3.68.94 \(\int (-1-6 x+e^{6 x+2 x^2} (-1-6 x-4 x^2)+e^{2 x^2} (1+4 x^2)) \, dx\)

Optimal. Leaf size=24 \[ \left (-1+e^{2 x^2}-e^{2 x (3+x)}-3 x\right ) x \]

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Rubi [A]  time = 0.08, antiderivative size = 47, normalized size of antiderivative = 1.96, number of steps used = 7, number of rules used = 4, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2288, 2226, 2204, 2212} \begin {gather*} -3 x^2+e^{2 x^2} x-\frac {e^{2 x^2+6 x} \left (2 x^2+3 x\right )}{2 x+3}-x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[-1 - 6*x + E^(6*x + 2*x^2)*(-1 - 6*x - 4*x^2) + E^(2*x^2)*(1 + 4*x^2),x]

[Out]

-x + E^(2*x^2)*x - 3*x^2 - (E^(6*x + 2*x^2)*(3*x + 2*x^2))/(3 + 2*x)

Rule 2204

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[(F^a*Sqrt[Pi]*Erfi[(c + d*x)*Rt[b*Log[F], 2
]])/(2*d*Rt[b*Log[F], 2]), x] /; FreeQ[{F, a, b, c, d}, x] && PosQ[b]

Rule 2212

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^(m
- n + 1)*F^(a + b*(c + d*x)^n))/(b*d*n*Log[F]), x] - Dist[(m - n + 1)/(b*n*Log[F]), Int[(c + d*x)^(m - n)*F^(
a + b*(c + d*x)^n), x], x] /; FreeQ[{F, a, b, c, d}, x] && IntegerQ[(2*(m + 1))/n] && LtQ[0, (m + 1)/n, 5] &&
IntegerQ[n] && (LtQ[0, n, m + 1] || LtQ[m, n, 0])

Rule 2226

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*(u_), x_Symbol] :> Int[ExpandLinearProduct[F^(a + b*(c + d*
x)^n), u, c, d, x], x] /; FreeQ[{F, a, b, c, d, n}, x] && PolynomialQ[u, x]

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} &=-x-3 x^2+\int e^{6 x+2 x^2} \left (-1-6 x-4 x^2\right ) \, dx+\int e^{2 x^2} \left (1+4 x^2\right ) \, dx\\ &=-x-3 x^2-\frac {e^{6 x+2 x^2} \left (3 x+2 x^2\right )}{3+2 x}+\int \left (e^{2 x^2}+4 e^{2 x^2} x^2\right ) \, dx\\ &=-x-3 x^2-\frac {e^{6 x+2 x^2} \left (3 x+2 x^2\right )}{3+2 x}+4 \int e^{2 x^2} x^2 \, dx+\int e^{2 x^2} \, dx\\ &=-x+e^{2 x^2} x-3 x^2-\frac {e^{6 x+2 x^2} \left (3 x+2 x^2\right )}{3+2 x}+\frac {1}{2} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} x\right )-\int e^{2 x^2} \, dx\\ &=-x+e^{2 x^2} x-3 x^2-\frac {e^{6 x+2 x^2} \left (3 x+2 x^2\right )}{3+2 x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.03, size = 25, normalized size = 1.04 \begin {gather*} -x \left (1-e^{2 x^2}+e^{2 x (3+x)}+3 x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[-1 - 6*x + E^(6*x + 2*x^2)*(-1 - 6*x - 4*x^2) + E^(2*x^2)*(1 + 4*x^2),x]

[Out]

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

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fricas [A]  time = 0.48, size = 30, normalized size = 1.25 \begin {gather*} -3 \, x^{2} + x e^{\left (2 \, x^{2}\right )} - x e^{\left (2 \, x^{2} + 6 \, x\right )} - x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x^2-6*x-1)*exp(2*x^2+6*x)+(4*x^2+1)*exp(2*x^2)-6*x-1,x, algorithm="fricas")

[Out]

-3*x^2 + x*e^(2*x^2) - x*e^(2*x^2 + 6*x) - x

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giac [A]  time = 0.26, size = 30, normalized size = 1.25 \begin {gather*} -3 \, x^{2} + x e^{\left (2 \, x^{2}\right )} - x e^{\left (2 \, x^{2} + 6 \, x\right )} - x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x^2-6*x-1)*exp(2*x^2+6*x)+(4*x^2+1)*exp(2*x^2)-6*x-1,x, algorithm="giac")

[Out]

-3*x^2 + x*e^(2*x^2) - x*e^(2*x^2 + 6*x) - x

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maple [A]  time = 0.04, size = 28, normalized size = 1.17

method	result	size
risch	\(x \,{\mathrm e}^{2 x^{2}}-x -3 x^{2}-{\mathrm e}^{2 \left (3+x \right ) x} x\)	\(28\)
default	\(x \,{\mathrm e}^{2 x^{2}}-x -3 x^{2}-{\mathrm e}^{2 x^{2}+6 x} x\)	\(31\)
norman	\(x \,{\mathrm e}^{2 x^{2}}-x -3 x^{2}-{\mathrm e}^{2 x^{2}+6 x} x\)	\(31\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-4*x^2-6*x-1)*exp(2*x^2+6*x)+(4*x^2+1)*exp(2*x^2)-6*x-1,x,method=_RETURNVERBOSE)

[Out]

x*exp(2*x^2)-x-3*x^2-exp(2*(3+x)*x)*x

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maxima [A]  time = 0.36, size = 30, normalized size = 1.25 \begin {gather*} -3 \, x^{2} + x e^{\left (2 \, x^{2}\right )} - x e^{\left (2 \, x^{2} + 6 \, x\right )} - x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x^2-6*x-1)*exp(2*x^2+6*x)+(4*x^2+1)*exp(2*x^2)-6*x-1,x, algorithm="maxima")

[Out]

-3*x^2 + x*e^(2*x^2) - x*e^(2*x^2 + 6*x) - x

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mupad [B]  time = 0.11, size = 30, normalized size = 1.25 \begin {gather*} x\,{\mathrm {e}}^{2\,x^2}-x-3\,x^2-x\,{\mathrm {e}}^{6\,x}\,{\mathrm {e}}^{2\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*exp(2*x^2) - x - 3*x^2 - x*exp(6*x)*exp(2*x^2)

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sympy [A]  time = 0.29, size = 26, normalized size = 1.08 \begin {gather*} - 3 x^{2} + x e^{2 x^{2}} - x e^{2 x^{2} + 6 x} - x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x**2-6*x-1)*exp(2*x**2+6*x)+(4*x**2+1)*exp(2*x**2)-6*x-1,x)

[Out]

-3*x**2 + x*exp(2*x**2) - x*exp(2*x**2 + 6*x) - x

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