3.35.16 \(\int e^{16-32 e^8 x^2+24 e^{16} x^4-8 e^{24} x^6+e^{32} x^8} (-1+64 e^8 x^2-96 e^{16} x^4+48 e^{24} x^6-8 e^{32} x^8) \, dx\)

Optimal. Leaf size=18 \[ -3-e^{\left (-2+e^8 x^2\right )^4} x \]

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Rubi [B]  time = 0.14, antiderivative size = 103, normalized size of antiderivative = 5.72, number of steps used = 1, number of rules used = 1, integrand size = 70, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.014, Rules used = {2288} \begin {gather*} -\frac {e^{e^{32} x^8-8 e^{24} x^6+24 e^{16} x^4-32 e^8 x^2+16} \left (-e^{32} x^8+6 e^{24} x^6-12 e^{16} x^4+8 e^8 x^2\right )}{-e^{32} x^7+6 e^{24} x^5-12 e^{16} x^3+8 e^8 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(16 - 32*E^8*x^2 + 24*E^16*x^4 - 8*E^24*x^6 + E^32*x^8)*(-1 + 64*E^8*x^2 - 96*E^16*x^4 + 48*E^24*x^6 - 8
*E^32*x^8),x]

[Out]

-((E^(16 - 32*E^8*x^2 + 24*E^16*x^4 - 8*E^24*x^6 + E^32*x^8)*(8*E^8*x^2 - 12*E^16*x^4 + 6*E^24*x^6 - E^32*x^8)
)/(8*E^8*x - 12*E^16*x^3 + 6*E^24*x^5 - E^32*x^7))

Rule 2288

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = (v*y)/(Log[F]*D[u, x])}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=-\frac {e^{16-32 e^8 x^2+24 e^{16} x^4-8 e^{24} x^6+e^{32} x^8} \left (8 e^8 x^2-12 e^{16} x^4+6 e^{24} x^6-e^{32} x^8\right )}{8 e^8 x-12 e^{16} x^3+6 e^{24} x^5-e^{32} x^7}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.04, size = 16, normalized size = 0.89 \begin {gather*} -e^{\left (-2+e^8 x^2\right )^4} x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(16 - 32*E^8*x^2 + 24*E^16*x^4 - 8*E^24*x^6 + E^32*x^8)*(-1 + 64*E^8*x^2 - 96*E^16*x^4 + 48*E^24*x
^6 - 8*E^32*x^8),x]

[Out]

-(E^(-2 + E^8*x^2)^4*x)

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fricas [B]  time = 1.20, size = 33, normalized size = 1.83 \begin {gather*} -x e^{\left (x^{8} e^{32} - 8 \, x^{6} e^{24} + 24 \, x^{4} e^{16} - 32 \, x^{2} e^{8} + 16\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-8*x^8*exp(4)^8+48*x^6*exp(4)^6-96*x^4*exp(4)^4+64*x^2*exp(4)^2-1)*exp(x^8*exp(4)^8-8*x^6*exp(4)^6+
24*x^4*exp(4)^4-32*x^2*exp(4)^2+16),x, algorithm="fricas")

[Out]

-x*e^(x^8*e^32 - 8*x^6*e^24 + 24*x^4*e^16 - 32*x^2*e^8 + 16)

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giac [B]  time = 0.17, size = 33, normalized size = 1.83 \begin {gather*} -x e^{\left (x^{8} e^{32} - 8 \, x^{6} e^{24} + 24 \, x^{4} e^{16} - 32 \, x^{2} e^{8} + 16\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-8*x^8*exp(4)^8+48*x^6*exp(4)^6-96*x^4*exp(4)^4+64*x^2*exp(4)^2-1)*exp(x^8*exp(4)^8-8*x^6*exp(4)^6+
24*x^4*exp(4)^4-32*x^2*exp(4)^2+16),x, algorithm="giac")

[Out]

-x*e^(x^8*e^32 - 8*x^6*e^24 + 24*x^4*e^16 - 32*x^2*e^8 + 16)

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maple [A]  time = 0.10, size = 34, normalized size = 1.89

method	result	size
risch	\(-x \,{\mathrm e}^{x^{8} {\mathrm e}^{32}-8 x^{6} {\mathrm e}^{24}+24 x^{4} {\mathrm e}^{16}-32 x^{2} {\mathrm e}^{8}+16}\)	\(34\)
gosper	\(-x \,{\mathrm e}^{x^{8} {\mathrm e}^{32}-8 x^{6} {\mathrm e}^{24}+24 x^{4} {\mathrm e}^{16}-32 x^{2} {\mathrm e}^{8}+16}\)	\(42\)
norman	\(-x \,{\mathrm e}^{x^{8} {\mathrm e}^{32}-8 x^{6} {\mathrm e}^{24}+24 x^{4} {\mathrm e}^{16}-32 x^{2} {\mathrm e}^{8}+16}\)	\(42\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-8*x^8*exp(4)^8+48*x^6*exp(4)^6-96*x^4*exp(4)^4+64*x^2*exp(4)^2-1)*exp(x^8*exp(4)^8-8*x^6*exp(4)^6+24*x^4
*exp(4)^4-32*x^2*exp(4)^2+16),x,method=_RETURNVERBOSE)

[Out]

-x*exp(x^8*exp(32)-8*x^6*exp(24)+24*x^4*exp(16)-32*x^2*exp(8)+16)

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maxima [B]  time = 0.95, size = 33, normalized size = 1.83 \begin {gather*} -x e^{\left (x^{8} e^{32} - 8 \, x^{6} e^{24} + 24 \, x^{4} e^{16} - 32 \, x^{2} e^{8} + 16\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-8*x^8*exp(4)^8+48*x^6*exp(4)^6-96*x^4*exp(4)^4+64*x^2*exp(4)^2-1)*exp(x^8*exp(4)^8-8*x^6*exp(4)^6+
24*x^4*exp(4)^4-32*x^2*exp(4)^2+16),x, algorithm="maxima")

[Out]

-x*e^(x^8*e^32 - 8*x^6*e^24 + 24*x^4*e^16 - 32*x^2*e^8 + 16)

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mupad [B]  time = 2.14, size = 36, normalized size = 2.00 \begin {gather*} -x\,{\mathrm {e}}^{-8\,x^6\,{\mathrm {e}}^{24}}\,{\mathrm {e}}^{x^8\,{\mathrm {e}}^{32}}\,{\mathrm {e}}^{-32\,x^2\,{\mathrm {e}}^8}\,{\mathrm {e}}^{24\,x^4\,{\mathrm {e}}^{16}}\,{\mathrm {e}}^{16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-exp(24*x^4*exp(16) - 32*x^2*exp(8) - 8*x^6*exp(24) + x^8*exp(32) + 16)*(96*x^4*exp(16) - 64*x^2*exp(8) -
48*x^6*exp(24) + 8*x^8*exp(32) + 1),x)

[Out]

-x*exp(-8*x^6*exp(24))*exp(x^8*exp(32))*exp(-32*x^2*exp(8))*exp(24*x^4*exp(16))*exp(16)

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sympy [B]  time = 0.17, size = 37, normalized size = 2.06 \begin {gather*} - x e^{x^{8} e^{32} - 8 x^{6} e^{24} + 24 x^{4} e^{16} - 32 x^{2} e^{8} + 16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-8*x**8*exp(4)**8+48*x**6*exp(4)**6-96*x**4*exp(4)**4+64*x**2*exp(4)**2-1)*exp(x**8*exp(4)**8-8*x**
6*exp(4)**6+24*x**4*exp(4)**4-32*x**2*exp(4)**2+16),x)

[Out]

-x*exp(x**8*exp(32) - 8*x**6*exp(24) + 24*x**4*exp(16) - 32*x**2*exp(8) + 16)

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