3.5.67 \(\int \frac {e^{\frac {108-120 x-4 x^2+16 x^3}{-1-x^2+x \log (5)}} (120+224 x-168 x^2-16 x^4+(-108-4 x^2+32 x^3) \log (5))}{1+2 x^2+x^4+(-2 x-2 x^3) \log (5)+x^2 \log ^2(5)} \, dx\)

Optimal. Leaf size=32 \[ e^{\frac {4 (3+x) \left (-(3-2 x)^2+x\right )}{x \left (\frac {1}{x}+x-\log (5)\right )}} \]

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Rubi [F]  time = 6.34, 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 {108-120 x-4 x^2+16 x^3}{-1-x^2+x \log (5)}} \left (120+224 x-168 x^2-16 x^4+\left (-108-4 x^2+32 x^3\right ) \log (5)\right )}{1+2 x^2+x^4+\left (-2 x-2 x^3\right ) \log (5)+x^2 \log ^2(5)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((108 - 120*x - 4*x^2 + 16*x^3)/(-1 - x^2 + x*Log[5]))*(120 + 224*x - 168*x^2 - 16*x^4 + (-108 - 4*x^2
+ 32*x^3)*Log[5]))/(1 + 2*x^2 + x^4 + (-2*x - 2*x^3)*Log[5] + x^2*Log[5]^2),x]

[Out]

-16*Defer[Int][E^((108 - 120*x - 4*x^2 + 16*x^3)/(-1 - x^2 + x*Log[5])), x] - (8*(28 - 7*Log[5] - 4*Log[5]^2)*
(Log[5] - I*Sqrt[4 - Log[5]^2])*Defer[Int][1/(E^((4*(27 - 30*x - x^2 + 4*x^3))/(1 + x^2 - x*Log[5]))*(2*x - Lo
g[5] + I*Sqrt[4 - Log[5]^2])^2), x])/(2 + Log[5]) - (32*(17 + Log[25])*Defer[Int][1/(E^((4*(27 - 30*x - x^2 +
4*x^3))/(1 + x^2 - x*Log[5]))*(2*x - Log[5] + I*Sqrt[4 - Log[5]^2])^2), x])/(2 + Log[5]) + ((8*I)*Log[5]*(28 -
7*Log[5] - 4*Log[5]^2)*Defer[Int][1/(E^((4*(27 - 30*x - x^2 + 4*x^3))/(1 + x^2 - x*Log[5]))*(2*x - Log[5] + I
*Sqrt[4 - Log[5]^2])), x])/((2 + Log[5])*Sqrt[4 - Log[5]^2]) - ((8*I)*(34 + Log[5] - 4*Log[5]^2)*Defer[Int][1/
(E^((4*(27 - 30*x - x^2 + 4*x^3))/(1 + x^2 - x*Log[5]))*(2*x - Log[5] + I*Sqrt[4 - Log[5]^2])), x])/Sqrt[4 - L
og[5]^2] + ((32*I)*(17 + Log[25])*Defer[Int][1/(E^((4*(27 - 30*x - x^2 + 4*x^3))/(1 + x^2 - x*Log[5]))*(2*x -
Log[5] + I*Sqrt[4 - Log[5]^2])), x])/((2 + Log[5])*Sqrt[4 - Log[5]^2]) - (8*(28 - 7*Log[5] - 4*Log[5]^2)*(Log[
5] + I*Sqrt[4 - Log[5]^2])*Defer[Int][1/(E^((4*(27 - 30*x - x^2 + 4*x^3))/(1 + x^2 - x*Log[5]))*(-2*x + Log[5]
+ I*Sqrt[4 - Log[5]^2])^2), x])/(2 + Log[5]) - (32*(17 + Log[25])*Defer[Int][1/(E^((4*(27 - 30*x - x^2 + 4*x^
3))/(1 + x^2 - x*Log[5]))*(-2*x + Log[5] + I*Sqrt[4 - Log[5]^2])^2), x])/(2 + Log[5]) + ((8*I)*Log[5]*(28 - 7*
Log[5] - 4*Log[5]^2)*Defer[Int][1/(E^((4*(27 - 30*x - x^2 + 4*x^3))/(1 + x^2 - x*Log[5]))*(-2*x + Log[5] + I*S
qrt[4 - Log[5]^2])), x])/((2 + Log[5])*Sqrt[4 - Log[5]^2]) - ((8*I)*(34 + Log[5] - 4*Log[5]^2)*Defer[Int][1/(E
^((4*(27 - 30*x - x^2 + 4*x^3))/(1 + x^2 - x*Log[5]))*(-2*x + Log[5] + I*Sqrt[4 - Log[5]^2])), x])/Sqrt[4 - Lo
g[5]^2] + ((32*I)*(17 + Log[25])*Defer[Int][1/(E^((4*(27 - 30*x - x^2 + 4*x^3))/(1 + x^2 - x*Log[5]))*(-2*x +
Log[5] + I*Sqrt[4 - Log[5]^2])), x])/((2 + Log[5])*Sqrt[4 - Log[5]^2])

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{\frac {108-120 x-4 x^2+16 x^3}{-1-x^2+x \log (5)}} \left (120+224 x-168 x^2-16 x^4+\left (-108-4 x^2+32 x^3\right ) \log (5)\right )}{1+x^4+\left (-2 x-2 x^3\right ) \log (5)+x^2 \left (2+\log ^2(5)\right )} \, dx\\ &=\int \frac {e^{\frac {108-120 x-4 x^2+16 x^3}{-1-x^2+x \log (5)}} \left (224 x-16 x^4+12 (10-9 \log (5))+32 x^3 \log (5)-4 x^2 (42+\log (5))\right )}{1+x^4-2 x \log (5)-2 x^3 \log (5)+x^2 \left (2+\log ^2(5)\right )} \, dx\\ &=\int \left (-16 e^{\frac {108-120 x-4 x^2+16 x^3}{-1-x^2+x \log (5)}}-\frac {4 e^{\frac {108-120 x-4 x^2+16 x^3}{-1-x^2+x \log (5)}} \left (-34-8 x (7-\log (5))+27 \log (5)+x^2 \left (34+\log (5)-4 \log ^2(5)\right )\right )}{1+x^4-2 x \log (5)-2 x^3 \log (5)+x^2 \left (2+\log ^2(5)\right )}\right ) \, dx\\ &=-\left (4 \int \frac {e^{\frac {108-120 x-4 x^2+16 x^3}{-1-x^2+x \log (5)}} \left (-34-8 x (7-\log (5))+27 \log (5)+x^2 \left (34+\log (5)-4 \log ^2(5)\right )\right )}{1+x^4-2 x \log (5)-2 x^3 \log (5)+x^2 \left (2+\log ^2(5)\right )} \, dx\right )-16 \int e^{\frac {108-120 x-4 x^2+16 x^3}{-1-x^2+x \log (5)}} \, dx\\ &=-\left (4 \int \frac {\exp \left (-\frac {4 \left (27-30 x-x^2+4 x^3\right )}{1+x^2-x \log (5)}\right ) \left (-34-8 x (7-\log (5))+27 \log (5)+x^2 \left (34+\log (5)-4 \log ^2(5)\right )\right )}{\left (1+x^2-x \log (5)\right )^2} \, dx\right )-16 \int e^{\frac {108-120 x-4 x^2+16 x^3}{-1-x^2+x \log (5)}} \, dx\\ &=-\left (4 \int \left (\frac {\exp \left (-\frac {4 \left (27-30 x-x^2+4 x^3\right )}{1+x^2-x \log (5)}\right ) \left (34+\log (5)-4 \log ^2(5)\right )}{1+x^2-x \log (5)}+\frac {\exp \left (-\frac {4 \left (27-30 x-x^2+4 x^3\right )}{1+x^2-x \log (5)}\right ) (2-\log (5)) \left (-2 (17+2 \log (5))-x \left (28-7 \log (5)-4 \log ^2(5)\right )\right )}{\left (1+x^2-x \log (5)\right )^2}\right ) \, dx\right )-16 \int e^{\frac {108-120 x-4 x^2+16 x^3}{-1-x^2+x \log (5)}} \, dx\\ &=-\left (16 \int e^{\frac {108-120 x-4 x^2+16 x^3}{-1-x^2+x \log (5)}} \, dx\right )-(4 (2-\log (5))) \int \frac {\exp \left (-\frac {4 \left (27-30 x-x^2+4 x^3\right )}{1+x^2-x \log (5)}\right ) \left (-2 (17+2 \log (5))-x \left (28-7 \log (5)-4 \log ^2(5)\right )\right )}{\left (1+x^2-x \log (5)\right )^2} \, dx-\left (4 \left (34+\log (5)-4 \log ^2(5)\right )\right ) \int \frac {\exp \left (-\frac {4 \left (27-30 x-x^2+4 x^3\right )}{1+x^2-x \log (5)}\right )}{1+x^2-x \log (5)} \, dx\\ &=-\left (16 \int e^{\frac {108-120 x-4 x^2+16 x^3}{-1-x^2+x \log (5)}} \, dx\right )-(4 (2-\log (5))) \int \left (\frac {\exp \left (-\frac {4 \left (27-30 x-x^2+4 x^3\right )}{1+x^2-x \log (5)}\right ) x \left (-28+7 \log (5)+4 \log ^2(5)\right )}{\left (1+x^2-x \log (5)\right )^2}-\frac {2 \exp \left (-\frac {4 \left (27-30 x-x^2+4 x^3\right )}{1+x^2-x \log (5)}\right ) (17+\log (25))}{\left (1+x^2-x \log (5)\right )^2}\right ) \, dx-\left (4 \left (34+\log (5)-4 \log ^2(5)\right )\right ) \int \left (\frac {2 i \exp \left (-\frac {4 \left (27-30 x-x^2+4 x^3\right )}{1+x^2-x \log (5)}\right )}{\sqrt {4-\log ^2(5)} \left (2 x-\log (5)+i \sqrt {4-\log ^2(5)}\right )}+\frac {2 i \exp \left (-\frac {4 \left (27-30 x-x^2+4 x^3\right )}{1+x^2-x \log (5)}\right )}{\sqrt {4-\log ^2(5)} \left (-2 x+\log (5)+i \sqrt {4-\log ^2(5)}\right )}\right ) \, dx\\ &=-\left (16 \int e^{\frac {108-120 x-4 x^2+16 x^3}{-1-x^2+x \log (5)}} \, dx\right )+\left (4 (2-\log (5)) \left (28-7 \log (5)-4 \log ^2(5)\right )\right ) \int \frac {\exp \left (-\frac {4 \left (27-30 x-x^2+4 x^3\right )}{1+x^2-x \log (5)}\right ) x}{\left (1+x^2-x \log (5)\right )^2} \, dx-\frac {\left (8 i \left (34+\log (5)-4 \log ^2(5)\right )\right ) \int \frac {\exp \left (-\frac {4 \left (27-30 x-x^2+4 x^3\right )}{1+x^2-x \log (5)}\right )}{2 x-\log (5)+i \sqrt {4-\log ^2(5)}} \, dx}{\sqrt {4-\log ^2(5)}}-\frac {\left (8 i \left (34+\log (5)-4 \log ^2(5)\right )\right ) \int \frac {\exp \left (-\frac {4 \left (27-30 x-x^2+4 x^3\right )}{1+x^2-x \log (5)}\right )}{-2 x+\log (5)+i \sqrt {4-\log ^2(5)}} \, dx}{\sqrt {4-\log ^2(5)}}+(8 (2-\log (5)) (17+\log (25))) \int \frac {\exp \left (-\frac {4 \left (27-30 x-x^2+4 x^3\right )}{1+x^2-x \log (5)}\right )}{\left (1+x^2-x \log (5)\right )^2} \, dx\\ &=-\left (16 \int e^{\frac {108-120 x-4 x^2+16 x^3}{-1-x^2+x \log (5)}} \, dx\right )+\left (4 (2-\log (5)) \left (28-7 \log (5)-4 \log ^2(5)\right )\right ) \int \left (-\frac {2 e^{-\frac {4 \left (27-30 x-x^2+4 x^3\right )}{1+x^2-x \log (5)}} \left (\log (5)-i \sqrt {4-\log ^2(5)}\right )}{\left (4-\log ^2(5)\right ) \left (2 x-\log (5)+i \sqrt {4-\log ^2(5)}\right )^2}+\frac {2 i e^{-\frac {4 \left (27-30 x-x^2+4 x^3\right )}{1+x^2-x \log (5)}} \log (5)}{\left (4-\log ^2(5)\right )^{3/2} \left (2 x-\log (5)+i \sqrt {4-\log ^2(5)}\right )}-\frac {2 e^{-\frac {4 \left (27-30 x-x^2+4 x^3\right )}{1+x^2-x \log (5)}} \left (\log (5)+i \sqrt {4-\log ^2(5)}\right )}{\left (4-\log ^2(5)\right ) \left (-2 x+\log (5)+i \sqrt {4-\log ^2(5)}\right )^2}+\frac {2 i e^{-\frac {4 \left (27-30 x-x^2+4 x^3\right )}{1+x^2-x \log (5)}} \log (5)}{\left (4-\log ^2(5)\right )^{3/2} \left (-2 x+\log (5)+i \sqrt {4-\log ^2(5)}\right )}\right ) \, dx-\frac {\left (8 i \left (34+\log (5)-4 \log ^2(5)\right )\right ) \int \frac {e^{-\frac {4 \left (27-30 x-x^2+4 x^3\right )}{1+x^2-x \log (5)}}}{2 x-\log (5)+i \sqrt {4-\log ^2(5)}} \, dx}{\sqrt {4-\log ^2(5)}}-\frac {\left (8 i \left (34+\log (5)-4 \log ^2(5)\right )\right ) \int \frac {e^{-\frac {4 \left (27-30 x-x^2+4 x^3\right )}{1+x^2-x \log (5)}}}{-2 x+\log (5)+i \sqrt {4-\log ^2(5)}} \, dx}{\sqrt {4-\log ^2(5)}}+(8 (2-\log (5)) (17+\log (25))) \int \left (-\frac {4 e^{-\frac {4 \left (27-30 x-x^2+4 x^3\right )}{1+x^2-x \log (5)}}}{\left (4-\log ^2(5)\right ) \left (2 x-\log (5)+i \sqrt {4-\log ^2(5)}\right )^2}+\frac {4 i e^{-\frac {4 \left (27-30 x-x^2+4 x^3\right )}{1+x^2-x \log (5)}}}{\left (4-\log ^2(5)\right )^{3/2} \left (2 x-\log (5)+i \sqrt {4-\log ^2(5)}\right )}-\frac {4 e^{-\frac {4 \left (27-30 x-x^2+4 x^3\right )}{1+x^2-x \log (5)}}}{\left (4-\log ^2(5)\right ) \left (-2 x+\log (5)+i \sqrt {4-\log ^2(5)}\right )^2}+\frac {4 i e^{-\frac {4 \left (27-30 x-x^2+4 x^3\right )}{1+x^2-x \log (5)}}}{\left (4-\log ^2(5)\right )^{3/2} \left (-2 x+\log (5)+i \sqrt {4-\log ^2(5)}\right )}\right ) \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]  time = 1.05, size = 31, normalized size = 0.97 \begin {gather*} e^{-\frac {4 \left (27-30 x-x^2+4 x^3\right )}{1+x^2-x \log (5)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((108 - 120*x - 4*x^2 + 16*x^3)/(-1 - x^2 + x*Log[5]))*(120 + 224*x - 168*x^2 - 16*x^4 + (-108 -
4*x^2 + 32*x^3)*Log[5]))/(1 + 2*x^2 + x^4 + (-2*x - 2*x^3)*Log[5] + x^2*Log[5]^2),x]

[Out]

E^((-4*(27 - 30*x - x^2 + 4*x^3))/(1 + x^2 - x*Log[5]))

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fricas [A]  time = 0.97, size = 30, normalized size = 0.94 \begin {gather*} e^{\left (-\frac {4 \, {\left (4 \, x^{3} - x^{2} - 30 \, x + 27\right )}}{x^{2} - x \log \relax (5) + 1}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((32*x^3-4*x^2-108)*log(5)-16*x^4-168*x^2+224*x+120)*exp((16*x^3-4*x^2-120*x+108)/(x*log(5)-x^2-1))/
(x^2*log(5)^2+(-2*x^3-2*x)*log(5)+x^4+2*x^2+1),x, algorithm="fricas")

[Out]

e^(-4*(4*x^3 - x^2 - 30*x + 27)/(x^2 - x*log(5) + 1))

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giac [B]  time = 0.57, size = 65, normalized size = 2.03 \begin {gather*} e^{\left (-\frac {16 \, x^{3}}{x^{2} - x \log \relax (5) + 1} + \frac {4 \, x^{2}}{x^{2} - x \log \relax (5) + 1} + \frac {120 \, x}{x^{2} - x \log \relax (5) + 1} - \frac {108}{x^{2} - x \log \relax (5) + 1}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((32*x^3-4*x^2-108)*log(5)-16*x^4-168*x^2+224*x+120)*exp((16*x^3-4*x^2-120*x+108)/(x*log(5)-x^2-1))/
(x^2*log(5)^2+(-2*x^3-2*x)*log(5)+x^4+2*x^2+1),x, algorithm="giac")

[Out]

e^(-16*x^3/(x^2 - x*log(5) + 1) + 4*x^2/(x^2 - x*log(5) + 1) + 120*x/(x^2 - x*log(5) + 1) - 108/(x^2 - x*log(5
) + 1))

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maple [A]  time = 0.22, size = 28, normalized size = 0.88

method	result	size
risch	\({\mathrm e}^{\frac {4 \left (x -1\right ) \left (4 x -9\right ) \left (3+x \right )}{x \ln \relax (5)-x^{2}-1}}\)	\(28\)
gosper	\({\mathrm e}^{\frac {16 x^{3}-4 x^{2}-120 x +108}{x \ln \relax (5)-x^{2}-1}}\)	\(32\)
norman	\(\frac {x \ln \relax (5) {\mathrm e}^{\frac {16 x^{3}-4 x^{2}-120 x +108}{x \ln \relax (5)-x^{2}-1}}-x^{2} {\mathrm e}^{\frac {16 x^{3}-4 x^{2}-120 x +108}{x \ln \relax (5)-x^{2}-1}}-{\mathrm e}^{\frac {16 x^{3}-4 x^{2}-120 x +108}{x \ln \relax (5)-x^{2}-1}}}{x \ln \relax (5)-x^{2}-1}\)	\(117\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((32*x^3-4*x^2-108)*ln(5)-16*x^4-168*x^2+224*x+120)*exp((16*x^3-4*x^2-120*x+108)/(x*ln(5)-x^2-1))/(x^2*ln(
5)^2+(-2*x^3-2*x)*ln(5)+x^4+2*x^2+1),x,method=_RETURNVERBOSE)

[Out]

exp(4*(x-1)*(4*x-9)*(3+x)/(x*ln(5)-x^2-1))

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maxima [B]  time = 1.17, size = 89, normalized size = 2.78 \begin {gather*} \frac {1}{152587890625} \, e^{\left (-\frac {16 \, x \log \relax (5)^{2}}{x^{2} - x \log \relax (5) + 1} - 16 \, x + \frac {4 \, x \log \relax (5)}{x^{2} - x \log \relax (5) + 1} + \frac {136 \, x}{x^{2} - x \log \relax (5) + 1} + \frac {16 \, \log \relax (5)}{x^{2} - x \log \relax (5) + 1} - \frac {112}{x^{2} - x \log \relax (5) + 1} + 4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((32*x^3-4*x^2-108)*log(5)-16*x^4-168*x^2+224*x+120)*exp((16*x^3-4*x^2-120*x+108)/(x*log(5)-x^2-1))/
(x^2*log(5)^2+(-2*x^3-2*x)*log(5)+x^4+2*x^2+1),x, algorithm="maxima")

[Out]

1/152587890625*e^(-16*x*log(5)^2/(x^2 - x*log(5) + 1) - 16*x + 4*x*log(5)/(x^2 - x*log(5) + 1) + 136*x/(x^2 -
x*log(5) + 1) + 16*log(5)/(x^2 - x*log(5) + 1) - 112/(x^2 - x*log(5) + 1) + 4)

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mupad [B]  time = 1.76, size = 68, normalized size = 2.12 \begin {gather*} {\mathrm {e}}^{\frac {120\,x}{x^2-\ln \relax (5)\,x+1}}\,{\mathrm {e}}^{\frac {4\,x^2}{x^2-\ln \relax (5)\,x+1}}\,{\mathrm {e}}^{-\frac {16\,x^3}{x^2-\ln \relax (5)\,x+1}}\,{\mathrm {e}}^{-\frac {108}{x^2-\ln \relax (5)\,x+1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp((120*x + 4*x^2 - 16*x^3 - 108)/(x^2 - x*log(5) + 1))*(log(5)*(4*x^2 - 32*x^3 + 108) - 224*x + 168*x^
2 + 16*x^4 - 120))/(x^2*log(5)^2 - log(5)*(2*x + 2*x^3) + 2*x^2 + x^4 + 1),x)

[Out]

exp((120*x)/(x^2 - x*log(5) + 1))*exp((4*x^2)/(x^2 - x*log(5) + 1))*exp(-(16*x^3)/(x^2 - x*log(5) + 1))*exp(-1
08/(x^2 - x*log(5) + 1))

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sympy [A]  time = 1.07, size = 26, normalized size = 0.81 \begin {gather*} e^{\frac {16 x^{3} - 4 x^{2} - 120 x + 108}{- x^{2} + x \log {\relax (5 )} - 1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((32*x**3-4*x**2-108)*ln(5)-16*x**4-168*x**2+224*x+120)*exp((16*x**3-4*x**2-120*x+108)/(x*ln(5)-x**2
-1))/(x**2*ln(5)**2+(-2*x**3-2*x)*ln(5)+x**4+2*x**2+1),x)

[Out]

exp((16*x**3 - 4*x**2 - 120*x + 108)/(-x**2 + x*log(5) - 1))

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