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

Optimal. Leaf size=449 \[ -\frac{b^{3/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{4 \sqrt{2} a^{3/4} (b c-a d)}+\frac{b^{3/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{4 \sqrt{2} a^{3/4} (b c-a d)}-\frac{b^{3/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{3/4} (b c-a d)}+\frac{b^{3/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{2 \sqrt{2} a^{3/4} (b c-a d)}+\frac{d^{3/4} \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{c}+\sqrt{d} x^2\right )}{4 \sqrt{2} c^{3/4} (b c-a d)}-\frac{d^{3/4} \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{c}+\sqrt{d} x^2\right )}{4 \sqrt{2} c^{3/4} (b c-a d)}+\frac{d^{3/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{2 \sqrt{2} c^{3/4} (b c-a d)}-\frac{d^{3/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}+1\right )}{2 \sqrt{2} c^{3/4} (b c-a d)} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.605709, antiderivative size = 449, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 7, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.368 \[ -\frac{b^{3/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{4 \sqrt{2} a^{3/4} (b c-a d)}+\frac{b^{3/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{4 \sqrt{2} a^{3/4} (b c-a d)}-\frac{b^{3/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{3/4} (b c-a d)}+\frac{b^{3/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{2 \sqrt{2} a^{3/4} (b c-a d)}+\frac{d^{3/4} \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{c}+\sqrt{d} x^2\right )}{4 \sqrt{2} c^{3/4} (b c-a d)}-\frac{d^{3/4} \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{c}+\sqrt{d} x^2\right )}{4 \sqrt{2} c^{3/4} (b c-a d)}+\frac{d^{3/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{2 \sqrt{2} c^{3/4} (b c-a d)}-\frac{d^{3/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}+1\right )}{2 \sqrt{2} c^{3/4} (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 114.059, size = 400, normalized size = 0.89 \[ - \frac{\sqrt{2} d^{\frac{3}{4}} \log{\left (- \sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x + \sqrt{c} + \sqrt{d} x^{2} \right )}}{8 c^{\frac{3}{4}} \left (a d - b c\right )} + \frac{\sqrt{2} d^{\frac{3}{4}} \log{\left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x + \sqrt{c} + \sqrt{d} x^{2} \right )}}{8 c^{\frac{3}{4}} \left (a d - b c\right )} - \frac{\sqrt{2} d^{\frac{3}{4}} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}} \right )}}{4 c^{\frac{3}{4}} \left (a d - b c\right )} + \frac{\sqrt{2} d^{\frac{3}{4}} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}} \right )}}{4 c^{\frac{3}{4}} \left (a d - b c\right )} + \frac{\sqrt{2} b^{\frac{3}{4}} \log{\left (- \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x + \sqrt{a} + \sqrt{b} x^{2} \right )}}{8 a^{\frac{3}{4}} \left (a d - b c\right )} - \frac{\sqrt{2} b^{\frac{3}{4}} \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x + \sqrt{a} + \sqrt{b} x^{2} \right )}}{8 a^{\frac{3}{4}} \left (a d - b c\right )} + \frac{\sqrt{2} b^{\frac{3}{4}} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}} \right )}}{4 a^{\frac{3}{4}} \left (a d - b c\right )} - \frac{\sqrt{2} b^{\frac{3}{4}} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}} \right )}}{4 a^{\frac{3}{4}} \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(2)*d**(3/4)*log(-sqrt(2)*c**(1/4)*d**(1/4)*x + sqrt(c) + sqrt(d)*x**2)/(8*
c**(3/4)*(a*d - b*c)) + sqrt(2)*d**(3/4)*log(sqrt(2)*c**(1/4)*d**(1/4)*x + sqrt(
c) + sqrt(d)*x**2)/(8*c**(3/4)*(a*d - b*c)) - sqrt(2)*d**(3/4)*atan(1 - sqrt(2)*
d**(1/4)*x/c**(1/4))/(4*c**(3/4)*(a*d - b*c)) + sqrt(2)*d**(3/4)*atan(1 + sqrt(2
)*d**(1/4)*x/c**(1/4))/(4*c**(3/4)*(a*d - b*c)) + sqrt(2)*b**(3/4)*log(-sqrt(2)*
a**(1/4)*b**(1/4)*x + sqrt(a) + sqrt(b)*x**2)/(8*a**(3/4)*(a*d - b*c)) - sqrt(2)
*b**(3/4)*log(sqrt(2)*a**(1/4)*b**(1/4)*x + sqrt(a) + sqrt(b)*x**2)/(8*a**(3/4)*
(a*d - b*c)) + sqrt(2)*b**(3/4)*atan(1 - sqrt(2)*b**(1/4)*x/a**(1/4))/(4*a**(3/4
)*(a*d - b*c)) - sqrt(2)*b**(3/4)*atan(1 + sqrt(2)*b**(1/4)*x/a**(1/4))/(4*a**(3
/4)*(a*d - b*c))

_______________________________________________________________________________________

Mathematica [A]  time = 0.266318, size = 340, normalized size = 0.76 \[ \frac{a^{3/4} d^{3/4} \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{c}+\sqrt{d} x^2\right )-a^{3/4} d^{3/4} \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{c}+\sqrt{d} x^2\right )+2 a^{3/4} d^{3/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )-2 a^{3/4} d^{3/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}+1\right )-b^{3/4} c^{3/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )+b^{3/4} c^{3/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )-2 b^{3/4} c^{3/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )+2 b^{3/4} c^{3/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} a^{3/4} c^{3/4} (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.001, size = 320, normalized size = 0.7 \[{\frac{d\sqrt{2}}{ \left ( 8\,ad-8\,bc \right ) c}\sqrt [4]{{\frac{c}{d}}}\ln \left ({1 \left ({x}^{2}+\sqrt [4]{{\frac{c}{d}}}x\sqrt{2}+\sqrt{{\frac{c}{d}}} \right ) \left ({x}^{2}-\sqrt [4]{{\frac{c}{d}}}x\sqrt{2}+\sqrt{{\frac{c}{d}}} \right ) ^{-1}} \right ) }+{\frac{d\sqrt{2}}{ \left ( 4\,ad-4\,bc \right ) c}\sqrt [4]{{\frac{c}{d}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}+1 \right ) }+{\frac{d\sqrt{2}}{ \left ( 4\,ad-4\,bc \right ) c}\sqrt [4]{{\frac{c}{d}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}-1 \right ) }-{\frac{b\sqrt{2}}{ \left ( 8\,ad-8\,bc \right ) a}\sqrt [4]{{\frac{a}{b}}}\ln \left ({1 \left ({x}^{2}+\sqrt [4]{{\frac{a}{b}}}x\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ({x}^{2}-\sqrt [4]{{\frac{a}{b}}}x\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ) }-{\frac{b\sqrt{2}}{ \left ( 4\,ad-4\,bc \right ) a}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ) }-{\frac{b\sqrt{2}}{ \left ( 4\,ad-4\,bc \right ) a}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x^4+a)/(d*x^4+c),x)

[Out]

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

_______________________________________________________________________________________

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, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.376943, size = 1472, normalized size = 3.28 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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)/(d*x**4+c),x)

[Out]

Timed out

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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