3.22.46 \(\int \frac {-x^2+e^{-x^5} (1+5 x^5)}{x^2} \, dx\)

Optimal. Leaf size=19 \[ -x+\frac {-e^{-x^5}+x}{x} \]

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Rubi [A]  time = 0.05, antiderivative size = 16, normalized size of antiderivative = 0.84, number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {14, 2288} \begin {gather*} -\frac {e^{-x^5}}{x}-x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-x^2 + (1 + 5*x^5)/E^x^5)/x^2,x]

[Out]

-(1/(E^x^5*x)) - 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 (-1+\frac {e^{-x^5} \left (1+5 x^5\right )}{x^2}\right ) \, dx\\ &=-x+\int \frac {e^{-x^5} \left (1+5 x^5\right )}{x^2} \, dx\\ &=-\frac {e^{-x^5}}{x}-x\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 16, normalized size = 0.84 \begin {gather*} -\frac {e^{-x^5}}{x}-x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-x^2 + (1 + 5*x^5)/E^x^5)/x^2,x]

[Out]

-(1/(E^x^5*x)) - x

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fricas [A]  time = 1.35, size = 15, normalized size = 0.79 \begin {gather*} -\frac {x^{2} + e^{\left (-x^{5}\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*x^5+1)*exp(-x^5)-x^2)/x^2,x, algorithm="fricas")

[Out]

-(x^2 + e^(-x^5))/x

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giac [A]  time = 0.24, size = 15, normalized size = 0.79 \begin {gather*} -\frac {x^{2} + e^{\left (-x^{5}\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*x^5+1)*exp(-x^5)-x^2)/x^2,x, algorithm="giac")

[Out]

-(x^2 + e^(-x^5))/x

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maple [A]  time = 0.04, size = 16, normalized size = 0.84

method	result	size
risch	\(-x -\frac {{\mathrm e}^{-x^{5}}}{x}\)	\(16\)
norman	\(\frac {-x^{2}-{\mathrm e}^{-x^{5}}}{x}\)	\(19\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((5*x^5+1)*exp(-x^5)-x^2)/x^2,x,method=_RETURNVERBOSE)

[Out]

-x-1/x*exp(-x^5)

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maxima [C]  time = 0.57, size = 34, normalized size = 1.79 \begin {gather*} -\frac {x^{4} \Gamma \left (\frac {4}{5}, x^{5}\right )}{{\left (x^{5}\right )}^{\frac {4}{5}}} - x - \frac {{\left (x^{5}\right )}^{\frac {1}{5}} \Gamma \left (-\frac {1}{5}, x^{5}\right )}{5 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*x^5+1)*exp(-x^5)-x^2)/x^2,x, algorithm="maxima")

[Out]

-x^4*gamma(4/5, x^5)/(x^5)^(4/5) - x - 1/5*(x^5)^(1/5)*gamma(-1/5, x^5)/x

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mupad [B]  time = 1.20, size = 15, normalized size = 0.79 \begin {gather*} -x-\frac {{\mathrm {e}}^{-x^5}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-x^5)*(5*x^5 + 1) - x^2)/x^2,x)

[Out]

- x - exp(-x^5)/x

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sympy [A]  time = 0.09, size = 10, normalized size = 0.53 \begin {gather*} - x - \frac {e^{- x^{5}}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*x**5+1)*exp(-x**5)-x**2)/x**2,x)

[Out]

-x - exp(-x**5)/x

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