3.2.72 \(\int \frac {1}{(a+b x^4)^2 (c+d x^4)^2} \, dx\) [172]

Optimal. Leaf size=596 \[ \frac {d (b c+a d) x}{4 a c (b c-a d)^2 \left (c+d x^4\right )}+\frac {b x}{4 a (b c-a d) \left (a+b x^4\right ) \left (c+d x^4\right )}-\frac {b^{7/4} (3 b c-11 a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} (b c-a d)^3}+\frac {b^{7/4} (3 b c-11 a d) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} (b c-a d)^3}-\frac {d^{7/4} (11 b c-3 a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{8 \sqrt {2} c^{7/4} (b c-a d)^3}+\frac {d^{7/4} (11 b c-3 a d) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{8 \sqrt {2} c^{7/4} (b c-a d)^3}-\frac {b^{7/4} (3 b c-11 a d) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{16 \sqrt {2} a^{7/4} (b c-a d)^3}+\frac {b^{7/4} (3 b c-11 a d) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{16 \sqrt {2} a^{7/4} (b c-a d)^3}-\frac {d^{7/4} (11 b c-3 a d) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt {d} x^2\right )}{16 \sqrt {2} c^{7/4} (b c-a d)^3}+\frac {d^{7/4} (11 b c-3 a d) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt {d} x^2\right )}{16 \sqrt {2} c^{7/4} (b c-a d)^3} \]

[Out]

1/4*d*(a*d+b*c)*x/a/c/(-a*d+b*c)^2/(d*x^4+c)+1/4*b*x/a/(-a*d+b*c)/(b*x^4+a)/(d*x^4+c)+1/16*b^(7/4)*(-11*a*d+3*
b*c)*arctan(-1+b^(1/4)*x*2^(1/2)/a^(1/4))/a^(7/4)/(-a*d+b*c)^3*2^(1/2)+1/16*b^(7/4)*(-11*a*d+3*b*c)*arctan(1+b
^(1/4)*x*2^(1/2)/a^(1/4))/a^(7/4)/(-a*d+b*c)^3*2^(1/2)+1/16*d^(7/4)*(-3*a*d+11*b*c)*arctan(-1+d^(1/4)*x*2^(1/2
)/c^(1/4))/c^(7/4)/(-a*d+b*c)^3*2^(1/2)+1/16*d^(7/4)*(-3*a*d+11*b*c)*arctan(1+d^(1/4)*x*2^(1/2)/c^(1/4))/c^(7/
4)/(-a*d+b*c)^3*2^(1/2)-1/32*b^(7/4)*(-11*a*d+3*b*c)*ln(-a^(1/4)*b^(1/4)*x*2^(1/2)+a^(1/2)+x^2*b^(1/2))/a^(7/4
)/(-a*d+b*c)^3*2^(1/2)+1/32*b^(7/4)*(-11*a*d+3*b*c)*ln(a^(1/4)*b^(1/4)*x*2^(1/2)+a^(1/2)+x^2*b^(1/2))/a^(7/4)/
(-a*d+b*c)^3*2^(1/2)-1/32*d^(7/4)*(-3*a*d+11*b*c)*ln(-c^(1/4)*d^(1/4)*x*2^(1/2)+c^(1/2)+x^2*d^(1/2))/c^(7/4)/(
-a*d+b*c)^3*2^(1/2)+1/32*d^(7/4)*(-3*a*d+11*b*c)*ln(c^(1/4)*d^(1/4)*x*2^(1/2)+c^(1/2)+x^2*d^(1/2))/c^(7/4)/(-a
*d+b*c)^3*2^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.50, antiderivative size = 596, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 9, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.474, Rules used = {425, 541, 536, 217, 1179, 642, 1176, 631, 210} \begin {gather*} -\frac {b^{7/4} \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) (3 b c-11 a d)}{8 \sqrt {2} a^{7/4} (b c-a d)^3}+\frac {b^{7/4} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) (3 b c-11 a d)}{8 \sqrt {2} a^{7/4} (b c-a d)^3}-\frac {b^{7/4} (3 b c-11 a d) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{16 \sqrt {2} a^{7/4} (b c-a d)^3}+\frac {b^{7/4} (3 b c-11 a d) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{16 \sqrt {2} a^{7/4} (b c-a d)^3}-\frac {d^{7/4} (11 b c-3 a d) \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{8 \sqrt {2} c^{7/4} (b c-a d)^3}+\frac {d^{7/4} (11 b c-3 a d) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}+1\right )}{8 \sqrt {2} c^{7/4} (b c-a d)^3}-\frac {d^{7/4} (11 b c-3 a d) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt {c}+\sqrt {d} x^2\right )}{16 \sqrt {2} c^{7/4} (b c-a d)^3}+\frac {d^{7/4} (11 b c-3 a d) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt {c}+\sqrt {d} x^2\right )}{16 \sqrt {2} c^{7/4} (b c-a d)^3}+\frac {b x}{4 a \left (a+b x^4\right ) \left (c+d x^4\right ) (b c-a d)}+\frac {d x (a d+b c)}{4 a c \left (c+d x^4\right ) (b c-a d)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(d*(b*c + a*d)*x)/(4*a*c*(b*c - a*d)^2*(c + d*x^4)) + (b*x)/(4*a*(b*c - a*d)*(a + b*x^4)*(c + d*x^4)) - (b^(7/
4)*(3*b*c - 11*a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(8*Sqrt[2]*a^(7/4)*(b*c - a*d)^3) + (b^(7/4)*(3*b
*c - 11*a*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(8*Sqrt[2]*a^(7/4)*(b*c - a*d)^3) - (d^(7/4)*(11*b*c - 3
*a*d)*ArcTan[1 - (Sqrt[2]*d^(1/4)*x)/c^(1/4)])/(8*Sqrt[2]*c^(7/4)*(b*c - a*d)^3) + (d^(7/4)*(11*b*c - 3*a*d)*A
rcTan[1 + (Sqrt[2]*d^(1/4)*x)/c^(1/4)])/(8*Sqrt[2]*c^(7/4)*(b*c - a*d)^3) - (b^(7/4)*(3*b*c - 11*a*d)*Log[Sqrt
[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(16*Sqrt[2]*a^(7/4)*(b*c - a*d)^3) + (b^(7/4)*(3*b*c - 11*a*d)
*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(16*Sqrt[2]*a^(7/4)*(b*c - a*d)^3) - (d^(7/4)*(11*b*c
- 3*a*d)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*x + Sqrt[d]*x^2])/(16*Sqrt[2]*c^(7/4)*(b*c - a*d)^3) + (d^(7/4
)*(11*b*c - 3*a*d)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*x + Sqrt[d]*x^2])/(16*Sqrt[2]*c^(7/4)*(b*c - a*d)^3)

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 217

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Di
st[1/(2*r), Int[(r - s*x^2)/(a + b*x^4), x], x] + Dist[1/(2*r), Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[
{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ, b
]]))

Rule 425

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(-b)*x*(a + b*x^n)^(p + 1)*
((c + d*x^n)^(q + 1)/(a*n*(p + 1)*(b*c - a*d))), x] + Dist[1/(a*n*(p + 1)*(b*c - a*d)), Int[(a + b*x^n)^(p + 1
)*(c + d*x^n)^q*Simp[b*c + n*(p + 1)*(b*c - a*d) + d*b*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c,
d, n, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] &&  !( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomi
alQ[a, b, c, d, n, p, q, x]

Rule 536

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

Rule 541

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[(
-(b*e - a*f))*x*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*n*(b*c - a*d)*(p + 1))), x] + Dist[1/(a*n*(b*c - a
*d)*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*
f)*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, q}, x] && LtQ[p, -1]

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1176

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[2*(d/e), 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 1179

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[-2*(d/e), 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rubi steps

\begin {align*} \int \frac {1}{\left (a+b x^4\right )^2 \left (c+d x^4\right )^2} \, dx &=\frac {b x}{4 a (b c-a d) \left (a+b x^4\right ) \left (c+d x^4\right )}-\frac {\int \frac {-3 b c+4 a d-7 b d x^4}{\left (a+b x^4\right ) \left (c+d x^4\right )^2} \, dx}{4 a (b c-a d)}\\ &=\frac {d (b c+a d) x}{4 a c (b c-a d)^2 \left (c+d x^4\right )}+\frac {b x}{4 a (b c-a d) \left (a+b x^4\right ) \left (c+d x^4\right )}-\frac {\int \frac {-4 \left (3 b^2 c^2-8 a b c d+3 a^2 d^2\right )-12 b d (b c+a d) x^4}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx}{16 a c (b c-a d)^2}\\ &=\frac {d (b c+a d) x}{4 a c (b c-a d)^2 \left (c+d x^4\right )}+\frac {b x}{4 a (b c-a d) \left (a+b x^4\right ) \left (c+d x^4\right )}+\frac {\left (b^2 (3 b c-11 a d)\right ) \int \frac {1}{a+b x^4} \, dx}{4 a (b c-a d)^3}+\frac {\left (d^2 (11 b c-3 a d)\right ) \int \frac {1}{c+d x^4} \, dx}{4 c (b c-a d)^3}\\ &=\frac {d (b c+a d) x}{4 a c (b c-a d)^2 \left (c+d x^4\right )}+\frac {b x}{4 a (b c-a d) \left (a+b x^4\right ) \left (c+d x^4\right )}+\frac {\left (b^2 (3 b c-11 a d)\right ) \int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx}{8 a^{3/2} (b c-a d)^3}+\frac {\left (b^2 (3 b c-11 a d)\right ) \int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx}{8 a^{3/2} (b c-a d)^3}+\frac {\left (d^2 (11 b c-3 a d)\right ) \int \frac {\sqrt {c}-\sqrt {d} x^2}{c+d x^4} \, dx}{8 c^{3/2} (b c-a d)^3}+\frac {\left (d^2 (11 b c-3 a d)\right ) \int \frac {\sqrt {c}+\sqrt {d} x^2}{c+d x^4} \, dx}{8 c^{3/2} (b c-a d)^3}\\ &=\frac {d (b c+a d) x}{4 a c (b c-a d)^2 \left (c+d x^4\right )}+\frac {b x}{4 a (b c-a d) \left (a+b x^4\right ) \left (c+d x^4\right )}+\frac {\left (b^{3/2} (3 b c-11 a d)\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx}{16 a^{3/2} (b c-a d)^3}+\frac {\left (b^{3/2} (3 b c-11 a d)\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx}{16 a^{3/2} (b c-a d)^3}-\frac {\left (b^{7/4} (3 b c-11 a d)\right ) \int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}+2 x}{-\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx}{16 \sqrt {2} a^{7/4} (b c-a d)^3}-\frac {\left (b^{7/4} (3 b c-11 a d)\right ) \int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}-2 x}{-\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx}{16 \sqrt {2} a^{7/4} (b c-a d)^3}+\frac {\left (d^{3/2} (11 b c-3 a d)\right ) \int \frac {1}{\frac {\sqrt {c}}{\sqrt {d}}-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}+x^2} \, dx}{16 c^{3/2} (b c-a d)^3}+\frac {\left (d^{3/2} (11 b c-3 a d)\right ) \int \frac {1}{\frac {\sqrt {c}}{\sqrt {d}}+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}+x^2} \, dx}{16 c^{3/2} (b c-a d)^3}-\frac {\left (d^{7/4} (11 b c-3 a d)\right ) \int \frac {\frac {\sqrt {2} \sqrt [4]{c}}{\sqrt [4]{d}}+2 x}{-\frac {\sqrt {c}}{\sqrt {d}}-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}-x^2} \, dx}{16 \sqrt {2} c^{7/4} (b c-a d)^3}-\frac {\left (d^{7/4} (11 b c-3 a d)\right ) \int \frac {\frac {\sqrt {2} \sqrt [4]{c}}{\sqrt [4]{d}}-2 x}{-\frac {\sqrt {c}}{\sqrt {d}}+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}-x^2} \, dx}{16 \sqrt {2} c^{7/4} (b c-a d)^3}\\ &=\frac {d (b c+a d) x}{4 a c (b c-a d)^2 \left (c+d x^4\right )}+\frac {b x}{4 a (b c-a d) \left (a+b x^4\right ) \left (c+d x^4\right )}-\frac {b^{7/4} (3 b c-11 a d) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{16 \sqrt {2} a^{7/4} (b c-a d)^3}+\frac {b^{7/4} (3 b c-11 a d) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{16 \sqrt {2} a^{7/4} (b c-a d)^3}-\frac {d^{7/4} (11 b c-3 a d) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt {d} x^2\right )}{16 \sqrt {2} c^{7/4} (b c-a d)^3}+\frac {d^{7/4} (11 b c-3 a d) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt {d} x^2\right )}{16 \sqrt {2} c^{7/4} (b c-a d)^3}+\frac {\left (b^{7/4} (3 b c-11 a d)\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} (b c-a d)^3}-\frac {\left (b^{7/4} (3 b c-11 a d)\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} (b c-a d)^3}+\frac {\left (d^{7/4} (11 b c-3 a d)\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{8 \sqrt {2} c^{7/4} (b c-a d)^3}-\frac {\left (d^{7/4} (11 b c-3 a d)\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{8 \sqrt {2} c^{7/4} (b c-a d)^3}\\ &=\frac {d (b c+a d) x}{4 a c (b c-a d)^2 \left (c+d x^4\right )}+\frac {b x}{4 a (b c-a d) \left (a+b x^4\right ) \left (c+d x^4\right )}-\frac {b^{7/4} (3 b c-11 a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} (b c-a d)^3}+\frac {b^{7/4} (3 b c-11 a d) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} (b c-a d)^3}-\frac {d^{7/4} (11 b c-3 a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{8 \sqrt {2} c^{7/4} (b c-a d)^3}+\frac {d^{7/4} (11 b c-3 a d) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{8 \sqrt {2} c^{7/4} (b c-a d)^3}-\frac {b^{7/4} (3 b c-11 a d) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{16 \sqrt {2} a^{7/4} (b c-a d)^3}+\frac {b^{7/4} (3 b c-11 a d) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{16 \sqrt {2} a^{7/4} (b c-a d)^3}-\frac {d^{7/4} (11 b c-3 a d) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt {d} x^2\right )}{16 \sqrt {2} c^{7/4} (b c-a d)^3}+\frac {d^{7/4} (11 b c-3 a d) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt {d} x^2\right )}{16 \sqrt {2} c^{7/4} (b c-a d)^3}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 6.12, size = 629, normalized size = 1.06 \begin {gather*} \frac {b^2 x}{4 a (-b c+a d)^2 \left (a+b x^4\right )}+\frac {d^2 x}{4 c (b c-a d)^2 \left (c+d x^4\right )}-\frac {b^{7/4} (-3 b c+11 a d) \tan ^{-1}\left (\frac {-\sqrt {2} \sqrt [4]{a}+2 \sqrt [4]{b} x}{\sqrt {2} \sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} (b c-a d)^3}-\frac {b^{7/4} (-3 b c+11 a d) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a}+2 \sqrt [4]{b} x}{\sqrt {2} \sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} (b c-a d)^3}-\frac {d^{7/4} (11 b c-3 a d) \tan ^{-1}\left (\frac {-\sqrt {2} \sqrt [4]{c}+2 \sqrt [4]{d} x}{\sqrt {2} \sqrt [4]{c}}\right )}{8 \sqrt {2} c^{7/4} (-b c+a d)^3}-\frac {d^{7/4} (11 b c-3 a d) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c}+2 \sqrt [4]{d} x}{\sqrt {2} \sqrt [4]{c}}\right )}{8 \sqrt {2} c^{7/4} (-b c+a d)^3}+\frac {b^{7/4} (-3 b c+11 a d) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{16 \sqrt {2} a^{7/4} (b c-a d)^3}-\frac {b^{7/4} (-3 b c+11 a d) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{16 \sqrt {2} a^{7/4} (b c-a d)^3}+\frac {d^{7/4} (11 b c-3 a d) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt {d} x^2\right )}{16 \sqrt {2} c^{7/4} (-b c+a d)^3}-\frac {d^{7/4} (11 b c-3 a d) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt {d} x^2\right )}{16 \sqrt {2} c^{7/4} (-b c+a d)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(b^2*x)/(4*a*(-(b*c) + a*d)^2*(a + b*x^4)) + (d^2*x)/(4*c*(b*c - a*d)^2*(c + d*x^4)) - (b^(7/4)*(-3*b*c + 11*a
*d)*ArcTan[(-(Sqrt[2]*a^(1/4)) + 2*b^(1/4)*x)/(Sqrt[2]*a^(1/4))])/(8*Sqrt[2]*a^(7/4)*(b*c - a*d)^3) - (b^(7/4)
*(-3*b*c + 11*a*d)*ArcTan[(Sqrt[2]*a^(1/4) + 2*b^(1/4)*x)/(Sqrt[2]*a^(1/4))])/(8*Sqrt[2]*a^(7/4)*(b*c - a*d)^3
) - (d^(7/4)*(11*b*c - 3*a*d)*ArcTan[(-(Sqrt[2]*c^(1/4)) + 2*d^(1/4)*x)/(Sqrt[2]*c^(1/4))])/(8*Sqrt[2]*c^(7/4)
*(-(b*c) + a*d)^3) - (d^(7/4)*(11*b*c - 3*a*d)*ArcTan[(Sqrt[2]*c^(1/4) + 2*d^(1/4)*x)/(Sqrt[2]*c^(1/4))])/(8*S
qrt[2]*c^(7/4)*(-(b*c) + a*d)^3) + (b^(7/4)*(-3*b*c + 11*a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b
]*x^2])/(16*Sqrt[2]*a^(7/4)*(b*c - a*d)^3) - (b^(7/4)*(-3*b*c + 11*a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*
x + Sqrt[b]*x^2])/(16*Sqrt[2]*a^(7/4)*(b*c - a*d)^3) + (d^(7/4)*(11*b*c - 3*a*d)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)
*d^(1/4)*x + Sqrt[d]*x^2])/(16*Sqrt[2]*c^(7/4)*(-(b*c) + a*d)^3) - (d^(7/4)*(11*b*c - 3*a*d)*Log[Sqrt[c] + Sqr
t[2]*c^(1/4)*d^(1/4)*x + Sqrt[d]*x^2])/(16*Sqrt[2]*c^(7/4)*(-(b*c) + a*d)^3)

________________________________________________________________________________________

Maple [A]
time = 0.46, size = 298, normalized size = 0.50

method	result	size
default	\(\frac {d^{2} \left (\frac {\left (a d -b c \right ) x}{4 c \left (d \,x^{4}+c \right )}+\frac {\left (3 a d -11 b c \right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {c}{d}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {c}{d}}}{x^{2}-\left (\frac {c}{d}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{32 c^{2}}\right )}{\left (a d -b c \right )^{3}}+\frac {b^{2} \left (\frac {\left (a d -b c \right ) x}{4 a \left (b \,x^{4}+a \right )}+\frac {\left (11 a d -3 b c \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}{x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 a^{2}}\right )}{\left (a d -b c \right )^{3}}\)	\(298\)
risch	\(\text {Expression too large to display}\)	\(2272\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x^4+a)^2/(d*x^4+c)^2,x,method=_RETURNVERBOSE)

[Out]

d^2/(a*d-b*c)^3*(1/4*(a*d-b*c)/c*x/(d*x^4+c)+1/32*(3*a*d-11*b*c)/c^2*(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)))+2*arctan(2^(1/2)/(c/d)^(1/4)*x+1)+2*arctan(2^(
1/2)/(c/d)^(1/4)*x-1)))+b^2/(a*d-b*c)^3*(1/4*(a*d-b*c)/a*x/(b*x^4+a)+1/32*(11*a*d-3*b*c)/a^2*(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)))+2*arctan(2^(1/2)/(a/b)
^(1/4)*x+1)+2*arctan(2^(1/2)/(a/b)^(1/4)*x-1)))

________________________________________________________________________________________

Maxima [A]
time = 0.54, size = 670, normalized size = 1.12 \begin {gather*} \frac {{\left (\frac {2 \, \sqrt {2} {\left (3 \, b c - 11 \, a d\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {2 \, \sqrt {2} {\left (3 \, b c - 11 \, a d\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {\sqrt {2} {\left (3 \, b c - 11 \, a d\right )} \log \left (\sqrt {b} x^{2} + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}} - \frac {\sqrt {2} {\left (3 \, b c - 11 \, a d\right )} \log \left (\sqrt {b} x^{2} - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}}\right )} b^{2}}{32 \, {\left (a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3}\right )}} + \frac {{\left (b^{2} c d + a b d^{2}\right )} x^{5} + {\left (b^{2} c^{2} + a^{2} d^{2}\right )} x}{4 \, {\left ({\left (a b^{3} c^{3} d - 2 \, a^{2} b^{2} c^{2} d^{2} + a^{3} b c d^{3}\right )} x^{8} + a^{2} b^{2} c^{4} - 2 \, a^{3} b c^{3} d + a^{4} c^{2} d^{2} + {\left (a b^{3} c^{4} - a^{2} b^{2} c^{3} d - a^{3} b c^{2} d^{2} + a^{4} c d^{3}\right )} x^{4}\right )}} + \frac {\frac {2 \, \sqrt {2} {\left (11 \, b c d^{2} - 3 \, a d^{3}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {d} x + \sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {c} \sqrt {\sqrt {c} \sqrt {d}}} + \frac {2 \, \sqrt {2} {\left (11 \, b c d^{2} - 3 \, a d^{3}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {d} x - \sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {c} \sqrt {\sqrt {c} \sqrt {d}}} + \frac {\sqrt {2} {\left (11 \, b c d^{2} - 3 \, a d^{3}\right )} \log \left (\sqrt {d} x^{2} + \sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} x + \sqrt {c}\right )}{c^{\frac {3}{4}} d^{\frac {1}{4}}} - \frac {\sqrt {2} {\left (11 \, b c d^{2} - 3 \, a d^{3}\right )} \log \left (\sqrt {d} x^{2} - \sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} x + \sqrt {c}\right )}{c^{\frac {3}{4}} d^{\frac {1}{4}}}}{32 \, {\left (b^{3} c^{4} - 3 \, a b^{2} c^{3} d + 3 \, a^{2} b c^{2} d^{2} - a^{3} c d^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/32*(2*sqrt(2)*(3*b*c - 11*a*d)*arctan(1/2*sqrt(2)*(2*sqrt(b)*x + sqrt(2)*a^(1/4)*b^(1/4))/sqrt(sqrt(a)*sqrt(
b)))/(sqrt(a)*sqrt(sqrt(a)*sqrt(b))) + 2*sqrt(2)*(3*b*c - 11*a*d)*arctan(1/2*sqrt(2)*(2*sqrt(b)*x - sqrt(2)*a^
(1/4)*b^(1/4))/sqrt(sqrt(a)*sqrt(b)))/(sqrt(a)*sqrt(sqrt(a)*sqrt(b))) + sqrt(2)*(3*b*c - 11*a*d)*log(sqrt(b)*x
^2 + sqrt(2)*a^(1/4)*b^(1/4)*x + sqrt(a))/(a^(3/4)*b^(1/4)) - sqrt(2)*(3*b*c - 11*a*d)*log(sqrt(b)*x^2 - sqrt(
2)*a^(1/4)*b^(1/4)*x + sqrt(a))/(a^(3/4)*b^(1/4)))*b^2/(a*b^3*c^3 - 3*a^2*b^2*c^2*d + 3*a^3*b*c*d^2 - a^4*d^3)
+ 1/4*((b^2*c*d + a*b*d^2)*x^5 + (b^2*c^2 + a^2*d^2)*x)/((a*b^3*c^3*d - 2*a^2*b^2*c^2*d^2 + a^3*b*c*d^3)*x^8
+ a^2*b^2*c^4 - 2*a^3*b*c^3*d + a^4*c^2*d^2 + (a*b^3*c^4 - a^2*b^2*c^3*d - a^3*b*c^2*d^2 + a^4*c*d^3)*x^4) + 1
/32*(2*sqrt(2)*(11*b*c*d^2 - 3*a*d^3)*arctan(1/2*sqrt(2)*(2*sqrt(d)*x + sqrt(2)*c^(1/4)*d^(1/4))/sqrt(sqrt(c)*
sqrt(d)))/(sqrt(c)*sqrt(sqrt(c)*sqrt(d))) + 2*sqrt(2)*(11*b*c*d^2 - 3*a*d^3)*arctan(1/2*sqrt(2)*(2*sqrt(d)*x -
sqrt(2)*c^(1/4)*d^(1/4))/sqrt(sqrt(c)*sqrt(d)))/(sqrt(c)*sqrt(sqrt(c)*sqrt(d))) + sqrt(2)*(11*b*c*d^2 - 3*a*d
^3)*log(sqrt(d)*x^2 + sqrt(2)*c^(1/4)*d^(1/4)*x + sqrt(c))/(c^(3/4)*d^(1/4)) - sqrt(2)*(11*b*c*d^2 - 3*a*d^3)*
log(sqrt(d)*x^2 - sqrt(2)*c^(1/4)*d^(1/4)*x + sqrt(c))/(c^(3/4)*d^(1/4)))/(b^3*c^4 - 3*a*b^2*c^3*d + 3*a^2*b*c
^2*d^2 - a^3*c*d^3)

________________________________________________________________________________________

Fricas [F(-1)] Timed out
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(1/(b*x^4+a)^2/(d*x^4+c)^2,x, algorithm="fricas")

[Out]

Timed out

________________________________________________________________________________________

Sympy [F(-1)] Timed out
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(1/(b*x**4+a)**2/(d*x**4+c)**2,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 967 vs. \(2 (462) = 924\).
time = 0.69, size = 967, normalized size = 1.62 \begin {gather*} \frac {{\left (3 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{2} c - 11 \, \left (a b^{3}\right )^{\frac {1}{4}} a b d\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{8 \, {\left (\sqrt {2} a^{2} b^{3} c^{3} - 3 \, \sqrt {2} a^{3} b^{2} c^{2} d + 3 \, \sqrt {2} a^{4} b c d^{2} - \sqrt {2} a^{5} d^{3}\right )}} + \frac {{\left (3 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{2} c - 11 \, \left (a b^{3}\right )^{\frac {1}{4}} a b d\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{8 \, {\left (\sqrt {2} a^{2} b^{3} c^{3} - 3 \, \sqrt {2} a^{3} b^{2} c^{2} d + 3 \, \sqrt {2} a^{4} b c d^{2} - \sqrt {2} a^{5} d^{3}\right )}} + \frac {{\left (11 \, \left (c d^{3}\right )^{\frac {1}{4}} b c d - 3 \, \left (c d^{3}\right )^{\frac {1}{4}} a d^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{8 \, {\left (\sqrt {2} b^{3} c^{5} - 3 \, \sqrt {2} a b^{2} c^{4} d + 3 \, \sqrt {2} a^{2} b c^{3} d^{2} - \sqrt {2} a^{3} c^{2} d^{3}\right )}} + \frac {{\left (11 \, \left (c d^{3}\right )^{\frac {1}{4}} b c d - 3 \, \left (c d^{3}\right )^{\frac {1}{4}} a d^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{8 \, {\left (\sqrt {2} b^{3} c^{5} - 3 \, \sqrt {2} a b^{2} c^{4} d + 3 \, \sqrt {2} a^{2} b c^{3} d^{2} - \sqrt {2} a^{3} c^{2} d^{3}\right )}} + \frac {{\left (3 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{2} c - 11 \, \left (a b^{3}\right )^{\frac {1}{4}} a b d\right )} \log \left (x^{2} + \sqrt {2} x \left (\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{b}}\right )}{16 \, {\left (\sqrt {2} a^{2} b^{3} c^{3} - 3 \, \sqrt {2} a^{3} b^{2} c^{2} d + 3 \, \sqrt {2} a^{4} b c d^{2} - \sqrt {2} a^{5} d^{3}\right )}} - \frac {{\left (3 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{2} c - 11 \, \left (a b^{3}\right )^{\frac {1}{4}} a b d\right )} \log \left (x^{2} - \sqrt {2} x \left (\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{b}}\right )}{16 \, {\left (\sqrt {2} a^{2} b^{3} c^{3} - 3 \, \sqrt {2} a^{3} b^{2} c^{2} d + 3 \, \sqrt {2} a^{4} b c d^{2} - \sqrt {2} a^{5} d^{3}\right )}} + \frac {{\left (11 \, \left (c d^{3}\right )^{\frac {1}{4}} b c d - 3 \, \left (c d^{3}\right )^{\frac {1}{4}} a d^{2}\right )} \log \left (x^{2} + \sqrt {2} x \left (\frac {c}{d}\right )^{\frac {1}{4}} + \sqrt {\frac {c}{d}}\right )}{16 \, {\left (\sqrt {2} b^{3} c^{5} - 3 \, \sqrt {2} a b^{2} c^{4} d + 3 \, \sqrt {2} a^{2} b c^{3} d^{2} - \sqrt {2} a^{3} c^{2} d^{3}\right )}} - \frac {{\left (11 \, \left (c d^{3}\right )^{\frac {1}{4}} b c d - 3 \, \left (c d^{3}\right )^{\frac {1}{4}} a d^{2}\right )} \log \left (x^{2} - \sqrt {2} x \left (\frac {c}{d}\right )^{\frac {1}{4}} + \sqrt {\frac {c}{d}}\right )}{16 \, {\left (\sqrt {2} b^{3} c^{5} - 3 \, \sqrt {2} a b^{2} c^{4} d + 3 \, \sqrt {2} a^{2} b c^{3} d^{2} - \sqrt {2} a^{3} c^{2} d^{3}\right )}} + \frac {b^{2} c d x^{5} + a b d^{2} x^{5} + b^{2} c^{2} x + a^{2} d^{2} x}{4 \, {\left (b d x^{8} + b c x^{4} + a d x^{4} + a c\right )} {\left (a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(3*(a*b^3)^(1/4)*b^2*c - 11*(a*b^3)^(1/4)*a*b*d)*arctan(1/2*sqrt(2)*(2*x + sqrt(2)*(a/b)^(1/4))/(a/b)^(1/4
))/(sqrt(2)*a^2*b^3*c^3 - 3*sqrt(2)*a^3*b^2*c^2*d + 3*sqrt(2)*a^4*b*c*d^2 - sqrt(2)*a^5*d^3) + 1/8*(3*(a*b^3)^
(1/4)*b^2*c - 11*(a*b^3)^(1/4)*a*b*d)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)*(a/b)^(1/4))/(a/b)^(1/4))/(sqrt(2)*a^2
*b^3*c^3 - 3*sqrt(2)*a^3*b^2*c^2*d + 3*sqrt(2)*a^4*b*c*d^2 - sqrt(2)*a^5*d^3) + 1/8*(11*(c*d^3)^(1/4)*b*c*d -
3*(c*d^3)^(1/4)*a*d^2)*arctan(1/2*sqrt(2)*(2*x + sqrt(2)*(c/d)^(1/4))/(c/d)^(1/4))/(sqrt(2)*b^3*c^5 - 3*sqrt(2
)*a*b^2*c^4*d + 3*sqrt(2)*a^2*b*c^3*d^2 - sqrt(2)*a^3*c^2*d^3) + 1/8*(11*(c*d^3)^(1/4)*b*c*d - 3*(c*d^3)^(1/4)
*a*d^2)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)*(c/d)^(1/4))/(c/d)^(1/4))/(sqrt(2)*b^3*c^5 - 3*sqrt(2)*a*b^2*c^4*d +
3*sqrt(2)*a^2*b*c^3*d^2 - sqrt(2)*a^3*c^2*d^3) + 1/16*(3*(a*b^3)^(1/4)*b^2*c - 11*(a*b^3)^(1/4)*a*b*d)*log(x^
2 + sqrt(2)*x*(a/b)^(1/4) + sqrt(a/b))/(sqrt(2)*a^2*b^3*c^3 - 3*sqrt(2)*a^3*b^2*c^2*d + 3*sqrt(2)*a^4*b*c*d^2
- sqrt(2)*a^5*d^3) - 1/16*(3*(a*b^3)^(1/4)*b^2*c - 11*(a*b^3)^(1/4)*a*b*d)*log(x^2 - sqrt(2)*x*(a/b)^(1/4) + s
qrt(a/b))/(sqrt(2)*a^2*b^3*c^3 - 3*sqrt(2)*a^3*b^2*c^2*d + 3*sqrt(2)*a^4*b*c*d^2 - sqrt(2)*a^5*d^3) + 1/16*(11
*(c*d^3)^(1/4)*b*c*d - 3*(c*d^3)^(1/4)*a*d^2)*log(x^2 + sqrt(2)*x*(c/d)^(1/4) + sqrt(c/d))/(sqrt(2)*b^3*c^5 -
3*sqrt(2)*a*b^2*c^4*d + 3*sqrt(2)*a^2*b*c^3*d^2 - sqrt(2)*a^3*c^2*d^3) - 1/16*(11*(c*d^3)^(1/4)*b*c*d - 3*(c*d
^3)^(1/4)*a*d^2)*log(x^2 - sqrt(2)*x*(c/d)^(1/4) + sqrt(c/d))/(sqrt(2)*b^3*c^5 - 3*sqrt(2)*a*b^2*c^4*d + 3*sqr
t(2)*a^2*b*c^3*d^2 - sqrt(2)*a^3*c^2*d^3) + 1/4*(b^2*c*d*x^5 + a*b*d^2*x^5 + b^2*c^2*x + a^2*d^2*x)/((b*d*x^8
+ b*c*x^4 + a*d*x^4 + a*c)*(a*b^2*c^3 - 2*a^2*b*c^2*d + a^3*c*d^2))

________________________________________________________________________________________

Mupad [B]
time = 5.62, size = 2500, normalized size = 4.19 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((x*(a^2*d^2 + b^2*c^2))/(4*a*c*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)) + (b*d*x^5*(a*d + b*c))/(4*a*c*(a^2*d^2 + b^2
*c^2 - 2*a*b*c*d)))/(a*c + x^4*(a*d + b*c) + b*d*x^8) - atan(((-(81*a^4*d^11 + 14641*b^4*c^4*d^7 - 15972*a*b^3
*c^3*d^8 + 6534*a^2*b^2*c^2*d^9 - 1188*a^3*b*c*d^10)/(65536*b^12*c^19 + 65536*a^12*c^7*d^12 - 786432*a^11*b*c^
8*d^11 + 4325376*a^2*b^10*c^17*d^2 - 14417920*a^3*b^9*c^16*d^3 + 32440320*a^4*b^8*c^15*d^4 - 51904512*a^5*b^7*
c^14*d^5 + 60555264*a^6*b^6*c^13*d^6 - 51904512*a^7*b^5*c^12*d^7 + 32440320*a^8*b^4*c^11*d^8 - 14417920*a^9*b^
3*c^10*d^9 + 4325376*a^10*b^2*c^9*d^10 - 786432*a*b^11*c^18*d))^(1/4)*((-(81*a^4*d^11 + 14641*b^4*c^4*d^7 - 15
972*a*b^3*c^3*d^8 + 6534*a^2*b^2*c^2*d^9 - 1188*a^3*b*c*d^10)/(65536*b^12*c^19 + 65536*a^12*c^7*d^12 - 786432*
a^11*b*c^8*d^11 + 4325376*a^2*b^10*c^17*d^2 - 14417920*a^3*b^9*c^16*d^3 + 32440320*a^4*b^8*c^15*d^4 - 51904512
*a^5*b^7*c^14*d^5 + 60555264*a^6*b^6*c^13*d^6 - 51904512*a^7*b^5*c^12*d^7 + 32440320*a^8*b^4*c^11*d^8 - 144179
20*a^9*b^3*c^10*d^9 + 4325376*a^10*b^2*c^9*d^10 - 786432*a*b^11*c^18*d))^(1/4)*(((891*a^8*b^7*d^15)/64 + (891*
b^15*c^8*d^7)/64 - (3105*a*b^14*c^7*d^8)/16 - (3105*a^7*b^8*c*d^14)/16 + (31509*a^2*b^13*c^6*d^9)/32 - (33069*
a^3*b^12*c^5*d^10)/16 + (60307*a^4*b^11*c^4*d^11)/32 - (33069*a^5*b^10*c^3*d^12)/16 + (31509*a^6*b^9*c^2*d^13)
/32)/(a^4*b^8*c^12 + a^12*c^4*d^8 - 8*a^5*b^7*c^11*d - 8*a^11*b*c^5*d^7 + 28*a^6*b^6*c^10*d^2 - 56*a^7*b^5*c^9
*d^3 + 70*a^8*b^4*c^8*d^4 - 56*a^9*b^3*c^7*d^5 + 28*a^10*b^2*c^6*d^6) + (-(81*a^4*d^11 + 14641*b^4*c^4*d^7 - 1
5972*a*b^3*c^3*d^8 + 6534*a^2*b^2*c^2*d^9 - 1188*a^3*b*c*d^10)/(65536*b^12*c^19 + 65536*a^12*c^7*d^12 - 786432
*a^11*b*c^8*d^11 + 4325376*a^2*b^10*c^17*d^2 - 14417920*a^3*b^9*c^16*d^3 + 32440320*a^4*b^8*c^15*d^4 - 5190451
2*a^5*b^7*c^14*d^5 + 60555264*a^6*b^6*c^13*d^6 - 51904512*a^7*b^5*c^12*d^7 + 32440320*a^8*b^4*c^11*d^8 - 14417
920*a^9*b^3*c^10*d^9 + 4325376*a^10*b^2*c^9*d^10 - 786432*a*b^11*c^18*d))^(3/4)*((x*(589824*a^2*b^23*c^21*d^4
- 11403264*a^3*b^22*c^20*d^5 + 98762752*a^4*b^21*c^19*d^6 - 510394368*a^5*b^20*c^18*d^7 + 1766916096*a^6*b^19*
c^17*d^8 - 4344840192*a^7*b^18*c^16*d^9 + 7796490240*a^8*b^17*c^15*d^10 - 10168369152*a^9*b^16*c^14*d^11 + 900
7726592*a^10*b^15*c^13*d^12 - 3635478528*a^11*b^14*c^12*d^13 - 3635478528*a^12*b^13*c^11*d^14 + 9007726592*a^1
3*b^12*c^10*d^15 - 10168369152*a^14*b^11*c^9*d^16 + 7796490240*a^15*b^10*c^8*d^17 - 4344840192*a^16*b^9*c^7*d^
18 + 1766916096*a^17*b^8*c^6*d^19 - 510394368*a^18*b^7*c^5*d^20 + 98762752*a^19*b^6*c^4*d^21 - 11403264*a^20*b
^5*c^3*d^22 + 589824*a^21*b^4*c^2*d^23))/(1024*(a^4*b^12*c^16 + a^16*c^4*d^12 - 12*a^5*b^11*c^15*d - 12*a^15*b
*c^5*d^11 + 66*a^6*b^10*c^14*d^2 - 220*a^7*b^9*c^13*d^3 + 495*a^8*b^8*c^12*d^4 - 792*a^9*b^7*c^11*d^5 + 924*a^
10*b^6*c^10*d^6 - 792*a^11*b^5*c^9*d^7 + 495*a^12*b^4*c^8*d^8 - 220*a^13*b^3*c^7*d^9 + 66*a^14*b^2*c^6*d^10))
+ ((-(81*a^4*d^11 + 14641*b^4*c^4*d^7 - 15972*a*b^3*c^3*d^8 + 6534*a^2*b^2*c^2*d^9 - 1188*a^3*b*c*d^10)/(65536
*b^12*c^19 + 65536*a^12*c^7*d^12 - 786432*a^11*b*c^8*d^11 + 4325376*a^2*b^10*c^17*d^2 - 14417920*a^3*b^9*c^16*
d^3 + 32440320*a^4*b^8*c^15*d^4 - 51904512*a^5*b^7*c^14*d^5 + 60555264*a^6*b^6*c^13*d^6 - 51904512*a^7*b^5*c^1
2*d^7 + 32440320*a^8*b^4*c^11*d^8 - 14417920*a^9*b^3*c^10*d^9 + 4325376*a^10*b^2*c^9*d^10 - 786432*a*b^11*c^18
*d))^(1/4)*(3072*a^4*b^19*c^19*d^4 - 45056*a^5*b^18*c^18*d^5 + 292864*a^6*b^17*c^17*d^6 - 1115136*a^7*b^16*c^1
6*d^7 + 2745344*a^8*b^15*c^15*d^8 - 4483072*a^9*b^14*c^14*d^9 + 4595712*a^10*b^13*c^13*d^10 - 1993728*a^11*b^1
2*c^12*d^11 - 1993728*a^12*b^11*c^11*d^12 + 4595712*a^13*b^10*c^10*d^13 - 4483072*a^14*b^9*c^9*d^14 + 2745344*
a^15*b^8*c^8*d^15 - 1115136*a^16*b^7*c^7*d^16 + 292864*a^17*b^6*c^6*d^17 - 45056*a^18*b^5*c^5*d^18 + 3072*a^19
*b^4*c^4*d^19))/(a^4*b^8*c^12 + a^12*c^4*d^8 - 8*a^5*b^7*c^11*d - 8*a^11*b*c^5*d^7 + 28*a^6*b^6*c^10*d^2 - 56*
a^7*b^5*c^9*d^3 + 70*a^8*b^4*c^8*d^4 - 56*a^9*b^3*c^7*d^5 + 28*a^10*b^2*c^6*d^6)))*1i + (x*(9801*a^8*b^9*d^17
+ 9801*b^17*c^8*d^9 - 149094*a*b^16*c^7*d^10 - 149094*a^7*b^10*c*d^16 + 1001520*a^2*b^15*c^6*d^11 - 3484602*a^
3*b^14*c^5*d^12 + 5769038*a^4*b^13*c^4*d^13 - 3484602*a^5*b^12*c^3*d^14 + 1001520*a^6*b^11*c^2*d^15)*1i)/(1024
*(a^4*b^12*c^16 + a^16*c^4*d^12 - 12*a^5*b^11*c^15*d - 12*a^15*b*c^5*d^11 + 66*a^6*b^10*c^14*d^2 - 220*a^7*b^9
*c^13*d^3 + 495*a^8*b^8*c^12*d^4 - 792*a^9*b^7*c^11*d^5 + 924*a^10*b^6*c^10*d^6 - 792*a^11*b^5*c^9*d^7 + 495*a
^12*b^4*c^8*d^8 - 220*a^13*b^3*c^7*d^9 + 66*a^14*b^2*c^6*d^10))) - (-(81*a^4*d^11 + 14641*b^4*c^4*d^7 - 15972*
a*b^3*c^3*d^8 + 6534*a^2*b^2*c^2*d^9 - 1188*a^3*b*c*d^10)/(65536*b^12*c^19 + 65536*a^12*c^7*d^12 - 786432*a^11
*b*c^8*d^11 + 4325376*a^2*b^10*c^17*d^2 - 14417920*a^3*b^9*c^16*d^3 + 32440320*a^4*b^8*c^15*d^4 - 51904512*a^5
*b^7*c^14*d^5 + 60555264*a^6*b^6*c^13*d^6 - 51904512*a^7*b^5*c^12*d^7 + 32440320*a^8*b^4*c^11*d^8 - 14417920*a
^9*b^3*c^10*d^9 + 4325376*a^10*b^2*c^9*d^10 - 786432*a*b^11*c^18*d))^(1/4)*((-(81*a^4*d^11 + 14641*b^4*c^4*d^7
- 15972*a*b^3*c^3*d^8 + 6534*a^2*b^2*c^2*d^9 -...

________________________________________________________________________________________