} - 162} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((162*x^5-729*x^4-1458*x^3+8019*x^2-5832*x)*exp(x)^2-1458*x^4+39366*x^2-104976*x)/((x^4-18*x^2+81)*e
xp(x)^4+(36*x^4-648*x^2+2916)*exp(x)^2+324*x^4-5832*x^2+26244),x, algorithm="maxima")

[Out]

-81*(x^3 - 4*x^2)/(18*x^2 + (x^2 - 9)*e^(2*x) - 162)

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mupad [B]  time = 4.44, size = 36, normalized size = 1.09 \begin {gather*} -\frac {81\,\left (x^5-4\,x^4-9\,x^3+36\,x^2\right )}{{\left (x^2-9\right )}^2\,\left ({\mathrm {e}}^{2\,x}+18\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(104976*x + exp(2*x)*(5832*x - 8019*x^2 + 1458*x^3 + 729*x^4 - 162*x^5) - 39366*x^2 + 1458*x^4)/(exp(4*x)
*(x^4 - 18*x^2 + 81) + exp(2*x)*(36*x^4 - 648*x^2 + 2916) - 5832*x^2 + 324*x^4 + 26244),x)

[Out]

-(81*(36*x^2 - 9*x^3 - 4*x^4 + x^5))/((x^2 - 9)^2*(exp(2*x) + 18))

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sympy [A]  time = 0.16, size = 26, normalized size = 0.79 \begin {gather*} \frac {- 81 x^{3} + 324 x^{2}}{18 x^{2} + \left (x^{2} - 9\right ) e^{2 x} - 162} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((162*x**5-729*x**4-1458*x**3+8019*x**2-5832*x)*exp(x)**2-1458*x**4+39366*x**2-104976*x)/((x**4-18*x
**2+81)*exp(x)**4+(36*x**4-648*x**2+2916)*exp(x)**2+324*x**4-5832*x**2+26244),x)

[Out]

(-81*x**3 + 324*x**2)/(18*x**2 + (x**2 - 9)*exp(2*x) - 162)

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