3.58.8 \(\int \frac {1}{8} e^6 (64 x+e^2 (-576 x-48 x^3)+e^4 (1728 x+288 x^3+12 x^5)+e^6 (-1728 x-432 x^3-36 x^5-x^7)) \, dx\)

Optimal. Leaf size=23 \[ 1-4 e^{12} \left (3-\frac {1}{e^2}+\frac {x^2}{4}\right )^4 \]

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Rubi [B]  time = 0.03, antiderivative size = 91, normalized size of antiderivative = 3.96, number of steps used = 5, number of rules used = 1, integrand size = 65, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.015, Rules used = {12} \begin {gather*} -\frac {1}{64} e^{12} x^8-\frac {3 e^{12} x^6}{4}+\frac {e^{10} x^6}{4}-\frac {27 e^{12} x^4}{2}+9 e^{10} x^4-\frac {3 e^8 x^4}{2}-108 e^{12} x^2+108 e^{10} x^2-36 e^8 x^2+4 e^6 x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^6*(64*x + E^2*(-576*x - 48*x^3) + E^4*(1728*x + 288*x^3 + 12*x^5) + E^6*(-1728*x - 432*x^3 - 36*x^5 - x
^7)))/8,x]

[Out]

4*E^6*x^2 - 36*E^8*x^2 + 108*E^10*x^2 - 108*E^12*x^2 - (3*E^8*x^4)/2 + 9*E^10*x^4 - (27*E^12*x^4)/2 + (E^10*x^
6)/4 - (3*E^12*x^6)/4 - (E^12*x^8)/64

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{8} e^6 \int \left (64 x+e^2 \left (-576 x-48 x^3\right )+e^4 \left (1728 x+288 x^3+12 x^5\right )+e^6 \left (-1728 x-432 x^3-36 x^5-x^7\right )\right ) \, dx\\ &=4 e^6 x^2+\frac {1}{8} e^8 \int \left (-576 x-48 x^3\right ) \, dx+\frac {1}{8} e^{10} \int \left (1728 x+288 x^3+12 x^5\right ) \, dx+\frac {1}{8} e^{12} \int \left (-1728 x-432 x^3-36 x^5-x^7\right ) \, dx\\ &=4 e^6 x^2-36 e^8 x^2+108 e^{10} x^2-108 e^{12} x^2-\frac {3 e^8 x^4}{2}+9 e^{10} x^4-\frac {27 e^{12} x^4}{2}+\frac {e^{10} x^6}{4}-\frac {3 e^{12} x^6}{4}-\frac {e^{12} x^8}{64}\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.01, size = 64, normalized size = 2.78 \begin {gather*} -\frac {1}{8} e^6 \left (32 \left (-1+3 e^2\right )^3 x^2+12 e^2 \left (-1+3 e^2\right )^2 x^4+2 e^4 \left (-1+3 e^2\right ) x^6+\frac {e^6 x^8}{8}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^6*(64*x + E^2*(-576*x - 48*x^3) + E^4*(1728*x + 288*x^3 + 12*x^5) + E^6*(-1728*x - 432*x^3 - 36*x
^5 - x^7)))/8,x]

[Out]

-1/8*(E^6*(32*(-1 + 3*E^2)^3*x^2 + 12*E^2*(-1 + 3*E^2)^2*x^4 + 2*E^4*(-1 + 3*E^2)*x^6 + (E^6*x^8)/8))

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fricas [B]  time = 0.54, size = 62, normalized size = 2.70 \begin {gather*} 4 \, x^{2} e^{6} - \frac {1}{64} \, {\left (x^{8} + 48 \, x^{6} + 864 \, x^{4} + 6912 \, x^{2}\right )} e^{12} + \frac {1}{4} \, {\left (x^{6} + 36 \, x^{4} + 432 \, x^{2}\right )} e^{10} - \frac {3}{2} \, {\left (x^{4} + 24 \, x^{2}\right )} e^{8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/8*((-x^7-36*x^5-432*x^3-1728*x)*exp(2)^3+(12*x^5+288*x^3+1728*x)*exp(2)^2+(-48*x^3-576*x)*exp(2)+6
4*x)*exp(3)^4/exp(2)^3,x, algorithm="fricas")

[Out]

4*x^2*e^6 - 1/64*(x^8 + 48*x^6 + 864*x^4 + 6912*x^2)*e^12 + 1/4*(x^6 + 36*x^4 + 432*x^2)*e^10 - 3/2*(x^4 + 24*
x^2)*e^8

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giac [B]  time = 0.16, size = 64, normalized size = 2.78 \begin {gather*} \frac {1}{64} \, {\left (256 \, x^{2} - {\left (x^{8} + 48 \, x^{6} + 864 \, x^{4} + 6912 \, x^{2}\right )} e^{6} + 16 \, {\left (x^{6} + 36 \, x^{4} + 432 \, x^{2}\right )} e^{4} - 96 \, {\left (x^{4} + 24 \, x^{2}\right )} e^{2}\right )} e^{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/8*((-x^7-36*x^5-432*x^3-1728*x)*exp(2)^3+(12*x^5+288*x^3+1728*x)*exp(2)^2+(-48*x^3-576*x)*exp(2)+6
4*x)*exp(3)^4/exp(2)^3,x, algorithm="giac")

[Out]

1/64*(256*x^2 - (x^8 + 48*x^6 + 864*x^4 + 6912*x^2)*e^6 + 16*(x^6 + 36*x^4 + 432*x^2)*e^4 - 96*(x^4 + 24*x^2)*
e^2)*e^6

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maple [B]  time = 0.06, size = 78, normalized size = 3.39

method	result	size
default	\(\frac {{\mathrm e}^{12} {\mathrm e}^{-6} \left ({\mathrm e}^{6} \left (-\frac {1}{8} x^{8}-6 x^{6}-108 x^{4}-864 x^{2}\right )+{\mathrm e}^{4} \left (2 x^{6}+72 x^{4}+864 x^{2}\right )+{\mathrm e}^{2} \left (-12 x^{4}-288 x^{2}\right )+32 x^{2}\right )}{8}\)	\(78\)
gosper	\(-\frac {{\mathrm e}^{12} \left (x^{6} {\mathrm e}^{6}+48 x^{4} {\mathrm e}^{6}-16 x^{4} {\mathrm e}^{4}+864 x^{2} {\mathrm e}^{6}-576 x^{2} {\mathrm e}^{4}+6912 \,{\mathrm e}^{6}+96 x^{2} {\mathrm e}^{2}-6912 \,{\mathrm e}^{4}+2304 \,{\mathrm e}^{2}-256\right ) x^{2} {\mathrm e}^{-6}}{64}\)	\(83\)
risch	\(-\frac {x^{8} {\mathrm e}^{12}}{64}+\frac {{\mathrm e}^{6} x^{6} {\mathrm e}^{4}}{4}-\frac {3 \,{\mathrm e}^{12} x^{6}}{4}-\frac {3 \,{\mathrm e}^{6} x^{4} {\mathrm e}^{2}}{2}+9 \,{\mathrm e}^{6} x^{4} {\mathrm e}^{4}-\frac {27 x^{4} {\mathrm e}^{12}}{2}-36 \,{\mathrm e}^{6} x^{2} {\mathrm e}^{2}+108 \,{\mathrm e}^{6} x^{2} {\mathrm e}^{4}-108 x^{2} {\mathrm e}^{12}+4 x^{2} {\mathrm e}^{6}\)	\(88\)
norman	\(\left (-\frac {3 \,{\mathrm e}^{12} \left (9 \,{\mathrm e}^{4}-6 \,{\mathrm e}^{2}+1\right ) x^{4}}{2}-\frac {{\mathrm e}^{12} {\mathrm e}^{4} x^{8}}{64}-\frac {{\mathrm e}^{2} {\mathrm e}^{12} \left (3 \,{\mathrm e}^{2}-1\right ) x^{6}}{4}-4 \,{\mathrm e}^{12} \left (27 \,{\mathrm e}^{6}-27 \,{\mathrm e}^{4}+9 \,{\mathrm e}^{2}-1\right ) {\mathrm e}^{-2} x^{2}\right ) {\mathrm e}^{-4}\)	\(89\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/8*((-x^7-36*x^5-432*x^3-1728*x)*exp(2)^3+(12*x^5+288*x^3+1728*x)*exp(2)^2+(-48*x^3-576*x)*exp(2)+64*x)*e
xp(3)^4/exp(2)^3,x,method=_RETURNVERBOSE)

[Out]

1/8*exp(3)^4/exp(2)^3*(exp(2)^3*(-1/8*x^8-6*x^6-108*x^4-864*x^2)+exp(2)^2*(2*x^6+72*x^4+864*x^2)+exp(2)*(-12*x
^4-288*x^2)+32*x^2)

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maxima [B]  time = 0.35, size = 64, normalized size = 2.78 \begin {gather*} \frac {1}{64} \, {\left (256 \, x^{2} - {\left (x^{8} + 48 \, x^{6} + 864 \, x^{4} + 6912 \, x^{2}\right )} e^{6} + 16 \, {\left (x^{6} + 36 \, x^{4} + 432 \, x^{2}\right )} e^{4} - 96 \, {\left (x^{4} + 24 \, x^{2}\right )} e^{2}\right )} e^{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/8*((-x^7-36*x^5-432*x^3-1728*x)*exp(2)^3+(12*x^5+288*x^3+1728*x)*exp(2)^2+(-48*x^3-576*x)*exp(2)+6
4*x)*exp(3)^4/exp(2)^3,x, algorithm="maxima")

[Out]

1/64*(256*x^2 - (x^8 + 48*x^6 + 864*x^4 + 6912*x^2)*e^6 + 16*(x^6 + 36*x^4 + 432*x^2)*e^4 - 96*(x^4 + 24*x^2)*
e^2)*e^6

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mupad [B]  time = 3.61, size = 51, normalized size = 2.22 \begin {gather*} -\frac {{\mathrm {e}}^{12}\,x^8}{64}-\frac {{\mathrm {e}}^{10}\,\left (3\,{\mathrm {e}}^2-1\right )\,x^6}{4}-\frac {3\,{\mathrm {e}}^8\,{\left (3\,{\mathrm {e}}^2-1\right )}^2\,x^4}{2}-4\,{\mathrm {e}}^6\,{\left (3\,{\mathrm {e}}^2-1\right )}^3\,x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(6)*(64*x - exp(2)*(576*x + 48*x^3) - exp(6)*(1728*x + 432*x^3 + 36*x^5 + x^7) + exp(4)*(1728*x + 288*
x^3 + 12*x^5)))/8,x)

[Out]

- (x^8*exp(12))/64 - 4*x^2*exp(6)*(3*exp(2) - 1)^3 - (3*x^4*exp(8)*(3*exp(2) - 1)^2)/2 - (x^6*exp(10)*(3*exp(2
) - 1))/4

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sympy [B]  time = 0.08, size = 68, normalized size = 2.96 \begin {gather*} - \frac {x^{8} e^{12}}{64} + x^{6} \left (- \frac {3 e^{12}}{4} + \frac {e^{10}}{4}\right ) + x^{4} \left (- \frac {27 e^{12}}{2} - \frac {3 e^{8}}{2} + 9 e^{10}\right ) + x^{2} \left (- 108 e^{12} - 36 e^{8} + 4 e^{6} + 108 e^{10}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/8*((-x**7-36*x**5-432*x**3-1728*x)*exp(2)**3+(12*x**5+288*x**3+1728*x)*exp(2)**2+(-48*x**3-576*x)*
exp(2)+64*x)*exp(3)**4/exp(2)**3,x)

[Out]

-x**8*exp(12)/64 + x**6*(-3*exp(12)/4 + exp(10)/4) + x**4*(-27*exp(12)/2 - 3*exp(8)/2 + 9*exp(10)) + x**2*(-10
8*exp(12) - 36*exp(8) + 4*exp(6) + 108*exp(10))

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