3.82.78 \(\int \frac {-3+e^{25+10 x+x^2+16 e^{2 x} x^2+e^x (40 x+8 x^2)} (-1+10 x+2 x^2+e^x (40 x+56 x^2+8 x^3)+e^{2 x} (32 x^2+32 x^3))}{9+6 e^{25+10 x+x^2+16 e^{2 x} x^2+e^x (40 x+8 x^2)}+e^{50+20 x+2 x^2+32 e^{2 x} x^2+2 e^x (40 x+8 x^2)}} \, dx\)

Optimal. Leaf size=22 \[ 9-\frac {x}{3+e^{\left (5+x+4 e^x x\right )^2}} \]

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Rubi [F]  time = 8.32, 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 {-3+\exp \left (25+10 x+x^2+16 e^{2 x} x^2+e^x \left (40 x+8 x^2\right )\right ) \left (-1+10 x+2 x^2+e^x \left (40 x+56 x^2+8 x^3\right )+e^{2 x} \left (32 x^2+32 x^3\right )\right )}{9+6 \exp \left (25+10 x+x^2+16 e^{2 x} x^2+e^x \left (40 x+8 x^2\right )\right )+\exp \left (50+20 x+2 x^2+32 e^{2 x} x^2+2 e^x \left (40 x+8 x^2\right )\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-3 + E^(25 + 10*x + x^2 + 16*E^(2*x)*x^2 + E^x*(40*x + 8*x^2))*(-1 + 10*x + 2*x^2 + E^x*(40*x + 56*x^2 +
8*x^3) + E^(2*x)*(32*x^2 + 32*x^3)))/(9 + 6*E^(25 + 10*x + x^2 + 16*E^(2*x)*x^2 + E^x*(40*x + 8*x^2)) + E^(50
+ 20*x + 2*x^2 + 32*E^(2*x)*x^2 + 2*E^x*(40*x + 8*x^2))),x]

[Out]

-Defer[Int][(3 + E^(5 + x + 4*E^x*x)^2)^(-1), x] - 30*Defer[Int][x/(3 + E^(5 + x + 4*E^x*x)^2)^2, x] - 120*Def
er[Int][(E^x*x)/(3 + E^(5 + x + 4*E^x*x)^2)^2, x] + 10*Defer[Int][x/(3 + E^(5 + x + 4*E^x*x)^2), x] + 40*Defer
[Int][(E^x*x)/(3 + E^(5 + x + 4*E^x*x)^2), x] - 6*Defer[Int][x^2/(3 + E^(5 + x + 4*E^x*x)^2)^2, x] - 168*Defer
[Int][(E^x*x^2)/(3 + E^(5 + x + 4*E^x*x)^2)^2, x] - 96*Defer[Int][(E^(2*x)*x^2)/(3 + E^(5 + x + 4*E^x*x)^2)^2,
x] + 2*Defer[Int][x^2/(3 + E^(5 + x + 4*E^x*x)^2), x] + 56*Defer[Int][(E^x*x^2)/(3 + E^(5 + x + 4*E^x*x)^2),
x] + 32*Defer[Int][(E^(2*x)*x^2)/(3 + E^(5 + x + 4*E^x*x)^2), x] - 24*Defer[Int][(E^x*x^3)/(3 + E^(5 + x + 4*E
^x*x)^2)^2, x] - 96*Defer[Int][(E^(2*x)*x^3)/(3 + E^(5 + x + 4*E^x*x)^2)^2, x] + 8*Defer[Int][(E^x*x^3)/(3 + E
^(5 + x + 4*E^x*x)^2), x] + 32*Defer[Int][(E^(2*x)*x^3)/(3 + E^(5 + x + 4*E^x*x)^2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-3+e^{\left (5+x+4 e^x x\right )^2} \left (-1+10 x+2 x^2+32 e^{2 x} x^2 (1+x)+8 e^x x \left (5+7 x+x^2\right )\right )}{\left (3+e^{\left (5+x+4 e^x x\right )^2}\right )^2} \, dx\\ &=\int \left (-\frac {6 x \left (5+20 e^x+x+28 e^x x+16 e^{2 x} x+4 e^x x^2+16 e^{2 x} x^2\right )}{\left (3+e^{\left (5+x+4 e^x x\right )^2}\right )^2}+\frac {-1+10 x+40 e^x x+2 x^2+56 e^x x^2+32 e^{2 x} x^2+8 e^x x^3+32 e^{2 x} x^3}{3+e^{\left (5+x+4 e^x x\right )^2}}\right ) \, dx\\ &=-\left (6 \int \frac {x \left (5+20 e^x+x+28 e^x x+16 e^{2 x} x+4 e^x x^2+16 e^{2 x} x^2\right )}{\left (3+e^{\left (5+x+4 e^x x\right )^2}\right )^2} \, dx\right )+\int \frac {-1+10 x+40 e^x x+2 x^2+56 e^x x^2+32 e^{2 x} x^2+8 e^x x^3+32 e^{2 x} x^3}{3+e^{\left (5+x+4 e^x x\right )^2}} \, dx\\ &=-\left (6 \int \frac {x \left (5+x+16 e^{2 x} x (1+x)+4 e^x \left (5+7 x+x^2\right )\right )}{\left (3+e^{\left (5+x+4 e^x x\right )^2}\right )^2} \, dx\right )+\int \left (-\frac {1}{3+e^{\left (5+x+4 e^x x\right )^2}}+\frac {10 x}{3+e^{\left (5+x+4 e^x x\right )^2}}+\frac {40 e^x x}{3+e^{\left (5+x+4 e^x x\right )^2}}+\frac {2 x^2}{3+e^{\left (5+x+4 e^x x\right )^2}}+\frac {56 e^x x^2}{3+e^{\left (5+x+4 e^x x\right )^2}}+\frac {32 e^{2 x} x^2}{3+e^{\left (5+x+4 e^x x\right )^2}}+\frac {8 e^x x^3}{3+e^{\left (5+x+4 e^x x\right )^2}}+\frac {32 e^{2 x} x^3}{3+e^{\left (5+x+4 e^x x\right )^2}}\right ) \, dx\\ &=2 \int \frac {x^2}{3+e^{\left (5+x+4 e^x x\right )^2}} \, dx-6 \int \left (\frac {5 x}{\left (3+e^{\left (5+x+4 e^x x\right )^2}\right )^2}+\frac {20 e^x x}{\left (3+e^{\left (5+x+4 e^x x\right )^2}\right )^2}+\frac {x^2}{\left (3+e^{\left (5+x+4 e^x x\right )^2}\right )^2}+\frac {28 e^x x^2}{\left (3+e^{\left (5+x+4 e^x x\right )^2}\right )^2}+\frac {16 e^{2 x} x^2}{\left (3+e^{\left (5+x+4 e^x x\right )^2}\right )^2}+\frac {4 e^x x^3}{\left (3+e^{\left (5+x+4 e^x x\right )^2}\right )^2}+\frac {16 e^{2 x} x^3}{\left (3+e^{\left (5+x+4 e^x x\right )^2}\right )^2}\right ) \, dx+8 \int \frac {e^x x^3}{3+e^{\left (5+x+4 e^x x\right )^2}} \, dx+10 \int \frac {x}{3+e^{\left (5+x+4 e^x x\right )^2}} \, dx+32 \int \frac {e^{2 x} x^2}{3+e^{\left (5+x+4 e^x x\right )^2}} \, dx+32 \int \frac {e^{2 x} x^3}{3+e^{\left (5+x+4 e^x x\right )^2}} \, dx+40 \int \frac {e^x x}{3+e^{\left (5+x+4 e^x x\right )^2}} \, dx+56 \int \frac {e^x x^2}{3+e^{\left (5+x+4 e^x x\right )^2}} \, dx-\int \frac {1}{3+e^{\left (5+x+4 e^x x\right )^2}} \, dx\\ &=2 \int \frac {x^2}{3+e^{\left (5+x+4 e^x x\right )^2}} \, dx-6 \int \frac {x^2}{\left (3+e^{\left (5+x+4 e^x x\right )^2}\right )^2} \, dx+8 \int \frac {e^x x^3}{3+e^{\left (5+x+4 e^x x\right )^2}} \, dx+10 \int \frac {x}{3+e^{\left (5+x+4 e^x x\right )^2}} \, dx-24 \int \frac {e^x x^3}{\left (3+e^{\left (5+x+4 e^x x\right )^2}\right )^2} \, dx-30 \int \frac {x}{\left (3+e^{\left (5+x+4 e^x x\right )^2}\right )^2} \, dx+32 \int \frac {e^{2 x} x^2}{3+e^{\left (5+x+4 e^x x\right )^2}} \, dx+32 \int \frac {e^{2 x} x^3}{3+e^{\left (5+x+4 e^x x\right )^2}} \, dx+40 \int \frac {e^x x}{3+e^{\left (5+x+4 e^x x\right )^2}} \, dx+56 \int \frac {e^x x^2}{3+e^{\left (5+x+4 e^x x\right )^2}} \, dx-96 \int \frac {e^{2 x} x^2}{\left (3+e^{\left (5+x+4 e^x x\right )^2}\right )^2} \, dx-96 \int \frac {e^{2 x} x^3}{\left (3+e^{\left (5+x+4 e^x x\right )^2}\right )^2} \, dx-120 \int \frac {e^x x}{\left (3+e^{\left (5+x+4 e^x x\right )^2}\right )^2} \, dx-168 \int \frac {e^x x^2}{\left (3+e^{\left (5+x+4 e^x x\right )^2}\right )^2} \, dx-\int \frac {1}{3+e^{\left (5+x+4 e^x x\right )^2}} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 1.20, size = 20, normalized size = 0.91 \begin {gather*} -\frac {x}{3+e^{\left (5+x+4 e^x x\right )^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-3 + E^(25 + 10*x + x^2 + 16*E^(2*x)*x^2 + E^x*(40*x + 8*x^2))*(-1 + 10*x + 2*x^2 + E^x*(40*x + 56*
x^2 + 8*x^3) + E^(2*x)*(32*x^2 + 32*x^3)))/(9 + 6*E^(25 + 10*x + x^2 + 16*E^(2*x)*x^2 + E^x*(40*x + 8*x^2)) +
E^(50 + 20*x + 2*x^2 + 32*E^(2*x)*x^2 + 2*E^x*(40*x + 8*x^2))),x]

[Out]

-(x/(3 + E^(5 + x + 4*E^x*x)^2))

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fricas [A]  time = 0.60, size = 36, normalized size = 1.64 \begin {gather*} -\frac {x}{e^{\left (16 \, x^{2} e^{\left (2 \, x\right )} + x^{2} + 8 \, {\left (x^{2} + 5 \, x\right )} e^{x} + 10 \, x + 25\right )} + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((32*x^3+32*x^2)*exp(x)^2+(8*x^3+56*x^2+40*x)*exp(x)+2*x^2+10*x-1)*exp(16*exp(x)^2*x^2+(8*x^2+40*x)
*exp(x)+x^2+10*x+25)-3)/(exp(16*exp(x)^2*x^2+(8*x^2+40*x)*exp(x)+x^2+10*x+25)^2+6*exp(16*exp(x)^2*x^2+(8*x^2+4
0*x)*exp(x)+x^2+10*x+25)+9),x, algorithm="fricas")

[Out]

-x/(e^(16*x^2*e^(2*x) + x^2 + 8*(x^2 + 5*x)*e^x + 10*x + 25) + 3)

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giac [A]  time = 0.68, size = 37, normalized size = 1.68 \begin {gather*} -\frac {x}{e^{\left (16 \, x^{2} e^{\left (2 \, x\right )} + 8 \, x^{2} e^{x} + x^{2} + 40 \, x e^{x} + 10 \, x + 25\right )} + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((32*x^3+32*x^2)*exp(x)^2+(8*x^3+56*x^2+40*x)*exp(x)+2*x^2+10*x-1)*exp(16*exp(x)^2*x^2+(8*x^2+40*x)
*exp(x)+x^2+10*x+25)-3)/(exp(16*exp(x)^2*x^2+(8*x^2+40*x)*exp(x)+x^2+10*x+25)^2+6*exp(16*exp(x)^2*x^2+(8*x^2+4
0*x)*exp(x)+x^2+10*x+25)+9),x, algorithm="giac")

[Out]

-x/(e^(16*x^2*e^(2*x) + 8*x^2*e^x + x^2 + 40*x*e^x + 10*x + 25) + 3)

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maple [A]  time = 0.08, size = 38, normalized size = 1.73

method	result	size
risch	\(-\frac {x}{{\mathrm e}^{16 \,{\mathrm e}^{2 x} x^{2}+8 \,{\mathrm e}^{x} x^{2}+40 \,{\mathrm e}^{x} x +x^{2}+10 x +25}+3}\)	\(38\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((32*x^3+32*x^2)*exp(x)^2+(8*x^3+56*x^2+40*x)*exp(x)+2*x^2+10*x-1)*exp(16*exp(x)^2*x^2+(8*x^2+40*x)*exp(x
)+x^2+10*x+25)-3)/(exp(16*exp(x)^2*x^2+(8*x^2+40*x)*exp(x)+x^2+10*x+25)^2+6*exp(16*exp(x)^2*x^2+(8*x^2+40*x)*e
xp(x)+x^2+10*x+25)+9),x,method=_RETURNVERBOSE)

[Out]

-x/(exp(16*exp(2*x)*x^2+8*exp(x)*x^2+40*exp(x)*x+x^2+10*x+25)+3)

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maxima [A]  time = 0.45, size = 37, normalized size = 1.68 \begin {gather*} -\frac {x}{e^{\left (16 \, x^{2} e^{\left (2 \, x\right )} + 8 \, x^{2} e^{x} + x^{2} + 40 \, x e^{x} + 10 \, x + 25\right )} + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((32*x^3+32*x^2)*exp(x)^2+(8*x^3+56*x^2+40*x)*exp(x)+2*x^2+10*x-1)*exp(16*exp(x)^2*x^2+(8*x^2+40*x)
*exp(x)+x^2+10*x+25)-3)/(exp(16*exp(x)^2*x^2+(8*x^2+40*x)*exp(x)+x^2+10*x+25)^2+6*exp(16*exp(x)^2*x^2+(8*x^2+4
0*x)*exp(x)+x^2+10*x+25)+9),x, algorithm="maxima")

[Out]

-x/(e^(16*x^2*e^(2*x) + 8*x^2*e^x + x^2 + 40*x*e^x + 10*x + 25) + 3)

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mupad [B]  time = 5.63, size = 42, normalized size = 1.91 \begin {gather*} -\frac {x}{{\mathrm {e}}^{40\,x\,{\mathrm {e}}^x}\,{\mathrm {e}}^{10\,x}\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^{25}\,{\mathrm {e}}^{8\,x^2\,{\mathrm {e}}^x}\,{\mathrm {e}}^{16\,x^2\,{\mathrm {e}}^{2\,x}}+3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(10*x + 16*x^2*exp(2*x) + exp(x)*(40*x + 8*x^2) + x^2 + 25)*(10*x + exp(2*x)*(32*x^2 + 32*x^3) + 2*x^2
+ exp(x)*(40*x + 56*x^2 + 8*x^3) - 1) - 3)/(6*exp(10*x + 16*x^2*exp(2*x) + exp(x)*(40*x + 8*x^2) + x^2 + 25)
+ exp(20*x + 32*x^2*exp(2*x) + 2*exp(x)*(40*x + 8*x^2) + 2*x^2 + 50) + 9),x)

[Out]

-x/(exp(40*x*exp(x))*exp(10*x)*exp(x^2)*exp(25)*exp(8*x^2*exp(x))*exp(16*x^2*exp(2*x)) + 3)

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sympy [B]  time = 0.28, size = 36, normalized size = 1.64 \begin {gather*} - \frac {x}{e^{16 x^{2} e^{2 x} + x^{2} + 10 x + \left (8 x^{2} + 40 x\right ) e^{x} + 25} + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((32*x**3+32*x**2)*exp(x)**2+(8*x**3+56*x**2+40*x)*exp(x)+2*x**2+10*x-1)*exp(16*exp(x)**2*x**2+(8*x
**2+40*x)*exp(x)+x**2+10*x+25)-3)/(exp(16*exp(x)**2*x**2+(8*x**2+40*x)*exp(x)+x**2+10*x+25)**2+6*exp(16*exp(x)
**2*x**2+(8*x**2+40*x)*exp(x)+x**2+10*x+25)+9),x)

[Out]

-x/(exp(16*x**2*exp(2*x) + x**2 + 10*x + (8*x**2 + 40*x)*exp(x) + 25) + 3)

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