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3.28.67 \(\int \frac {1+x^4}{(-1+x^4) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx\) [2767]

Optimal. Leaf size=262 \[ \frac {\text {ArcTan}\left (\frac {\sqrt {2 a-c} x}{\sqrt {a}+\sqrt {a} x^2-\sqrt {a+b x+c x^2+b x^3+a x^4}}\right )}{\sqrt {2 a-c}}-\frac {\sqrt {-2 a-2 b-c} \text {ArcTan}\left (\frac {\sqrt {-2 a-2 b-c} x}{\sqrt {a}-2 \sqrt {a} x+\sqrt {a} x^2-\sqrt {a+b x+c x^2+b x^3+a x^4}}\right )}{2 (2 a+2 b+c)}-\frac {\sqrt {-2 a+2 b-c} \text {ArcTan}\left (\frac {\sqrt {-2 a+2 b-c} x}{\sqrt {a}+2 \sqrt {a} x+\sqrt {a} x^2-\sqrt {a+b x+c x^2+b x^3+a x^4}}\right )}{2 (2 a-2 b+c)} \]

[Out]

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

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Rubi [F]
time = 0.95, 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] - (I/2)*Defer[Int][1/((I - x)*Sqrt[a + b*x + c*x^2 + b*
x^3 + a*x^4]), x] + Defer[Int][1/((-1 + x)*Sqrt[a + b*x + c*x^2 + b*x^3 + a*x^4]), x]/2 - (I/2)*Defer[Int][1/(
(I + x)*Sqrt[a + b*x + c*x^2 + b*x^3 + a*x^4]), x] - Defer[Int][1/((1 + 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\\ &=2 \int \frac {1}{\left (-1+x^4\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\\ &=2 \int \left (\frac {1}{2 \left (-1+x^2\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}}-\frac {1}{2 \left (1+x^2\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\\ &=\int \frac {1}{\sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx+\int \frac {1}{\left (-1+x^2\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx-\int \frac {1}{\left (1+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-\int \left (\frac {i}{2 (i-x) \sqrt {a+b x+c x^2+b x^3+a x^4}}+\frac {i}{2 (i+x) \sqrt {a+b x+c x^2+b x^3+a x^4}}\right ) \, dx+\int \left (\frac {1}{2 (-1+x) \sqrt {a+b x+c x^2+b x^3+a x^4}}-\frac {1}{2 (1+x) \sqrt {a+b x+c x^2+b x^3+a x^4}}\right ) \, dx\\ &=-\left (\frac {1}{2} i \int \frac {1}{(i-x) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx\right )-\frac {1}{2} i \int \frac {1}{(i+x) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx+\frac {1}{2} \int \frac {1}{(-1+x) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx-\frac {1}{2} \int \frac {1}{(1+x) \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 [A]
time = 1.63, size = 216, normalized size = 0.82 \begin {gather*} \frac {1}{2} \left (\frac {\text {ArcTan}\left (\frac {\sqrt {-2 a-2 b-c} x}{\sqrt {a} (-1+x)^2-\sqrt {a+b x+c x^2+b x^3+a x^4}}\right )}{\sqrt {-2 a-2 b-c}}+\frac {\text {ArcTan}\left (\frac {\sqrt {-2 a+2 b-c} x}{\sqrt {a} (1+x)^2-\sqrt {a+b x+c x^2+b x^3+a x^4}}\right )}{\sqrt {-2 a+2 b-c}}+\frac {2 \text {ArcTan}\left (\frac {\sqrt {2 a-c} x}{\sqrt {a} \left (1+x^2\right )-\sqrt {a+b x+c x^2+b x^3+a x^4}}\right )}{\sqrt {2 a-c}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.27, size = 82800, normalized size = 316.03

method result size
default \(\text {Expression too large to display}\) \(82800\)

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,method=_RETURNVERBOSE)

[Out]

result too large to display

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \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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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 536 vs. \(2 (224) = 448\).
time = 2.14, size = 4887, normalized size = 18.65 \begin {gather*} \text {Too large to display} \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="fricas")

[Out]

[1/8*((4*a^2 + 4*a*b - 2*b*c - c^2)*sqrt(2*a - 2*b + c)*log(((24*a^2 - 16*a*b + b^2 + 4*a*c)*x^4 + 4*(8*a^2 +
4*a*b - 3*b^2 - 2*(2*a - b)*c)*x^3 + 2*(24*a^2 + 3*b^2 - 4*(a + 2*b)*c + 4*c^2)*x^2 + 4*sqrt(a*x^4 + b*x^3 + c
*x^2 + b*x + a)*((4*a - b)*x^2 + 2*(2*a + b - c)*x + 4*a - b)*sqrt(2*a - 2*b + c) + 24*a^2 - 16*a*b + b^2 + 4*
a*c + 4*(8*a^2 + 4*a*b - 3*b^2 - 2*(2*a - b)*c)*x)/(x^4 + 4*x^3 + 6*x^2 + 4*x + 1)) + (4*a^2 - 4*a*b + 2*b*c -
 c^2)*sqrt(2*a + 2*b + c)*log(((24*a^2 + 16*a*b + b^2 + 4*a*c)*x^4 - 4*(8*a^2 - 4*a*b - 3*b^2 - 2*(2*a + b)*c)
*x^3 + 2*(24*a^2 + 3*b^2 - 4*(a - 2*b)*c + 4*c^2)*x^2 - 4*sqrt(a*x^4 + b*x^3 + c*x^2 + b*x + a)*((4*a + b)*x^2
 - 2*(2*a - b - c)*x + 4*a + b)*sqrt(2*a + 2*b + c) + 24*a^2 + 16*a*b + b^2 + 4*a*c - 4*(8*a^2 - 4*a*b - 3*b^2
 - 2*(2*a + b)*c)*x)/(x^4 - 4*x^3 + 6*x^2 - 4*x + 1)) - 2*(4*a^2 - 4*b^2 + 4*a*c + c^2)*sqrt(-2*a + c)*log(-((
8*a^2 - b^2 - 4*a*c)*x^4 + 8*(2*a*b - b*c)*x^3 - 2*(8*a^2 + b^2 - 12*a*c + 4*c^2)*x^2 + 8*a^2 + 4*sqrt(a*x^4 +
 b*x^3 + c*x^2 + b*x + a)*(b*x^2 - 2*(2*a - c)*x + b)*sqrt(-2*a + c) - b^2 - 4*a*c + 8*(2*a*b - b*c)*x)/(x^4 +
 2*x^2 + 1)))/(8*a^3 - 8*a*b^2 - 2*a*c^2 - c^3 + 4*(a^2 + b^2)*c), -1/8*(4*(4*a^2 - 4*b^2 + 4*a*c + c^2)*sqrt(
2*a - c)*arctan(-1/2*sqrt(a*x^4 + b*x^3 + c*x^2 + b*x + a)*(b*x^2 - 2*(2*a - c)*x + b)*sqrt(2*a - c)/((2*a^2 -
 a*c)*x^4 + (2*a*b - b*c)*x^3 + (2*a*c - c^2)*x^2 + 2*a^2 - a*c + (2*a*b - b*c)*x)) - (4*a^2 + 4*a*b - 2*b*c -
 c^2)*sqrt(2*a - 2*b + c)*log(((24*a^2 - 16*a*b + b^2 + 4*a*c)*x^4 + 4*(8*a^2 + 4*a*b - 3*b^2 - 2*(2*a - b)*c)
*x^3 + 2*(24*a^2 + 3*b^2 - 4*(a + 2*b)*c + 4*c^2)*x^2 + 4*sqrt(a*x^4 + b*x^3 + c*x^2 + b*x + a)*((4*a - b)*x^2
 + 2*(2*a + b - c)*x + 4*a - b)*sqrt(2*a - 2*b + c) + 24*a^2 - 16*a*b + b^2 + 4*a*c + 4*(8*a^2 + 4*a*b - 3*b^2
 - 2*(2*a - b)*c)*x)/(x^4 + 4*x^3 + 6*x^2 + 4*x + 1)) - (4*a^2 - 4*a*b + 2*b*c - c^2)*sqrt(2*a + 2*b + c)*log(
((24*a^2 + 16*a*b + b^2 + 4*a*c)*x^4 - 4*(8*a^2 - 4*a*b - 3*b^2 - 2*(2*a + b)*c)*x^3 + 2*(24*a^2 + 3*b^2 - 4*(
a - 2*b)*c + 4*c^2)*x^2 - 4*sqrt(a*x^4 + b*x^3 + c*x^2 + b*x + a)*((4*a + b)*x^2 - 2*(2*a - b - c)*x + 4*a + b
)*sqrt(2*a + 2*b + c) + 24*a^2 + 16*a*b + b^2 + 4*a*c - 4*(8*a^2 - 4*a*b - 3*b^2 - 2*(2*a + b)*c)*x)/(x^4 - 4*
x^3 + 6*x^2 - 4*x + 1)))/(8*a^3 - 8*a*b^2 - 2*a*c^2 - c^3 + 4*(a^2 + b^2)*c), 1/8*(2*(4*a^2 - 4*a*b + 2*b*c -
c^2)*sqrt(-2*a - 2*b - c)*arctan(1/2*sqrt(a*x^4 + b*x^3 + c*x^2 + b*x + a)*((4*a + b)*x^2 - 2*(2*a - b - c)*x
+ 4*a + b)*sqrt(-2*a - 2*b - c)/((2*a^2 + 2*a*b + a*c)*x^4 + (2*a*b + 2*b^2 + b*c)*x^3 + (2*(a + b)*c + c^2)*x
^2 + 2*a^2 + 2*a*b + a*c + (2*a*b + 2*b^2 + b*c)*x)) + (4*a^2 + 4*a*b - 2*b*c - c^2)*sqrt(2*a - 2*b + c)*log((
(24*a^2 - 16*a*b + b^2 + 4*a*c)*x^4 + 4*(8*a^2 + 4*a*b - 3*b^2 - 2*(2*a - b)*c)*x^3 + 2*(24*a^2 + 3*b^2 - 4*(a
 + 2*b)*c + 4*c^2)*x^2 + 4*sqrt(a*x^4 + b*x^3 + c*x^2 + b*x + a)*((4*a - b)*x^2 + 2*(2*a + b - c)*x + 4*a - b)
*sqrt(2*a - 2*b + c) + 24*a^2 - 16*a*b + b^2 + 4*a*c + 4*(8*a^2 + 4*a*b - 3*b^2 - 2*(2*a - b)*c)*x)/(x^4 + 4*x
^3 + 6*x^2 + 4*x + 1)) - 2*(4*a^2 - 4*b^2 + 4*a*c + c^2)*sqrt(-2*a + c)*log(-((8*a^2 - b^2 - 4*a*c)*x^4 + 8*(2
*a*b - b*c)*x^3 - 2*(8*a^2 + b^2 - 12*a*c + 4*c^2)*x^2 + 8*a^2 + 4*sqrt(a*x^4 + b*x^3 + c*x^2 + b*x + a)*(b*x^
2 - 2*(2*a - c)*x + b)*sqrt(-2*a + c) - b^2 - 4*a*c + 8*(2*a*b - b*c)*x)/(x^4 + 2*x^2 + 1)))/(8*a^3 - 8*a*b^2
- 2*a*c^2 - c^3 + 4*(a^2 + b^2)*c), -1/8*(4*(4*a^2 - 4*b^2 + 4*a*c + c^2)*sqrt(2*a - c)*arctan(-1/2*sqrt(a*x^4
 + b*x^3 + c*x^2 + b*x + a)*(b*x^2 - 2*(2*a - c)*x + b)*sqrt(2*a - c)/((2*a^2 - a*c)*x^4 + (2*a*b - b*c)*x^3 +
 (2*a*c - c^2)*x^2 + 2*a^2 - a*c + (2*a*b - b*c)*x)) - 2*(4*a^2 - 4*a*b + 2*b*c - c^2)*sqrt(-2*a - 2*b - c)*ar
ctan(1/2*sqrt(a*x^4 + b*x^3 + c*x^2 + b*x + a)*((4*a + b)*x^2 - 2*(2*a - b - c)*x + 4*a + b)*sqrt(-2*a - 2*b -
 c)/((2*a^2 + 2*a*b + a*c)*x^4 + (2*a*b + 2*b^2 + b*c)*x^3 + (2*(a + b)*c + c^2)*x^2 + 2*a^2 + 2*a*b + a*c + (
2*a*b + 2*b^2 + b*c)*x)) - (4*a^2 + 4*a*b - 2*b*c - c^2)*sqrt(2*a - 2*b + c)*log(((24*a^2 - 16*a*b + b^2 + 4*a
*c)*x^4 + 4*(8*a^2 + 4*a*b - 3*b^2 - 2*(2*a - b)*c)*x^3 + 2*(24*a^2 + 3*b^2 - 4*(a + 2*b)*c + 4*c^2)*x^2 + 4*s
qrt(a*x^4 + b*x^3 + c*x^2 + b*x + a)*((4*a - b)*x^2 + 2*(2*a + b - c)*x + 4*a - b)*sqrt(2*a - 2*b + c) + 24*a^
2 - 16*a*b + b^2 + 4*a*c + 4*(8*a^2 + 4*a*b - 3*b^2 - 2*(2*a - b)*c)*x)/(x^4 + 4*x^3 + 6*x^2 + 4*x + 1)))/(8*a
^3 - 8*a*b^2 - 2*a*c^2 - c^3 + 4*(a^2 + b^2)*c), 1/8*(2*(4*a^2 + 4*a*b - 2*b*c - c^2)*sqrt(-2*a + 2*b - c)*arc
tan(-1/2*sqrt(a*x^4 + b*x^3 + c*x^2 + b*x + a)*((4*a - b)*x^2 + 2*(2*a + b - c)*x + 4*a - b)*sqrt(-2*a + 2*b -
 c)/((2*a^2 - 2*a*b + a*c)*x^4 + (2*a*b - 2*b^2 + b*c)*x^3 + (2*(a - b)*c + c^2)*x^2 + 2*a^2 - 2*a*b + a*c + (
2*a*b - 2*b^2 + b*c)*x)) + (4*a^2 - 4*a*b + 2*b*c - c^2)*sqrt(2*a + 2*b + c)*log(((24*a^2 + 16*a*b + b^2 + 4*a
*c)*x^4 - 4*(8*a^2 - 4*a*b - 3*b^2 - 2*(2*a + b)*c)*x^3 + 2*(24*a^2 + 3*b^2 - 4*(a - 2*b)*c + 4*c^2)*x^2 - 4*s
qrt(a*x^4 + b*x^3 + c*x^2 + b*x + a)*((4*a + b)*x^2 - 2*(2*a - b - c)*x + 4*a + b)*sqrt(2*a + 2*b + c) + 24*a^
2 + 16*a*b + b^2 + 4*a*c - 4*(8*a^2 - 4*a*b - 3...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{4} + 1}{\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 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**4 + 1)/((x - 1)*(x + 1)*(x**2 + 1)*sqrt(a*x**4 + a + b*x**3 + b*x + c*x**2)), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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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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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