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

Optimal. Leaf size=79 \[ \frac{\sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{2 \sqrt{a} (b c-a d)}-\frac{\sqrt{d} \tan ^{-1}\left (\frac{\sqrt{d} x^2}{\sqrt{c}}\right )}{2 \sqrt{c} (b c-a d)} \]

[Out]

(Sqrt[b]*ArcTan[(Sqrt[b]*x^2)/Sqrt[a]])/(2*Sqrt[a]*(b*c - a*d)) - (Sqrt[d]*ArcTa
n[(Sqrt[d]*x^2)/Sqrt[c]])/(2*Sqrt[c]*(b*c - a*d))

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Rubi [A]  time = 0.128169, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ \frac{\sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{2 \sqrt{a} (b c-a d)}-\frac{\sqrt{d} \tan ^{-1}\left (\frac{\sqrt{d} x^2}{\sqrt{c}}\right )}{2 \sqrt{c} (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[b]*ArcTan[(Sqrt[b]*x^2)/Sqrt[a]])/(2*Sqrt[a]*(b*c - a*d)) - (Sqrt[d]*ArcTa
n[(Sqrt[d]*x^2)/Sqrt[c]])/(2*Sqrt[c]*(b*c - a*d))

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Rubi in Sympy [A]  time = 23.0259, size = 66, normalized size = 0.84 \[ \frac{\sqrt{d} \operatorname{atan}{\left (\frac{\sqrt{d} x^{2}}{\sqrt{c}} \right )}}{2 \sqrt{c} \left (a d - b c\right )} - \frac{\sqrt{b} \operatorname{atan}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{2 \sqrt{a} \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(d)*atan(sqrt(d)*x**2/sqrt(c))/(2*sqrt(c)*(a*d - b*c)) - sqrt(b)*atan(sqrt(b
)*x**2/sqrt(a))/(2*sqrt(a)*(a*d - b*c))

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Mathematica [A]  time = 0.0783699, size = 66, normalized size = 0.84 \[ \frac{\frac{\sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{\sqrt{a}}-\frac{\sqrt{d} \tan ^{-1}\left (\frac{\sqrt{d} x^2}{\sqrt{c}}\right )}{\sqrt{c}}}{2 b c-2 a d} \]

Antiderivative was successfully verified.

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

[Out]

((Sqrt[b]*ArcTan[(Sqrt[b]*x^2)/Sqrt[a]])/Sqrt[a] - (Sqrt[d]*ArcTan[(Sqrt[d]*x^2)
/Sqrt[c]])/Sqrt[c])/(2*b*c - 2*a*d)

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Maple [A]  time = 0.008, size = 60, normalized size = 0.8 \[{\frac{d}{2\,ad-2\,bc}\arctan \left ({d{x}^{2}{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}-{\frac{b}{2\,ad-2\,bc}\arctan \left ({b{x}^{2}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(b*x^4+a)/(d*x^4+c),x)

[Out]

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

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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(x/((b*x^4 + a)*(d*x^4 + c)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.275073, size = 1, normalized size = 0.01 \[ \left [-\frac{\sqrt{-\frac{b}{a}} \log \left (\frac{b x^{4} - 2 \, a x^{2} \sqrt{-\frac{b}{a}} - a}{b x^{4} + a}\right ) + \sqrt{-\frac{d}{c}} \log \left (\frac{d x^{4} + 2 \, c x^{2} \sqrt{-\frac{d}{c}} - c}{d x^{4} + c}\right )}{4 \,{\left (b c - a d\right )}}, \frac{2 \, \sqrt{\frac{d}{c}} \arctan \left (\frac{c \sqrt{\frac{d}{c}}}{d x^{2}}\right ) - \sqrt{-\frac{b}{a}} \log \left (\frac{b x^{4} - 2 \, a x^{2} \sqrt{-\frac{b}{a}} - a}{b x^{4} + a}\right )}{4 \,{\left (b c - a d\right )}}, -\frac{2 \, \sqrt{\frac{b}{a}} \arctan \left (\frac{a \sqrt{\frac{b}{a}}}{b x^{2}}\right ) + \sqrt{-\frac{d}{c}} \log \left (\frac{d x^{4} + 2 \, c x^{2} \sqrt{-\frac{d}{c}} - c}{d x^{4} + c}\right )}{4 \,{\left (b c - a d\right )}}, -\frac{\sqrt{\frac{b}{a}} \arctan \left (\frac{a \sqrt{\frac{b}{a}}}{b x^{2}}\right ) - \sqrt{\frac{d}{c}} \arctan \left (\frac{c \sqrt{\frac{d}{c}}}{d x^{2}}\right )}{2 \,{\left (b c - a d\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/4*(sqrt(-b/a)*log((b*x^4 - 2*a*x^2*sqrt(-b/a) - a)/(b*x^4 + a)) + sqrt(-d/c)
*log((d*x^4 + 2*c*x^2*sqrt(-d/c) - c)/(d*x^4 + c)))/(b*c - a*d), 1/4*(2*sqrt(d/c
)*arctan(c*sqrt(d/c)/(d*x^2)) - sqrt(-b/a)*log((b*x^4 - 2*a*x^2*sqrt(-b/a) - a)/
(b*x^4 + a)))/(b*c - a*d), -1/4*(2*sqrt(b/a)*arctan(a*sqrt(b/a)/(b*x^2)) + sqrt(
-d/c)*log((d*x^4 + 2*c*x^2*sqrt(-d/c) - c)/(d*x^4 + c)))/(b*c - a*d), -1/2*(sqrt
(b/a)*arctan(a*sqrt(b/a)/(b*x^2)) - sqrt(d/c)*arctan(c*sqrt(d/c)/(d*x^2)))/(b*c
- a*d)]

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Sympy [A]  time = 31.2626, size = 719, normalized size = 9.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.244262, size = 261, normalized size = 3.3 \[ -\frac{\sqrt{c d} b{\left | d \right |} \arctan \left (\frac{2 \, \sqrt{\frac{1}{2}} x^{2}}{\sqrt{\frac{b c + a d + \sqrt{-4 \, a b c d +{\left (b c + a d\right )}^{2}}}{b d}}}\right )}{b c d{\left | b c - a d \right |} + a d^{2}{\left | b c - a d \right |} +{\left (b c - a d\right )}^{2} d} + \frac{\sqrt{a b} d{\left | b \right |} \arctan \left (\frac{2 \, \sqrt{\frac{1}{2}} x^{2}}{\sqrt{\frac{b c + a d - \sqrt{-4 \, a b c d +{\left (b c + a d\right )}^{2}}}{b d}}}\right )}{b^{2} c{\left | b c - a d \right |} + a b d{\left | b c - a d \right |} -{\left (b c - a d\right )}^{2} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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