3.1.14 \(\int \frac {8 x+15 x^3+2 x^5+e^{5 x} (-2 x^2-2 x^3-4 x^4-4 x^5)+e^{4 x} (-6 x^3-6 x^4-12 x^5-12 x^6)+e^x (8 x-3 x^2+16 x^3+10 x^4+8 x^6)+e^{3 x} (-1+4 x+2 x^2+8 x^3-6 x^4-6 x^5-12 x^6-12 x^7)+e^{2 x} (-3 x+4 x^2+10 x^3+8 x^4+6 x^5-2 x^6-4 x^7-4 x^8)}{x^4+2 x^6+e^{3 x} (x+2 x^3)+e^{2 x} (3 x^2+6 x^4)+e^x (3 x^3+6 x^5)} \, dx\)

Optimal. Leaf size=36 \[ -\left (e^x x-\frac {2}{e^x+x}\right )^2+\log \left (-x+\frac {1}{5} \left (-\frac {2}{x}+x\right )\right ) \]

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Rubi [F]  time = 2.39, 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 {8 x+15 x^3+2 x^5+e^{5 x} \left (-2 x^2-2 x^3-4 x^4-4 x^5\right )+e^{4 x} \left (-6 x^3-6 x^4-12 x^5-12 x^6\right )+e^x \left (8 x-3 x^2+16 x^3+10 x^4+8 x^6\right )+e^{3 x} \left (-1+4 x+2 x^2+8 x^3-6 x^4-6 x^5-12 x^6-12 x^7\right )+e^{2 x} \left (-3 x+4 x^2+10 x^3+8 x^4+6 x^5-2 x^6-4 x^7-4 x^8\right )}{x^4+2 x^6+e^{3 x} \left (x+2 x^3\right )+e^{2 x} \left (3 x^2+6 x^4\right )+e^x \left (3 x^3+6 x^5\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(8*x + 15*x^3 + 2*x^5 + E^(5*x)*(-2*x^2 - 2*x^3 - 4*x^4 - 4*x^5) + E^(4*x)*(-6*x^3 - 6*x^4 - 12*x^5 - 12*x
^6) + E^x*(8*x - 3*x^2 + 16*x^3 + 10*x^4 + 8*x^6) + E^(3*x)*(-1 + 4*x + 2*x^2 + 8*x^3 - 6*x^4 - 6*x^5 - 12*x^6
- 12*x^7) + E^(2*x)*(-3*x + 4*x^2 + 10*x^3 + 8*x^4 + 6*x^5 - 2*x^6 - 4*x^7 - 4*x^8))/(x^4 + 2*x^6 + E^(3*x)*(
x + 2*x^3) + E^(2*x)*(3*x^2 + 6*x^4) + E^x*(3*x^3 + 6*x^5)),x]

[Out]

4*x - E^(2*x)*x^2 - 4/(E^x + x)^2 - Log[x] + Log[1 + 2*x^2] + 4*Defer[Int][x^2/(E^x + x)^2, x] - 4*Defer[Int][
x^3/(E^x + x)^2, x] - 8*Defer[Int][x/(E^x + x), x] + 4*Defer[Int][x^2/(E^x + x), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {8 x+15 x^3+2 x^5+e^{5 x} \left (-2 x^2-2 x^3-4 x^4-4 x^5\right )+e^{4 x} \left (-6 x^3-6 x^4-12 x^5-12 x^6\right )+e^x \left (8 x-3 x^2+16 x^3+10 x^4+8 x^6\right )+e^{3 x} \left (-1+4 x+2 x^2+8 x^3-6 x^4-6 x^5-12 x^6-12 x^7\right )+e^{2 x} \left (-3 x+4 x^2+10 x^3+8 x^4+6 x^5-2 x^6-4 x^7-4 x^8\right )}{x \left (e^x+x\right )^3 \left (1+2 x^2\right )} \, dx\\ &=\int \left (-2 e^{2 x} x (1+x)-\frac {8 (-1+x)}{\left (e^x+x\right )^3}+\frac {4 (-2+x) x}{e^x+x}-\frac {4 \left (-2-x^2+x^3\right )}{\left (e^x+x\right )^2}+\frac {-1+4 x+2 x^2+8 x^3}{x \left (1+2 x^2\right )}\right ) \, dx\\ &=-\left (2 \int e^{2 x} x (1+x) \, dx\right )+4 \int \frac {(-2+x) x}{e^x+x} \, dx-4 \int \frac {-2-x^2+x^3}{\left (e^x+x\right )^2} \, dx-8 \int \frac {-1+x}{\left (e^x+x\right )^3} \, dx+\int \frac {-1+4 x+2 x^2+8 x^3}{x \left (1+2 x^2\right )} \, dx\\ &=-\left (2 \int \left (e^{2 x} x+e^{2 x} x^2\right ) \, dx\right )-4 \int \left (-\frac {2}{\left (e^x+x\right )^2}-\frac {x^2}{\left (e^x+x\right )^2}+\frac {x^3}{\left (e^x+x\right )^2}\right ) \, dx+4 \int \left (-\frac {2 x}{e^x+x}+\frac {x^2}{e^x+x}\right ) \, dx-8 \int \left (-\frac {1}{\left (e^x+x\right )^3}+\frac {x}{\left (e^x+x\right )^3}\right ) \, dx+\int \left (4-\frac {1}{x}+\frac {4 x}{1+2 x^2}\right ) \, dx\\ &=4 x-\log (x)-2 \int e^{2 x} x \, dx-2 \int e^{2 x} x^2 \, dx+4 \int \frac {x^2}{\left (e^x+x\right )^2} \, dx-4 \int \frac {x^3}{\left (e^x+x\right )^2} \, dx+4 \int \frac {x^2}{e^x+x} \, dx+4 \int \frac {x}{1+2 x^2} \, dx+8 \int \frac {1}{\left (e^x+x\right )^3} \, dx-8 \int \frac {x}{\left (e^x+x\right )^3} \, dx+8 \int \frac {1}{\left (e^x+x\right )^2} \, dx-8 \int \frac {x}{e^x+x} \, dx\\ &=4 x-e^{2 x} x-e^{2 x} x^2-\frac {4}{\left (e^x+x\right )^2}-\log (x)+\log \left (1+2 x^2\right )+2 \int e^{2 x} x \, dx+4 \int \frac {x^2}{\left (e^x+x\right )^2} \, dx-4 \int \frac {x^3}{\left (e^x+x\right )^2} \, dx+4 \int \frac {x^2}{e^x+x} \, dx-8 \int \frac {x}{e^x+x} \, dx+\int e^{2 x} \, dx\\ &=\frac {e^{2 x}}{2}+4 x-e^{2 x} x^2-\frac {4}{\left (e^x+x\right )^2}-\log (x)+\log \left (1+2 x^2\right )+4 \int \frac {x^2}{\left (e^x+x\right )^2} \, dx-4 \int \frac {x^3}{\left (e^x+x\right )^2} \, dx+4 \int \frac {x^2}{e^x+x} \, dx-8 \int \frac {x}{e^x+x} \, dx-\int e^{2 x} \, dx\\ &=4 x-e^{2 x} x^2-\frac {4}{\left (e^x+x\right )^2}-\log (x)+\log \left (1+2 x^2\right )+4 \int \frac {x^2}{\left (e^x+x\right )^2} \, dx-4 \int \frac {x^3}{\left (e^x+x\right )^2} \, dx+4 \int \frac {x^2}{e^x+x} \, dx-8 \int \frac {x}{e^x+x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.11, size = 40, normalized size = 1.11 \begin {gather*} -\frac {\left (-2+e^{2 x} x+e^x x^2\right )^2}{\left (e^x+x\right )^2}-\log (x)+\log \left (1+2 x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(8*x + 15*x^3 + 2*x^5 + E^(5*x)*(-2*x^2 - 2*x^3 - 4*x^4 - 4*x^5) + E^(4*x)*(-6*x^3 - 6*x^4 - 12*x^5
- 12*x^6) + E^x*(8*x - 3*x^2 + 16*x^3 + 10*x^4 + 8*x^6) + E^(3*x)*(-1 + 4*x + 2*x^2 + 8*x^3 - 6*x^4 - 6*x^5 -
12*x^6 - 12*x^7) + E^(2*x)*(-3*x + 4*x^2 + 10*x^3 + 8*x^4 + 6*x^5 - 2*x^6 - 4*x^7 - 4*x^8))/(x^4 + 2*x^6 + E^(
3*x)*(x + 2*x^3) + E^(2*x)*(3*x^2 + 6*x^4) + E^x*(3*x^3 + 6*x^5)),x]

[Out]

-((-2 + E^(2*x)*x + E^x*x^2)^2/(E^x + x)^2) - Log[x] + Log[1 + 2*x^2]

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fricas [B]  time = 0.65, size = 94, normalized size = 2.61 \begin {gather*} -\frac {2 \, x^{3} e^{\left (3 \, x\right )} + x^{2} e^{\left (4 \, x\right )} - 4 \, x^{2} e^{x} + {\left (x^{4} - 4 \, x\right )} e^{\left (2 \, x\right )} - {\left (x^{2} + 2 \, x e^{x} + e^{\left (2 \, x\right )}\right )} \log \left (2 \, x^{2} + 1\right ) + {\left (x^{2} + 2 \, x e^{x} + e^{\left (2 \, x\right )}\right )} \log \relax (x) + 4}{x^{2} + 2 \, x e^{x} + e^{\left (2 \, x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x^5-4*x^4-2*x^3-2*x^2)*exp(x)^5+(-12*x^6-12*x^5-6*x^4-6*x^3)*exp(x)^4+(-12*x^7-12*x^6-6*x^5-6*x
^4+8*x^3+2*x^2+4*x-1)*exp(x)^3+(-4*x^8-4*x^7-2*x^6+6*x^5+8*x^4+10*x^3+4*x^2-3*x)*exp(x)^2+(8*x^6+10*x^4+16*x^3
-3*x^2+8*x)*exp(x)+2*x^5+15*x^3+8*x)/((2*x^3+x)*exp(x)^3+(6*x^4+3*x^2)*exp(x)^2+(6*x^5+3*x^3)*exp(x)+2*x^6+x^4
),x, algorithm="fricas")

[Out]

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

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giac [B]  time = 0.28, size = 118, normalized size = 3.28 \begin {gather*} -\frac {x^{4} e^{\left (2 \, x\right )} + 2 \, x^{3} e^{\left (3 \, x\right )} + x^{2} e^{\left (4 \, x\right )} - 4 \, x^{2} e^{x} - x^{2} \log \left (2 \, x^{2} + 1\right ) - 2 \, x e^{x} \log \left (2 \, x^{2} + 1\right ) + x^{2} \log \relax (x) + 2 \, x e^{x} \log \relax (x) - 4 \, x e^{\left (2 \, x\right )} - e^{\left (2 \, x\right )} \log \left (2 \, x^{2} + 1\right ) + e^{\left (2 \, x\right )} \log \relax (x) + 4}{x^{2} + 2 \, x e^{x} + e^{\left (2 \, x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x^5-4*x^4-2*x^3-2*x^2)*exp(x)^5+(-12*x^6-12*x^5-6*x^4-6*x^3)*exp(x)^4+(-12*x^7-12*x^6-6*x^5-6*x
^4+8*x^3+2*x^2+4*x-1)*exp(x)^3+(-4*x^8-4*x^7-2*x^6+6*x^5+8*x^4+10*x^3+4*x^2-3*x)*exp(x)^2+(8*x^6+10*x^4+16*x^3
-3*x^2+8*x)*exp(x)+2*x^5+15*x^3+8*x)/((2*x^3+x)*exp(x)^3+(6*x^4+3*x^2)*exp(x)^2+(6*x^5+3*x^3)*exp(x)+2*x^6+x^4
),x, algorithm="giac")

[Out]

-(x^4*e^(2*x) + 2*x^3*e^(3*x) + x^2*e^(4*x) - 4*x^2*e^x - x^2*log(2*x^2 + 1) - 2*x*e^x*log(2*x^2 + 1) + x^2*lo
g(x) + 2*x*e^x*log(x) - 4*x*e^(2*x) - e^(2*x)*log(2*x^2 + 1) + e^(2*x)*log(x) + 4)/(x^2 + 2*x*e^x + e^(2*x))

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maple [A]  time = 0.10, size = 45, normalized size = 1.25

method	result	size
risch	\(4 x -\ln \relax (x )+\ln \left (2 x^{2}+1\right )-{\mathrm e}^{2 x} x^{2}-\frac {4 \left (x^{3}+{\mathrm e}^{x} x^{2}+1\right )}{\left ({\mathrm e}^{x}+x \right )^{2}}\)	\(45\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-4*x^5-4*x^4-2*x^3-2*x^2)*exp(x)^5+(-12*x^6-12*x^5-6*x^4-6*x^3)*exp(x)^4+(-12*x^7-12*x^6-6*x^5-6*x^4+8*x
^3+2*x^2+4*x-1)*exp(x)^3+(-4*x^8-4*x^7-2*x^6+6*x^5+8*x^4+10*x^3+4*x^2-3*x)*exp(x)^2+(8*x^6+10*x^4+16*x^3-3*x^2
+8*x)*exp(x)+2*x^5+15*x^3+8*x)/((2*x^3+x)*exp(x)^3+(6*x^4+3*x^2)*exp(x)^2+(6*x^5+3*x^3)*exp(x)+2*x^6+x^4),x,me
thod=_RETURNVERBOSE)

[Out]

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

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maxima [B]  time = 0.60, size = 68, normalized size = 1.89 \begin {gather*} -\frac {2 \, x^{3} e^{\left (3 \, x\right )} + x^{2} e^{\left (4 \, x\right )} - 4 \, x^{2} e^{x} + {\left (x^{4} - 4 \, x\right )} e^{\left (2 \, x\right )} + 4}{x^{2} + 2 \, x e^{x} + e^{\left (2 \, x\right )}} + \log \left (2 \, x^{2} + 1\right ) - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x^5-4*x^4-2*x^3-2*x^2)*exp(x)^5+(-12*x^6-12*x^5-6*x^4-6*x^3)*exp(x)^4+(-12*x^7-12*x^6-6*x^5-6*x
^4+8*x^3+2*x^2+4*x-1)*exp(x)^3+(-4*x^8-4*x^7-2*x^6+6*x^5+8*x^4+10*x^3+4*x^2-3*x)*exp(x)^2+(8*x^6+10*x^4+16*x^3
-3*x^2+8*x)*exp(x)+2*x^5+15*x^3+8*x)/((2*x^3+x)*exp(x)^3+(6*x^4+3*x^2)*exp(x)^2+(6*x^5+3*x^3)*exp(x)+2*x^6+x^4
),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.45, size = 62, normalized size = 1.72 \begin {gather*} 4\,x+\ln \left (x^2+\frac {1}{2}\right )-\ln \relax (x)-x^2\,{\mathrm {e}}^{2\,x}-\frac {4}{{\mathrm {e}}^{2\,x}+2\,x\,{\mathrm {e}}^x+x^2}+\frac {4\,\left (x^2-x^3\right )}{\left (x+{\mathrm {e}}^x\right )\,\left (x-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((8*x - exp(3*x)*(6*x^4 - 2*x^2 - 8*x^3 - 4*x + 6*x^5 + 12*x^6 + 12*x^7 + 1) + exp(x)*(8*x - 3*x^2 + 16*x^3
+ 10*x^4 + 8*x^6) - exp(2*x)*(3*x - 4*x^2 - 10*x^3 - 8*x^4 - 6*x^5 + 2*x^6 + 4*x^7 + 4*x^8) - exp(5*x)*(2*x^2
+ 2*x^3 + 4*x^4 + 4*x^5) - exp(4*x)*(6*x^3 + 6*x^4 + 12*x^5 + 12*x^6) + 15*x^3 + 2*x^5)/(exp(x)*(3*x^3 + 6*x^
5) + exp(2*x)*(3*x^2 + 6*x^4) + exp(3*x)*(x + 2*x^3) + x^4 + 2*x^6),x)

[Out]

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

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sympy [B]  time = 0.28, size = 54, normalized size = 1.50 \begin {gather*} - x^{2} e^{2 x} + 4 x - \log {\relax (x )} + \log {\left (2 x^{2} + 1 \right )} + \frac {- 4 x^{3} - 4 x^{2} e^{x} - 4}{x^{2} + 2 x e^{x} + e^{2 x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x**5-4*x**4-2*x**3-2*x**2)*exp(x)**5+(-12*x**6-12*x**5-6*x**4-6*x**3)*exp(x)**4+(-12*x**7-12*x*
*6-6*x**5-6*x**4+8*x**3+2*x**2+4*x-1)*exp(x)**3+(-4*x**8-4*x**7-2*x**6+6*x**5+8*x**4+10*x**3+4*x**2-3*x)*exp(x
)**2+(8*x**6+10*x**4+16*x**3-3*x**2+8*x)*exp(x)+2*x**5+15*x**3+8*x)/((2*x**3+x)*exp(x)**3+(6*x**4+3*x**2)*exp(
x)**2+(6*x**5+3*x**3)*exp(x)+2*x**6+x**4),x)

[Out]

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

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