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

Optimal. Leaf size=460 \[ -\frac{b^{5/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{4 \sqrt{2} a^{5/4} (b c-a d)}+\frac{b^{5/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{4 \sqrt{2} a^{5/4} (b c-a d)}+\frac{b^{5/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{5/4} (b c-a d)}-\frac{b^{5/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{2 \sqrt{2} a^{5/4} (b c-a d)}+\frac{d^{5/4} \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{c}+\sqrt{d} x^2\right )}{4 \sqrt{2} c^{5/4} (b c-a d)}-\frac{d^{5/4} \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{c}+\sqrt{d} x^2\right )}{4 \sqrt{2} c^{5/4} (b c-a d)}-\frac{d^{5/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{2 \sqrt{2} c^{5/4} (b c-a d)}+\frac{d^{5/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}+1\right )}{2 \sqrt{2} c^{5/4} (b c-a d)}-\frac{1}{a c x} \]

[Out]

-(1/(a*c*x)) + (b^(5/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^(5
/4)*(b*c - a*d)) - (b^(5/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*
a^(5/4)*(b*c - a*d)) - (d^(5/4)*ArcTan[1 - (Sqrt[2]*d^(1/4)*x)/c^(1/4)])/(2*Sqrt
[2]*c^(5/4)*(b*c - a*d)) + (d^(5/4)*ArcTan[1 + (Sqrt[2]*d^(1/4)*x)/c^(1/4)])/(2*
Sqrt[2]*c^(5/4)*(b*c - a*d)) - (b^(5/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x
+ Sqrt[b]*x^2])/(4*Sqrt[2]*a^(5/4)*(b*c - a*d)) + (b^(5/4)*Log[Sqrt[a] + Sqrt[2]
*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(4*Sqrt[2]*a^(5/4)*(b*c - a*d)) + (d^(5/4)*Lo
g[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*x + Sqrt[d]*x^2])/(4*Sqrt[2]*c^(5/4)*(b*c -
a*d)) - (d^(5/4)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*x + Sqrt[d]*x^2])/(4*Sqrt
[2]*c^(5/4)*(b*c - a*d))

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Rubi [A]  time = 1.03999, antiderivative size = 460, normalized size of antiderivative = 1., number of steps used = 21, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364 \[ -\frac{b^{5/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{4 \sqrt{2} a^{5/4} (b c-a d)}+\frac{b^{5/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{4 \sqrt{2} a^{5/4} (b c-a d)}+\frac{b^{5/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{5/4} (b c-a d)}-\frac{b^{5/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{2 \sqrt{2} a^{5/4} (b c-a d)}+\frac{d^{5/4} \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{c}+\sqrt{d} x^2\right )}{4 \sqrt{2} c^{5/4} (b c-a d)}-\frac{d^{5/4} \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{c}+\sqrt{d} x^2\right )}{4 \sqrt{2} c^{5/4} (b c-a d)}-\frac{d^{5/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{2 \sqrt{2} c^{5/4} (b c-a d)}+\frac{d^{5/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}+1\right )}{2 \sqrt{2} c^{5/4} (b c-a d)}-\frac{1}{a c x} \]

Antiderivative was successfully verified.

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

[Out]

-(1/(a*c*x)) + (b^(5/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^(5
/4)*(b*c - a*d)) - (b^(5/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*
a^(5/4)*(b*c - a*d)) - (d^(5/4)*ArcTan[1 - (Sqrt[2]*d^(1/4)*x)/c^(1/4)])/(2*Sqrt
[2]*c^(5/4)*(b*c - a*d)) + (d^(5/4)*ArcTan[1 + (Sqrt[2]*d^(1/4)*x)/c^(1/4)])/(2*
Sqrt[2]*c^(5/4)*(b*c - a*d)) - (b^(5/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x
+ Sqrt[b]*x^2])/(4*Sqrt[2]*a^(5/4)*(b*c - a*d)) + (b^(5/4)*Log[Sqrt[a] + Sqrt[2]
*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(4*Sqrt[2]*a^(5/4)*(b*c - a*d)) + (d^(5/4)*Lo
g[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*x + Sqrt[d]*x^2])/(4*Sqrt[2]*c^(5/4)*(b*c -
a*d)) - (d^(5/4)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*x + Sqrt[d]*x^2])/(4*Sqrt
[2]*c^(5/4)*(b*c - a*d))

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

[Out]

Timed out

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Mathematica [A]  time = 0.514008, size = 385, normalized size = 0.84 \[ \frac{\frac{\sqrt{2} b^{5/4} x \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{a^{5/4}}-\frac{\sqrt{2} b^{5/4} x \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{a^{5/4}}-\frac{2 \sqrt{2} b^{5/4} x \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{a^{5/4}}+\frac{2 \sqrt{2} b^{5/4} x \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{a^{5/4}}+\frac{8 b}{a}-\frac{\sqrt{2} d^{5/4} x \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{c}+\sqrt{d} x^2\right )}{c^{5/4}}+\frac{\sqrt{2} d^{5/4} x \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{c}+\sqrt{d} x^2\right )}{c^{5/4}}+\frac{2 \sqrt{2} d^{5/4} x \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{c^{5/4}}-\frac{2 \sqrt{2} d^{5/4} x \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}+1\right )}{c^{5/4}}-\frac{8 d}{c}}{8 a d x-8 b c x} \]

Antiderivative was successfully verified.

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

[Out]

((8*b)/a - (8*d)/c - (2*Sqrt[2]*b^(5/4)*x*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)
])/a^(5/4) + (2*Sqrt[2]*b^(5/4)*x*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/a^(5/
4) + (2*Sqrt[2]*d^(5/4)*x*ArcTan[1 - (Sqrt[2]*d^(1/4)*x)/c^(1/4)])/c^(5/4) - (2*
Sqrt[2]*d^(5/4)*x*ArcTan[1 + (Sqrt[2]*d^(1/4)*x)/c^(1/4)])/c^(5/4) + (Sqrt[2]*b^
(5/4)*x*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/a^(5/4) - (Sqrt[
2]*b^(5/4)*x*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/a^(5/4) - (
Sqrt[2]*d^(5/4)*x*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*x + Sqrt[d]*x^2])/c^(5/4
) + (Sqrt[2]*d^(5/4)*x*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*x + Sqrt[d]*x^2])/c
^(5/4))/(-8*b*c*x + 8*a*d*x)

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Maple [A]  time = 0.003, size = 331, normalized size = 0.7 \[ -{\frac{d\sqrt{2}}{8\,c \left ( ad-bc \right ) }\ln \left ({1 \left ({x}^{2}-\sqrt [4]{{\frac{c}{d}}}x\sqrt{2}+\sqrt{{\frac{c}{d}}} \right ) \left ({x}^{2}+\sqrt [4]{{\frac{c}{d}}}x\sqrt{2}+\sqrt{{\frac{c}{d}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}-{\frac{d\sqrt{2}}{4\,c \left ( ad-bc \right ) }\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}-{\frac{d\sqrt{2}}{4\,c \left ( ad-bc \right ) }\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}+{\frac{b\sqrt{2}}{8\,a \left ( ad-bc \right ) }\ln \left ({1 \left ({x}^{2}-\sqrt [4]{{\frac{a}{b}}}x\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ({x}^{2}+\sqrt [4]{{\frac{a}{b}}}x\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{b\sqrt{2}}{4\,a \left ( ad-bc \right ) }\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{b\sqrt{2}}{4\,a \left ( ad-bc \right ) }\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-{\frac{1}{acx}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/8*d/c/(a*d-b*c)/(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)))-1/4*d/c/(a*d-b*c)/(c/d)^(1/4)*2^(1/2)
*arctan(2^(1/2)/(c/d)^(1/4)*x+1)-1/4*d/c/(a*d-b*c)/(c/d)^(1/4)*2^(1/2)*arctan(2^
(1/2)/(c/d)^(1/4)*x-1)+1/8*b/a/(a*d-b*c)/(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)))+1/4*b/a/(a*d-b*
c)/(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x+1)+1/4*b/a/(a*d-b*c)/(a/b)^(
1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x-1)-1/a/c/x

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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)*(d*x^4 + c)*x^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.59273, size = 1778, normalized size = 3.87 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*(4*(-b^5/(a^5*b^4*c^4 - 4*a^6*b^3*c^3*d + 6*a^7*b^2*c^2*d^2 - 4*a^8*b*c*d^3
+ a^9*d^4))^(1/4)*a*c*x*arctan(-(a^4*b^3*c^3 - 3*a^5*b^2*c^2*d + 3*a^6*b*c*d^2 -
a^7*d^3)*(-b^5/(a^5*b^4*c^4 - 4*a^6*b^3*c^3*d + 6*a^7*b^2*c^2*d^2 - 4*a^8*b*c*d
^3 + a^9*d^4))^(3/4)/(b^4*x + b^4*sqrt((b^3*x^2 - (a^3*b^2*c^2 - 2*a^4*b*c*d + a
^5*d^2)*sqrt(-b^5/(a^5*b^4*c^4 - 4*a^6*b^3*c^3*d + 6*a^7*b^2*c^2*d^2 - 4*a^8*b*c
*d^3 + a^9*d^4)))/b^3))) - 4*(-d^5/(b^4*c^9 - 4*a*b^3*c^8*d + 6*a^2*b^2*c^7*d^2
- 4*a^3*b*c^6*d^3 + a^4*c^5*d^4))^(1/4)*a*c*x*arctan(-(b^3*c^7 - 3*a*b^2*c^6*d +
3*a^2*b*c^5*d^2 - a^3*c^4*d^3)*(-d^5/(b^4*c^9 - 4*a*b^3*c^8*d + 6*a^2*b^2*c^7*d
^2 - 4*a^3*b*c^6*d^3 + a^4*c^5*d^4))^(3/4)/(d^4*x + d^4*sqrt((d^3*x^2 - (b^2*c^5
- 2*a*b*c^4*d + a^2*c^3*d^2)*sqrt(-d^5/(b^4*c^9 - 4*a*b^3*c^8*d + 6*a^2*b^2*c^7
*d^2 - 4*a^3*b*c^6*d^3 + a^4*c^5*d^4)))/d^3))) - (-b^5/(a^5*b^4*c^4 - 4*a^6*b^3*
c^3*d + 6*a^7*b^2*c^2*d^2 - 4*a^8*b*c*d^3 + a^9*d^4))^(1/4)*a*c*x*log(b^4*x + (a
^4*b^3*c^3 - 3*a^5*b^2*c^2*d + 3*a^6*b*c*d^2 - a^7*d^3)*(-b^5/(a^5*b^4*c^4 - 4*a
^6*b^3*c^3*d + 6*a^7*b^2*c^2*d^2 - 4*a^8*b*c*d^3 + a^9*d^4))^(3/4)) + (-b^5/(a^5
*b^4*c^4 - 4*a^6*b^3*c^3*d + 6*a^7*b^2*c^2*d^2 - 4*a^8*b*c*d^3 + a^9*d^4))^(1/4)
*a*c*x*log(b^4*x - (a^4*b^3*c^3 - 3*a^5*b^2*c^2*d + 3*a^6*b*c*d^2 - a^7*d^3)*(-b
^5/(a^5*b^4*c^4 - 4*a^6*b^3*c^3*d + 6*a^7*b^2*c^2*d^2 - 4*a^8*b*c*d^3 + a^9*d^4)
)^(3/4)) + (-d^5/(b^4*c^9 - 4*a*b^3*c^8*d + 6*a^2*b^2*c^7*d^2 - 4*a^3*b*c^6*d^3
+ a^4*c^5*d^4))^(1/4)*a*c*x*log(d^4*x + (b^3*c^7 - 3*a*b^2*c^6*d + 3*a^2*b*c^5*d
^2 - a^3*c^4*d^3)*(-d^5/(b^4*c^9 - 4*a*b^3*c^8*d + 6*a^2*b^2*c^7*d^2 - 4*a^3*b*c
^6*d^3 + a^4*c^5*d^4))^(3/4)) - (-d^5/(b^4*c^9 - 4*a*b^3*c^8*d + 6*a^2*b^2*c^7*d
^2 - 4*a^3*b*c^6*d^3 + a^4*c^5*d^4))^(1/4)*a*c*x*log(d^4*x - (b^3*c^7 - 3*a*b^2*
c^6*d + 3*a^2*b*c^5*d^2 - a^3*c^4*d^3)*(-d^5/(b^4*c^9 - 4*a*b^3*c^8*d + 6*a^2*b^
2*c^7*d^2 - 4*a^3*b*c^6*d^3 + a^4*c^5*d^4))^(3/4)) - 4)/(a*c*x)

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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