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

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

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Rubi [F]  time = 1.63, 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+x^4}{\left (1+x^4\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx \end {gather*}

Verification is not applicable to the result.

[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*} \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 \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*}

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Mathematica [C]  time = 6.56, size = 15389, normalized size = 63.85 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Result too large to show

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IntegrateAlgebraic [A]  time = 0.65, size = 241, normalized size = 1.00 \begin {gather*} -\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 ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*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*Lo
g[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) +
Sqrt[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) & ]

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fricas [B]  time = 8.75, size = 4319, normalized size = 17.92

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[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))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [F]  time = 0.10, size = 0, normalized size = 0.00 \[\int \frac {x^{4}-1}{\left (x^{4}+1\right ) \sqrt {a \,x^{4}+b \,x^{3}+c \,x^{2}+b x +a}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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