3.72 \(\int \frac {1}{1+\cos ^4(x)} \, dx\)

Optimal. Leaf size=292 \[ \frac {x}{2 \sqrt {\sqrt {2}-1}}+\frac {1}{8} \sqrt {\sqrt {2}-1} \log \left (2 \cot ^2(x)-2 \sqrt {\sqrt {2}-1} \cot (x)+\sqrt {2}\right )-\frac {1}{8} \sqrt {\sqrt {2}-1} \log \left (\sqrt {2} \cot ^2(x)+\sqrt {2 \left (\sqrt {2}-1\right )} \cot (x)+1\right )+\frac {\tan ^{-1}\left (\frac {\sqrt {\sqrt {2}-1} \left (1-2 \sin ^2(x)\right )+\left (\sqrt {2}-2\right ) \sin (x) \cos (x)}{\left (\sqrt {2}-2\right ) \sin ^2(x)+2 \sqrt {\sqrt {2}-1} \sin (x) \cos (x)+\sqrt {1+\sqrt {2}}+2}\right )}{4 \sqrt {\sqrt {2}-1}}+\frac {\tan ^{-1}\left (\frac {\sqrt {\sqrt {2}-1} \left (2 \sin ^2(x)-1\right )+\left (\sqrt {2}-2\right ) \sin (x) \cos (x)}{\left (\sqrt {2}-2\right ) \sin ^2(x)-2 \sqrt {\sqrt {2}-1} \sin (x) \cos (x)+\sqrt {1+\sqrt {2}}+2}\right )}{4 \sqrt {\sqrt {2}-1}} \]

[Out]

1/2*x/(2^(1/2)-1)^(1/2)+1/4*arctan((cos(x)*sin(x)*(-2+2^(1/2))+(-1+2*sin(x)^2)*(2^(1/2)-1)^(1/2))/(2+sin(x)^2*
(-2+2^(1/2))-2*cos(x)*sin(x)*(2^(1/2)-1)^(1/2)+(1+2^(1/2))^(1/2)))/(2^(1/2)-1)^(1/2)+1/4*arctan((cos(x)*sin(x)
*(-2+2^(1/2))+(1-2*sin(x)^2)*(2^(1/2)-1)^(1/2))/(2+sin(x)^2*(-2+2^(1/2))+2*cos(x)*sin(x)*(2^(1/2)-1)^(1/2)+(1+
2^(1/2))^(1/2)))/(2^(1/2)-1)^(1/2)+1/8*ln(2*cot(x)^2+2^(1/2)-2*cot(x)*(2^(1/2)-1)^(1/2))*(2^(1/2)-1)^(1/2)-1/8
*ln(1+cot(x)^2*2^(1/2)+cot(x)*(-2+2*2^(1/2))^(1/2))*(2^(1/2)-1)^(1/2)

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Rubi [A]  time = 0.19, antiderivative size = 292, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {3209, 1169, 634, 618, 204, 628} \[ \frac {x}{2 \sqrt {\sqrt {2}-1}}+\frac {1}{8} \sqrt {\sqrt {2}-1} \log \left (2 \cot ^2(x)-2 \sqrt {\sqrt {2}-1} \cot (x)+\sqrt {2}\right )-\frac {1}{8} \sqrt {\sqrt {2}-1} \log \left (\sqrt {2} \cot ^2(x)+\sqrt {2 \left (\sqrt {2}-1\right )} \cot (x)+1\right )+\frac {\tan ^{-1}\left (\frac {\sqrt {\sqrt {2}-1} \left (1-2 \sin ^2(x)\right )+\left (\sqrt {2}-2\right ) \sin (x) \cos (x)}{\left (\sqrt {2}-2\right ) \sin ^2(x)+2 \sqrt {\sqrt {2}-1} \sin (x) \cos (x)+\sqrt {1+\sqrt {2}}+2}\right )}{4 \sqrt {\sqrt {2}-1}}+\frac {\tan ^{-1}\left (\frac {\sqrt {\sqrt {2}-1} \left (2 \sin ^2(x)-1\right )+\left (\sqrt {2}-2\right ) \sin (x) \cos (x)}{\left (\sqrt {2}-2\right ) \sin ^2(x)-2 \sqrt {\sqrt {2}-1} \sin (x) \cos (x)+\sqrt {1+\sqrt {2}}+2}\right )}{4 \sqrt {\sqrt {2}-1}} \]

Antiderivative was successfully verified.

[In]

Int[(1 + Cos[x]^4)^(-1),x]

[Out]

x/(2*Sqrt[-1 + Sqrt[2]]) + ArcTan[((-2 + Sqrt[2])*Cos[x]*Sin[x] + Sqrt[-1 + Sqrt[2]]*(1 - 2*Sin[x]^2))/(2 + Sq
rt[1 + Sqrt[2]] + 2*Sqrt[-1 + Sqrt[2]]*Cos[x]*Sin[x] + (-2 + Sqrt[2])*Sin[x]^2)]/(4*Sqrt[-1 + Sqrt[2]]) + ArcT
an[((-2 + Sqrt[2])*Cos[x]*Sin[x] + Sqrt[-1 + Sqrt[2]]*(-1 + 2*Sin[x]^2))/(2 + Sqrt[1 + Sqrt[2]] - 2*Sqrt[-1 +
Sqrt[2]]*Cos[x]*Sin[x] + (-2 + Sqrt[2])*Sin[x]^2)]/(4*Sqrt[-1 + Sqrt[2]]) + (Sqrt[-1 + Sqrt[2]]*Log[Sqrt[2] -
2*Sqrt[-1 + Sqrt[2]]*Cot[x] + 2*Cot[x]^2])/8 - (Sqrt[-1 + Sqrt[2]]*Log[1 + Sqrt[2*(-1 + Sqrt[2])]*Cot[x] + Sqr
t[2]*Cot[x]^2])/8

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}{1+\cos ^4(x)} \, dx &=-\operatorname {Subst}\left (\int \frac {1+x^2}{1+2 x^2+2 x^4} \, dx,x,\cot (x)\right )\\ &=-\frac {\operatorname {Subst}\left (\int \frac {\sqrt {-1+\sqrt {2}}-\left (1-\frac {1}{\sqrt {2}}\right ) x}{\frac {1}{\sqrt {2}}-\sqrt {-1+\sqrt {2}} x+x^2} \, dx,x,\cot (x)\right )}{2 \sqrt {2 \left (-1+\sqrt {2}\right )}}-\frac {\operatorname {Subst}\left (\int \frac {\sqrt {-1+\sqrt {2}}+\left (1-\frac {1}{\sqrt {2}}\right ) x}{\frac {1}{\sqrt {2}}+\sqrt {-1+\sqrt {2}} x+x^2} \, dx,x,\cot (x)\right )}{2 \sqrt {2 \left (-1+\sqrt {2}\right )}}\\ &=\frac {1}{8} \sqrt {-1+\sqrt {2}} \operatorname {Subst}\left (\int \frac {-\sqrt {-1+\sqrt {2}}+2 x}{\frac {1}{\sqrt {2}}-\sqrt {-1+\sqrt {2}} x+x^2} \, dx,x,\cot (x)\right )-\frac {1}{8} \sqrt {-1+\sqrt {2}} \operatorname {Subst}\left (\int \frac {\sqrt {-1+\sqrt {2}}+2 x}{\frac {1}{\sqrt {2}}+\sqrt {-1+\sqrt {2}} x+x^2} \, dx,x,\cot (x)\right )-\frac {1}{8} \sqrt {3+2 \sqrt {2}} \operatorname {Subst}\left (\int \frac {1}{\frac {1}{\sqrt {2}}-\sqrt {-1+\sqrt {2}} x+x^2} \, dx,x,\cot (x)\right )-\frac {1}{8} \sqrt {3+2 \sqrt {2}} \operatorname {Subst}\left (\int \frac {1}{\frac {1}{\sqrt {2}}+\sqrt {-1+\sqrt {2}} x+x^2} \, dx,x,\cot (x)\right )\\ &=\frac {1}{8} \sqrt {-1+\sqrt {2}} \log \left (\sqrt {2}-2 \sqrt {-1+\sqrt {2}} \cot (x)+2 \cot ^2(x)\right )-\frac {1}{8} \sqrt {-1+\sqrt {2}} \log \left (1+\sqrt {2 \left (-1+\sqrt {2}\right )} \cot (x)+\sqrt {2} \cot ^2(x)\right )+\frac {1}{4} \sqrt {3+2 \sqrt {2}} \operatorname {Subst}\left (\int \frac {1}{-1-\sqrt {2}-x^2} \, dx,x,-\sqrt {-1+\sqrt {2}}+2 \cot (x)\right )+\frac {1}{4} \sqrt {3+2 \sqrt {2}} \operatorname {Subst}\left (\int \frac {1}{-1-\sqrt {2}-x^2} \, dx,x,\sqrt {-1+\sqrt {2}}+2 \cot (x)\right )\\ &=\frac {1}{2} \sqrt {1+\sqrt {2}} x-\frac {1}{4} \sqrt {1+\sqrt {2}} \tan ^{-1}\left (\frac {\left (2-\sqrt {2}\right ) \cos (x) \sin (x)-\sqrt {-1+\sqrt {2}} \left (1-2 \sin ^2(x)\right )}{2+\sqrt {1+\sqrt {2}}+2 \sqrt {-1+\sqrt {2}} \cos (x) \sin (x)-\left (2-\sqrt {2}\right ) \sin ^2(x)}\right )-\frac {1}{4} \sqrt {1+\sqrt {2}} \tan ^{-1}\left (\frac {\left (2-\sqrt {2}\right ) \cos (x) \sin (x)+\sqrt {-1+\sqrt {2}} \left (1-2 \sin ^2(x)\right )}{2+\sqrt {1+\sqrt {2}}-2 \sqrt {-1+\sqrt {2}} \cos (x) \sin (x)-\left (2-\sqrt {2}\right ) \sin ^2(x)}\right )+\frac {1}{8} \sqrt {-1+\sqrt {2}} \log \left (\sqrt {2}-2 \sqrt {-1+\sqrt {2}} \cot (x)+2 \cot ^2(x)\right )-\frac {1}{8} \sqrt {-1+\sqrt {2}} \log \left (1+\sqrt {2 \left (-1+\sqrt {2}\right )} \cot (x)+\sqrt {2} \cot ^2(x)\right )\\ \end {align*}

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Mathematica [C]  time = 0.08, size = 45, normalized size = 0.15 \[ \frac {\tan ^{-1}\left (\frac {\tan (x)}{\sqrt {1-i}}\right )}{2 \sqrt {1-i}}+\frac {\tan ^{-1}\left (\frac {\tan (x)}{\sqrt {1+i}}\right )}{2 \sqrt {1+i}} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + Cos[x]^4)^(-1),x]

[Out]

ArcTan[Tan[x]/Sqrt[1 - I]]/(2*Sqrt[1 - I]) + ArcTan[Tan[x]/Sqrt[1 + I]]/(2*Sqrt[1 + I])

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fricas [B]  time = 42.42, size = 3830, normalized size = 13.12 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/32*2^(1/4)*sqrt(2*sqrt(2) + 4)*(sqrt(2) - 1)*log(-(4*sqrt(2) - 5)*cos(x)^4 + 4*(sqrt(2) - 1)*cos(x)^2 + (2^
(1/4)*(3*sqrt(2) - 4)*cos(x)^3 - 2^(1/4)*(sqrt(2) - 2)*cos(x))*sqrt(2*sqrt(2) + 4)*sin(x) + 1) + 1/32*2^(1/4)*
sqrt(2*sqrt(2) + 4)*(sqrt(2) - 1)*log(-(4*sqrt(2) - 5)*cos(x)^4 + 4*(sqrt(2) - 1)*cos(x)^2 - (2^(1/4)*(3*sqrt(
2) - 4)*cos(x)^3 - 2^(1/4)*(sqrt(2) - 2)*cos(x))*sqrt(2*sqrt(2) + 4)*sin(x) + 1) - 1/16*2^(1/4)*sqrt(2*sqrt(2)
+ 4)*arctan(1/4*(32*(sqrt(2)*(3*sqrt(2) + 2) - 2*sqrt(2) - 6)*cos(x)^16 - 16*(sqrt(2)*(19*sqrt(2) + 22) - 8*s
qrt(2) - 52)*cos(x)^14 + 32*(sqrt(2)*(8*sqrt(2) + 19) + 2*sqrt(2) - 37)*cos(x)^12 + 16*(2*sqrt(2)*(4*sqrt(2) -
13) - 22*sqrt(2) + 39)*cos(x)^10 - 8*(sqrt(2)*(41*sqrt(2) - 10) - 42*sqrt(2) - 2)*cos(x)^8 + 4*(sqrt(2)*(49*s
qrt(2) + 6) - 32*sqrt(2) - 32)*cos(x)^6 - 8*(sqrt(2)*(6*sqrt(2) + 1) - 2*sqrt(2) - 5)*cos(x)^4 + 2*(8*(2^(3/4)
*(2*sqrt(2) - 1) - 2*2^(1/4)*(3*sqrt(2) + 2))*cos(x)^15 - 8*(2^(3/4)*(3*sqrt(2) + 2) - 4*2^(1/4)*(4*sqrt(2) +
5))*cos(x)^13 - 4*(2*2^(3/4)*(3*sqrt(2) - 10) + 2^(1/4)*(19*sqrt(2) + 58))*cos(x)^11 + 4*(6*2^(3/4)*(3*sqrt(2)
- 4) - 2^(1/4)*(19*sqrt(2) - 32))*cos(x)^9 - 2*(2^(3/4)*(28*sqrt(2) - 27) - 4*2^(1/4)*(15*sqrt(2) - 2))*cos(x
)^7 + 2*(2^(3/4)*(9*sqrt(2) - 8) - 2*2^(1/4)*(15*sqrt(2) + 2))*cos(x)^5 - (2*2^(3/4)*(sqrt(2) - 1) - 2^(1/4)*(
13*sqrt(2) + 2))*cos(x)^3 - 2^(3/4)*cos(x))*sqrt(2*sqrt(2) + 4)*sin(x) + 4*cos(x)^2 + (16*(sqrt(2)*(5*sqrt(2)
- 6) - 8*sqrt(2) + 4)*cos(x)^14 - 56*(sqrt(2)*(5*sqrt(2) - 6) - 8*sqrt(2) + 4)*cos(x)^12 + 8*(sqrt(2)*(49*sqrt
(2) - 62) - 76*sqrt(2) + 54)*cos(x)^10 - 40*(sqrt(2)*(7*sqrt(2) - 10) - 10*sqrt(2) + 13)*cos(x)^8 + 4*(sqrt(2)
*(27*sqrt(2) - 46) - 32*sqrt(2) + 92)*cos(x)^6 - 2*(11*sqrt(2)*(sqrt(2) - 2) - 8*sqrt(2) + 72)*cos(x)^4 + 2*(s
qrt(2)*(sqrt(2) - 2) + 14)*cos(x)^2 + (8*(2^(3/4)*(8*sqrt(2) - 11) - 2*2^(1/4)*(5*sqrt(2) - 6))*cos(x)^13 - 24
*(2^(3/4)*(8*sqrt(2) - 11) - 2*2^(1/4)*(5*sqrt(2) - 6))*cos(x)^11 + 4*(2*2^(3/4)*(28*sqrt(2) - 39) - 2^(1/4)*(
73*sqrt(2) - 94))*cos(x)^9 - 8*(2^(3/4)*(16*sqrt(2) - 23) - 2^(1/4)*(23*sqrt(2) - 34))*cos(x)^7 + 2*(9*2^(3/4)
*(2*sqrt(2) - 3) - 8*2^(1/4)*(4*sqrt(2) - 7))*cos(x)^5 - 2*(2^(3/4)*(2*sqrt(2) - 3) - 6*2^(1/4)*(sqrt(2) - 2))
*cos(x)^3 - 2^(1/4)*(sqrt(2) - 2)*cos(x))*sqrt(2*sqrt(2) + 4)*sin(x) - 2)*sqrt(-4*(4*sqrt(2) - 5)*cos(x)^4 + 1
6*(sqrt(2) - 1)*cos(x)^2 + 4*(2^(1/4)*(3*sqrt(2) - 4)*cos(x)^3 - 2^(1/4)*(sqrt(2) - 2)*cos(x))*sqrt(2*sqrt(2)
+ 4)*sin(x) + 4))/(112*cos(x)^16 - 448*cos(x)^14 + 608*cos(x)^12 - 256*cos(x)^10 - 152*cos(x)^8 + 208*cos(x)^6
- 88*cos(x)^4 + 16*cos(x)^2 - 1)) + 1/16*2^(1/4)*sqrt(2*sqrt(2) + 4)*arctan(-1/4*(32*(sqrt(2)*(3*sqrt(2) + 2)
- 2*sqrt(2) - 6)*cos(x)^16 - 16*(sqrt(2)*(19*sqrt(2) + 22) - 8*sqrt(2) - 52)*cos(x)^14 + 32*(sqrt(2)*(8*sqrt(
2) + 19) + 2*sqrt(2) - 37)*cos(x)^12 + 16*(2*sqrt(2)*(4*sqrt(2) - 13) - 22*sqrt(2) + 39)*cos(x)^10 - 8*(sqrt(2
)*(41*sqrt(2) - 10) - 42*sqrt(2) - 2)*cos(x)^8 + 4*(sqrt(2)*(49*sqrt(2) + 6) - 32*sqrt(2) - 32)*cos(x)^6 - 8*(
sqrt(2)*(6*sqrt(2) + 1) - 2*sqrt(2) - 5)*cos(x)^4 + 2*(8*(2^(3/4)*(2*sqrt(2) - 1) - 2*2^(1/4)*(3*sqrt(2) + 2))
*cos(x)^15 - 8*(2^(3/4)*(3*sqrt(2) + 2) - 4*2^(1/4)*(4*sqrt(2) + 5))*cos(x)^13 - 4*(2*2^(3/4)*(3*sqrt(2) - 10)
+ 2^(1/4)*(19*sqrt(2) + 58))*cos(x)^11 + 4*(6*2^(3/4)*(3*sqrt(2) - 4) - 2^(1/4)*(19*sqrt(2) - 32))*cos(x)^9 -
2*(2^(3/4)*(28*sqrt(2) - 27) - 4*2^(1/4)*(15*sqrt(2) - 2))*cos(x)^7 + 2*(2^(3/4)*(9*sqrt(2) - 8) - 2*2^(1/4)*
(15*sqrt(2) + 2))*cos(x)^5 - (2*2^(3/4)*(sqrt(2) - 1) - 2^(1/4)*(13*sqrt(2) + 2))*cos(x)^3 - 2^(3/4)*cos(x))*s
qrt(2*sqrt(2) + 4)*sin(x) + 4*cos(x)^2 - (16*(sqrt(2)*(5*sqrt(2) - 6) - 8*sqrt(2) + 4)*cos(x)^14 - 56*(sqrt(2)
*(5*sqrt(2) - 6) - 8*sqrt(2) + 4)*cos(x)^12 + 8*(sqrt(2)*(49*sqrt(2) - 62) - 76*sqrt(2) + 54)*cos(x)^10 - 40*(
sqrt(2)*(7*sqrt(2) - 10) - 10*sqrt(2) + 13)*cos(x)^8 + 4*(sqrt(2)*(27*sqrt(2) - 46) - 32*sqrt(2) + 92)*cos(x)^
6 - 2*(11*sqrt(2)*(sqrt(2) - 2) - 8*sqrt(2) + 72)*cos(x)^4 + 2*(sqrt(2)*(sqrt(2) - 2) + 14)*cos(x)^2 + (8*(2^(
3/4)*(8*sqrt(2) - 11) - 2*2^(1/4)*(5*sqrt(2) - 6))*cos(x)^13 - 24*(2^(3/4)*(8*sqrt(2) - 11) - 2*2^(1/4)*(5*sqr
t(2) - 6))*cos(x)^11 + 4*(2*2^(3/4)*(28*sqrt(2) - 39) - 2^(1/4)*(73*sqrt(2) - 94))*cos(x)^9 - 8*(2^(3/4)*(16*s
qrt(2) - 23) - 2^(1/4)*(23*sqrt(2) - 34))*cos(x)^7 + 2*(9*2^(3/4)*(2*sqrt(2) - 3) - 8*2^(1/4)*(4*sqrt(2) - 7))
*cos(x)^5 - 2*(2^(3/4)*(2*sqrt(2) - 3) - 6*2^(1/4)*(sqrt(2) - 2))*cos(x)^3 - 2^(1/4)*(sqrt(2) - 2)*cos(x))*sqr
t(2*sqrt(2) + 4)*sin(x) - 2)*sqrt(-4*(4*sqrt(2) - 5)*cos(x)^4 + 16*(sqrt(2) - 1)*cos(x)^2 + 4*(2^(1/4)*(3*sqrt
(2) - 4)*cos(x)^3 - 2^(1/4)*(sqrt(2) - 2)*cos(x))*sqrt(2*sqrt(2) + 4)*sin(x) + 4))/(112*cos(x)^16 - 448*cos(x)
^14 + 608*cos(x)^12 - 256*cos(x)^10 - 152*cos(x)^8 + 208*cos(x)^6 - 88*cos(x)^4 + 16*cos(x)^2 - 1)) - 1/16*2^(
1/4)*sqrt(2*sqrt(2) + 4)*arctan(-1/4*(32*(sqrt(2)*(3*sqrt(2) + 2) - 2*sqrt(2) - 6)*cos(x)^16 - 16*(sqrt(2)*(19
*sqrt(2) + 22) - 8*sqrt(2) - 52)*cos(x)^14 + 32*(sqrt(2)*(8*sqrt(2) + 19) + 2*sqrt(2) - 37)*cos(x)^12 + 16*(2*
sqrt(2)*(4*sqrt(2) - 13) - 22*sqrt(2) + 39)*cos(x)^10 - 8*(sqrt(2)*(41*sqrt(2) - 10) - 42*sqrt(2) - 2)*cos(x)^
8 + 4*(sqrt(2)*(49*sqrt(2) + 6) - 32*sqrt(2) - 32)*cos(x)^6 - 8*(sqrt(2)*(6*sqrt(2) + 1) - 2*sqrt(2) - 5)*cos(
x)^4 - 2*(8*(2^(3/4)*(2*sqrt(2) - 1) - 2*2^(1/4)*(3*sqrt(2) + 2))*cos(x)^15 - 8*(2^(3/4)*(3*sqrt(2) + 2) - 4*2
^(1/4)*(4*sqrt(2) + 5))*cos(x)^13 - 4*(2*2^(3/4)*(3*sqrt(2) - 10) + 2^(1/4)*(19*sqrt(2) + 58))*cos(x)^11 + 4*(
6*2^(3/4)*(3*sqrt(2) - 4) - 2^(1/4)*(19*sqrt(2) - 32))*cos(x)^9 - 2*(2^(3/4)*(28*sqrt(2) - 27) - 4*2^(1/4)*(15
*sqrt(2) - 2))*cos(x)^7 + 2*(2^(3/4)*(9*sqrt(2) - 8) - 2*2^(1/4)*(15*sqrt(2) + 2))*cos(x)^5 - (2*2^(3/4)*(sqrt
(2) - 1) - 2^(1/4)*(13*sqrt(2) + 2))*cos(x)^3 - 2^(3/4)*cos(x))*sqrt(2*sqrt(2) + 4)*sin(x) + 4*cos(x)^2 + (16*
(sqrt(2)*(5*sqrt(2) - 6) - 8*sqrt(2) + 4)*cos(x)^14 - 56*(sqrt(2)*(5*sqrt(2) - 6) - 8*sqrt(2) + 4)*cos(x)^12 +
8*(sqrt(2)*(49*sqrt(2) - 62) - 76*sqrt(2) + 54)*cos(x)^10 - 40*(sqrt(2)*(7*sqrt(2) - 10) - 10*sqrt(2) + 13)*c
os(x)^8 + 4*(sqrt(2)*(27*sqrt(2) - 46) - 32*sqrt(2) + 92)*cos(x)^6 - 2*(11*sqrt(2)*(sqrt(2) - 2) - 8*sqrt(2) +
72)*cos(x)^4 + 2*(sqrt(2)*(sqrt(2) - 2) + 14)*cos(x)^2 - (8*(2^(3/4)*(8*sqrt(2) - 11) - 2*2^(1/4)*(5*sqrt(2)
- 6))*cos(x)^13 - 24*(2^(3/4)*(8*sqrt(2) - 11) - 2*2^(1/4)*(5*sqrt(2) - 6))*cos(x)^11 + 4*(2*2^(3/4)*(28*sqrt(
2) - 39) - 2^(1/4)*(73*sqrt(2) - 94))*cos(x)^9 - 8*(2^(3/4)*(16*sqrt(2) - 23) - 2^(1/4)*(23*sqrt(2) - 34))*cos
(x)^7 + 2*(9*2^(3/4)*(2*sqrt(2) - 3) - 8*2^(1/4)*(4*sqrt(2) - 7))*cos(x)^5 - 2*(2^(3/4)*(2*sqrt(2) - 3) - 6*2^
(1/4)*(sqrt(2) - 2))*cos(x)^3 - 2^(1/4)*(sqrt(2) - 2)*cos(x))*sqrt(2*sqrt(2) + 4)*sin(x) - 2)*sqrt(-4*(4*sqrt(
2) - 5)*cos(x)^4 + 16*(sqrt(2) - 1)*cos(x)^2 - 4*(2^(1/4)*(3*sqrt(2) - 4)*cos(x)^3 - 2^(1/4)*(sqrt(2) - 2)*cos
(x))*sqrt(2*sqrt(2) + 4)*sin(x) + 4))/(112*cos(x)^16 - 448*cos(x)^14 + 608*cos(x)^12 - 256*cos(x)^10 - 152*cos
(x)^8 + 208*cos(x)^6 - 88*cos(x)^4 + 16*cos(x)^2 - 1)) + 1/16*2^(1/4)*sqrt(2*sqrt(2) + 4)*arctan(1/4*(32*(sqrt
(2)*(3*sqrt(2) + 2) - 2*sqrt(2) - 6)*cos(x)^16 - 16*(sqrt(2)*(19*sqrt(2) + 22) - 8*sqrt(2) - 52)*cos(x)^14 + 3
2*(sqrt(2)*(8*sqrt(2) + 19) + 2*sqrt(2) - 37)*cos(x)^12 + 16*(2*sqrt(2)*(4*sqrt(2) - 13) - 22*sqrt(2) + 39)*co
s(x)^10 - 8*(sqrt(2)*(41*sqrt(2) - 10) - 42*sqrt(2) - 2)*cos(x)^8 + 4*(sqrt(2)*(49*sqrt(2) + 6) - 32*sqrt(2) -
32)*cos(x)^6 - 8*(sqrt(2)*(6*sqrt(2) + 1) - 2*sqrt(2) - 5)*cos(x)^4 - 2*(8*(2^(3/4)*(2*sqrt(2) - 1) - 2*2^(1/
4)*(3*sqrt(2) + 2))*cos(x)^15 - 8*(2^(3/4)*(3*sqrt(2) + 2) - 4*2^(1/4)*(4*sqrt(2) + 5))*cos(x)^13 - 4*(2*2^(3/
4)*(3*sqrt(2) - 10) + 2^(1/4)*(19*sqrt(2) + 58))*cos(x)^11 + 4*(6*2^(3/4)*(3*sqrt(2) - 4) - 2^(1/4)*(19*sqrt(2
) - 32))*cos(x)^9 - 2*(2^(3/4)*(28*sqrt(2) - 27) - 4*2^(1/4)*(15*sqrt(2) - 2))*cos(x)^7 + 2*(2^(3/4)*(9*sqrt(2
) - 8) - 2*2^(1/4)*(15*sqrt(2) + 2))*cos(x)^5 - (2*2^(3/4)*(sqrt(2) - 1) - 2^(1/4)*(13*sqrt(2) + 2))*cos(x)^3
- 2^(3/4)*cos(x))*sqrt(2*sqrt(2) + 4)*sin(x) + 4*cos(x)^2 - (16*(sqrt(2)*(5*sqrt(2) - 6) - 8*sqrt(2) + 4)*cos(
x)^14 - 56*(sqrt(2)*(5*sqrt(2) - 6) - 8*sqrt(2) + 4)*cos(x)^12 + 8*(sqrt(2)*(49*sqrt(2) - 62) - 76*sqrt(2) + 5
4)*cos(x)^10 - 40*(sqrt(2)*(7*sqrt(2) - 10) - 10*sqrt(2) + 13)*cos(x)^8 + 4*(sqrt(2)*(27*sqrt(2) - 46) - 32*sq
rt(2) + 92)*cos(x)^6 - 2*(11*sqrt(2)*(sqrt(2) - 2) - 8*sqrt(2) + 72)*cos(x)^4 + 2*(sqrt(2)*(sqrt(2) - 2) + 14)
*cos(x)^2 - (8*(2^(3/4)*(8*sqrt(2) - 11) - 2*2^(1/4)*(5*sqrt(2) - 6))*cos(x)^13 - 24*(2^(3/4)*(8*sqrt(2) - 11)
- 2*2^(1/4)*(5*sqrt(2) - 6))*cos(x)^11 + 4*(2*2^(3/4)*(28*sqrt(2) - 39) - 2^(1/4)*(73*sqrt(2) - 94))*cos(x)^9
- 8*(2^(3/4)*(16*sqrt(2) - 23) - 2^(1/4)*(23*sqrt(2) - 34))*cos(x)^7 + 2*(9*2^(3/4)*(2*sqrt(2) - 3) - 8*2^(1/
4)*(4*sqrt(2) - 7))*cos(x)^5 - 2*(2^(3/4)*(2*sqrt(2) - 3) - 6*2^(1/4)*(sqrt(2) - 2))*cos(x)^3 - 2^(1/4)*(sqrt(
2) - 2)*cos(x))*sqrt(2*sqrt(2) + 4)*sin(x) - 2)*sqrt(-4*(4*sqrt(2) - 5)*cos(x)^4 + 16*(sqrt(2) - 1)*cos(x)^2 -
4*(2^(1/4)*(3*sqrt(2) - 4)*cos(x)^3 - 2^(1/4)*(sqrt(2) - 2)*cos(x))*sqrt(2*sqrt(2) + 4)*sin(x) + 4))/(112*cos
(x)^16 - 448*cos(x)^14 + 608*cos(x)^12 - 256*cos(x)^10 - 152*cos(x)^8 + 208*cos(x)^6 - 88*cos(x)^4 + 16*cos(x)
^2 - 1))

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giac [A]  time = 0.67, size = 170, normalized size = 0.58 \[ \frac {1}{4} \, {\left (\pi \left \lfloor \frac {x}{\pi } + \frac {1}{2} \right \rfloor + \arctan \left (\frac {2^{\frac {3}{4}} {\left (2^{\frac {1}{4}} \sqrt {-\sqrt {2} + 2} + 2 \, \tan \relax (x)\right )}}{2 \, \sqrt {\sqrt {2} + 2}}\right )\right )} \sqrt {\sqrt {2} + 1} + \frac {1}{4} \, {\left (\pi \left \lfloor \frac {x}{\pi } + \frac {1}{2} \right \rfloor + \arctan \left (-\frac {2^{\frac {3}{4}} {\left (2^{\frac {1}{4}} \sqrt {-\sqrt {2} + 2} - 2 \, \tan \relax (x)\right )}}{2 \, \sqrt {\sqrt {2} + 2}}\right )\right )} \sqrt {\sqrt {2} + 1} - \frac {1}{8} \, \sqrt {\sqrt {2} - 1} \log \left (\tan \relax (x)^{2} + 2^{\frac {1}{4}} \sqrt {-\sqrt {2} + 2} \tan \relax (x) + \sqrt {2}\right ) + \frac {1}{8} \, \sqrt {\sqrt {2} - 1} \log \left (\tan \relax (x)^{2} - 2^{\frac {1}{4}} \sqrt {-\sqrt {2} + 2} \tan \relax (x) + \sqrt {2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(pi*floor(x/pi + 1/2) + arctan(1/2*2^(3/4)*(2^(1/4)*sqrt(-sqrt(2) + 2) + 2*tan(x))/sqrt(sqrt(2) + 2)))*sqr
t(sqrt(2) + 1) + 1/4*(pi*floor(x/pi + 1/2) + arctan(-1/2*2^(3/4)*(2^(1/4)*sqrt(-sqrt(2) + 2) - 2*tan(x))/sqrt(
sqrt(2) + 2)))*sqrt(sqrt(2) + 1) - 1/8*sqrt(sqrt(2) - 1)*log(tan(x)^2 + 2^(1/4)*sqrt(-sqrt(2) + 2)*tan(x) + sq
rt(2)) + 1/8*sqrt(sqrt(2) - 1)*log(tan(x)^2 - 2^(1/4)*sqrt(-sqrt(2) + 2)*tan(x) + sqrt(2))

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maple [A]  time = 0.22, size = 227, normalized size = 0.78 \[ \frac {\sqrt {2}\, \sqrt {-2+2 \sqrt {2}}\, \ln \left (\tan ^{2}\relax (x )-\tan \relax (x ) \sqrt {-2+2 \sqrt {2}}+\sqrt {2}\right )}{16}+\frac {\arctan \left (\frac {2 \tan \relax (x )-\sqrt {-2+2 \sqrt {2}}}{\sqrt {2 \sqrt {2}+2}}\right ) \sqrt {2}}{4 \sqrt {2 \sqrt {2}+2}}+\frac {\arctan \left (\frac {2 \tan \relax (x )-\sqrt {-2+2 \sqrt {2}}}{\sqrt {2 \sqrt {2}+2}}\right )}{2 \sqrt {2 \sqrt {2}+2}}-\frac {\sqrt {2}\, \sqrt {-2+2 \sqrt {2}}\, \ln \left (\tan ^{2}\relax (x )+\tan \relax (x ) \sqrt {-2+2 \sqrt {2}}+\sqrt {2}\right )}{16}+\frac {\arctan \left (\frac {2 \tan \relax (x )+\sqrt {-2+2 \sqrt {2}}}{\sqrt {2 \sqrt {2}+2}}\right ) \sqrt {2}}{4 \sqrt {2 \sqrt {2}+2}}+\frac {\arctan \left (\frac {2 \tan \relax (x )+\sqrt {-2+2 \sqrt {2}}}{\sqrt {2 \sqrt {2}+2}}\right )}{2 \sqrt {2 \sqrt {2}+2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1+cos(x)^4),x)

[Out]

1/16*2^(1/2)*(-2+2*2^(1/2))^(1/2)*ln(tan(x)^2-tan(x)*(-2+2*2^(1/2))^(1/2)+2^(1/2))+1/4/(2*2^(1/2)+2)^(1/2)*arc
tan((2*tan(x)-(-2+2*2^(1/2))^(1/2))/(2*2^(1/2)+2)^(1/2))*2^(1/2)+1/2/(2*2^(1/2)+2)^(1/2)*arctan((2*tan(x)-(-2+
2*2^(1/2))^(1/2))/(2*2^(1/2)+2)^(1/2))-1/16*2^(1/2)*(-2+2*2^(1/2))^(1/2)*ln(tan(x)^2+tan(x)*(-2+2*2^(1/2))^(1/
2)+2^(1/2))+1/4/(2*2^(1/2)+2)^(1/2)*arctan((2*tan(x)+(-2+2*2^(1/2))^(1/2))/(2*2^(1/2)+2)^(1/2))*2^(1/2)+1/2/(2
*2^(1/2)+2)^(1/2)*arctan((2*tan(x)+(-2+2*2^(1/2))^(1/2))/(2*2^(1/2)+2)^(1/2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(cos(x)^4 + 1), x)

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mupad [B]  time = 2.73, size = 214, normalized size = 0.73 \[ \mathrm {atanh}\left (\frac {4\,\sqrt {2}\,\mathrm {tan}\relax (x)\,\sqrt {-\frac {\sqrt {2}}{64}-\frac {1}{64}}}{64\,\sqrt {\frac {\sqrt {2}}{64}-\frac {1}{64}}\,\sqrt {-\frac {\sqrt {2}}{64}-\frac {1}{64}}-1}+\frac {4\,\sqrt {2}\,\mathrm {tan}\relax (x)\,\sqrt {\frac {\sqrt {2}}{64}-\frac {1}{64}}}{64\,\sqrt {\frac {\sqrt {2}}{64}-\frac {1}{64}}\,\sqrt {-\frac {\sqrt {2}}{64}-\frac {1}{64}}-1}\right )\,\left (2\,\sqrt {-\frac {\sqrt {2}}{64}-\frac {1}{64}}-2\,\sqrt {\frac {\sqrt {2}}{64}-\frac {1}{64}}\right )-\mathrm {atanh}\left (\frac {4\,\sqrt {2}\,\mathrm {tan}\relax (x)\,\sqrt {-\frac {\sqrt {2}}{64}-\frac {1}{64}}}{64\,\sqrt {\frac {\sqrt {2}}{64}-\frac {1}{64}}\,\sqrt {-\frac {\sqrt {2}}{64}-\frac {1}{64}}+1}-\frac {4\,\sqrt {2}\,\mathrm {tan}\relax (x)\,\sqrt {\frac {\sqrt {2}}{64}-\frac {1}{64}}}{64\,\sqrt {\frac {\sqrt {2}}{64}-\frac {1}{64}}\,\sqrt {-\frac {\sqrt {2}}{64}-\frac {1}{64}}+1}\right )\,\left (2\,\sqrt {-\frac {\sqrt {2}}{64}-\frac {1}{64}}+2\,\sqrt {\frac {\sqrt {2}}{64}-\frac {1}{64}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cos(x)^4 + 1),x)

[Out]

atanh((4*2^(1/2)*tan(x)*(- 2^(1/2)/64 - 1/64)^(1/2))/(64*(2^(1/2)/64 - 1/64)^(1/2)*(- 2^(1/2)/64 - 1/64)^(1/2)
- 1) + (4*2^(1/2)*tan(x)*(2^(1/2)/64 - 1/64)^(1/2))/(64*(2^(1/2)/64 - 1/64)^(1/2)*(- 2^(1/2)/64 - 1/64)^(1/2)
- 1))*(2*(- 2^(1/2)/64 - 1/64)^(1/2) - 2*(2^(1/2)/64 - 1/64)^(1/2)) - atanh((4*2^(1/2)*tan(x)*(- 2^(1/2)/64 -
1/64)^(1/2))/(64*(2^(1/2)/64 - 1/64)^(1/2)*(- 2^(1/2)/64 - 1/64)^(1/2) + 1) - (4*2^(1/2)*tan(x)*(2^(1/2)/64 -
1/64)^(1/2))/(64*(2^(1/2)/64 - 1/64)^(1/2)*(- 2^(1/2)/64 - 1/64)^(1/2) + 1))*(2*(- 2^(1/2)/64 - 1/64)^(1/2) +
2*(2^(1/2)/64 - 1/64)^(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/(1+cos(x)**4),x)

[Out]

Timed out

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