3.14.37 \(\int \frac {-14 x-16 x^2-4 x^3+\frac {e^{2 x} (-2 e^x-x)}{(2 e^x+x)^2}+e^x (-28-32 x-8 x^2)+\frac {e^x (1+7 x+4 x^2+e^x (16+8 x))}{2 e^x+x}}{16 x+16 x^2+4 x^3+\frac {e^{2 x}}{2 e^x+x}+\frac {e^x (e^x (-16-8 x)-8 x-4 x^2)}{2 e^x+x}+e^x (32+32 x+8 x^2)} \, dx\)

Optimal. Leaf size=28 \[ 40-x-\frac {1}{4+2 x-\frac {e^x}{2 e^x+x}} \]

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Rubi [F]  time = 2.01, 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 {-14 x-16 x^2-4 x^3+\frac {e^{2 x} \left (-2 e^x-x\right )}{\left (2 e^x+x\right )^2}+e^x \left (-28-32 x-8 x^2\right )+\frac {e^x \left (1+7 x+4 x^2+e^x (16+8 x)\right )}{2 e^x+x}}{16 x+16 x^2+4 x^3+\frac {e^{2 x}}{2 e^x+x}+\frac {e^x \left (e^x (-16-8 x)-8 x-4 x^2\right )}{2 e^x+x}+e^x \left (32+32 x+8 x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-14*x - 16*x^2 - 4*x^3 + (E^(2*x)*(-2*E^x - x))/(2*E^x + x)^2 + E^x*(-28 - 32*x - 8*x^2) + (E^x*(1 + 7*x
+ 4*x^2 + E^x*(16 + 8*x)))/(2*E^x + x))/(16*x + 16*x^2 + 4*x^3 + E^(2*x)/(2*E^x + x) + (E^x*(E^x*(-16 - 8*x) -
8*x - 4*x^2))/(2*E^x + x) + E^x*(32 + 32*x + 8*x^2)),x]

[Out]

-x - 2/(7 + 4*x) - (7*Defer[Int][(7*E^x + 4*x + 4*E^x*x + 2*x^2)^(-2), x])/32 - (3*Defer[Int][x/(7*E^x + 4*x +
4*E^x*x + 2*x^2)^2, x])/8 + Defer[Int][x^2/(7*E^x + 4*x + 4*E^x*x + 2*x^2)^2, x]/2 + (49*Defer[Int][1/((7 + 4
*x)^2*(7*E^x + 4*x + 4*E^x*x + 2*x^2)^2), x])/8 + (21*Defer[Int][1/((7 + 4*x)*(7*E^x + 4*x + 4*E^x*x + 2*x^2)^
2), x])/32 - Defer[Int][(7*E^x + 4*x + 4*E^x*x + 2*x^2)^(-1), x]/4 + 14*Defer[Int][1/((7 + 4*x)^2*(7*E^x + 4*x
+ 4*E^x*x + 2*x^2)), x] + (3*Defer[Int][1/((7 + 4*x)*(7*E^x + 4*x + 4*E^x*x + 2*x^2)), x])/4

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-2 x^2 \left (7+8 x+2 x^2\right )-e^{2 x} \left (41+56 x+16 x^2\right )-e^x \left (-1+49 x+60 x^2+16 x^3\right )}{\left (2 x (2+x)+e^x (7+4 x)\right )^2} \, dx\\ &=\int \left (\frac {-41-56 x-16 x^2}{(7+4 x)^2}-\frac {-7+11 x+4 x^2}{(7+4 x)^2 \left (7 e^x+4 x+4 e^x x+2 x^2\right )}+\frac {2 x \left (-14+11 x^2+4 x^3\right )}{(7+4 x)^2 \left (7 e^x+4 x+4 e^x x+2 x^2\right )^2}\right ) \, dx\\ &=2 \int \frac {x \left (-14+11 x^2+4 x^3\right )}{(7+4 x)^2 \left (7 e^x+4 x+4 e^x x+2 x^2\right )^2} \, dx+\int \frac {-41-56 x-16 x^2}{(7+4 x)^2} \, dx-\int \frac {-7+11 x+4 x^2}{(7+4 x)^2 \left (7 e^x+4 x+4 e^x x+2 x^2\right )} \, dx\\ &=2 \int \left (-\frac {7}{64 \left (7 e^x+4 x+4 e^x x+2 x^2\right )^2}-\frac {3 x}{16 \left (7 e^x+4 x+4 e^x x+2 x^2\right )^2}+\frac {x^2}{4 \left (7 e^x+4 x+4 e^x x+2 x^2\right )^2}+\frac {49}{16 (7+4 x)^2 \left (7 e^x+4 x+4 e^x x+2 x^2\right )^2}+\frac {21}{64 (7+4 x) \left (7 e^x+4 x+4 e^x x+2 x^2\right )^2}\right ) \, dx+\int \left (-1+\frac {8}{(7+4 x)^2}\right ) \, dx-\int \left (\frac {1}{4 \left (7 e^x+4 x+4 e^x x+2 x^2\right )}-\frac {14}{(7+4 x)^2 \left (7 e^x+4 x+4 e^x x+2 x^2\right )}-\frac {3}{4 (7+4 x) \left (7 e^x+4 x+4 e^x x+2 x^2\right )}\right ) \, dx\\ &=-x-\frac {2}{7+4 x}-\frac {7}{32} \int \frac {1}{\left (7 e^x+4 x+4 e^x x+2 x^2\right )^2} \, dx-\frac {1}{4} \int \frac {1}{7 e^x+4 x+4 e^x x+2 x^2} \, dx-\frac {3}{8} \int \frac {x}{\left (7 e^x+4 x+4 e^x x+2 x^2\right )^2} \, dx+\frac {1}{2} \int \frac {x^2}{\left (7 e^x+4 x+4 e^x x+2 x^2\right )^2} \, dx+\frac {21}{32} \int \frac {1}{(7+4 x) \left (7 e^x+4 x+4 e^x x+2 x^2\right )^2} \, dx+\frac {3}{4} \int \frac {1}{(7+4 x) \left (7 e^x+4 x+4 e^x x+2 x^2\right )} \, dx+\frac {49}{8} \int \frac {1}{(7+4 x)^2 \left (7 e^x+4 x+4 e^x x+2 x^2\right )^2} \, dx+14 \int \frac {1}{(7+4 x)^2 \left (7 e^x+4 x+4 e^x x+2 x^2\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.46, size = 40, normalized size = 1.43 \begin {gather*} -x-\frac {2}{7+4 x}+\frac {x}{(7+4 x) \left (2 x (2+x)+e^x (7+4 x)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-14*x - 16*x^2 - 4*x^3 + (E^(2*x)*(-2*E^x - x))/(2*E^x + x)^2 + E^x*(-28 - 32*x - 8*x^2) + (E^x*(1
+ 7*x + 4*x^2 + E^x*(16 + 8*x)))/(2*E^x + x))/(16*x + 16*x^2 + 4*x^3 + E^(2*x)/(2*E^x + x) + (E^x*(E^x*(-16 -
8*x) - 8*x - 4*x^2))/(2*E^x + x) + E^x*(32 + 32*x + 8*x^2)),x]

[Out]

-x - 2/(7 + 4*x) + x/((7 + 4*x)*(2*x*(2 + x) + E^x*(7 + 4*x)))

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fricas [A]  time = 0.77, size = 46, normalized size = 1.64 \begin {gather*} -\frac {2 \, x^{3} + 4 \, x^{2} + {\left (4 \, x^{2} + 7 \, x + 2\right )} e^{x} + x}{2 \, x^{2} + {\left (4 \, x + 7\right )} e^{x} + 4 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*exp(x)-x)*exp(-log(2*exp(x)+x)+x)^2+((8*x+16)*exp(x)+4*x^2+7*x+1)*exp(-log(2*exp(x)+x)+x)+(-8*x
^2-32*x-28)*exp(x)-4*x^3-16*x^2-14*x)/((2*exp(x)+x)*exp(-log(2*exp(x)+x)+x)^2+((-8*x-16)*exp(x)-4*x^2-8*x)*exp
(-log(2*exp(x)+x)+x)+(8*x^2+32*x+32)*exp(x)+4*x^3+16*x^2+16*x),x, algorithm="fricas")

[Out]

-(2*x^3 + 4*x^2 + (4*x^2 + 7*x + 2)*e^x + x)/(2*x^2 + (4*x + 7)*e^x + 4*x)

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giac [A]  time = 1.37, size = 50, normalized size = 1.79 \begin {gather*} -\frac {2 \, x^{3} + 4 \, x^{2} e^{x} + 4 \, x^{2} + 7 \, x e^{x} + x + 2 \, e^{x}}{2 \, x^{2} + 4 \, x e^{x} + 4 \, x + 7 \, e^{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*exp(x)-x)*exp(-log(2*exp(x)+x)+x)^2+((8*x+16)*exp(x)+4*x^2+7*x+1)*exp(-log(2*exp(x)+x)+x)+(-8*x
^2-32*x-28)*exp(x)-4*x^3-16*x^2-14*x)/((2*exp(x)+x)*exp(-log(2*exp(x)+x)+x)^2+((-8*x-16)*exp(x)-4*x^2-8*x)*exp
(-log(2*exp(x)+x)+x)+(8*x^2+32*x+32)*exp(x)+4*x^3+16*x^2+16*x),x, algorithm="giac")

[Out]

-(2*x^3 + 4*x^2*e^x + 4*x^2 + 7*x*e^x + x + 2*e^x)/(2*x^2 + 4*x*e^x + 4*x + 7*e^x)

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maple [A]  time = 0.12, size = 41, normalized size = 1.46

method	result	size
risch	\(-x -\frac {1}{2 \left (x +\frac {7}{4}\right )}+\frac {x}{\left (4 x +7\right ) \left (2 x^{2}+4 \,{\mathrm e}^{x} x +4 x +7 \,{\mathrm e}^{x}\right )}\)	\(41\)
norman	\(\frac {-\frac {52 x^{3}}{15}-\frac {4 \,{\mathrm e}^{2 x}}{15}+\frac {x^{2}}{15}-\frac {193 \,{\mathrm e}^{x} x^{2}}{15}-\frac {178 x \,{\mathrm e}^{2 x}}{15}-2 x^{4}-8 \,{\mathrm e}^{x} x^{3}-8 \,{\mathrm e}^{2 x} x^{2}}{\left (2 \,{\mathrm e}^{x}+x \right ) \left (2 x^{2}+4 \,{\mathrm e}^{x} x +4 x +7 \,{\mathrm e}^{x}\right )}\)	\(82\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-2*exp(x)-x)*exp(-ln(2*exp(x)+x)+x)^2+((8*x+16)*exp(x)+4*x^2+7*x+1)*exp(-ln(2*exp(x)+x)+x)+(-8*x^2-32*x-
28)*exp(x)-4*x^3-16*x^2-14*x)/((2*exp(x)+x)*exp(-ln(2*exp(x)+x)+x)^2+((-8*x-16)*exp(x)-4*x^2-8*x)*exp(-ln(2*ex
p(x)+x)+x)+(8*x^2+32*x+32)*exp(x)+4*x^3+16*x^2+16*x),x,method=_RETURNVERBOSE)

[Out]

-x-1/2/(x+7/4)+x/(4*x+7)/(2*x^2+4*exp(x)*x+4*x+7*exp(x))

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maxima [A]  time = 0.61, size = 46, normalized size = 1.64 \begin {gather*} -\frac {2 \, x^{3} + 4 \, x^{2} + {\left (4 \, x^{2} + 7 \, x + 2\right )} e^{x} + x}{2 \, x^{2} + {\left (4 \, x + 7\right )} e^{x} + 4 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*exp(x)-x)*exp(-log(2*exp(x)+x)+x)^2+((8*x+16)*exp(x)+4*x^2+7*x+1)*exp(-log(2*exp(x)+x)+x)+(-8*x
^2-32*x-28)*exp(x)-4*x^3-16*x^2-14*x)/((2*exp(x)+x)*exp(-log(2*exp(x)+x)+x)^2+((-8*x-16)*exp(x)-4*x^2-8*x)*exp
(-log(2*exp(x)+x)+x)+(8*x^2+32*x+32)*exp(x)+4*x^3+16*x^2+16*x),x, algorithm="maxima")

[Out]

-(2*x^3 + 4*x^2 + (4*x^2 + 7*x + 2)*e^x + x)/(2*x^2 + (4*x + 7)*e^x + 4*x)

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mupad [B]  time = 1.14, size = 32, normalized size = 1.14 \begin {gather*} -x-\frac {x+2\,{\mathrm {e}}^x}{4\,x+7\,{\mathrm {e}}^x+4\,x\,{\mathrm {e}}^x+2\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(14*x - exp(x - log(x + 2*exp(x)))*(7*x + exp(x)*(8*x + 16) + 4*x^2 + 1) + exp(x)*(32*x + 8*x^2 + 28) + 1
6*x^2 + 4*x^3 + exp(2*x - 2*log(x + 2*exp(x)))*(x + 2*exp(x)))/(16*x - exp(x - log(x + 2*exp(x)))*(8*x + exp(x
)*(8*x + 16) + 4*x^2) + exp(x)*(32*x + 8*x^2 + 32) + 16*x^2 + 4*x^3 + exp(2*x - 2*log(x + 2*exp(x)))*(x + 2*ex
p(x))),x)

[Out]

- x - (x + 2*exp(x))/(4*x + 7*exp(x) + 4*x*exp(x) + 2*x^2)

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sympy [A]  time = 0.36, size = 36, normalized size = 1.29 \begin {gather*} - x + \frac {x}{8 x^{3} + 30 x^{2} + 28 x + \left (16 x^{2} + 56 x + 49\right ) e^{x}} - \frac {2}{4 x + 7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*exp(x)-x)*exp(-ln(2*exp(x)+x)+x)**2+((8*x+16)*exp(x)+4*x**2+7*x+1)*exp(-ln(2*exp(x)+x)+x)+(-8*x
**2-32*x-28)*exp(x)-4*x**3-16*x**2-14*x)/((2*exp(x)+x)*exp(-ln(2*exp(x)+x)+x)**2+((-8*x-16)*exp(x)-4*x**2-8*x)
*exp(-ln(2*exp(x)+x)+x)+(8*x**2+32*x+32)*exp(x)+4*x**3+16*x**2+16*x),x)

[Out]

-x + x/(8*x**3 + 30*x**2 + 28*x + (16*x**2 + 56*x + 49)*exp(x)) - 2/(4*x + 7)

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