3.71.54 \(\int \frac {e^{\frac {x^2}{-3-x^2+4 x^3}} (6 x^2+4 x^5+e^{-\frac {x^2}{-3-x^2+4 x^3}} (-9-6 x^2+24 x^3-x^4+8 x^5-16 x^6))}{9 x+6 x^3-24 x^4+x^5-8 x^6+16 x^7} \, dx\) [7054]

Optimal. Leaf size=27 \[ 3-e^{-\frac {x}{\frac {3}{x}+x-4 x^2}}-\log (x) \]

[Out]

3-1/exp(x/(3/x-4*x^2+x))-ln(x)

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Rubi [F]
time = 3.38, 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^{\frac {x^2}{-3-x^2+4 x^3}} \left (6 x^2+4 x^5+e^{-\frac {x^2}{-3-x^2+4 x^3}} \left (-9-6 x^2+24 x^3-x^4+8 x^5-16 x^6\right )\right )}{9 x+6 x^3-24 x^4+x^5-8 x^6+16 x^7} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(x^2/(-3 - x^2 + 4*x^3))*(6*x^2 + 4*x^5 + (-9 - 6*x^2 + 24*x^3 - x^4 + 8*x^5 - 16*x^6)/E^(x^2/(-3 - x^2
+ 4*x^3))))/(9*x + 6*x^3 - 24*x^4 + x^5 - 8*x^6 + 16*x^7),x]

[Out]

-Log[x] + (288*Defer[Int][E^(x^2/(-3 - x^2 + 4*x^3))/(-3 + I*Sqrt[39] - 8*x)^2, x])/65 + (8*(3 - I*Sqrt[39])*D
efer[Int][E^(x^2/(-3 - x^2 + 4*x^3))/(-3 + I*Sqrt[39] - 8*x)^2, x])/65 - ((8*I)/5)*Sqrt[3/13]*Defer[Int][E^(x^
2/(-3 - x^2 + 4*x^3))/(-3 + I*Sqrt[39] - 8*x), x] + Defer[Int][E^(x^2/(-3 - x^2 + 4*x^3))/(-1 + x)^2, x]/10 +
(4*(9 - I*Sqrt[39])*Defer[Int][E^(x^2/(-3 - x^2 + 4*x^3))/(3 - I*Sqrt[39] + 8*x), x])/125 - (4*(117 + (37*I)*S
qrt[39])*Defer[Int][E^(x^2/(-3 - x^2 + 4*x^3))/(3 - I*Sqrt[39] + 8*x), x])/1625 + (288*Defer[Int][E^(x^2/(-3 -
x^2 + 4*x^3))/(3 + I*Sqrt[39] + 8*x)^2, x])/65 + (8*(3 + I*Sqrt[39])*Defer[Int][E^(x^2/(-3 - x^2 + 4*x^3))/(3
+ I*Sqrt[39] + 8*x)^2, x])/65 - ((8*I)/5)*Sqrt[3/13]*Defer[Int][E^(x^2/(-3 - x^2 + 4*x^3))/(3 + I*Sqrt[39] +
8*x), x] + (4*(9 + I*Sqrt[39])*Defer[Int][E^(x^2/(-3 - x^2 + 4*x^3))/(3 + I*Sqrt[39] + 8*x), x])/125 - (4*(117
- (37*I)*Sqrt[39])*Defer[Int][E^(x^2/(-3 - x^2 + 4*x^3))/(3 + I*Sqrt[39] + 8*x), x])/1625

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{\frac {x^2}{-3-x^2+4 x^3}} \left (6 x^2+4 x^5+e^{-\frac {x^2}{-3-x^2+4 x^3}} \left (-9-6 x^2+24 x^3-x^4+8 x^5-16 x^6\right )\right )}{x \left (3+x^2-4 x^3\right )^2} \, dx\\ &=\int \left (-\frac {e^{\frac {x^2}{3+x^2-4 x^3}+\frac {x^2}{-3-x^2+4 x^3}}}{x}+\frac {6 e^{\frac {x^2}{-3-x^2+4 x^3}} x}{\left (-3-x^2+4 x^3\right )^2}+\frac {4 e^{\frac {x^2}{-3-x^2+4 x^3}} x^4}{\left (-3-x^2+4 x^3\right )^2}\right ) \, dx\\ &=4 \int \frac {e^{\frac {x^2}{-3-x^2+4 x^3}} x^4}{\left (-3-x^2+4 x^3\right )^2} \, dx+6 \int \frac {e^{\frac {x^2}{-3-x^2+4 x^3}} x}{\left (-3-x^2+4 x^3\right )^2} \, dx-\int \frac {e^{\frac {x^2}{3+x^2-4 x^3}+\frac {x^2}{-3-x^2+4 x^3}}}{x} \, dx\\ &=4 \int \left (\frac {e^{\frac {x^2}{-3-x^2+4 x^3}}}{100 (-1+x)^2}+\frac {9 e^{\frac {x^2}{-3-x^2+4 x^3}}}{500 (-1+x)}+\frac {9 e^{\frac {x^2}{-3-x^2+4 x^3}} (-2+x)}{100 \left (3+3 x+4 x^2\right )^2}-\frac {3 e^{\frac {x^2}{-3-x^2+4 x^3}} (-7+6 x)}{250 \left (3+3 x+4 x^2\right )}\right ) \, dx+6 \int \left (\frac {e^{\frac {x^2}{-3-x^2+4 x^3}}}{100 (-1+x)^2}-\frac {3 e^{\frac {x^2}{-3-x^2+4 x^3}}}{250 (-1+x)}+\frac {e^{\frac {x^2}{-3-x^2+4 x^3}} (-33+4 x)}{100 \left (3+3 x+4 x^2\right )^2}+\frac {e^{\frac {x^2}{-3-x^2+4 x^3}} (11+12 x)}{250 \left (3+3 x+4 x^2\right )}\right ) \, dx-\int \frac {1}{x} \, dx\\ &=-\log (x)+\frac {3}{125} \int \frac {e^{\frac {x^2}{-3-x^2+4 x^3}} (11+12 x)}{3+3 x+4 x^2} \, dx+\frac {1}{25} \int \frac {e^{\frac {x^2}{-3-x^2+4 x^3}}}{(-1+x)^2} \, dx-\frac {6}{125} \int \frac {e^{\frac {x^2}{-3-x^2+4 x^3}} (-7+6 x)}{3+3 x+4 x^2} \, dx+\frac {3}{50} \int \frac {e^{\frac {x^2}{-3-x^2+4 x^3}}}{(-1+x)^2} \, dx+\frac {3}{50} \int \frac {e^{\frac {x^2}{-3-x^2+4 x^3}} (-33+4 x)}{\left (3+3 x+4 x^2\right )^2} \, dx+\frac {9}{25} \int \frac {e^{\frac {x^2}{-3-x^2+4 x^3}} (-2+x)}{\left (3+3 x+4 x^2\right )^2} \, dx\\ &=-\log (x)+\frac {3}{125} \int \left (\frac {\left (12-4 i \sqrt {\frac {13}{3}}\right ) e^{\frac {x^2}{-3-x^2+4 x^3}}}{3-i \sqrt {39}+8 x}+\frac {\left (12+4 i \sqrt {\frac {13}{3}}\right ) e^{\frac {x^2}{-3-x^2+4 x^3}}}{3+i \sqrt {39}+8 x}\right ) \, dx+\frac {1}{25} \int \frac {e^{\frac {x^2}{-3-x^2+4 x^3}}}{(-1+x)^2} \, dx-\frac {6}{125} \int \left (\frac {\left (6+\frac {74 i}{\sqrt {39}}\right ) e^{\frac {x^2}{-3-x^2+4 x^3}}}{3-i \sqrt {39}+8 x}+\frac {\left (6-\frac {74 i}{\sqrt {39}}\right ) e^{\frac {x^2}{-3-x^2+4 x^3}}}{3+i \sqrt {39}+8 x}\right ) \, dx+\frac {3}{50} \int \frac {e^{\frac {x^2}{-3-x^2+4 x^3}}}{(-1+x)^2} \, dx+\frac {3}{50} \int \left (-\frac {33 e^{\frac {x^2}{-3-x^2+4 x^3}}}{\left (3+3 x+4 x^2\right )^2}+\frac {4 e^{\frac {x^2}{-3-x^2+4 x^3}} x}{\left (3+3 x+4 x^2\right )^2}\right ) \, dx+\frac {9}{25} \int \left (-\frac {2 e^{\frac {x^2}{-3-x^2+4 x^3}}}{\left (3+3 x+4 x^2\right )^2}+\frac {e^{\frac {x^2}{-3-x^2+4 x^3}} x}{\left (3+3 x+4 x^2\right )^2}\right ) \, dx\\ &=-\log (x)+\frac {1}{25} \int \frac {e^{\frac {x^2}{-3-x^2+4 x^3}}}{(-1+x)^2} \, dx+\frac {3}{50} \int \frac {e^{\frac {x^2}{-3-x^2+4 x^3}}}{(-1+x)^2} \, dx+\frac {6}{25} \int \frac {e^{\frac {x^2}{-3-x^2+4 x^3}} x}{\left (3+3 x+4 x^2\right )^2} \, dx+\frac {9}{25} \int \frac {e^{\frac {x^2}{-3-x^2+4 x^3}} x}{\left (3+3 x+4 x^2\right )^2} \, dx-\frac {18}{25} \int \frac {e^{\frac {x^2}{-3-x^2+4 x^3}}}{\left (3+3 x+4 x^2\right )^2} \, dx-\frac {99}{50} \int \frac {e^{\frac {x^2}{-3-x^2+4 x^3}}}{\left (3+3 x+4 x^2\right )^2} \, dx+\frac {1}{125} \left (4 \left (9-i \sqrt {39}\right )\right ) \int \frac {e^{\frac {x^2}{-3-x^2+4 x^3}}}{3-i \sqrt {39}+8 x} \, dx+\frac {1}{125} \left (4 \left (9+i \sqrt {39}\right )\right ) \int \frac {e^{\frac {x^2}{-3-x^2+4 x^3}}}{3+i \sqrt {39}+8 x} \, dx-\frac {\left (4 \left (117-37 i \sqrt {39}\right )\right ) \int \frac {e^{\frac {x^2}{-3-x^2+4 x^3}}}{3+i \sqrt {39}+8 x} \, dx}{1625}-\frac {\left (4 \left (117+37 i \sqrt {39}\right )\right ) \int \frac {e^{\frac {x^2}{-3-x^2+4 x^3}}}{3-i \sqrt {39}+8 x} \, dx}{1625}\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.08, size = 27, normalized size = 1.00 \begin {gather*} -e^{\frac {x^2}{-3-x^2+4 x^3}}-\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(x^2/(-3 - x^2 + 4*x^3))*(6*x^2 + 4*x^5 + (-9 - 6*x^2 + 24*x^3 - x^4 + 8*x^5 - 16*x^6)/E^(x^2/(-3
- x^2 + 4*x^3))))/(9*x + 6*x^3 - 24*x^4 + x^5 - 8*x^6 + 16*x^7),x]

[Out]

-E^(x^2/(-3 - x^2 + 4*x^3)) - Log[x]

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Maple [A]
time = 0.24, size = 30, normalized size = 1.11

method	result	size
risch	\(-\ln \left (x \right )-{\mathrm e}^{\frac {x^{2}}{\left (x -1\right ) \left (4 x^{2}+3 x +3\right )}}\)	\(30\)
norman	\(\frac {\left (-4 x^{3}+x^{2}+3\right ) {\mathrm e}^{\frac {x^{2}}{4 x^{3}-x^{2}-3}}}{4 x^{3}-x^{2}-3}-\ln \left (x \right )\)	\(53\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-16*x^6+8*x^5-x^4+24*x^3-6*x^2-9)*exp(-x^2/(4*x^3-x^2-3))+4*x^5+6*x^2)/(16*x^7-8*x^6+x^5-24*x^4+6*x^3+9*
x)/exp(-x^2/(4*x^3-x^2-3)),x,method=_RETURNVERBOSE)

[Out]

-ln(x)-exp(x^2/(x-1)/(4*x^2+3*x+3))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 201 vs. \(2 (27) = 54\).
time = 0.57, size = 201, normalized size = 7.44 \begin {gather*} \frac {181 \, x^{2} + 12 \, x + 327}{325 \, {\left (4 \, x^{3} - x^{2} - 3\right )}} - \frac {2 \, {\left (109 \, x^{2} + 18 \, x + 3\right )}}{325 \, {\left (4 \, x^{3} - x^{2} - 3\right )}} + \frac {3 \, {\left (72 \, x^{2} - 6 \, x - 1\right )}}{325 \, {\left (4 \, x^{3} - x^{2} - 3\right )}} - \frac {3 \, {\left (24 \, x^{2} - 2 \, x - 217\right )}}{650 \, {\left (4 \, x^{3} - x^{2} - 3\right )}} - \frac {12 \, {\left (12 \, x^{2} - x + 54\right )}}{325 \, {\left (4 \, x^{3} - x^{2} - 3\right )}} + \frac {2 \, x^{2} + 54 \, x + 9}{650 \, {\left (4 \, x^{3} - x^{2} - 3\right )}} - e^{\left (\frac {3 \, x}{5 \, {\left (4 \, x^{2} + 3 \, x + 3\right )}} + \frac {3}{10 \, {\left (4 \, x^{2} + 3 \, x + 3\right )}} + \frac {1}{10 \, {\left (x - 1\right )}}\right )} - \log \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-16*x^6+8*x^5-x^4+24*x^3-6*x^2-9)*exp(-x^2/(4*x^3-x^2-3))+4*x^5+6*x^2)/(16*x^7-8*x^6+x^5-24*x^4+6*
x^3+9*x)/exp(-x^2/(4*x^3-x^2-3)),x, algorithm="maxima")

[Out]

1/325*(181*x^2 + 12*x + 327)/(4*x^3 - x^2 - 3) - 2/325*(109*x^2 + 18*x + 3)/(4*x^3 - x^2 - 3) + 3/325*(72*x^2
- 6*x - 1)/(4*x^3 - x^2 - 3) - 3/650*(24*x^2 - 2*x - 217)/(4*x^3 - x^2 - 3) - 12/325*(12*x^2 - x + 54)/(4*x^3
- x^2 - 3) + 1/650*(2*x^2 + 54*x + 9)/(4*x^3 - x^2 - 3) - e^(3/5*x/(4*x^2 + 3*x + 3) + 3/10/(4*x^2 + 3*x + 3)
+ 1/10/(x - 1)) - log(x)

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Fricas [A]
time = 0.39, size = 26, normalized size = 0.96 \begin {gather*} -e^{\left (\frac {x^{2}}{4 \, x^{3} - x^{2} - 3}\right )} - \log \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-16*x^6+8*x^5-x^4+24*x^3-6*x^2-9)*exp(-x^2/(4*x^3-x^2-3))+4*x^5+6*x^2)/(16*x^7-8*x^6+x^5-24*x^4+6*
x^3+9*x)/exp(-x^2/(4*x^3-x^2-3)),x, algorithm="fricas")

[Out]

-e^(x^2/(4*x^3 - x^2 - 3)) - log(x)

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Sympy [A]
time = 0.09, size = 19, normalized size = 0.70 \begin {gather*} - e^{\frac {x^{2}}{4 x^{3} - x^{2} - 3}} - \log {\left (x \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-16*x**6+8*x**5-x**4+24*x**3-6*x**2-9)*exp(-x**2/(4*x**3-x**2-3))+4*x**5+6*x**2)/(16*x**7-8*x**6+x
**5-24*x**4+6*x**3+9*x)/exp(-x**2/(4*x**3-x**2-3)),x)

[Out]

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

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Giac [A]
time = 0.40, size = 26, normalized size = 0.96 \begin {gather*} -e^{\left (\frac {x^{2}}{4 \, x^{3} - x^{2} - 3}\right )} - \log \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-16*x^6+8*x^5-x^4+24*x^3-6*x^2-9)*exp(-x^2/(4*x^3-x^2-3))+4*x^5+6*x^2)/(16*x^7-8*x^6+x^5-24*x^4+6*
x^3+9*x)/exp(-x^2/(4*x^3-x^2-3)),x, algorithm="giac")

[Out]

-e^(x^2/(4*x^3 - x^2 - 3)) - log(x)

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Mupad [B]
time = 4.45, size = 25, normalized size = 0.93 \begin {gather*} -{\mathrm {e}}^{-\frac {x^2}{-4\,x^3+x^2+3}}-\ln \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-x^2/(x^2 - 4*x^3 + 3))*(6*x^2 + 4*x^5 - exp(x^2/(x^2 - 4*x^3 + 3))*(6*x^2 - 24*x^3 + x^4 - 8*x^5 + 1
6*x^6 + 9)))/(9*x + 6*x^3 - 24*x^4 + x^5 - 8*x^6 + 16*x^7),x)

[Out]

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

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