\(\int \frac {1+4 x^2+e^{\frac {84+32 x+4 x^2+(21 x+8 x^2+x^3) \log (x)}{x}} (168-42 x-24 x^2-2 x^3+(-16 x^2-4 x^3) \log (x))}{2 x^2} \, dx\) [7899]

Optimal result

Integrand size = 75, antiderivative size = 32 \[ \int \frac {1+4 x^2+e^{\frac {84+32 x+4 x^2+\left (21 x+8 x^2+x^3\right ) \log (x)}{x}} \left (168-42 x-24 x^2-2 x^3+\left (-16 x^2-4 x^3\right ) \log (x)\right )}{2 x^2} \, dx=5-e^{\left (5+(4+x)^2\right ) \left (\frac {4}{x}+\log (x)\right )}-\frac {1}{2 x}+2 x \]

[Out]

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

Rubi [F]

\[ \int \frac {1+4 x^2+e^{\frac {84+32 x+4 x^2+\left (21 x+8 x^2+x^3\right ) \log (x)}{x}} \left (168-42 x-24 x^2-2 x^3+\left (-16 x^2-4 x^3\right ) \log (x)\right )}{2 x^2} \, dx=\int \frac {1+4 x^2+\exp \left (\frac {84+32 x+4 x^2+\left (21 x+8 x^2+x^3\right ) \log (x)}{x}\right ) \left (168-42 x-24 x^2-2 x^3+\left (-16 x^2-4 x^3\right ) \log (x)\right )}{2 x^2} \, dx \]

[In]

Int[(1 + 4*x^2 + E^((84 + 32*x + 4*x^2 + (21*x + 8*x^2 + x^3)*Log[x])/x)*(168 - 42*x - 24*x^2 - 2*x^3 + (-16*x
^2 - 4*x^3)*Log[x]))/(2*x^2),x]

[Out]

-1/2*1/x + 2*x + 84*Defer[Int][E^(32 + 84/x + 4*x)*x^(19 + 8*x + x^2), x] - 21*Defer[Int][E^(32 + 84/x + 4*x)*
x^(20 + 8*x + x^2), x] - 12*Defer[Int][E^(32 + 84/x + 4*x)*x^(21 + 8*x + x^2), x] - 8*Log[x]*Defer[Int][E^(32
+ 84/x + 4*x)*x^(21 + 8*x + x^2), x] - Defer[Int][E^(32 + 84/x + 4*x)*x^(22 + 8*x + x^2), x] - 2*Log[x]*Defer[
Int][E^(32 + 84/x + 4*x)*x^(22 + 8*x + x^2), x] + 8*Defer[Int][Defer[Int][E^(4*(8 + 21/x + x))*x^(21 + 8*x + x
^2), x]/x, x] + 2*Defer[Int][Defer[Int][E^(4*(8 + 21/x + x))*x^(22 + 8*x + x^2), x]/x, x]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \int \frac {1+4 x^2+\exp \left (\frac {84+32 x+4 x^2+\left (21 x+8 x^2+x^3\right ) \log (x)}{x}\right ) \left (168-42 x-24 x^2-2 x^3+\left (-16 x^2-4 x^3\right ) \log (x)\right )}{x^2} \, dx \\ & = \frac {1}{2} \int \left (\frac {1+4 x^2}{x^2}-2 e^{32+\frac {84}{x}+4 x} x^{19+8 x+x^2} \left (-84+21 x+12 x^2+x^3+8 x^2 \log (x)+2 x^3 \log (x)\right )\right ) \, dx \\ & = \frac {1}{2} \int \frac {1+4 x^2}{x^2} \, dx-\int e^{32+\frac {84}{x}+4 x} x^{19+8 x+x^2} \left (-84+21 x+12 x^2+x^3+8 x^2 \log (x)+2 x^3 \log (x)\right ) \, dx \\ & = \frac {1}{2} \int \left (4+\frac {1}{x^2}\right ) \, dx-\int \left (-84 e^{32+\frac {84}{x}+4 x} x^{19+8 x+x^2}+21 e^{32+\frac {84}{x}+4 x} x^{20+8 x+x^2}+12 e^{32+\frac {84}{x}+4 x} x^{21+8 x+x^2}+e^{32+\frac {84}{x}+4 x} x^{22+8 x+x^2}+8 e^{32+\frac {84}{x}+4 x} x^{21+8 x+x^2} \log (x)+2 e^{32+\frac {84}{x}+4 x} x^{22+8 x+x^2} \log (x)\right ) \, dx \\ & = -\frac {1}{2 x}+2 x-2 \int e^{32+\frac {84}{x}+4 x} x^{22+8 x+x^2} \log (x) \, dx-8 \int e^{32+\frac {84}{x}+4 x} x^{21+8 x+x^2} \log (x) \, dx-12 \int e^{32+\frac {84}{x}+4 x} x^{21+8 x+x^2} \, dx-21 \int e^{32+\frac {84}{x}+4 x} x^{20+8 x+x^2} \, dx+84 \int e^{32+\frac {84}{x}+4 x} x^{19+8 x+x^2} \, dx-\int e^{32+\frac {84}{x}+4 x} x^{22+8 x+x^2} \, dx \\ & = -\frac {1}{2 x}+2 x+2 \int \frac {\int e^{4 \left (8+\frac {21}{x}+x\right )} x^{22+8 x+x^2} \, dx}{x} \, dx+8 \int \frac {\int e^{4 \left (8+\frac {21}{x}+x\right )} x^{21+8 x+x^2} \, dx}{x} \, dx-12 \int e^{32+\frac {84}{x}+4 x} x^{21+8 x+x^2} \, dx-21 \int e^{32+\frac {84}{x}+4 x} x^{20+8 x+x^2} \, dx+84 \int e^{32+\frac {84}{x}+4 x} x^{19+8 x+x^2} \, dx-(2 \log (x)) \int e^{32+\frac {84}{x}+4 x} x^{22+8 x+x^2} \, dx-(8 \log (x)) \int e^{32+\frac {84}{x}+4 x} x^{21+8 x+x^2} \, dx-\int e^{32+\frac {84}{x}+4 x} x^{22+8 x+x^2} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 3.60 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.06 \[ \int \frac {1+4 x^2+e^{\frac {84+32 x+4 x^2+\left (21 x+8 x^2+x^3\right ) \log (x)}{x}} \left (168-42 x-24 x^2-2 x^3+\left (-16 x^2-4 x^3\right ) \log (x)\right )}{2 x^2} \, dx=-\frac {1}{2 x}+2 x-e^{32+\frac {84}{x}+4 x} x^{21+x (8+x)} \]

[In]

Integrate[(1 + 4*x^2 + E^((84 + 32*x + 4*x^2 + (21*x + 8*x^2 + x^3)*Log[x])/x)*(168 - 42*x - 24*x^2 - 2*x^3 +
(-16*x^2 - 4*x^3)*Log[x]))/(2*x^2),x]

[Out]

-1/2*1/x + 2*x - E^(32 + 84/x + 4*x)*x^(21 + x*(8 + x))

Maple [A] (verified)

Time = 0.12 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.97

method	result	size
risch	\(-{\mathrm e}^{\frac {\left (x^{2}+8 x +21\right ) \left (x \ln \left (x \right )+4\right )}{x}}+2 x -\frac {1}{2 x}\)	\(31\)
default	\(-{\mathrm e}^{\frac {\left (x^{3}+8 x^{2}+21 x \right ) \ln \left (x \right )+4 x^{2}+32 x +84}{x}}+2 x -\frac {1}{2 x}\)	\(42\)
parts	\(-{\mathrm e}^{\frac {\left (x^{3}+8 x^{2}+21 x \right ) \ln \left (x \right )+4 x^{2}+32 x +84}{x}}+2 x -\frac {1}{2 x}\)	\(42\)
norman	\(\frac {-\frac {1}{2}+2 x^{2}-x \,{\mathrm e}^{\frac {\left (x^{3}+8 x^{2}+21 x \right ) \ln \left (x \right )+4 x^{2}+32 x +84}{x}}}{x}\)	\(45\)
parallelrisch	\(-\frac {-4 x^{2}+2 x \,{\mathrm e}^{\frac {\left (x^{3}+8 x^{2}+21 x \right ) \ln \left (x \right )+4 x^{2}+32 x +84}{x}}+1}{2 x}\)	\(46\)

[In]

int(1/2*(((-4*x^3-16*x^2)*ln(x)-2*x^3-24*x^2-42*x+168)*exp(((x^3+8*x^2+21*x)*ln(x)+4*x^2+32*x+84)/x)+4*x^2+1)/
x^2,x,method=_RETURNVERBOSE)

[Out]

-exp((x^2+8*x+21)*(x*ln(x)+4)/x)+2*x-1/2/x

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.41 \[ \int \frac {1+4 x^2+e^{\frac {84+32 x+4 x^2+\left (21 x+8 x^2+x^3\right ) \log (x)}{x}} \left (168-42 x-24 x^2-2 x^3+\left (-16 x^2-4 x^3\right ) \log (x)\right )}{2 x^2} \, dx=\frac {4 \, x^{2} - 2 \, x e^{\left (\frac {4 \, x^{2} + {\left (x^{3} + 8 \, x^{2} + 21 \, x\right )} \log \left (x\right ) + 32 \, x + 84}{x}\right )} - 1}{2 \, x} \]

[In]

integrate(1/2*(((-4*x^3-16*x^2)*log(x)-2*x^3-24*x^2-42*x+168)*exp(((x^3+8*x^2+21*x)*log(x)+4*x^2+32*x+84)/x)+4
*x^2+1)/x^2,x, algorithm="fricas")

[Out]

1/2*(4*x^2 - 2*x*e^((4*x^2 + (x^3 + 8*x^2 + 21*x)*log(x) + 32*x + 84)/x) - 1)/x

Sympy [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.12 \[ \int \frac {1+4 x^2+e^{\frac {84+32 x+4 x^2+\left (21 x+8 x^2+x^3\right ) \log (x)}{x}} \left (168-42 x-24 x^2-2 x^3+\left (-16 x^2-4 x^3\right ) \log (x)\right )}{2 x^2} \, dx=2 x - e^{\frac {4 x^{2} + 32 x + \left (x^{3} + 8 x^{2} + 21 x\right ) \log {\left (x \right )} + 84}{x}} - \frac {1}{2 x} \]

[In]

integrate(1/2*(((-4*x**3-16*x**2)*ln(x)-2*x**3-24*x**2-42*x+168)*exp(((x**3+8*x**2+21*x)*ln(x)+4*x**2+32*x+84)
/x)+4*x**2+1)/x**2,x)

[Out]

2*x - exp((4*x**2 + 32*x + (x**3 + 8*x**2 + 21*x)*log(x) + 84)/x) - 1/(2*x)

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.12 \[ \int \frac {1+4 x^2+e^{\frac {84+32 x+4 x^2+\left (21 x+8 x^2+x^3\right ) \log (x)}{x}} \left (168-42 x-24 x^2-2 x^3+\left (-16 x^2-4 x^3\right ) \log (x)\right )}{2 x^2} \, dx=-x^{21} e^{\left (x^{2} \log \left (x\right ) + 8 \, x \log \left (x\right ) + 4 \, x + \frac {84}{x} + 32\right )} + 2 \, x - \frac {1}{2 \, x} \]

[In]

integrate(1/2*(((-4*x^3-16*x^2)*log(x)-2*x^3-24*x^2-42*x+168)*exp(((x^3+8*x^2+21*x)*log(x)+4*x^2+32*x+84)/x)+4
*x^2+1)/x^2,x, algorithm="maxima")

[Out]

-x^21*e^(x^2*log(x) + 8*x*log(x) + 4*x + 84/x + 32) + 2*x - 1/2/x

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.16 \[ \int \frac {1+4 x^2+e^{\frac {84+32 x+4 x^2+\left (21 x+8 x^2+x^3\right ) \log (x)}{x}} \left (168-42 x-24 x^2-2 x^3+\left (-16 x^2-4 x^3\right ) \log (x)\right )}{2 x^2} \, dx=2 \, x - \frac {1}{2 \, x} - e^{\left (x^{2} \log \left (x\right ) + 8 \, x \log \left (x\right ) + 4 \, x + \frac {84}{x} + 21 \, \log \left (x\right ) + 32\right )} \]

[In]

integrate(1/2*(((-4*x^3-16*x^2)*log(x)-2*x^3-24*x^2-42*x+168)*exp(((x^3+8*x^2+21*x)*log(x)+4*x^2+32*x+84)/x)+4
*x^2+1)/x^2,x, algorithm="giac")

[Out]

2*x - 1/2/x - e^(x^2*log(x) + 8*x*log(x) + 4*x + 84/x + 21*log(x) + 32)

Mupad [B] (verification not implemented)

Time = 13.01 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.12 \[ \int \frac {1+4 x^2+e^{\frac {84+32 x+4 x^2+\left (21 x+8 x^2+x^3\right ) \log (x)}{x}} \left (168-42 x-24 x^2-2 x^3+\left (-16 x^2-4 x^3\right ) \log (x)\right )}{2 x^2} \, dx=2\,x-\frac {1}{2\,x}-x^{8\,x}\,x^{x^2}\,x^{21}\,{\mathrm {e}}^{4\,x}\,{\mathrm {e}}^{32}\,{\mathrm {e}}^{84/x} \]

[In]

int((2*x^2 - (exp((32*x + log(x)*(21*x + 8*x^2 + x^3) + 4*x^2 + 84)/x)*(42*x + log(x)*(16*x^2 + 4*x^3) + 24*x^
2 + 2*x^3 - 168))/2 + 1/2)/x^2,x)

[Out]

2*x - 1/(2*x) - x^(8*x)*x^(x^2)*x^21*exp(4*x)*exp(32)*exp(84/x)