3.80.82 \(\int e^{4096 e^{8 x^2} x^2} (1+e^{8 x^2} (8192 x^2+65536 x^4)) \, dx\)

Optimal. Leaf size=16 \[ e^{4096 e^{8 x^2} x^2} x \]

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Rubi [B]  time = 0.03, antiderivative size = 54, normalized size of antiderivative = 3.38, number of steps used = 1, number of rules used = 1, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.028, Rules used = {2288} \begin {gather*} \frac {e^{4096 e^{8 x^2} x^2+8 x^2} \left (8 x^4+x^2\right )}{e^{8 x^2} x+8 e^{8 x^2} x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(4096*E^(8*x^2)*x^2)*(1 + E^(8*x^2)*(8192*x^2 + 65536*x^4)),x]

[Out]

(E^(8*x^2 + 4096*E^(8*x^2)*x^2)*(x^2 + 8*x^4))/(E^(8*x^2)*x + 8*E^(8*x^2)*x^3)

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^{8 x^2+4096 e^{8 x^2} x^2} \left (x^2+8 x^4\right )}{e^{8 x^2} x+8 e^{8 x^2} x^3}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 16, normalized size = 1.00 \begin {gather*} e^{4096 e^{8 x^2} x^2} x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(4096*E^(8*x^2)*x^2)*(1 + E^(8*x^2)*(8192*x^2 + 65536*x^4)),x]

[Out]

E^(4096*E^(8*x^2)*x^2)*x

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fricas [A]  time = 1.06, size = 14, normalized size = 0.88 \begin {gather*} x e^{\left (4096 \, x^{2} e^{\left (8 \, x^{2}\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((65536*x^4+8192*x^2)*exp(4*x^2)^2+1)*exp(4096*x^2*exp(4*x^2)^2),x, algorithm="fricas")

[Out]

x*e^(4096*x^2*e^(8*x^2))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int {\left (8192 \, {\left (8 \, x^{4} + x^{2}\right )} e^{\left (8 \, x^{2}\right )} + 1\right )} e^{\left (4096 \, x^{2} e^{\left (8 \, x^{2}\right )}\right )}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((65536*x^4+8192*x^2)*exp(4*x^2)^2+1)*exp(4096*x^2*exp(4*x^2)^2),x, algorithm="giac")

[Out]

integrate((8192*(8*x^4 + x^2)*e^(8*x^2) + 1)*e^(4096*x^2*e^(8*x^2)), x)

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maple [A]  time = 0.12, size = 15, normalized size = 0.94

method	result	size
risch	\({\mathrm e}^{4096 x^{2} {\mathrm e}^{8 x^{2}}} x\)	\(15\)
norman	\({\mathrm e}^{4096 x^{2} {\mathrm e}^{8 x^{2}}} x\)	\(17\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((65536*x^4+8192*x^2)*exp(4*x^2)^2+1)*exp(4096*x^2*exp(4*x^2)^2),x,method=_RETURNVERBOSE)

[Out]

exp(4096*x^2*exp(8*x^2))*x

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maxima [A]  time = 0.40, size = 14, normalized size = 0.88 \begin {gather*} x e^{\left (4096 \, x^{2} e^{\left (8 \, x^{2}\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((65536*x^4+8192*x^2)*exp(4*x^2)^2+1)*exp(4096*x^2*exp(4*x^2)^2),x, algorithm="maxima")

[Out]

x*e^(4096*x^2*e^(8*x^2))

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mupad [B]  time = 5.52, size = 14, normalized size = 0.88 \begin {gather*} x\,{\mathrm {e}}^{4096\,x^2\,{\mathrm {e}}^{8\,x^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(4096*x^2*exp(8*x^2))*(exp(8*x^2)*(8192*x^2 + 65536*x^4) + 1),x)

[Out]

x*exp(4096*x^2*exp(8*x^2))

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sympy [A]  time = 10.11, size = 14, normalized size = 0.88 \begin {gather*} x e^{4096 x^{2} e^{8 x^{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((65536*x**4+8192*x**2)*exp(4*x**2)**2+1)*exp(4096*x**2*exp(4*x**2)**2),x)

[Out]

x*exp(4096*x**2*exp(8*x**2))

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