\(\int \frac {-1+x^4}{(1+x^4) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx\) [2674]

Optimal result

Integrand size = 37, antiderivative size = 241 \[ \int \frac {-1+x^4}{\left (1+x^4\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx=-\frac {1}{2} \text {RootSum}\left [4 a^2-2 b^2-4 a c+c^2+8 \sqrt {a} b \text {$\#$1}-4 a \text {$\#$1}^2-2 c \text {$\#$1}^2+\text {$\#$1}^4\&,\frac {-2 a \log (x)+c \log (x)+2 a \log \left (\sqrt {a} \left (-1-x^2\right )+\sqrt {a+b x+c x^2+b x^3+a x^4}-x \text {$\#$1}\right )-c \log \left (\sqrt {a} \left (-1-x^2\right )+\sqrt {a+b x+c x^2+b x^3+a x^4}-x \text {$\#$1}\right )-\log (x) \text {$\#$1}^2+\log \left (\sqrt {a} \left (-1-x^2\right )+\sqrt {a+b x+c x^2+b x^3+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^2}{-2 \sqrt {a} b+2 a \text {$\#$1}+c \text {$\#$1}-\text {$\#$1}^3}\&\right ] \]

[Out]

Unintegrable

Rubi [F]

\[ \int \frac {-1+x^4}{\left (1+x^4\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx=\int \frac {-1+x^4}{\left (1+x^4\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx \]

[In]

Int[(-1 + x^4)/((1 + x^4)*Sqrt[a + b*x + c*x^2 + b*x^3 + a*x^4]),x]

[Out]

Defer[Int][1/Sqrt[a + b*x + c*x^2 + b*x^3 + a*x^4], x] - ((-1)^(1/4)*Defer[Int][1/(((-1)^(1/4) - x)*Sqrt[a + b
*x + c*x^2 + b*x^3 + a*x^4]), x])/2 + ((-1)^(3/4)*Defer[Int][1/((-(-1)^(3/4) - x)*Sqrt[a + b*x + c*x^2 + b*x^3
+ a*x^4]), x])/2 - ((-1)^(1/4)*Defer[Int][1/(((-1)^(1/4) + x)*Sqrt[a + b*x + c*x^2 + b*x^3 + a*x^4]), x])/2 +
((-1)^(3/4)*Defer[Int][1/((-(-1)^(3/4) + x)*Sqrt[a + b*x + c*x^2 + b*x^3 + a*x^4]), x])/2

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{\sqrt {a+b x+c x^2+b x^3+a x^4}}-\frac {2}{\left (1+x^4\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}}\right ) \, dx \\ & = -\left (2 \int \frac {1}{\left (1+x^4\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx\right )+\int \frac {1}{\sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx \\ & = -\left (2 \int \left (\frac {i}{2 \left (i-x^2\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}}+\frac {i}{2 \left (i+x^2\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}}\right ) \, dx\right )+\int \frac {1}{\sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx \\ & = -\left (i \int \frac {1}{\left (i-x^2\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx\right )-i \int \frac {1}{\left (i+x^2\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx+\int \frac {1}{\sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx \\ & = -\left (i \int \left (-\frac {(-1)^{3/4}}{2 \left (\sqrt [4]{-1}-x\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}}-\frac {(-1)^{3/4}}{2 \left (\sqrt [4]{-1}+x\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}}\right ) \, dx\right )-i \int \left (-\frac {\sqrt [4]{-1}}{2 \left (-(-1)^{3/4}-x\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}}-\frac {\sqrt [4]{-1}}{2 \left (-(-1)^{3/4}+x\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}}\right ) \, dx+\int \frac {1}{\sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx \\ & = -\left (\frac {1}{2} \sqrt [4]{-1} \int \frac {1}{\left (\sqrt [4]{-1}-x\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx\right )-\frac {1}{2} \sqrt [4]{-1} \int \frac {1}{\left (\sqrt [4]{-1}+x\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx+\frac {1}{2} (-1)^{3/4} \int \frac {1}{\left (-(-1)^{3/4}-x\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx+\frac {1}{2} (-1)^{3/4} \int \frac {1}{\left (-(-1)^{3/4}+x\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx+\int \frac {1}{\sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.42 (sec) , antiderivative size = 238, normalized size of antiderivative = 0.99 \[ \int \frac {-1+x^4}{\left (1+x^4\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx=\frac {1}{2} \text {RootSum}\left [4 a^2-2 b^2-4 a c+c^2+8 \sqrt {a} b \text {$\#$1}-4 a \text {$\#$1}^2-2 c \text {$\#$1}^2+\text {$\#$1}^4\&,\frac {2 a \log (x)-c \log (x)-2 a \log \left (-\sqrt {a} \left (1+x^2\right )+\sqrt {a+b x+c x^2+b x^3+a x^4}-x \text {$\#$1}\right )+c \log \left (-\sqrt {a} \left (1+x^2\right )+\sqrt {a+b x+c x^2+b x^3+a x^4}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^2-\log \left (-\sqrt {a} \left (1+x^2\right )+\sqrt {a+b x+c x^2+b x^3+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^2}{-2 \sqrt {a} b+2 a \text {$\#$1}+c \text {$\#$1}-\text {$\#$1}^3}\&\right ] \]

[In]

Integrate[(-1 + x^4)/((1 + x^4)*Sqrt[a + b*x + c*x^2 + b*x^3 + a*x^4]),x]

[Out]

RootSum[4*a^2 - 2*b^2 - 4*a*c + c^2 + 8*Sqrt[a]*b*#1 - 4*a*#1^2 - 2*c*#1^2 + #1^4 & , (2*a*Log[x] - c*Log[x] -
2*a*Log[-(Sqrt[a]*(1 + x^2)) + Sqrt[a + b*x + c*x^2 + b*x^3 + a*x^4] - x*#1] + c*Log[-(Sqrt[a]*(1 + x^2)) + S
qrt[a + b*x + c*x^2 + b*x^3 + a*x^4] - x*#1] + Log[x]*#1^2 - Log[-(Sqrt[a]*(1 + x^2)) + Sqrt[a + b*x + c*x^2 +
b*x^3 + a*x^4] - x*#1]*#1^2)/(-2*Sqrt[a]*b + 2*a*#1 + c*#1 - #1^3) & ]/2

Maple [N/A] (verified)

Time = 0.87 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.86

method	result	size
pseudoelliptic	\(-\frac {\ln \left (\frac {2 \sqrt {\sqrt {2}\, b +c}\, \sqrt {a \,x^{4}+b \,x^{3}+c \,x^{2}+b x +a}+\left (2 a \,x^{2}+b x +2 a \right ) \sqrt {2}+b \,x^{2}+\left (-4 a +2 c \right ) x +b}{-x \sqrt {2}+x^{2}+1}\right ) \sqrt {-\sqrt {2}\, b +c}+\ln \left (\frac {2 \sqrt {-\sqrt {2}\, b +c}\, \sqrt {a \,x^{4}+b \,x^{3}+c \,x^{2}+b x +a}+\left (-2 a \,x^{2}-b x -2 a \right ) \sqrt {2}+b \,x^{2}+\left (-4 a +2 c \right ) x +b}{x \sqrt {2}+x^{2}+1}\right ) \sqrt {\sqrt {2}\, b +c}}{2 \sqrt {-\sqrt {2}\, b +c}\, \sqrt {\sqrt {2}\, b +c}}\)	\(207\)

[In]

int((x^4-1)/(x^4+1)/(a*x^4+b*x^3+c*x^2+b*x+a)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/2*(ln((2*(2^(1/2)*b+c)^(1/2)*(a*x^4+b*x^3+c*x^2+b*x+a)^(1/2)+(2*a*x^2+b*x+2*a)*2^(1/2)+b*x^2+(-4*a+2*c)*x+b
)/(-x*2^(1/2)+x^2+1))*(-2^(1/2)*b+c)^(1/2)+ln((2*(-2^(1/2)*b+c)^(1/2)*(a*x^4+b*x^3+c*x^2+b*x+a)^(1/2)+(-2*a*x^
2-b*x-2*a)*2^(1/2)+b*x^2+(-4*a+2*c)*x+b)/(x*2^(1/2)+x^2+1))*(2^(1/2)*b+c)^(1/2))/(-2^(1/2)*b+c)^(1/2)/(2^(1/2)
*b+c)^(1/2)

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 3 vs. order 1.

Time = 5.00 (sec) , antiderivative size = 4319, normalized size of antiderivative = 17.92 \[ \int \frac {-1+x^4}{\left (1+x^4\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx=\text {Too large to display} \]

[In]

integrate((x^4-1)/(x^4+1)/(a*x^4+b*x^3+c*x^2+b*x+a)^(1/2),x, algorithm="fricas")

[Out]

1/4*sqrt(-(2*sqrt(1/2)*(2*b^2 - c^2)*sqrt(b^2/(4*b^4 - 4*b^2*c^2 + c^4)) + c)/(2*b^2 - c^2))*log(-(2*(16*a^2*b
^3 + 2*b^5 - 16*a*b^3*c + (8*a^2*b + b^3)*c^2 + (16*a^2*b^3 + 2*b^5 - 16*a*b^3*c + (8*a^2*b + b^3)*c^2)*x^2 +
2*sqrt(1/2)*(16*a*b^5 - 4*a*b*c^4 + 2*(8*a^2*b + b^3)*c^3 + 2*(8*a*b^5 - 2*a*b*c^4 + (8*a^2*b + b^3)*c^3 - 2*(
8*a^2*b^3 + b^5)*c)*x^2 - 4*(8*a^2*b^3 + b^5)*c - (32*a^2*b^4 + 4*b^6 - 32*a*b^4*c + 16*a*b^2*c^3 - (8*a^2 + b
^2)*c^4)*x)*sqrt(b^2/(4*b^4 - 4*b^2*c^2 + c^4)) - 4*(4*a*b^4 + 2*a*b^2*c^2 - (8*a^2*b^2 + b^4)*c)*x)*sqrt(a*x^
4 + b*x^3 + c*x^2 + b*x + a) + (16*a^2*b^4 - 2*b^6 + (16*a^2*b^4 - 2*b^6 + (8*a^2*b^2 - b^4)*c^2 - 8*(8*a^3*b^
2 - a*b^4)*c)*x^4 + 2*(16*a*b^5 + 24*a*b^3*c^2 - (8*a^2*b + b^3)*c^3 - 6*(8*a^2*b^3 + b^5)*c)*x^3 + (8*a^2*b^2
- b^4)*c^2 - 4*(16*a^2*b^4 + 2*b^6 - 24*a*b^4*c - 4*a*b^2*c^3 + 3*(8*a^2*b^2 + b^4)*c^2)*x^2 + 4*sqrt(1/2)*(3
2*a^3*b^4 - 4*a*b^6 - (8*a^3 - a*b^2)*c^4 + (32*a^3*b^4 - 4*a*b^6 - (8*a^3 - a*b^2)*c^4 + (8*a^2*b^2 - b^4)*c^
3 - 2*(8*a^2*b^4 - b^6)*c)*x^4 + (8*a^2*b^2 - b^4)*c^3 + (32*a^2*b^5 + 4*b^7 - 48*a*b^5*c + 16*a*b^3*c^3 + 4*a
*b*c^5 - 3*(8*a^2*b + b^3)*c^4 + 4*(8*a^2*b^3 + b^5)*c^2)*x^3 - (32*a*b^6 + 32*a*b^4*c^2 - 24*a*b^2*c^4 + (8*a
^2 + b^2)*c^5 + 4*(8*a^2*b^2 + b^4)*c^3 - 12*(8*a^2*b^4 + b^6)*c)*x^2 - 2*(8*a^2*b^4 - b^6)*c + (32*a^2*b^5 +
4*b^7 - 48*a*b^5*c + 16*a*b^3*c^3 + 4*a*b*c^5 - 3*(8*a^2*b + b^3)*c^4 + 4*(8*a^2*b^3 + b^5)*c^2)*x)*sqrt(b^2/(
4*b^4 - 4*b^2*c^2 + c^4)) - 8*(8*a^3*b^2 - a*b^4)*c + 2*(16*a*b^5 + 24*a*b^3*c^2 - (8*a^2*b + b^3)*c^3 - 6*(8*
a^2*b^3 + b^5)*c)*x)*sqrt(-(2*sqrt(1/2)*(2*b^2 - c^2)*sqrt(b^2/(4*b^4 - 4*b^2*c^2 + c^4)) + c)/(2*b^2 - c^2)))
/(x^4 + 1)) - 1/4*sqrt(-(2*sqrt(1/2)*(2*b^2 - c^2)*sqrt(b^2/(4*b^4 - 4*b^2*c^2 + c^4)) + c)/(2*b^2 - c^2))*log
(-(2*(16*a^2*b^3 + 2*b^5 - 16*a*b^3*c + (8*a^2*b + b^3)*c^2 + (16*a^2*b^3 + 2*b^5 - 16*a*b^3*c + (8*a^2*b + b^
3)*c^2)*x^2 + 2*sqrt(1/2)*(16*a*b^5 - 4*a*b*c^4 + 2*(8*a^2*b + b^3)*c^3 + 2*(8*a*b^5 - 2*a*b*c^4 + (8*a^2*b +
b^3)*c^3 - 2*(8*a^2*b^3 + b^5)*c)*x^2 - 4*(8*a^2*b^3 + b^5)*c - (32*a^2*b^4 + 4*b^6 - 32*a*b^4*c + 16*a*b^2*c^
3 - (8*a^2 + b^2)*c^4)*x)*sqrt(b^2/(4*b^4 - 4*b^2*c^2 + c^4)) - 4*(4*a*b^4 + 2*a*b^2*c^2 - (8*a^2*b^2 + b^4)*c
)*x)*sqrt(a*x^4 + b*x^3 + c*x^2 + b*x + a) - (16*a^2*b^4 - 2*b^6 + (16*a^2*b^4 - 2*b^6 + (8*a^2*b^2 - b^4)*c^2
- 8*(8*a^3*b^2 - a*b^4)*c)*x^4 + 2*(16*a*b^5 + 24*a*b^3*c^2 - (8*a^2*b + b^3)*c^3 - 6*(8*a^2*b^3 + b^5)*c)*x^
3 + (8*a^2*b^2 - b^4)*c^2 - 4*(16*a^2*b^4 + 2*b^6 - 24*a*b^4*c - 4*a*b^2*c^3 + 3*(8*a^2*b^2 + b^4)*c^2)*x^2 +
4*sqrt(1/2)*(32*a^3*b^4 - 4*a*b^6 - (8*a^3 - a*b^2)*c^4 + (32*a^3*b^4 - 4*a*b^6 - (8*a^3 - a*b^2)*c^4 + (8*a^2
*b^2 - b^4)*c^3 - 2*(8*a^2*b^4 - b^6)*c)*x^4 + (8*a^2*b^2 - b^4)*c^3 + (32*a^2*b^5 + 4*b^7 - 48*a*b^5*c + 16*a
*b^3*c^3 + 4*a*b*c^5 - 3*(8*a^2*b + b^3)*c^4 + 4*(8*a^2*b^3 + b^5)*c^2)*x^3 - (32*a*b^6 + 32*a*b^4*c^2 - 24*a*
b^2*c^4 + (8*a^2 + b^2)*c^5 + 4*(8*a^2*b^2 + b^4)*c^3 - 12*(8*a^2*b^4 + b^6)*c)*x^2 - 2*(8*a^2*b^4 - b^6)*c +
(32*a^2*b^5 + 4*b^7 - 48*a*b^5*c + 16*a*b^3*c^3 + 4*a*b*c^5 - 3*(8*a^2*b + b^3)*c^4 + 4*(8*a^2*b^3 + b^5)*c^2)
*x)*sqrt(b^2/(4*b^4 - 4*b^2*c^2 + c^4)) - 8*(8*a^3*b^2 - a*b^4)*c + 2*(16*a*b^5 + 24*a*b^3*c^2 - (8*a^2*b + b^
3)*c^3 - 6*(8*a^2*b^3 + b^5)*c)*x)*sqrt(-(2*sqrt(1/2)*(2*b^2 - c^2)*sqrt(b^2/(4*b^4 - 4*b^2*c^2 + c^4)) + c)/(
2*b^2 - c^2)))/(x^4 + 1)) + 1/4*sqrt((2*sqrt(1/2)*(2*b^2 - c^2)*sqrt(b^2/(4*b^4 - 4*b^2*c^2 + c^4)) - c)/(2*b^
2 - c^2))*log(-(2*(16*a^2*b^3 + 2*b^5 - 16*a*b^3*c + (8*a^2*b + b^3)*c^2 + (16*a^2*b^3 + 2*b^5 - 16*a*b^3*c +
(8*a^2*b + b^3)*c^2)*x^2 - 2*sqrt(1/2)*(16*a*b^5 - 4*a*b*c^4 + 2*(8*a^2*b + b^3)*c^3 + 2*(8*a*b^5 - 2*a*b*c^4
+ (8*a^2*b + b^3)*c^3 - 2*(8*a^2*b^3 + b^5)*c)*x^2 - 4*(8*a^2*b^3 + b^5)*c - (32*a^2*b^4 + 4*b^6 - 32*a*b^4*c
+ 16*a*b^2*c^3 - (8*a^2 + b^2)*c^4)*x)*sqrt(b^2/(4*b^4 - 4*b^2*c^2 + c^4)) - 4*(4*a*b^4 + 2*a*b^2*c^2 - (8*a^2
*b^2 + b^4)*c)*x)*sqrt(a*x^4 + b*x^3 + c*x^2 + b*x + a) + (16*a^2*b^4 - 2*b^6 + (16*a^2*b^4 - 2*b^6 + (8*a^2*b
^2 - b^4)*c^2 - 8*(8*a^3*b^2 - a*b^4)*c)*x^4 + 2*(16*a*b^5 + 24*a*b^3*c^2 - (8*a^2*b + b^3)*c^3 - 6*(8*a^2*b^3
+ b^5)*c)*x^3 + (8*a^2*b^2 - b^4)*c^2 - 4*(16*a^2*b^4 + 2*b^6 - 24*a*b^4*c - 4*a*b^2*c^3 + 3*(8*a^2*b^2 + b^4
)*c^2)*x^2 - 4*sqrt(1/2)*(32*a^3*b^4 - 4*a*b^6 - (8*a^3 - a*b^2)*c^4 + (32*a^3*b^4 - 4*a*b^6 - (8*a^3 - a*b^2)
*c^4 + (8*a^2*b^2 - b^4)*c^3 - 2*(8*a^2*b^4 - b^6)*c)*x^4 + (8*a^2*b^2 - b^4)*c^3 + (32*a^2*b^5 + 4*b^7 - 48*a
*b^5*c + 16*a*b^3*c^3 + 4*a*b*c^5 - 3*(8*a^2*b + b^3)*c^4 + 4*(8*a^2*b^3 + b^5)*c^2)*x^3 - (32*a*b^6 + 32*a*b^
4*c^2 - 24*a*b^2*c^4 + (8*a^2 + b^2)*c^5 + 4*(8*a^2*b^2 + b^4)*c^3 - 12*(8*a^2*b^4 + b^6)*c)*x^2 - 2*(8*a^2*b^
4 - b^6)*c + (32*a^2*b^5 + 4*b^7 - 48*a*b^5*c + 16*a*b^3*c^3 + 4*a*b*c^5 - 3*(8*a^2*b + b^3)*c^4 + 4*(8*a^2*b^
3 + b^5)*c^2)*x)*sqrt(b^2/(4*b^4 - 4*b^2*c^2 + c^4)) - 8*(8*a^3*b^2 - a*b^4)*c + 2*(16*a*b^5 + 24*a*b^3*c^2 -
(8*a^2*b + b^3)*c^3 - 6*(8*a^2*b^3 + b^5)*c)*x)*sqrt((2*sqrt(1/2)*(2*b^2 - c^2)*sqrt(b^2/(4*b^4 - 4*b^2*c^2 +
c^4)) - c)/(2*b^2 - c^2)))/(x^4 + 1)) - 1/4*sqrt((2*sqrt(1/2)*(2*b^2 - c^2)*sqrt(b^2/(4*b^4 - 4*b^2*c^2 + c^4)
) - c)/(2*b^2 - c^2))*log(-(2*(16*a^2*b^3 + 2*b^5 - 16*a*b^3*c + (8*a^2*b + b^3)*c^2 + (16*a^2*b^3 + 2*b^5 - 1
6*a*b^3*c + (8*a^2*b + b^3)*c^2)*x^2 - 2*sqrt(1/2)*(16*a*b^5 - 4*a*b*c^4 + 2*(8*a^2*b + b^3)*c^3 + 2*(8*a*b^5
- 2*a*b*c^4 + (8*a^2*b + b^3)*c^3 - 2*(8*a^2*b^3 + b^5)*c)*x^2 - 4*(8*a^2*b^3 + b^5)*c - (32*a^2*b^4 + 4*b^6 -
32*a*b^4*c + 16*a*b^2*c^3 - (8*a^2 + b^2)*c^4)*x)*sqrt(b^2/(4*b^4 - 4*b^2*c^2 + c^4)) - 4*(4*a*b^4 + 2*a*b^2*
c^2 - (8*a^2*b^2 + b^4)*c)*x)*sqrt(a*x^4 + b*x^3 + c*x^2 + b*x + a) - (16*a^2*b^4 - 2*b^6 + (16*a^2*b^4 - 2*b^
6 + (8*a^2*b^2 - b^4)*c^2 - 8*(8*a^3*b^2 - a*b^4)*c)*x^4 + 2*(16*a*b^5 + 24*a*b^3*c^2 - (8*a^2*b + b^3)*c^3 -
6*(8*a^2*b^3 + b^5)*c)*x^3 + (8*a^2*b^2 - b^4)*c^2 - 4*(16*a^2*b^4 + 2*b^6 - 24*a*b^4*c - 4*a*b^2*c^3 + 3*(8*a
^2*b^2 + b^4)*c^2)*x^2 - 4*sqrt(1/2)*(32*a^3*b^4 - 4*a*b^6 - (8*a^3 - a*b^2)*c^4 + (32*a^3*b^4 - 4*a*b^6 - (8*
a^3 - a*b^2)*c^4 + (8*a^2*b^2 - b^4)*c^3 - 2*(8*a^2*b^4 - b^6)*c)*x^4 + (8*a^2*b^2 - b^4)*c^3 + (32*a^2*b^5 +
4*b^7 - 48*a*b^5*c + 16*a*b^3*c^3 + 4*a*b*c^5 - 3*(8*a^2*b + b^3)*c^4 + 4*(8*a^2*b^3 + b^5)*c^2)*x^3 - (32*a*b
^6 + 32*a*b^4*c^2 - 24*a*b^2*c^4 + (8*a^2 + b^2)*c^5 + 4*(8*a^2*b^2 + b^4)*c^3 - 12*(8*a^2*b^4 + b^6)*c)*x^2 -
2*(8*a^2*b^4 - b^6)*c + (32*a^2*b^5 + 4*b^7 - 48*a*b^5*c + 16*a*b^3*c^3 + 4*a*b*c^5 - 3*(8*a^2*b + b^3)*c^4 +
4*(8*a^2*b^3 + b^5)*c^2)*x)*sqrt(b^2/(4*b^4 - 4*b^2*c^2 + c^4)) - 8*(8*a^3*b^2 - a*b^4)*c + 2*(16*a*b^5 + 24*
a*b^3*c^2 - (8*a^2*b + b^3)*c^3 - 6*(8*a^2*b^3 + b^5)*c)*x)*sqrt((2*sqrt(1/2)*(2*b^2 - c^2)*sqrt(b^2/(4*b^4 -
4*b^2*c^2 + c^4)) - c)/(2*b^2 - c^2)))/(x^4 + 1))

Sympy [N/A]

Not integrable

Time = 12.21 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.17 \[ \int \frac {-1+x^4}{\left (1+x^4\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx=\int \frac {\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}{\left (x^{4} + 1\right ) \sqrt {a x^{4} + a + b x^{3} + b x + c x^{2}}}\, dx \]

[In]

integrate((x**4-1)/(x**4+1)/(a*x**4+b*x**3+c*x**2+b*x+a)**(1/2),x)

[Out]

Integral((x - 1)*(x + 1)*(x**2 + 1)/((x**4 + 1)*sqrt(a*x**4 + a + b*x**3 + b*x + c*x**2)), x)

Maxima [N/A]

Not integrable

Time = 0.37 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.15 \[ \int \frac {-1+x^4}{\left (1+x^4\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx=\int { \frac {x^{4} - 1}{\sqrt {a x^{4} + b x^{3} + c x^{2} + b x + a} {\left (x^{4} + 1\right )}} \,d x } \]

[In]

integrate((x^4-1)/(x^4+1)/(a*x^4+b*x^3+c*x^2+b*x+a)^(1/2),x, algorithm="maxima")

[Out]

integrate((x^4 - 1)/(sqrt(a*x^4 + b*x^3 + c*x^2 + b*x + a)*(x^4 + 1)), x)

Giac [N/A]

Not integrable

Time = 0.53 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.15 \[ \int \frac {-1+x^4}{\left (1+x^4\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx=\int { \frac {x^{4} - 1}{\sqrt {a x^{4} + b x^{3} + c x^{2} + b x + a} {\left (x^{4} + 1\right )}} \,d x } \]

[In]

integrate((x^4-1)/(x^4+1)/(a*x^4+b*x^3+c*x^2+b*x+a)^(1/2),x, algorithm="giac")

[Out]

integrate((x^4 - 1)/(sqrt(a*x^4 + b*x^3 + c*x^2 + b*x + a)*(x^4 + 1)), x)

Mupad [N/A]

Not integrable

Time = 7.13 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.15 \[ \int \frac {-1+x^4}{\left (1+x^4\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx=\int \frac {x^4-1}{\left (x^4+1\right )\,\sqrt {a\,x^4+b\,x^3+c\,x^2+b\,x+a}} \,d x \]

[In]

int((x^4 - 1)/((x^4 + 1)*(a + b*x + a*x^4 + b*x^3 + c*x^2)^(1/2)),x)

[Out]

int((x^4 - 1)/((x^4 + 1)*(a + b*x + a*x^4 + b*x^3 + c*x^2)^(1/2)), x)