3.38.6 \(\int \frac {1+2 x+x^2+x^{\frac {-x-2 x^2}{1+x}} (-1-x+2 x^2+2 x^3+(x+4 x^2+2 x^3) \log (x))}{1+2 x+x^2} \, dx\)

Optimal. Leaf size=23 \[ -5+x-x^{1-x+\frac {x^2}{-1-x}} \]

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Rubi [F]  time = 2.02, 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+2 x+x^2+x^{\frac {-x-2 x^2}{1+x}} \left (-1-x+2 x^2+2 x^3+\left (x+4 x^2+2 x^3\right ) \log (x)\right )}{1+2 x+x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(1 + 2*x + x^2 + x^((-x - 2*x^2)/(1 + x))*(-1 - x + 2*x^2 + 2*x^3 + (x + 4*x^2 + 2*x^3)*Log[x]))/(1 + 2*x
+ x^2),x]

[Out]

x - x^(1 - (x*(1 + 2*x))/(1 + x))/(1 + x) + 1/(x^((x*(1 + 2*x))/(1 + x))*(1 + 2*x)) + (2*x^(2 - (x*(1 + 2*x))/
(1 + x)))/((1 + x)*(1 - (x*(1 + 2*x))/(1 + x))) - (1 - (x*(1 + 2*x))/(1 + x))/(x^((x*(1 + 2*x))/(1 + x))*(1 +
2*x)) + (2*x^(3 - (x*(1 + 2*x))/(1 + x)))/((1 + x)*(2 - (x*(1 + 2*x))/(1 + x))) - (2*x^(1 + (1 - 2*x^2)/(1 + x
))*(2 - (x*(1 + 2*x))/(1 + x)))/((1 + x)*(1 - (x*(1 + 2*x))/(1 + x))) - (2*x^(1 + (2 + x - 2*x^2)/(1 + x))*(3
- (x*(1 + 2*x))/(1 + x)))/((1 + x)*(2 - (x*(1 + 2*x))/(1 + x))) - x^(1 - (x*(1 + 2*x))/(1 + x))*Hypergeometric
2F1[1, 1 - (x*(1 + 2*x))/(1 + x), 2 - (x*(1 + 2*x))/(1 + x), -x] - (x^(2 - (x*(1 + 2*x))/(1 + x))*(1 + 2*x)*Hy
pergeometric2F1[1, 1 - (x*(1 + 2*x))/(1 + x), 2 - (x*(1 + 2*x))/(1 + x), -x])/((1 + x)*(1 - (x*(1 + 2*x))/(1 +
x))) + (2*x^(1 + (1 - 2*x^2)/(1 + x))*(1 - 2*x^2)*(2 - (x*(1 + 2*x))/(1 + x))*Hypergeometric2F1[1, 1 + (1 - 2
*x^2)/(1 + x), 2 + (1 - 2*x^2)/(1 + x), -x])/((1 + x)*(1 - (x*(1 + 2*x))/(1 + x))*(1 + (1 - 2*x^2)/(1 + x))) +
(2*x^(1 + (2 + x - 2*x^2)/(1 + x))*(2 + x - 2*x^2)*(3 - (x*(1 + 2*x))/(1 + x))*Hypergeometric2F1[1, 1 + (2 +
x - 2*x^2)/(1 + x), 2 + (2 + x - 2*x^2)/(1 + x), -x])/((1 + x)*(2 - (x*(1 + 2*x))/(1 + x))*(1 + (2 + x - 2*x^2
)/(1 + x))) + 2*Log[x]*Defer[Int][x^((1 - 2*x^2)/(1 + x)), x] - Defer[Int][(x^((1 - 2*x^2)/(1 + x))*Log[x])/(1
+ x)^2, x] - 2*Defer[Int][Defer[Int][x^((1 - 2*x^2)/(1 + x)), x]/x, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {1+2 x+x^2+x^{\frac {-x-2 x^2}{1+x}} \left (-1-x+2 x^2+2 x^3+\left (x+4 x^2+2 x^3\right ) \log (x)\right )}{(1+x)^2} \, dx\\ &=\int \left (1+\frac {x^{-\frac {x (1+2 x)}{1+x}} \left (-1-x+2 x^2+2 x^3+x \log (x)+4 x^2 \log (x)+2 x^3 \log (x)\right )}{(1+x)^2}\right ) \, dx\\ &=x+\int \frac {x^{-\frac {x (1+2 x)}{1+x}} \left (-1-x+2 x^2+2 x^3+x \log (x)+4 x^2 \log (x)+2 x^3 \log (x)\right )}{(1+x)^2} \, dx\\ &=x+\int \left (-\frac {x^{-\frac {x (1+2 x)}{1+x}}}{(1+x)^2}-\frac {x^{1-\frac {x (1+2 x)}{1+x}}}{(1+x)^2}+\frac {2 x^{2-\frac {x (1+2 x)}{1+x}}}{(1+x)^2}+\frac {2 x^{3-\frac {x (1+2 x)}{1+x}}}{(1+x)^2}+\frac {x^{1-\frac {x (1+2 x)}{1+x}} \left (1+4 x+2 x^2\right ) \log (x)}{(1+x)^2}\right ) \, dx\\ &=x+2 \int \frac {x^{2-\frac {x (1+2 x)}{1+x}}}{(1+x)^2} \, dx+2 \int \frac {x^{3-\frac {x (1+2 x)}{1+x}}}{(1+x)^2} \, dx-\int \frac {x^{-\frac {x (1+2 x)}{1+x}}}{(1+x)^2} \, dx-\int \frac {x^{1-\frac {x (1+2 x)}{1+x}}}{(1+x)^2} \, dx+\int \frac {x^{1-\frac {x (1+2 x)}{1+x}} \left (1+4 x+2 x^2\right ) \log (x)}{(1+x)^2} \, dx\\ &=x-\frac {x^{1-\frac {x (1+2 x)}{1+x}}}{1+x}+\frac {x^{-\frac {x (1+2 x)}{1+x}}}{1+2 x}+\frac {2 x^{2-\frac {x (1+2 x)}{1+x}}}{(1+x) \left (1-\frac {x (1+2 x)}{1+x}\right )}+\frac {2 x^{3-\frac {x (1+2 x)}{1+x}}}{(1+x) \left (2-\frac {x (1+2 x)}{1+x}\right )}-\frac {(x (1+2 x)) \int \frac {x^{-\frac {x (1+2 x)}{1+x}}}{1+x} \, dx}{1+x}-\frac {\left ((1+x) \left (1-\frac {x (1+2 x)}{1+x}\right )\right ) \int \frac {x^{-\frac {x (1+2 x)}{1+x}}}{(1+x)^2} \, dx}{x (1+2 x)}-\frac {\left (2 \left (2-\frac {x (1+2 x)}{1+x}\right )\right ) \int \frac {x^{\frac {1-2 x^2}{1+x}}}{(1+x)^2} \, dx}{1-\frac {x (1+2 x)}{1+x}}-\frac {\left (2 \left (3-\frac {x (1+2 x)}{1+x}\right )\right ) \int \frac {x^{\frac {2+x-2 x^2}{1+x}}}{(1+x)^2} \, dx}{2-\frac {x (1+2 x)}{1+x}}+\int \frac {x^{\frac {1-2 x^2}{1+x}} \left (1+4 x+2 x^2\right ) \log (x)}{(1+x)^2} \, dx\\ &=x-\frac {x^{1-\frac {x (1+2 x)}{1+x}}}{1+x}+\frac {x^{-\frac {x (1+2 x)}{1+x}}}{1+2 x}+\frac {2 x^{2-\frac {x (1+2 x)}{1+x}}}{(1+x) \left (1-\frac {x (1+2 x)}{1+x}\right )}-\frac {x^{-\frac {x (1+2 x)}{1+x}} \left (1-\frac {x (1+2 x)}{1+x}\right )}{1+2 x}+\frac {2 x^{3-\frac {x (1+2 x)}{1+x}}}{(1+x) \left (2-\frac {x (1+2 x)}{1+x}\right )}-\frac {2 x^{1+\frac {1-2 x^2}{1+x}} \left (2-\frac {x (1+2 x)}{1+x}\right )}{(1+x) \left (1-\frac {x (1+2 x)}{1+x}\right )}-\frac {2 x^{1+\frac {2+x-2 x^2}{1+x}} \left (3-\frac {x (1+2 x)}{1+x}\right )}{(1+x) \left (2-\frac {x (1+2 x)}{1+x}\right )}-\frac {x^{2-\frac {x (1+2 x)}{1+x}} (1+2 x) \, _2F_1\left (1,1-\frac {x (1+2 x)}{1+x};2-\frac {x (1+2 x)}{1+x};-x\right )}{(1+x) \left (1-\frac {x (1+2 x)}{1+x}\right )}-\left (1-\frac {x (1+2 x)}{1+x}\right ) \int \frac {x^{-\frac {x (1+2 x)}{1+x}}}{1+x} \, dx+\frac {\left (2 \left (1-2 x^2\right ) \left (2-\frac {x (1+2 x)}{1+x}\right )\right ) \int \frac {x^{\frac {1-2 x^2}{1+x}}}{1+x} \, dx}{(1+x) \left (1-\frac {x (1+2 x)}{1+x}\right )}+\frac {\left (2 \left (2+x-2 x^2\right ) \left (3-\frac {x (1+2 x)}{1+x}\right )\right ) \int \frac {x^{\frac {2+x-2 x^2}{1+x}}}{1+x} \, dx}{(1+x) \left (2-\frac {x (1+2 x)}{1+x}\right )}+\int \left (2 x^{\frac {1-2 x^2}{1+x}} \log (x)-\frac {x^{\frac {1-2 x^2}{1+x}} \log (x)}{(1+x)^2}\right ) \, dx\\ &=x-\frac {x^{1-\frac {x (1+2 x)}{1+x}}}{1+x}+\frac {x^{-\frac {x (1+2 x)}{1+x}}}{1+2 x}+\frac {2 x^{2-\frac {x (1+2 x)}{1+x}}}{(1+x) \left (1-\frac {x (1+2 x)}{1+x}\right )}-\frac {x^{-\frac {x (1+2 x)}{1+x}} \left (1-\frac {x (1+2 x)}{1+x}\right )}{1+2 x}+\frac {2 x^{3-\frac {x (1+2 x)}{1+x}}}{(1+x) \left (2-\frac {x (1+2 x)}{1+x}\right )}-\frac {2 x^{1+\frac {1-2 x^2}{1+x}} \left (2-\frac {x (1+2 x)}{1+x}\right )}{(1+x) \left (1-\frac {x (1+2 x)}{1+x}\right )}-\frac {2 x^{1+\frac {2+x-2 x^2}{1+x}} \left (3-\frac {x (1+2 x)}{1+x}\right )}{(1+x) \left (2-\frac {x (1+2 x)}{1+x}\right )}-x^{1-\frac {x (1+2 x)}{1+x}} \, _2F_1\left (1,1-\frac {x (1+2 x)}{1+x};2-\frac {x (1+2 x)}{1+x};-x\right )-\frac {x^{2-\frac {x (1+2 x)}{1+x}} (1+2 x) \, _2F_1\left (1,1-\frac {x (1+2 x)}{1+x};2-\frac {x (1+2 x)}{1+x};-x\right )}{(1+x) \left (1-\frac {x (1+2 x)}{1+x}\right )}+\frac {2 x^{1+\frac {1-2 x^2}{1+x}} \left (1-2 x^2\right ) \left (2-\frac {x (1+2 x)}{1+x}\right ) \, _2F_1\left (1,1+\frac {1-2 x^2}{1+x};2+\frac {1-2 x^2}{1+x};-x\right )}{(1+x) \left (1-\frac {x (1+2 x)}{1+x}\right ) \left (1+\frac {1-2 x^2}{1+x}\right )}+\frac {2 x^{1+\frac {2+x-2 x^2}{1+x}} \left (2+x-2 x^2\right ) \left (3-\frac {x (1+2 x)}{1+x}\right ) \, _2F_1\left (1,1+\frac {2+x-2 x^2}{1+x};2+\frac {2+x-2 x^2}{1+x};-x\right )}{(1+x) \left (2-\frac {x (1+2 x)}{1+x}\right ) \left (1+\frac {2+x-2 x^2}{1+x}\right )}+2 \int x^{\frac {1-2 x^2}{1+x}} \log (x) \, dx-\int \frac {x^{\frac {1-2 x^2}{1+x}} \log (x)}{(1+x)^2} \, dx\\ &=x-\frac {x^{1-\frac {x (1+2 x)}{1+x}}}{1+x}+\frac {x^{-\frac {x (1+2 x)}{1+x}}}{1+2 x}+\frac {2 x^{2-\frac {x (1+2 x)}{1+x}}}{(1+x) \left (1-\frac {x (1+2 x)}{1+x}\right )}-\frac {x^{-\frac {x (1+2 x)}{1+x}} \left (1-\frac {x (1+2 x)}{1+x}\right )}{1+2 x}+\frac {2 x^{3-\frac {x (1+2 x)}{1+x}}}{(1+x) \left (2-\frac {x (1+2 x)}{1+x}\right )}-\frac {2 x^{1+\frac {1-2 x^2}{1+x}} \left (2-\frac {x (1+2 x)}{1+x}\right )}{(1+x) \left (1-\frac {x (1+2 x)}{1+x}\right )}-\frac {2 x^{1+\frac {2+x-2 x^2}{1+x}} \left (3-\frac {x (1+2 x)}{1+x}\right )}{(1+x) \left (2-\frac {x (1+2 x)}{1+x}\right )}-x^{1-\frac {x (1+2 x)}{1+x}} \, _2F_1\left (1,1-\frac {x (1+2 x)}{1+x};2-\frac {x (1+2 x)}{1+x};-x\right )-\frac {x^{2-\frac {x (1+2 x)}{1+x}} (1+2 x) \, _2F_1\left (1,1-\frac {x (1+2 x)}{1+x};2-\frac {x (1+2 x)}{1+x};-x\right )}{(1+x) \left (1-\frac {x (1+2 x)}{1+x}\right )}+\frac {2 x^{1+\frac {1-2 x^2}{1+x}} \left (1-2 x^2\right ) \left (2-\frac {x (1+2 x)}{1+x}\right ) \, _2F_1\left (1,1+\frac {1-2 x^2}{1+x};2+\frac {1-2 x^2}{1+x};-x\right )}{(1+x) \left (1-\frac {x (1+2 x)}{1+x}\right ) \left (1+\frac {1-2 x^2}{1+x}\right )}+\frac {2 x^{1+\frac {2+x-2 x^2}{1+x}} \left (2+x-2 x^2\right ) \left (3-\frac {x (1+2 x)}{1+x}\right ) \, _2F_1\left (1,1+\frac {2+x-2 x^2}{1+x};2+\frac {2+x-2 x^2}{1+x};-x\right )}{(1+x) \left (2-\frac {x (1+2 x)}{1+x}\right ) \left (1+\frac {2+x-2 x^2}{1+x}\right )}-2 \int \frac {\int x^{\frac {1-2 x^2}{1+x}} \, dx}{x} \, dx+(2 \log (x)) \int x^{\frac {1-2 x^2}{1+x}} \, dx-\int \frac {x^{\frac {1-2 x^2}{1+x}} \log (x)}{(1+x)^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.38, size = 19, normalized size = 0.83 \begin {gather*} x-x^{\frac {1-2 x^2}{1+x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + 2*x + x^2 + x^((-x - 2*x^2)/(1 + x))*(-1 - x + 2*x^2 + 2*x^3 + (x + 4*x^2 + 2*x^3)*Log[x]))/(1
+ 2*x + x^2),x]

[Out]

x - x^((1 - 2*x^2)/(1 + x))

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fricas [A]  time = 0.80, size = 39, normalized size = 1.70 \begin {gather*} \frac {x x^{\frac {2 \, x^{2} + x}{x + 1}} - x}{x^{\frac {2 \, x^{2} + x}{x + 1}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x^3+4*x^2+x)*log(x)+2*x^3+2*x^2-x-1)*exp((-2*x^2-x)*log(x)/(x+1))+x^2+2*x+1)/(x^2+2*x+1),x, alg
orithm="fricas")

[Out]

(x*x^((2*x^2 + x)/(x + 1)) - x)/x^((2*x^2 + x)/(x + 1))

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giac [A]  time = 1.11, size = 25, normalized size = 1.09 \begin {gather*} x - \frac {x^{\left (\frac {1}{x + 1}\right )}}{x^{\frac {2 \, x^{2}}{x + 1}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x^3+4*x^2+x)*log(x)+2*x^3+2*x^2-x-1)*exp((-2*x^2-x)*log(x)/(x+1))+x^2+2*x+1)/(x^2+2*x+1),x, alg
orithm="giac")

[Out]

x - x^(1/(x + 1))/x^(2*x^2/(x + 1))

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maple [A]  time = 0.11, size = 21, normalized size = 0.91

method	result	size
risch	\(-x \,x^{-\frac {\left (2 x +1\right ) x}{x +1}}+x\)	\(21\)
norman	\(\frac {x^{2}-x \,{\mathrm e}^{\frac {\left (-2 x^{2}-x \right ) \ln \relax (x )}{x +1}}-x^{2} {\mathrm e}^{\frac {\left (-2 x^{2}-x \right ) \ln \relax (x )}{x +1}}-1}{x +1}\)	\(56\)
default	\(x +\frac {-x \,{\mathrm e}^{\frac {\left (-2 x^{2}-x \right ) \ln \relax (x )}{x +1}}-2 x^{2} {\mathrm e}^{\frac {\left (-2 x^{2}-x \right ) \ln \relax (x )}{x +1}}-x^{3} {\mathrm e}^{\frac {\left (-2 x^{2}-x \right ) \ln \relax (x )}{x +1}}}{\left (x +1\right )^{2}}\)	\(77\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((2*x^3+4*x^2+x)*ln(x)+2*x^3+2*x^2-x-1)*exp((-2*x^2-x)*ln(x)/(x+1))+x^2+2*x+1)/(x^2+2*x+1),x,method=_RETU
RNVERBOSE)

[Out]

-x*x^(-(2*x+1)*x/(x+1))+x

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maxima [A]  time = 0.45, size = 23, normalized size = 1.00 \begin {gather*} -x^{2} e^{\left (-2 \, x \log \relax (x) - \frac {\log \relax (x)}{x + 1}\right )} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x^3+4*x^2+x)*log(x)+2*x^3+2*x^2-x-1)*exp((-2*x^2-x)*log(x)/(x+1))+x^2+2*x+1)/(x^2+2*x+1),x, alg
orithm="maxima")

[Out]

-x^2*e^(-2*x*log(x) - log(x)/(x + 1)) + x

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mupad [B]  time = 2.41, size = 22, normalized size = 0.96 \begin {gather*} -x\,\left (\frac {1}{x^{\frac {2\,x^2+x}{x+1}}}-1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x + exp(-(log(x)*(x + 2*x^2))/(x + 1))*(log(x)*(x + 4*x^2 + 2*x^3) - x + 2*x^2 + 2*x^3 - 1) + x^2 + 1)/
(2*x + x^2 + 1),x)

[Out]

-x*(1/x^((x + 2*x^2)/(x + 1)) - 1)

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sympy [A]  time = 159.28, size = 19, normalized size = 0.83 \begin {gather*} - x e^{\frac {\left (- 2 x^{2} - x\right ) \log {\relax (x )}}{x + 1}} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x*exp((-2*x**2 - x)*log(x)/(x + 1)) + x

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