3.165 \(\int \frac{1}{(a+b x^4) (c+d x^4)^2} \, dx\)

Optimal. Leaf size=513 \[ -\frac{b^{7/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{4 \sqrt{2} a^{3/4} (b c-a d)^2}+\frac{b^{7/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{4 \sqrt{2} a^{3/4} (b c-a d)^2}-\frac{b^{7/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{3/4} (b c-a d)^2}+\frac{b^{7/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{2 \sqrt{2} a^{3/4} (b c-a d)^2}+\frac{d^{3/4} (7 b c-3 a d) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{c}+\sqrt{d} x^2\right )}{16 \sqrt{2} c^{7/4} (b c-a d)^2}-\frac{d^{3/4} (7 b c-3 a d) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{c}+\sqrt{d} x^2\right )}{16 \sqrt{2} c^{7/4} (b c-a d)^2}+\frac{d^{3/4} (7 b c-3 a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{8 \sqrt{2} c^{7/4} (b c-a d)^2}-\frac{d^{3/4} (7 b c-3 a d) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}+1\right )}{8 \sqrt{2} c^{7/4} (b c-a d)^2}-\frac{d x}{4 c \left (c+d x^4\right ) (b c-a d)} \]

[Out]

-(d*x)/(4*c*(b*c - a*d)*(c + d*x^4)) - (b^(7/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^(3/4)*(b
*c - a*d)^2) + (b^(7/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^(3/4)*(b*c - a*d)^2) + (d^(3/4)*
(7*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)^2) - (d^(3/4)*(7*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)^2) - (b^(7/4)*Log[Sqrt[a] - Sq
rt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(4*Sqrt[2]*a^(3/4)*(b*c - a*d)^2) + (b^(7/4)*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)^2) + (d^(3/4)*(7*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)^2) - (d^(3/4)*(7*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)^2)

________________________________________________________________________________________

Rubi [A]  time = 0.419693, antiderivative size = 513, normalized size of antiderivative = 1., number of steps used = 20, number of rules used = 8, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.421, Rules used = {414, 522, 211, 1165, 628, 1162, 617, 204} \[ -\frac{b^{7/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{4 \sqrt{2} a^{3/4} (b c-a d)^2}+\frac{b^{7/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{4 \sqrt{2} a^{3/4} (b c-a d)^2}-\frac{b^{7/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{3/4} (b c-a d)^2}+\frac{b^{7/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{2 \sqrt{2} a^{3/4} (b c-a d)^2}+\frac{d^{3/4} (7 b c-3 a d) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{c}+\sqrt{d} x^2\right )}{16 \sqrt{2} c^{7/4} (b c-a d)^2}-\frac{d^{3/4} (7 b c-3 a d) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{c}+\sqrt{d} x^2\right )}{16 \sqrt{2} c^{7/4} (b c-a d)^2}+\frac{d^{3/4} (7 b c-3 a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{8 \sqrt{2} c^{7/4} (b c-a d)^2}-\frac{d^{3/4} (7 b c-3 a d) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}+1\right )}{8 \sqrt{2} c^{7/4} (b c-a d)^2}-\frac{d x}{4 c \left (c+d x^4\right ) (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b*x^4)*(c + d*x^4)^2),x]

[Out]

-(d*x)/(4*c*(b*c - a*d)*(c + d*x^4)) - (b^(7/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^(3/4)*(b
*c - a*d)^2) + (b^(7/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^(3/4)*(b*c - a*d)^2) + (d^(3/4)*
(7*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)^2) - (d^(3/4)*(7*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)^2) - (b^(7/4)*Log[Sqrt[a] - Sq
rt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(4*Sqrt[2]*a^(3/4)*(b*c - a*d)^2) + (b^(7/4)*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)^2) + (d^(3/4)*(7*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)^2) - (d^(3/4)*(7*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)^2)

Rule 414

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]) && IntBinomial
Q[a, b, c, d, n, p, q, x]

Rule 522

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 211

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 1165

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]

Rule 628

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 1162

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 617

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 204

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

Rubi steps

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

Mathematica [A]  time = 0.31656, size = 498, normalized size = 0.97 \[ \frac{8 a^{3/4} c^{3/4} d x (a d-b c)+\sqrt{2} a^{3/4} d^{3/4} \left (c+d x^4\right ) (7 b c-3 a d) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{c}+\sqrt{d} x^2\right )+\sqrt{2} a^{3/4} d^{3/4} \left (c+d x^4\right ) (3 a d-7 b c) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{c}+\sqrt{d} x^2\right )-2 \sqrt{2} a^{3/4} d^{3/4} \left (c+d x^4\right ) (3 a d-7 b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )+2 \sqrt{2} a^{3/4} d^{3/4} \left (c+d x^4\right ) (3 a d-7 b c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}+1\right )-4 \sqrt{2} b^{7/4} c^{7/4} \left (c+d x^4\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )+4 \sqrt{2} b^{7/4} c^{7/4} \left (c+d x^4\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )-8 \sqrt{2} b^{7/4} c^{7/4} \left (c+d x^4\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )+8 \sqrt{2} b^{7/4} c^{7/4} \left (c+d x^4\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{32 a^{3/4} c^{7/4} \left (c+d x^4\right ) (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(8*a^(3/4)*c^(3/4)*d*(-(b*c) + a*d)*x - 8*Sqrt[2]*b^(7/4)*c^(7/4)*(c + d*x^4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a
^(1/4)] + 8*Sqrt[2]*b^(7/4)*c^(7/4)*(c + d*x^4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)] - 2*Sqrt[2]*a^(3/4)*d^
(3/4)*(-7*b*c + 3*a*d)*(c + d*x^4)*ArcTan[1 - (Sqrt[2]*d^(1/4)*x)/c^(1/4)] + 2*Sqrt[2]*a^(3/4)*d^(3/4)*(-7*b*c
+ 3*a*d)*(c + d*x^4)*ArcTan[1 + (Sqrt[2]*d^(1/4)*x)/c^(1/4)] - 4*Sqrt[2]*b^(7/4)*c^(7/4)*(c + d*x^4)*Log[Sqrt
[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2] + 4*Sqrt[2]*b^(7/4)*c^(7/4)*(c + d*x^4)*Log[Sqrt[a] + Sqrt[2]*a
^(1/4)*b^(1/4)*x + Sqrt[b]*x^2] + Sqrt[2]*a^(3/4)*d^(3/4)*(7*b*c - 3*a*d)*(c + d*x^4)*Log[Sqrt[c] - Sqrt[2]*c^
(1/4)*d^(1/4)*x + Sqrt[d]*x^2] + Sqrt[2]*a^(3/4)*d^(3/4)*(-7*b*c + 3*a*d)*(c + d*x^4)*Log[Sqrt[c] + Sqrt[2]*c^
(1/4)*d^(1/4)*x + Sqrt[d]*x^2])/(32*a^(3/4)*c^(7/4)*(b*c - a*d)^2*(c + d*x^4))

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Maple [A]  time = 0.01, size = 550, normalized size = 1.1 \begin{align*}{\frac{{d}^{2}xa}{4\, \left ( ad-bc \right ) ^{2}c \left ( d{x}^{4}+c \right ) }}-{\frac{bdx}{4\, \left ( ad-bc \right ) ^{2} \left ( d{x}^{4}+c \right ) }}+{\frac{3\,{d}^{2}\sqrt{2}a}{16\, \left ( ad-bc \right ) ^{2}{c}^{2}}\sqrt [4]{{\frac{c}{d}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}+1 \right ) }-{\frac{7\,d\sqrt{2}b}{16\, \left ( ad-bc \right ) ^{2}c}\sqrt [4]{{\frac{c}{d}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}+1 \right ) }+{\frac{3\,{d}^{2}\sqrt{2}a}{16\, \left ( ad-bc \right ) ^{2}{c}^{2}}\sqrt [4]{{\frac{c}{d}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}-1 \right ) }-{\frac{7\,d\sqrt{2}b}{16\, \left ( ad-bc \right ) ^{2}c}\sqrt [4]{{\frac{c}{d}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}-1 \right ) }+{\frac{3\,{d}^{2}\sqrt{2}a}{32\, \left ( ad-bc \right ) ^{2}{c}^{2}}\sqrt [4]{{\frac{c}{d}}}\ln \left ({ \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{7\,d\sqrt{2}b}{32\, \left ( ad-bc \right ) ^{2}c}\sqrt [4]{{\frac{c}{d}}}\ln \left ({ \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{{b}^{2}\sqrt{2}}{8\, \left ( ad-bc \right ) ^{2}a}\sqrt [4]{{\frac{a}{b}}}\ln \left ({ \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}^{2}\sqrt{2}}{4\, \left ( ad-bc \right ) ^{2}a}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ) }+{\frac{{b}^{2}\sqrt{2}}{4\, \left ( ad-bc \right ) ^{2}a}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*d^2/(a*d-b*c)^2/c*x/(d*x^4+c)*a-1/4*d/(a*d-b*c)^2*x/(d*x^4+c)*b+3/16*d^2/(a*d-b*c)^2/c^2*(c/d)^(1/4)*2^(1/
2)*arctan(2^(1/2)/(c/d)^(1/4)*x+1)*a-7/16*d/(a*d-b*c)^2/c*(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x+1)*
b+3/16*d^2/(a*d-b*c)^2/c^2*(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x-1)*a-7/16*d/(a*d-b*c)^2/c*(c/d)^(1
/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x-1)*b+3/32*d^2/(a*d-b*c)^2/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)))*a-7/32*d/(a*d-b*c)^2/c*(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)))*b+1/8*b^2/(a*d-b*c)^2*(1/b*a
)^(1/4)/a*2^(1/2)*ln((x^2+(1/b*a)^(1/4)*x*2^(1/2)+(1/b*a)^(1/2))/(x^2-(1/b*a)^(1/4)*x*2^(1/2)+(1/b*a)^(1/2)))+
1/4*b^2/(a*d-b*c)^2*(1/b*a)^(1/4)/a*2^(1/2)*arctan(2^(1/2)/(1/b*a)^(1/4)*x+1)+1/4*b^2/(a*d-b*c)^2*(1/b*a)^(1/4
)/a*2^(1/2)*arctan(2^(1/2)/(1/b*a)^(1/4)*x-1)

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 117.466, size = 6808, normalized size = 13.27 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*(4*((b*c^2*d - a*c*d^2)*x^4 + b*c^3 - a*c^2*d)*(-(2401*b^4*c^4*d^3 - 4116*a*b^3*c^3*d^4 + 2646*a^2*b^2*c^
2*d^5 - 756*a^3*b*c*d^6 + 81*a^4*d^7)/(b^8*c^15 - 8*a*b^7*c^14*d + 28*a^2*b^6*c^13*d^2 - 56*a^3*b^5*c^12*d^3 +
70*a^4*b^4*c^11*d^4 - 56*a^5*b^3*c^10*d^5 + 28*a^6*b^2*c^9*d^6 - 8*a^7*b*c^8*d^7 + a^8*c^7*d^8))^(1/4)*arctan
(((b^6*c^11 - 6*a*b^5*c^10*d + 15*a^2*b^4*c^9*d^2 - 20*a^3*b^3*c^8*d^3 + 15*a^4*b^2*c^7*d^4 - 6*a^5*b*c^6*d^5
+ a^6*c^5*d^6)*x*(-(2401*b^4*c^4*d^3 - 4116*a*b^3*c^3*d^4 + 2646*a^2*b^2*c^2*d^5 - 756*a^3*b*c*d^6 + 81*a^4*d^
7)/(b^8*c^15 - 8*a*b^7*c^14*d + 28*a^2*b^6*c^13*d^2 - 56*a^3*b^5*c^12*d^3 + 70*a^4*b^4*c^11*d^4 - 56*a^5*b^3*c
^10*d^5 + 28*a^6*b^2*c^9*d^6 - 8*a^7*b*c^8*d^7 + a^8*c^7*d^8))^(3/4) - (b^6*c^11 - 6*a*b^5*c^10*d + 15*a^2*b^4
*c^9*d^2 - 20*a^3*b^3*c^8*d^3 + 15*a^4*b^2*c^7*d^4 - 6*a^5*b*c^6*d^5 + a^6*c^5*d^6)*(-(2401*b^4*c^4*d^3 - 4116
*a*b^3*c^3*d^4 + 2646*a^2*b^2*c^2*d^5 - 756*a^3*b*c*d^6 + 81*a^4*d^7)/(b^8*c^15 - 8*a*b^7*c^14*d + 28*a^2*b^6*
c^13*d^2 - 56*a^3*b^5*c^12*d^3 + 70*a^4*b^4*c^11*d^4 - 56*a^5*b^3*c^10*d^5 + 28*a^6*b^2*c^9*d^6 - 8*a^7*b*c^8*
d^7 + a^8*c^7*d^8))^(3/4)*sqrt(((49*b^2*c^2*d^2 - 42*a*b*c*d^3 + 9*a^2*d^4)*x^2 + (b^4*c^8 - 4*a*b^3*c^7*d + 6
*a^2*b^2*c^6*d^2 - 4*a^3*b*c^5*d^3 + a^4*c^4*d^4)*sqrt(-(2401*b^4*c^4*d^3 - 4116*a*b^3*c^3*d^4 + 2646*a^2*b^2*
c^2*d^5 - 756*a^3*b*c*d^6 + 81*a^4*d^7)/(b^8*c^15 - 8*a*b^7*c^14*d + 28*a^2*b^6*c^13*d^2 - 56*a^3*b^5*c^12*d^3
+ 70*a^4*b^4*c^11*d^4 - 56*a^5*b^3*c^10*d^5 + 28*a^6*b^2*c^9*d^6 - 8*a^7*b*c^8*d^7 + a^8*c^7*d^8)))/(49*b^2*c
^2*d^2 - 42*a*b*c*d^3 + 9*a^2*d^4)))/(343*b^3*c^3*d^2 - 441*a*b^2*c^2*d^3 + 189*a^2*b*c*d^4 - 27*a^3*d^5)) + 1
6*(-b^7/(a^3*b^8*c^8 - 8*a^4*b^7*c^7*d + 28*a^5*b^6*c^6*d^2 - 56*a^6*b^5*c^5*d^3 + 70*a^7*b^4*c^4*d^4 - 56*a^8
*b^3*c^3*d^5 + 28*a^9*b^2*c^2*d^6 - 8*a^10*b*c*d^7 + a^11*d^8))^(1/4)*((b*c^2*d - a*c*d^2)*x^4 + b*c^3 - a*c^2
*d)*arctan(-((a^2*b^6*c^6 - 6*a^3*b^5*c^5*d + 15*a^4*b^4*c^4*d^2 - 20*a^5*b^3*c^3*d^3 + 15*a^6*b^2*c^2*d^4 - 6
*a^7*b*c*d^5 + a^8*d^6)*(-b^7/(a^3*b^8*c^8 - 8*a^4*b^7*c^7*d + 28*a^5*b^6*c^6*d^2 - 56*a^6*b^5*c^5*d^3 + 70*a^
7*b^4*c^4*d^4 - 56*a^8*b^3*c^3*d^5 + 28*a^9*b^2*c^2*d^6 - 8*a^10*b*c*d^7 + a^11*d^8))^(3/4)*x - (a^2*b^6*c^6 -
6*a^3*b^5*c^5*d + 15*a^4*b^4*c^4*d^2 - 20*a^5*b^3*c^3*d^3 + 15*a^6*b^2*c^2*d^4 - 6*a^7*b*c*d^5 + a^8*d^6)*(-b
^7/(a^3*b^8*c^8 - 8*a^4*b^7*c^7*d + 28*a^5*b^6*c^6*d^2 - 56*a^6*b^5*c^5*d^3 + 70*a^7*b^4*c^4*d^4 - 56*a^8*b^3*
c^3*d^5 + 28*a^9*b^2*c^2*d^6 - 8*a^10*b*c*d^7 + a^11*d^8))^(3/4)*sqrt((b^4*x^2 + (a^2*b^4*c^4 - 4*a^3*b^3*c^3*
d + 6*a^4*b^2*c^2*d^2 - 4*a^5*b*c*d^3 + a^6*d^4)*sqrt(-b^7/(a^3*b^8*c^8 - 8*a^4*b^7*c^7*d + 28*a^5*b^6*c^6*d^2
- 56*a^6*b^5*c^5*d^3 + 70*a^7*b^4*c^4*d^4 - 56*a^8*b^3*c^3*d^5 + 28*a^9*b^2*c^2*d^6 - 8*a^10*b*c*d^7 + a^11*d
^8)))/b^4))/b^5) + 4*(-b^7/(a^3*b^8*c^8 - 8*a^4*b^7*c^7*d + 28*a^5*b^6*c^6*d^2 - 56*a^6*b^5*c^5*d^3 + 70*a^7*b
^4*c^4*d^4 - 56*a^8*b^3*c^3*d^5 + 28*a^9*b^2*c^2*d^6 - 8*a^10*b*c*d^7 + a^11*d^8))^(1/4)*((b*c^2*d - a*c*d^2)*
x^4 + b*c^3 - a*c^2*d)*log(b^2*x + (-b^7/(a^3*b^8*c^8 - 8*a^4*b^7*c^7*d + 28*a^5*b^6*c^6*d^2 - 56*a^6*b^5*c^5*
d^3 + 70*a^7*b^4*c^4*d^4 - 56*a^8*b^3*c^3*d^5 + 28*a^9*b^2*c^2*d^6 - 8*a^10*b*c*d^7 + a^11*d^8))^(1/4)*(a*b^2*
c^2 - 2*a^2*b*c*d + a^3*d^2)) - 4*(-b^7/(a^3*b^8*c^8 - 8*a^4*b^7*c^7*d + 28*a^5*b^6*c^6*d^2 - 56*a^6*b^5*c^5*d
^3 + 70*a^7*b^4*c^4*d^4 - 56*a^8*b^3*c^3*d^5 + 28*a^9*b^2*c^2*d^6 - 8*a^10*b*c*d^7 + a^11*d^8))^(1/4)*((b*c^2*
d - a*c*d^2)*x^4 + b*c^3 - a*c^2*d)*log(b^2*x - (-b^7/(a^3*b^8*c^8 - 8*a^4*b^7*c^7*d + 28*a^5*b^6*c^6*d^2 - 56
*a^6*b^5*c^5*d^3 + 70*a^7*b^4*c^4*d^4 - 56*a^8*b^3*c^3*d^5 + 28*a^9*b^2*c^2*d^6 - 8*a^10*b*c*d^7 + a^11*d^8))^
(1/4)*(a*b^2*c^2 - 2*a^2*b*c*d + a^3*d^2)) + ((b*c^2*d - a*c*d^2)*x^4 + b*c^3 - a*c^2*d)*(-(2401*b^4*c^4*d^3 -
4116*a*b^3*c^3*d^4 + 2646*a^2*b^2*c^2*d^5 - 756*a^3*b*c*d^6 + 81*a^4*d^7)/(b^8*c^15 - 8*a*b^7*c^14*d + 28*a^2
*b^6*c^13*d^2 - 56*a^3*b^5*c^12*d^3 + 70*a^4*b^4*c^11*d^4 - 56*a^5*b^3*c^10*d^5 + 28*a^6*b^2*c^9*d^6 - 8*a^7*b
*c^8*d^7 + a^8*c^7*d^8))^(1/4)*log(-(7*b*c*d - 3*a*d^2)*x + (b^2*c^4 - 2*a*b*c^3*d + a^2*c^2*d^2)*(-(2401*b^4*
c^4*d^3 - 4116*a*b^3*c^3*d^4 + 2646*a^2*b^2*c^2*d^5 - 756*a^3*b*c*d^6 + 81*a^4*d^7)/(b^8*c^15 - 8*a*b^7*c^14*d
+ 28*a^2*b^6*c^13*d^2 - 56*a^3*b^5*c^12*d^3 + 70*a^4*b^4*c^11*d^4 - 56*a^5*b^3*c^10*d^5 + 28*a^6*b^2*c^9*d^6
- 8*a^7*b*c^8*d^7 + a^8*c^7*d^8))^(1/4)) - ((b*c^2*d - a*c*d^2)*x^4 + b*c^3 - a*c^2*d)*(-(2401*b^4*c^4*d^3 - 4
116*a*b^3*c^3*d^4 + 2646*a^2*b^2*c^2*d^5 - 756*a^3*b*c*d^6 + 81*a^4*d^7)/(b^8*c^15 - 8*a*b^7*c^14*d + 28*a^2*b
^6*c^13*d^2 - 56*a^3*b^5*c^12*d^3 + 70*a^4*b^4*c^11*d^4 - 56*a^5*b^3*c^10*d^5 + 28*a^6*b^2*c^9*d^6 - 8*a^7*b*c
^8*d^7 + a^8*c^7*d^8))^(1/4)*log(-(7*b*c*d - 3*a*d^2)*x - (b^2*c^4 - 2*a*b*c^3*d + a^2*c^2*d^2)*(-(2401*b^4*c^
4*d^3 - 4116*a*b^3*c^3*d^4 + 2646*a^2*b^2*c^2*d^5 - 756*a^3*b*c*d^6 + 81*a^4*d^7)/(b^8*c^15 - 8*a*b^7*c^14*d +
28*a^2*b^6*c^13*d^2 - 56*a^3*b^5*c^12*d^3 + 70*a^4*b^4*c^11*d^4 - 56*a^5*b^3*c^10*d^5 + 28*a^6*b^2*c^9*d^6 -
8*a^7*b*c^8*d^7 + a^8*c^7*d^8))^(1/4)) - 4*d*x)/((b*c^2*d - a*c*d^2)*x^4 + b*c^3 - a*c^2*d)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 3.01974, size = 900, normalized size = 1.75 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(a*b^3)^(1/4)*b*arctan(1/2*sqrt(2)*(2*x + sqrt(2)*(a/b)^(1/4))/(a/b)^(1/4))/(sqrt(2)*a*b^2*c^2 - 2*sqrt(2)
*a^2*b*c*d + sqrt(2)*a^3*d^2) + 1/2*(a*b^3)^(1/4)*b*arctan(1/2*sqrt(2)*(2*x - sqrt(2)*(a/b)^(1/4))/(a/b)^(1/4)
)/(sqrt(2)*a*b^2*c^2 - 2*sqrt(2)*a^2*b*c*d + sqrt(2)*a^3*d^2) + 1/4*(a*b^3)^(1/4)*b*log(x^2 + sqrt(2)*x*(a/b)^
(1/4) + sqrt(a/b))/(sqrt(2)*a*b^2*c^2 - 2*sqrt(2)*a^2*b*c*d + sqrt(2)*a^3*d^2) - 1/4*(a*b^3)^(1/4)*b*log(x^2 -
sqrt(2)*x*(a/b)^(1/4) + sqrt(a/b))/(sqrt(2)*a*b^2*c^2 - 2*sqrt(2)*a^2*b*c*d + sqrt(2)*a^3*d^2) - 1/8*(7*(c*d^
3)^(1/4)*b*c - 3*(c*d^3)^(1/4)*a*d)*arctan(1/2*sqrt(2)*(2*x + sqrt(2)*(c/d)^(1/4))/(c/d)^(1/4))/(sqrt(2)*b^2*c
^4 - 2*sqrt(2)*a*b*c^3*d + sqrt(2)*a^2*c^2*d^2) - 1/8*(7*(c*d^3)^(1/4)*b*c - 3*(c*d^3)^(1/4)*a*d)*arctan(1/2*s
qrt(2)*(2*x - sqrt(2)*(c/d)^(1/4))/(c/d)^(1/4))/(sqrt(2)*b^2*c^4 - 2*sqrt(2)*a*b*c^3*d + sqrt(2)*a^2*c^2*d^2)
- 1/16*(7*(c*d^3)^(1/4)*b*c - 3*(c*d^3)^(1/4)*a*d)*log(x^2 + sqrt(2)*x*(c/d)^(1/4) + sqrt(c/d))/(sqrt(2)*b^2*c
^4 - 2*sqrt(2)*a*b*c^3*d + sqrt(2)*a^2*c^2*d^2) + 1/16*(7*(c*d^3)^(1/4)*b*c - 3*(c*d^3)^(1/4)*a*d)*log(x^2 - s
qrt(2)*x*(c/d)^(1/4) + sqrt(c/d))/(sqrt(2)*b^2*c^4 - 2*sqrt(2)*a*b*c^3*d + sqrt(2)*a^2*c^2*d^2) - 1/4*d*x/((d*
x^4 + c)*(b*c^2 - a*c*d))