3.708 \(\int \frac {1}{1+a+(-1+a) x^4} \, dx\)

Optimal. Leaf size=83 \[ \frac {\tan ^{-1}\left (\frac {\sqrt [4]{1-a} x}{\sqrt [4]{a+1}}\right )}{2 \sqrt {a+1} \sqrt [4]{1-a^2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{1-a} x}{\sqrt [4]{a+1}}\right )}{2 \sqrt {a+1} \sqrt [4]{1-a^2}} \]

[Out]

1/2*arctan((1-a)^(1/4)*x/(1+a)^(1/4))/(-a^2+1)^(1/4)/(1+a)^(1/2)+1/2*arctanh((1-a)^(1/4)*x/(1+a)^(1/4))/(-a^2+
1)^(1/4)/(1+a)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.07, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {212, 208, 205} \[ \frac {\tan ^{-1}\left (\frac {\sqrt [4]{1-a} x}{\sqrt [4]{a+1}}\right )}{2 \sqrt {a+1} \sqrt [4]{1-a^2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{1-a} x}{\sqrt [4]{a+1}}\right )}{2 \sqrt {a+1} \sqrt [4]{1-a^2}} \]

Antiderivative was successfully verified.

[In]

Int[(1 + a + (-1 + a)*x^4)^(-1),x]

[Out]

ArcTan[((1 - a)^(1/4)*x)/(1 + a)^(1/4)]/(2*Sqrt[1 + a]*(1 - a^2)^(1/4)) + ArcTanh[((1 - a)^(1/4)*x)/(1 + a)^(1
/4)]/(2*Sqrt[1 + a]*(1 - a^2)^(1/4))

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rule 212

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-(a/b), 2]], s = Denominator[Rt[-(a/b), 2]
]}, Dist[r/(2*a), Int[1/(r - s*x^2), x], x] + Dist[r/(2*a), Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&
!GtQ[a/b, 0]

Rubi steps

\begin {align*} \int \frac {1}{1+a+(-1+a) x^4} \, dx &=\frac {\int \frac {1}{\sqrt {1+a}-\sqrt {1-a} x^2} \, dx}{2 \sqrt {1+a}}+\frac {\int \frac {1}{\sqrt {1+a}+\sqrt {1-a} x^2} \, dx}{2 \sqrt {1+a}}\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt [4]{1-a} x}{\sqrt [4]{1+a}}\right )}{2 \sqrt {1+a} \sqrt [4]{1-a^2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{1-a} x}{\sqrt [4]{1+a}}\right )}{2 \sqrt {1+a} \sqrt [4]{1-a^2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.08, size = 160, normalized size = 1.93 \[ \frac {-\log \left (\sqrt {a-1} x^2-\sqrt {2} \sqrt [4]{a-1} \sqrt [4]{a+1} x+\sqrt {a+1}\right )+\log \left (\sqrt {a-1} x^2+\sqrt {2} \sqrt [4]{a-1} \sqrt [4]{a+1} x+\sqrt {a+1}\right )-2 \tan ^{-1}\left (1-\sqrt {2} \sqrt [4]{\frac {a-1}{a+1}} x\right )+2 \tan ^{-1}\left (\sqrt {2} \sqrt [4]{\frac {a-1}{a+1}} x+1\right )}{4 \sqrt {2} \sqrt [4]{a-1} (a+1)^{3/4}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(1 + a + (-1 + a)*x^4)^(-1),x]

[Out]

(-2*ArcTan[1 - Sqrt[2]*((-1 + a)/(1 + a))^(1/4)*x] + 2*ArcTan[1 + Sqrt[2]*((-1 + a)/(1 + a))^(1/4)*x] - Log[Sq
rt[1 + a] - Sqrt[2]*(-1 + a)^(1/4)*(1 + a)^(1/4)*x + Sqrt[-1 + a]*x^2] + Log[Sqrt[1 + a] + Sqrt[2]*(-1 + a)^(1
/4)*(1 + a)^(1/4)*x + Sqrt[-1 + a]*x^2])/(4*Sqrt[2]*(-1 + a)^(1/4)*(1 + a)^(3/4))

________________________________________________________________________________________

fricas [B]  time = 0.73, size = 216, normalized size = 2.60 \[ \left (-\frac {1}{a^{4} + 2 \, a^{3} - 2 \, a - 1}\right )^{\frac {1}{4}} \arctan \left (-{\left (a^{3} + a^{2} - a - 1\right )} x \left (-\frac {1}{a^{4} + 2 \, a^{3} - 2 \, a - 1}\right )^{\frac {3}{4}} + {\left (a^{3} + a^{2} - a - 1\right )} \sqrt {x^{2} + {\left (a^{2} + 2 \, a + 1\right )} \sqrt {-\frac {1}{a^{4} + 2 \, a^{3} - 2 \, a - 1}}} \left (-\frac {1}{a^{4} + 2 \, a^{3} - 2 \, a - 1}\right )^{\frac {3}{4}}\right ) + \frac {1}{4} \, \left (-\frac {1}{a^{4} + 2 \, a^{3} - 2 \, a - 1}\right )^{\frac {1}{4}} \log \left ({\left (a + 1\right )} \left (-\frac {1}{a^{4} + 2 \, a^{3} - 2 \, a - 1}\right )^{\frac {1}{4}} + x\right ) - \frac {1}{4} \, \left (-\frac {1}{a^{4} + 2 \, a^{3} - 2 \, a - 1}\right )^{\frac {1}{4}} \log \left (-{\left (a + 1\right )} \left (-\frac {1}{a^{4} + 2 \, a^{3} - 2 \, a - 1}\right )^{\frac {1}{4}} + x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+a+(-1+a)*x^4),x, algorithm="fricas")

[Out]

(-1/(a^4 + 2*a^3 - 2*a - 1))^(1/4)*arctan(-(a^3 + a^2 - a - 1)*x*(-1/(a^4 + 2*a^3 - 2*a - 1))^(3/4) + (a^3 + a
^2 - a - 1)*sqrt(x^2 + (a^2 + 2*a + 1)*sqrt(-1/(a^4 + 2*a^3 - 2*a - 1)))*(-1/(a^4 + 2*a^3 - 2*a - 1))^(3/4)) +
1/4*(-1/(a^4 + 2*a^3 - 2*a - 1))^(1/4)*log((a + 1)*(-1/(a^4 + 2*a^3 - 2*a - 1))^(1/4) + x) - 1/4*(-1/(a^4 + 2
*a^3 - 2*a - 1))^(1/4)*log(-(a + 1)*(-1/(a^4 + 2*a^3 - 2*a - 1))^(1/4) + x)

________________________________________________________________________________________

giac [B]  time = 0.17, size = 267, normalized size = 3.22 \[ \frac {{\left (a^{4} - 2 \, a^{3} + 2 \, a - 1\right )}^{\frac {1}{4}} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (\frac {a + 1}{a - 1}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a + 1}{a - 1}\right )^{\frac {1}{4}}}\right )}{2 \, {\left (\sqrt {2} a^{2} - \sqrt {2}\right )}} + \frac {{\left (a^{4} - 2 \, a^{3} + 2 \, a - 1\right )}^{\frac {1}{4}} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (\frac {a + 1}{a - 1}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a + 1}{a - 1}\right )^{\frac {1}{4}}}\right )}{2 \, {\left (\sqrt {2} a^{2} - \sqrt {2}\right )}} + \frac {{\left (a^{4} - 2 \, a^{3} + 2 \, a - 1\right )}^{\frac {1}{4}} \log \left (x^{2} + \sqrt {2} x \left (\frac {a + 1}{a - 1}\right )^{\frac {1}{4}} + \sqrt {\frac {a + 1}{a - 1}}\right )}{4 \, {\left (\sqrt {2} a^{2} - \sqrt {2}\right )}} - \frac {{\left (a^{4} - 2 \, a^{3} + 2 \, a - 1\right )}^{\frac {1}{4}} \log \left (x^{2} - \sqrt {2} x \left (\frac {a + 1}{a - 1}\right )^{\frac {1}{4}} + \sqrt {\frac {a + 1}{a - 1}}\right )}{4 \, {\left (\sqrt {2} a^{2} - \sqrt {2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+a+(-1+a)*x^4),x, algorithm="giac")

[Out]

1/2*(a^4 - 2*a^3 + 2*a - 1)^(1/4)*arctan(1/2*sqrt(2)*(2*x + sqrt(2)*((a + 1)/(a - 1))^(1/4))/((a + 1)/(a - 1))
^(1/4))/(sqrt(2)*a^2 - sqrt(2)) + 1/2*(a^4 - 2*a^3 + 2*a - 1)^(1/4)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)*((a + 1)
/(a - 1))^(1/4))/((a + 1)/(a - 1))^(1/4))/(sqrt(2)*a^2 - sqrt(2)) + 1/4*(a^4 - 2*a^3 + 2*a - 1)^(1/4)*log(x^2
+ sqrt(2)*x*((a + 1)/(a - 1))^(1/4) + sqrt((a + 1)/(a - 1)))/(sqrt(2)*a^2 - sqrt(2)) - 1/4*(a^4 - 2*a^3 + 2*a
- 1)^(1/4)*log(x^2 - sqrt(2)*x*((a + 1)/(a - 1))^(1/4) + sqrt((a + 1)/(a - 1)))/(sqrt(2)*a^2 - sqrt(2))

________________________________________________________________________________________

maple [B]  time = 0.01, size = 170, normalized size = 2.05 \[ \frac {\left (\frac {a +1}{a -1}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a +1}{a -1}\right )^{\frac {1}{4}}}-1\right )}{4 a +4}+\frac {\left (\frac {a +1}{a -1}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a +1}{a -1}\right )^{\frac {1}{4}}}+1\right )}{4 a +4}+\frac {\left (\frac {a +1}{a -1}\right )^{\frac {1}{4}} \sqrt {2}\, \ln \left (\frac {x^{2}+\left (\frac {a +1}{a -1}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a +1}{a -1}}}{x^{2}-\left (\frac {a +1}{a -1}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a +1}{a -1}}}\right )}{8 a +8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1+a+(a-1)*x^4),x)

[Out]

1/8*((a+1)/(a-1))^(1/4)/(a+1)*2^(1/2)*ln((x^2+((a+1)/(a-1))^(1/4)*x*2^(1/2)+((a+1)/(a-1))^(1/2))/(x^2-((a+1)/(
a-1))^(1/4)*x*2^(1/2)+((a+1)/(a-1))^(1/2)))+1/4*((a+1)/(a-1))^(1/4)/(a+1)*2^(1/2)*arctan(2^(1/2)/((a+1)/(a-1))
^(1/4)*x+1)+1/4*((a+1)/(a-1))^(1/4)/(a+1)*2^(1/2)*arctan(2^(1/2)/((a+1)/(a-1))^(1/4)*x-1)

________________________________________________________________________________________

maxima [B]  time = 3.16, size = 318, normalized size = 3.83 \[ \frac {\sqrt {2} \log \left (\sqrt {a - 1} x^{2} + \sqrt {2} {\left (a + 1\right )}^{\frac {1}{4}} {\left (a - 1\right )}^{\frac {1}{4}} x + \sqrt {a + 1}\right )}{8 \, {\left (a + 1\right )}^{\frac {3}{4}} {\left (a - 1\right )}^{\frac {1}{4}}} - \frac {\sqrt {2} \log \left (\sqrt {a - 1} x^{2} - \sqrt {2} {\left (a + 1\right )}^{\frac {1}{4}} {\left (a - 1\right )}^{\frac {1}{4}} x + \sqrt {a + 1}\right )}{8 \, {\left (a + 1\right )}^{\frac {3}{4}} {\left (a - 1\right )}^{\frac {1}{4}}} + \frac {\sqrt {2} \log \left (\frac {2 \, \sqrt {a - 1} x - \sqrt {2} \sqrt {-\sqrt {a + 1} \sqrt {a - 1}} + \sqrt {2} {\left (a + 1\right )}^{\frac {1}{4}} {\left (a - 1\right )}^{\frac {1}{4}}}{2 \, \sqrt {a - 1} x + \sqrt {2} \sqrt {-\sqrt {a + 1} \sqrt {a - 1}} + \sqrt {2} {\left (a + 1\right )}^{\frac {1}{4}} {\left (a - 1\right )}^{\frac {1}{4}}}\right )}{8 \, \sqrt {-\sqrt {a + 1} \sqrt {a - 1}} \sqrt {a + 1}} + \frac {\sqrt {2} \log \left (\frac {2 \, \sqrt {a - 1} x - \sqrt {2} \sqrt {-\sqrt {a + 1} \sqrt {a - 1}} - \sqrt {2} {\left (a + 1\right )}^{\frac {1}{4}} {\left (a - 1\right )}^{\frac {1}{4}}}{2 \, \sqrt {a - 1} x + \sqrt {2} \sqrt {-\sqrt {a + 1} \sqrt {a - 1}} - \sqrt {2} {\left (a + 1\right )}^{\frac {1}{4}} {\left (a - 1\right )}^{\frac {1}{4}}}\right )}{8 \, \sqrt {-\sqrt {a + 1} \sqrt {a - 1}} \sqrt {a + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+a+(-1+a)*x^4),x, algorithm="maxima")

[Out]

1/8*sqrt(2)*log(sqrt(a - 1)*x^2 + sqrt(2)*(a + 1)^(1/4)*(a - 1)^(1/4)*x + sqrt(a + 1))/((a + 1)^(3/4)*(a - 1)^
(1/4)) - 1/8*sqrt(2)*log(sqrt(a - 1)*x^2 - sqrt(2)*(a + 1)^(1/4)*(a - 1)^(1/4)*x + sqrt(a + 1))/((a + 1)^(3/4)
*(a - 1)^(1/4)) + 1/8*sqrt(2)*log((2*sqrt(a - 1)*x - sqrt(2)*sqrt(-sqrt(a + 1)*sqrt(a - 1)) + sqrt(2)*(a + 1)^
(1/4)*(a - 1)^(1/4))/(2*sqrt(a - 1)*x + sqrt(2)*sqrt(-sqrt(a + 1)*sqrt(a - 1)) + sqrt(2)*(a + 1)^(1/4)*(a - 1)
^(1/4)))/(sqrt(-sqrt(a + 1)*sqrt(a - 1))*sqrt(a + 1)) + 1/8*sqrt(2)*log((2*sqrt(a - 1)*x - sqrt(2)*sqrt(-sqrt(
a + 1)*sqrt(a - 1)) - sqrt(2)*(a + 1)^(1/4)*(a - 1)^(1/4))/(2*sqrt(a - 1)*x + sqrt(2)*sqrt(-sqrt(a + 1)*sqrt(a
- 1)) - sqrt(2)*(a + 1)^(1/4)*(a - 1)^(1/4)))/(sqrt(-sqrt(a + 1)*sqrt(a - 1))*sqrt(a + 1))

________________________________________________________________________________________

mupad [B]  time = 1.18, size = 543, normalized size = 6.54 \[ \frac {\mathrm {atan}\left (\frac {\frac {\left (\frac {x\,\left (4\,a^3-12\,a^2+12\,a-4\right )}{4}-\frac {4\,a^4-8\,a^3+8\,a-4}{4\,{\left (1-a\right )}^{1/4}\,{\left (a+1\right )}^{3/4}}\right )\,1{}\mathrm {i}}{{\left (1-a\right )}^{1/4}\,{\left (a+1\right )}^{3/4}}+\frac {\left (\frac {x\,\left (4\,a^3-12\,a^2+12\,a-4\right )}{4}+\frac {4\,a^4-8\,a^3+8\,a-4}{4\,{\left (1-a\right )}^{1/4}\,{\left (a+1\right )}^{3/4}}\right )\,1{}\mathrm {i}}{{\left (1-a\right )}^{1/4}\,{\left (a+1\right )}^{3/4}}}{\frac {\frac {x\,\left (4\,a^3-12\,a^2+12\,a-4\right )}{4}-\frac {4\,a^4-8\,a^3+8\,a-4}{4\,{\left (1-a\right )}^{1/4}\,{\left (a+1\right )}^{3/4}}}{{\left (1-a\right )}^{1/4}\,{\left (a+1\right )}^{3/4}}-\frac {\frac {x\,\left (4\,a^3-12\,a^2+12\,a-4\right )}{4}+\frac {4\,a^4-8\,a^3+8\,a-4}{4\,{\left (1-a\right )}^{1/4}\,{\left (a+1\right )}^{3/4}}}{{\left (1-a\right )}^{1/4}\,{\left (a+1\right )}^{3/4}}}\right )\,1{}\mathrm {i}}{2\,{\left (1-a\right )}^{1/4}\,{\left (a+1\right )}^{3/4}}+\frac {\mathrm {atan}\left (\frac {\frac {\frac {x\,\left (4\,a^3-12\,a^2+12\,a-4\right )}{4}-\frac {\left (4\,a^4-8\,a^3+8\,a-4\right )\,1{}\mathrm {i}}{4\,{\left (1-a\right )}^{1/4}\,{\left (a+1\right )}^{3/4}}}{{\left (1-a\right )}^{1/4}\,{\left (a+1\right )}^{3/4}}+\frac {\frac {x\,\left (4\,a^3-12\,a^2+12\,a-4\right )}{4}+\frac {\left (4\,a^4-8\,a^3+8\,a-4\right )\,1{}\mathrm {i}}{4\,{\left (1-a\right )}^{1/4}\,{\left (a+1\right )}^{3/4}}}{{\left (1-a\right )}^{1/4}\,{\left (a+1\right )}^{3/4}}}{\frac {\left (\frac {x\,\left (4\,a^3-12\,a^2+12\,a-4\right )}{4}-\frac {\left (4\,a^4-8\,a^3+8\,a-4\right )\,1{}\mathrm {i}}{4\,{\left (1-a\right )}^{1/4}\,{\left (a+1\right )}^{3/4}}\right )\,1{}\mathrm {i}}{{\left (1-a\right )}^{1/4}\,{\left (a+1\right )}^{3/4}}-\frac {\left (\frac {x\,\left (4\,a^3-12\,a^2+12\,a-4\right )}{4}+\frac {\left (4\,a^4-8\,a^3+8\,a-4\right )\,1{}\mathrm {i}}{4\,{\left (1-a\right )}^{1/4}\,{\left (a+1\right )}^{3/4}}\right )\,1{}\mathrm {i}}{{\left (1-a\right )}^{1/4}\,{\left (a+1\right )}^{3/4}}}\right )}{2\,{\left (1-a\right )}^{1/4}\,{\left (a+1\right )}^{3/4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + x^4*(a - 1) + 1),x)

[Out]

(atan(((((x*(12*a - 12*a^2 + 4*a^3 - 4))/4 - (8*a - 8*a^3 + 4*a^4 - 4)/(4*(1 - a)^(1/4)*(a + 1)^(3/4)))*1i)/((
1 - a)^(1/4)*(a + 1)^(3/4)) + (((x*(12*a - 12*a^2 + 4*a^3 - 4))/4 + (8*a - 8*a^3 + 4*a^4 - 4)/(4*(1 - a)^(1/4)
*(a + 1)^(3/4)))*1i)/((1 - a)^(1/4)*(a + 1)^(3/4)))/(((x*(12*a - 12*a^2 + 4*a^3 - 4))/4 - (8*a - 8*a^3 + 4*a^4
- 4)/(4*(1 - a)^(1/4)*(a + 1)^(3/4)))/((1 - a)^(1/4)*(a + 1)^(3/4)) - ((x*(12*a - 12*a^2 + 4*a^3 - 4))/4 + (8
*a - 8*a^3 + 4*a^4 - 4)/(4*(1 - a)^(1/4)*(a + 1)^(3/4)))/((1 - a)^(1/4)*(a + 1)^(3/4))))*1i)/(2*(1 - a)^(1/4)*
(a + 1)^(3/4)) + atan((((x*(12*a - 12*a^2 + 4*a^3 - 4))/4 - ((8*a - 8*a^3 + 4*a^4 - 4)*1i)/(4*(1 - a)^(1/4)*(a
+ 1)^(3/4)))/((1 - a)^(1/4)*(a + 1)^(3/4)) + ((x*(12*a - 12*a^2 + 4*a^3 - 4))/4 + ((8*a - 8*a^3 + 4*a^4 - 4)*
1i)/(4*(1 - a)^(1/4)*(a + 1)^(3/4)))/((1 - a)^(1/4)*(a + 1)^(3/4)))/((((x*(12*a - 12*a^2 + 4*a^3 - 4))/4 - ((8
*a - 8*a^3 + 4*a^4 - 4)*1i)/(4*(1 - a)^(1/4)*(a + 1)^(3/4)))*1i)/((1 - a)^(1/4)*(a + 1)^(3/4)) - (((x*(12*a -
12*a^2 + 4*a^3 - 4))/4 + ((8*a - 8*a^3 + 4*a^4 - 4)*1i)/(4*(1 - a)^(1/4)*(a + 1)^(3/4)))*1i)/((1 - a)^(1/4)*(a
+ 1)^(3/4))))/(2*(1 - a)^(1/4)*(a + 1)^(3/4))

________________________________________________________________________________________

sympy [A]  time = 0.49, size = 32, normalized size = 0.39 \[ \operatorname {RootSum} {\left (t^{4} \left (256 a^{4} + 512 a^{3} - 512 a - 256\right ) + 1, \left (t \mapsto t \log {\left (4 t a + 4 t + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+a+(-1+a)*x**4),x)

[Out]

RootSum(_t**4*(256*a**4 + 512*a**3 - 512*a - 256) + 1, Lambda(_t, _t*log(4*_t*a + 4*_t + x)))

________________________________________________________________________________________