3.95.53 \(\int \frac {-x^9+6 x^{14}+e^{\frac {1-2 x^5+x^{10}}{x^8}} (8-6 x^5-2 x^{10})}{x^9} \, dx\)

Optimal. Leaf size=21 \[ 3-e^{\left (-\frac {1}{x^4}+x\right )^2}-x+x^6 \]

________________________________________________________________________________________

Rubi [A]  time = 0.40, antiderivative size = 24, normalized size of antiderivative = 1.14, number of steps used = 3, number of rules used = 2, integrand size = 44, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {14, 6706} \begin {gather*} x^6-e^{\frac {\left (1-x^5\right )^2}{x^8}}-x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-x^9 + 6*x^14 + E^((1 - 2*x^5 + x^10)/x^8)*(8 - 6*x^5 - 2*x^10))/x^9,x]

[Out]

-E^((1 - x^5)^2/x^8) - x + x^6

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 6706

Int[(F_)^(v_)*(u_), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Simp[(q*F^v)/Log[F], x] /;  !FalseQ[q]
] /; FreeQ[F, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-1+6 x^5-\frac {2 e^{\frac {\left (-1+x^5\right )^2}{x^8}} (-1+x) \left (1+x+x^2+x^3+x^4\right ) \left (4+x^5\right )}{x^9}\right ) \, dx\\ &=-x+x^6-2 \int \frac {e^{\frac {\left (-1+x^5\right )^2}{x^8}} (-1+x) \left (1+x+x^2+x^3+x^4\right ) \left (4+x^5\right )}{x^9} \, dx\\ &=-e^{\frac {\left (1-x^5\right )^2}{x^8}}-x+x^6\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.09, size = 23, normalized size = 1.10 \begin {gather*} -e^{\frac {1}{x^8}-\frac {2}{x^3}+x^2}-x+x^6 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-x^9 + 6*x^14 + E^((1 - 2*x^5 + x^10)/x^8)*(8 - 6*x^5 - 2*x^10))/x^9,x]

[Out]

-E^(x^(-8) - 2/x^3 + x^2) - x + x^6

________________________________________________________________________________________

fricas [A]  time = 0.60, size = 24, normalized size = 1.14 \begin {gather*} x^{6} - x - e^{\left (\frac {x^{10} - 2 \, x^{5} + 1}{x^{8}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^10-6*x^5+8)*exp((x^10-2*x^5+1)/x^8)+6*x^14-x^9)/x^9,x, algorithm="fricas")

[Out]

x^6 - x - e^((x^10 - 2*x^5 + 1)/x^8)

________________________________________________________________________________________

giac [A]  time = 0.20, size = 24, normalized size = 1.14 \begin {gather*} x^{6} - x - e^{\left (\frac {x^{10} - 2 \, x^{5} + 1}{x^{8}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^10-6*x^5+8)*exp((x^10-2*x^5+1)/x^8)+6*x^14-x^9)/x^9,x, algorithm="giac")

[Out]

x^6 - x - e^((x^10 - 2*x^5 + 1)/x^8)

________________________________________________________________________________________

maple [A]  time = 0.07, size = 34, normalized size = 1.62

method	result	size
norman	\(\frac {x^{14}-x^{9}-{\mathrm e}^{\frac {x^{10}-2 x^{5}+1}{x^{8}}} x^{8}}{x^{8}}\)	\(34\)
risch	\(x^{6}-x -{\mathrm e}^{\frac {\left (x -1\right )^{2} \left (x^{4}+x^{3}+x^{2}+x +1\right )^{2}}{x^{8}}}\)	\(34\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-2*x^10-6*x^5+8)*exp((x^10-2*x^5+1)/x^8)+6*x^14-x^9)/x^9,x,method=_RETURNVERBOSE)

[Out]

(x^14-x^9-exp((x^10-2*x^5+1)/x^8)*x^8)/x^8

________________________________________________________________________________________

maxima [A]  time = 0.42, size = 22, normalized size = 1.05 \begin {gather*} x^{6} - x - e^{\left (x^{2} - \frac {2}{x^{3}} + \frac {1}{x^{8}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^10-6*x^5+8)*exp((x^10-2*x^5+1)/x^8)+6*x^14-x^9)/x^9,x, algorithm="maxima")

[Out]

x^6 - x - e^(x^2 - 2/x^3 + 1/x^8)

________________________________________________________________________________________

mupad [B]  time = 7.85, size = 23, normalized size = 1.10 \begin {gather*} x^6-x-{\mathrm {e}}^{x^2}\,{\mathrm {e}}^{\frac {1}{x^8}}\,{\mathrm {e}}^{-\frac {2}{x^3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp((x^10 - 2*x^5 + 1)/x^8)*(6*x^5 + 2*x^10 - 8) + x^9 - 6*x^14)/x^9,x)

[Out]

x^6 - x - exp(x^2)*exp(1/x^8)*exp(-2/x^3)

________________________________________________________________________________________

sympy [A]  time = 0.18, size = 19, normalized size = 0.90 \begin {gather*} x^{6} - x - e^{\frac {x^{10} - 2 x^{5} + 1}{x^{8}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x**10-6*x**5+8)*exp((x**10-2*x**5+1)/x**8)+6*x**14-x**9)/x**9,x)

[Out]

x**6 - x - exp((x**10 - 2*x**5 + 1)/x**8)

________________________________________________________________________________________