3.25.2 \(\int \frac {e^{\frac {e^{1+x}}{2 x^6}+e^{\frac {e^{1+x}}{2 x^6}} (16-8 x-7 x^2+2 x^3+x^4)} (-16 x^7-28 x^8+12 x^9+8 x^{10}+e^{1+x} (-96+64 x+34 x^2-19 x^3-4 x^4+x^5))}{2 x^7} \, dx\)

Optimal. Leaf size=25 \[ e^{e^{\frac {e^{1+x}}{2 x^6}} \left (-4+x+x^2\right )^2} \]

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Rubi [F]  time = 18.26, 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 {e^{1+x}}{2 x^6}+e^{\frac {e^{1+x}}{2 x^6}} \left (16-8 x-7 x^2+2 x^3+x^4\right )\right ) \left (-16 x^7-28 x^8+12 x^9+8 x^{10}+e^{1+x} \left (-96+64 x+34 x^2-19 x^3-4 x^4+x^5\right )\right )}{2 x^7} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(E^(1 + x)/(2*x^6) + E^(E^(1 + x)/(2*x^6))*(16 - 8*x - 7*x^2 + 2*x^3 + x^4))*(-16*x^7 - 28*x^8 + 12*x^9
+ 8*x^10 + E^(1 + x)*(-96 + 64*x + 34*x^2 - 19*x^3 - 4*x^4 + x^5)))/(2*x^7),x]

[Out]

-8*Defer[Int][E^(E^(1 + x)/(2*x^6) + E^(E^(1 + x)/(2*x^6))*(-4 + x + x^2)^2), x] - 48*Defer[Int][E^(1 + E^(1 +
x)/(2*x^6) + x + E^(E^(1 + x)/(2*x^6))*(-4 + x + x^2)^2)/x^7, x] + 32*Defer[Int][E^(1 + E^(1 + x)/(2*x^6) + x
+ E^(E^(1 + x)/(2*x^6))*(-4 + x + x^2)^2)/x^6, x] + 17*Defer[Int][E^(1 + E^(1 + x)/(2*x^6) + x + E^(E^(1 + x)
/(2*x^6))*(-4 + x + x^2)^2)/x^5, x] - (19*Defer[Int][E^(1 + E^(1 + x)/(2*x^6) + x + E^(E^(1 + x)/(2*x^6))*(-4
+ x + x^2)^2)/x^4, x])/2 - 2*Defer[Int][E^(1 + E^(1 + x)/(2*x^6) + x + E^(E^(1 + x)/(2*x^6))*(-4 + x + x^2)^2)
/x^3, x] + Defer[Int][E^(1 + E^(1 + x)/(2*x^6) + x + E^(E^(1 + x)/(2*x^6))*(-4 + x + x^2)^2)/x^2, x]/2 - 14*De
fer[Int][E^(E^(1 + x)/(2*x^6) + E^(E^(1 + x)/(2*x^6))*(-4 + x + x^2)^2)*x, x] + 6*Defer[Int][E^(E^(1 + x)/(2*x
^6) + E^(E^(1 + x)/(2*x^6))*(-4 + x + x^2)^2)*x^2, x] + 4*Defer[Int][E^(E^(1 + x)/(2*x^6) + E^(E^(1 + x)/(2*x^
6))*(-4 + x + x^2)^2)*x^3, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{2} \int \frac {\exp \left (\frac {e^{1+x}}{2 x^6}+e^{\frac {e^{1+x}}{2 x^6}} \left (16-8 x-7 x^2+2 x^3+x^4\right )\right ) \left (-16 x^7-28 x^8+12 x^9+8 x^{10}+e^{1+x} \left (-96+64 x+34 x^2-19 x^3-4 x^4+x^5\right )\right )}{x^7} \, dx\\ &=\frac {1}{2} \int \frac {\exp \left (\frac {e^{1+x}}{2 x^6}+e^{\frac {e^{1+x}}{2 x^6}} \left (16-8 x-7 x^2+2 x^3+x^4\right )\right ) \left (4-x-x^2\right ) \left (-24 e^{1+x}+10 e^{1+x} x+5 e^{1+x} x^2-e^{1+x} x^3-4 x^7-8 x^8\right )}{x^7} \, dx\\ &=\frac {1}{2} \int \frac {\exp \left (\frac {e^{1+x}}{2 x^6}+e^{\frac {e^{1+x}}{2 x^6}} \left (-4+x+x^2\right )^2\right ) \left (4-x-x^2\right ) \left (-4 x^7 (1+2 x)-e^{1+x} \left (24-10 x-5 x^2+x^3\right )\right )}{x^7} \, dx\\ &=\frac {1}{2} \int \left (4 \exp \left (\frac {e^{1+x}}{2 x^6}+e^{\frac {e^{1+x}}{2 x^6}} \left (-4+x+x^2\right )^2\right ) (1+2 x) \left (-4+x+x^2\right )+\frac {\exp \left (1+\frac {e^{1+x}}{2 x^6}+x+e^{\frac {e^{1+x}}{2 x^6}} \left (-4+x+x^2\right )^2\right ) (-6+x) \left (-4+x+x^2\right )^2}{x^7}\right ) \, dx\\ &=\frac {1}{2} \int \frac {\exp \left (1+\frac {e^{1+x}}{2 x^6}+x+e^{\frac {e^{1+x}}{2 x^6}} \left (-4+x+x^2\right )^2\right ) (-6+x) \left (-4+x+x^2\right )^2}{x^7} \, dx+2 \int \exp \left (\frac {e^{1+x}}{2 x^6}+e^{\frac {e^{1+x}}{2 x^6}} \left (-4+x+x^2\right )^2\right ) (1+2 x) \left (-4+x+x^2\right ) \, dx\\ &=\frac {1}{2} \int \left (-\frac {96 \exp \left (1+\frac {e^{1+x}}{2 x^6}+x+e^{\frac {e^{1+x}}{2 x^6}} \left (-4+x+x^2\right )^2\right )}{x^7}+\frac {64 \exp \left (1+\frac {e^{1+x}}{2 x^6}+x+e^{\frac {e^{1+x}}{2 x^6}} \left (-4+x+x^2\right )^2\right )}{x^6}+\frac {34 \exp \left (1+\frac {e^{1+x}}{2 x^6}+x+e^{\frac {e^{1+x}}{2 x^6}} \left (-4+x+x^2\right )^2\right )}{x^5}-\frac {19 \exp \left (1+\frac {e^{1+x}}{2 x^6}+x+e^{\frac {e^{1+x}}{2 x^6}} \left (-4+x+x^2\right )^2\right )}{x^4}-\frac {4 \exp \left (1+\frac {e^{1+x}}{2 x^6}+x+e^{\frac {e^{1+x}}{2 x^6}} \left (-4+x+x^2\right )^2\right )}{x^3}+\frac {\exp \left (1+\frac {e^{1+x}}{2 x^6}+x+e^{\frac {e^{1+x}}{2 x^6}} \left (-4+x+x^2\right )^2\right )}{x^2}\right ) \, dx+2 \int \left (-4 \exp \left (\frac {e^{1+x}}{2 x^6}+e^{\frac {e^{1+x}}{2 x^6}} \left (-4+x+x^2\right )^2\right )-7 \exp \left (\frac {e^{1+x}}{2 x^6}+e^{\frac {e^{1+x}}{2 x^6}} \left (-4+x+x^2\right )^2\right ) x+3 \exp \left (\frac {e^{1+x}}{2 x^6}+e^{\frac {e^{1+x}}{2 x^6}} \left (-4+x+x^2\right )^2\right ) x^2+2 \exp \left (\frac {e^{1+x}}{2 x^6}+e^{\frac {e^{1+x}}{2 x^6}} \left (-4+x+x^2\right )^2\right ) x^3\right ) \, dx\\ &=\frac {1}{2} \int \frac {\exp \left (1+\frac {e^{1+x}}{2 x^6}+x+e^{\frac {e^{1+x}}{2 x^6}} \left (-4+x+x^2\right )^2\right )}{x^2} \, dx-2 \int \frac {\exp \left (1+\frac {e^{1+x}}{2 x^6}+x+e^{\frac {e^{1+x}}{2 x^6}} \left (-4+x+x^2\right )^2\right )}{x^3} \, dx+4 \int \exp \left (\frac {e^{1+x}}{2 x^6}+e^{\frac {e^{1+x}}{2 x^6}} \left (-4+x+x^2\right )^2\right ) x^3 \, dx+6 \int \exp \left (\frac {e^{1+x}}{2 x^6}+e^{\frac {e^{1+x}}{2 x^6}} \left (-4+x+x^2\right )^2\right ) x^2 \, dx-8 \int \exp \left (\frac {e^{1+x}}{2 x^6}+e^{\frac {e^{1+x}}{2 x^6}} \left (-4+x+x^2\right )^2\right ) \, dx-\frac {19}{2} \int \frac {\exp \left (1+\frac {e^{1+x}}{2 x^6}+x+e^{\frac {e^{1+x}}{2 x^6}} \left (-4+x+x^2\right )^2\right )}{x^4} \, dx-14 \int \exp \left (\frac {e^{1+x}}{2 x^6}+e^{\frac {e^{1+x}}{2 x^6}} \left (-4+x+x^2\right )^2\right ) x \, dx+17 \int \frac {\exp \left (1+\frac {e^{1+x}}{2 x^6}+x+e^{\frac {e^{1+x}}{2 x^6}} \left (-4+x+x^2\right )^2\right )}{x^5} \, dx+32 \int \frac {\exp \left (1+\frac {e^{1+x}}{2 x^6}+x+e^{\frac {e^{1+x}}{2 x^6}} \left (-4+x+x^2\right )^2\right )}{x^6} \, dx-48 \int \frac {\exp \left (1+\frac {e^{1+x}}{2 x^6}+x+e^{\frac {e^{1+x}}{2 x^6}} \left (-4+x+x^2\right )^2\right )}{x^7} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 2.36, size = 25, normalized size = 1.00 \begin {gather*} e^{e^{\frac {e^{1+x}}{2 x^6}} \left (-4+x+x^2\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(E^(1 + x)/(2*x^6) + E^(E^(1 + x)/(2*x^6))*(16 - 8*x - 7*x^2 + 2*x^3 + x^4))*(-16*x^7 - 28*x^8 +
12*x^9 + 8*x^10 + E^(1 + x)*(-96 + 64*x + 34*x^2 - 19*x^3 - 4*x^4 + x^5)))/(2*x^7),x]

[Out]

E^(E^(E^(1 + x)/(2*x^6))*(-4 + x + x^2)^2)

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fricas [B]  time = 0.53, size = 57, normalized size = 2.28 \begin {gather*} e^{\left (\frac {2 \, {\left (x^{10} + 2 \, x^{9} - 7 \, x^{8} - 8 \, x^{7} + 16 \, x^{6}\right )} e^{\left (\frac {e^{\left (x + 1\right )}}{2 \, x^{6}}\right )} + e^{\left (x + 1\right )}}{2 \, x^{6}} - \frac {e^{\left (x + 1\right )}}{2 \, x^{6}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*((x^5-4*x^4-19*x^3+34*x^2+64*x-96)*exp(x+1)+8*x^10+12*x^9-28*x^8-16*x^7)*exp(1/4*exp(x+1)/x^6)^2
*exp((x^4+2*x^3-7*x^2-8*x+16)*exp(1/4*exp(x+1)/x^6)^2)/x^7,x, algorithm="fricas")

[Out]

e^(1/2*(2*(x^10 + 2*x^9 - 7*x^8 - 8*x^7 + 16*x^6)*e^(1/2*e^(x + 1)/x^6) + e^(x + 1))/x^6 - 1/2*e^(x + 1)/x^6)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (8 \, x^{10} + 12 \, x^{9} - 28 \, x^{8} - 16 \, x^{7} + {\left (x^{5} - 4 \, x^{4} - 19 \, x^{3} + 34 \, x^{2} + 64 \, x - 96\right )} e^{\left (x + 1\right )}\right )} e^{\left ({\left (x^{4} + 2 \, x^{3} - 7 \, x^{2} - 8 \, x + 16\right )} e^{\left (\frac {e^{\left (x + 1\right )}}{2 \, x^{6}}\right )} + \frac {e^{\left (x + 1\right )}}{2 \, x^{6}}\right )}}{2 \, x^{7}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*((x^5-4*x^4-19*x^3+34*x^2+64*x-96)*exp(x+1)+8*x^10+12*x^9-28*x^8-16*x^7)*exp(1/4*exp(x+1)/x^6)^2
*exp((x^4+2*x^3-7*x^2-8*x+16)*exp(1/4*exp(x+1)/x^6)^2)/x^7,x, algorithm="giac")

[Out]

integrate(1/2*(8*x^10 + 12*x^9 - 28*x^8 - 16*x^7 + (x^5 - 4*x^4 - 19*x^3 + 34*x^2 + 64*x - 96)*e^(x + 1))*e^((
x^4 + 2*x^3 - 7*x^2 - 8*x + 16)*e^(1/2*e^(x + 1)/x^6) + 1/2*e^(x + 1)/x^6)/x^7, x)

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maple [A]  time = 0.16, size = 21, normalized size = 0.84

method	result	size
risch	\({\mathrm e}^{\left (x^{2}+x -4\right )^{2} {\mathrm e}^{\frac {{\mathrm e}^{x +1}}{2 x^{6}}}}\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/2*((x^5-4*x^4-19*x^3+34*x^2+64*x-96)*exp(x+1)+8*x^10+12*x^9-28*x^8-16*x^7)*exp(1/4*exp(x+1)/x^6)^2*exp((
x^4+2*x^3-7*x^2-8*x+16)*exp(1/4*exp(x+1)/x^6)^2)/x^7,x,method=_RETURNVERBOSE)

[Out]

exp((x^2+x-4)^2*exp(1/2*exp(x+1)/x^6))

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maxima [B]  time = 0.92, size = 71, normalized size = 2.84 \begin {gather*} e^{\left (x^{4} e^{\left (\frac {e^{\left (x + 1\right )}}{2 \, x^{6}}\right )} + 2 \, x^{3} e^{\left (\frac {e^{\left (x + 1\right )}}{2 \, x^{6}}\right )} - 7 \, x^{2} e^{\left (\frac {e^{\left (x + 1\right )}}{2 \, x^{6}}\right )} - 8 \, x e^{\left (\frac {e^{\left (x + 1\right )}}{2 \, x^{6}}\right )} + 16 \, e^{\left (\frac {e^{\left (x + 1\right )}}{2 \, x^{6}}\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*((x^5-4*x^4-19*x^3+34*x^2+64*x-96)*exp(x+1)+8*x^10+12*x^9-28*x^8-16*x^7)*exp(1/4*exp(x+1)/x^6)^2
*exp((x^4+2*x^3-7*x^2-8*x+16)*exp(1/4*exp(x+1)/x^6)^2)/x^7,x, algorithm="maxima")

[Out]

e^(x^4*e^(1/2*e^(x + 1)/x^6) + 2*x^3*e^(1/2*e^(x + 1)/x^6) - 7*x^2*e^(1/2*e^(x + 1)/x^6) - 8*x*e^(1/2*e^(x + 1
)/x^6) + 16*e^(1/2*e^(x + 1)/x^6))

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mupad [B]  time = 1.57, size = 75, normalized size = 3.00 \begin {gather*} {\mathrm {e}}^{-8\,x\,{\mathrm {e}}^{\frac {\mathrm {e}\,{\mathrm {e}}^x}{2\,x^6}}}\,{\mathrm {e}}^{x^4\,{\mathrm {e}}^{\frac {\mathrm {e}\,{\mathrm {e}}^x}{2\,x^6}}}\,{\mathrm {e}}^{2\,x^3\,{\mathrm {e}}^{\frac {\mathrm {e}\,{\mathrm {e}}^x}{2\,x^6}}}\,{\mathrm {e}}^{-7\,x^2\,{\mathrm {e}}^{\frac {\mathrm {e}\,{\mathrm {e}}^x}{2\,x^6}}}\,{\mathrm {e}}^{16\,{\mathrm {e}}^{\frac {\mathrm {e}\,{\mathrm {e}}^x}{2\,x^6}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(exp(exp(x + 1)/(2*x^6))*(2*x^3 - 7*x^2 - 8*x + x^4 + 16))*exp(exp(x + 1)/(2*x^6))*(exp(x + 1)*(64*x +
34*x^2 - 19*x^3 - 4*x^4 + x^5 - 96) - 16*x^7 - 28*x^8 + 12*x^9 + 8*x^10))/(2*x^7),x)

[Out]

exp(-8*x*exp((exp(1)*exp(x))/(2*x^6)))*exp(x^4*exp((exp(1)*exp(x))/(2*x^6)))*exp(2*x^3*exp((exp(1)*exp(x))/(2*
x^6)))*exp(-7*x^2*exp((exp(1)*exp(x))/(2*x^6)))*exp(16*exp((exp(1)*exp(x))/(2*x^6)))

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sympy [A]  time = 8.58, size = 31, normalized size = 1.24 \begin {gather*} e^{\left (x^{4} + 2 x^{3} - 7 x^{2} - 8 x + 16\right ) e^{\frac {e^{x + 1}}{2 x^{6}}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*((x**5-4*x**4-19*x**3+34*x**2+64*x-96)*exp(x+1)+8*x**10+12*x**9-28*x**8-16*x**7)*exp(1/4*exp(x+1
)/x**6)**2*exp((x**4+2*x**3-7*x**2-8*x+16)*exp(1/4*exp(x+1)/x**6)**2)/x**7,x)

[Out]

exp((x**4 + 2*x**3 - 7*x**2 - 8*x + 16)*exp(exp(x + 1)/(2*x**6)))

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