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

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

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Rubi [A]  time = 0.05, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {465, 391, 205} \begin {gather*} \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 {gather*}

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 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]

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 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]

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*}

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Mathematica [A]  time = 0.05, size = 66, normalized size = 0.84 \begin {gather*} \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} \end {gather*}

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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.50, size = 325, normalized size = 4.11 \begin {gather*} \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 {gather*}

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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giac [A]  time = 0.18, size = 59, normalized size = 0.75 \begin {gather*} \frac {b \arctan \left (\frac {b x^{2}}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} {\left (b c - a d\right )}} - \frac {d \arctan \left (\frac {d x^{2}}{\sqrt {c d}}\right )}{2 \, {\left (b c - a d\right )} \sqrt {c d}} \end {gather*}

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]

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

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maple [A]  time = 0.06, size = 60, normalized size = 0.76 \begin {gather*} -\frac {b \arctan \left (\frac {b \,x^{2}}{\sqrt {a b}}\right )}{2 \left (a d -b c \right ) \sqrt {a b}}+\frac {d \arctan \left (\frac {d \,x^{2}}{\sqrt {c d}}\right )}{2 \left (a d -b c \right ) \sqrt {c d}} \end {gather*}

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

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maxima [A]  time = 1.28, size = 59, normalized size = 0.75 \begin {gather*} \frac {b \arctan \left (\frac {b x^{2}}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} {\left (b c - a d\right )}} - \frac {d \arctan \left (\frac {d x^{2}}{\sqrt {c d}}\right )}{2 \, {\left (b c - a d\right )} \sqrt {c d}} \end {gather*}

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]

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

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mupad [B]  time = 5.29, size = 399, normalized size = 5.05 \begin {gather*} \frac {\ln \left (a^2\,d^2\,{\left (-a\,b\right )}^{5/2}+b^2\,c^2\,{\left (-a\,b\right )}^{5/2}+2\,c\,d\,{\left (-a\,b\right )}^{7/2}-a^2\,b^5\,c^2\,x^2-a^4\,b^3\,d^2\,x^2+2\,a^3\,b^4\,c\,d\,x^2\right )\,\sqrt {-a\,b}}{4\,a^2\,d-4\,a\,b\,c}-\frac {\ln \left (a^2\,d^2\,{\left (-a\,b\right )}^{5/2}+b^2\,c^2\,{\left (-a\,b\right )}^{5/2}+2\,c\,d\,{\left (-a\,b\right )}^{7/2}+a^2\,b^5\,c^2\,x^2+a^4\,b^3\,d^2\,x^2-2\,a^3\,b^4\,c\,d\,x^2\right )\,\sqrt {-a\,b}}{4\,\left (a^2\,d-a\,b\,c\right )}-\frac {\ln \left (a^2\,d^2\,{\left (-c\,d\right )}^{5/2}+b^2\,c^2\,{\left (-c\,d\right )}^{5/2}+2\,a\,b\,{\left (-c\,d\right )}^{7/2}+a^2\,c^2\,d^5\,x^2+b^2\,c^4\,d^3\,x^2-2\,a\,b\,c^3\,d^4\,x^2\right )\,\sqrt {-c\,d}}{4\,\left (b\,c^2-a\,c\,d\right )}+\frac {\ln \left (a^2\,d^2\,{\left (-c\,d\right )}^{5/2}+b^2\,c^2\,{\left (-c\,d\right )}^{5/2}+2\,a\,b\,{\left (-c\,d\right )}^{7/2}-a^2\,c^2\,d^5\,x^2-b^2\,c^4\,d^3\,x^2+2\,a\,b\,c^3\,d^4\,x^2\right )\,\sqrt {-c\,d}}{4\,b\,c^2-4\,a\,c\,d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/((a + b*x^4)*(c + d*x^4)),x)

[Out]

(log(a^2*d^2*(-a*b)^(5/2) + b^2*c^2*(-a*b)^(5/2) + 2*c*d*(-a*b)^(7/2) - a^2*b^5*c^2*x^2 - a^4*b^3*d^2*x^2 + 2*
a^3*b^4*c*d*x^2)*(-a*b)^(1/2))/(4*a^2*d - 4*a*b*c) - (log(a^2*d^2*(-a*b)^(5/2) + b^2*c^2*(-a*b)^(5/2) + 2*c*d*
(-a*b)^(7/2) + a^2*b^5*c^2*x^2 + a^4*b^3*d^2*x^2 - 2*a^3*b^4*c*d*x^2)*(-a*b)^(1/2))/(4*(a^2*d - a*b*c)) - (log
(a^2*d^2*(-c*d)^(5/2) + b^2*c^2*(-c*d)^(5/2) + 2*a*b*(-c*d)^(7/2) + a^2*c^2*d^5*x^2 + b^2*c^4*d^3*x^2 - 2*a*b*
c^3*d^4*x^2)*(-c*d)^(1/2))/(4*(b*c^2 - a*c*d)) + (log(a^2*d^2*(-c*d)^(5/2) + b^2*c^2*(-c*d)^(5/2) + 2*a*b*(-c*
d)^(7/2) - a^2*c^2*d^5*x^2 - b^2*c^4*d^3*x^2 + 2*a*b*c^3*d^4*x^2)*(-c*d)^(1/2))/(4*b*c^2 - 4*a*c*d)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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