\(\int \frac {4+8 x^4+4 x^8+e^{\frac {11 x+4 x^4+11 x^5}{4+4 x^4}} (11+16 x^3+22 x^4+11 x^8)}{4+8 x^4+4 x^8} \, dx\) [5221]

Optimal result

Integrand size = 71, antiderivative size = 22 \[ \int \frac {4+8 x^4+4 x^8+e^{\frac {11 x+4 x^4+11 x^5}{4+4 x^4}} \left (11+16 x^3+22 x^4+11 x^8\right )}{4+8 x^4+4 x^8} \, dx=-5+e^{\frac {11 x}{4}+\frac {x^4}{1+x^4}}+x \]

[Out]

x-5+exp(x^4/(x^4+1)+11/4*x)

Rubi [F]

\[ \int \frac {4+8 x^4+4 x^8+e^{\frac {11 x+4 x^4+11 x^5}{4+4 x^4}} \left (11+16 x^3+22 x^4+11 x^8\right )}{4+8 x^4+4 x^8} \, dx=\int \frac {4+8 x^4+4 x^8+e^{\frac {11 x+4 x^4+11 x^5}{4+4 x^4}} \left (11+16 x^3+22 x^4+11 x^8\right )}{4+8 x^4+4 x^8} \, dx \]

[In]

Int[(4 + 8*x^4 + 4*x^8 + E^((11*x + 4*x^4 + 11*x^5)/(4 + 4*x^4))*(11 + 16*x^3 + 22*x^4 + 11*x^8))/(4 + 8*x^4 +
4*x^8),x]

[Out]

x + (11*Defer[Int][E^((x*(11 + 4*x^3 + 11*x^4))/(4*(1 + x^4))), x])/4 - I*Defer[Int][(E^((x*(11 + 4*x^3 + 11*x
^4))/(4*(1 + x^4)))*x)/(I - x^2)^2, x] + I*Defer[Int][(E^((x*(11 + 4*x^3 + 11*x^4))/(4*(1 + x^4)))*x)/(I + x^2
)^2, x]

Rubi steps \begin{align*} \text {integral}& = 4 \int \frac {4+8 x^4+4 x^8+e^{\frac {11 x+4 x^4+11 x^5}{4+4 x^4}} \left (11+16 x^3+22 x^4+11 x^8\right )}{\left (4+4 x^4\right )^2} \, dx \\ & = 4 \int \left (\frac {1}{4}+\frac {e^{\frac {x \left (11+4 x^3+11 x^4\right )}{4 \left (1+x^4\right )}} \left (11+16 x^3+22 x^4+11 x^8\right )}{16 \left (1+x^4\right )^2}\right ) \, dx \\ & = x+\frac {1}{4} \int \frac {e^{\frac {x \left (11+4 x^3+11 x^4\right )}{4 \left (1+x^4\right )}} \left (11+16 x^3+22 x^4+11 x^8\right )}{\left (1+x^4\right )^2} \, dx \\ & = x+\frac {1}{4} \int \left (11 e^{\frac {x \left (11+4 x^3+11 x^4\right )}{4 \left (1+x^4\right )}}+\frac {16 e^{\frac {x \left (11+4 x^3+11 x^4\right )}{4 \left (1+x^4\right )}} x^3}{\left (1+x^4\right )^2}\right ) \, dx \\ & = x+\frac {11}{4} \int e^{\frac {x \left (11+4 x^3+11 x^4\right )}{4 \left (1+x^4\right )}} \, dx+4 \int \frac {e^{\frac {x \left (11+4 x^3+11 x^4\right )}{4 \left (1+x^4\right )}} x^3}{\left (1+x^4\right )^2} \, dx \\ & = x+\frac {11}{4} \int e^{\frac {x \left (11+4 x^3+11 x^4\right )}{4 \left (1+x^4\right )}} \, dx+4 \int \left (-\frac {i e^{\frac {x \left (11+4 x^3+11 x^4\right )}{4 \left (1+x^4\right )}} x}{4 \left (i-x^2\right )^2}+\frac {i e^{\frac {x \left (11+4 x^3+11 x^4\right )}{4 \left (1+x^4\right )}} x}{4 \left (i+x^2\right )^2}\right ) \, dx \\ & = x-i \int \frac {e^{\frac {x \left (11+4 x^3+11 x^4\right )}{4 \left (1+x^4\right )}} x}{\left (i-x^2\right )^2} \, dx+i \int \frac {e^{\frac {x \left (11+4 x^3+11 x^4\right )}{4 \left (1+x^4\right )}} x}{\left (i+x^2\right )^2} \, dx+\frac {11}{4} \int e^{\frac {x \left (11+4 x^3+11 x^4\right )}{4 \left (1+x^4\right )}} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.56 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.27 \[ \int \frac {4+8 x^4+4 x^8+e^{\frac {11 x+4 x^4+11 x^5}{4+4 x^4}} \left (11+16 x^3+22 x^4+11 x^8\right )}{4+8 x^4+4 x^8} \, dx=\frac {1}{4} \left (4 e^{1+\frac {11 x}{4}-\frac {1}{1+x^4}}+4 x\right ) \]

[In]

Integrate[(4 + 8*x^4 + 4*x^8 + E^((11*x + 4*x^4 + 11*x^5)/(4 + 4*x^4))*(11 + 16*x^3 + 22*x^4 + 11*x^8))/(4 + 8
*x^4 + 4*x^8),x]

[Out]

(4*E^(1 + (11*x)/4 - (1 + x^4)^(-1)) + 4*x)/4

Maple [A] (verified)

Time = 0.30 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.18

method	result	size
risch	\(x +{\mathrm e}^{\frac {x \left (11 x^{4}+4 x^{3}+11\right )}{4 x^{4}+4}}\)	\(26\)
parallelrisch	\(x +{\mathrm e}^{\frac {11 x^{5}+4 x^{4}+11 x}{4 x^{4}+4}}\)	\(27\)
parts	\(x +\frac {x^{4} {\mathrm e}^{\frac {11 x^{5}+4 x^{4}+11 x}{4 x^{4}+4}}+{\mathrm e}^{\frac {11 x^{5}+4 x^{4}+11 x}{4 x^{4}+4}}}{x^{4}+1}\)	\(66\)
norman	\(\frac {x +x^{5}+x^{4} {\mathrm e}^{\frac {11 x^{5}+4 x^{4}+11 x}{4 x^{4}+4}}+{\mathrm e}^{\frac {11 x^{5}+4 x^{4}+11 x}{4 x^{4}+4}}}{x^{4}+1}\)	\(68\)

[In]

int(((11*x^8+22*x^4+16*x^3+11)*exp((11*x^5+4*x^4+11*x)/(4*x^4+4))+4*x^8+8*x^4+4)/(4*x^8+8*x^4+4),x,method=_RET
URNVERBOSE)

[Out]

x+exp(1/4*x*(11*x^4+4*x^3+11)/(x^4+1))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.18 \[ \int \frac {4+8 x^4+4 x^8+e^{\frac {11 x+4 x^4+11 x^5}{4+4 x^4}} \left (11+16 x^3+22 x^4+11 x^8\right )}{4+8 x^4+4 x^8} \, dx=x + e^{\left (\frac {11 \, x^{5} + 4 \, x^{4} + 11 \, x}{4 \, {\left (x^{4} + 1\right )}}\right )} \]

[In]

integrate(((11*x^8+22*x^4+16*x^3+11)*exp((11*x^5+4*x^4+11*x)/(4*x^4+4))+4*x^8+8*x^4+4)/(4*x^8+8*x^4+4),x, algo
rithm="fricas")

[Out]

x + e^(1/4*(11*x^5 + 4*x^4 + 11*x)/(x^4 + 1))

Sympy [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {4+8 x^4+4 x^8+e^{\frac {11 x+4 x^4+11 x^5}{4+4 x^4}} \left (11+16 x^3+22 x^4+11 x^8\right )}{4+8 x^4+4 x^8} \, dx=x + e^{\frac {11 x^{5} + 4 x^{4} + 11 x}{4 x^{4} + 4}} \]

[In]

integrate(((11*x**8+22*x**4+16*x**3+11)*exp((11*x**5+4*x**4+11*x)/(4*x**4+4))+4*x**8+8*x**4+4)/(4*x**8+8*x**4+
4),x)

[Out]

x + exp((11*x**5 + 4*x**4 + 11*x)/(4*x**4 + 4))

Maxima [A] (verification not implemented)

none

Time = 0.37 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.77 \[ \int \frac {4+8 x^4+4 x^8+e^{\frac {11 x+4 x^4+11 x^5}{4+4 x^4}} \left (11+16 x^3+22 x^4+11 x^8\right )}{4+8 x^4+4 x^8} \, dx=x + e^{\left (\frac {11}{4} \, x - \frac {1}{x^{4} + 1} + 1\right )} \]

[In]

integrate(((11*x^8+22*x^4+16*x^3+11)*exp((11*x^5+4*x^4+11*x)/(4*x^4+4))+4*x^8+8*x^4+4)/(4*x^8+8*x^4+4),x, algo
rithm="maxima")

[Out]

x + e^(11/4*x - 1/(x^4 + 1) + 1)

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.18 \[ \int \frac {4+8 x^4+4 x^8+e^{\frac {11 x+4 x^4+11 x^5}{4+4 x^4}} \left (11+16 x^3+22 x^4+11 x^8\right )}{4+8 x^4+4 x^8} \, dx=x + e^{\left (\frac {11 \, x^{5} + 4 \, x^{4} + 11 \, x}{4 \, {\left (x^{4} + 1\right )}}\right )} \]

[In]

integrate(((11*x^8+22*x^4+16*x^3+11)*exp((11*x^5+4*x^4+11*x)/(4*x^4+4))+4*x^8+8*x^4+4)/(4*x^8+8*x^4+4),x, algo
rithm="giac")

[Out]

x + e^(1/4*(11*x^5 + 4*x^4 + 11*x)/(x^4 + 1))

Mupad [B] (verification not implemented)

Time = 12.11 (sec) , antiderivative size = 46, normalized size of antiderivative = 2.09 \[ \int \frac {4+8 x^4+4 x^8+e^{\frac {11 x+4 x^4+11 x^5}{4+4 x^4}} \left (11+16 x^3+22 x^4+11 x^8\right )}{4+8 x^4+4 x^8} \, dx=x+{\mathrm {e}}^{\frac {11\,x}{4\,x^4+4}}\,{\mathrm {e}}^{\frac {4\,x^4}{4\,x^4+4}}\,{\mathrm {e}}^{\frac {11\,x^5}{4\,x^4+4}} \]

[In]

int((exp((11*x + 4*x^4 + 11*x^5)/(4*x^4 + 4))*(16*x^3 + 22*x^4 + 11*x^8 + 11) + 8*x^4 + 4*x^8 + 4)/(8*x^4 + 4*
x^8 + 4),x)

[Out]

x + exp((11*x)/(4*x^4 + 4))*exp((4*x^4)/(4*x^4 + 4))*exp((11*x^5)/(4*x^4 + 4))