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

Optimal. Leaf size=462 \[ \frac{b^{7/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{4 \sqrt{2} a^{7/4} (b c-a d)}-\frac{b^{7/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{4 \sqrt{2} a^{7/4} (b c-a d)}+\frac{b^{7/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{7/4} (b c-a d)}-\frac{b^{7/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{2 \sqrt{2} a^{7/4} (b c-a d)}-\frac{d^{7/4} \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{c}+\sqrt{d} x^2\right )}{4 \sqrt{2} c^{7/4} (b c-a d)}+\frac{d^{7/4} \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{c}+\sqrt{d} x^2\right )}{4 \sqrt{2} c^{7/4} (b c-a d)}-\frac{d^{7/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{2 \sqrt{2} c^{7/4} (b c-a d)}+\frac{d^{7/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}+1\right )}{2 \sqrt{2} c^{7/4} (b c-a d)}-\frac{1}{3 a c x^3} \]

[Out]

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

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Rubi [A]  time = 0.432014, antiderivative size = 462, normalized size of antiderivative = 1., number of steps used = 20, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {480, 522, 211, 1165, 628, 1162, 617, 204} \[ \frac{b^{7/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{4 \sqrt{2} a^{7/4} (b c-a d)}-\frac{b^{7/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{4 \sqrt{2} a^{7/4} (b c-a d)}+\frac{b^{7/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{7/4} (b c-a d)}-\frac{b^{7/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{2 \sqrt{2} a^{7/4} (b c-a d)}-\frac{d^{7/4} \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{c}+\sqrt{d} x^2\right )}{4 \sqrt{2} c^{7/4} (b c-a d)}+\frac{d^{7/4} \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{c}+\sqrt{d} x^2\right )}{4 \sqrt{2} c^{7/4} (b c-a d)}-\frac{d^{7/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{2 \sqrt{2} c^{7/4} (b c-a d)}+\frac{d^{7/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}+1\right )}{2 \sqrt{2} c^{7/4} (b c-a d)}-\frac{1}{3 a c x^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 480

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[((e*x)^(m
+ 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*c*e*(m + 1)), x] - Dist[1/(a*c*e^n*(m + 1)), Int[(e*x)^(m +
n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[(b*c + a*d)*(m + n + 1) + n*(b*c*p + a*d*q) + b*d*(m + n*(p + q + 2) + 1)*
x^n, x], x], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[m, -1] && IntBino
mialQ[a, b, c, d, e, m, 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}{x^4 \left (a+b x^4\right ) \left (c+d x^4\right )} \, dx &=-\frac{1}{3 a c x^3}+\frac{\int \frac{-3 (b c+a d)-3 b d x^4}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx}{3 a c}\\ &=-\frac{1}{3 a c x^3}-\frac{b^2 \int \frac{1}{a+b x^4} \, dx}{a (b c-a d)}+\frac{d^2 \int \frac{1}{c+d x^4} \, dx}{c (b c-a d)}\\ &=-\frac{1}{3 a c x^3}-\frac{b^2 \int \frac{\sqrt{a}-\sqrt{b} x^2}{a+b x^4} \, dx}{2 a^{3/2} (b c-a d)}-\frac{b^2 \int \frac{\sqrt{a}+\sqrt{b} x^2}{a+b x^4} \, dx}{2 a^{3/2} (b c-a d)}+\frac{d^2 \int \frac{\sqrt{c}-\sqrt{d} x^2}{c+d x^4} \, dx}{2 c^{3/2} (b c-a d)}+\frac{d^2 \int \frac{\sqrt{c}+\sqrt{d} x^2}{c+d x^4} \, dx}{2 c^{3/2} (b c-a d)}\\ &=-\frac{1}{3 a c x^3}-\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 a^{3/2} (b c-a d)}-\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 a^{3/2} (b c-a d)}+\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^{7/4} (b c-a d)}+\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^{7/4} (b c-a d)}+\frac{d^{3/2} \int \frac{1}{\frac{\sqrt{c}}{\sqrt{d}}-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{d}}+x^2} \, dx}{4 c^{3/2} (b c-a d)}+\frac{d^{3/2} \int \frac{1}{\frac{\sqrt{c}}{\sqrt{d}}+\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{d}}+x^2} \, dx}{4 c^{3/2} (b c-a d)}-\frac{d^{7/4} \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}{4 \sqrt{2} c^{7/4} (b c-a d)}-\frac{d^{7/4} \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}{4 \sqrt{2} c^{7/4} (b c-a d)}\\ &=-\frac{1}{3 a c x^3}+\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^{7/4} (b c-a d)}-\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^{7/4} (b c-a d)}-\frac{d^{7/4} \log \left (\sqrt{c}-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{d} x^2\right )}{4 \sqrt{2} c^{7/4} (b c-a d)}+\frac{d^{7/4} \log \left (\sqrt{c}+\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{d} x^2\right )}{4 \sqrt{2} c^{7/4} (b c-a d)}-\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^{7/4} (b c-a d)}+\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^{7/4} (b c-a d)}+\frac{d^{7/4} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{2 \sqrt{2} c^{7/4} (b c-a d)}-\frac{d^{7/4} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{2 \sqrt{2} c^{7/4} (b c-a d)}\\ &=-\frac{1}{3 a c x^3}+\frac{b^{7/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{7/4} (b c-a d)}-\frac{b^{7/4} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{7/4} (b c-a d)}-\frac{d^{7/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{2 \sqrt{2} c^{7/4} (b c-a d)}+\frac{d^{7/4} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{2 \sqrt{2} c^{7/4} (b c-a d)}+\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^{7/4} (b c-a d)}-\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^{7/4} (b c-a d)}-\frac{d^{7/4} \log \left (\sqrt{c}-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{d} x^2\right )}{4 \sqrt{2} c^{7/4} (b c-a d)}+\frac{d^{7/4} \log \left (\sqrt{c}+\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{d} x^2\right )}{4 \sqrt{2} c^{7/4} (b c-a d)}\\ \end{align*}

Mathematica [A]  time = 0.225688, size = 406, normalized size = 0.88 \[ \frac{-\frac{3 \sqrt{2} b^{7/4} x^3 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{a^{7/4}}+\frac{3 \sqrt{2} b^{7/4} x^3 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{a^{7/4}}-\frac{6 \sqrt{2} b^{7/4} x^3 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{a^{7/4}}+\frac{6 \sqrt{2} b^{7/4} x^3 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{a^{7/4}}+\frac{8 b}{a}+\frac{3 \sqrt{2} d^{7/4} x^3 \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{c}+\sqrt{d} x^2\right )}{c^{7/4}}-\frac{3 \sqrt{2} d^{7/4} x^3 \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{c}+\sqrt{d} x^2\right )}{c^{7/4}}+\frac{6 \sqrt{2} d^{7/4} x^3 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{c^{7/4}}-\frac{6 \sqrt{2} d^{7/4} x^3 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}+1\right )}{c^{7/4}}-\frac{8 d}{c}}{24 x^3 (a d-b c)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((8*b)/a - (8*d)/c - (6*Sqrt[2]*b^(7/4)*x^3*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/a^(7/4) + (6*Sqrt[2]*b^(7
/4)*x^3*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/a^(7/4) + (6*Sqrt[2]*d^(7/4)*x^3*ArcTan[1 - (Sqrt[2]*d^(1/4)*
x)/c^(1/4)])/c^(7/4) - (6*Sqrt[2]*d^(7/4)*x^3*ArcTan[1 + (Sqrt[2]*d^(1/4)*x)/c^(1/4)])/c^(7/4) - (3*Sqrt[2]*b^
(7/4)*x^3*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/a^(7/4) + (3*Sqrt[2]*b^(7/4)*x^3*Log[Sqrt[a]
+ Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/a^(7/4) + (3*Sqrt[2]*d^(7/4)*x^3*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^
(1/4)*x + Sqrt[d]*x^2])/c^(7/4) - (3*Sqrt[2]*d^(7/4)*x^3*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*x + Sqrt[d]*x^2
])/c^(7/4))/(24*(-(b*c) + a*d)*x^3)

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Maple [A]  time = 0.01, size = 343, normalized size = 0.7 \begin{align*} -{\frac{{d}^{2}\sqrt{2}}{8\,{c}^{2} \left ( ad-bc \right ) }\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{{d}^{2}\sqrt{2}}{4\,{c}^{2} \left ( ad-bc \right ) }\sqrt [4]{{\frac{c}{d}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}+1 \right ) }-{\frac{{d}^{2}\sqrt{2}}{4\,{c}^{2} \left ( ad-bc \right ) }\sqrt [4]{{\frac{c}{d}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}-1 \right ) }+{\frac{{b}^{2}\sqrt{2}}{8\,{a}^{2} \left ( ad-bc \right ) }\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\,{a}^{2} \left ( ad-bc \right ) }\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\,{a}^{2} \left ( ad-bc \right ) }\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ) }-{\frac{1}{3\,ac{x}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 20.4758, size = 2822, normalized size = 6.11 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(12*(-b^7/(a^7*b^4*c^4 - 4*a^8*b^3*c^3*d + 6*a^9*b^2*c^2*d^2 - 4*a^10*b*c*d^3 + a^11*d^4))^(1/4)*a*c*x^3*
arctan(((a^5*b^3*c^3 - 3*a^6*b^2*c^2*d + 3*a^7*b*c*d^2 - a^8*d^3)*(-b^7/(a^7*b^4*c^4 - 4*a^8*b^3*c^3*d + 6*a^9
*b^2*c^2*d^2 - 4*a^10*b*c*d^3 + a^11*d^4))^(3/4)*x - (a^5*b^3*c^3 - 3*a^6*b^2*c^2*d + 3*a^7*b*c*d^2 - a^8*d^3)
*(-b^7/(a^7*b^4*c^4 - 4*a^8*b^3*c^3*d + 6*a^9*b^2*c^2*d^2 - 4*a^10*b*c*d^3 + a^11*d^4))^(3/4)*sqrt((b^4*x^2 +
(a^4*b^2*c^2 - 2*a^5*b*c*d + a^6*d^2)*sqrt(-b^7/(a^7*b^4*c^4 - 4*a^8*b^3*c^3*d + 6*a^9*b^2*c^2*d^2 - 4*a^10*b*
c*d^3 + a^11*d^4)))/b^4))/b^5) - 12*(-d^7/(b^4*c^11 - 4*a*b^3*c^10*d + 6*a^2*b^2*c^9*d^2 - 4*a^3*b*c^8*d^3 + a
^4*c^7*d^4))^(1/4)*a*c*x^3*arctan(((b^3*c^8 - 3*a*b^2*c^7*d + 3*a^2*b*c^6*d^2 - a^3*c^5*d^3)*(-d^7/(b^4*c^11 -
4*a*b^3*c^10*d + 6*a^2*b^2*c^9*d^2 - 4*a^3*b*c^8*d^3 + a^4*c^7*d^4))^(3/4)*x - (b^3*c^8 - 3*a*b^2*c^7*d + 3*a
^2*b*c^6*d^2 - a^3*c^5*d^3)*(-d^7/(b^4*c^11 - 4*a*b^3*c^10*d + 6*a^2*b^2*c^9*d^2 - 4*a^3*b*c^8*d^3 + a^4*c^7*d
^4))^(3/4)*sqrt((d^4*x^2 + (b^2*c^6 - 2*a*b*c^5*d + a^2*c^4*d^2)*sqrt(-d^7/(b^4*c^11 - 4*a*b^3*c^10*d + 6*a^2*
b^2*c^9*d^2 - 4*a^3*b*c^8*d^3 + a^4*c^7*d^4)))/d^4))/d^5) - 3*(-b^7/(a^7*b^4*c^4 - 4*a^8*b^3*c^3*d + 6*a^9*b^2
*c^2*d^2 - 4*a^10*b*c*d^3 + a^11*d^4))^(1/4)*a*c*x^3*log(b^2*x + (-b^7/(a^7*b^4*c^4 - 4*a^8*b^3*c^3*d + 6*a^9*
b^2*c^2*d^2 - 4*a^10*b*c*d^3 + a^11*d^4))^(1/4)*(a^2*b*c - a^3*d)) + 3*(-b^7/(a^7*b^4*c^4 - 4*a^8*b^3*c^3*d +
6*a^9*b^2*c^2*d^2 - 4*a^10*b*c*d^3 + a^11*d^4))^(1/4)*a*c*x^3*log(b^2*x - (-b^7/(a^7*b^4*c^4 - 4*a^8*b^3*c^3*d
+ 6*a^9*b^2*c^2*d^2 - 4*a^10*b*c*d^3 + a^11*d^4))^(1/4)*(a^2*b*c - a^3*d)) + 3*(-d^7/(b^4*c^11 - 4*a*b^3*c^10
*d + 6*a^2*b^2*c^9*d^2 - 4*a^3*b*c^8*d^3 + a^4*c^7*d^4))^(1/4)*a*c*x^3*log(d^2*x + (-d^7/(b^4*c^11 - 4*a*b^3*c
^10*d + 6*a^2*b^2*c^9*d^2 - 4*a^3*b*c^8*d^3 + a^4*c^7*d^4))^(1/4)*(b*c^3 - a*c^2*d)) - 3*(-d^7/(b^4*c^11 - 4*a
*b^3*c^10*d + 6*a^2*b^2*c^9*d^2 - 4*a^3*b*c^8*d^3 + a^4*c^7*d^4))^(1/4)*a*c*x^3*log(d^2*x - (-d^7/(b^4*c^11 -
4*a*b^3*c^10*d + 6*a^2*b^2*c^9*d^2 - 4*a^3*b*c^8*d^3 + a^4*c^7*d^4))^(1/4)*(b*c^3 - a*c^2*d)) - 4)/(a*c*x^3)

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

[Out]

Timed out

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Giac [C]  time = 5.52145, size = 2225, normalized size = 4.82 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*I*2^(1/4)*(1/2)^(1/4)*(-(b^11*c^11 - 4*a*b^10*c^10*d + 6*a^2*b^9*c^9*d^2 - 4*a^3*b^8*c^8*d^3 + a^4*b^7*c^
7*d^4)/(a^7*b^8*c^15 - 8*a^8*b^7*c^14*d + 28*a^9*b^6*c^13*d^2 - 56*a^10*b^5*c^12*d^3 + 70*a^11*b^4*c^11*d^4 -
56*a^12*b^3*c^10*d^5 + 28*a^13*b^2*c^9*d^6 - 8*a^14*b*c^8*d^7 + a^15*c^7*d^8))^(1/4)*log(2*I*b*x + 2*(-a*b^3)^
(1/4)) + 1/4*I*2^(1/4)*(1/2)^(1/4)*(-(b^11*c^11 - 4*a*b^10*c^10*d + 6*a^2*b^9*c^9*d^2 - 4*a^3*b^8*c^8*d^3 + a^
4*b^7*c^7*d^4)/(a^7*b^8*c^15 - 8*a^8*b^7*c^14*d + 28*a^9*b^6*c^13*d^2 - 56*a^10*b^5*c^12*d^3 + 70*a^11*b^4*c^1
1*d^4 - 56*a^12*b^3*c^10*d^5 + 28*a^13*b^2*c^9*d^6 - 8*a^14*b*c^8*d^7 + a^15*c^7*d^8))^(1/4)*log(-2*I*b*x + 2*
(-a*b^3)^(1/4)) - 1/4*I*2^(1/4)*(1/2)^(1/4)*(-(a^7*b^4*c^4*d^7 - 4*a^8*b^3*c^3*d^8 + 6*a^9*b^2*c^2*d^9 - 4*a^1
0*b*c*d^10 + a^11*d^11)/(a^7*b^8*c^15 - 8*a^8*b^7*c^14*d + 28*a^9*b^6*c^13*d^2 - 56*a^10*b^5*c^12*d^3 + 70*a^1
1*b^4*c^11*d^4 - 56*a^12*b^3*c^10*d^5 + 28*a^13*b^2*c^9*d^6 - 8*a^14*b*c^8*d^7 + a^15*c^7*d^8))^(1/4)*log(2*I*
d*x + 2*(-c*d^3)^(1/4)) + 1/4*I*2^(1/4)*(1/2)^(1/4)*(-(a^7*b^4*c^4*d^7 - 4*a^8*b^3*c^3*d^8 + 6*a^9*b^2*c^2*d^9
- 4*a^10*b*c*d^10 + a^11*d^11)/(a^7*b^8*c^15 - 8*a^8*b^7*c^14*d + 28*a^9*b^6*c^13*d^2 - 56*a^10*b^5*c^12*d^3
+ 70*a^11*b^4*c^11*d^4 - 56*a^12*b^3*c^10*d^5 + 28*a^13*b^2*c^9*d^6 - 8*a^14*b*c^8*d^7 + a^15*c^7*d^8))^(1/4)*
log(-2*I*d*x + 2*(-c*d^3)^(1/4)) + 1/4*2^(1/4)*(1/2)^(1/4)*(-(b^11*c^11 - 4*a*b^10*c^10*d + 6*a^2*b^9*c^9*d^2
- 4*a^3*b^8*c^8*d^3 + a^4*b^7*c^7*d^4)/(a^7*b^8*c^15 - 8*a^8*b^7*c^14*d + 28*a^9*b^6*c^13*d^2 - 56*a^10*b^5*c^
12*d^3 + 70*a^11*b^4*c^11*d^4 - 56*a^12*b^3*c^10*d^5 + 28*a^13*b^2*c^9*d^6 - 8*a^14*b*c^8*d^7 + a^15*c^7*d^8))
^(1/4)*log(abs(2*b*x + 2*(-a*b^3)^(1/4))) - 1/4*2^(1/4)*(1/2)^(1/4)*(-(b^11*c^11 - 4*a*b^10*c^10*d + 6*a^2*b^9
*c^9*d^2 - 4*a^3*b^8*c^8*d^3 + a^4*b^7*c^7*d^4)/(a^7*b^8*c^15 - 8*a^8*b^7*c^14*d + 28*a^9*b^6*c^13*d^2 - 56*a^
10*b^5*c^12*d^3 + 70*a^11*b^4*c^11*d^4 - 56*a^12*b^3*c^10*d^5 + 28*a^13*b^2*c^9*d^6 - 8*a^14*b*c^8*d^7 + a^15*
c^7*d^8))^(1/4)*log(abs(-2*b*x + 2*(-a*b^3)^(1/4))) + 1/4*2^(1/4)*(1/2)^(1/4)*(-(a^7*b^4*c^4*d^7 - 4*a^8*b^3*c
^3*d^8 + 6*a^9*b^2*c^2*d^9 - 4*a^10*b*c*d^10 + a^11*d^11)/(a^7*b^8*c^15 - 8*a^8*b^7*c^14*d + 28*a^9*b^6*c^13*d
^2 - 56*a^10*b^5*c^12*d^3 + 70*a^11*b^4*c^11*d^4 - 56*a^12*b^3*c^10*d^5 + 28*a^13*b^2*c^9*d^6 - 8*a^14*b*c^8*d
^7 + a^15*c^7*d^8))^(1/4)*log(abs(2*d*x + 2*(-c*d^3)^(1/4))) - 1/4*2^(1/4)*(1/2)^(1/4)*(-(a^7*b^4*c^4*d^7 - 4*
a^8*b^3*c^3*d^8 + 6*a^9*b^2*c^2*d^9 - 4*a^10*b*c*d^10 + a^11*d^11)/(a^7*b^8*c^15 - 8*a^8*b^7*c^14*d + 28*a^9*b
^6*c^13*d^2 - 56*a^10*b^5*c^12*d^3 + 70*a^11*b^4*c^11*d^4 - 56*a^12*b^3*c^10*d^5 + 28*a^13*b^2*c^9*d^6 - 8*a^1
4*b*c^8*d^7 + a^15*c^7*d^8))^(1/4)*log(abs(-2*d*x + 2*(-c*d^3)^(1/4))) - 1/3/(a*c*x^3)