3.88.7 \(\int \frac {1+x^2+e^{16-40 x+33 x^2-10 x^3+x^4+(-32+56 x-28 x^2+4 x^3) \log (2)+(24-26 x+6 x^2) \log ^2(2)+(-8+4 x) \log ^3(2)+\log ^4(2)} (-1-40 x+66 x^2-30 x^3+4 x^4+(56 x-56 x^2+12 x^3) \log (2)+(-26 x+12 x^2) \log ^2(2)+4 x \log ^3(2))}{x^2} \, dx\)

Optimal. Leaf size=27 \[ e-\frac {1-e^{\left (-x+(-2+x+\log (2))^2\right )^2}}{x}+x \]

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Rubi [F]  time = 21.37, 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 {1+x^2+\exp \left (16-40 x+33 x^2-10 x^3+x^4+\left (-32+56 x-28 x^2+4 x^3\right ) \log (2)+\left (24-26 x+6 x^2\right ) \log ^2(2)+(-8+4 x) \log ^3(2)+\log ^4(2)\right ) \left (-1-40 x+66 x^2-30 x^3+4 x^4+\left (56 x-56 x^2+12 x^3\right ) \log (2)+\left (-26 x+12 x^2\right ) \log ^2(2)+4 x \log ^3(2)\right )}{x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(1 + x^2 + E^(16 - 40*x + 33*x^2 - 10*x^3 + x^4 + (-32 + 56*x - 28*x^2 + 4*x^3)*Log[2] + (24 - 26*x + 6*x^
2)*Log[2]^2 + (-8 + 4*x)*Log[2]^3 + Log[2]^4)*(-1 - 40*x + 66*x^2 - 30*x^3 + 4*x^4 + (56*x - 56*x^2 + 12*x^3)*
Log[2] + (-26*x + 12*x^2)*Log[2]^2 + 4*x*Log[2]^3))/x^2,x]

[Out]

-x^(-1) + x + (33 - 28*Log[2] + 6*Log[2]^2)*Defer[Int][2^(-31 + 56*x - 28*x^2 + 4*x^3)*E^(16 - 10*x^3 + x^4 +
24*Log[2]^2 - 8*Log[2]^3 + Log[2]^4 + 3*x^2*(11 + 2*Log[2]^2) - 2*x*(20 + 13*Log[2]^2 - 2*Log[2]^3)), x] - Def
er[Int][(16^(-8 + 14*x - 7*x^2 + x^3)*E^(16 - 10*x^3 + x^4 + 24*Log[2]^2 - 8*Log[2]^3 + Log[2]^4 + 3*x^2*(11 +
2*Log[2]^2) - 2*x*(20 + 13*Log[2]^2 - 2*Log[2]^3)))/x^2, x] - (2 - Log[2])^2*(5 - Log[4])*Defer[Int][(2^(-31
+ 56*x - 28*x^2 + 4*x^3)*E^(16 - 10*x^3 + x^4 + 24*Log[2]^2 - 8*Log[2]^3 + Log[2]^4 + 3*x^2*(11 + 2*Log[2]^2)
- 2*x*(20 + 13*Log[2]^2 - 2*Log[2]^3)))/x, x] - 3*(5 - Log[4])*Defer[Int][2^(-31 + 56*x - 28*x^2 + 4*x^3)*E^(1
6 - 10*x^3 + x^4 + 24*Log[2]^2 - 8*Log[2]^3 + Log[2]^4 + 3*x^2*(11 + 2*Log[2]^2) - 2*x*(20 + 13*Log[2]^2 - 2*L
og[2]^3))*x, x] + Defer[Int][4^(-15 + 28*x - 14*x^2 + 2*x^3)*E^(16 - 10*x^3 + x^4 + 24*Log[2]^2 - 8*Log[2]^3 +
Log[2]^4 + 3*x^2*(11 + 2*Log[2]^2) - 2*x*(20 + 13*Log[2]^2 - 2*Log[2]^3))*x^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {1+x^2}{x^2}+\frac {16^{-8+14 x-7 x^2+x^3} \exp \left (16-10 x^3+x^4+24 \log ^2(2)-8 \log ^3(2)+\log ^4(2)+3 x^2 \left (11+2 \log ^2(2)\right )-2 x \left (20+13 \log ^2(2)-2 \log ^3(2)\right )\right ) \left (-1+4 x^4+2 x^2 \left (33-28 \log (2)+6 \log ^2(2)\right )-6 x^3 (5-\log (4))-2 x (2-\log (2))^2 (5-\log (4))\right )}{x^2}\right ) \, dx\\ &=\int \frac {1+x^2}{x^2} \, dx+\int \frac {16^{-8+14 x-7 x^2+x^3} \exp \left (16-10 x^3+x^4+24 \log ^2(2)-8 \log ^3(2)+\log ^4(2)+3 x^2 \left (11+2 \log ^2(2)\right )-2 x \left (20+13 \log ^2(2)-2 \log ^3(2)\right )\right ) \left (-1+4 x^4+2 x^2 \left (33-28 \log (2)+6 \log ^2(2)\right )-6 x^3 (5-\log (4))-2 x (2-\log (2))^2 (5-\log (4))\right )}{x^2} \, dx\\ &=\int \left (1+\frac {1}{x^2}\right ) \, dx+\int \left (-\frac {16^{-8+14 x-7 x^2+x^3} \exp \left (16-10 x^3+x^4+24 \log ^2(2)-8 \log ^3(2)+\log ^4(2)+3 x^2 \left (11+2 \log ^2(2)\right )-2 x \left (20+13 \log ^2(2)-2 \log ^3(2)\right )\right )}{x^2}+4^{-15+28 x-14 x^2+2 x^3} \exp \left (16-10 x^3+x^4+24 \log ^2(2)-8 \log ^3(2)+\log ^4(2)+3 x^2 \left (11+2 \log ^2(2)\right )-2 x \left (20+13 \log ^2(2)-2 \log ^3(2)\right )\right ) x^2+2^{-31+56 x-28 x^2+4 x^3} \exp \left (16-10 x^3+x^4+24 \log ^2(2)-8 \log ^3(2)+\log ^4(2)+3 x^2 \left (11+2 \log ^2(2)\right )-2 x \left (20+13 \log ^2(2)-2 \log ^3(2)\right )\right ) \left (33-28 \log (2)+6 \log ^2(2)\right )+3\ 2^{-31+56 x-28 x^2+4 x^3} \exp \left (16-10 x^3+x^4+24 \log ^2(2)-8 \log ^3(2)+\log ^4(2)+3 x^2 \left (11+2 \log ^2(2)\right )-2 x \left (20+13 \log ^2(2)-2 \log ^3(2)\right )\right ) x (-5+\log (4))+\frac {2^{-31+56 x-28 x^2+4 x^3} \exp \left (16-10 x^3+x^4+24 \log ^2(2)-8 \log ^3(2)+\log ^4(2)+3 x^2 \left (11+2 \log ^2(2)\right )-2 x \left (20+13 \log ^2(2)-2 \log ^3(2)\right )\right ) (-2+\log (2))^2 (-5+\log (4))}{x}\right ) \, dx\\ &=-\frac {1}{x}+x+\left (33-28 \log (2)+6 \log ^2(2)\right ) \int 2^{-31+56 x-28 x^2+4 x^3} \exp \left (16-10 x^3+x^4+24 \log ^2(2)-8 \log ^3(2)+\log ^4(2)+3 x^2 \left (11+2 \log ^2(2)\right )-2 x \left (20+13 \log ^2(2)-2 \log ^3(2)\right )\right ) \, dx-\left ((2-\log (2))^2 (5-\log (4))\right ) \int \frac {2^{-31+56 x-28 x^2+4 x^3} \exp \left (16-10 x^3+x^4+24 \log ^2(2)-8 \log ^3(2)+\log ^4(2)+3 x^2 \left (11+2 \log ^2(2)\right )-2 x \left (20+13 \log ^2(2)-2 \log ^3(2)\right )\right )}{x} \, dx+(3 (-5+\log (4))) \int 2^{-31+56 x-28 x^2+4 x^3} \exp \left (16-10 x^3+x^4+24 \log ^2(2)-8 \log ^3(2)+\log ^4(2)+3 x^2 \left (11+2 \log ^2(2)\right )-2 x \left (20+13 \log ^2(2)-2 \log ^3(2)\right )\right ) x \, dx-\int \frac {16^{-8+14 x-7 x^2+x^3} \exp \left (16-10 x^3+x^4+24 \log ^2(2)-8 \log ^3(2)+\log ^4(2)+3 x^2 \left (11+2 \log ^2(2)\right )-2 x \left (20+13 \log ^2(2)-2 \log ^3(2)\right )\right )}{x^2} \, dx+\int 4^{-15+28 x-14 x^2+2 x^3} \exp \left (16-10 x^3+x^4+24 \log ^2(2)-8 \log ^3(2)+\log ^4(2)+3 x^2 \left (11+2 \log ^2(2)\right )-2 x \left (20+13 \log ^2(2)-2 \log ^3(2)\right )\right ) x^2 \, dx\\ \end {aligned} \end {gather*}

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Mathematica [F]  time = 5.63, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1+x^2+e^{16-40 x+33 x^2-10 x^3+x^4+\left (-32+56 x-28 x^2+4 x^3\right ) \log (2)+\left (24-26 x+6 x^2\right ) \log ^2(2)+(-8+4 x) \log ^3(2)+\log ^4(2)} \left (-1-40 x+66 x^2-30 x^3+4 x^4+\left (56 x-56 x^2+12 x^3\right ) \log (2)+\left (-26 x+12 x^2\right ) \log ^2(2)+4 x \log ^3(2)\right )}{x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(1 + x^2 + E^(16 - 40*x + 33*x^2 - 10*x^3 + x^4 + (-32 + 56*x - 28*x^2 + 4*x^3)*Log[2] + (24 - 26*x
+ 6*x^2)*Log[2]^2 + (-8 + 4*x)*Log[2]^3 + Log[2]^4)*(-1 - 40*x + 66*x^2 - 30*x^3 + 4*x^4 + (56*x - 56*x^2 + 12
*x^3)*Log[2] + (-26*x + 12*x^2)*Log[2]^2 + 4*x*Log[2]^3))/x^2,x]

[Out]

Integrate[(1 + x^2 + E^(16 - 40*x + 33*x^2 - 10*x^3 + x^4 + (-32 + 56*x - 28*x^2 + 4*x^3)*Log[2] + (24 - 26*x
+ 6*x^2)*Log[2]^2 + (-8 + 4*x)*Log[2]^3 + Log[2]^4)*(-1 - 40*x + 66*x^2 - 30*x^3 + 4*x^4 + (56*x - 56*x^2 + 12
*x^3)*Log[2] + (-26*x + 12*x^2)*Log[2]^2 + 4*x*Log[2]^3))/x^2, x]

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fricas [B]  time = 0.56, size = 74, normalized size = 2.74 \begin {gather*} \frac {x^{2} + e^{\left (x^{4} + 4 \, {\left (x - 2\right )} \log \relax (2)^{3} + \log \relax (2)^{4} - 10 \, x^{3} + 2 \, {\left (3 \, x^{2} - 13 \, x + 12\right )} \log \relax (2)^{2} + 33 \, x^{2} + 4 \, {\left (x^{3} - 7 \, x^{2} + 14 \, x - 8\right )} \log \relax (2) - 40 \, x + 16\right )} - 1}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x*log(2)^3+(12*x^2-26*x)*log(2)^2+(12*x^3-56*x^2+56*x)*log(2)+4*x^4-30*x^3+66*x^2-40*x-1)*exp(lo
g(2)^4+(4*x-8)*log(2)^3+(6*x^2-26*x+24)*log(2)^2+(4*x^3-28*x^2+56*x-32)*log(2)+x^4-10*x^3+33*x^2-40*x+16)+x^2+
1)/x^2,x, algorithm="fricas")

[Out]

(x^2 + e^(x^4 + 4*(x - 2)*log(2)^3 + log(2)^4 - 10*x^3 + 2*(3*x^2 - 13*x + 12)*log(2)^2 + 33*x^2 + 4*(x^3 - 7*
x^2 + 14*x - 8)*log(2) - 40*x + 16) - 1)/x

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giac [B]  time = 0.68, size = 89, normalized size = 3.30 \begin {gather*} \frac {4294967296 \, x^{2} + e^{\left (x^{4} + 4 \, x^{3} \log \relax (2) + 6 \, x^{2} \log \relax (2)^{2} + 4 \, x \log \relax (2)^{3} + \log \relax (2)^{4} - 10 \, x^{3} - 28 \, x^{2} \log \relax (2) - 26 \, x \log \relax (2)^{2} - 8 \, \log \relax (2)^{3} + 33 \, x^{2} + 56 \, x \log \relax (2) + 24 \, \log \relax (2)^{2} - 40 \, x + 16\right )} - 4294967296}{4294967296 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x*log(2)^3+(12*x^2-26*x)*log(2)^2+(12*x^3-56*x^2+56*x)*log(2)+4*x^4-30*x^3+66*x^2-40*x-1)*exp(lo
g(2)^4+(4*x-8)*log(2)^3+(6*x^2-26*x+24)*log(2)^2+(4*x^3-28*x^2+56*x-32)*log(2)+x^4-10*x^3+33*x^2-40*x+16)+x^2+
1)/x^2,x, algorithm="giac")

[Out]

1/4294967296*(4294967296*x^2 + e^(x^4 + 4*x^3*log(2) + 6*x^2*log(2)^2 + 4*x*log(2)^3 + log(2)^4 - 10*x^3 - 28*
x^2*log(2) - 26*x*log(2)^2 - 8*log(2)^3 + 33*x^2 + 56*x*log(2) + 24*log(2)^2 - 40*x + 16) - 4294967296)/x

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maple [B]  time = 10.02, size = 76, normalized size = 2.81

method	result	size
norman	\(\frac {-1+x^{2}+{\mathrm e}^{\ln \relax (2)^{4}+\left (4 x -8\right ) \ln \relax (2)^{3}+\left (6 x^{2}-26 x +24\right ) \ln \relax (2)^{2}+\left (4 x^{3}-28 x^{2}+56 x -32\right ) \ln \relax (2)+x^{4}-10 x^{3}+33 x^{2}-40 x +16}}{x}\)	\(76\)
risch	\(x -\frac {1}{x}+\frac {16^{\left (x -1\right ) \left (x -2\right ) \left (x -4\right )} {\mathrm e}^{\ln \relax (2)^{4}+4 x \ln \relax (2)^{3}+6 x^{2} \ln \relax (2)^{2}+x^{4}-8 \ln \relax (2)^{3}-26 x \ln \relax (2)^{2}-10 x^{3}+24 \ln \relax (2)^{2}+33 x^{2}-40 x +16}}{x}\)	\(82\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((4*x*ln(2)^3+(12*x^2-26*x)*ln(2)^2+(12*x^3-56*x^2+56*x)*ln(2)+4*x^4-30*x^3+66*x^2-40*x-1)*exp(ln(2)^4+(4*
x-8)*ln(2)^3+(6*x^2-26*x+24)*ln(2)^2+(4*x^3-28*x^2+56*x-32)*ln(2)+x^4-10*x^3+33*x^2-40*x+16)+x^2+1)/x^2,x,meth
od=_RETURNVERBOSE)

[Out]

(-1+x^2+exp(ln(2)^4+(4*x-8)*ln(2)^3+(6*x^2-26*x+24)*ln(2)^2+(4*x^3-28*x^2+56*x-32)*ln(2)+x^4-10*x^3+33*x^2-40*
x+16))/x

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maxima [B]  time = 0.67, size = 89, normalized size = 3.30 \begin {gather*} x + \frac {e^{\left (x^{4} + 4 \, x^{3} \log \relax (2) + 6 \, x^{2} \log \relax (2)^{2} + 4 \, x \log \relax (2)^{3} + \log \relax (2)^{4} - 10 \, x^{3} - 28 \, x^{2} \log \relax (2) - 26 \, x \log \relax (2)^{2} - 8 \, \log \relax (2)^{3} + 33 \, x^{2} + 56 \, x \log \relax (2) + 24 \, \log \relax (2)^{2} - 40 \, x + 16\right )}}{4294967296 \, x} - \frac {1}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x*log(2)^3+(12*x^2-26*x)*log(2)^2+(12*x^3-56*x^2+56*x)*log(2)+4*x^4-30*x^3+66*x^2-40*x-1)*exp(lo
g(2)^4+(4*x-8)*log(2)^3+(6*x^2-26*x+24)*log(2)^2+(4*x^3-28*x^2+56*x-32)*log(2)+x^4-10*x^3+33*x^2-40*x+16)+x^2+
1)/x^2,x, algorithm="maxima")

[Out]

x + 1/4294967296*e^(x^4 + 4*x^3*log(2) + 6*x^2*log(2)^2 + 4*x*log(2)^3 + log(2)^4 - 10*x^3 - 28*x^2*log(2) - 2
6*x*log(2)^2 - 8*log(2)^3 + 33*x^2 + 56*x*log(2) + 24*log(2)^2 - 40*x + 16)/x - 1/x

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mupad [B]  time = 0.37, size = 100, normalized size = 3.70 \begin {gather*} x-\frac {1}{x}+\frac {2^{56\,x}\,2^{4\,x^3}\,{\mathrm {e}}^{4\,x\,{\ln \relax (2)}^3}\,{\mathrm {e}}^{-26\,x\,{\ln \relax (2)}^2}\,{\mathrm {e}}^{{\ln \relax (2)}^4}\,{\mathrm {e}}^{-40\,x}\,{\mathrm {e}}^{x^4}\,{\mathrm {e}}^{16}\,{\mathrm {e}}^{6\,x^2\,{\ln \relax (2)}^2}\,{\mathrm {e}}^{-8\,{\ln \relax (2)}^3}\,{\mathrm {e}}^{24\,{\ln \relax (2)}^2}\,{\mathrm {e}}^{-10\,x^3}\,{\mathrm {e}}^{33\,x^2}}{4294967296\,2^{28\,x^2}\,x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2 - exp(log(2)^3*(4*x - 8) - 40*x + log(2)*(56*x - 28*x^2 + 4*x^3 - 32) + log(2)^2*(6*x^2 - 26*x + 24)
+ log(2)^4 + 33*x^2 - 10*x^3 + x^4 + 16)*(40*x - log(2)*(56*x - 56*x^2 + 12*x^3) + log(2)^2*(26*x - 12*x^2) -
4*x*log(2)^3 - 66*x^2 + 30*x^3 - 4*x^4 + 1) + 1)/x^2,x)

[Out]

x - 1/x + (2^(56*x)*2^(4*x^3)*exp(4*x*log(2)^3)*exp(-26*x*log(2)^2)*exp(log(2)^4)*exp(-40*x)*exp(x^4)*exp(16)*
exp(6*x^2*log(2)^2)*exp(-8*log(2)^3)*exp(24*log(2)^2)*exp(-10*x^3)*exp(33*x^2))/(4294967296*2^(28*x^2)*x)

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sympy [B]  time = 0.29, size = 75, normalized size = 2.78 \begin {gather*} x + \frac {e^{x^{4} - 10 x^{3} + 33 x^{2} - 40 x + \left (4 x - 8\right ) \log {\relax (2 )}^{3} + \left (6 x^{2} - 26 x + 24\right ) \log {\relax (2 )}^{2} + \left (4 x^{3} - 28 x^{2} + 56 x - 32\right ) \log {\relax (2 )} + \log {\relax (2 )}^{4} + 16}}{x} - \frac {1}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x*ln(2)**3+(12*x**2-26*x)*ln(2)**2+(12*x**3-56*x**2+56*x)*ln(2)+4*x**4-30*x**3+66*x**2-40*x-1)*e
xp(ln(2)**4+(4*x-8)*ln(2)**3+(6*x**2-26*x+24)*ln(2)**2+(4*x**3-28*x**2+56*x-32)*ln(2)+x**4-10*x**3+33*x**2-40*
x+16)+x**2+1)/x**2,x)

[Out]

x + exp(x**4 - 10*x**3 + 33*x**2 - 40*x + (4*x - 8)*log(2)**3 + (6*x**2 - 26*x + 24)*log(2)**2 + (4*x**3 - 28*
x**2 + 56*x - 32)*log(2) + log(2)**4 + 16)/x - 1/x

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