3.221 \(\int \frac{1+x}{1+x^5} \, dx\)

Optimal. Leaf size=109 \[ -\frac{1}{5} \sqrt [5]{-1} \left (1+\sqrt [5]{-1}\right ) \log \left (\sqrt [5]{-1}-x\right )+\frac{1}{5} (-1)^{4/5} \left (1-(-1)^{4/5}\right ) \log \left (-x-(-1)^{4/5}\right )+\frac{1}{5} (-1)^{2/5} \left (1-(-1)^{2/5}\right ) \log \left (x+(-1)^{2/5}\right )-\frac{1}{5} (-1)^{3/5} \left (1+(-1)^{3/5}\right ) \log \left (x-(-1)^{3/5}\right ) \]

[Out]

-((-1)^(1/5)*(1 + (-1)^(1/5))*Log[(-1)^(1/5) - x])/5 + ((-1)^(4/5)*(1 - (-1)^(4/5))*Log[-(-1)^(4/5) - x])/5 +
((-1)^(2/5)*(1 - (-1)^(2/5))*Log[(-1)^(2/5) + x])/5 - ((-1)^(3/5)*(1 + (-1)^(3/5))*Log[-(-1)^(3/5) + x])/5

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Rubi [A]  time = 0.0641871, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {1586, 2068} \[ -\frac{1}{5} \sqrt [5]{-1} \left (1+\sqrt [5]{-1}\right ) \log \left (\sqrt [5]{-1}-x\right )+\frac{1}{5} (-1)^{4/5} \left (1-(-1)^{4/5}\right ) \log \left (-x-(-1)^{4/5}\right )+\frac{1}{5} (-1)^{2/5} \left (1-(-1)^{2/5}\right ) \log \left (x+(-1)^{2/5}\right )-\frac{1}{5} (-1)^{3/5} \left (1+(-1)^{3/5}\right ) \log \left (x-(-1)^{3/5}\right ) \]

Antiderivative was successfully verified.

[In]

Int[(1 + x)/(1 + x^5),x]

[Out]

-((-1)^(1/5)*(1 + (-1)^(1/5))*Log[(-1)^(1/5) - x])/5 + ((-1)^(4/5)*(1 - (-1)^(4/5))*Log[-(-1)^(4/5) - x])/5 +
((-1)^(2/5)*(1 - (-1)^(2/5))*Log[(-1)^(2/5) + x])/5 - ((-1)^(3/5)*(1 + (-1)^(3/5))*Log[-(-1)^(3/5) + x])/5

Rule 1586

Int[(u_.)*(Px_)^(p_.)*(Qx_)^(q_.), x_Symbol] :> Int[u*PolynomialQuotient[Px, Qx, x]^p*Qx^(p + q), x] /; FreeQ[
q, x] && PolyQ[Px, x] && PolyQ[Qx, x] && EqQ[PolynomialRemainder[Px, Qx, x], 0] && IntegerQ[p] && LtQ[p*q, 0]

Rule 2068

Int[(P4_)^(p_), x_Symbol] :> With[{a = Coeff[P4, x, 0], b = Coeff[P4, x, 1], c = Coeff[P4, x, 2], d = Coeff[P4
, x, 3], e = Coeff[P4, x, 4]}, Dist[1/a^(3*p), Int[ExpandIntegrand[1/((a - b*x)^p/(a^5 - b^5*x^5)^p), x], x],
x] /; NeQ[a, 0] && EqQ[c, b^2/a] && EqQ[d, b^3/a^2] && EqQ[e, b^4/a^3]] /; FreeQ[p, x] && PolyQ[P4, x, 4] && I
LtQ[p, 0]

Rubi steps

\begin{align*} \int \frac{1+x}{1+x^5} \, dx &=\int \frac{1}{1-x+x^2-x^3+x^4} \, dx\\ &=\int \left (\frac{-1+(-1)^{4/5}}{5 \left (-1+\sqrt [5]{-1} x\right )}+\frac{-1-(-1)^{3/5}}{5 \left (-1-(-1)^{2/5} x\right )}+\frac{-1+(-1)^{2/5}}{5 \left (-1+(-1)^{3/5} x\right )}+\frac{-1-\sqrt [5]{-1}}{5 \left (-1-(-1)^{4/5} x\right )}\right ) \, dx\\ &=-\frac{1}{5} \sqrt [5]{-1} \left (1+\sqrt [5]{-1}\right ) \log \left (\sqrt [5]{-1}-x\right )+\frac{1}{5} (-1)^{4/5} \left (1-(-1)^{4/5}\right ) \log \left (-(-1)^{4/5}-x\right )+\frac{1}{5} (-1)^{2/5} \left (1-(-1)^{2/5}\right ) \log \left ((-1)^{2/5}+x\right )-\frac{1}{5} (-1)^{3/5} \left (1+(-1)^{3/5}\right ) \log \left (-(-1)^{3/5}+x\right )\\ \end{align*}

Mathematica [C]  time = 0.0098981, size = 51, normalized size = 0.47 \[ \text{RootSum}\left [\text{$\#$1}^4-\text{$\#$1}^3+\text{$\#$1}^2-\text{$\#$1}+1\& ,\frac{\log (x-\text{$\#$1})}{4 \text{$\#$1}^3-3 \text{$\#$1}^2+2 \text{$\#$1}-1}\& \right ] \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + x)/(1 + x^5),x]

[Out]

RootSum[1 - #1 + #1^2 - #1^3 + #1^4 & , Log[x - #1]/(-1 + 2*#1 - 3*#1^2 + 4*#1^3) & ]

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Maple [B]  time = 0.019, size = 173, normalized size = 1.6 \begin{align*} -{\frac{\sqrt{5}\ln \left ( -x\sqrt{5}+2\,{x}^{2}-x+2 \right ) }{10}}+{\frac{1}{\sqrt{10-2\,\sqrt{5}}}\arctan \left ({\frac{-\sqrt{5}+4\,x-1}{\sqrt{10-2\,\sqrt{5}}}} \right ) }+{\frac{\sqrt{5}}{5\,\sqrt{10-2\,\sqrt{5}}}\arctan \left ({\frac{-\sqrt{5}+4\,x-1}{\sqrt{10-2\,\sqrt{5}}}} \right ) }+{\frac{\sqrt{5}\ln \left ( x\sqrt{5}+2\,{x}^{2}-x+2 \right ) }{10}}+{\frac{1}{\sqrt{10+2\,\sqrt{5}}}\arctan \left ({\frac{\sqrt{5}+4\,x-1}{\sqrt{10+2\,\sqrt{5}}}} \right ) }-{\frac{\sqrt{5}}{5\,\sqrt{10+2\,\sqrt{5}}}\arctan \left ({\frac{\sqrt{5}+4\,x-1}{\sqrt{10+2\,\sqrt{5}}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1+x)/(x^5+1),x)

[Out]

-1/10*5^(1/2)*ln(-x*5^(1/2)+2*x^2-x+2)+1/(10-2*5^(1/2))^(1/2)*arctan((-5^(1/2)+4*x-1)/(10-2*5^(1/2))^(1/2))+1/
5/(10-2*5^(1/2))^(1/2)*arctan((-5^(1/2)+4*x-1)/(10-2*5^(1/2))^(1/2))*5^(1/2)+1/10*5^(1/2)*ln(x*5^(1/2)+2*x^2-x
+2)+1/(10+2*5^(1/2))^(1/2)*arctan((5^(1/2)+4*x-1)/(10+2*5^(1/2))^(1/2))-1/5/(10+2*5^(1/2))^(1/2)*arctan((5^(1/
2)+4*x-1)/(10+2*5^(1/2))^(1/2))*5^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x + 1}{x^{5} + 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)/(x^5+1),x, algorithm="maxima")

[Out]

integrate((x + 1)/(x^5 + 1), x)

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Fricas [B]  time = 8.73671, size = 2955, normalized size = 27.11 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)/(x^5+1),x, algorithm="fricas")

[Out]

-1/10*(sqrt(5) - 5*sqrt(-2/25*sqrt(5) - 1/5))*log(3/8*(sqrt(5) + 5*sqrt(-2/25*sqrt(5) - 1/5))^3 + 1/8*(3*sqrt(
5) + 15*sqrt(-2/25*sqrt(5) - 1/5) + 8)*(sqrt(5) - 5*sqrt(-2/25*sqrt(5) - 1/5))^2 + 3/8*((sqrt(5) + 5*sqrt(-2/2
5*sqrt(5) - 1/5))^2 - 12)*(sqrt(5) - 5*sqrt(-2/25*sqrt(5) - 1/5)) + 11*x + 1) - 1/10*(sqrt(5) + 5*sqrt(-2/25*s
qrt(5) - 1/5))*log(-3/8*(sqrt(5) + 5*sqrt(-2/25*sqrt(5) - 1/5))^3 + (sqrt(5) + 5*sqrt(-2/25*sqrt(5) - 1/5))^2
+ 11*x - 9/2*sqrt(5) - 45/2*sqrt(-2/25*sqrt(5) - 1/5) - 14) + 1/10*(sqrt(5) + 5*sqrt(-3/100*(sqrt(5) + 5*sqrt(
-2/25*sqrt(5) - 1/5))^2 - 1/50*(sqrt(5) + 5*sqrt(-2/25*sqrt(5) - 1/5))*(sqrt(5) - 5*sqrt(-2/25*sqrt(5) - 1/5))
- 3/100*(sqrt(5) - 5*sqrt(-2/25*sqrt(5) - 1/5))^2))*log(-1/8*(3*sqrt(5) + 15*sqrt(-2/25*sqrt(5) - 1/5) + 8)*(
sqrt(5) - 5*sqrt(-2/25*sqrt(5) - 1/5))^2 - (sqrt(5) + 5*sqrt(-2/25*sqrt(5) - 1/5))^2 - 3/8*((sqrt(5) + 5*sqrt(
-2/25*sqrt(5) - 1/5))^2 - 12)*(sqrt(5) - 5*sqrt(-2/25*sqrt(5) - 1/5)) + 5/4*sqrt(-3/100*(sqrt(5) + 5*sqrt(-2/2
5*sqrt(5) - 1/5))^2 - 1/50*(sqrt(5) + 5*sqrt(-2/25*sqrt(5) - 1/5))*(sqrt(5) - 5*sqrt(-2/25*sqrt(5) - 1/5)) - 3
/100*(sqrt(5) - 5*sqrt(-2/25*sqrt(5) - 1/5))^2)*((3*sqrt(5) + 15*sqrt(-2/25*sqrt(5) - 1/5) + 8)*(sqrt(5) - 5*s
qrt(-2/25*sqrt(5) - 1/5)) + 8*sqrt(5) + 40*sqrt(-2/25*sqrt(5) - 1/5) + 36) + 22*x + 9/2*sqrt(5) + 45/2*sqrt(-2
/25*sqrt(5) - 1/5) + 2) + 1/10*(sqrt(5) - 5*sqrt(-3/100*(sqrt(5) + 5*sqrt(-2/25*sqrt(5) - 1/5))^2 - 1/50*(sqrt
(5) + 5*sqrt(-2/25*sqrt(5) - 1/5))*(sqrt(5) - 5*sqrt(-2/25*sqrt(5) - 1/5)) - 3/100*(sqrt(5) - 5*sqrt(-2/25*sqr
t(5) - 1/5))^2))*log(-1/8*(3*sqrt(5) + 15*sqrt(-2/25*sqrt(5) - 1/5) + 8)*(sqrt(5) - 5*sqrt(-2/25*sqrt(5) - 1/5
))^2 - (sqrt(5) + 5*sqrt(-2/25*sqrt(5) - 1/5))^2 - 3/8*((sqrt(5) + 5*sqrt(-2/25*sqrt(5) - 1/5))^2 - 12)*(sqrt(
5) - 5*sqrt(-2/25*sqrt(5) - 1/5)) - 5/4*sqrt(-3/100*(sqrt(5) + 5*sqrt(-2/25*sqrt(5) - 1/5))^2 - 1/50*(sqrt(5)
+ 5*sqrt(-2/25*sqrt(5) - 1/5))*(sqrt(5) - 5*sqrt(-2/25*sqrt(5) - 1/5)) - 3/100*(sqrt(5) - 5*sqrt(-2/25*sqrt(5)
- 1/5))^2)*((3*sqrt(5) + 15*sqrt(-2/25*sqrt(5) - 1/5) + 8)*(sqrt(5) - 5*sqrt(-2/25*sqrt(5) - 1/5)) + 8*sqrt(5
) + 40*sqrt(-2/25*sqrt(5) - 1/5) + 36) + 22*x + 9/2*sqrt(5) + 45/2*sqrt(-2/25*sqrt(5) - 1/5) + 2)

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Sympy [A]  time = 0.372474, size = 36, normalized size = 0.33 \begin{align*} \operatorname{RootSum}{\left (125 t^{4} - 5 t + 1, \left ( t \mapsto t \log{\left (\frac{375 t^{3}}{11} + \frac{100 t^{2}}{11} + \frac{45 t}{11} + x - \frac{14}{11} \right )} \right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)/(x**5+1),x)

[Out]

RootSum(125*_t**4 - 5*_t + 1, Lambda(_t, _t*log(375*_t**3/11 + 100*_t**2/11 + 45*_t/11 + x - 14/11)))

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Giac [A]  time = 1.10738, size = 136, normalized size = 1.25 \begin{align*} \frac{1}{5} \, \sqrt{-2 \, \sqrt{5} + 5} \arctan \left (\frac{4 \, x + \sqrt{5} - 1}{\sqrt{2 \, \sqrt{5} + 10}}\right ) + \frac{1}{5} \, \sqrt{2 \, \sqrt{5} + 5} \arctan \left (\frac{4 \, x - \sqrt{5} - 1}{\sqrt{-2 \, \sqrt{5} + 10}}\right ) - \frac{1}{10} \, \sqrt{5} \log \left (x^{2} - \frac{1}{2} \, x{\left (\sqrt{5} + 1\right )} + 1\right ) + \frac{1}{10} \, \sqrt{5} \log \left (x^{2} + \frac{1}{2} \, x{\left (\sqrt{5} - 1\right )} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)/(x^5+1),x, algorithm="giac")

[Out]

1/5*sqrt(-2*sqrt(5) + 5)*arctan((4*x + sqrt(5) - 1)/sqrt(2*sqrt(5) + 10)) + 1/5*sqrt(2*sqrt(5) + 5)*arctan((4*
x - sqrt(5) - 1)/sqrt(-2*sqrt(5) + 10)) - 1/10*sqrt(5)*log(x^2 - 1/2*x*(sqrt(5) + 1) + 1) + 1/10*sqrt(5)*log(x
^2 + 1/2*x*(sqrt(5) - 1) + 1)