3.35.53 \(\int \frac {-1-x^2+e^{256+32 x+x^2} (1-32 x-2 x^2)}{x^2} \, dx\)

Optimal. Leaf size=20 \[ \frac {1-e^{(16+x)^2}-x^2}{x} \]

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Rubi [A]  time = 0.08, antiderivative size = 34, normalized size of antiderivative = 1.70, number of steps used = 5, number of rules used = 2, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {14, 2288} \begin {gather*} -\frac {e^{x^2+32 x+256} \left (x^2+16 x\right )}{(x+16) x^2}-x+\frac {1}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-1 - x^2 + E^(256 + 32*x + x^2)*(1 - 32*x - 2*x^2))/x^2,x]

[Out]

x^(-1) - x - (E^(256 + 32*x + x^2)*(16*x + x^2))/(x^2*(16 + x))

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

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} &=\int \left (\frac {e^{256+32 x+x^2} \left (1-32 x-2 x^2\right )}{x^2}+\frac {-1-x^2}{x^2}\right ) \, dx\\ &=\int \frac {e^{256+32 x+x^2} \left (1-32 x-2 x^2\right )}{x^2} \, dx+\int \frac {-1-x^2}{x^2} \, dx\\ &=-\frac {e^{256+32 x+x^2} \left (16 x+x^2\right )}{x^2 (16+x)}+\int \left (-1-\frac {1}{x^2}\right ) \, dx\\ &=\frac {1}{x}-x-\frac {e^{256+32 x+x^2} \left (16 x+x^2\right )}{x^2 (16+x)}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.04, size = 17, normalized size = 0.85 \begin {gather*} -\frac {-1+e^{(16+x)^2}+x^2}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1 - x^2 + E^(256 + 32*x + x^2)*(1 - 32*x - 2*x^2))/x^2,x]

[Out]

-((-1 + E^(16 + x)^2 + x^2)/x)

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fricas [A]  time = 1.44, size = 19, normalized size = 0.95 \begin {gather*} -\frac {x^{2} + e^{\left (x^{2} + 32 \, x + 256\right )} - 1}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^2-32*x+1)*exp(x^2+32*x+256)-x^2-1)/x^2,x, algorithm="fricas")

[Out]

-(x^2 + e^(x^2 + 32*x + 256) - 1)/x

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giac [A]  time = 0.22, size = 19, normalized size = 0.95 \begin {gather*} -\frac {x^{2} + e^{\left (x^{2} + 32 \, x + 256\right )} - 1}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^2-32*x+1)*exp(x^2+32*x+256)-x^2-1)/x^2,x, algorithm="giac")

[Out]

-(x^2 + e^(x^2 + 32*x + 256) - 1)/x

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maple [A]  time = 0.05, size = 19, normalized size = 0.95

method	result	size
risch	\(\frac {1}{x}-x -\frac {{\mathrm e}^{\left (x +16\right )^{2}}}{x}\)	\(19\)
norman	\(\frac {1-x^{2}-{\mathrm e}^{x^{2}+32 x +256}}{x}\)	\(23\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-2*x^2-32*x+1)*exp(x^2+32*x+256)-x^2-1)/x^2,x,method=_RETURNVERBOSE)

[Out]

1/x-x-exp((x+16)^2)/x

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} i \, \sqrt {\pi } \operatorname {erf}\left (i \, x + 16 i\right ) - x + \frac {1}{x} - \int \frac {{\left (32 \, x e^{256} - e^{256}\right )} e^{\left (x^{2} + 32 \, x\right )}}{x^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^2-32*x+1)*exp(x^2+32*x+256)-x^2-1)/x^2,x, algorithm="maxima")

[Out]

I*sqrt(pi)*erf(I*x + 16*I) - x + 1/x - integrate((32*x*e^256 - e^256)*e^(x^2 + 32*x)/x^2, x)

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mupad [B]  time = 0.08, size = 20, normalized size = 1.00 \begin {gather*} -x-\frac {{\mathrm {e}}^{x^2+32\,x+256}-1}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(32*x + x^2 + 256)*(32*x + 2*x^2 - 1) + x^2 + 1)/x^2,x)

[Out]

- x - (exp(32*x + x^2 + 256) - 1)/x

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sympy [A]  time = 0.11, size = 15, normalized size = 0.75 \begin {gather*} - x - \frac {e^{x^{2} + 32 x + 256}}{x} + \frac {1}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x**2-32*x+1)*exp(x**2+32*x+256)-x**2-1)/x**2,x)

[Out]

-x - exp(x**2 + 32*x + 256)/x + 1/x

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