\(\int \frac {e^6-4 x^2-2 e^8 x^2-6 x^3+e^4 (6 x^2+3 x^3)}{e^6 x} \, dx\) [4987]

Optimal result

Integrand size = 44, antiderivative size = 24 \[ \int \frac {e^6-4 x^2-2 e^8 x^2-6 x^3+e^4 \left (6 x^2+3 x^3\right )}{e^6 x} \, dx=-\frac {\left (-2+e^4\right ) \left (-1+e^4-x\right ) x^2}{e^6}+\log (x) \]

[Out]

ln(x)-(exp(4)-x-1)/exp(3)^2*x^2*(exp(4)-2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.50, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.068, Rules used = {6, 12, 14} \[ \int \frac {e^6-4 x^2-2 e^8 x^2-6 x^3+e^4 \left (6 x^2+3 x^3\right )}{e^6 x} \, dx=-\frac {\left (2-e^4\right ) x^3}{e^6}-\frac {\left (2-3 e^4+e^8\right ) x^2}{e^6}+\log (x) \]

[In]

Int[(E^6 - 4*x^2 - 2*E^8*x^2 - 6*x^3 + E^4*(6*x^2 + 3*x^3))/(E^6*x),x]

[Out]

-(((2 - 3*E^4 + E^8)*x^2)/E^6) - ((2 - E^4)*x^3)/E^6 + Log[x]

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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]]

Rubi steps \begin{align*} \text {integral}& = \int \frac {e^6+\left (-4-2 e^8\right ) x^2-6 x^3+e^4 \left (6 x^2+3 x^3\right )}{e^6 x} \, dx \\ & = \frac {\int \frac {e^6+\left (-4-2 e^8\right ) x^2-6 x^3+e^4 \left (6 x^2+3 x^3\right )}{x} \, dx}{e^6} \\ & = \frac {\int \left (\frac {e^6}{x}-2 \left (2-3 e^4+e^8\right ) x-3 \left (2-e^4\right ) x^2\right ) \, dx}{e^6} \\ & = -\frac {\left (2-3 e^4+e^8\right ) x^2}{e^6}-\frac {\left (2-e^4\right ) x^3}{e^6}+\log (x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.75 \[ \int \frac {e^6-4 x^2-2 e^8 x^2-6 x^3+e^4 \left (6 x^2+3 x^3\right )}{e^6 x} \, dx=-\frac {2 x^2}{e^6}+\frac {3 x^2}{e^2}-e^2 x^2-\frac {2 x^3}{e^6}+\frac {x^3}{e^2}+\log (x) \]

[In]

Integrate[(E^6 - 4*x^2 - 2*E^8*x^2 - 6*x^3 + E^4*(6*x^2 + 3*x^3))/(E^6*x),x]

[Out]

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

Maple [A] (verified)

Time = 0.08 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.71

method	result	size
norman	\(\left (\left ({\mathrm e}^{4}-2\right ) {\mathrm e}^{-3} x^{3}-\left ({\mathrm e}^{8}-3 \,{\mathrm e}^{4}+2\right ) {\mathrm e}^{-3} x^{2}\right ) {\mathrm e}^{-3}+\ln \left (x \right )\)	\(41\)
default	\({\mathrm e}^{-6} \left (x^{3} {\mathrm e}^{4}-x^{2} {\mathrm e}^{8}+3 x^{2} {\mathrm e}^{4}-2 x^{3}-2 x^{2}+{\mathrm e}^{6} \ln \left (x \right )\right )\)	\(42\)
risch	\(-{\mathrm e}^{-6} {\mathrm e}^{8} x^{2}+{\mathrm e}^{-6} {\mathrm e}^{4} x^{3}+3 \,{\mathrm e}^{-6} {\mathrm e}^{4} x^{2}-2 \,{\mathrm e}^{-6} x^{3}-2 \,{\mathrm e}^{-6} x^{2}+\ln \left (x \right )\)	\(46\)
parallelrisch	\({\mathrm e}^{-6} \left (x^{3} {\mathrm e}^{4}-x^{2} {\mathrm e}^{8}+3 x^{2} {\mathrm e}^{4}-2 x^{3}-2 x^{2}+{\mathrm e}^{6} \ln \left (x \right )\right )\)	\(46\)

[In]

int((-2*x^2*exp(4)^2+(3*x^3+6*x^2)*exp(4)+exp(3)^2-6*x^3-4*x^2)/x/exp(3)^2,x,method=_RETURNVERBOSE)

[Out]

((exp(4)-2)/exp(3)*x^3-(exp(4)^2-3*exp(4)+2)/exp(3)*x^2)/exp(3)+ln(x)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.67 \[ \int \frac {e^6-4 x^2-2 e^8 x^2-6 x^3+e^4 \left (6 x^2+3 x^3\right )}{e^6 x} \, dx=-{\left (2 \, x^{3} + x^{2} e^{8} + 2 \, x^{2} - {\left (x^{3} + 3 \, x^{2}\right )} e^{4} - e^{6} \log \left (x\right )\right )} e^{\left (-6\right )} \]

[In]

integrate((-2*x^2*exp(4)^2+(3*x^3+6*x^2)*exp(4)+exp(3)^2-6*x^3-4*x^2)/x/exp(3)^2,x, algorithm="fricas")

[Out]

-(2*x^3 + x^2*e^8 + 2*x^2 - (x^3 + 3*x^2)*e^4 - e^6*log(x))*e^(-6)

Sympy [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.29 \[ \int \frac {e^6-4 x^2-2 e^8 x^2-6 x^3+e^4 \left (6 x^2+3 x^3\right )}{e^6 x} \, dx=\frac {- x^{3} \cdot \left (2 - e^{4}\right ) - x^{2} \left (- 3 e^{4} + 2 + e^{8}\right ) + e^{6} \log {\left (x \right )}}{e^{6}} \]

[In]

integrate((-2*x**2*exp(4)**2+(3*x**3+6*x**2)*exp(4)+exp(3)**2-6*x**3-4*x**2)/x/exp(3)**2,x)

[Out]

(-x**3*(2 - exp(4)) - x**2*(-3*exp(4) + 2 + exp(8)) + exp(6)*log(x))*exp(-6)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.25 \[ \int \frac {e^6-4 x^2-2 e^8 x^2-6 x^3+e^4 \left (6 x^2+3 x^3\right )}{e^6 x} \, dx={\left (x^{3} {\left (e^{4} - 2\right )} - x^{2} {\left (e^{8} - 3 \, e^{4} + 2\right )} + e^{6} \log \left (x\right )\right )} e^{\left (-6\right )} \]

[In]

integrate((-2*x^2*exp(4)^2+(3*x^3+6*x^2)*exp(4)+exp(3)^2-6*x^3-4*x^2)/x/exp(3)^2,x, algorithm="maxima")

[Out]

(x^3*(e^4 - 2) - x^2*(e^8 - 3*e^4 + 2) + e^6*log(x))*e^(-6)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.67 \[ \int \frac {e^6-4 x^2-2 e^8 x^2-6 x^3+e^4 \left (6 x^2+3 x^3\right )}{e^6 x} \, dx={\left (x^{3} e^{4} - 2 \, x^{3} - x^{2} e^{8} + 3 \, x^{2} e^{4} - 2 \, x^{2} + e^{6} \log \left ({\left | x \right |}\right )\right )} e^{\left (-6\right )} \]

[In]

integrate((-2*x^2*exp(4)^2+(3*x^3+6*x^2)*exp(4)+exp(3)^2-6*x^3-4*x^2)/x/exp(3)^2,x, algorithm="giac")

[Out]

(x^3*e^4 - 2*x^3 - x^2*e^8 + 3*x^2*e^4 - 2*x^2 + e^6*log(abs(x)))*e^(-6)

Mupad [B] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.38 \[ \int \frac {e^6-4 x^2-2 e^8 x^2-6 x^3+e^4 \left (6 x^2+3 x^3\right )}{e^6 x} \, dx=\ln \left (x\right )-\frac {x^2\,{\mathrm {e}}^{-6}\,\left (2\,{\mathrm {e}}^8-6\,{\mathrm {e}}^4+4\right )}{2}+\frac {x^3\,{\mathrm {e}}^{-6}\,\left (3\,{\mathrm {e}}^4-6\right )}{3} \]

[In]

int(-(exp(-6)*(2*x^2*exp(8) - exp(4)*(6*x^2 + 3*x^3) - exp(6) + 4*x^2 + 6*x^3))/x,x)

[Out]

log(x) - (x^2*exp(-6)*(2*exp(8) - 6*exp(4) + 4))/2 + (x^3*exp(-6)*(3*exp(4) - 6))/3