\(\int \frac {1}{81} (162+e^{-1+4 e^{\frac {256 x^4}{81}}+5 x} (405+4096 e^{\frac {256 x^4}{81}} x^3)) \, dx\) [9238]

Optimal result

Integrand size = 41, antiderivative size = 23 \[ \int \frac {1}{81} \left (162+e^{-1+4 e^{\frac {256 x^4}{81}}+5 x} \left (405+4096 e^{\frac {256 x^4}{81}} x^3\right )\right ) \, dx=2+e^{-1+4 e^{\frac {256 x^4}{81}}+5 x}+2 x \]

[Out]

2*x+exp(4*exp(256/81*x^4)+5*x-1)+2

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.049, Rules used = {12, 6838} \[ \int \frac {1}{81} \left (162+e^{-1+4 e^{\frac {256 x^4}{81}}+5 x} \left (405+4096 e^{\frac {256 x^4}{81}} x^3\right )\right ) \, dx=e^{4 e^{\frac {256 x^4}{81}}+5 x-1}+2 x \]

[In]

Int[(162 + E^(-1 + 4*E^((256*x^4)/81) + 5*x)*(405 + 4096*E^((256*x^4)/81)*x^3))/81,x]

[Out]

E^(-1 + 4*E^((256*x^4)/81) + 5*x) + 2*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 6838

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{align*} \text {integral}& = \frac {1}{81} \int \left (162+e^{-1+4 e^{\frac {256 x^4}{81}}+5 x} \left (405+4096 e^{\frac {256 x^4}{81}} x^3\right )\right ) \, dx \\ & = 2 x+\frac {1}{81} \int e^{-1+4 e^{\frac {256 x^4}{81}}+5 x} \left (405+4096 e^{\frac {256 x^4}{81}} x^3\right ) \, dx \\ & = e^{-1+4 e^{\frac {256 x^4}{81}}+5 x}+2 x \\ \end{align*}

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int \frac {1}{81} \left (162+e^{-1+4 e^{\frac {256 x^4}{81}}+5 x} \left (405+4096 e^{\frac {256 x^4}{81}} x^3\right )\right ) \, dx=e^{-1+4 e^{\frac {256 x^4}{81}}+5 x}+2 x \]

[In]

Integrate[(162 + E^(-1 + 4*E^((256*x^4)/81) + 5*x)*(405 + 4096*E^((256*x^4)/81)*x^3))/81,x]

[Out]

E^(-1 + 4*E^((256*x^4)/81) + 5*x) + 2*x

Maple [A] (verified)

Time = 0.15 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.83

method	result	size
default	\(2 x +{\mathrm e}^{4 \,{\mathrm e}^{\frac {256 x^{4}}{81}}+5 x -1}\)	\(19\)
norman	\(2 x +{\mathrm e}^{4 \,{\mathrm e}^{\frac {256 x^{4}}{81}}+5 x -1}\)	\(19\)
risch	\(2 x +{\mathrm e}^{4 \,{\mathrm e}^{\frac {256 x^{4}}{81}}+5 x -1}\)	\(19\)
parallelrisch	\(2 x +{\mathrm e}^{4 \,{\mathrm e}^{\frac {256 x^{4}}{81}}+5 x -1}\)	\(19\)

[In]

int(1/81*(4096*x^3*exp(256/81*x^4)+405)*exp(4*exp(256/81*x^4)+5*x-1)+2,x,method=_RETURNVERBOSE)

[Out]

2*x+exp(4*exp(256/81*x^4)+5*x-1)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.78 \[ \int \frac {1}{81} \left (162+e^{-1+4 e^{\frac {256 x^4}{81}}+5 x} \left (405+4096 e^{\frac {256 x^4}{81}} x^3\right )\right ) \, dx=2 \, x + e^{\left (5 \, x + 4 \, e^{\left (\frac {256}{81} \, x^{4}\right )} - 1\right )} \]

[In]

integrate(1/81*(4096*x^3*exp(256/81*x^4)+405)*exp(4*exp(256/81*x^4)+5*x-1)+2,x, algorithm="fricas")

[Out]

2*x + e^(5*x + 4*e^(256/81*x^4) - 1)

Sympy [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.83 \[ \int \frac {1}{81} \left (162+e^{-1+4 e^{\frac {256 x^4}{81}}+5 x} \left (405+4096 e^{\frac {256 x^4}{81}} x^3\right )\right ) \, dx=2 x + e^{5 x + 4 e^{\frac {256 x^{4}}{81}} - 1} \]

[In]

integrate(1/81*(4096*x**3*exp(256/81*x**4)+405)*exp(4*exp(256/81*x**4)+5*x-1)+2,x)

[Out]

2*x + exp(5*x + 4*exp(256*x**4/81) - 1)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.78 \[ \int \frac {1}{81} \left (162+e^{-1+4 e^{\frac {256 x^4}{81}}+5 x} \left (405+4096 e^{\frac {256 x^4}{81}} x^3\right )\right ) \, dx=2 \, x + e^{\left (5 \, x + 4 \, e^{\left (\frac {256}{81} \, x^{4}\right )} - 1\right )} \]

[In]

integrate(1/81*(4096*x^3*exp(256/81*x^4)+405)*exp(4*exp(256/81*x^4)+5*x-1)+2,x, algorithm="maxima")

[Out]

2*x + e^(5*x + 4*e^(256/81*x^4) - 1)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.78 \[ \int \frac {1}{81} \left (162+e^{-1+4 e^{\frac {256 x^4}{81}}+5 x} \left (405+4096 e^{\frac {256 x^4}{81}} x^3\right )\right ) \, dx=2 \, x + e^{\left (5 \, x + 4 \, e^{\left (\frac {256}{81} \, x^{4}\right )} - 1\right )} \]

[In]

integrate(1/81*(4096*x^3*exp(256/81*x^4)+405)*exp(4*exp(256/81*x^4)+5*x-1)+2,x, algorithm="giac")

[Out]

2*x + e^(5*x + 4*e^(256/81*x^4) - 1)

Mupad [B] (verification not implemented)

Time = 13.74 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {1}{81} \left (162+e^{-1+4 e^{\frac {256 x^4}{81}}+5 x} \left (405+4096 e^{\frac {256 x^4}{81}} x^3\right )\right ) \, dx={\mathrm {e}}^{-1}\,\left (2\,x\,\mathrm {e}+{\mathrm {e}}^{5\,x}\,{\mathrm {e}}^{4\,{\mathrm {e}}^{\frac {256\,x^4}{81}}}\right ) \]

[In]

int((exp(5*x + 4*exp((256*x^4)/81) - 1)*(4096*x^3*exp((256*x^4)/81) + 405))/81 + 2,x)

[Out]

exp(-1)*(2*x*exp(1) + exp(5*x)*exp(4*exp((256*x^4)/81)))