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

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

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Rubi [A]  time = 0.0480111, 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, Rules used = {465, 391, 205} \[ \frac{\sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{2 \sqrt{a} (b c-a d)}-\frac{\sqrt{d} \tan ^{-1}\left (\frac{\sqrt{d} x^2}{\sqrt{c}}\right )}{2 \sqrt{c} (b c-a d)} \]

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

Rule 465

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> With[{k = GCD[m + 1,
n]}, Dist[1/k, Subst[Int[x^((m + 1)/k - 1)*(a + b*x^(n/k))^p*(c + d*x^(n/k))^q, x], x, x^k], x] /; k != 1] /;
FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && IntegerQ[m]

Rule 391

Int[1/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^(n_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x^n),
x], x] - Dist[d/(b*c - a*d), Int[1/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0]

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rubi steps

\begin{align*} \int \frac{x}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx,x,x^2\right )\\ &=\frac{b \operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,x^2\right )}{2 (b c-a d)}-\frac{d \operatorname{Subst}\left (\int \frac{1}{c+d x^2} \, dx,x,x^2\right )}{2 (b c-a d)}\\ &=\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)}\\ \end{align*}

Mathematica [A]  time = 0.0459221, 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 \begin{align*}{\frac{d}{2\,ad-2\,bc}\arctan \left ({{x}^{2}d{\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}}}} \end{align*}

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(b*x^2/(a*b)^(1/2))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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 = 1.3518, size = 668, normalized size = 8.46 \begin{align*} \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 ] \end{align*}

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 [B]  time = 21.4931, size = 719, normalized size = 9.1 \begin{align*} \text{result too large to display} \end{align*}

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 [B]  time = 1.17518, size = 261, normalized size = 3.3 \begin{align*} -\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} \end{align*}

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)