3.49.7 \(\int \frac {e^{\frac {1}{16} (1-8 x+24 x^2-32 x^3+16 x^4+(-4+24 x-48 x^2+32 x^3) \log (\frac {x^2}{1+x})+(6-24 x+24 x^2) \log ^2(\frac {x^2}{1+x})+(-4+8 x) \log ^3(\frac {x^2}{1+x})+\log ^4(\frac {x^2}{1+x}))} (-2+9 x-8 x^2-8 x^3+16 x^5+(6-15 x-6 x^2+12 x^3+24 x^4) \log (\frac {x^2}{1+x})+(-6+3 x+12 x^2+12 x^3) \log ^2(\frac {x^2}{1+x})+(2+3 x+2 x^2) \log ^3(\frac {x^2}{1+x}))}{4 x+4 x^2} \, dx\)

Optimal. Leaf size=23 \[ e^{\frac {1}{16} \left (-1+2 x+\log \left (\frac {x^2}{1+x}\right )\right )^4} \]

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Rubi [F]  time = 62.22, 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 {\exp \left (\frac {1}{16} \left (1-8 x+24 x^2-32 x^3+16 x^4+\left (-4+24 x-48 x^2+32 x^3\right ) \log \left (\frac {x^2}{1+x}\right )+\left (6-24 x+24 x^2\right ) \log ^2\left (\frac {x^2}{1+x}\right )+(-4+8 x) \log ^3\left (\frac {x^2}{1+x}\right )+\log ^4\left (\frac {x^2}{1+x}\right )\right )\right ) \left (-2+9 x-8 x^2-8 x^3+16 x^5+\left (6-15 x-6 x^2+12 x^3+24 x^4\right ) \log \left (\frac {x^2}{1+x}\right )+\left (-6+3 x+12 x^2+12 x^3\right ) \log ^2\left (\frac {x^2}{1+x}\right )+\left (2+3 x+2 x^2\right ) \log ^3\left (\frac {x^2}{1+x}\right )\right )}{4 x+4 x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((1 - 8*x + 24*x^2 - 32*x^3 + 16*x^4 + (-4 + 24*x - 48*x^2 + 32*x^3)*Log[x^2/(1 + x)] + (6 - 24*x + 24*
x^2)*Log[x^2/(1 + x)]^2 + (-4 + 8*x)*Log[x^2/(1 + x)]^3 + Log[x^2/(1 + x)]^4)/16)*(-2 + 9*x - 8*x^2 - 8*x^3 +
16*x^5 + (6 - 15*x - 6*x^2 + 12*x^3 + 24*x^4)*Log[x^2/(1 + x)] + (-6 + 3*x + 12*x^2 + 12*x^3)*Log[x^2/(1 + x)]
^2 + (2 + 3*x + 2*x^2)*Log[x^2/(1 + x)]^3))/(4*x + 4*x^2),x]

[Out]

-2*Defer[Int][E^(((1 - 2*x)^4 + 6*(1 - 2*x)^2*Log[x^2/(1 + x)]^2 + (-4 + 8*x)*Log[x^2/(1 + x)]^3 + Log[x^2/(1
+ x)]^4)/16)*(x^2/(1 + x))^(3/4 + (3*x)/2 - 3*x^2 + 2*x^3), x] - Defer[Int][(E^(((1 - 2*x)^4 + 6*(1 - 2*x)^2*L
og[x^2/(1 + x)]^2 + (-4 + 8*x)*Log[x^2/(1 + x)]^3 + Log[x^2/(1 + x)]^4)/16)*(x^2/(1 + x))^(3/4 + (3*x)/2 - 3*x
^2 + 2*x^3))/x^3, x]/2 + (9*Defer[Int][(E^(((1 - 2*x)^4 + 6*(1 - 2*x)^2*Log[x^2/(1 + x)]^2 + (-4 + 8*x)*Log[x^
2/(1 + x)]^3 + Log[x^2/(1 + x)]^4)/16)*(x^2/(1 + x))^(3/4 + (3*x)/2 - 3*x^2 + 2*x^3))/x^2, x])/4 - 2*Defer[Int
][(E^(((1 - 2*x)^4 + 6*(1 - 2*x)^2*Log[x^2/(1 + x)]^2 + (-4 + 8*x)*Log[x^2/(1 + x)]^3 + Log[x^2/(1 + x)]^4)/16
)*(x^2/(1 + x))^(3/4 + (3*x)/2 - 3*x^2 + 2*x^3))/x, x] + 4*Defer[Int][E^(((1 - 2*x)^4 + 6*(1 - 2*x)^2*Log[x^2/
(1 + x)]^2 + (-4 + 8*x)*Log[x^2/(1 + x)]^3 + Log[x^2/(1 + x)]^4)/16)*x^2*(x^2/(1 + x))^(3/4 + (3*x)/2 - 3*x^2
+ 2*x^3), x] + 3*Defer[Int][E^(((1 - 2*x)^4 + 6*(1 - 2*x)^2*Log[x^2/(1 + x)]^2 + (-4 + 8*x)*Log[x^2/(1 + x)]^3
+ Log[x^2/(1 + x)]^4)/16)*(x^2/(1 + x))^(3/4 + (3*x)/2 - 3*x^2 + 2*x^3)*Log[x^2/(1 + x)], x] + (3*Defer[Int][
(E^(((1 - 2*x)^4 + 6*(1 - 2*x)^2*Log[x^2/(1 + x)]^2 + (-4 + 8*x)*Log[x^2/(1 + x)]^3 + Log[x^2/(1 + x)]^4)/16)*
(x^2/(1 + x))^(3/4 + (3*x)/2 - 3*x^2 + 2*x^3)*Log[x^2/(1 + x)])/x^3, x])/2 - (15*Defer[Int][(E^(((1 - 2*x)^4 +
6*(1 - 2*x)^2*Log[x^2/(1 + x)]^2 + (-4 + 8*x)*Log[x^2/(1 + x)]^3 + Log[x^2/(1 + x)]^4)/16)*(x^2/(1 + x))^(3/4
+ (3*x)/2 - 3*x^2 + 2*x^3)*Log[x^2/(1 + x)])/x^2, x])/4 - (3*Defer[Int][(E^(((1 - 2*x)^4 + 6*(1 - 2*x)^2*Log[
x^2/(1 + x)]^2 + (-4 + 8*x)*Log[x^2/(1 + x)]^3 + Log[x^2/(1 + x)]^4)/16)*(x^2/(1 + x))^(3/4 + (3*x)/2 - 3*x^2
+ 2*x^3)*Log[x^2/(1 + x)])/x, x])/2 + 6*Defer[Int][E^(((1 - 2*x)^4 + 6*(1 - 2*x)^2*Log[x^2/(1 + x)]^2 + (-4 +
8*x)*Log[x^2/(1 + x)]^3 + Log[x^2/(1 + x)]^4)/16)*x*(x^2/(1 + x))^(3/4 + (3*x)/2 - 3*x^2 + 2*x^3)*Log[x^2/(1 +
x)], x] + 3*Defer[Int][E^(((1 - 2*x)^4 + 6*(1 - 2*x)^2*Log[x^2/(1 + x)]^2 + (-4 + 8*x)*Log[x^2/(1 + x)]^3 + L
og[x^2/(1 + x)]^4)/16)*(x^2/(1 + x))^(3/4 + (3*x)/2 - 3*x^2 + 2*x^3)*Log[x^2/(1 + x)]^2, x] - (3*Defer[Int][(E
^(((1 - 2*x)^4 + 6*(1 - 2*x)^2*Log[x^2/(1 + x)]^2 + (-4 + 8*x)*Log[x^2/(1 + x)]^3 + Log[x^2/(1 + x)]^4)/16)*(x
^2/(1 + x))^(3/4 + (3*x)/2 - 3*x^2 + 2*x^3)*Log[x^2/(1 + x)]^2)/x^3, x])/2 + (3*Defer[Int][(E^(((1 - 2*x)^4 +
6*(1 - 2*x)^2*Log[x^2/(1 + x)]^2 + (-4 + 8*x)*Log[x^2/(1 + x)]^3 + Log[x^2/(1 + x)]^4)/16)*(x^2/(1 + x))^(3/4
+ (3*x)/2 - 3*x^2 + 2*x^3)*Log[x^2/(1 + x)]^2)/x^2, x])/4 + 3*Defer[Int][(E^(((1 - 2*x)^4 + 6*(1 - 2*x)^2*Log[
x^2/(1 + x)]^2 + (-4 + 8*x)*Log[x^2/(1 + x)]^3 + Log[x^2/(1 + x)]^4)/16)*(x^2/(1 + x))^(3/4 + (3*x)/2 - 3*x^2
+ 2*x^3)*Log[x^2/(1 + x)]^2)/x, x] + Defer[Int][(E^(((1 - 2*x)^4 + 6*(1 - 2*x)^2*Log[x^2/(1 + x)]^2 + (-4 + 8*
x)*Log[x^2/(1 + x)]^3 + Log[x^2/(1 + x)]^4)/16)*(x^2/(1 + x))^(3/4 + (3*x)/2 - 3*x^2 + 2*x^3)*Log[x^2/(1 + x)]
^3)/x^3, x]/2 + (3*Defer[Int][(E^(((1 - 2*x)^4 + 6*(1 - 2*x)^2*Log[x^2/(1 + x)]^2 + (-4 + 8*x)*Log[x^2/(1 + x)
]^3 + Log[x^2/(1 + x)]^4)/16)*(x^2/(1 + x))^(3/4 + (3*x)/2 - 3*x^2 + 2*x^3)*Log[x^2/(1 + x)]^3)/x^2, x])/4 + D
efer[Int][(E^(((1 - 2*x)^4 + 6*(1 - 2*x)^2*Log[x^2/(1 + x)]^2 + (-4 + 8*x)*Log[x^2/(1 + x)]^3 + Log[x^2/(1 + x
)]^4)/16)*(x^2/(1 + x))^(3/4 + (3*x)/2 - 3*x^2 + 2*x^3)*Log[x^2/(1 + x)]^3)/x, x]/2

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\exp \left (\frac {1}{16} \left (1-8 x+24 x^2-32 x^3+16 x^4+\left (-4+24 x-48 x^2+32 x^3\right ) \log \left (\frac {x^2}{1+x}\right )+\left (6-24 x+24 x^2\right ) \log ^2\left (\frac {x^2}{1+x}\right )+(-4+8 x) \log ^3\left (\frac {x^2}{1+x}\right )+\log ^4\left (\frac {x^2}{1+x}\right )\right )\right ) \left (-2+9 x-8 x^2-8 x^3+16 x^5+\left (6-15 x-6 x^2+12 x^3+24 x^4\right ) \log \left (\frac {x^2}{1+x}\right )+\left (-6+3 x+12 x^2+12 x^3\right ) \log ^2\left (\frac {x^2}{1+x}\right )+\left (2+3 x+2 x^2\right ) \log ^3\left (\frac {x^2}{1+x}\right )\right )}{x (4+4 x)} \, dx\\ &=\int \frac {\exp \left (\frac {1}{16} \left ((1-2 x)^4+6 (1-2 x)^2 \log ^2\left (\frac {x^2}{1+x}\right )+(-4+8 x) \log ^3\left (\frac {x^2}{1+x}\right )+\log ^4\left (\frac {x^2}{1+x}\right )\right )\right ) \left (\frac {x^2}{1+x}\right )^{\frac {3}{4}+\frac {3 x}{2}-3 x^2+2 x^3} \left (-2-3 x-2 x^2\right ) \left (1-2 x-\log \left (\frac {x^2}{1+x}\right )\right )^3}{4 x^3} \, dx\\ &=\frac {1}{4} \int \frac {\exp \left (\frac {1}{16} \left ((1-2 x)^4+6 (1-2 x)^2 \log ^2\left (\frac {x^2}{1+x}\right )+(-4+8 x) \log ^3\left (\frac {x^2}{1+x}\right )+\log ^4\left (\frac {x^2}{1+x}\right )\right )\right ) \left (\frac {x^2}{1+x}\right )^{\frac {3}{4}+\frac {3 x}{2}-3 x^2+2 x^3} \left (-2-3 x-2 x^2\right ) \left (1-2 x-\log \left (\frac {x^2}{1+x}\right )\right )^3}{x^3} \, dx\\ &=\frac {1}{4} \int \left (\frac {\exp \left (\frac {1}{16} \left ((1-2 x)^4+6 (1-2 x)^2 \log ^2\left (\frac {x^2}{1+x}\right )+(-4+8 x) \log ^3\left (\frac {x^2}{1+x}\right )+\log ^4\left (\frac {x^2}{1+x}\right )\right )\right ) \left (\frac {x^2}{1+x}\right )^{\frac {3}{4}+\frac {3 x}{2}-3 x^2+2 x^3} (-1+2 x)^3 \left (2+3 x+2 x^2\right )}{x^3}+\frac {3 \exp \left (\frac {1}{16} \left ((1-2 x)^4+6 (1-2 x)^2 \log ^2\left (\frac {x^2}{1+x}\right )+(-4+8 x) \log ^3\left (\frac {x^2}{1+x}\right )+\log ^4\left (\frac {x^2}{1+x}\right )\right )\right ) \left (\frac {x^2}{1+x}\right )^{\frac {3}{4}+\frac {3 x}{2}-3 x^2+2 x^3} (-1+2 x)^2 \left (2+3 x+2 x^2\right ) \log \left (\frac {x^2}{1+x}\right )}{x^3}+\frac {3 \exp \left (\frac {1}{16} \left ((1-2 x)^4+6 (1-2 x)^2 \log ^2\left (\frac {x^2}{1+x}\right )+(-4+8 x) \log ^3\left (\frac {x^2}{1+x}\right )+\log ^4\left (\frac {x^2}{1+x}\right )\right )\right ) \left (\frac {x^2}{1+x}\right )^{\frac {3}{4}+\frac {3 x}{2}-3 x^2+2 x^3} (-1+2 x) \left (2+3 x+2 x^2\right ) \log ^2\left (\frac {x^2}{1+x}\right )}{x^3}+\frac {\exp \left (\frac {1}{16} \left ((1-2 x)^4+6 (1-2 x)^2 \log ^2\left (\frac {x^2}{1+x}\right )+(-4+8 x) \log ^3\left (\frac {x^2}{1+x}\right )+\log ^4\left (\frac {x^2}{1+x}\right )\right )\right ) \left (\frac {x^2}{1+x}\right )^{\frac {3}{4}+\frac {3 x}{2}-3 x^2+2 x^3} \left (2+3 x+2 x^2\right ) \log ^3\left (\frac {x^2}{1+x}\right )}{x^3}\right ) \, dx\\ &=\frac {1}{4} \int \frac {\exp \left (\frac {1}{16} \left ((1-2 x)^4+6 (1-2 x)^2 \log ^2\left (\frac {x^2}{1+x}\right )+(-4+8 x) \log ^3\left (\frac {x^2}{1+x}\right )+\log ^4\left (\frac {x^2}{1+x}\right )\right )\right ) \left (\frac {x^2}{1+x}\right )^{\frac {3}{4}+\frac {3 x}{2}-3 x^2+2 x^3} (-1+2 x)^3 \left (2+3 x+2 x^2\right )}{x^3} \, dx+\frac {1}{4} \int \frac {\exp \left (\frac {1}{16} \left ((1-2 x)^4+6 (1-2 x)^2 \log ^2\left (\frac {x^2}{1+x}\right )+(-4+8 x) \log ^3\left (\frac {x^2}{1+x}\right )+\log ^4\left (\frac {x^2}{1+x}\right )\right )\right ) \left (\frac {x^2}{1+x}\right )^{\frac {3}{4}+\frac {3 x}{2}-3 x^2+2 x^3} \left (2+3 x+2 x^2\right ) \log ^3\left (\frac {x^2}{1+x}\right )}{x^3} \, dx+\frac {3}{4} \int \frac {\exp \left (\frac {1}{16} \left ((1-2 x)^4+6 (1-2 x)^2 \log ^2\left (\frac {x^2}{1+x}\right )+(-4+8 x) \log ^3\left (\frac {x^2}{1+x}\right )+\log ^4\left (\frac {x^2}{1+x}\right )\right )\right ) \left (\frac {x^2}{1+x}\right )^{\frac {3}{4}+\frac {3 x}{2}-3 x^2+2 x^3} (-1+2 x)^2 \left (2+3 x+2 x^2\right ) \log \left (\frac {x^2}{1+x}\right )}{x^3} \, dx+\frac {3}{4} \int \frac {\exp \left (\frac {1}{16} \left ((1-2 x)^4+6 (1-2 x)^2 \log ^2\left (\frac {x^2}{1+x}\right )+(-4+8 x) \log ^3\left (\frac {x^2}{1+x}\right )+\log ^4\left (\frac {x^2}{1+x}\right )\right )\right ) \left (\frac {x^2}{1+x}\right )^{\frac {3}{4}+\frac {3 x}{2}-3 x^2+2 x^3} (-1+2 x) \left (2+3 x+2 x^2\right ) \log ^2\left (\frac {x^2}{1+x}\right )}{x^3} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.39, size = 87, normalized size = 3.78 \begin {gather*} e^{\frac {1}{16} \left ((1-2 x)^4+6 (1-2 x)^2 \log ^2\left (\frac {x^2}{1+x}\right )+(-4+8 x) \log ^3\left (\frac {x^2}{1+x}\right )+\log ^4\left (\frac {x^2}{1+x}\right )\right )} \left (\frac {x^2}{1+x}\right )^{\frac {1}{4} (-1+2 x)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((1 - 8*x + 24*x^2 - 32*x^3 + 16*x^4 + (-4 + 24*x - 48*x^2 + 32*x^3)*Log[x^2/(1 + x)] + (6 - 24*x
+ 24*x^2)*Log[x^2/(1 + x)]^2 + (-4 + 8*x)*Log[x^2/(1 + x)]^3 + Log[x^2/(1 + x)]^4)/16)*(-2 + 9*x - 8*x^2 - 8*
x^3 + 16*x^5 + (6 - 15*x - 6*x^2 + 12*x^3 + 24*x^4)*Log[x^2/(1 + x)] + (-6 + 3*x + 12*x^2 + 12*x^3)*Log[x^2/(1
+ x)]^2 + (2 + 3*x + 2*x^2)*Log[x^2/(1 + x)]^3))/(4*x + 4*x^2),x]

[Out]

E^(((1 - 2*x)^4 + 6*(1 - 2*x)^2*Log[x^2/(1 + x)]^2 + (-4 + 8*x)*Log[x^2/(1 + x)]^3 + Log[x^2/(1 + x)]^4)/16)*(
x^2/(1 + x))^((-1 + 2*x)^3/4)

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fricas [B]  time = 0.85, size = 103, normalized size = 4.48 \begin {gather*} e^{\left (x^{4} + \frac {1}{4} \, {\left (2 \, x - 1\right )} \log \left (\frac {x^{2}}{x + 1}\right )^{3} + \frac {1}{16} \, \log \left (\frac {x^{2}}{x + 1}\right )^{4} - 2 \, x^{3} + \frac {3}{8} \, {\left (4 \, x^{2} - 4 \, x + 1\right )} \log \left (\frac {x^{2}}{x + 1}\right )^{2} + \frac {3}{2} \, x^{2} + \frac {1}{4} \, {\left (8 \, x^{3} - 12 \, x^{2} + 6 \, x - 1\right )} \log \left (\frac {x^{2}}{x + 1}\right ) - \frac {1}{2} \, x + \frac {1}{16}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^2+3*x+2)*log(x^2/(x+1))^3+(12*x^3+12*x^2+3*x-6)*log(x^2/(x+1))^2+(24*x^4+12*x^3-6*x^2-15*x+6)*
log(x^2/(x+1))+16*x^5-8*x^3-8*x^2+9*x-2)*exp(1/16*log(x^2/(x+1))^4+1/16*(8*x-4)*log(x^2/(x+1))^3+1/16*(24*x^2-
24*x+6)*log(x^2/(x+1))^2+1/16*(32*x^3-48*x^2+24*x-4)*log(x^2/(x+1))+x^4-2*x^3+3/2*x^2-1/2*x+1/16)/(4*x^2+4*x),
x, algorithm="fricas")

[Out]

e^(x^4 + 1/4*(2*x - 1)*log(x^2/(x + 1))^3 + 1/16*log(x^2/(x + 1))^4 - 2*x^3 + 3/8*(4*x^2 - 4*x + 1)*log(x^2/(x
+ 1))^2 + 3/2*x^2 + 1/4*(8*x^3 - 12*x^2 + 6*x - 1)*log(x^2/(x + 1)) - 1/2*x + 1/16)

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giac [B]  time = 18.45, size = 163, normalized size = 7.09 \begin {gather*} e^{\left (x^{4} + 2 \, x^{3} \log \left (\frac {x^{2}}{x + 1}\right ) + \frac {3}{2} \, x^{2} \log \left (\frac {x^{2}}{x + 1}\right )^{2} + \frac {1}{2} \, x \log \left (\frac {x^{2}}{x + 1}\right )^{3} + \frac {1}{16} \, \log \left (\frac {x^{2}}{x + 1}\right )^{4} - 2 \, x^{3} - 3 \, x^{2} \log \left (\frac {x^{2}}{x + 1}\right ) - \frac {3}{2} \, x \log \left (\frac {x^{2}}{x + 1}\right )^{2} - \frac {1}{4} \, \log \left (\frac {x^{2}}{x + 1}\right )^{3} + \frac {3}{2} \, x^{2} + \frac {3}{2} \, x \log \left (\frac {x^{2}}{x + 1}\right ) + \frac {3}{8} \, \log \left (\frac {x^{2}}{x + 1}\right )^{2} - \frac {1}{2} \, x - \frac {1}{4} \, \log \left (\frac {x^{2}}{x + 1}\right ) + \frac {1}{16}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^2+3*x+2)*log(x^2/(x+1))^3+(12*x^3+12*x^2+3*x-6)*log(x^2/(x+1))^2+(24*x^4+12*x^3-6*x^2-15*x+6)*
log(x^2/(x+1))+16*x^5-8*x^3-8*x^2+9*x-2)*exp(1/16*log(x^2/(x+1))^4+1/16*(8*x-4)*log(x^2/(x+1))^3+1/16*(24*x^2-
24*x+6)*log(x^2/(x+1))^2+1/16*(32*x^3-48*x^2+24*x-4)*log(x^2/(x+1))+x^4-2*x^3+3/2*x^2-1/2*x+1/16)/(4*x^2+4*x),
x, algorithm="giac")

[Out]

e^(x^4 + 2*x^3*log(x^2/(x + 1)) + 3/2*x^2*log(x^2/(x + 1))^2 + 1/2*x*log(x^2/(x + 1))^3 + 1/16*log(x^2/(x + 1)
)^4 - 2*x^3 - 3*x^2*log(x^2/(x + 1)) - 3/2*x*log(x^2/(x + 1))^2 - 1/4*log(x^2/(x + 1))^3 + 3/2*x^2 + 3/2*x*log
(x^2/(x + 1)) + 3/8*log(x^2/(x + 1))^2 - 1/2*x - 1/4*log(x^2/(x + 1)) + 1/16)

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maple [B]  time = 0.27, size = 129, normalized size = 5.61

method	result	size
risch	\(\left (\frac {x^{2}}{x +1}\right )^{\frac {\left (2 x -1\right )^{3}}{4}} {\mathrm e}^{\frac {\ln \left (\frac {x^{2}}{x +1}\right )^{4}}{16}+\frac {\ln \left (\frac {x^{2}}{x +1}\right )^{3} x}{2}-\frac {\ln \left (\frac {x^{2}}{x +1}\right )^{3}}{4}+\frac {3 \ln \left (\frac {x^{2}}{x +1}\right )^{2} x^{2}}{2}-\frac {3 \ln \left (\frac {x^{2}}{x +1}\right )^{2} x}{2}+\frac {3 \ln \left (\frac {x^{2}}{x +1}\right )^{2}}{8}+\frac {1}{16}+x^{4}-2 x^{3}+\frac {3 x^{2}}{2}-\frac {x}{2}}\)	\(129\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x^2+3*x+2)*ln(x^2/(x+1))^3+(12*x^3+12*x^2+3*x-6)*ln(x^2/(x+1))^2+(24*x^4+12*x^3-6*x^2-15*x+6)*ln(x^2/(
x+1))+16*x^5-8*x^3-8*x^2+9*x-2)*exp(1/16*ln(x^2/(x+1))^4+1/16*(8*x-4)*ln(x^2/(x+1))^3+1/16*(24*x^2-24*x+6)*ln(
x^2/(x+1))^2+1/16*(32*x^3-48*x^2+24*x-4)*ln(x^2/(x+1))+x^4-2*x^3+3/2*x^2-1/2*x+1/16)/(4*x^2+4*x),x,method=_RET
URNVERBOSE)

[Out]

(x^2/(x+1))^(1/4*(2*x-1)^3)*exp(1/16*ln(x^2/(x+1))^4+1/2*ln(x^2/(x+1))^3*x-1/4*ln(x^2/(x+1))^3+3/2*ln(x^2/(x+1
))^2*x^2-3/2*ln(x^2/(x+1))^2*x+3/8*ln(x^2/(x+1))^2+1/16+x^4-2*x^3+3/2*x^2-1/2*x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {1}{4} \, \int \frac {{\left (16 \, x^{5} + {\left (2 \, x^{2} + 3 \, x + 2\right )} \log \left (\frac {x^{2}}{x + 1}\right )^{3} - 8 \, x^{3} + 3 \, {\left (4 \, x^{3} + 4 \, x^{2} + x - 2\right )} \log \left (\frac {x^{2}}{x + 1}\right )^{2} - 8 \, x^{2} + 3 \, {\left (8 \, x^{4} + 4 \, x^{3} - 2 \, x^{2} - 5 \, x + 2\right )} \log \left (\frac {x^{2}}{x + 1}\right ) + 9 \, x - 2\right )} e^{\left (x^{4} + \frac {1}{4} \, {\left (2 \, x - 1\right )} \log \left (\frac {x^{2}}{x + 1}\right )^{3} + \frac {1}{16} \, \log \left (\frac {x^{2}}{x + 1}\right )^{4} - 2 \, x^{3} + \frac {3}{8} \, {\left (4 \, x^{2} - 4 \, x + 1\right )} \log \left (\frac {x^{2}}{x + 1}\right )^{2} + \frac {3}{2} \, x^{2} + \frac {1}{4} \, {\left (8 \, x^{3} - 12 \, x^{2} + 6 \, x - 1\right )} \log \left (\frac {x^{2}}{x + 1}\right ) - \frac {1}{2} \, x + \frac {1}{16}\right )}}{x^{2} + x}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^2+3*x+2)*log(x^2/(x+1))^3+(12*x^3+12*x^2+3*x-6)*log(x^2/(x+1))^2+(24*x^4+12*x^3-6*x^2-15*x+6)*
log(x^2/(x+1))+16*x^5-8*x^3-8*x^2+9*x-2)*exp(1/16*log(x^2/(x+1))^4+1/16*(8*x-4)*log(x^2/(x+1))^3+1/16*(24*x^2-
24*x+6)*log(x^2/(x+1))^2+1/16*(32*x^3-48*x^2+24*x-4)*log(x^2/(x+1))+x^4-2*x^3+3/2*x^2-1/2*x+1/16)/(4*x^2+4*x),
x, algorithm="maxima")

[Out]

1/4*integrate((16*x^5 + (2*x^2 + 3*x + 2)*log(x^2/(x + 1))^3 - 8*x^3 + 3*(4*x^3 + 4*x^2 + x - 2)*log(x^2/(x +
1))^2 - 8*x^2 + 3*(8*x^4 + 4*x^3 - 2*x^2 - 5*x + 2)*log(x^2/(x + 1)) + 9*x - 2)*e^(x^4 + 1/4*(2*x - 1)*log(x^2
/(x + 1))^3 + 1/16*log(x^2/(x + 1))^4 - 2*x^3 + 3/8*(4*x^2 - 4*x + 1)*log(x^2/(x + 1))^2 + 3/2*x^2 + 1/4*(8*x^
3 - 12*x^2 + 6*x - 1)*log(x^2/(x + 1)) - 1/2*x + 1/16)/(x^2 + x), x)

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mupad [B]  time = 3.81, size = 338, normalized size = 14.70 \begin {gather*} {\mathrm {e}}^{\frac {\ln \left (x^2\right )\,{\ln \left (\frac {1}{x+1}\right )}^3}{4}}\,{\mathrm {e}}^{\frac {{\ln \left (x^2\right )}^3\,\ln \left (\frac {1}{x+1}\right )}{4}}\,{\mathrm {e}}^{-\frac {3\,\ln \left (x^2\right )\,{\ln \left (\frac {1}{x+1}\right )}^2}{4}}\,{\mathrm {e}}^{-\frac {3\,{\ln \left (x^2\right )}^2\,\ln \left (\frac {1}{x+1}\right )}{4}}\,{\mathrm {e}}^{-\frac {x}{2}}\,{\mathrm {e}}^{x^4}\,{\mathrm {e}}^{\frac {x\,{\ln \left (\frac {1}{x+1}\right )}^3}{2}}\,{\mathrm {e}}^{-\frac {3\,x\,{\ln \left (\frac {1}{x+1}\right )}^2}{2}}\,{\mathrm {e}}^{\frac {3\,x^2\,{\ln \left (x^2\right )}^2}{2}}\,{\mathrm {e}}^{1/16}\,{\mathrm {e}}^{-\frac {{\ln \left (x^2\right )}^3}{4}}\,{\mathrm {e}}^{\frac {3\,{\ln \left (x^2\right )}^2}{8}}\,{\mathrm {e}}^{\frac {{\ln \left (x^2\right )}^4}{16}}\,{\mathrm {e}}^{\frac {3\,{\ln \left (x^2\right )}^2\,{\ln \left (\frac {1}{x+1}\right )}^2}{8}}\,{\mathrm {e}}^{\frac {3\,x\,\ln \left (x^2\right )\,{\ln \left (\frac {1}{x+1}\right )}^2}{2}}\,{\mathrm {e}}^{\frac {3\,x\,{\ln \left (x^2\right )}^2\,\ln \left (\frac {1}{x+1}\right )}{2}}\,{\mathrm {e}}^{-2\,x^3}\,{\mathrm {e}}^{\frac {3\,x^2}{2}}\,{\mathrm {e}}^{\frac {3\,x^2\,{\ln \left (\frac {1}{x+1}\right )}^2}{2}}\,{\mathrm {e}}^{\frac {3\,\ln \left (x^2\right )\,\ln \left (\frac {1}{x+1}\right )}{4}}\,{\mathrm {e}}^{-\frac {{\ln \left (\frac {1}{x+1}\right )}^3}{4}}\,{\mathrm {e}}^{\frac {3\,{\ln \left (\frac {1}{x+1}\right )}^2}{8}}\,{\mathrm {e}}^{\frac {{\ln \left (\frac {1}{x+1}\right )}^4}{16}}\,{\mathrm {e}}^{\frac {3\,x\,\ln \left (\frac {1}{x+1}\right )}{2}}\,{\mathrm {e}}^{\frac {x\,{\ln \left (x^2\right )}^3}{2}}\,{\mathrm {e}}^{-\frac {3\,x\,{\ln \left (x^2\right )}^2}{2}}\,{\left (\frac {1}{x+1}\right )}^{3\,x^2\,\ln \left (x^2\right )-3\,x\,\ln \left (x^2\right )-3\,x^2+2\,x^3-\frac {1}{4}}\,{\left (x^2\right )}^{2\,x^3-3\,x^2+\frac {3\,x}{2}-\frac {1}{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp((log(x^2/(x + 1))^2*(24*x^2 - 24*x + 6))/16 - x/2 + log(x^2/(x + 1))^4/16 + (log(x^2/(x + 1))^3*(8*x
- 4))/16 + (3*x^2)/2 - 2*x^3 + x^4 + (log(x^2/(x + 1))*(24*x - 48*x^2 + 32*x^3 - 4))/16 + 1/16)*(9*x + log(x^2
/(x + 1))^3*(3*x + 2*x^2 + 2) + log(x^2/(x + 1))*(12*x^3 - 6*x^2 - 15*x + 24*x^4 + 6) + log(x^2/(x + 1))^2*(3*
x + 12*x^2 + 12*x^3 - 6) - 8*x^2 - 8*x^3 + 16*x^5 - 2))/(4*x + 4*x^2),x)

[Out]

exp((log(x^2)*log(1/(x + 1))^3)/4)*exp((log(x^2)^3*log(1/(x + 1)))/4)*exp(-(3*log(x^2)*log(1/(x + 1))^2)/4)*ex
p(-(3*log(x^2)^2*log(1/(x + 1)))/4)*exp(-x/2)*exp(x^4)*exp((x*log(1/(x + 1))^3)/2)*exp(-(3*x*log(1/(x + 1))^2)
/2)*exp((3*x^2*log(x^2)^2)/2)*exp(1/16)*exp(-log(x^2)^3/4)*exp((3*log(x^2)^2)/8)*exp(log(x^2)^4/16)*exp((3*log
(x^2)^2*log(1/(x + 1))^2)/8)*exp((3*x*log(x^2)*log(1/(x + 1))^2)/2)*exp((3*x*log(x^2)^2*log(1/(x + 1)))/2)*exp
(-2*x^3)*exp((3*x^2)/2)*exp((3*x^2*log(1/(x + 1))^2)/2)*exp((3*log(x^2)*log(1/(x + 1)))/4)*exp(-log(1/(x + 1))
^3/4)*exp((3*log(1/(x + 1))^2)/8)*exp(log(1/(x + 1))^4/16)*exp((3*x*log(1/(x + 1)))/2)*exp((x*log(x^2)^3)/2)*e
xp(-(3*x*log(x^2)^2)/2)*(1/(x + 1))^(3*x^2*log(x^2) - 3*x*log(x^2) - 3*x^2 + 2*x^3 - 1/4)*(x^2)^((3*x)/2 - 3*x
^2 + 2*x^3 - 1/4)

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sympy [B]  time = 1.08, size = 104, normalized size = 4.52 \begin {gather*} e^{x^{4} - 2 x^{3} + \frac {3 x^{2}}{2} - \frac {x}{2} + \left (\frac {x}{2} - \frac {1}{4}\right ) \log {\left (\frac {x^{2}}{x + 1} \right )}^{3} + \left (\frac {3 x^{2}}{2} - \frac {3 x}{2} + \frac {3}{8}\right ) \log {\left (\frac {x^{2}}{x + 1} \right )}^{2} + \left (2 x^{3} - 3 x^{2} + \frac {3 x}{2} - \frac {1}{4}\right ) \log {\left (\frac {x^{2}}{x + 1} \right )} + \frac {\log {\left (\frac {x^{2}}{x + 1} \right )}^{4}}{16} + \frac {1}{16}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x**2+3*x+2)*ln(x**2/(x+1))**3+(12*x**3+12*x**2+3*x-6)*ln(x**2/(x+1))**2+(24*x**4+12*x**3-6*x**2-
15*x+6)*ln(x**2/(x+1))+16*x**5-8*x**3-8*x**2+9*x-2)*exp(1/16*ln(x**2/(x+1))**4+1/16*(8*x-4)*ln(x**2/(x+1))**3+
1/16*(24*x**2-24*x+6)*ln(x**2/(x+1))**2+1/16*(32*x**3-48*x**2+24*x-4)*ln(x**2/(x+1))+x**4-2*x**3+3/2*x**2-1/2*
x+1/16)/(4*x**2+4*x),x)

[Out]

exp(x**4 - 2*x**3 + 3*x**2/2 - x/2 + (x/2 - 1/4)*log(x**2/(x + 1))**3 + (3*x**2/2 - 3*x/2 + 3/8)*log(x**2/(x +
1))**2 + (2*x**3 - 3*x**2 + 3*x/2 - 1/4)*log(x**2/(x + 1)) + log(x**2/(x + 1))**4/16 + 1/16)

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