3.73 \(\int \frac{1}{\left (a+b x^4\right )^2 \left (c+d x^4\right )^2} \, dx\)

Optimal. Leaf size=596 \[ -\frac{b^{7/4} (3 b c-11 a d) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{16 \sqrt{2} a^{7/4} (b c-a d)^3}+\frac{b^{7/4} (3 b c-11 a d) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{16 \sqrt{2} a^{7/4} (b c-a d)^3}-\frac{b^{7/4} (3 b c-11 a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{7/4} (b c-a d)^3}+\frac{b^{7/4} (3 b c-11 a d) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{8 \sqrt{2} a^{7/4} (b c-a d)^3}-\frac{d^{7/4} (11 b c-3 a d) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{c}+\sqrt{d} x^2\right )}{16 \sqrt{2} c^{7/4} (b c-a d)^3}+\frac{d^{7/4} (11 b c-3 a d) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{c}+\sqrt{d} x^2\right )}{16 \sqrt{2} c^{7/4} (b c-a d)^3}-\frac{d^{7/4} (11 b c-3 a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{8 \sqrt{2} c^{7/4} (b c-a d)^3}+\frac{d^{7/4} (11 b c-3 a d) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}+1\right )}{8 \sqrt{2} c^{7/4} (b c-a d)^3}+\frac{b x}{4 a \left (a+b x^4\right ) \left (c+d x^4\right ) (b c-a d)}+\frac{d x (a d+b c)}{4 a c \left (c+d x^4\right ) (b c-a d)^2} \]

[Out]

(d*(b*c + a*d)*x)/(4*a*c*(b*c - a*d)^2*(c + d*x^4)) + (b*x)/(4*a*(b*c - a*d)*(a
+ b*x^4)*(c + d*x^4)) - (b^(7/4)*(3*b*c - 11*a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)
/a^(1/4)])/(8*Sqrt[2]*a^(7/4)*(b*c - a*d)^3) + (b^(7/4)*(3*b*c - 11*a*d)*ArcTan[
1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(8*Sqrt[2]*a^(7/4)*(b*c - a*d)^3) - (d^(7/4)*(
11*b*c - 3*a*d)*ArcTan[1 - (Sqrt[2]*d^(1/4)*x)/c^(1/4)])/(8*Sqrt[2]*c^(7/4)*(b*c
- a*d)^3) + (d^(7/4)*(11*b*c - 3*a*d)*ArcTan[1 + (Sqrt[2]*d^(1/4)*x)/c^(1/4)])/
(8*Sqrt[2]*c^(7/4)*(b*c - a*d)^3) - (b^(7/4)*(3*b*c - 11*a*d)*Log[Sqrt[a] - Sqrt
[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(16*Sqrt[2]*a^(7/4)*(b*c - a*d)^3) + (b^(7
/4)*(3*b*c - 11*a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(16
*Sqrt[2]*a^(7/4)*(b*c - a*d)^3) - (d^(7/4)*(11*b*c - 3*a*d)*Log[Sqrt[c] - Sqrt[2
]*c^(1/4)*d^(1/4)*x + Sqrt[d]*x^2])/(16*Sqrt[2]*c^(7/4)*(b*c - a*d)^3) + (d^(7/4
)*(11*b*c - 3*a*d)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*x + Sqrt[d]*x^2])/(16*S
qrt[2]*c^(7/4)*(b*c - a*d)^3)

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Rubi [A]  time = 1.46784, antiderivative size = 596, normalized size of antiderivative = 1., number of steps used = 21, number of rules used = 9, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.474 \[ -\frac{b^{7/4} (3 b c-11 a d) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{16 \sqrt{2} a^{7/4} (b c-a d)^3}+\frac{b^{7/4} (3 b c-11 a d) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{16 \sqrt{2} a^{7/4} (b c-a d)^3}-\frac{b^{7/4} (3 b c-11 a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{7/4} (b c-a d)^3}+\frac{b^{7/4} (3 b c-11 a d) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{8 \sqrt{2} a^{7/4} (b c-a d)^3}-\frac{d^{7/4} (11 b c-3 a d) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{c}+\sqrt{d} x^2\right )}{16 \sqrt{2} c^{7/4} (b c-a d)^3}+\frac{d^{7/4} (11 b c-3 a d) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{c}+\sqrt{d} x^2\right )}{16 \sqrt{2} c^{7/4} (b c-a d)^3}-\frac{d^{7/4} (11 b c-3 a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{8 \sqrt{2} c^{7/4} (b c-a d)^3}+\frac{d^{7/4} (11 b c-3 a d) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}+1\right )}{8 \sqrt{2} c^{7/4} (b c-a d)^3}+\frac{b x}{4 a \left (a+b x^4\right ) \left (c+d x^4\right ) (b c-a d)}+\frac{d x (a d+b c)}{4 a c \left (c+d x^4\right ) (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b*x^4)^2*(c + d*x^4)^2),x]

[Out]

(d*(b*c + a*d)*x)/(4*a*c*(b*c - a*d)^2*(c + d*x^4)) + (b*x)/(4*a*(b*c - a*d)*(a
+ b*x^4)*(c + d*x^4)) - (b^(7/4)*(3*b*c - 11*a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)
/a^(1/4)])/(8*Sqrt[2]*a^(7/4)*(b*c - a*d)^3) + (b^(7/4)*(3*b*c - 11*a*d)*ArcTan[
1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(8*Sqrt[2]*a^(7/4)*(b*c - a*d)^3) - (d^(7/4)*(
11*b*c - 3*a*d)*ArcTan[1 - (Sqrt[2]*d^(1/4)*x)/c^(1/4)])/(8*Sqrt[2]*c^(7/4)*(b*c
- a*d)^3) + (d^(7/4)*(11*b*c - 3*a*d)*ArcTan[1 + (Sqrt[2]*d^(1/4)*x)/c^(1/4)])/
(8*Sqrt[2]*c^(7/4)*(b*c - a*d)^3) - (b^(7/4)*(3*b*c - 11*a*d)*Log[Sqrt[a] - Sqrt
[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(16*Sqrt[2]*a^(7/4)*(b*c - a*d)^3) + (b^(7
/4)*(3*b*c - 11*a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(16
*Sqrt[2]*a^(7/4)*(b*c - a*d)^3) - (d^(7/4)*(11*b*c - 3*a*d)*Log[Sqrt[c] - Sqrt[2
]*c^(1/4)*d^(1/4)*x + Sqrt[d]*x^2])/(16*Sqrt[2]*c^(7/4)*(b*c - a*d)^3) + (d^(7/4
)*(11*b*c - 3*a*d)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*x + Sqrt[d]*x^2])/(16*S
qrt[2]*c^(7/4)*(b*c - a*d)^3)

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x**4+a)**2/(d*x**4+c)**2,x)

[Out]

Timed out

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Mathematica [A]  time = 3.49316, size = 561, normalized size = 0.94 \[ \frac{1}{32} \left (\frac{\sqrt{2} b^{7/4} (11 a d-3 b c) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{a^{7/4} (b c-a d)^3}+\frac{\sqrt{2} b^{7/4} (11 a d-3 b c) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{a^{7/4} (a d-b c)^3}+\frac{2 \sqrt{2} b^{7/4} (11 a d-3 b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{a^{7/4} (b c-a d)^3}+\frac{2 \sqrt{2} b^{7/4} (11 a d-3 b c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{a^{7/4} (a d-b c)^3}+\frac{8 b^2 x}{a \left (a+b x^4\right ) (b c-a d)^2}+\frac{\sqrt{2} d^{7/4} (11 b c-3 a d) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{c}+\sqrt{d} x^2\right )}{c^{7/4} (a d-b c)^3}+\frac{\sqrt{2} d^{7/4} (11 b c-3 a d) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{c}+\sqrt{d} x^2\right )}{c^{7/4} (b c-a d)^3}+\frac{2 \sqrt{2} d^{7/4} (3 a d-11 b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{c^{7/4} (b c-a d)^3}+\frac{2 \sqrt{2} d^{7/4} (11 b c-3 a d) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}+1\right )}{c^{7/4} (b c-a d)^3}+\frac{8 d^2 x}{c \left (c+d x^4\right ) (b c-a d)^2}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b*x^4)^2*(c + d*x^4)^2),x]

[Out]

((8*b^2*x)/(a*(b*c - a*d)^2*(a + b*x^4)) + (8*d^2*x)/(c*(b*c - a*d)^2*(c + d*x^4
)) + (2*Sqrt[2]*b^(7/4)*(-3*b*c + 11*a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)
])/(a^(7/4)*(b*c - a*d)^3) + (2*Sqrt[2]*b^(7/4)*(-3*b*c + 11*a*d)*ArcTan[1 + (Sq
rt[2]*b^(1/4)*x)/a^(1/4)])/(a^(7/4)*(-(b*c) + a*d)^3) + (2*Sqrt[2]*d^(7/4)*(-11*
b*c + 3*a*d)*ArcTan[1 - (Sqrt[2]*d^(1/4)*x)/c^(1/4)])/(c^(7/4)*(b*c - a*d)^3) +
(2*Sqrt[2]*d^(7/4)*(11*b*c - 3*a*d)*ArcTan[1 + (Sqrt[2]*d^(1/4)*x)/c^(1/4)])/(c^
(7/4)*(b*c - a*d)^3) + (Sqrt[2]*b^(7/4)*(-3*b*c + 11*a*d)*Log[Sqrt[a] - Sqrt[2]*
a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(a^(7/4)*(b*c - a*d)^3) + (Sqrt[2]*b^(7/4)*(-3
*b*c + 11*a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(a^(7/4)*
(-(b*c) + a*d)^3) + (Sqrt[2]*d^(7/4)*(11*b*c - 3*a*d)*Log[Sqrt[c] - Sqrt[2]*c^(1
/4)*d^(1/4)*x + Sqrt[d]*x^2])/(c^(7/4)*(-(b*c) + a*d)^3) + (Sqrt[2]*d^(7/4)*(11*
b*c - 3*a*d)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*x + Sqrt[d]*x^2])/(c^(7/4)*(b
*c - a*d)^3))/32

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Maple [A]  time = 0.003, size = 784, normalized size = 1.3 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x^4+a)^2/(d*x^4+c)^2,x)

[Out]

1/4*d^3/(a*d-b*c)^3/c*x/(d*x^4+c)*a-1/4*d^2/(a*d-b*c)^3*x/(d*x^4+c)*b+3/16*d^3/(
a*d-b*c)^3/c^2*(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x+1)*a-11/16*d^2/(
a*d-b*c)^3/c*(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x+1)*b+3/16*d^3/(a*d
-b*c)^3/c^2*(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x-1)*a-11/16*d^2/(a*d
-b*c)^3/c*(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x-1)*b+3/32*d^3/(a*d-b*
c)^3/c^2*(c/d)^(1/4)*2^(1/2)*ln((x^2+(c/d)^(1/4)*x*2^(1/2)+(c/d)^(1/2))/(x^2-(c/
d)^(1/4)*x*2^(1/2)+(c/d)^(1/2)))*a-11/32*d^2/(a*d-b*c)^3/c*(c/d)^(1/4)*2^(1/2)*l
n((x^2+(c/d)^(1/4)*x*2^(1/2)+(c/d)^(1/2))/(x^2-(c/d)^(1/4)*x*2^(1/2)+(c/d)^(1/2)
))*b+1/4*b^2/(a*d-b*c)^3*x/(b*x^4+a)*d-1/4*b^3/(a*d-b*c)^3*x/a/(b*x^4+a)*c+11/16
*b^2/(a*d-b*c)^3/a*(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x-1)*d-3/16*b^
3/(a*d-b*c)^3/a^2*(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x-1)*c+11/32*b^
2/(a*d-b*c)^3/a*(a/b)^(1/4)*2^(1/2)*ln((x^2+(a/b)^(1/4)*x*2^(1/2)+(a/b)^(1/2))/(
x^2-(a/b)^(1/4)*x*2^(1/2)+(a/b)^(1/2)))*d-3/32*b^3/(a*d-b*c)^3/a^2*(a/b)^(1/4)*2
^(1/2)*ln((x^2+(a/b)^(1/4)*x*2^(1/2)+(a/b)^(1/2))/(x^2-(a/b)^(1/4)*x*2^(1/2)+(a/
b)^(1/2)))*c+11/16*b^2/(a*d-b*c)^3/a*(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1
/4)*x+1)*d-3/16*b^3/(a*d-b*c)^3/a^2*(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/
4)*x+1)*c

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)^2*(d*x^4 + c)^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)^2*(d*x^4 + c)^2),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b*x**4+a)**2/(d*x**4+c)**2,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{4} + a\right )}^{2}{\left (d x^{4} + c\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)^2*(d*x^4 + c)^2),x, algorithm="giac")

[Out]

integrate(1/((b*x^4 + a)^2*(d*x^4 + c)^2), x)