3.238 \(\int \frac {1}{1+\sin ^4(x)} \, dx\)

Optimal. Leaf size=309 \[ \frac {x}{2 \sqrt {\sqrt {2}-1}}-\frac {1}{8} \sqrt {\sqrt {2}-1} \log \left (2 \tan ^2(x)-2 \sqrt {\sqrt {2}-1} \tan (x)+\sqrt {2}\right )+\frac {1}{8} \sqrt {\sqrt {2}-1} \log \left (\sqrt {2} \tan ^2(x)+\sqrt {2 \left (\sqrt {2}-1\right )} \tan (x)+1\right )+\frac {\tan ^{-1}\left (\frac {-2 \sqrt {\sqrt {2}-1} \cos ^2(x)-\left (\sqrt {2}-2\right ) \sin (x) \cos (x)+\sqrt {\sqrt {2}-1}}{\left (\sqrt {2}-2\right ) \cos ^2(x)-2 \sqrt {\sqrt {2}-1} \sin (x) \cos (x)+\sqrt {1+\sqrt {2}}+2}\right )}{4 \sqrt {\sqrt {2}-1}}-\frac {\tan ^{-1}\left (\frac {-2 \sqrt {\sqrt {2}-1} \cos ^2(x)+\left (\sqrt {2}-2\right ) \sin (x) \cos (x)+\sqrt {\sqrt {2}-1}}{\left (\sqrt {2}-2\right ) \cos ^2(x)+2 \sqrt {\sqrt {2}-1} \sin (x) \cos (x)+\sqrt {1+\sqrt {2}}+2}\right )}{4 \sqrt {\sqrt {2}-1}} \]

[Out]

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

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Rubi [A]  time = 0.20, antiderivative size = 309, 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} \[ \frac {x}{2 \sqrt {\sqrt {2}-1}}-\frac {1}{8} \sqrt {\sqrt {2}-1} \log \left (2 \tan ^2(x)-2 \sqrt {\sqrt {2}-1} \tan (x)+\sqrt {2}\right )+\frac {1}{8} \sqrt {\sqrt {2}-1} \log \left (\sqrt {2} \tan ^2(x)+\sqrt {2 \left (\sqrt {2}-1\right )} \tan (x)+1\right )+\frac {\tan ^{-1}\left (\frac {-2 \sqrt {\sqrt {2}-1} \cos ^2(x)-\left (\sqrt {2}-2\right ) \sin (x) \cos (x)+\sqrt {\sqrt {2}-1}}{\left (\sqrt {2}-2\right ) \cos ^2(x)-2 \sqrt {\sqrt {2}-1} \sin (x) \cos (x)+\sqrt {1+\sqrt {2}}+2}\right )}{4 \sqrt {\sqrt {2}-1}}-\frac {\tan ^{-1}\left (\frac {-2 \sqrt {\sqrt {2}-1} \cos ^2(x)+\left (\sqrt {2}-2\right ) \sin (x) \cos (x)+\sqrt {\sqrt {2}-1}}{\left (\sqrt {2}-2\right ) \cos ^2(x)+2 \sqrt {\sqrt {2}-1} \sin (x) \cos (x)+\sqrt {1+\sqrt {2}}+2}\right )}{4 \sqrt {\sqrt {2}-1}} \]

Antiderivative was successfully verified.

[In]

Int[(1 + Sin[x]^4)^(-1),x]

[Out]

x/(2*Sqrt[-1 + Sqrt[2]]) + ArcTan[(Sqrt[-1 + Sqrt[2]] - 2*Sqrt[-1 + Sqrt[2]]*Cos[x]^2 - (-2 + Sqrt[2])*Cos[x]*
Sin[x])/(2 + Sqrt[1 + Sqrt[2]] + (-2 + Sqrt[2])*Cos[x]^2 - 2*Sqrt[-1 + Sqrt[2]]*Cos[x]*Sin[x])]/(4*Sqrt[-1 + S
qrt[2]]) - ArcTan[(Sqrt[-1 + Sqrt[2]] - 2*Sqrt[-1 + Sqrt[2]]*Cos[x]^2 + (-2 + Sqrt[2])*Cos[x]*Sin[x])/(2 + Sqr
t[1 + Sqrt[2]] + (-2 + Sqrt[2])*Cos[x]^2 + 2*Sqrt[-1 + Sqrt[2]]*Cos[x]*Sin[x])]/(4*Sqrt[-1 + Sqrt[2]]) - (Sqrt
[-1 + Sqrt[2]]*Log[Sqrt[2] - 2*Sqrt[-1 + Sqrt[2]]*Tan[x] + 2*Tan[x]^2])/8 + (Sqrt[-1 + Sqrt[2]]*Log[1 + Sqrt[2
*(-1 + Sqrt[2])]*Tan[x] + Sqrt[2]*Tan[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+\sin ^4(x)} \, dx &=\operatorname {Subst}\left (\int \frac {1+x^2}{1+2 x^2+2 x^4} \, dx,x,\tan (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,\tan (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,\tan (x)\right )}{2 \sqrt {2 \left (-1+\sqrt {2}\right )}}\\ &=-\left (\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,\tan (x)\right )\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,\tan (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,\tan (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,\tan (x)\right )\\ &=-\frac {1}{8} \sqrt {-1+\sqrt {2}} \log \left (\sqrt {2}-2 \sqrt {-1+\sqrt {2}} \tan (x)+2 \tan ^2(x)\right )+\frac {1}{8} \sqrt {-1+\sqrt {2}} \log \left (1+\sqrt {2 \left (-1+\sqrt {2}\right )} \tan (x)+\sqrt {2} \tan ^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 \tan (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 \tan (x)\right )\\ &=\frac {1}{2} \sqrt {1+\sqrt {2}} x+\frac {1}{4} \sqrt {1+\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {-1+\sqrt {2}}-2 \sqrt {-1+\sqrt {2}} \cos ^2(x)+\left (2-\sqrt {2}\right ) \cos (x) \sin (x)}{2+\sqrt {1+\sqrt {2}}-\left (2-\sqrt {2}\right ) \cos ^2(x)-2 \sqrt {-1+\sqrt {2}} \cos (x) \sin (x)}\right )-\frac {1}{4} \sqrt {1+\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {-1+\sqrt {2}}-2 \sqrt {-1+\sqrt {2}} \cos ^2(x)-\left (2-\sqrt {2}\right ) \cos (x) \sin (x)}{2+\sqrt {1+\sqrt {2}}-\left (2-\sqrt {2}\right ) \cos ^2(x)+2 \sqrt {-1+\sqrt {2}} \cos (x) \sin (x)}\right )-\frac {1}{8} \sqrt {-1+\sqrt {2}} \log \left (\sqrt {2}-2 \sqrt {-1+\sqrt {2}} \tan (x)+2 \tan ^2(x)\right )+\frac {1}{8} \sqrt {-1+\sqrt {2}} \log \left (1+\sqrt {2 \left (-1+\sqrt {2}\right )} \tan (x)+\sqrt {2} \tan ^2(x)\right )\\ \end {align*}

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Mathematica [C]  time = 0.07, size = 45, normalized size = 0.15 \[ \frac {\tan ^{-1}\left (\sqrt {1-i} \tan (x)\right )}{2 \sqrt {1-i}}+\frac {\tan ^{-1}\left (\sqrt {1+i} \tan (x)\right )}{2 \sqrt {1+i}} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + Sin[x]^4)^(-1),x]

[Out]

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

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fricas [B]  time = 32.59, size = 3830, normalized size = 12.39 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+sin(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 + 2*(2*sqrt(2) - 3)*cos(x)^2 + (
2^(1/4)*(3*sqrt(2) - 4)*cos(x)^3 - 2*2^(1/4)*(sqrt(2) - 1)*cos(x))*sqrt(2*sqrt(2) + 4)*sin(x) + 2) + 1/32*2^(1
/4)*sqrt(2*sqrt(2) + 4)*(sqrt(2) - 1)*log(-(4*sqrt(2) - 5)*cos(x)^4 + 2*(2*sqrt(2) - 3)*cos(x)^2 - (2^(1/4)*(3
*sqrt(2) - 4)*cos(x)^3 - 2*2^(1/4)*(sqrt(2) - 1)*cos(x))*sqrt(2*sqrt(2) + 4)*sin(x) + 2) - 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)*(29*sqrt(2) + 1
0) - 24*sqrt(2) - 44)*cos(x)^14 + 16*(sqrt(2)*(51*sqrt(2) - 4) - 52*sqrt(2) - 46)*cos(x)^12 - 16*(sqrt(2)*(41*
sqrt(2) - 36) - 54*sqrt(2) + 15)*cos(x)^10 + 8*(sqrt(2)*(29*sqrt(2) - 90) - 58*sqrt(2) + 132)*cos(x)^8 - 4*(sq
rt(2)*(5*sqrt(2) - 98) - 32*sqrt(2) + 216)*cos(x)^6 - 4*(sqrt(2)*(sqrt(2) + 24) + 4*sqrt(2) - 82)*cos(x)^4 + 4
*(2*sqrt(2) - 15)*cos(x)^2 + 2*(8*(2^(3/4)*(2*sqrt(2) - 1) - 2*2^(1/4)*(3*sqrt(2) + 2))*cos(x)^15 - 8*(2^(3/4)
*(11*sqrt(2) - 9) - 2*2^(1/4)*(13*sqrt(2) + 4))*cos(x)^13 + 4*(2*2^(3/4)*(21*sqrt(2) - 23) - 2^(1/4)*(79*sqrt(
2) - 14))*cos(x)^11 - 8*(2^(3/4)*(19*sqrt(2) - 27) - 2^(1/4)*(27*sqrt(2) - 31))*cos(x)^9 + 2*(2^(3/4)*(36*sqrt
(2) - 65) - 32*2^(1/4)*(sqrt(2) - 4))*cos(x)^7 - 2*(2^(3/4)*(9*sqrt(2) - 19) - 2*2^(1/4)*(sqrt(2) - 30))*cos(x
)^5 + (2*2^(3/4)*(sqrt(2) - 2) + 2^(1/4)*(sqrt(2) + 26))*cos(x)^3 - 2*2^(1/4)*cos(x))*sqrt(2*sqrt(2) + 4)*sin(
x) + (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)*co
s(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*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 + 8*(2*sqrt(2) - 3)*cos(x)^2 + 4*(2^(1/4)*(3*sqrt(2) - 4)*cos(x)^3 - 2*2^(1/4)*(sqrt
(2) - 1)*cos(x))*sqrt(2*sqrt(2) + 4)*sin(x) + 8) + 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)*arct
an(-1/4*(32*(sqrt(2)*(3*sqrt(2) + 2) - 2*sqrt(2) - 6)*cos(x)^16 - 16*(sqrt(2)*(29*sqrt(2) + 10) - 24*sqrt(2) -
44)*cos(x)^14 + 16*(sqrt(2)*(51*sqrt(2) - 4) - 52*sqrt(2) - 46)*cos(x)^12 - 16*(sqrt(2)*(41*sqrt(2) - 36) - 5
4*sqrt(2) + 15)*cos(x)^10 + 8*(sqrt(2)*(29*sqrt(2) - 90) - 58*sqrt(2) + 132)*cos(x)^8 - 4*(sqrt(2)*(5*sqrt(2)
- 98) - 32*sqrt(2) + 216)*cos(x)^6 - 4*(sqrt(2)*(sqrt(2) + 24) + 4*sqrt(2) - 82)*cos(x)^4 + 4*(2*sqrt(2) - 15)
*cos(x)^2 + 2*(8*(2^(3/4)*(2*sqrt(2) - 1) - 2*2^(1/4)*(3*sqrt(2) + 2))*cos(x)^15 - 8*(2^(3/4)*(11*sqrt(2) - 9)
- 2*2^(1/4)*(13*sqrt(2) + 4))*cos(x)^13 + 4*(2*2^(3/4)*(21*sqrt(2) - 23) - 2^(1/4)*(79*sqrt(2) - 14))*cos(x)^
11 - 8*(2^(3/4)*(19*sqrt(2) - 27) - 2^(1/4)*(27*sqrt(2) - 31))*cos(x)^9 + 2*(2^(3/4)*(36*sqrt(2) - 65) - 32*2^
(1/4)*(sqrt(2) - 4))*cos(x)^7 - 2*(2^(3/4)*(9*sqrt(2) - 19) - 2*2^(1/4)*(sqrt(2) - 30))*cos(x)^5 + (2*2^(3/4)*
(sqrt(2) - 2) + 2^(1/4)*(sqrt(2) + 26))*cos(x)^3 - 2*2^(1/4)*cos(x))*sqrt(2*sqrt(2) + 4)*sin(x) - (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))*co
s(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)*(s
qrt(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 + 8*(2*sqrt(2) - 3)*cos(x)^2 + 4*(2^(1/4)*(3*sqrt(2) - 4)*cos(x)^3 - 2*2^(1/4)*(sqrt(2) - 1)*cos(x))*
sqrt(2*sqrt(2) + 4)*sin(x) + 8) + 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)*(29*sqrt(2) + 10) - 24*sqrt(2) - 44)*cos(x)^14 +
16*(sqrt(2)*(51*sqrt(2) - 4) - 52*sqrt(2) - 46)*cos(x)^12 - 16*(sqrt(2)*(41*sqrt(2) - 36) - 54*sqrt(2) + 15)*c
os(x)^10 + 8*(sqrt(2)*(29*sqrt(2) - 90) - 58*sqrt(2) + 132)*cos(x)^8 - 4*(sqrt(2)*(5*sqrt(2) - 98) - 32*sqrt(2
) + 216)*cos(x)^6 - 4*(sqrt(2)*(sqrt(2) + 24) + 4*sqrt(2) - 82)*cos(x)^4 + 4*(2*sqrt(2) - 15)*cos(x)^2 - 2*(8*
(2^(3/4)*(2*sqrt(2) - 1) - 2*2^(1/4)*(3*sqrt(2) + 2))*cos(x)^15 - 8*(2^(3/4)*(11*sqrt(2) - 9) - 2*2^(1/4)*(13*
sqrt(2) + 4))*cos(x)^13 + 4*(2*2^(3/4)*(21*sqrt(2) - 23) - 2^(1/4)*(79*sqrt(2) - 14))*cos(x)^11 - 8*(2^(3/4)*(
19*sqrt(2) - 27) - 2^(1/4)*(27*sqrt(2) - 31))*cos(x)^9 + 2*(2^(3/4)*(36*sqrt(2) - 65) - 32*2^(1/4)*(sqrt(2) -
4))*cos(x)^7 - 2*(2^(3/4)*(9*sqrt(2) - 19) - 2*2^(1/4)*(sqrt(2) - 30))*cos(x)^5 + (2*2^(3/4)*(sqrt(2) - 2) + 2
^(1/4)*(sqrt(2) + 26))*cos(x)^3 - 2*2^(1/4)*cos(x))*sqrt(2*sqrt(2) + 4)*sin(x) + (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*sqrt(2) - 6))*cos(x)^11 + 4*(2*2^(3/4)*(28*sqrt(2) - 39) - 2^(1/4)*(73*sq
rt(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*s
qrt(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 + 8*(2*s
qrt(2) - 3)*cos(x)^2 - 4*(2^(1/4)*(3*sqrt(2) - 4)*cos(x)^3 - 2*2^(1/4)*(sqrt(2) - 1)*cos(x))*sqrt(2*sqrt(2) +
4)*sin(x) + 8) + 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)*(29*sqrt(2) + 10) - 24*sqrt(2) - 44)*cos(x)^14 + 16*(sqrt(2)*(51*sq
rt(2) - 4) - 52*sqrt(2) - 46)*cos(x)^12 - 16*(sqrt(2)*(41*sqrt(2) - 36) - 54*sqrt(2) + 15)*cos(x)^10 + 8*(sqrt
(2)*(29*sqrt(2) - 90) - 58*sqrt(2) + 132)*cos(x)^8 - 4*(sqrt(2)*(5*sqrt(2) - 98) - 32*sqrt(2) + 216)*cos(x)^6
- 4*(sqrt(2)*(sqrt(2) + 24) + 4*sqrt(2) - 82)*cos(x)^4 + 4*(2*sqrt(2) - 15)*cos(x)^2 - 2*(8*(2^(3/4)*(2*sqrt(2
) - 1) - 2*2^(1/4)*(3*sqrt(2) + 2))*cos(x)^15 - 8*(2^(3/4)*(11*sqrt(2) - 9) - 2*2^(1/4)*(13*sqrt(2) + 4))*cos(
x)^13 + 4*(2*2^(3/4)*(21*sqrt(2) - 23) - 2^(1/4)*(79*sqrt(2) - 14))*cos(x)^11 - 8*(2^(3/4)*(19*sqrt(2) - 27) -
2^(1/4)*(27*sqrt(2) - 31))*cos(x)^9 + 2*(2^(3/4)*(36*sqrt(2) - 65) - 32*2^(1/4)*(sqrt(2) - 4))*cos(x)^7 - 2*(
2^(3/4)*(9*sqrt(2) - 19) - 2*2^(1/4)*(sqrt(2) - 30))*cos(x)^5 + (2*2^(3/4)*(sqrt(2) - 2) + 2^(1/4)*(sqrt(2) +
26))*cos(x)^3 - 2*2^(1/4)*cos(x))*sqrt(2*sqrt(2) + 4)*sin(x) - (16*(sqrt(2)*(5*sqrt(2) - 6) - 8*sqrt(2) + 4)*c
os(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*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)*(sq
rt(2) - 2)*cos(x))*sqrt(2*sqrt(2) + 4)*sin(x) - 2)*sqrt(-4*(4*sqrt(2) - 5)*cos(x)^4 + 8*(2*sqrt(2) - 3)*cos(x)
^2 - 4*(2^(1/4)*(3*sqrt(2) - 4)*cos(x)^3 - 2*2^(1/4)*(sqrt(2) - 1)*cos(x))*sqrt(2*sqrt(2) + 4)*sin(x) + 8) + 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 = 0.36, size = 170, normalized size = 0.55 \[ \frac {1}{4} \, {\left (\pi \left \lfloor \frac {x}{\pi } + \frac {1}{2} \right \rfloor + \arctan \left (\frac {2 \, \left (\frac {1}{2}\right )^{\frac {3}{4}} {\left (\left (\frac {1}{2}\right )^{\frac {1}{4}} \sqrt {-\sqrt {2} + 2} + 2 \, \tan \relax (x)\right )}}{\sqrt {\sqrt {2} + 2}}\right )\right )} \sqrt {\sqrt {2} + 1} + \frac {1}{4} \, {\left (\pi \left \lfloor \frac {x}{\pi } + \frac {1}{2} \right \rfloor + \arctan \left (-\frac {2 \, \left (\frac {1}{2}\right )^{\frac {3}{4}} {\left (\left (\frac {1}{2}\right )^{\frac {1}{4}} \sqrt {-\sqrt {2} + 2} - 2 \, \tan \relax (x)\right )}}{\sqrt {\sqrt {2} + 2}}\right )\right )} \sqrt {\sqrt {2} + 1} + \frac {1}{8} \, \sqrt {\sqrt {2} - 1} \log \left (\tan \relax (x)^{2} + \left (\frac {1}{2}\right )^{\frac {1}{4}} \sqrt {-\sqrt {2} + 2} \tan \relax (x) + \sqrt {\frac {1}{2}}\right ) - \frac {1}{8} \, \sqrt {\sqrt {2} - 1} \log \left (\tan \relax (x)^{2} - \left (\frac {1}{2}\right )^{\frac {1}{4}} \sqrt {-\sqrt {2} + 2} \tan \relax (x) + \sqrt {\frac {1}{2}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.45, size = 239, normalized size = 0.77 \[ -\frac {\sqrt {2}\, \sqrt {-2+2 \sqrt {2}}\, \ln \left (-\sqrt {-2+2 \sqrt {2}}\, \sqrt {2}\, \tan \relax (x )+2 \left (\tan ^{2}\relax (x )\right )+\sqrt {2}\right )}{16}+\frac {\arctan \left (\frac {-\sqrt {2}\, \sqrt {-2+2 \sqrt {2}}+4 \tan \relax (x )}{2 \sqrt {1+\sqrt {2}}}\right ) \sqrt {2}}{4 \sqrt {1+\sqrt {2}}}+\frac {\arctan \left (\frac {-\sqrt {2}\, \sqrt {-2+2 \sqrt {2}}+4 \tan \relax (x )}{2 \sqrt {1+\sqrt {2}}}\right )}{4 \sqrt {1+\sqrt {2}}}+\frac {\sqrt {2}\, \sqrt {-2+2 \sqrt {2}}\, \ln \left (\sqrt {2}+2 \left (\tan ^{2}\relax (x )\right )+\sqrt {-2+2 \sqrt {2}}\, \sqrt {2}\, \tan \relax (x )\right )}{16}+\frac {\arctan \left (\frac {4 \tan \relax (x )+\sqrt {2}\, \sqrt {-2+2 \sqrt {2}}}{2 \sqrt {1+\sqrt {2}}}\right ) \sqrt {2}}{4 \sqrt {1+\sqrt {2}}}+\frac {\arctan \left (\frac {4 \tan \relax (x )+\sqrt {2}\, \sqrt {-2+2 \sqrt {2}}}{2 \sqrt {1+\sqrt {2}}}\right )}{4 \sqrt {1+\sqrt {2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(sin(x)^4 + 1), x)

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mupad [B]  time = 14.35, size = 236, normalized size = 0.76 \[ \mathrm {atanh}\left (\frac {\mathrm {tan}\relax (x)}{8\,\sqrt {-\frac {\sqrt {2}}{64}-\frac {1}{64}}}-\frac {\mathrm {tan}\relax (x)}{8\,\sqrt {\frac {\sqrt {2}}{64}-\frac {1}{64}}}+\frac {\sqrt {2}\,\mathrm {tan}\relax (x)}{16\,\sqrt {-\frac {\sqrt {2}}{64}-\frac {1}{64}}}+\frac {\sqrt {2}\,\mathrm {tan}\relax (x)}{16\,\sqrt {\frac {\sqrt {2}}{64}-\frac {1}{64}}}\right )\,\left (2\,\sqrt {-\frac {\sqrt {2}}{64}-\frac {1}{64}}-2\,\sqrt {\frac {\sqrt {2}}{64}-\frac {1}{64}}\right )+\mathrm {atanh}\left (\frac {\mathrm {tan}\relax (x)}{8\,\sqrt {-\frac {\sqrt {2}}{64}-\frac {1}{64}}}+\frac {\mathrm {tan}\relax (x)}{8\,\sqrt {\frac {\sqrt {2}}{64}-\frac {1}{64}}}+\frac {\sqrt {2}\,\mathrm {tan}\relax (x)}{16\,\sqrt {-\frac {\sqrt {2}}{64}-\frac {1}{64}}}-\frac {\sqrt {2}\,\mathrm {tan}\relax (x)}{16\,\sqrt {\frac {\sqrt {2}}{64}-\frac {1}{64}}}\right )\,\left (2\,\sqrt {-\frac {\sqrt {2}}{64}-\frac {1}{64}}+2\,\sqrt {\frac {\sqrt {2}}{64}-\frac {1}{64}}\right )-\frac {\left (x-\mathrm {atan}\left (\mathrm {tan}\relax (x)\right )\right )\,\left (\pi \,\left (2\,\sqrt {-\frac {\sqrt {2}}{64}-\frac {1}{64}}-2\,\sqrt {\frac {\sqrt {2}}{64}-\frac {1}{64}}\right )\,1{}\mathrm {i}+\pi \,\left (2\,\sqrt {-\frac {\sqrt {2}}{64}-\frac {1}{64}}+2\,\sqrt {\frac {\sqrt {2}}{64}-\frac {1}{64}}\right )\,1{}\mathrm {i}\right )}{\pi } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(sin(x)^4 + 1),x)

[Out]

atanh(tan(x)/(8*(- 2^(1/2)/64 - 1/64)^(1/2)) - tan(x)/(8*(2^(1/2)/64 - 1/64)^(1/2)) + (2^(1/2)*tan(x))/(16*(-
2^(1/2)/64 - 1/64)^(1/2)) + (2^(1/2)*tan(x))/(16*(2^(1/2)/64 - 1/64)^(1/2)))*(2*(- 2^(1/2)/64 - 1/64)^(1/2) -
2*(2^(1/2)/64 - 1/64)^(1/2)) + atanh(tan(x)/(8*(- 2^(1/2)/64 - 1/64)^(1/2)) + tan(x)/(8*(2^(1/2)/64 - 1/64)^(1
/2)) + (2^(1/2)*tan(x))/(16*(- 2^(1/2)/64 - 1/64)^(1/2)) - (2^(1/2)*tan(x))/(16*(2^(1/2)/64 - 1/64)^(1/2)))*(2
*(- 2^(1/2)/64 - 1/64)^(1/2) + 2*(2^(1/2)/64 - 1/64)^(1/2)) - ((x - atan(tan(x)))*(pi*(2*(- 2^(1/2)/64 - 1/64)
^(1/2) - 2*(2^(1/2)/64 - 1/64)^(1/2))*1i + pi*(2*(- 2^(1/2)/64 - 1/64)^(1/2) + 2*(2^(1/2)/64 - 1/64)^(1/2))*1i
))/pi

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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+sin(x)**4),x)

[Out]

Timed out

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