3.70 \(\int \frac{1}{a+b \cos ^4(x)} \, dx\)

Optimal. Leaf size=487 \[ -\frac{\left (\sqrt{a}-\sqrt{a+b}\right ) \log \left ((a+b)^{3/4} \cot ^2(x)-\sqrt{2} \sqrt [4]{a} \sqrt{-\sqrt{a} \sqrt{a+b}+a+b} \cot (x)+\sqrt{a} \sqrt [4]{a+b}\right )}{4 \sqrt{2} a^{3/4} \sqrt [4]{a+b} \sqrt{-\sqrt{a} \sqrt{a+b}+a+b}}+\frac{\left (\sqrt{a}-\sqrt{a+b}\right ) \log \left ((a+b)^{3/4} \cot ^2(x)+\sqrt{2} \sqrt [4]{a} \sqrt{-\sqrt{a} \sqrt{a+b}+a+b} \cot (x)+\sqrt{a} \sqrt [4]{a+b}\right )}{4 \sqrt{2} a^{3/4} \sqrt [4]{a+b} \sqrt{-\sqrt{a} \sqrt{a+b}+a+b}}+\frac{\left (\sqrt{a+b}+\sqrt{a}\right ) \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt{-\sqrt{a} \sqrt{a+b}+a+b}-\sqrt{2} (a+b)^{3/4} \cot (x)}{\sqrt [4]{a} \sqrt{\sqrt{a} \sqrt{a+b}+a+b}}\right )}{2 \sqrt{2} a^{3/4} \sqrt [4]{a+b} \sqrt{\sqrt{a} \sqrt{a+b}+a+b}}-\frac{\left (\sqrt{a+b}+\sqrt{a}\right ) \tan ^{-1}\left (\frac{\sqrt{2} (a+b)^{3/4} \cot (x)+\sqrt [4]{a} \sqrt{-\sqrt{a} \sqrt{a+b}+a+b}}{\sqrt [4]{a} \sqrt{\sqrt{a} \sqrt{a+b}+a+b}}\right )}{2 \sqrt{2} a^{3/4} \sqrt [4]{a+b} \sqrt{\sqrt{a} \sqrt{a+b}+a+b}} \]

[Out]

((Sqrt[a] + Sqrt[a + b])*ArcTan[(a^(1/4)*Sqrt[a + b - Sqrt[a]*Sqrt[a + b]] - Sqrt[2]*(a + b)^(3/4)*Cot[x])/(a^
(1/4)*Sqrt[a + b + Sqrt[a]*Sqrt[a + b]])])/(2*Sqrt[2]*a^(3/4)*(a + b)^(1/4)*Sqrt[a + b + Sqrt[a]*Sqrt[a + b]])
- ((Sqrt[a] + Sqrt[a + b])*ArcTan[(a^(1/4)*Sqrt[a + b - Sqrt[a]*Sqrt[a + b]] + Sqrt[2]*(a + b)^(3/4)*Cot[x])/
(a^(1/4)*Sqrt[a + b + Sqrt[a]*Sqrt[a + b]])])/(2*Sqrt[2]*a^(3/4)*(a + b)^(1/4)*Sqrt[a + b + Sqrt[a]*Sqrt[a + b
]]) - ((Sqrt[a] - Sqrt[a + b])*Log[Sqrt[a]*(a + b)^(1/4) - Sqrt[2]*a^(1/4)*Sqrt[a + b - Sqrt[a]*Sqrt[a + b]]*C
ot[x] + (a + b)^(3/4)*Cot[x]^2])/(4*Sqrt[2]*a^(3/4)*(a + b)^(1/4)*Sqrt[a + b - Sqrt[a]*Sqrt[a + b]]) + ((Sqrt[
a] - Sqrt[a + b])*Log[Sqrt[a]*(a + b)^(1/4) + Sqrt[2]*a^(1/4)*Sqrt[a + b - Sqrt[a]*Sqrt[a + b]]*Cot[x] + (a +
b)^(3/4)*Cot[x]^2])/(4*Sqrt[2]*a^(3/4)*(a + b)^(1/4)*Sqrt[a + b - Sqrt[a]*Sqrt[a + b]])

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Rubi [A]  time = 1.10087, antiderivative size = 487, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {3209, 1169, 634, 618, 204, 628} \[ -\frac{\left (\sqrt{a}-\sqrt{a+b}\right ) \log \left ((a+b)^{3/4} \cot ^2(x)-\sqrt{2} \sqrt [4]{a} \sqrt{-\sqrt{a} \sqrt{a+b}+a+b} \cot (x)+\sqrt{a} \sqrt [4]{a+b}\right )}{4 \sqrt{2} a^{3/4} \sqrt [4]{a+b} \sqrt{-\sqrt{a} \sqrt{a+b}+a+b}}+\frac{\left (\sqrt{a}-\sqrt{a+b}\right ) \log \left ((a+b)^{3/4} \cot ^2(x)+\sqrt{2} \sqrt [4]{a} \sqrt{-\sqrt{a} \sqrt{a+b}+a+b} \cot (x)+\sqrt{a} \sqrt [4]{a+b}\right )}{4 \sqrt{2} a^{3/4} \sqrt [4]{a+b} \sqrt{-\sqrt{a} \sqrt{a+b}+a+b}}+\frac{\left (\sqrt{a+b}+\sqrt{a}\right ) \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt{-\sqrt{a} \sqrt{a+b}+a+b}-\sqrt{2} (a+b)^{3/4} \cot (x)}{\sqrt [4]{a} \sqrt{\sqrt{a} \sqrt{a+b}+a+b}}\right )}{2 \sqrt{2} a^{3/4} \sqrt [4]{a+b} \sqrt{\sqrt{a} \sqrt{a+b}+a+b}}-\frac{\left (\sqrt{a+b}+\sqrt{a}\right ) \tan ^{-1}\left (\frac{\sqrt{2} (a+b)^{3/4} \cot (x)+\sqrt [4]{a} \sqrt{-\sqrt{a} \sqrt{a+b}+a+b}}{\sqrt [4]{a} \sqrt{\sqrt{a} \sqrt{a+b}+a+b}}\right )}{2 \sqrt{2} a^{3/4} \sqrt [4]{a+b} \sqrt{\sqrt{a} \sqrt{a+b}+a+b}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Cos[x]^4)^(-1),x]

[Out]

((Sqrt[a] + Sqrt[a + b])*ArcTan[(a^(1/4)*Sqrt[a + b - Sqrt[a]*Sqrt[a + b]] - Sqrt[2]*(a + b)^(3/4)*Cot[x])/(a^
(1/4)*Sqrt[a + b + Sqrt[a]*Sqrt[a + b]])])/(2*Sqrt[2]*a^(3/4)*(a + b)^(1/4)*Sqrt[a + b + Sqrt[a]*Sqrt[a + b]])
- ((Sqrt[a] + Sqrt[a + b])*ArcTan[(a^(1/4)*Sqrt[a + b - Sqrt[a]*Sqrt[a + b]] + Sqrt[2]*(a + b)^(3/4)*Cot[x])/
(a^(1/4)*Sqrt[a + b + Sqrt[a]*Sqrt[a + b]])])/(2*Sqrt[2]*a^(3/4)*(a + b)^(1/4)*Sqrt[a + b + Sqrt[a]*Sqrt[a + b
]]) - ((Sqrt[a] - Sqrt[a + b])*Log[Sqrt[a]*(a + b)^(1/4) - Sqrt[2]*a^(1/4)*Sqrt[a + b - Sqrt[a]*Sqrt[a + b]]*C
ot[x] + (a + b)^(3/4)*Cot[x]^2])/(4*Sqrt[2]*a^(3/4)*(a + b)^(1/4)*Sqrt[a + b - Sqrt[a]*Sqrt[a + b]]) + ((Sqrt[
a] - Sqrt[a + b])*Log[Sqrt[a]*(a + b)^(1/4) + Sqrt[2]*a^(1/4)*Sqrt[a + b - Sqrt[a]*Sqrt[a + b]]*Cot[x] + (a +
b)^(3/4)*Cot[x]^2])/(4*Sqrt[2]*a^(3/4)*(a + b)^(1/4)*Sqrt[a + b - Sqrt[a]*Sqrt[a + b]])

Rule 3209

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^4)^(p_.), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dis
t[ff/f, Subst[Int[(a + 2*a*ff^2*x^2 + (a + b)*ff^4*x^4)^p/(1 + ff^2*x^2)^(2*p + 1), x], x, Tan[e + f*x]/ff], x
]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[p]

Rule 1169

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[a/c, 2]}, With[{r =
Rt[2*q - b/c, 2]}, Dist[1/(2*c*q*r), Int[(d*r - (d - e*q)*x)/(q - r*x + x^2), x], x] + Dist[1/(2*c*q*r), Int[(
d*r + (d - e*q)*x)/(q + r*x + x^2), x], x]]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2
- b*d*e + a*e^2, 0] && NegQ[b^2 - 4*a*c]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; 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])

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]

Rubi steps

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

Mathematica [C]  time = 0.222464, size = 121, normalized size = 0.25 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{a} \tan (x)}{\sqrt{a+i \sqrt{a} \sqrt{b}}}\right )}{2 \sqrt{a} \sqrt{a+i \sqrt{a} \sqrt{b}}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{a} \tan (x)}{\sqrt{-a+i \sqrt{a} \sqrt{b}}}\right )}{2 \sqrt{a} \sqrt{-a+i \sqrt{a} \sqrt{b}}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Cos[x]^4)^(-1),x]

[Out]

ArcTan[(Sqrt[a]*Tan[x])/Sqrt[a + I*Sqrt[a]*Sqrt[b]]]/(2*Sqrt[a]*Sqrt[a + I*Sqrt[a]*Sqrt[b]]) - ArcTanh[(Sqrt[a
]*Tan[x])/Sqrt[-a + I*Sqrt[a]*Sqrt[b]]]/(2*Sqrt[a]*Sqrt[-a + I*Sqrt[a]*Sqrt[b]])

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Maple [B]  time = 0.153, size = 3348, normalized size = 6.9 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4/a/(a+b)^(3/2)/(4*a^(1/2)*(a+b)^(1/2)-2*((a+b)*a)^(1/2)+2*a)^(1/2)*arctan((2*a^(1/2)*tan(x)+(2*((a+b)*a)^(
1/2)-2*a)^(1/2))/(4*a^(1/2)*(a+b)^(1/2)-2*((a+b)*a)^(1/2)+2*a)^(1/2))*(2*((a+b)*a)^(1/2)-2*a)^(1/2)*(2*(a^2+a*
b)^(1/2)-2*a)^(1/2)*(a^2+a*b)^(1/2)+1/4*a/b/(a+b)^(3/2)/(4*a^(1/2)*(a+b)^(1/2)-2*((a+b)*a)^(1/2)+2*a)^(1/2)*ar
ctan((-2*a^(1/2)*tan(x)+(2*((a+b)*a)^(1/2)-2*a)^(1/2))/(4*a^(1/2)*(a+b)^(1/2)-2*((a+b)*a)^(1/2)+2*a)^(1/2))*(2
*((a+b)*a)^(1/2)-2*a)^(1/2)*(2*(a^2+a*b)^(1/2)-2*a)^(1/2)+1/4/b/(a+b)^(3/2)/(4*a^(1/2)*(a+b)^(1/2)-2*((a+b)*a)
^(1/2)+2*a)^(1/2)*arctan((-2*a^(1/2)*tan(x)+(2*((a+b)*a)^(1/2)-2*a)^(1/2))/(4*a^(1/2)*(a+b)^(1/2)-2*((a+b)*a)^
(1/2)+2*a)^(1/2))*(2*((a+b)*a)^(1/2)-2*a)^(1/2)*(2*(a^2+a*b)^(1/2)-2*a)^(1/2)*(a^2+a*b)^(1/2)+1/4/a/(a+b)^(3/2
)/(4*a^(1/2)*(a+b)^(1/2)-2*((a+b)*a)^(1/2)+2*a)^(1/2)*arctan((-2*a^(1/2)*tan(x)+(2*((a+b)*a)^(1/2)-2*a)^(1/2))
/(4*a^(1/2)*(a+b)^(1/2)-2*((a+b)*a)^(1/2)+2*a)^(1/2))*(2*((a+b)*a)^(1/2)-2*a)^(1/2)*(2*(a^2+a*b)^(1/2)-2*a)^(1
/2)*(a^2+a*b)^(1/2)-1/4*a/b/(a+b)^(3/2)/(4*a^(1/2)*(a+b)^(1/2)-2*((a+b)*a)^(1/2)+2*a)^(1/2)*arctan((2*a^(1/2)*
tan(x)+(2*((a+b)*a)^(1/2)-2*a)^(1/2))/(4*a^(1/2)*(a+b)^(1/2)-2*((a+b)*a)^(1/2)+2*a)^(1/2))*(2*((a+b)*a)^(1/2)-
2*a)^(1/2)*(2*(a^2+a*b)^(1/2)-2*a)^(1/2)-1/4/b/(a+b)^(3/2)/(4*a^(1/2)*(a+b)^(1/2)-2*((a+b)*a)^(1/2)+2*a)^(1/2)
*arctan((2*a^(1/2)*tan(x)+(2*((a+b)*a)^(1/2)-2*a)^(1/2))/(4*a^(1/2)*(a+b)^(1/2)-2*((a+b)*a)^(1/2)+2*a)^(1/2))*
(2*((a+b)*a)^(1/2)-2*a)^(1/2)*(2*(a^2+a*b)^(1/2)-2*a)^(1/2)*(a^2+a*b)^(1/2)+1/4/a^(3/2)/(a+b)/(4*a^(1/2)*(a+b)
^(1/2)-2*((a+b)*a)^(1/2)+2*a)^(1/2)*arctan((2*a^(1/2)*tan(x)+(2*((a+b)*a)^(1/2)-2*a)^(1/2))/(4*a^(1/2)*(a+b)^(
1/2)-2*((a+b)*a)^(1/2)+2*a)^(1/2))*(2*((a+b)*a)^(1/2)-2*a)^(1/2)*(2*(a^2+a*b)^(1/2)-2*a)^(1/2)*(a^2+a*b)^(1/2)
+1/4*a^(1/2)/b/(a+b)/(4*a^(1/2)*(a+b)^(1/2)-2*((a+b)*a)^(1/2)+2*a)^(1/2)*arctan((2*a^(1/2)*tan(x)+(2*((a+b)*a)
^(1/2)-2*a)^(1/2))/(4*a^(1/2)*(a+b)^(1/2)-2*((a+b)*a)^(1/2)+2*a)^(1/2))*(2*((a+b)*a)^(1/2)-2*a)^(1/2)*(2*(a^2+
a*b)^(1/2)-2*a)^(1/2)-1/4/a^(3/2)/(a+b)/(4*a^(1/2)*(a+b)^(1/2)-2*((a+b)*a)^(1/2)+2*a)^(1/2)*arctan((-2*a^(1/2)
*tan(x)+(2*((a+b)*a)^(1/2)-2*a)^(1/2))/(4*a^(1/2)*(a+b)^(1/2)-2*((a+b)*a)^(1/2)+2*a)^(1/2))*(2*((a+b)*a)^(1/2)
-2*a)^(1/2)*(2*(a^2+a*b)^(1/2)-2*a)^(1/2)*(a^2+a*b)^(1/2)-1/4*a^(1/2)/b/(a+b)/(4*a^(1/2)*(a+b)^(1/2)-2*((a+b)*
a)^(1/2)+2*a)^(1/2)*arctan((-2*a^(1/2)*tan(x)+(2*((a+b)*a)^(1/2)-2*a)^(1/2))/(4*a^(1/2)*(a+b)^(1/2)-2*((a+b)*a
)^(1/2)+2*a)^(1/2))*(2*((a+b)*a)^(1/2)-2*a)^(1/2)*(2*(a^2+a*b)^(1/2)-2*a)^(1/2)-1/4/(a+b)^(3/2)/(4*a^(1/2)*(a+
b)^(1/2)-2*((a+b)*a)^(1/2)+2*a)^(1/2)*arctan((2*a^(1/2)*tan(x)+(2*((a+b)*a)^(1/2)-2*a)^(1/2))/(4*a^(1/2)*(a+b)
^(1/2)-2*((a+b)*a)^(1/2)+2*a)^(1/2))*(2*((a+b)*a)^(1/2)-2*a)^(1/2)*(2*(a^2+a*b)^(1/2)-2*a)^(1/2)+1/8/a^(3/2)/(
a+b)*ln(-a^(1/2)*tan(x)^2+tan(x)*(2*((a+b)*a)^(1/2)-2*a)^(1/2)-(a+b)^(1/2))*(2*(a^2+a*b)^(1/2)-2*a)^(1/2)*(a^2
+a*b)^(1/2)+1/8*a^(1/2)/b/(a+b)*ln(-a^(1/2)*tan(x)^2+tan(x)*(2*((a+b)*a)^(1/2)-2*a)^(1/2)-(a+b)^(1/2))*(2*(a^2
+a*b)^(1/2)-2*a)^(1/2)-1/8/a^(3/2)/(a+b)*ln(a^(1/2)*tan(x)^2+tan(x)*(2*((a+b)*a)^(1/2)-2*a)^(1/2)+(a+b)^(1/2))
*(2*(a^2+a*b)^(1/2)-2*a)^(1/2)*(a^2+a*b)^(1/2)-1/8*a^(1/2)/b/(a+b)*ln(a^(1/2)*tan(x)^2+tan(x)*(2*((a+b)*a)^(1/
2)-2*a)^(1/2)+(a+b)^(1/2))*(2*(a^2+a*b)^(1/2)-2*a)^(1/2)-1/8*a/b/(a+b)^(3/2)*ln(-a^(1/2)*tan(x)^2+tan(x)*(2*((
a+b)*a)^(1/2)-2*a)^(1/2)-(a+b)^(1/2))*(2*(a^2+a*b)^(1/2)-2*a)^(1/2)-1/8/b/(a+b)^(3/2)*ln(-a^(1/2)*tan(x)^2+tan
(x)*(2*((a+b)*a)^(1/2)-2*a)^(1/2)-(a+b)^(1/2))*(2*(a^2+a*b)^(1/2)-2*a)^(1/2)*(a^2+a*b)^(1/2)-1/8/a/(a+b)^(3/2)
*ln(-a^(1/2)*tan(x)^2+tan(x)*(2*((a+b)*a)^(1/2)-2*a)^(1/2)-(a+b)^(1/2))*(2*(a^2+a*b)^(1/2)-2*a)^(1/2)*(a^2+a*b
)^(1/2)+1/4/(a+b)^(3/2)/(4*a^(1/2)*(a+b)^(1/2)-2*((a+b)*a)^(1/2)+2*a)^(1/2)*arctan((-2*a^(1/2)*tan(x)+(2*((a+b
)*a)^(1/2)-2*a)^(1/2))/(4*a^(1/2)*(a+b)^(1/2)-2*((a+b)*a)^(1/2)+2*a)^(1/2))*(2*((a+b)*a)^(1/2)-2*a)^(1/2)*(2*(
a^2+a*b)^(1/2)-2*a)^(1/2)+1/8*a/b/(a+b)^(3/2)*ln(a^(1/2)*tan(x)^2+tan(x)*(2*((a+b)*a)^(1/2)-2*a)^(1/2)+(a+b)^(
1/2))*(2*(a^2+a*b)^(1/2)-2*a)^(1/2)+1/8/b/(a+b)^(3/2)*ln(a^(1/2)*tan(x)^2+tan(x)*(2*((a+b)*a)^(1/2)-2*a)^(1/2)
+(a+b)^(1/2))*(2*(a^2+a*b)^(1/2)-2*a)^(1/2)*(a^2+a*b)^(1/2)+1/8/a/(a+b)^(3/2)*ln(a^(1/2)*tan(x)^2+tan(x)*(2*((
a+b)*a)^(1/2)-2*a)^(1/2)+(a+b)^(1/2))*(2*(a^2+a*b)^(1/2)-2*a)^(1/2)*(a^2+a*b)^(1/2)+1/8/(a+b)^(3/2)*ln(a^(1/2)
*tan(x)^2+tan(x)*(2*((a+b)*a)^(1/2)-2*a)^(1/2)+(a+b)^(1/2))*(2*(a^2+a*b)^(1/2)-2*a)^(1/2)-1/8/(a+b)^(3/2)*ln(-
a^(1/2)*tan(x)^2+tan(x)*(2*((a+b)*a)^(1/2)-2*a)^(1/2)-(a+b)^(1/2))*(2*(a^2+a*b)^(1/2)-2*a)^(1/2)+1/8/a^(1/2)/(
a+b)*ln(-a^(1/2)*tan(x)^2+tan(x)*(2*((a+b)*a)^(1/2)-2*a)^(1/2)-(a+b)^(1/2))*(2*(a^2+a*b)^(1/2)-2*a)^(1/2)-1/8/
a^(1/2)/(a+b)*ln(a^(1/2)*tan(x)^2+tan(x)*(2*((a+b)*a)^(1/2)-2*a)^(1/2)+(a+b)^(1/2))*(2*(a^2+a*b)^(1/2)-2*a)^(1
/2)+a/(a+b)^(3/2)/(4*a^(1/2)*(a+b)^(1/2)-2*((a+b)*a)^(1/2)+2*a)^(1/2)*arctan((2*a^(1/2)*tan(x)+(2*((a+b)*a)^(1
/2)-2*a)^(1/2))/(4*a^(1/2)*(a+b)^(1/2)-2*((a+b)*a)^(1/2)+2*a)^(1/2))+b/(a+b)^(3/2)/(4*a^(1/2)*(a+b)^(1/2)-2*((
a+b)*a)^(1/2)+2*a)^(1/2)*arctan((2*a^(1/2)*tan(x)+(2*((a+b)*a)^(1/2)-2*a)^(1/2))/(4*a^(1/2)*(a+b)^(1/2)-2*((a+
b)*a)^(1/2)+2*a)^(1/2))-a/(a+b)^(3/2)/(4*a^(1/2)*(a+b)^(1/2)-2*((a+b)*a)^(1/2)+2*a)^(1/2)*arctan((-2*a^(1/2)*t
an(x)+(2*((a+b)*a)^(1/2)-2*a)^(1/2))/(4*a^(1/2)*(a+b)^(1/2)-2*((a+b)*a)^(1/2)+2*a)^(1/2))-b/(a+b)^(3/2)/(4*a^(
1/2)*(a+b)^(1/2)-2*((a+b)*a)^(1/2)+2*a)^(1/2)*arctan((-2*a^(1/2)*tan(x)+(2*((a+b)*a)^(1/2)-2*a)^(1/2))/(4*a^(1
/2)*(a+b)^(1/2)-2*((a+b)*a)^(1/2)+2*a)^(1/2))+1/4/a^(1/2)/(a+b)/(4*a^(1/2)*(a+b)^(1/2)-2*((a+b)*a)^(1/2)+2*a)^
(1/2)*arctan((2*a^(1/2)*tan(x)+(2*((a+b)*a)^(1/2)-2*a)^(1/2))/(4*a^(1/2)*(a+b)^(1/2)-2*((a+b)*a)^(1/2)+2*a)^(1
/2))*(2*((a+b)*a)^(1/2)-2*a)^(1/2)*(2*(a^2+a*b)^(1/2)-2*a)^(1/2)+1/8/a^(1/2)/b/(a+b)*ln(-a^(1/2)*tan(x)^2+tan(
x)*(2*((a+b)*a)^(1/2)-2*a)^(1/2)-(a+b)^(1/2))*(2*(a^2+a*b)^(1/2)-2*a)^(1/2)*(a^2+a*b)^(1/2)-1/4/a^(1/2)/(a+b)/
(4*a^(1/2)*(a+b)^(1/2)-2*((a+b)*a)^(1/2)+2*a)^(1/2)*arctan((-2*a^(1/2)*tan(x)+(2*((a+b)*a)^(1/2)-2*a)^(1/2))/(
4*a^(1/2)*(a+b)^(1/2)-2*((a+b)*a)^(1/2)+2*a)^(1/2))*(2*((a+b)*a)^(1/2)-2*a)^(1/2)*(2*(a^2+a*b)^(1/2)-2*a)^(1/2
)-1/8/a^(1/2)/b/(a+b)*ln(a^(1/2)*tan(x)^2+tan(x)*(2*((a+b)*a)^(1/2)-2*a)^(1/2)+(a+b)^(1/2))*(2*(a^2+a*b)^(1/2)
-2*a)^(1/2)*(a^2+a*b)^(1/2)-1/4/a^(1/2)/b/(a+b)/(4*a^(1/2)*(a+b)^(1/2)-2*((a+b)*a)^(1/2)+2*a)^(1/2)*arctan((-2
*a^(1/2)*tan(x)+(2*((a+b)*a)^(1/2)-2*a)^(1/2))/(4*a^(1/2)*(a+b)^(1/2)-2*((a+b)*a)^(1/2)+2*a)^(1/2))*(2*((a+b)*
a)^(1/2)-2*a)^(1/2)*(2*(a^2+a*b)^(1/2)-2*a)^(1/2)*(a^2+a*b)^(1/2)+1/4/a^(1/2)/b/(a+b)/(4*a^(1/2)*(a+b)^(1/2)-2
*((a+b)*a)^(1/2)+2*a)^(1/2)*arctan((2*a^(1/2)*tan(x)+(2*((a+b)*a)^(1/2)-2*a)^(1/2))/(4*a^(1/2)*(a+b)^(1/2)-2*(
(a+b)*a)^(1/2)+2*a)^(1/2))*(2*((a+b)*a)^(1/2)-2*a)^(1/2)*(2*(a^2+a*b)^(1/2)-2*a)^(1/2)*(a^2+a*b)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{b \cos \left (x\right )^{4} + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(b*cos(x)^4 + a), x)

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Fricas [B]  time = 2.44499, size = 1796, normalized size = 3.69 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*sqrt(-((a^2 + a*b)*sqrt(-b/(a^5 + 2*a^4*b + a^3*b^2)) + 1)/(a^2 + a*b))*log(b*cos(x)^2 + 2*(a*b*cos(x)*si
n(x) + (a^4 + a^3*b)*sqrt(-b/(a^5 + 2*a^4*b + a^3*b^2))*cos(x)*sin(x))*sqrt(-((a^2 + a*b)*sqrt(-b/(a^5 + 2*a^4
*b + a^3*b^2)) + 1)/(a^2 + a*b)) - (a^3 + a^2*b - 2*(a^3 + a^2*b)*cos(x)^2)*sqrt(-b/(a^5 + 2*a^4*b + a^3*b^2))
) + 1/8*sqrt(-((a^2 + a*b)*sqrt(-b/(a^5 + 2*a^4*b + a^3*b^2)) + 1)/(a^2 + a*b))*log(b*cos(x)^2 - 2*(a*b*cos(x)
*sin(x) + (a^4 + a^3*b)*sqrt(-b/(a^5 + 2*a^4*b + a^3*b^2))*cos(x)*sin(x))*sqrt(-((a^2 + a*b)*sqrt(-b/(a^5 + 2*
a^4*b + a^3*b^2)) + 1)/(a^2 + a*b)) - (a^3 + a^2*b - 2*(a^3 + a^2*b)*cos(x)^2)*sqrt(-b/(a^5 + 2*a^4*b + a^3*b^
2))) + 1/8*sqrt(((a^2 + a*b)*sqrt(-b/(a^5 + 2*a^4*b + a^3*b^2)) - 1)/(a^2 + a*b))*log(-b*cos(x)^2 + 2*(a*b*cos
(x)*sin(x) - (a^4 + a^3*b)*sqrt(-b/(a^5 + 2*a^4*b + a^3*b^2))*cos(x)*sin(x))*sqrt(((a^2 + a*b)*sqrt(-b/(a^5 +
2*a^4*b + a^3*b^2)) - 1)/(a^2 + a*b)) - (a^3 + a^2*b - 2*(a^3 + a^2*b)*cos(x)^2)*sqrt(-b/(a^5 + 2*a^4*b + a^3*
b^2))) - 1/8*sqrt(((a^2 + a*b)*sqrt(-b/(a^5 + 2*a^4*b + a^3*b^2)) - 1)/(a^2 + a*b))*log(-b*cos(x)^2 - 2*(a*b*c
os(x)*sin(x) - (a^4 + a^3*b)*sqrt(-b/(a^5 + 2*a^4*b + a^3*b^2))*cos(x)*sin(x))*sqrt(((a^2 + a*b)*sqrt(-b/(a^5
+ 2*a^4*b + a^3*b^2)) - 1)/(a^2 + a*b)) - (a^3 + a^2*b - 2*(a^3 + a^2*b)*cos(x)^2)*sqrt(-b/(a^5 + 2*a^4*b + a^
3*b^2)))

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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/(a+b*cos(x)**4),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError