3.94.11 \(\int \frac {-\frac {e^{-4+4 x}}{x^4}+\frac {4 e^{-3+3 x}}{x^2}-17 x^2+34 x^3-x^4+\frac {e^{-2+2 x} (-30 x+26 x^2)}{x^2}+\frac {e^{-1+x} (-4 x^2-28 x^3)}{x}}{-4 e^{-3+3 x}+\frac {e^{-4+4 x}}{x^2}+x^4-2 x^5+x^6+\frac {e^{-2+2 x} (-2 x^3+6 x^4)}{x^2}+\frac {e^{-1+x} (4 x^4-4 x^5)}{x}} \, dx\)

Optimal. Leaf size=31 \[ 1+\frac {1}{x}+4 \left (4+\frac {4}{x-\left (-\frac {e^{-1+x}}{x}+x\right )^2}\right ) \]

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Rubi [F]  time = 3.54, 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 {-\frac {e^{-4+4 x}}{x^4}+\frac {4 e^{-3+3 x}}{x^2}-17 x^2+34 x^3-x^4+\frac {e^{-2+2 x} \left (-30 x+26 x^2\right )}{x^2}+\frac {e^{-1+x} \left (-4 x^2-28 x^3\right )}{x}}{-4 e^{-3+3 x}+\frac {e^{-4+4 x}}{x^2}+x^4-2 x^5+x^6+\frac {e^{-2+2 x} \left (-2 x^3+6 x^4\right )}{x^2}+\frac {e^{-1+x} \left (4 x^4-4 x^5\right )}{x}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-(E^(-4 + 4*x)/x^4) + (4*E^(-3 + 3*x))/x^2 - 17*x^2 + 34*x^3 - x^4 + (E^(-2 + 2*x)*(-30*x + 26*x^2))/x^2
+ (E^(-1 + x)*(-4*x^2 - 28*x^3))/x)/(-4*E^(-3 + 3*x) + E^(-4 + 4*x)/x^2 + x^4 - 2*x^5 + x^6 + (E^(-2 + 2*x)*(-
2*x^3 + 6*x^4))/x^2 + (E^(-1 + x)*(4*x^4 - 4*x^5))/x),x]

[Out]

x^(-1) - 64*E^3*Defer[Int][(E^x*x^3)/(E^(2*x) - 2*E^(1 + x)*x^2 - E^2*x^3 + E^2*x^4)^2, x] - 48*E^4*Defer[Int]
[x^4/(E^(2*x) - 2*E^(1 + x)*x^2 - E^2*x^3 + E^2*x^4)^2, x] + 32*E^3*Defer[Int][(E^x*x^4)/(E^(2*x) - 2*E^(1 + x
)*x^2 - E^2*x^3 + E^2*x^4)^2, x] + 96*E^4*Defer[Int][x^5/(E^(2*x) - 2*E^(1 + x)*x^2 - E^2*x^3 + E^2*x^4)^2, x]
- 32*E^4*Defer[Int][x^6/(E^(2*x) - 2*E^(1 + x)*x^2 - E^2*x^3 + E^2*x^4)^2, x] - 32*E^2*Defer[Int][x/(E^(2*x)
- 2*E^(1 + x)*x^2 - E^2*x^3 + E^2*x^4), x] + 32*E^2*Defer[Int][x^2/(E^(2*x) - 2*E^(1 + x)*x^2 - E^2*x^3 + E^2*
x^4), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-e^{4 x}+4 e^{1+3 x} x^2-4 e^{3+x} x^5 (1+7 x)+2 e^{2+2 x} x^3 (-15+13 x)-e^4 x^6 \left (17-34 x+x^2\right )}{x^2 \left (e^{2 x}-2 e^{1+x} x^2+e^2 (-1+x) x^3\right )^2} \, dx\\ &=\int \left (-\frac {1}{x^2}-\frac {16 e^3 x^3 \left (4 e^x+3 e x-2 e^x x-6 e x^2+2 e x^3\right )}{\left (e^{2 x}-2 e^{1+x} x^2-e^2 x^3+e^2 x^4\right )^2}+\frac {32 e^2 (-1+x) x}{e^{2 x}-2 e^{1+x} x^2-e^2 x^3+e^2 x^4}\right ) \, dx\\ &=\frac {1}{x}+\left (32 e^2\right ) \int \frac {(-1+x) x}{e^{2 x}-2 e^{1+x} x^2-e^2 x^3+e^2 x^4} \, dx-\left (16 e^3\right ) \int \frac {x^3 \left (4 e^x+3 e x-2 e^x x-6 e x^2+2 e x^3\right )}{\left (e^{2 x}-2 e^{1+x} x^2-e^2 x^3+e^2 x^4\right )^2} \, dx\\ &=\frac {1}{x}+\left (32 e^2\right ) \int \left (-\frac {x}{e^{2 x}-2 e^{1+x} x^2-e^2 x^3+e^2 x^4}+\frac {x^2}{e^{2 x}-2 e^{1+x} x^2-e^2 x^3+e^2 x^4}\right ) \, dx-\left (16 e^3\right ) \int \left (\frac {4 e^x x^3}{\left (e^{2 x}-2 e^{1+x} x^2-e^2 x^3+e^2 x^4\right )^2}+\frac {3 e x^4}{\left (e^{2 x}-2 e^{1+x} x^2-e^2 x^3+e^2 x^4\right )^2}-\frac {2 e^x x^4}{\left (e^{2 x}-2 e^{1+x} x^2-e^2 x^3+e^2 x^4\right )^2}-\frac {6 e x^5}{\left (e^{2 x}-2 e^{1+x} x^2-e^2 x^3+e^2 x^4\right )^2}+\frac {2 e x^6}{\left (e^{2 x}-2 e^{1+x} x^2-e^2 x^3+e^2 x^4\right )^2}\right ) \, dx\\ &=\frac {1}{x}-\left (32 e^2\right ) \int \frac {x}{e^{2 x}-2 e^{1+x} x^2-e^2 x^3+e^2 x^4} \, dx+\left (32 e^2\right ) \int \frac {x^2}{e^{2 x}-2 e^{1+x} x^2-e^2 x^3+e^2 x^4} \, dx+\left (32 e^3\right ) \int \frac {e^x x^4}{\left (e^{2 x}-2 e^{1+x} x^2-e^2 x^3+e^2 x^4\right )^2} \, dx-\left (64 e^3\right ) \int \frac {e^x x^3}{\left (e^{2 x}-2 e^{1+x} x^2-e^2 x^3+e^2 x^4\right )^2} \, dx-\left (32 e^4\right ) \int \frac {x^6}{\left (e^{2 x}-2 e^{1+x} x^2-e^2 x^3+e^2 x^4\right )^2} \, dx-\left (48 e^4\right ) \int \frac {x^4}{\left (e^{2 x}-2 e^{1+x} x^2-e^2 x^3+e^2 x^4\right )^2} \, dx+\left (96 e^4\right ) \int \frac {x^5}{\left (e^{2 x}-2 e^{1+x} x^2-e^2 x^3+e^2 x^4\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.09, size = 40, normalized size = 1.29 \begin {gather*} \frac {1}{x}-\frac {16 e^2 x^2}{e^{2 x}-2 e^{1+x} x^2+e^2 (-1+x) x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-(E^(-4 + 4*x)/x^4) + (4*E^(-3 + 3*x))/x^2 - 17*x^2 + 34*x^3 - x^4 + (E^(-2 + 2*x)*(-30*x + 26*x^2)
)/x^2 + (E^(-1 + x)*(-4*x^2 - 28*x^3))/x)/(-4*E^(-3 + 3*x) + E^(-4 + 4*x)/x^2 + x^4 - 2*x^5 + x^6 + (E^(-2 + 2
*x)*(-2*x^3 + 6*x^4))/x^2 + (E^(-1 + x)*(4*x^4 - 4*x^5))/x),x]

[Out]

x^(-1) - (16*E^2*x^2)/(E^(2*x) - 2*E^(1 + x)*x^2 + E^2*(-1 + x)*x^3)

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fricas [B]  time = 0.57, size = 65, normalized size = 2.10 \begin {gather*} \frac {x^{2} - 2 \, x e^{\left (x - \log \relax (x) - 1\right )} - 17 \, x + e^{\left (2 \, x - 2 \, \log \relax (x) - 2\right )}}{x^{3} - 2 \, x^{2} e^{\left (x - \log \relax (x) - 1\right )} - x^{2} + x e^{\left (2 \, x - 2 \, \log \relax (x) - 2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-exp(x-log(x)-1)^4+4*x*exp(x-log(x)-1)^3+(26*x^2-30*x)*exp(x-log(x)-1)^2+(-28*x^3-4*x^2)*exp(x-log(
x)-1)-x^4+34*x^3-17*x^2)/(x^2*exp(x-log(x)-1)^4-4*x^3*exp(x-log(x)-1)^3+(6*x^4-2*x^3)*exp(x-log(x)-1)^2+(-4*x^
5+4*x^4)*exp(x-log(x)-1)+x^6-2*x^5+x^4),x, algorithm="fricas")

[Out]

(x^2 - 2*x*e^(x - log(x) - 1) - 17*x + e^(2*x - 2*log(x) - 2))/(x^3 - 2*x^2*e^(x - log(x) - 1) - x^2 + x*e^(2*
x - 2*log(x) - 2))

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giac [B]  time = 0.21, size = 59, normalized size = 1.90 \begin {gather*} \frac {x^{4} e^{2} - 33 \, x^{3} e^{2} - 2 \, x^{2} e^{\left (x + 1\right )} + e^{\left (2 \, x\right )}}{x^{5} e^{2} - x^{4} e^{2} - 2 \, x^{3} e^{\left (x + 1\right )} + x e^{\left (2 \, x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-exp(x-log(x)-1)^4+4*x*exp(x-log(x)-1)^3+(26*x^2-30*x)*exp(x-log(x)-1)^2+(-28*x^3-4*x^2)*exp(x-log(
x)-1)-x^4+34*x^3-17*x^2)/(x^2*exp(x-log(x)-1)^4-4*x^3*exp(x-log(x)-1)^3+(6*x^4-2*x^3)*exp(x-log(x)-1)^2+(-4*x^
5+4*x^4)*exp(x-log(x)-1)+x^6-2*x^5+x^4),x, algorithm="giac")

[Out]

(x^4*e^2 - 33*x^3*e^2 - 2*x^2*e^(x + 1) + e^(2*x))/(x^5*e^2 - x^4*e^2 - 2*x^3*e^(x + 1) + x*e^(2*x))

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maple [A]  time = 0.08, size = 32, normalized size = 1.03

method	result	size
risch	\(\frac {1}{x}-\frac {16}{\frac {{\mathrm e}^{2 x -2}}{x^{2}}-2 \,{\mathrm e}^{x -1}+x^{2}-x}\)	\(32\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-exp(x-ln(x)-1)^4+4*x*exp(x-ln(x)-1)^3+(26*x^2-30*x)*exp(x-ln(x)-1)^2+(-28*x^3-4*x^2)*exp(x-ln(x)-1)-x^4+
34*x^3-17*x^2)/(x^2*exp(x-ln(x)-1)^4-4*x^3*exp(x-ln(x)-1)^3+(6*x^4-2*x^3)*exp(x-ln(x)-1)^2+(-4*x^5+4*x^4)*exp(
x-ln(x)-1)+x^6-2*x^5+x^4),x,method=_RETURNVERBOSE)

[Out]

1/x-16/(1/x^2*exp(2*x-2)-2*exp(x-1)+x^2-x)

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maxima [B]  time = 3.01, size = 59, normalized size = 1.90 \begin {gather*} \frac {x^{4} e^{2} - 17 \, x^{3} e^{2} - 2 \, x^{2} e^{\left (x + 1\right )} + e^{\left (2 \, x\right )}}{x^{5} e^{2} - x^{4} e^{2} - 2 \, x^{3} e^{\left (x + 1\right )} + x e^{\left (2 \, x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-exp(x-log(x)-1)^4+4*x*exp(x-log(x)-1)^3+(26*x^2-30*x)*exp(x-log(x)-1)^2+(-28*x^3-4*x^2)*exp(x-log(
x)-1)-x^4+34*x^3-17*x^2)/(x^2*exp(x-log(x)-1)^4-4*x^3*exp(x-log(x)-1)^3+(6*x^4-2*x^3)*exp(x-log(x)-1)^2+(-4*x^
5+4*x^4)*exp(x-log(x)-1)+x^6-2*x^5+x^4),x, algorithm="maxima")

[Out]

(x^4*e^2 - 17*x^3*e^2 - 2*x^2*e^(x + 1) + e^(2*x))/(x^5*e^2 - x^4*e^2 - 2*x^3*e^(x + 1) + x*e^(2*x))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int -\frac {{\mathrm {e}}^{4\,x-4\,\ln \relax (x)-4}+{\mathrm {e}}^{2\,x-2\,\ln \relax (x)-2}\,\left (30\,x-26\,x^2\right )+{\mathrm {e}}^{x-\ln \relax (x)-1}\,\left (28\,x^3+4\,x^2\right )-4\,x\,{\mathrm {e}}^{3\,x-3\,\ln \relax (x)-3}+17\,x^2-34\,x^3+x^4}{x^2\,{\mathrm {e}}^{4\,x-4\,\ln \relax (x)-4}-4\,x^3\,{\mathrm {e}}^{3\,x-3\,\ln \relax (x)-3}-{\mathrm {e}}^{2\,x-2\,\ln \relax (x)-2}\,\left (2\,x^3-6\,x^4\right )+{\mathrm {e}}^{x-\ln \relax (x)-1}\,\left (4\,x^4-4\,x^5\right )+x^4-2\,x^5+x^6} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(4*x - 4*log(x) - 4) + exp(2*x - 2*log(x) - 2)*(30*x - 26*x^2) + exp(x - log(x) - 1)*(4*x^2 + 28*x^3)
- 4*x*exp(3*x - 3*log(x) - 3) + 17*x^2 - 34*x^3 + x^4)/(x^2*exp(4*x - 4*log(x) - 4) - 4*x^3*exp(3*x - 3*log(x
) - 3) - exp(2*x - 2*log(x) - 2)*(2*x^3 - 6*x^4) + exp(x - log(x) - 1)*(4*x^4 - 4*x^5) + x^4 - 2*x^5 + x^6),x)

[Out]

int(-(exp(4*x - 4*log(x) - 4) + exp(2*x - 2*log(x) - 2)*(30*x - 26*x^2) + exp(x - log(x) - 1)*(4*x^2 + 28*x^3)
- 4*x*exp(3*x - 3*log(x) - 3) + 17*x^2 - 34*x^3 + x^4)/(x^2*exp(4*x - 4*log(x) - 4) - 4*x^3*exp(3*x - 3*log(x
) - 3) - exp(2*x - 2*log(x) - 2)*(2*x^3 - 6*x^4) + exp(x - log(x) - 1)*(4*x^4 - 4*x^5) + x^4 - 2*x^5 + x^6), x
)

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sympy [A]  time = 0.26, size = 31, normalized size = 1.00 \begin {gather*} - \frac {16 x^{2}}{x^{4} - x^{3} - 2 x^{2} e^{x - 1} + e^{2 x - 2}} + \frac {1}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-exp(x-ln(x)-1)**4+4*x*exp(x-ln(x)-1)**3+(26*x**2-30*x)*exp(x-ln(x)-1)**2+(-28*x**3-4*x**2)*exp(x-l
n(x)-1)-x**4+34*x**3-17*x**2)/(x**2*exp(x-ln(x)-1)**4-4*x**3*exp(x-ln(x)-1)**3+(6*x**4-2*x**3)*exp(x-ln(x)-1)*
*2+(-4*x**5+4*x**4)*exp(x-ln(x)-1)+x**6-2*x**5+x**4),x)

[Out]

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

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