3.80.67 \(\int \frac {-1-3 x-6 x^2-7 x^3-6 x^4-3 x^5-x^6+e^{\frac {2+4 x+6 x^2+5 x^3+2 x^4}{1+2 x+3 x^2+2 x^3+x^4}} (3 x^2+x^3-x^4)}{1+3 x+6 x^2+7 x^3+6 x^4+3 x^5+x^6} \, dx\)

Optimal. Leaf size=18 \[ e^{2+\frac {x}{\left (1+\frac {1}{x}+x\right )^2}}-x \]

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Rubi [F]  time = 6.44, 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 {-1-3 x-6 x^2-7 x^3-6 x^4-3 x^5-x^6+\exp \left (\frac {2+4 x+6 x^2+5 x^3+2 x^4}{1+2 x+3 x^2+2 x^3+x^4}\right ) \left (3 x^2+x^3-x^4\right )}{1+3 x+6 x^2+7 x^3+6 x^4+3 x^5+x^6} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-1 - 3*x - 6*x^2 - 7*x^3 - 6*x^4 - 3*x^5 - x^6 + E^((2 + 4*x + 6*x^2 + 5*x^3 + 2*x^4)/(1 + 2*x + 3*x^2 +
2*x^3 + x^4))*(3*x^2 + x^3 - x^4))/(1 + 3*x + 6*x^2 + 7*x^3 + 6*x^4 + 3*x^5 + x^6),x]

[Out]

(-10*x)/3 - (5*x^2)/6 + (2 + x)/(2*(1 + x + x^2)^2) + (x*(2 + x))/(1 + x + x^2)^2 + (x^3*(2 + x))/(1 + x + x^2
)^2 + (x^4*(2 + x))/(2*(1 + x + x^2)^2) + (x^5*(2 + x))/(6*(1 + x + x^2)^2) - (1 + 2*x)/(6*(1 + x + x^2)^2) -
(7*x^3*(1 + 2*x))/(6*(1 + x + x^2)^2) - (2*(1 + x))/(1 + x + x^2) + (5*x*(2 + x))/(6*(1 + x + x^2)) + (1 + 2*x
)/(6*(1 + x + x^2)) + (x^3*(3 + 2*x))/(3*(1 + x + x^2)) + (x^2*(5 + 3*x))/(2*(1 + x + x^2)) + (((16*I)/3)*Defe
r[Int][E^((2 + 4*x + 6*x^2 + 5*x^3 + 2*x^4)/(1 + x + x^2)^2)/(-1 + I*Sqrt[3] - 2*x)^3, x])/Sqrt[3] - (16*(3 +
I*Sqrt[3])*Defer[Int][E^((2 + 4*x + 6*x^2 + 5*x^3 + 2*x^4)/(1 + x + x^2)^2)/(-1 + I*Sqrt[3] - 2*x)^3, x])/9 -
(4*Defer[Int][E^((2 + 4*x + 6*x^2 + 5*x^3 + 2*x^4)/(1 + x + x^2)^2)/(-1 + I*Sqrt[3] - 2*x)^2, x])/3 + 2*(1 - I
*Sqrt[3])*Defer[Int][E^((2 + 4*x + 6*x^2 + 5*x^3 + 2*x^4)/(1 + x + x^2)^2)/(-1 + I*Sqrt[3] - 2*x)^2, x] - (8*(
3 - I*Sqrt[3])*Defer[Int][E^((2 + 4*x + 6*x^2 + 5*x^3 + 2*x^4)/(1 + x + x^2)^2)/(-1 + I*Sqrt[3] - 2*x)^2, x])/
9 + (((16*I)/3)*Defer[Int][E^((2 + 4*x + 6*x^2 + 5*x^3 + 2*x^4)/(1 + x + x^2)^2)/(1 + I*Sqrt[3] + 2*x)^3, x])/
Sqrt[3] + (16*(3 - I*Sqrt[3])*Defer[Int][E^((2 + 4*x + 6*x^2 + 5*x^3 + 2*x^4)/(1 + x + x^2)^2)/(1 + I*Sqrt[3]
+ 2*x)^3, x])/9 - (4*Defer[Int][E^((2 + 4*x + 6*x^2 + 5*x^3 + 2*x^4)/(1 + x + x^2)^2)/(1 + I*Sqrt[3] + 2*x)^2,
x])/3 + 2*(1 + I*Sqrt[3])*Defer[Int][E^((2 + 4*x + 6*x^2 + 5*x^3 + 2*x^4)/(1 + x + x^2)^2)/(1 + I*Sqrt[3] + 2
*x)^2, x] - (8*(3 + I*Sqrt[3])*Defer[Int][E^((2 + 4*x + 6*x^2 + 5*x^3 + 2*x^4)/(1 + x + x^2)^2)/(1 + I*Sqrt[3]
+ 2*x)^2, x])/9

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-1-3 x-6 x^2-7 x^3-6 x^4-3 x^5-x^6+e^{\frac {2+4 x+6 x^2+5 x^3+2 x^4}{\left (1+x+x^2\right )^2}} x^2 \left (3+x-x^2\right )}{\left (1+x+x^2\right )^3} \, dx\\ &=\int \left (-\frac {1}{\left (1+x+x^2\right )^3}-\frac {3 x}{\left (1+x+x^2\right )^3}-\frac {6 x^2}{\left (1+x+x^2\right )^3}-\frac {7 x^3}{\left (1+x+x^2\right )^3}-\frac {6 x^4}{\left (1+x+x^2\right )^3}-\frac {3 x^5}{\left (1+x+x^2\right )^3}-\frac {x^6}{\left (1+x+x^2\right )^3}+\frac {e^{\frac {2+4 x+6 x^2+5 x^3+2 x^4}{\left (1+x+x^2\right )^2}} x^2 \left (3+x-x^2\right )}{\left (1+x+x^2\right )^3}\right ) \, dx\\ &=-\left (3 \int \frac {x}{\left (1+x+x^2\right )^3} \, dx\right )-3 \int \frac {x^5}{\left (1+x+x^2\right )^3} \, dx-6 \int \frac {x^2}{\left (1+x+x^2\right )^3} \, dx-6 \int \frac {x^4}{\left (1+x+x^2\right )^3} \, dx-7 \int \frac {x^3}{\left (1+x+x^2\right )^3} \, dx-\int \frac {1}{\left (1+x+x^2\right )^3} \, dx-\int \frac {x^6}{\left (1+x+x^2\right )^3} \, dx+\int \frac {e^{\frac {2+4 x+6 x^2+5 x^3+2 x^4}{\left (1+x+x^2\right )^2}} x^2 \left (3+x-x^2\right )}{\left (1+x+x^2\right )^3} \, dx\\ &=\frac {2+x}{2 \left (1+x+x^2\right )^2}+\frac {x (2+x)}{\left (1+x+x^2\right )^2}+\frac {x^3 (2+x)}{\left (1+x+x^2\right )^2}+\frac {x^4 (2+x)}{2 \left (1+x+x^2\right )^2}+\frac {x^5 (2+x)}{6 \left (1+x+x^2\right )^2}-\frac {1+2 x}{6 \left (1+x+x^2\right )^2}-\frac {7 x^3 (1+2 x)}{6 \left (1+x+x^2\right )^2}-\frac {1}{6} \int \frac {x^4 (10+2 x)}{\left (1+x+x^2\right )^2} \, dx-\frac {1}{2} \int \frac {x^3 (8+x)}{\left (1+x+x^2\right )^2} \, dx+\frac {3}{2} \int \frac {1}{\left (1+x+x^2\right )^2} \, dx+\frac {7}{2} \int \frac {x^2}{\left (1+x+x^2\right )^2} \, dx-6 \int \frac {x^2}{\left (1+x+x^2\right )^2} \, dx-\int \frac {1}{\left (1+x+x^2\right )^2} \, dx-\int \frac {2-2 x}{\left (1+x+x^2\right )^2} \, dx+\int \left (\frac {e^{\frac {2+4 x+6 x^2+5 x^3+2 x^4}{\left (1+x+x^2\right )^2}}}{-1-x-x^2}-\frac {2 e^{\frac {2+4 x+6 x^2+5 x^3+2 x^4}{\left (1+x+x^2\right )^2}} (1+2 x)}{\left (1+x+x^2\right )^3}+\frac {3 e^{\frac {2+4 x+6 x^2+5 x^3+2 x^4}{\left (1+x+x^2\right )^2}} (1+x)}{\left (1+x+x^2\right )^2}\right ) \, dx\\ &=\frac {2+x}{2 \left (1+x+x^2\right )^2}+\frac {x (2+x)}{\left (1+x+x^2\right )^2}+\frac {x^3 (2+x)}{\left (1+x+x^2\right )^2}+\frac {x^4 (2+x)}{2 \left (1+x+x^2\right )^2}+\frac {x^5 (2+x)}{6 \left (1+x+x^2\right )^2}-\frac {1+2 x}{6 \left (1+x+x^2\right )^2}-\frac {7 x^3 (1+2 x)}{6 \left (1+x+x^2\right )^2}-\frac {2 (1+x)}{1+x+x^2}+\frac {5 x (2+x)}{6 \left (1+x+x^2\right )}+\frac {1+2 x}{6 \left (1+x+x^2\right )}+\frac {x^3 (3+2 x)}{3 \left (1+x+x^2\right )}+\frac {x^2 (5+3 x)}{2 \left (1+x+x^2\right )}+\frac {1}{18} \int \frac {(-54-30 x) x^2}{1+x+x^2} \, dx+\frac {1}{6} \int \frac {(-30-12 x) x}{1+x+x^2} \, dx-\frac {2}{3} \int \frac {1}{1+x+x^2} \, dx-2 \int \frac {e^{\frac {2+4 x+6 x^2+5 x^3+2 x^4}{\left (1+x+x^2\right )^2}} (1+2 x)}{\left (1+x+x^2\right )^3} \, dx-2 \int \frac {1}{1+x+x^2} \, dx+\frac {7}{3} \int \frac {1}{1+x+x^2} \, dx+3 \int \frac {e^{\frac {2+4 x+6 x^2+5 x^3+2 x^4}{\left (1+x+x^2\right )^2}} (1+x)}{\left (1+x+x^2\right )^2} \, dx-4 \int \frac {1}{1+x+x^2} \, dx+\int \frac {e^{\frac {2+4 x+6 x^2+5 x^3+2 x^4}{\left (1+x+x^2\right )^2}}}{-1-x-x^2} \, dx+\int \frac {1}{1+x+x^2} \, dx\\ &=-2 x+\frac {2+x}{2 \left (1+x+x^2\right )^2}+\frac {x (2+x)}{\left (1+x+x^2\right )^2}+\frac {x^3 (2+x)}{\left (1+x+x^2\right )^2}+\frac {x^4 (2+x)}{2 \left (1+x+x^2\right )^2}+\frac {x^5 (2+x)}{6 \left (1+x+x^2\right )^2}-\frac {1+2 x}{6 \left (1+x+x^2\right )^2}-\frac {7 x^3 (1+2 x)}{6 \left (1+x+x^2\right )^2}-\frac {2 (1+x)}{1+x+x^2}+\frac {5 x (2+x)}{6 \left (1+x+x^2\right )}+\frac {1+2 x}{6 \left (1+x+x^2\right )}+\frac {x^3 (3+2 x)}{3 \left (1+x+x^2\right )}+\frac {x^2 (5+3 x)}{2 \left (1+x+x^2\right )}+\frac {1}{18} \int \left (-24-30 x+\frac {6 (4+9 x)}{1+x+x^2}\right ) \, dx+\frac {1}{6} \int \frac {12-18 x}{1+x+x^2} \, dx+\frac {4}{3} \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 x\right )-2 \int \left (\frac {e^{\frac {2+4 x+6 x^2+5 x^3+2 x^4}{\left (1+x+x^2\right )^2}}}{\left (1+x+x^2\right )^3}+\frac {2 e^{\frac {2+4 x+6 x^2+5 x^3+2 x^4}{\left (1+x+x^2\right )^2}} x}{\left (1+x+x^2\right )^3}\right ) \, dx-2 \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 x\right )+3 \int \left (\frac {e^{\frac {2+4 x+6 x^2+5 x^3+2 x^4}{\left (1+x+x^2\right )^2}}}{\left (1+x+x^2\right )^2}+\frac {e^{\frac {2+4 x+6 x^2+5 x^3+2 x^4}{\left (1+x+x^2\right )^2}} x}{\left (1+x+x^2\right )^2}\right ) \, dx+4 \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 x\right )-\frac {14}{3} \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 x\right )+8 \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 x\right )+\int \left (-\frac {2 i e^{\frac {2+4 x+6 x^2+5 x^3+2 x^4}{\left (1+x+x^2\right )^2}}}{\sqrt {3} \left (-1+i \sqrt {3}-2 x\right )}-\frac {2 i e^{\frac {2+4 x+6 x^2+5 x^3+2 x^4}{\left (1+x+x^2\right )^2}}}{\sqrt {3} \left (1+i \sqrt {3}+2 x\right )}\right ) \, dx\\ &=-\frac {10 x}{3}-\frac {5 x^2}{6}+\frac {2+x}{2 \left (1+x+x^2\right )^2}+\frac {x (2+x)}{\left (1+x+x^2\right )^2}+\frac {x^3 (2+x)}{\left (1+x+x^2\right )^2}+\frac {x^4 (2+x)}{2 \left (1+x+x^2\right )^2}+\frac {x^5 (2+x)}{6 \left (1+x+x^2\right )^2}-\frac {1+2 x}{6 \left (1+x+x^2\right )^2}-\frac {7 x^3 (1+2 x)}{6 \left (1+x+x^2\right )^2}-\frac {2 (1+x)}{1+x+x^2}+\frac {5 x (2+x)}{6 \left (1+x+x^2\right )}+\frac {1+2 x}{6 \left (1+x+x^2\right )}+\frac {x^3 (3+2 x)}{3 \left (1+x+x^2\right )}+\frac {x^2 (5+3 x)}{2 \left (1+x+x^2\right )}-\frac {20 \tan ^{-1}\left (\frac {1+2 x}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {1}{3} \int \frac {4+9 x}{1+x+x^2} \, dx-\frac {3}{2} \int \frac {1+2 x}{1+x+x^2} \, dx-2 \int \frac {e^{\frac {2+4 x+6 x^2+5 x^3+2 x^4}{\left (1+x+x^2\right )^2}}}{\left (1+x+x^2\right )^3} \, dx+3 \int \frac {e^{\frac {2+4 x+6 x^2+5 x^3+2 x^4}{\left (1+x+x^2\right )^2}}}{\left (1+x+x^2\right )^2} \, dx+3 \int \frac {e^{\frac {2+4 x+6 x^2+5 x^3+2 x^4}{\left (1+x+x^2\right )^2}} x}{\left (1+x+x^2\right )^2} \, dx+\frac {7}{2} \int \frac {1}{1+x+x^2} \, dx-4 \int \frac {e^{\frac {2+4 x+6 x^2+5 x^3+2 x^4}{\left (1+x+x^2\right )^2}} x}{\left (1+x+x^2\right )^3} \, dx-\frac {(2 i) \int \frac {e^{\frac {2+4 x+6 x^2+5 x^3+2 x^4}{\left (1+x+x^2\right )^2}}}{-1+i \sqrt {3}-2 x} \, dx}{\sqrt {3}}-\frac {(2 i) \int \frac {e^{\frac {2+4 x+6 x^2+5 x^3+2 x^4}{\left (1+x+x^2\right )^2}}}{1+i \sqrt {3}+2 x} \, dx}{\sqrt {3}}\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.10, size = 28, normalized size = 1.56 \begin {gather*} e^{2+\frac {1}{\left (1+x+x^2\right )^2}+\frac {-1+x}{1+x+x^2}}-x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1 - 3*x - 6*x^2 - 7*x^3 - 6*x^4 - 3*x^5 - x^6 + E^((2 + 4*x + 6*x^2 + 5*x^3 + 2*x^4)/(1 + 2*x + 3*
x^2 + 2*x^3 + x^4))*(3*x^2 + x^3 - x^4))/(1 + 3*x + 6*x^2 + 7*x^3 + 6*x^4 + 3*x^5 + x^6),x]

[Out]

E^(2 + (1 + x + x^2)^(-2) + (-1 + x)/(1 + x + x^2)) - x

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fricas [B]  time = 1.17, size = 46, normalized size = 2.56 \begin {gather*} -x + e^{\left (\frac {2 \, x^{4} + 5 \, x^{3} + 6 \, x^{2} + 4 \, x + 2}{x^{4} + 2 \, x^{3} + 3 \, x^{2} + 2 \, x + 1}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^4+x^3+3*x^2)*exp((2*x^4+5*x^3+6*x^2+4*x+2)/(x^4+2*x^3+3*x^2+2*x+1))-x^6-3*x^5-6*x^4-7*x^3-6*x^2
-3*x-1)/(x^6+3*x^5+6*x^4+7*x^3+6*x^2+3*x+1),x, algorithm="fricas")

[Out]

-x + e^((2*x^4 + 5*x^3 + 6*x^2 + 4*x + 2)/(x^4 + 2*x^3 + 3*x^2 + 2*x + 1))

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giac [B]  time = 0.44, size = 126, normalized size = 7.00 \begin {gather*} -x + e^{\left (\frac {2 \, x^{4}}{x^{4} + 2 \, x^{3} + 3 \, x^{2} + 2 \, x + 1} + \frac {5 \, x^{3}}{x^{4} + 2 \, x^{3} + 3 \, x^{2} + 2 \, x + 1} + \frac {6 \, x^{2}}{x^{4} + 2 \, x^{3} + 3 \, x^{2} + 2 \, x + 1} + \frac {4 \, x}{x^{4} + 2 \, x^{3} + 3 \, x^{2} + 2 \, x + 1} + \frac {2}{x^{4} + 2 \, x^{3} + 3 \, x^{2} + 2 \, x + 1}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^4+x^3+3*x^2)*exp((2*x^4+5*x^3+6*x^2+4*x+2)/(x^4+2*x^3+3*x^2+2*x+1))-x^6-3*x^5-6*x^4-7*x^3-6*x^2
-3*x-1)/(x^6+3*x^5+6*x^4+7*x^3+6*x^2+3*x+1),x, algorithm="giac")

[Out]

-x + e^(2*x^4/(x^4 + 2*x^3 + 3*x^2 + 2*x + 1) + 5*x^3/(x^4 + 2*x^3 + 3*x^2 + 2*x + 1) + 6*x^2/(x^4 + 2*x^3 + 3
*x^2 + 2*x + 1) + 4*x/(x^4 + 2*x^3 + 3*x^2 + 2*x + 1) + 2/(x^4 + 2*x^3 + 3*x^2 + 2*x + 1))

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maple [A]  time = 0.27, size = 35, normalized size = 1.94

method	result	size
risch	\(-x +{\mathrm e}^{\frac {2 x^{4}+5 x^{3}+6 x^{2}+4 x +2}{\left (x^{2}+x +1\right )^{2}}}\)	\(35\)
norman	\(\frac {x^{4} {\mathrm e}^{\frac {2 x^{4}+5 x^{3}+6 x^{2}+4 x +2}{x^{4}+2 x^{3}+3 x^{2}+2 x +1}}-\frac {5 x^{3}}{3}-\frac {4 x^{4}}{3}+\frac {x}{3}-x^{5}+2 x \,{\mathrm e}^{\frac {2 x^{4}+5 x^{3}+6 x^{2}+4 x +2}{x^{4}+2 x^{3}+3 x^{2}+2 x +1}}+3 \,{\mathrm e}^{\frac {2 x^{4}+5 x^{3}+6 x^{2}+4 x +2}{x^{4}+2 x^{3}+3 x^{2}+2 x +1}} x^{2}+2 \,{\mathrm e}^{\frac {2 x^{4}+5 x^{3}+6 x^{2}+4 x +2}{x^{4}+2 x^{3}+3 x^{2}+2 x +1}} x^{3}+\frac {2}{3}+{\mathrm e}^{\frac {2 x^{4}+5 x^{3}+6 x^{2}+4 x +2}{x^{4}+2 x^{3}+3 x^{2}+2 x +1}}}{\left (x^{2}+x +1\right )^{2}}\)	\(257\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-x^4+x^3+3*x^2)*exp((2*x^4+5*x^3+6*x^2+4*x+2)/(x^4+2*x^3+3*x^2+2*x+1))-x^6-3*x^5-6*x^4-7*x^3-6*x^2-3*x-1
)/(x^6+3*x^5+6*x^4+7*x^3+6*x^2+3*x+1),x,method=_RETURNVERBOSE)

[Out]

-x+exp((2*x^4+5*x^3+6*x^2+4*x+2)/(x^2+x+1)^2)

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maxima [B]  time = 0.71, size = 305, normalized size = 16.94 \begin {gather*} -x + \frac {8 \, x^{3} + 18 \, x^{2} + 16 \, x + 9}{6 \, {\left (x^{4} + 2 \, x^{3} + 3 \, x^{2} + 2 \, x + 1\right )}} - \frac {8 \, x^{3} + 9 \, x^{2} + 8 \, x + 2}{2 \, {\left (x^{4} + 2 \, x^{3} + 3 \, x^{2} + 2 \, x + 1\right )}} - \frac {4 \, x^{3} + 6 \, x^{2} + 8 \, x + 3}{6 \, {\left (x^{4} + 2 \, x^{3} + 3 \, x^{2} + 2 \, x + 1\right )}} + \frac {7 \, {\left (2 \, x^{3} + 6 \, x^{2} + 4 \, x + 3\right )}}{6 \, {\left (x^{4} + 2 \, x^{3} + 3 \, x^{2} + 2 \, x + 1\right )}} + \frac {2 \, x^{3} + 3 \, x^{2} + 4 \, x + 3}{2 \, {\left (x^{4} + 2 \, x^{3} + 3 \, x^{2} + 2 \, x + 1\right )}} - \frac {2 \, x^{3} + 3 \, x^{2} + 2 \, x + 2}{x^{4} + 2 \, x^{3} + 3 \, x^{2} + 2 \, x + 1} + \frac {2 \, x^{3} - 3 \, x^{2} - 2 \, x - 3}{x^{4} + 2 \, x^{3} + 3 \, x^{2} + 2 \, x + 1} + e^{\left (\frac {x}{x^{2} + x + 1} + \frac {1}{x^{4} + 2 \, x^{3} + 3 \, x^{2} + 2 \, x + 1} - \frac {1}{x^{2} + x + 1} + 2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^4+x^3+3*x^2)*exp((2*x^4+5*x^3+6*x^2+4*x+2)/(x^4+2*x^3+3*x^2+2*x+1))-x^6-3*x^5-6*x^4-7*x^3-6*x^2
-3*x-1)/(x^6+3*x^5+6*x^4+7*x^3+6*x^2+3*x+1),x, algorithm="maxima")

[Out]

-x + 1/6*(8*x^3 + 18*x^2 + 16*x + 9)/(x^4 + 2*x^3 + 3*x^2 + 2*x + 1) - 1/2*(8*x^3 + 9*x^2 + 8*x + 2)/(x^4 + 2*
x^3 + 3*x^2 + 2*x + 1) - 1/6*(4*x^3 + 6*x^2 + 8*x + 3)/(x^4 + 2*x^3 + 3*x^2 + 2*x + 1) + 7/6*(2*x^3 + 6*x^2 +
4*x + 3)/(x^4 + 2*x^3 + 3*x^2 + 2*x + 1) + 1/2*(2*x^3 + 3*x^2 + 4*x + 3)/(x^4 + 2*x^3 + 3*x^2 + 2*x + 1) - (2*
x^3 + 3*x^2 + 2*x + 2)/(x^4 + 2*x^3 + 3*x^2 + 2*x + 1) + (2*x^3 - 3*x^2 - 2*x - 3)/(x^4 + 2*x^3 + 3*x^2 + 2*x
+ 1) + e^(x/(x^2 + x + 1) + 1/(x^4 + 2*x^3 + 3*x^2 + 2*x + 1) - 1/(x^2 + x + 1) + 2)

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mupad [B]  time = 7.72, size = 130, normalized size = 7.22 \begin {gather*} {\mathrm {e}}^{\frac {4\,x}{x^4+2\,x^3+3\,x^2+2\,x+1}}\,{\mathrm {e}}^{\frac {2\,x^4}{x^4+2\,x^3+3\,x^2+2\,x+1}}\,{\mathrm {e}}^{\frac {6\,x^2}{x^4+2\,x^3+3\,x^2+2\,x+1}}\,{\mathrm {e}}^{\frac {5\,x^3}{x^4+2\,x^3+3\,x^2+2\,x+1}}\,{\mathrm {e}}^{\frac {2}{x^4+2\,x^3+3\,x^2+2\,x+1}}-x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(3*x - exp((4*x + 6*x^2 + 5*x^3 + 2*x^4 + 2)/(2*x + 3*x^2 + 2*x^3 + x^4 + 1))*(3*x^2 + x^3 - x^4) + 6*x^2
+ 7*x^3 + 6*x^4 + 3*x^5 + x^6 + 1)/(3*x + 6*x^2 + 7*x^3 + 6*x^4 + 3*x^5 + x^6 + 1),x)

[Out]

exp((4*x)/(2*x + 3*x^2 + 2*x^3 + x^4 + 1))*exp((2*x^4)/(2*x + 3*x^2 + 2*x^3 + x^4 + 1))*exp((6*x^2)/(2*x + 3*x
^2 + 2*x^3 + x^4 + 1))*exp((5*x^3)/(2*x + 3*x^2 + 2*x^3 + x^4 + 1))*exp(2/(2*x + 3*x^2 + 2*x^3 + x^4 + 1)) - x

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sympy [B]  time = 0.26, size = 41, normalized size = 2.28 \begin {gather*} - x + e^{\frac {2 x^{4} + 5 x^{3} + 6 x^{2} + 4 x + 2}{x^{4} + 2 x^{3} + 3 x^{2} + 2 x + 1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x**4+x**3+3*x**2)*exp((2*x**4+5*x**3+6*x**2+4*x+2)/(x**4+2*x**3+3*x**2+2*x+1))-x**6-3*x**5-6*x**4
-7*x**3-6*x**2-3*x-1)/(x**6+3*x**5+6*x**4+7*x**3+6*x**2+3*x+1),x)

[Out]

-x + exp((2*x**4 + 5*x**3 + 6*x**2 + 4*x + 2)/(x**4 + 2*x**3 + 3*x**2 + 2*x + 1))

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