3.82.52 \(\int \frac {e^{\frac {45 x-8 x^3+(-24+4 x^2) \log (4)}{-24+4 x^2}} (126-270 x-51 x^2+99 x^3+4 x^4-8 x^5)}{288-576 x+192 x^2+192 x^3-88 x^4-16 x^5+8 x^6} \, dx\)

Optimal. Leaf size=33 \[ 1+\frac {4 e^{-2 x-\frac {x}{4 \left (-2+\frac {x^2}{3}\right )}}}{-2+2 x} \]

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Rubi [F]  time = 2.04, 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 {45 x-8 x^3+\left (-24+4 x^2\right ) \log (4)}{-24+4 x^2}\right ) \left (126-270 x-51 x^2+99 x^3+4 x^4-8 x^5\right )}{288-576 x+192 x^2+192 x^3-88 x^4-16 x^5+8 x^6} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((45*x - 8*x^3 + (-24 + 4*x^2)*Log[4])/(-24 + 4*x^2))*(126 - 270*x - 51*x^2 + 99*x^3 + 4*x^4 - 8*x^5))/
(288 - 576*x + 192*x^2 + 192*x^3 - 88*x^4 - 16*x^5 + 8*x^6),x]

[Out]

(3*Defer[Int][E^((x*(45 - 8*x^2))/(4*(-6 + x^2)))/(Sqrt[6] - x)^2, x])/20 + (Sqrt[3/2]*Defer[Int][E^((x*(45 -
8*x^2))/(4*(-6 + x^2)))/(Sqrt[6] - x), x])/20 + (7*(6 + Sqrt[6])*Defer[Int][E^((x*(45 - 8*x^2))/(4*(-6 + x^2))
)/(Sqrt[6] - x), x])/200 - 2*Defer[Int][E^((x*(45 - 8*x^2))/(4*(-6 + x^2)))/(-1 + x)^2, x] - (179*Defer[Int][E
^((x*(45 - 8*x^2))/(4*(-6 + x^2)))/(-1 + x), x])/50 + (3*Defer[Int][E^((x*(45 - 8*x^2))/(4*(-6 + x^2)))/(Sqrt[
6] + x)^2, x])/20 + (Sqrt[3/2]*Defer[Int][E^((x*(45 - 8*x^2))/(4*(-6 + x^2)))/(Sqrt[6] + x), x])/20 - (7*(6 -
Sqrt[6])*Defer[Int][E^((x*(45 - 8*x^2))/(4*(-6 + x^2)))/(Sqrt[6] + x), x])/200 + (18*Defer[Int][(E^((x*(45 - 8
*x^2))/(4*(-6 + x^2)))*x)/(-6 + x^2)^2, x])/5

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{\frac {x \left (45-8 x^2\right )}{4 \left (-6+x^2\right )}} \left (126-270 x-51 x^2+99 x^3+4 x^4-8 x^5\right )}{2 \left (6-6 x-x^2+x^3\right )^2} \, dx\\ &=\frac {1}{2} \int \frac {e^{\frac {x \left (45-8 x^2\right )}{4 \left (-6+x^2\right )}} \left (126-270 x-51 x^2+99 x^3+4 x^4-8 x^5\right )}{\left (6-6 x-x^2+x^3\right )^2} \, dx\\ &=\frac {1}{2} \int \left (-\frac {4 e^{\frac {x \left (45-8 x^2\right )}{4 \left (-6+x^2\right )}}}{(-1+x)^2}-\frac {179 e^{\frac {x \left (45-8 x^2\right )}{4 \left (-6+x^2\right )}}}{25 (-1+x)}+\frac {36 e^{\frac {x \left (45-8 x^2\right )}{4 \left (-6+x^2\right )}} (1+x)}{5 \left (-6+x^2\right )^2}-\frac {21 e^{\frac {x \left (45-8 x^2\right )}{4 \left (-6+x^2\right )}} (1+x)}{25 \left (-6+x^2\right )}\right ) \, dx\\ &=-\left (\frac {21}{50} \int \frac {e^{\frac {x \left (45-8 x^2\right )}{4 \left (-6+x^2\right )}} (1+x)}{-6+x^2} \, dx\right )-2 \int \frac {e^{\frac {x \left (45-8 x^2\right )}{4 \left (-6+x^2\right )}}}{(-1+x)^2} \, dx-\frac {179}{50} \int \frac {e^{\frac {x \left (45-8 x^2\right )}{4 \left (-6+x^2\right )}}}{-1+x} \, dx+\frac {18}{5} \int \frac {e^{\frac {x \left (45-8 x^2\right )}{4 \left (-6+x^2\right )}} (1+x)}{\left (-6+x^2\right )^2} \, dx\\ &=-\left (\frac {21}{50} \int \left (-\frac {\left (6+\sqrt {6}\right ) e^{\frac {x \left (45-8 x^2\right )}{4 \left (-6+x^2\right )}}}{12 \left (\sqrt {6}-x\right )}-\frac {\left (-6+\sqrt {6}\right ) e^{\frac {x \left (45-8 x^2\right )}{4 \left (-6+x^2\right )}}}{12 \left (\sqrt {6}+x\right )}\right ) \, dx\right )-2 \int \frac {e^{\frac {x \left (45-8 x^2\right )}{4 \left (-6+x^2\right )}}}{(-1+x)^2} \, dx-\frac {179}{50} \int \frac {e^{\frac {x \left (45-8 x^2\right )}{4 \left (-6+x^2\right )}}}{-1+x} \, dx+\frac {18}{5} \int \left (\frac {e^{\frac {x \left (45-8 x^2\right )}{4 \left (-6+x^2\right )}}}{\left (-6+x^2\right )^2}+\frac {e^{\frac {x \left (45-8 x^2\right )}{4 \left (-6+x^2\right )}} x}{\left (-6+x^2\right )^2}\right ) \, dx\\ &=-\left (2 \int \frac {e^{\frac {x \left (45-8 x^2\right )}{4 \left (-6+x^2\right )}}}{(-1+x)^2} \, dx\right )-\frac {179}{50} \int \frac {e^{\frac {x \left (45-8 x^2\right )}{4 \left (-6+x^2\right )}}}{-1+x} \, dx+\frac {18}{5} \int \frac {e^{\frac {x \left (45-8 x^2\right )}{4 \left (-6+x^2\right )}}}{\left (-6+x^2\right )^2} \, dx+\frac {18}{5} \int \frac {e^{\frac {x \left (45-8 x^2\right )}{4 \left (-6+x^2\right )}} x}{\left (-6+x^2\right )^2} \, dx-\frac {1}{200} \left (7 \left (6-\sqrt {6}\right )\right ) \int \frac {e^{\frac {x \left (45-8 x^2\right )}{4 \left (-6+x^2\right )}}}{\sqrt {6}+x} \, dx+\frac {1}{200} \left (7 \left (6+\sqrt {6}\right )\right ) \int \frac {e^{\frac {x \left (45-8 x^2\right )}{4 \left (-6+x^2\right )}}}{\sqrt {6}-x} \, dx\\ &=-\left (2 \int \frac {e^{\frac {x \left (45-8 x^2\right )}{4 \left (-6+x^2\right )}}}{(-1+x)^2} \, dx\right )-\frac {179}{50} \int \frac {e^{\frac {x \left (45-8 x^2\right )}{4 \left (-6+x^2\right )}}}{-1+x} \, dx+\frac {18}{5} \int \frac {e^{\frac {x \left (45-8 x^2\right )}{4 \left (-6+x^2\right )}} x}{\left (-6+x^2\right )^2} \, dx+\frac {18}{5} \int \left (\frac {e^{\frac {x \left (45-8 x^2\right )}{4 \left (-6+x^2\right )}}}{24 \left (\sqrt {6}-x\right )^2}+\frac {e^{\frac {x \left (45-8 x^2\right )}{4 \left (-6+x^2\right )}}}{24 \left (\sqrt {6}+x\right )^2}+\frac {e^{\frac {x \left (45-8 x^2\right )}{4 \left (-6+x^2\right )}}}{12 \left (6-x^2\right )}\right ) \, dx-\frac {1}{200} \left (7 \left (6-\sqrt {6}\right )\right ) \int \frac {e^{\frac {x \left (45-8 x^2\right )}{4 \left (-6+x^2\right )}}}{\sqrt {6}+x} \, dx+\frac {1}{200} \left (7 \left (6+\sqrt {6}\right )\right ) \int \frac {e^{\frac {x \left (45-8 x^2\right )}{4 \left (-6+x^2\right )}}}{\sqrt {6}-x} \, dx\\ &=\frac {3}{20} \int \frac {e^{\frac {x \left (45-8 x^2\right )}{4 \left (-6+x^2\right )}}}{\left (\sqrt {6}-x\right )^2} \, dx+\frac {3}{20} \int \frac {e^{\frac {x \left (45-8 x^2\right )}{4 \left (-6+x^2\right )}}}{\left (\sqrt {6}+x\right )^2} \, dx+\frac {3}{10} \int \frac {e^{\frac {x \left (45-8 x^2\right )}{4 \left (-6+x^2\right )}}}{6-x^2} \, dx-2 \int \frac {e^{\frac {x \left (45-8 x^2\right )}{4 \left (-6+x^2\right )}}}{(-1+x)^2} \, dx-\frac {179}{50} \int \frac {e^{\frac {x \left (45-8 x^2\right )}{4 \left (-6+x^2\right )}}}{-1+x} \, dx+\frac {18}{5} \int \frac {e^{\frac {x \left (45-8 x^2\right )}{4 \left (-6+x^2\right )}} x}{\left (-6+x^2\right )^2} \, dx-\frac {1}{200} \left (7 \left (6-\sqrt {6}\right )\right ) \int \frac {e^{\frac {x \left (45-8 x^2\right )}{4 \left (-6+x^2\right )}}}{\sqrt {6}+x} \, dx+\frac {1}{200} \left (7 \left (6+\sqrt {6}\right )\right ) \int \frac {e^{\frac {x \left (45-8 x^2\right )}{4 \left (-6+x^2\right )}}}{\sqrt {6}-x} \, dx\\ &=\frac {3}{20} \int \frac {e^{\frac {x \left (45-8 x^2\right )}{4 \left (-6+x^2\right )}}}{\left (\sqrt {6}-x\right )^2} \, dx+\frac {3}{20} \int \frac {e^{\frac {x \left (45-8 x^2\right )}{4 \left (-6+x^2\right )}}}{\left (\sqrt {6}+x\right )^2} \, dx+\frac {3}{10} \int \left (\frac {e^{\frac {x \left (45-8 x^2\right )}{4 \left (-6+x^2\right )}}}{2 \sqrt {6} \left (\sqrt {6}-x\right )}+\frac {e^{\frac {x \left (45-8 x^2\right )}{4 \left (-6+x^2\right )}}}{2 \sqrt {6} \left (\sqrt {6}+x\right )}\right ) \, dx-2 \int \frac {e^{\frac {x \left (45-8 x^2\right )}{4 \left (-6+x^2\right )}}}{(-1+x)^2} \, dx-\frac {179}{50} \int \frac {e^{\frac {x \left (45-8 x^2\right )}{4 \left (-6+x^2\right )}}}{-1+x} \, dx+\frac {18}{5} \int \frac {e^{\frac {x \left (45-8 x^2\right )}{4 \left (-6+x^2\right )}} x}{\left (-6+x^2\right )^2} \, dx-\frac {1}{200} \left (7 \left (6-\sqrt {6}\right )\right ) \int \frac {e^{\frac {x \left (45-8 x^2\right )}{4 \left (-6+x^2\right )}}}{\sqrt {6}+x} \, dx+\frac {1}{200} \left (7 \left (6+\sqrt {6}\right )\right ) \int \frac {e^{\frac {x \left (45-8 x^2\right )}{4 \left (-6+x^2\right )}}}{\sqrt {6}-x} \, dx\\ &=\frac {3}{20} \int \frac {e^{\frac {x \left (45-8 x^2\right )}{4 \left (-6+x^2\right )}}}{\left (\sqrt {6}-x\right )^2} \, dx+\frac {3}{20} \int \frac {e^{\frac {x \left (45-8 x^2\right )}{4 \left (-6+x^2\right )}}}{\left (\sqrt {6}+x\right )^2} \, dx-2 \int \frac {e^{\frac {x \left (45-8 x^2\right )}{4 \left (-6+x^2\right )}}}{(-1+x)^2} \, dx-\frac {179}{50} \int \frac {e^{\frac {x \left (45-8 x^2\right )}{4 \left (-6+x^2\right )}}}{-1+x} \, dx+\frac {18}{5} \int \frac {e^{\frac {x \left (45-8 x^2\right )}{4 \left (-6+x^2\right )}} x}{\left (-6+x^2\right )^2} \, dx+\frac {1}{20} \sqrt {\frac {3}{2}} \int \frac {e^{\frac {x \left (45-8 x^2\right )}{4 \left (-6+x^2\right )}}}{\sqrt {6}-x} \, dx+\frac {1}{20} \sqrt {\frac {3}{2}} \int \frac {e^{\frac {x \left (45-8 x^2\right )}{4 \left (-6+x^2\right )}}}{\sqrt {6}+x} \, dx-\frac {1}{200} \left (7 \left (6-\sqrt {6}\right )\right ) \int \frac {e^{\frac {x \left (45-8 x^2\right )}{4 \left (-6+x^2\right )}}}{\sqrt {6}+x} \, dx+\frac {1}{200} \left (7 \left (6+\sqrt {6}\right )\right ) \int \frac {e^{\frac {x \left (45-8 x^2\right )}{4 \left (-6+x^2\right )}}}{\sqrt {6}-x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.64, size = 25, normalized size = 0.76 \begin {gather*} \frac {2 e^{-x \left (2+\frac {3}{4 \left (-6+x^2\right )}\right )}}{-1+x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((45*x - 8*x^3 + (-24 + 4*x^2)*Log[4])/(-24 + 4*x^2))*(126 - 270*x - 51*x^2 + 99*x^3 + 4*x^4 - 8*
x^5))/(288 - 576*x + 192*x^2 + 192*x^3 - 88*x^4 - 16*x^5 + 8*x^6),x]

[Out]

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

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fricas [A]  time = 0.66, size = 35, normalized size = 1.06 \begin {gather*} \frac {e^{\left (-\frac {8 \, x^{3} - 8 \, {\left (x^{2} - 6\right )} \log \relax (2) - 45 \, x}{4 \, {\left (x^{2} - 6\right )}}\right )}}{2 \, {\left (x - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-8*x^5+4*x^4+99*x^3-51*x^2-270*x+126)*exp((2*(4*x^2-24)*log(2)-8*x^3+45*x)/(4*x^2-24))/(8*x^6-16*x^
5-88*x^4+192*x^3+192*x^2-576*x+288),x, algorithm="fricas")

[Out]

1/2*e^(-1/4*(8*x^3 - 8*(x^2 - 6)*log(2) - 45*x)/(x^2 - 6))/(x - 1)

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giac [A]  time = 0.27, size = 26, normalized size = 0.79 \begin {gather*} \frac {2 \, e^{\left (-\frac {8 \, x^{3} - 45 \, x}{4 \, {\left (x^{2} - 6\right )}}\right )}}{x - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-8*x^5+4*x^4+99*x^3-51*x^2-270*x+126)*exp((2*(4*x^2-24)*log(2)-8*x^3+45*x)/(4*x^2-24))/(8*x^6-16*x^
5-88*x^4+192*x^3+192*x^2-576*x+288),x, algorithm="giac")

[Out]

2*e^(-1/4*(8*x^3 - 45*x)/(x^2 - 6))/(x - 1)

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maple [A]  time = 0.12, size = 38, normalized size = 1.15

method	result	size
gosper	\(\frac {{\mathrm e}^{\frac {8 x^{2} \ln \relax (2)-8 x^{3}-48 \ln \relax (2)+45 x}{4 x^{2}-24}}}{2 x -2}\)	\(38\)
risch	\(\frac {{\mathrm e}^{\frac {8 x^{2} \ln \relax (2)-8 x^{3}-48 \ln \relax (2)+45 x}{4 x^{2}-24}}}{2 x -2}\)	\(38\)
norman	\(\frac {\frac {x^{2} {\mathrm e}^{\frac {2 \left (4 x^{2}-24\right ) \ln \relax (2)-8 x^{3}+45 x}{4 x^{2}-24}}}{2}-3 \,{\mathrm e}^{\frac {2 \left (4 x^{2}-24\right ) \ln \relax (2)-8 x^{3}+45 x}{4 x^{2}-24}}}{x^{3}-x^{2}-6 x +6}\)	\(87\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-8*x^5+4*x^4+99*x^3-51*x^2-270*x+126)*exp((2*(4*x^2-24)*ln(2)-8*x^3+45*x)/(4*x^2-24))/(8*x^6-16*x^5-88*x^
4+192*x^3+192*x^2-576*x+288),x,method=_RETURNVERBOSE)

[Out]

1/2*exp(1/4*(8*x^2*ln(2)-8*x^3-48*ln(2)+45*x)/(x^2-6))/(x-1)

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maxima [A]  time = 0.61, size = 22, normalized size = 0.67 \begin {gather*} \frac {2 \, e^{\left (-2 \, x - \frac {3 \, x}{4 \, {\left (x^{2} - 6\right )}}\right )}}{x - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-8*x^5+4*x^4+99*x^3-51*x^2-270*x+126)*exp((2*(4*x^2-24)*log(2)-8*x^3+45*x)/(4*x^2-24))/(8*x^6-16*x^
5-88*x^4+192*x^3+192*x^2-576*x+288),x, algorithm="maxima")

[Out]

2*e^(-2*x - 3/4*x/(x^2 - 6))/(x - 1)

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mupad [B]  time = 6.15, size = 33, normalized size = 1.00 \begin {gather*} \frac {2\,{\mathrm {e}}^{-\frac {2\,x^3}{x^2-6}}\,{\mathrm {e}}^{\frac {45\,x}{4\,\left (x^2-6\right )}}}{x-1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp((45*x + 2*log(2)*(4*x^2 - 24) - 8*x^3)/(4*x^2 - 24))*(270*x + 51*x^2 - 99*x^3 - 4*x^4 + 8*x^5 - 126)
)/(192*x^2 - 576*x + 192*x^3 - 88*x^4 - 16*x^5 + 8*x^6 + 288),x)

[Out]

(2*exp(-(2*x^3)/(x^2 - 6))*exp((45*x)/(4*(x^2 - 6))))/(x - 1)

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sympy [A]  time = 0.24, size = 31, normalized size = 0.94 \begin {gather*} \frac {e^{\frac {- 8 x^{3} + 45 x + \left (8 x^{2} - 48\right ) \log {\relax (2 )}}{4 x^{2} - 24}}}{2 x - 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-8*x**5+4*x**4+99*x**3-51*x**2-270*x+126)*exp((2*(4*x**2-24)*ln(2)-8*x**3+45*x)/(4*x**2-24))/(8*x**
6-16*x**5-88*x**4+192*x**3+192*x**2-576*x+288),x)

[Out]

exp((-8*x**3 + 45*x + (8*x**2 - 48)*log(2))/(4*x**2 - 24))/(2*x - 2)

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