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

Optimal. Leaf size=21 \[ \frac {e^{1-e^4+e^{e^4}+x^2}}{x} \]

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Rubi [A]  time = 0.07, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {2288} \begin {gather*} \frac {e^{x^2+e^{e^4}-e^4+1}}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

E^(1 - E^4 + E^E^4 + x^2)/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} &=\frac {e^{1-e^4+e^{e^4}+x^2}}{x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 21, normalized size = 1.00 \begin {gather*} \frac {e^{1-e^4+e^{e^4}+x^2}}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

E^(1 - E^4 + E^E^4 + x^2)/x

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fricas [A]  time = 0.51, size = 17, normalized size = 0.81 \begin {gather*} e^{\left (x^{2} - e^{4} + e^{\left (e^{4}\right )} - \log \relax (x) + 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^(x^2 - e^4 + e^(e^4) - log(x) + 1)

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giac [A]  time = 0.13, size = 17, normalized size = 0.81 \begin {gather*} \frac {e^{\left (x^{2} - e^{4} + e^{\left (e^{4}\right )} + 1\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^(x^2 - e^4 + e^(e^4) + 1)/x

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maple [A]  time = 0.03, size = 18, normalized size = 0.86

method	result	size
gosper	\({\mathrm e}^{-\ln \relax (x )+{\mathrm e}^{{\mathrm e}^{4}}-{\mathrm e}^{4}+x^{2}+1}\)	\(18\)
norman	\({\mathrm e}^{-\ln \relax (x )+{\mathrm e}^{{\mathrm e}^{4}}-{\mathrm e}^{4}+x^{2}+1}\)	\(18\)
risch	\(\frac {{\mathrm e}^{1+{\mathrm e}^{{\mathrm e}^{4}}-{\mathrm e}^{4}+x^{2}}}{x}\)	\(18\)
default	\({\mathrm e} \,{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{4}}} {\mathrm e}^{-{\mathrm e}^{4}} \sqrt {\pi }\, \erfi \relax (x )-{\mathrm e} \,{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{4}}} {\mathrm e}^{-{\mathrm e}^{4}} \left (-\frac {{\mathrm e}^{x^{2}}}{x}+\sqrt {\pi }\, \erfi \relax (x )\right )\)	\(48\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp(-ln(x)+exp(exp(4))-exp(4)+x^2+1)

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maxima [C]  time = 0.41, size = 49, normalized size = 2.33 \begin {gather*} -i \, \sqrt {\pi } \operatorname {erf}\left (i \, x\right ) e^{\left (-e^{4} + e^{\left (e^{4}\right )} + 1\right )} + \frac {\sqrt {-x^{2}} e^{\left (-e^{4} + e^{\left (e^{4}\right )} + 1\right )} \Gamma \left (-\frac {1}{2}, -x^{2}\right )}{2 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-I*sqrt(pi)*erf(I*x)*e^(-e^4 + e^(e^4) + 1) + 1/2*sqrt(-x^2)*e^(-e^4 + e^(e^4) + 1)*gamma(-1/2, -x^2)/x

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mupad [B]  time = 5.36, size = 19, normalized size = 0.90 \begin {gather*} \frac {{\mathrm {e}}^{-{\mathrm {e}}^4}\,{\mathrm {e}}^{{\mathrm {e}}^{{\mathrm {e}}^4}}\,{\mathrm {e}}^{x^2}\,\mathrm {e}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(exp(-exp(4))*exp(exp(exp(4)))*exp(x^2)*exp(1))/x

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sympy [A]  time = 0.10, size = 15, normalized size = 0.71 \begin {gather*} \frac {e^{x^{2} - e^{4} + 1 + e^{e^{4}}}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp(x**2 - exp(4) + 1 + exp(exp(4)))/x

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