\(\int \frac {1}{1+a+(-1+a) x^4} \, dx\) [8]

Optimal result

Integrand size = 12, antiderivative size = 83 \[ \int \frac {1}{1+a+(-1+a) x^4} \, dx=\frac {\arctan \left (\frac {\sqrt [4]{1-a} x}{\sqrt [4]{1+a}}\right )}{2 \sqrt {1+a} \sqrt [4]{1-a^2}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{1-a} x}{\sqrt [4]{1+a}}\right )}{2 \sqrt {1+a} \sqrt [4]{1-a^2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.93 \[ \int \frac {1}{1+a+(-1+a) x^4} \, dx=\frac {-2 \arctan \left (1-\sqrt {2} \sqrt [4]{\frac {-1+a}{1+a}} x\right )+2 \arctan \left (1+\sqrt {2} \sqrt [4]{\frac {-1+a}{1+a}} x\right )-\log \left (\sqrt {1+a}-\sqrt {2} \sqrt [4]{-1+a} \sqrt [4]{1+a} x+\sqrt {-1+a} x^2\right )+\log \left (\sqrt {1+a}+\sqrt {2} \sqrt [4]{-1+a} \sqrt [4]{1+a} x+\sqrt {-1+a} x^2\right )}{4 \sqrt {2} \sqrt [4]{-1+a} (1+a)^{3/4}} \] Input:

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

Output:

(-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[Sqrt[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))
 

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {756, 218, 221}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 756

\(\displaystyle \frac {\int \frac {1}{\sqrt {a+1}-\sqrt {1-a} x^2}dx}{2 \sqrt {a+1}}+\frac {\int \frac {1}{\sqrt {1-a} x^2+\sqrt {a+1}}dx}{2 \sqrt {a+1}}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {\int \frac {1}{\sqrt {a+1}-\sqrt {1-a} x^2}dx}{2 \sqrt {a+1}}+\frac {\arctan \left (\frac {\sqrt [4]{1-a} x}{\sqrt [4]{a+1}}\right )}{2 \sqrt {a+1} \sqrt [4]{1-a^2}}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\arctan \left (\frac {\sqrt [4]{1-a} x}{\sqrt [4]{a+1}}\right )}{2 \sqrt {a+1} \sqrt [4]{1-a^2}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{1-a} x}{\sqrt [4]{a+1}}\right )}{2 \sqrt {a+1} \sqrt [4]{1-a^2}}\)

Input:

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

Output:

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))
 

Defintions of rubi rules used

rule 218

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

rule 221

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

rule 756

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

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.50 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.45

method	result	size
risch	\(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (a -1\right ) \textit {\_Z}^{4}+1+a \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3} a -\textit {\_R}^{3}}\right )}{4}\)	\(37\)
default	\(\frac {\left (\frac {1+a}{a -1}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {1+a}{a -1}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {1+a}{a -1}}}{x^{2}-\left (\frac {1+a}{a -1}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {1+a}{a -1}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {1+a}{a -1}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {1+a}{a -1}\right )^{\frac {1}{4}}}-1\right )\right )}{8+8 a}\)	\(132\)

Input:

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

Output:

1/4*sum(1/(_R^3*a-_R^3)*ln(x-_R),_R=RootOf((a-1)*_Z^4+1+a))
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.08 (sec) , antiderivative size = 196, normalized size of antiderivative = 2.36 \[ \int \frac {1}{1+a+(-1+a) x^4} \, dx=\frac {1}{4} \, \left (-\frac {1}{a^{4} + 2 \, a^{3} - 2 \, a - 1}\right )^{\frac {1}{4}} \log \left ({\left (a + 1\right )} \left (-\frac {1}{a^{4} + 2 \, a^{3} - 2 \, a - 1}\right )^{\frac {1}{4}} + x\right ) - \frac {1}{4} \, \left (-\frac {1}{a^{4} + 2 \, a^{3} - 2 \, a - 1}\right )^{\frac {1}{4}} \log \left (-{\left (a + 1\right )} \left (-\frac {1}{a^{4} + 2 \, a^{3} - 2 \, a - 1}\right )^{\frac {1}{4}} + x\right ) - \frac {1}{4} i \, \left (-\frac {1}{a^{4} + 2 \, a^{3} - 2 \, a - 1}\right )^{\frac {1}{4}} \log \left (-{\left (i \, a + i\right )} \left (-\frac {1}{a^{4} + 2 \, a^{3} - 2 \, a - 1}\right )^{\frac {1}{4}} + x\right ) + \frac {1}{4} i \, \left (-\frac {1}{a^{4} + 2 \, a^{3} - 2 \, a - 1}\right )^{\frac {1}{4}} \log \left (-{\left (-i \, a - i\right )} \left (-\frac {1}{a^{4} + 2 \, a^{3} - 2 \, a - 1}\right )^{\frac {1}{4}} + x\right ) \] Input:

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

Output:

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

Sympy [A] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.39 \[ \int \frac {1}{1+a+(-1+a) x^4} \, dx=\operatorname {RootSum} {\left (t^{4} \cdot \left (256 a^{4} + 512 a^{3} - 512 a - 256\right ) + 1, \left ( t \mapsto t \log {\left (4 t a + 4 t + x \right )} \right )\right )} \] Input:

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

Output:

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 318 vs. \(2 (63) = 126\).

Time = 0.12 (sec) , antiderivative size = 318, normalized size of antiderivative = 3.83 \[ \int \frac {1}{1+a+(-1+a) x^4} \, dx=\frac {\sqrt {2} \log \left (\sqrt {a - 1} x^{2} + \sqrt {2} {\left (a + 1\right )}^{\frac {1}{4}} {\left (a - 1\right )}^{\frac {1}{4}} x + \sqrt {a + 1}\right )}{8 \, {\left (a + 1\right )}^{\frac {3}{4}} {\left (a - 1\right )}^{\frac {1}{4}}} - \frac {\sqrt {2} \log \left (\sqrt {a - 1} x^{2} - \sqrt {2} {\left (a + 1\right )}^{\frac {1}{4}} {\left (a - 1\right )}^{\frac {1}{4}} x + \sqrt {a + 1}\right )}{8 \, {\left (a + 1\right )}^{\frac {3}{4}} {\left (a - 1\right )}^{\frac {1}{4}}} + \frac {\sqrt {2} \log \left (\frac {2 \, \sqrt {a - 1} x - \sqrt {2} \sqrt {-\sqrt {a + 1} \sqrt {a - 1}} + \sqrt {2} {\left (a + 1\right )}^{\frac {1}{4}} {\left (a - 1\right )}^{\frac {1}{4}}}{2 \, \sqrt {a - 1} x + \sqrt {2} \sqrt {-\sqrt {a + 1} \sqrt {a - 1}} + \sqrt {2} {\left (a + 1\right )}^{\frac {1}{4}} {\left (a - 1\right )}^{\frac {1}{4}}}\right )}{8 \, \sqrt {-\sqrt {a + 1} \sqrt {a - 1}} \sqrt {a + 1}} + \frac {\sqrt {2} \log \left (\frac {2 \, \sqrt {a - 1} x - \sqrt {2} \sqrt {-\sqrt {a + 1} \sqrt {a - 1}} - \sqrt {2} {\left (a + 1\right )}^{\frac {1}{4}} {\left (a - 1\right )}^{\frac {1}{4}}}{2 \, \sqrt {a - 1} x + \sqrt {2} \sqrt {-\sqrt {a + 1} \sqrt {a - 1}} - \sqrt {2} {\left (a + 1\right )}^{\frac {1}{4}} {\left (a - 1\right )}^{\frac {1}{4}}}\right )}{8 \, \sqrt {-\sqrt {a + 1} \sqrt {a - 1}} \sqrt {a + 1}} \] Input:

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

Output:

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*sqr 
t(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))*s 
qrt(a + 1))
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 267 vs. \(2 (63) = 126\).

Time = 0.12 (sec) , antiderivative size = 267, normalized size of antiderivative = 3.22 \[ \int \frac {1}{1+a+(-1+a) x^4} \, dx=\frac {{\left (a^{4} - 2 \, a^{3} + 2 \, a - 1\right )}^{\frac {1}{4}} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (\frac {a + 1}{a - 1}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a + 1}{a - 1}\right )^{\frac {1}{4}}}\right )}{2 \, {\left (\sqrt {2} a^{2} - \sqrt {2}\right )}} + \frac {{\left (a^{4} - 2 \, a^{3} + 2 \, a - 1\right )}^{\frac {1}{4}} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (\frac {a + 1}{a - 1}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a + 1}{a - 1}\right )^{\frac {1}{4}}}\right )}{2 \, {\left (\sqrt {2} a^{2} - \sqrt {2}\right )}} + \frac {{\left (a^{4} - 2 \, a^{3} + 2 \, a - 1\right )}^{\frac {1}{4}} \log \left (x^{2} + \sqrt {2} x \left (\frac {a + 1}{a - 1}\right )^{\frac {1}{4}} + \sqrt {\frac {a + 1}{a - 1}}\right )}{4 \, {\left (\sqrt {2} a^{2} - \sqrt {2}\right )}} - \frac {{\left (a^{4} - 2 \, a^{3} + 2 \, a - 1\right )}^{\frac {1}{4}} \log \left (x^{2} - \sqrt {2} x \left (\frac {a + 1}{a - 1}\right )^{\frac {1}{4}} + \sqrt {\frac {a + 1}{a - 1}}\right )}{4 \, {\left (\sqrt {2} a^{2} - \sqrt {2}\right )}} \] Input:

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

Output:

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) + sq 
rt((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))
 

Mupad [B] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 543, normalized size of antiderivative = 6.54 \[ \int \frac {1}{1+a+(-1+a) x^4} \, dx=\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}} \] Input:

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

Output:

(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))
 

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.93 \[ \int \frac {1}{1+a+(-1+a) x^4} \, dx=\frac {\left (a +1\right )^{\frac {1}{4}} \left (a -1\right )^{\frac {3}{4}} \sqrt {2}\, \left (-2 \mathit {atan} \left (\frac {\left (a +1\right )^{\frac {1}{4}} \left (a -1\right )^{\frac {1}{4}} \sqrt {2}-2 \sqrt {a -1}\, x}{\left (a +1\right )^{\frac {1}{4}} \left (a -1\right )^{\frac {1}{4}} \sqrt {2}}\right )+2 \mathit {atan} \left (\frac {\left (a +1\right )^{\frac {1}{4}} \left (a -1\right )^{\frac {1}{4}} \sqrt {2}+2 \sqrt {a -1}\, x}{\left (a +1\right )^{\frac {1}{4}} \left (a -1\right )^{\frac {1}{4}} \sqrt {2}}\right )-\mathrm {log}\left (-\left (a +1\right )^{\frac {1}{4}} \left (a -1\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {a -1}\, x^{2}+\sqrt {a +1}\right )+\mathrm {log}\left (\left (a +1\right )^{\frac {1}{4}} \left (a -1\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {a -1}\, x^{2}+\sqrt {a +1}\right )\right )}{8 a^{2}-8} \] Input:

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

Output:

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