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

Optimal. Leaf size=22 \[ -5+e^{\frac {11 x}{4}+\frac {x^4}{1+x^4}}+x \]

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Rubi [F]  time = 1.12, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \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 \end {gather*}

Verification is not applicable to the result.

[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 {gather*} \begin {aligned} \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 {aligned} \end {gather*}

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Mathematica [A]  time = 0.23, size = 28, normalized size = 1.27 \begin {gather*} \frac {1}{4} \left (4 e^{1+\frac {11 x}{4}-\frac {1}{1+x^4}}+4 x\right ) \end {gather*}

Antiderivative was successfully verified.

[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

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fricas [A]  time = 0.61, size = 26, normalized size = 1.18 \begin {gather*} x + e^{\left (\frac {11 \, x^{5} + 4 \, x^{4} + 11 \, x}{4 \, {\left (x^{4} + 1\right )}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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giac [A]  time = 0.30, size = 26, normalized size = 1.18 \begin {gather*} x + e^{\left (\frac {11 \, x^{5} + 4 \, x^{4} + 11 \, x}{4 \, {\left (x^{4} + 1\right )}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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maple [A]  time = 0.17, size = 26, normalized size = 1.18

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

Verification of antiderivative is not currently implemented for this CAS.

[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))

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maxima [A]  time = 0.56, size = 17, normalized size = 0.77 \begin {gather*} x + e^{\left (\frac {11}{4} \, x - \frac {1}{x^{4} + 1} + 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [B]  time = 4.03, size = 46, normalized size = 2.09 \begin {gather*} 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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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sympy [A]  time = 0.19, size = 22, normalized size = 1.00 \begin {gather*} x + e^{\frac {11 x^{5} + 4 x^{4} + 11 x}{4 x^{4} + 4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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