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]

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

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Rubi [A]  time = 1.10, antiderivative size = 487, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, 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 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 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 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 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 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 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]

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*}

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Mathematica [C]  time = 0.24, 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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fricas [B]  time = 0.86, size = 809, normalized size = 1.66 \[ -\frac {1}{8} \, \sqrt {-\frac {{\left (a^{2} + a b\right )} \sqrt {-\frac {b}{a^{5} + 2 \, a^{4} b + a^{3} b^{2}}} + 1}{a^{2} + a b}} \log \left (b \cos \relax (x)^{2} + 2 \, {\left (a b \cos \relax (x) \sin \relax (x) + {\left (a^{4} + a^{3} b\right )} \sqrt {-\frac {b}{a^{5} + 2 \, a^{4} b + a^{3} b^{2}}} \cos \relax (x) \sin \relax (x)\right )} \sqrt {-\frac {{\left (a^{2} + a b\right )} \sqrt {-\frac {b}{a^{5} + 2 \, a^{4} b + a^{3} b^{2}}} + 1}{a^{2} + a b}} - {\left (a^{3} + a^{2} b - 2 \, {\left (a^{3} + a^{2} b\right )} \cos \relax (x)^{2}\right )} \sqrt {-\frac {b}{a^{5} + 2 \, a^{4} b + a^{3} b^{2}}}\right ) + \frac {1}{8} \, \sqrt {-\frac {{\left (a^{2} + a b\right )} \sqrt {-\frac {b}{a^{5} + 2 \, a^{4} b + a^{3} b^{2}}} + 1}{a^{2} + a b}} \log \left (b \cos \relax (x)^{2} - 2 \, {\left (a b \cos \relax (x) \sin \relax (x) + {\left (a^{4} + a^{3} b\right )} \sqrt {-\frac {b}{a^{5} + 2 \, a^{4} b + a^{3} b^{2}}} \cos \relax (x) \sin \relax (x)\right )} \sqrt {-\frac {{\left (a^{2} + a b\right )} \sqrt {-\frac {b}{a^{5} + 2 \, a^{4} b + a^{3} b^{2}}} + 1}{a^{2} + a b}} - {\left (a^{3} + a^{2} b - 2 \, {\left (a^{3} + a^{2} b\right )} \cos \relax (x)^{2}\right )} \sqrt {-\frac {b}{a^{5} + 2 \, a^{4} b + a^{3} b^{2}}}\right ) + \frac {1}{8} \, \sqrt {\frac {{\left (a^{2} + a b\right )} \sqrt {-\frac {b}{a^{5} + 2 \, a^{4} b + a^{3} b^{2}}} - 1}{a^{2} + a b}} \log \left (-b \cos \relax (x)^{2} + 2 \, {\left (a b \cos \relax (x) \sin \relax (x) - {\left (a^{4} + a^{3} b\right )} \sqrt {-\frac {b}{a^{5} + 2 \, a^{4} b + a^{3} b^{2}}} \cos \relax (x) \sin \relax (x)\right )} \sqrt {\frac {{\left (a^{2} + a b\right )} \sqrt {-\frac {b}{a^{5} + 2 \, a^{4} b + a^{3} b^{2}}} - 1}{a^{2} + a b}} - {\left (a^{3} + a^{2} b - 2 \, {\left (a^{3} + a^{2} b\right )} \cos \relax (x)^{2}\right )} \sqrt {-\frac {b}{a^{5} + 2 \, a^{4} b + a^{3} b^{2}}}\right ) - \frac {1}{8} \, \sqrt {\frac {{\left (a^{2} + a b\right )} \sqrt {-\frac {b}{a^{5} + 2 \, a^{4} b + a^{3} b^{2}}} - 1}{a^{2} + a b}} \log \left (-b \cos \relax (x)^{2} - 2 \, {\left (a b \cos \relax (x) \sin \relax (x) - {\left (a^{4} + a^{3} b\right )} \sqrt {-\frac {b}{a^{5} + 2 \, a^{4} b + a^{3} b^{2}}} \cos \relax (x) \sin \relax (x)\right )} \sqrt {\frac {{\left (a^{2} + a b\right )} \sqrt {-\frac {b}{a^{5} + 2 \, a^{4} b + a^{3} b^{2}}} - 1}{a^{2} + a b}} - {\left (a^{3} + a^{2} b - 2 \, {\left (a^{3} + a^{2} b\right )} \cos \relax (x)^{2}\right )} \sqrt {-\frac {b}{a^{5} + 2 \, a^{4} b + a^{3} b^{2}}}\right ) \]

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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giac [A]  time = 1.03, size = 307, normalized size = 0.63 \[ \frac {{\left (3 \, \sqrt {a^{2} + \sqrt {-a b} a} a^{2} + 4 \, \sqrt {a^{2} + \sqrt {-a b} a} a b - 3 \, \sqrt {a^{2} + \sqrt {-a b} a} \sqrt {-a b} a - 4 \, \sqrt {a^{2} + \sqrt {-a b} a} \sqrt {-a b} b\right )} {\left (\pi \left \lfloor \frac {x}{\pi } + \frac {1}{2} \right \rfloor + \arctan \left (\frac {2 \, \tan \relax (x)}{\sqrt {\frac {4 \, a + \sqrt {-16 \, {\left (a + b\right )} a + 16 \, a^{2}}}{a}}}\right )\right )} {\left | a \right |}}{2 \, {\left (3 \, a^{5} + 7 \, a^{4} b + 4 \, a^{3} b^{2}\right )}} + \frac {{\left (3 \, \sqrt {a^{2} - \sqrt {-a b} a} a^{2} + 4 \, \sqrt {a^{2} - \sqrt {-a b} a} a b + 3 \, \sqrt {a^{2} - \sqrt {-a b} a} \sqrt {-a b} a + 4 \, \sqrt {a^{2} - \sqrt {-a b} a} \sqrt {-a b} b\right )} {\left (\pi \left \lfloor \frac {x}{\pi } + \frac {1}{2} \right \rfloor + \arctan \left (\frac {2 \, \tan \relax (x)}{\sqrt {\frac {4 \, a - \sqrt {-16 \, {\left (a + b\right )} a + 16 \, a^{2}}}{a}}}\right )\right )} {\left | a \right |}}{2 \, {\left (3 \, a^{5} + 7 \, a^{4} b + 4 \, a^{3} b^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(3*sqrt(a^2 + sqrt(-a*b)*a)*a^2 + 4*sqrt(a^2 + sqrt(-a*b)*a)*a*b - 3*sqrt(a^2 + sqrt(-a*b)*a)*sqrt(-a*b)*a
- 4*sqrt(a^2 + sqrt(-a*b)*a)*sqrt(-a*b)*b)*(pi*floor(x/pi + 1/2) + arctan(2*tan(x)/sqrt((4*a + sqrt(-16*(a +
b)*a + 16*a^2))/a)))*abs(a)/(3*a^5 + 7*a^4*b + 4*a^3*b^2) + 1/2*(3*sqrt(a^2 - sqrt(-a*b)*a)*a^2 + 4*sqrt(a^2 -
sqrt(-a*b)*a)*a*b + 3*sqrt(a^2 - sqrt(-a*b)*a)*sqrt(-a*b)*a + 4*sqrt(a^2 - sqrt(-a*b)*a)*sqrt(-a*b)*b)*(pi*fl
oor(x/pi + 1/2) + arctan(2*tan(x)/sqrt((4*a - sqrt(-16*(a + b)*a + 16*a^2))/a)))*abs(a)/(3*a^5 + 7*a^4*b + 4*a
^3*b^2)

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maple [B]  time = 0.23, size = 3350, normalized size = 6.88 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-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)+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)*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))-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)+1/8*a/b/(a+b)^(3/2)*ln(a^(1/2)*tan(x)^2+t
an(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)*ta
n(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^(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)-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^(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/(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)*ta
n(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)*a
rctan((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/(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)*ta
n(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)-1/8/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)-1/4/(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^(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)+1/4/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/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^(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)*(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)*(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)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{b \cos \relax (x)^{4} + a}\,{d x} \]

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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mupad [B]  time = 2.66, size = 926, normalized size = 1.90 \[ -2\,\mathrm {atanh}\left (\frac {8\,a^6\,b\,\mathrm {tan}\relax (x)\,\sqrt {-\frac {a^2}{16\,\left (a^4+b\,a^3\right )}-\frac {\sqrt {-a^3\,b}}{16\,\left (a^4+b\,a^3\right )}}}{\frac {2\,a^9\,b}{a^4+b\,a^3}-2\,a^4\,b^2-2\,a^5\,b+\frac {2\,a^8\,b^2}{a^4+b\,a^3}+\frac {2\,a^7\,b\,\sqrt {-a^3\,b}}{a^4+b\,a^3}+\frac {2\,a^6\,b^2\,\sqrt {-a^3\,b}}{a^4+b\,a^3}}-\frac {8\,a^2\,b\,\mathrm {tan}\relax (x)\,\sqrt {-\frac {a^2}{16\,\left (a^4+b\,a^3\right )}-\frac {\sqrt {-a^3\,b}}{16\,\left (a^4+b\,a^3\right )}}}{\frac {2\,a^5\,b}{a^4+b\,a^3}-2\,a\,b+\frac {2\,a^3\,b\,\sqrt {-a^3\,b}}{a^4+b\,a^3}}+\frac {8\,a^4\,b\,\mathrm {tan}\relax (x)\,\sqrt {-a^3\,b}\,\sqrt {-\frac {a^2}{16\,\left (a^4+b\,a^3\right )}-\frac {\sqrt {-a^3\,b}}{16\,\left (a^4+b\,a^3\right )}}}{\frac {2\,a^9\,b}{a^4+b\,a^3}-2\,a^4\,b^2-2\,a^5\,b+\frac {2\,a^8\,b^2}{a^4+b\,a^3}+\frac {2\,a^7\,b\,\sqrt {-a^3\,b}}{a^4+b\,a^3}+\frac {2\,a^6\,b^2\,\sqrt {-a^3\,b}}{a^4+b\,a^3}}\right )\,\sqrt {-\frac {a^2+\sqrt {-a^3\,b}}{16\,\left (a^4+b\,a^3\right )}}-2\,\mathrm {atanh}\left (\frac {8\,a^2\,b\,\mathrm {tan}\relax (x)\,\sqrt {\frac {\sqrt {-a^3\,b}}{16\,\left (a^4+b\,a^3\right )}-\frac {a^2}{16\,\left (a^4+b\,a^3\right )}}}{2\,a\,b-\frac {2\,a^5\,b}{a^4+b\,a^3}+\frac {2\,a^3\,b\,\sqrt {-a^3\,b}}{a^4+b\,a^3}}-\frac {8\,a^6\,b\,\mathrm {tan}\relax (x)\,\sqrt {\frac {\sqrt {-a^3\,b}}{16\,\left (a^4+b\,a^3\right )}-\frac {a^2}{16\,\left (a^4+b\,a^3\right )}}}{2\,a^5\,b+2\,a^4\,b^2-\frac {2\,a^9\,b}{a^4+b\,a^3}-\frac {2\,a^8\,b^2}{a^4+b\,a^3}+\frac {2\,a^7\,b\,\sqrt {-a^3\,b}}{a^4+b\,a^3}+\frac {2\,a^6\,b^2\,\sqrt {-a^3\,b}}{a^4+b\,a^3}}+\frac {8\,a^4\,b\,\mathrm {tan}\relax (x)\,\sqrt {-a^3\,b}\,\sqrt {\frac {\sqrt {-a^3\,b}}{16\,\left (a^4+b\,a^3\right )}-\frac {a^2}{16\,\left (a^4+b\,a^3\right )}}}{2\,a^5\,b+2\,a^4\,b^2-\frac {2\,a^9\,b}{a^4+b\,a^3}-\frac {2\,a^8\,b^2}{a^4+b\,a^3}+\frac {2\,a^7\,b\,\sqrt {-a^3\,b}}{a^4+b\,a^3}+\frac {2\,a^6\,b^2\,\sqrt {-a^3\,b}}{a^4+b\,a^3}}\right )\,\sqrt {-\frac {a^2-\sqrt {-a^3\,b}}{16\,\left (a^4+b\,a^3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + b*cos(x)^4),x)

[Out]

- 2*atanh((8*a^6*b*tan(x)*(- a^2/(16*(a^3*b + a^4)) - (-a^3*b)^(1/2)/(16*(a^3*b + a^4)))^(1/2))/((2*a^9*b)/(a^
3*b + a^4) - 2*a^4*b^2 - 2*a^5*b + (2*a^8*b^2)/(a^3*b + a^4) + (2*a^7*b*(-a^3*b)^(1/2))/(a^3*b + a^4) + (2*a^6
*b^2*(-a^3*b)^(1/2))/(a^3*b + a^4)) - (8*a^2*b*tan(x)*(- a^2/(16*(a^3*b + a^4)) - (-a^3*b)^(1/2)/(16*(a^3*b +
a^4)))^(1/2))/((2*a^5*b)/(a^3*b + a^4) - 2*a*b + (2*a^3*b*(-a^3*b)^(1/2))/(a^3*b + a^4)) + (8*a^4*b*tan(x)*(-a
^3*b)^(1/2)*(- a^2/(16*(a^3*b + a^4)) - (-a^3*b)^(1/2)/(16*(a^3*b + a^4)))^(1/2))/((2*a^9*b)/(a^3*b + a^4) - 2
*a^4*b^2 - 2*a^5*b + (2*a^8*b^2)/(a^3*b + a^4) + (2*a^7*b*(-a^3*b)^(1/2))/(a^3*b + a^4) + (2*a^6*b^2*(-a^3*b)^
(1/2))/(a^3*b + a^4)))*(-(a^2 + (-a^3*b)^(1/2))/(16*(a^3*b + a^4)))^(1/2) - 2*atanh((8*a^2*b*tan(x)*((-a^3*b)^
(1/2)/(16*(a^3*b + a^4)) - a^2/(16*(a^3*b + a^4)))^(1/2))/(2*a*b - (2*a^5*b)/(a^3*b + a^4) + (2*a^3*b*(-a^3*b)
^(1/2))/(a^3*b + a^4)) - (8*a^6*b*tan(x)*((-a^3*b)^(1/2)/(16*(a^3*b + a^4)) - a^2/(16*(a^3*b + a^4)))^(1/2))/(
2*a^5*b + 2*a^4*b^2 - (2*a^9*b)/(a^3*b + a^4) - (2*a^8*b^2)/(a^3*b + a^4) + (2*a^7*b*(-a^3*b)^(1/2))/(a^3*b +
a^4) + (2*a^6*b^2*(-a^3*b)^(1/2))/(a^3*b + a^4)) + (8*a^4*b*tan(x)*(-a^3*b)^(1/2)*((-a^3*b)^(1/2)/(16*(a^3*b +
a^4)) - a^2/(16*(a^3*b + a^4)))^(1/2))/(2*a^5*b + 2*a^4*b^2 - (2*a^9*b)/(a^3*b + a^4) - (2*a^8*b^2)/(a^3*b +
a^4) + (2*a^7*b*(-a^3*b)^(1/2))/(a^3*b + a^4) + (2*a^6*b^2*(-a^3*b)^(1/2))/(a^3*b + a^4)))*(-(a^2 - (-a^3*b)^(
1/2))/(16*(a^3*b + a^4)))^(1/2)

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

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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