3.255 \(\int \frac {1}{1+\sin ^5(x)} \, dx\)
Optimal. Leaf size=195 \[ \frac {2 \tan ^{-1}\left (\frac {\tan \left (\frac {x}{2}\right )+(-1)^{2/5}}{\sqrt {1-(-1)^{4/5}}}\right )}{5 \sqrt {1-(-1)^{4/5}}}+\frac {2 \tan ^{-1}\left (\frac {\tan \left (\frac {x}{2}\right )+(-1)^{4/5}}{\sqrt {1+(-1)^{3/5}}}\right )}{5 \sqrt {1+(-1)^{3/5}}}-\frac {2 \tan ^{-1}\left (\frac {(-1)^{3/5} \left ((-1)^{2/5} \tan \left (\frac {x}{2}\right )+1\right )}{\sqrt {1+\sqrt [5]{-1}}}\right )}{5 \sqrt {1+\sqrt [5]{-1}}}-\frac {2 \tan ^{-1}\left (\frac {\sqrt [5]{-1} \left ((-1)^{4/5} \tan \left (\frac {x}{2}\right )+1\right )}{\sqrt {1-(-1)^{2/5}}}\right )}{5 \sqrt {1-(-1)^{2/5}}}-\frac {\cos (x)}{5 (\sin (x)+1)} \]
[Out]
-1/5*cos(x)/(1+sin(x))-2/5*arctan((-1)^(3/5)*(1+(-1)^(2/5)*tan(1/2*x))/(1+(-1)^(1/5))^(1/2))/(1+(-1)^(1/5))^(1
/2)-2/5*arctan((-1)^(1/5)*(1+(-1)^(4/5)*tan(1/2*x))/(1-(-1)^(2/5))^(1/2))/(1-(-1)^(2/5))^(1/2)+2/5*arctan(((-1
)^(4/5)+tan(1/2*x))/(1+(-1)^(3/5))^(1/2))/(1+(-1)^(3/5))^(1/2)+2/5*arctan(((-1)^(2/5)+tan(1/2*x))/(1-(-1)^(4/5
))^(1/2))/(1-(-1)^(4/5))^(1/2)
________________________________________________________________________________________
Rubi [A] time = 0.38, antiderivative size = 195, normalized size of antiderivative = 1.00,
number of steps used = 15, number of rules used = 5, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used =
{3213, 2648, 2660, 618, 204} \[ \frac {2 \tan ^{-1}\left (\frac {\tan \left (\frac {x}{2}\right )+(-1)^{2/5}}{\sqrt {1-(-1)^{4/5}}}\right )}{5 \sqrt {1-(-1)^{4/5}}}+\frac {2 \tan ^{-1}\left (\frac {\tan \left (\frac {x}{2}\right )+(-1)^{4/5}}{\sqrt {1+(-1)^{3/5}}}\right )}{5 \sqrt {1+(-1)^{3/5}}}-\frac {2 \tan ^{-1}\left (\frac {(-1)^{3/5} \left ((-1)^{2/5} \tan \left (\frac {x}{2}\right )+1\right )}{\sqrt {1+\sqrt [5]{-1}}}\right )}{5 \sqrt {1+\sqrt [5]{-1}}}-\frac {2 \tan ^{-1}\left (\frac {\sqrt [5]{-1} \left ((-1)^{4/5} \tan \left (\frac {x}{2}\right )+1\right )}{\sqrt {1-(-1)^{2/5}}}\right )}{5 \sqrt {1-(-1)^{2/5}}}-\frac {\cos (x)}{5 (\sin (x)+1)} \]
Antiderivative was successfully verified.
[In]
Int[(1 + Sin[x]^5)^(-1),x]
[Out]
(2*ArcTan[((-1)^(2/5) + Tan[x/2])/Sqrt[1 - (-1)^(4/5)]])/(5*Sqrt[1 - (-1)^(4/5)]) + (2*ArcTan[((-1)^(4/5) + Ta
n[x/2])/Sqrt[1 + (-1)^(3/5)]])/(5*Sqrt[1 + (-1)^(3/5)]) - (2*ArcTan[((-1)^(3/5)*(1 + (-1)^(2/5)*Tan[x/2]))/Sqr
t[1 + (-1)^(1/5)]])/(5*Sqrt[1 + (-1)^(1/5)]) - (2*ArcTan[((-1)^(1/5)*(1 + (-1)^(4/5)*Tan[x/2]))/Sqrt[1 - (-1)^
(2/5)]])/(5*Sqrt[1 - (-1)^(2/5)]) - Cos[x]/(5*(1 + Sin[x]))
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 2648
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> -Simp[Cos[c + d*x]/(d*(b + a*Sin[c + d*x])), x]
/; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]
Rule 2660
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x]}, Dis
t[(2*e)/d, Subst[Int[1/(a + 2*b*e*x + a*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] &&
NeQ[a^2 - b^2, 0]
Rule 3213
Int[((a_) + (b_.)*((c_.)*sin[(e_.) + (f_.)*(x_)])^(n_))^(p_), x_Symbol] :> Int[ExpandTrig[(a + b*(c*sin[e + f*
x])^n)^p, x], x] /; FreeQ[{a, b, c, e, f, n}, x] && (IGtQ[p, 0] || (EqQ[p, -1] && IntegerQ[n]))
Rubi steps
\begin {align*} \int \frac {1}{1+\sin ^5(x)} \, dx &=\int \left (-\frac {1}{5 (-1-\sin (x))}-\frac {1}{5 \left (-1+\sqrt [5]{-1} \sin (x)\right )}-\frac {1}{5 \left (-1-(-1)^{2/5} \sin (x)\right )}-\frac {1}{5 \left (-1+(-1)^{3/5} \sin (x)\right )}-\frac {1}{5 \left (-1-(-1)^{4/5} \sin (x)\right )}\right ) \, dx\\ &=-\left (\frac {1}{5} \int \frac {1}{-1-\sin (x)} \, dx\right )-\frac {1}{5} \int \frac {1}{-1+\sqrt [5]{-1} \sin (x)} \, dx-\frac {1}{5} \int \frac {1}{-1-(-1)^{2/5} \sin (x)} \, dx-\frac {1}{5} \int \frac {1}{-1+(-1)^{3/5} \sin (x)} \, dx-\frac {1}{5} \int \frac {1}{-1-(-1)^{4/5} \sin (x)} \, dx\\ &=-\frac {\cos (x)}{5 (1+\sin (x))}-\frac {2}{5} \operatorname {Subst}\left (\int \frac {1}{-1+2 \sqrt [5]{-1} x-x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )-\frac {2}{5} \operatorname {Subst}\left (\int \frac {1}{-1-2 (-1)^{2/5} x-x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )-\frac {2}{5} \operatorname {Subst}\left (\int \frac {1}{-1+2 (-1)^{3/5} x-x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )-\frac {2}{5} \operatorname {Subst}\left (\int \frac {1}{-1-2 (-1)^{4/5} x-x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )\\ &=-\frac {\cos (x)}{5 (1+\sin (x))}+\frac {4}{5} \operatorname {Subst}\left (\int \frac {1}{-4 \left (1+\sqrt [5]{-1}\right )-x^2} \, dx,x,2 (-1)^{3/5}-2 \tan \left (\frac {x}{2}\right )\right )+\frac {4}{5} \operatorname {Subst}\left (\int \frac {1}{-4 \left (1-(-1)^{2/5}\right )-x^2} \, dx,x,2 \sqrt [5]{-1}-2 \tan \left (\frac {x}{2}\right )\right )+\frac {4}{5} \operatorname {Subst}\left (\int \frac {1}{-4 \left (1+(-1)^{3/5}\right )-x^2} \, dx,x,-2 (-1)^{4/5}-2 \tan \left (\frac {x}{2}\right )\right )+\frac {4}{5} \operatorname {Subst}\left (\int \frac {1}{-4 \left (1-(-1)^{4/5}\right )-x^2} \, dx,x,-2 (-1)^{2/5}-2 \tan \left (\frac {x}{2}\right )\right )\\ &=-\frac {2 \tan ^{-1}\left (\frac {\sqrt [5]{-1}-\tan \left (\frac {x}{2}\right )}{\sqrt {1-(-1)^{2/5}}}\right )}{5 \sqrt {1-(-1)^{2/5}}}-\frac {2 \tan ^{-1}\left (\frac {(-1)^{3/5}-\tan \left (\frac {x}{2}\right )}{\sqrt {1+\sqrt [5]{-1}}}\right )}{5 \sqrt {1+\sqrt [5]{-1}}}+\frac {2 \tan ^{-1}\left (\frac {(-1)^{2/5}+\tan \left (\frac {x}{2}\right )}{\sqrt {1-(-1)^{4/5}}}\right )}{5 \sqrt {1-(-1)^{4/5}}}+\frac {2 \tan ^{-1}\left (\frac {(-1)^{4/5}+\tan \left (\frac {x}{2}\right )}{\sqrt {1+(-1)^{3/5}}}\right )}{5 \sqrt {1+(-1)^{3/5}}}-\frac {\cos (x)}{5 (1+\sin (x))}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.15, size = 411, normalized size = 2.11 \[ \frac {2 \sin \left (\frac {x}{2}\right )}{5 \left (\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )\right )}-\frac {1}{10} i \text {RootSum}\left [\text {$\#$1}^8-2 i \text {$\#$1}^7-8 \text {$\#$1}^6+14 i \text {$\#$1}^5+30 \text {$\#$1}^4-14 i \text {$\#$1}^3-8 \text {$\#$1}^2+2 i \text {$\#$1}+1\& ,\frac {2 \text {$\#$1}^6 \tan ^{-1}\left (\frac {\sin (x)}{\cos (x)-\text {$\#$1}}\right )-8 i \text {$\#$1}^5 \tan ^{-1}\left (\frac {\sin (x)}{\cos (x)-\text {$\#$1}}\right )-30 \text {$\#$1}^4 \tan ^{-1}\left (\frac {\sin (x)}{\cos (x)-\text {$\#$1}}\right )+80 i \text {$\#$1}^3 \tan ^{-1}\left (\frac {\sin (x)}{\cos (x)-\text {$\#$1}}\right )-15 i \text {$\#$1}^2 \log \left (\text {$\#$1}^2-2 \text {$\#$1} \cos (x)+1\right )-4 \text {$\#$1} \log \left (\text {$\#$1}^2-2 \text {$\#$1} \cos (x)+1\right )+i \log \left (\text {$\#$1}^2-2 \text {$\#$1} \cos (x)+1\right )+30 \text {$\#$1}^2 \tan ^{-1}\left (\frac {\sin (x)}{\cos (x)-\text {$\#$1}}\right )-i \text {$\#$1}^6 \log \left (\text {$\#$1}^2-2 \text {$\#$1} \cos (x)+1\right )-4 \text {$\#$1}^5 \log \left (\text {$\#$1}^2-2 \text {$\#$1} \cos (x)+1\right )+15 i \text {$\#$1}^4 \log \left (\text {$\#$1}^2-2 \text {$\#$1} \cos (x)+1\right )+40 \text {$\#$1}^3 \log \left (\text {$\#$1}^2-2 \text {$\#$1} \cos (x)+1\right )-8 i \text {$\#$1} \tan ^{-1}\left (\frac {\sin (x)}{\cos (x)-\text {$\#$1}}\right )-2 \tan ^{-1}\left (\frac {\sin (x)}{\cos (x)-\text {$\#$1}}\right )}{4 \text {$\#$1}^7-7 i \text {$\#$1}^6-24 \text {$\#$1}^5+35 i \text {$\#$1}^4+60 \text {$\#$1}^3-21 i \text {$\#$1}^2-8 \text {$\#$1}+i}\& \right ] \]
Antiderivative was successfully verified.
[In]
Integrate[(1 + Sin[x]^5)^(-1),x]
[Out]
(-1/10*I)*RootSum[1 + (2*I)*#1 - 8*#1^2 - (14*I)*#1^3 + 30*#1^4 + (14*I)*#1^5 - 8*#1^6 - (2*I)*#1^7 + #1^8 & ,
(-2*ArcTan[Sin[x]/(Cos[x] - #1)] + I*Log[1 - 2*Cos[x]*#1 + #1^2] - (8*I)*ArcTan[Sin[x]/(Cos[x] - #1)]*#1 - 4*
Log[1 - 2*Cos[x]*#1 + #1^2]*#1 + 30*ArcTan[Sin[x]/(Cos[x] - #1)]*#1^2 - (15*I)*Log[1 - 2*Cos[x]*#1 + #1^2]*#1^
2 + (80*I)*ArcTan[Sin[x]/(Cos[x] - #1)]*#1^3 + 40*Log[1 - 2*Cos[x]*#1 + #1^2]*#1^3 - 30*ArcTan[Sin[x]/(Cos[x]
- #1)]*#1^4 + (15*I)*Log[1 - 2*Cos[x]*#1 + #1^2]*#1^4 - (8*I)*ArcTan[Sin[x]/(Cos[x] - #1)]*#1^5 - 4*Log[1 - 2*
Cos[x]*#1 + #1^2]*#1^5 + 2*ArcTan[Sin[x]/(Cos[x] - #1)]*#1^6 - I*Log[1 - 2*Cos[x]*#1 + #1^2]*#1^6)/(I - 8*#1 -
(21*I)*#1^2 + 60*#1^3 + (35*I)*#1^4 - 24*#1^5 - (7*I)*#1^6 + 4*#1^7) & ] + (2*Sin[x/2])/(5*(Cos[x/2] + Sin[x/
2]))
________________________________________________________________________________________
fricas [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)^5),x, algorithm="fricas")
[Out]
Timed out
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(1+sin(x)^5),x, algorithm="giac")
[Out]
sage0*x
________________________________________________________________________________________
maple [C] time = 0.22, size = 133, normalized size = 0.68 \[ \frac {2 \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{8}-2 \textit {\_Z}^{7}+8 \textit {\_Z}^{6}-14 \textit {\_Z}^{5}+30 \textit {\_Z}^{4}-14 \textit {\_Z}^{3}+8 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right )}{\sum }\frac {\left (2 \textit {\_R}^{6}-3 \textit {\_R}^{5}+10 \textit {\_R}^{4}-10 \textit {\_R}^{3}+10 \textit {\_R}^{2}-3 \textit {\_R} +2\right ) \ln \left (\tan \left (\frac {x}{2}\right )-\textit {\_R} \right )}{4 \textit {\_R}^{7}-7 \textit {\_R}^{6}+24 \textit {\_R}^{5}-35 \textit {\_R}^{4}+60 \textit {\_R}^{3}-21 \textit {\_R}^{2}+8 \textit {\_R} -1}\right )}{5}-\frac {2}{5 \left (\tan \left (\frac {x}{2}\right )+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(1+sin(x)^5),x)
[Out]
2/5*sum((2*_R^6-3*_R^5+10*_R^4-10*_R^3+10*_R^2-3*_R+2)/(4*_R^7-7*_R^6+24*_R^5-35*_R^4+60*_R^3-21*_R^2+8*_R-1)*
ln(tan(1/2*x)-_R),_R=RootOf(_Z^8-2*_Z^7+8*_Z^6-14*_Z^5+30*_Z^4-14*_Z^3+8*_Z^2-2*_Z+1))-2/5/(tan(1/2*x)+1)
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(1+sin(x)^5),x, algorithm="maxima")
[Out]
-1/5*(5*(cos(x)^2 + sin(x)^2 + 2*sin(x) + 1)*integrate(-2/5*((4*cos(6*x) - 40*cos(4*x) + 4*cos(2*x) - sin(7*x)
+ 15*sin(5*x) - 15*sin(3*x) + sin(x))*cos(8*x) + 2*(22*cos(5*x) - 22*cos(3*x) + 2*cos(x) - 8*sin(6*x) + 55*si
n(4*x) - 8*sin(2*x))*cos(7*x) - 2*cos(7*x)^2 + 4*(110*cos(4*x) - 16*cos(2*x) - 44*sin(5*x) + 44*sin(3*x) - 4*s
in(x) + 1)*cos(6*x) - 32*cos(6*x)^2 + 2*(210*cos(3*x) - 22*cos(x) - 505*sin(4*x) + 88*sin(2*x))*cos(5*x) - 210
*cos(5*x)^2 + 10*(44*cos(2*x) - 101*sin(3*x) + 11*sin(x) - 4)*cos(4*x) - 1200*cos(4*x)^2 + 44*(cos(x) - 4*sin(
2*x))*cos(3*x) - 210*cos(3*x)^2 - 4*(4*sin(x) - 1)*cos(2*x) - 32*cos(2*x)^2 - 2*cos(x)^2 + (cos(7*x) - 15*cos(
5*x) + 15*cos(3*x) - cos(x) + 4*sin(6*x) - 40*sin(4*x) + 4*sin(2*x))*sin(8*x) + (16*cos(6*x) - 110*cos(4*x) +
16*cos(2*x) + 44*sin(5*x) - 44*sin(3*x) + 4*sin(x) - 1)*sin(7*x) - 2*sin(7*x)^2 + 8*(22*cos(5*x) - 22*cos(3*x)
+ 2*cos(x) + 55*sin(4*x) - 8*sin(2*x))*sin(6*x) - 32*sin(6*x)^2 + (1010*cos(4*x) - 176*cos(2*x) + 420*sin(3*x
) - 44*sin(x) + 15)*sin(5*x) - 210*sin(5*x)^2 + 10*(101*cos(3*x) - 11*cos(x) + 44*sin(2*x))*sin(4*x) - 1200*si
n(4*x)^2 + (176*cos(2*x) + 44*sin(x) - 15)*sin(3*x) - 210*sin(3*x)^2 + 16*cos(x)*sin(2*x) - 32*sin(2*x)^2 - 2*
sin(x)^2 + sin(x))/(2*(8*cos(6*x) - 30*cos(4*x) + 8*cos(2*x) - 2*sin(7*x) + 14*sin(5*x) - 14*sin(3*x) + 2*sin(
x) - 1)*cos(8*x) - cos(8*x)^2 + 8*(7*cos(5*x) - 7*cos(3*x) + cos(x) - 4*sin(6*x) + 15*sin(4*x) - 4*sin(2*x))*c
os(7*x) - 4*cos(7*x)^2 + 16*(30*cos(4*x) - 8*cos(2*x) - 14*sin(5*x) + 14*sin(3*x) - 2*sin(x) + 1)*cos(6*x) - 6
4*cos(6*x)^2 + 56*(7*cos(3*x) - cos(x) - 15*sin(4*x) + 4*sin(2*x))*cos(5*x) - 196*cos(5*x)^2 + 60*(8*cos(2*x)
- 14*sin(3*x) + 2*sin(x) - 1)*cos(4*x) - 900*cos(4*x)^2 + 56*(cos(x) - 4*sin(2*x))*cos(3*x) - 196*cos(3*x)^2 -
16*(2*sin(x) - 1)*cos(2*x) - 64*cos(2*x)^2 - 4*cos(x)^2 + 4*(cos(7*x) - 7*cos(5*x) + 7*cos(3*x) - cos(x) + 4*
sin(6*x) - 15*sin(4*x) + 4*sin(2*x))*sin(8*x) - sin(8*x)^2 + 4*(8*cos(6*x) - 30*cos(4*x) + 8*cos(2*x) + 14*sin
(5*x) - 14*sin(3*x) + 2*sin(x) - 1)*sin(7*x) - 4*sin(7*x)^2 + 32*(7*cos(5*x) - 7*cos(3*x) + cos(x) + 15*sin(4*
x) - 4*sin(2*x))*sin(6*x) - 64*sin(6*x)^2 + 28*(30*cos(4*x) - 8*cos(2*x) + 14*sin(3*x) - 2*sin(x) + 1)*sin(5*x
) - 196*sin(5*x)^2 + 120*(7*cos(3*x) - cos(x) + 4*sin(2*x))*sin(4*x) - 900*sin(4*x)^2 + 28*(8*cos(2*x) + 2*sin
(x) - 1)*sin(3*x) - 196*sin(3*x)^2 + 32*cos(x)*sin(2*x) - 64*sin(2*x)^2 - 4*sin(x)^2 + 4*sin(x) - 1), x) + 2*c
os(x))/(cos(x)^2 + sin(x)^2 + 2*sin(x) + 1)
________________________________________________________________________________________
mupad [B] time = 15.25, size = 3513, normalized size = 18.02 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(sin(x)^5 + 1),x)
[Out]
2*atanh((989855744*((- (2*5^(1/2))/5 - 1)^(1/2)/50 - 1/50)^(1/2))/(5*((301989888*tan(x/2))/5 + (2382364672*5^(
1/2)*tan(x/2))/125 + (1308622848*tan(x/2)*(- (2*5^(1/2))/5 - 1)^(1/2))/25 - (452984832*5^(1/2)*(- (2*5^(1/2))/
5 - 1)^(1/2))/25 + (16777216*5^(1/2))/5 - 16777216*(- (2*5^(1/2))/5 - 1)^(1/2) + (436207616*5^(1/2)*tan(x/2)*(
- (2*5^(1/2))/5 - 1)^(1/2))/25 + 184549376/25)) - (2030043136*tan(x/2)*((- (2*5^(1/2))/5 - 1)^(1/2)/50 - 1/50)
^(1/2))/(5*((301989888*tan(x/2))/5 + (2382364672*5^(1/2)*tan(x/2))/125 + (1308622848*tan(x/2)*(- (2*5^(1/2))/5
- 1)^(1/2))/25 - (452984832*5^(1/2)*(- (2*5^(1/2))/5 - 1)^(1/2))/25 + (16777216*5^(1/2))/5 - 16777216*(- (2*5
^(1/2))/5 - 1)^(1/2) + (436207616*5^(1/2)*tan(x/2)*(- (2*5^(1/2))/5 - 1)^(1/2))/25 + 184549376/25)) + (1627389
952*5^(1/2)*((- (2*5^(1/2))/5 - 1)^(1/2)/50 - 1/50)^(1/2))/(25*((301989888*tan(x/2))/5 + (2382364672*5^(1/2)*t
an(x/2))/125 + (1308622848*tan(x/2)*(- (2*5^(1/2))/5 - 1)^(1/2))/25 - (452984832*5^(1/2)*(- (2*5^(1/2))/5 - 1)
^(1/2))/25 + (16777216*5^(1/2))/5 - 16777216*(- (2*5^(1/2))/5 - 1)^(1/2) + (436207616*5^(1/2)*tan(x/2)*(- (2*5
^(1/2))/5 - 1)^(1/2))/25 + 184549376/25)) + (553648128*(- (2*5^(1/2))/5 - 1)^(1/2)*((- (2*5^(1/2))/5 - 1)^(1/2
)/50 - 1/50)^(1/2))/(5*((301989888*tan(x/2))/5 + (2382364672*5^(1/2)*tan(x/2))/125 + (1308622848*tan(x/2)*(- (
2*5^(1/2))/5 - 1)^(1/2))/25 - (452984832*5^(1/2)*(- (2*5^(1/2))/5 - 1)^(1/2))/25 + (16777216*5^(1/2))/5 - 1677
7216*(- (2*5^(1/2))/5 - 1)^(1/2) + (436207616*5^(1/2)*tan(x/2)*(- (2*5^(1/2))/5 - 1)^(1/2))/25 + 184549376/25)
) + (184549376*5^(1/2)*(- (2*5^(1/2))/5 - 1)^(1/2)*((- (2*5^(1/2))/5 - 1)^(1/2)/50 - 1/50)^(1/2))/(5*((3019898
88*tan(x/2))/5 + (2382364672*5^(1/2)*tan(x/2))/125 + (1308622848*tan(x/2)*(- (2*5^(1/2))/5 - 1)^(1/2))/25 - (4
52984832*5^(1/2)*(- (2*5^(1/2))/5 - 1)^(1/2))/25 + (16777216*5^(1/2))/5 - 16777216*(- (2*5^(1/2))/5 - 1)^(1/2)
+ (436207616*5^(1/2)*tan(x/2)*(- (2*5^(1/2))/5 - 1)^(1/2))/25 + 184549376/25)) - (5083496448*5^(1/2)*tan(x/2)
*((- (2*5^(1/2))/5 - 1)^(1/2)/50 - 1/50)^(1/2))/(25*((301989888*tan(x/2))/5 + (2382364672*5^(1/2)*tan(x/2))/12
5 + (1308622848*tan(x/2)*(- (2*5^(1/2))/5 - 1)^(1/2))/25 - (452984832*5^(1/2)*(- (2*5^(1/2))/5 - 1)^(1/2))/25
+ (16777216*5^(1/2))/5 - 16777216*(- (2*5^(1/2))/5 - 1)^(1/2) + (436207616*5^(1/2)*tan(x/2)*(- (2*5^(1/2))/5 -
1)^(1/2))/25 + 184549376/25)) - (553648128*tan(x/2)*(- (2*5^(1/2))/5 - 1)^(1/2)*((- (2*5^(1/2))/5 - 1)^(1/2)/
50 - 1/50)^(1/2))/(5*((301989888*tan(x/2))/5 + (2382364672*5^(1/2)*tan(x/2))/125 + (1308622848*tan(x/2)*(- (2*
5^(1/2))/5 - 1)^(1/2))/25 - (452984832*5^(1/2)*(- (2*5^(1/2))/5 - 1)^(1/2))/25 + (16777216*5^(1/2))/5 - 167772
16*(- (2*5^(1/2))/5 - 1)^(1/2) + (436207616*5^(1/2)*tan(x/2)*(- (2*5^(1/2))/5 - 1)^(1/2))/25 + 184549376/25))
+ (553648128*5^(1/2)*tan(x/2)*(- (2*5^(1/2))/5 - 1)^(1/2)*((- (2*5^(1/2))/5 - 1)^(1/2)/50 - 1/50)^(1/2))/(5*((
301989888*tan(x/2))/5 + (2382364672*5^(1/2)*tan(x/2))/125 + (1308622848*tan(x/2)*(- (2*5^(1/2))/5 - 1)^(1/2))/
25 - (452984832*5^(1/2)*(- (2*5^(1/2))/5 - 1)^(1/2))/25 + (16777216*5^(1/2))/5 - 16777216*(- (2*5^(1/2))/5 - 1
)^(1/2) + (436207616*5^(1/2)*tan(x/2)*(- (2*5^(1/2))/5 - 1)^(1/2))/25 + 184549376/25)))*((- (2*5^(1/2))/5 - 1)
^(1/2)/50 - 1/50)^(1/2) - 2*atanh((2030043136*tan(x/2)*(- (- (2*5^(1/2))/5 - 1)^(1/2)/50 - 1/50)^(1/2))/(5*((3
01989888*tan(x/2))/5 + (2382364672*5^(1/2)*tan(x/2))/125 - (1308622848*tan(x/2)*(- (2*5^(1/2))/5 - 1)^(1/2))/2
5 + (452984832*5^(1/2)*(- (2*5^(1/2))/5 - 1)^(1/2))/25 + (16777216*5^(1/2))/5 + 16777216*(- (2*5^(1/2))/5 - 1)
^(1/2) - (436207616*5^(1/2)*tan(x/2)*(- (2*5^(1/2))/5 - 1)^(1/2))/25 + 184549376/25)) - (989855744*(- (- (2*5^
(1/2))/5 - 1)^(1/2)/50 - 1/50)^(1/2))/(5*((301989888*tan(x/2))/5 + (2382364672*5^(1/2)*tan(x/2))/125 - (130862
2848*tan(x/2)*(- (2*5^(1/2))/5 - 1)^(1/2))/25 + (452984832*5^(1/2)*(- (2*5^(1/2))/5 - 1)^(1/2))/25 + (16777216
*5^(1/2))/5 + 16777216*(- (2*5^(1/2))/5 - 1)^(1/2) - (436207616*5^(1/2)*tan(x/2)*(- (2*5^(1/2))/5 - 1)^(1/2))/
25 + 184549376/25)) - (1627389952*5^(1/2)*(- (- (2*5^(1/2))/5 - 1)^(1/2)/50 - 1/50)^(1/2))/(25*((301989888*tan
(x/2))/5 + (2382364672*5^(1/2)*tan(x/2))/125 - (1308622848*tan(x/2)*(- (2*5^(1/2))/5 - 1)^(1/2))/25 + (4529848
32*5^(1/2)*(- (2*5^(1/2))/5 - 1)^(1/2))/25 + (16777216*5^(1/2))/5 + 16777216*(- (2*5^(1/2))/5 - 1)^(1/2) - (43
6207616*5^(1/2)*tan(x/2)*(- (2*5^(1/2))/5 - 1)^(1/2))/25 + 184549376/25)) + (553648128*(- (2*5^(1/2))/5 - 1)^(
1/2)*(- (- (2*5^(1/2))/5 - 1)^(1/2)/50 - 1/50)^(1/2))/(5*((301989888*tan(x/2))/5 + (2382364672*5^(1/2)*tan(x/2
))/125 - (1308622848*tan(x/2)*(- (2*5^(1/2))/5 - 1)^(1/2))/25 + (452984832*5^(1/2)*(- (2*5^(1/2))/5 - 1)^(1/2)
)/25 + (16777216*5^(1/2))/5 + 16777216*(- (2*5^(1/2))/5 - 1)^(1/2) - (436207616*5^(1/2)*tan(x/2)*(- (2*5^(1/2)
)/5 - 1)^(1/2))/25 + 184549376/25)) + (184549376*5^(1/2)*(- (2*5^(1/2))/5 - 1)^(1/2)*(- (- (2*5^(1/2))/5 - 1)^
(1/2)/50 - 1/50)^(1/2))/(5*((301989888*tan(x/2))/5 + (2382364672*5^(1/2)*tan(x/2))/125 - (1308622848*tan(x/2)*
(- (2*5^(1/2))/5 - 1)^(1/2))/25 + (452984832*5^(1/2)*(- (2*5^(1/2))/5 - 1)^(1/2))/25 + (16777216*5^(1/2))/5 +
16777216*(- (2*5^(1/2))/5 - 1)^(1/2) - (436207616*5^(1/2)*tan(x/2)*(- (2*5^(1/2))/5 - 1)^(1/2))/25 + 184549376
/25)) + (5083496448*5^(1/2)*tan(x/2)*(- (- (2*5^(1/2))/5 - 1)^(1/2)/50 - 1/50)^(1/2))/(25*((301989888*tan(x/2)
)/5 + (2382364672*5^(1/2)*tan(x/2))/125 - (1308622848*tan(x/2)*(- (2*5^(1/2))/5 - 1)^(1/2))/25 + (452984832*5^
(1/2)*(- (2*5^(1/2))/5 - 1)^(1/2))/25 + (16777216*5^(1/2))/5 + 16777216*(- (2*5^(1/2))/5 - 1)^(1/2) - (4362076
16*5^(1/2)*tan(x/2)*(- (2*5^(1/2))/5 - 1)^(1/2))/25 + 184549376/25)) - (553648128*tan(x/2)*(- (2*5^(1/2))/5 -
1)^(1/2)*(- (- (2*5^(1/2))/5 - 1)^(1/2)/50 - 1/50)^(1/2))/(5*((301989888*tan(x/2))/5 + (2382364672*5^(1/2)*tan
(x/2))/125 - (1308622848*tan(x/2)*(- (2*5^(1/2))/5 - 1)^(1/2))/25 + (452984832*5^(1/2)*(- (2*5^(1/2))/5 - 1)^(
1/2))/25 + (16777216*5^(1/2))/5 + 16777216*(- (2*5^(1/2))/5 - 1)^(1/2) - (436207616*5^(1/2)*tan(x/2)*(- (2*5^(
1/2))/5 - 1)^(1/2))/25 + 184549376/25)) + (553648128*5^(1/2)*tan(x/2)*(- (2*5^(1/2))/5 - 1)^(1/2)*(- (- (2*5^(
1/2))/5 - 1)^(1/2)/50 - 1/50)^(1/2))/(5*((301989888*tan(x/2))/5 + (2382364672*5^(1/2)*tan(x/2))/125 - (1308622
848*tan(x/2)*(- (2*5^(1/2))/5 - 1)^(1/2))/25 + (452984832*5^(1/2)*(- (2*5^(1/2))/5 - 1)^(1/2))/25 + (16777216*
5^(1/2))/5 + 16777216*(- (2*5^(1/2))/5 - 1)^(1/2) - (436207616*5^(1/2)*tan(x/2)*(- (2*5^(1/2))/5 - 1)^(1/2))/2
5 + 184549376/25)))*(- (- (2*5^(1/2))/5 - 1)^(1/2)/50 - 1/50)^(1/2) - 2/(5*(tan(x/2) + 1)) + 2*atanh((98985574
4*(- ((2*5^(1/2))/5 - 1)^(1/2)/50 - 1/50)^(1/2))/(5*((301989888*tan(x/2))/5 - (2382364672*5^(1/2)*tan(x/2))/12
5 - (1308622848*tan(x/2)*((2*5^(1/2))/5 - 1)^(1/2))/25 - (452984832*5^(1/2)*((2*5^(1/2))/5 - 1)^(1/2))/25 - (1
6777216*5^(1/2))/5 + 16777216*((2*5^(1/2))/5 - 1)^(1/2) + (436207616*5^(1/2)*tan(x/2)*((2*5^(1/2))/5 - 1)^(1/2
))/25 + 184549376/25)) - (2030043136*tan(x/2)*(- ((2*5^(1/2))/5 - 1)^(1/2)/50 - 1/50)^(1/2))/(5*((301989888*ta
n(x/2))/5 - (2382364672*5^(1/2)*tan(x/2))/125 - (1308622848*tan(x/2)*((2*5^(1/2))/5 - 1)^(1/2))/25 - (45298483
2*5^(1/2)*((2*5^(1/2))/5 - 1)^(1/2))/25 - (16777216*5^(1/2))/5 + 16777216*((2*5^(1/2))/5 - 1)^(1/2) + (4362076
16*5^(1/2)*tan(x/2)*((2*5^(1/2))/5 - 1)^(1/2))/25 + 184549376/25)) - (1627389952*5^(1/2)*(- ((2*5^(1/2))/5 - 1
)^(1/2)/50 - 1/50)^(1/2))/(25*((301989888*tan(x/2))/5 - (2382364672*5^(1/2)*tan(x/2))/125 - (1308622848*tan(x/
2)*((2*5^(1/2))/5 - 1)^(1/2))/25 - (452984832*5^(1/2)*((2*5^(1/2))/5 - 1)^(1/2))/25 - (16777216*5^(1/2))/5 + 1
6777216*((2*5^(1/2))/5 - 1)^(1/2) + (436207616*5^(1/2)*tan(x/2)*((2*5^(1/2))/5 - 1)^(1/2))/25 + 184549376/25))
- (553648128*((2*5^(1/2))/5 - 1)^(1/2)*(- ((2*5^(1/2))/5 - 1)^(1/2)/50 - 1/50)^(1/2))/(5*((301989888*tan(x/2)
)/5 - (2382364672*5^(1/2)*tan(x/2))/125 - (1308622848*tan(x/2)*((2*5^(1/2))/5 - 1)^(1/2))/25 - (452984832*5^(1
/2)*((2*5^(1/2))/5 - 1)^(1/2))/25 - (16777216*5^(1/2))/5 + 16777216*((2*5^(1/2))/5 - 1)^(1/2) + (436207616*5^(
1/2)*tan(x/2)*((2*5^(1/2))/5 - 1)^(1/2))/25 + 184549376/25)) + (184549376*5^(1/2)*((2*5^(1/2))/5 - 1)^(1/2)*(-
((2*5^(1/2))/5 - 1)^(1/2)/50 - 1/50)^(1/2))/(5*((301989888*tan(x/2))/5 - (2382364672*5^(1/2)*tan(x/2))/125 -
(1308622848*tan(x/2)*((2*5^(1/2))/5 - 1)^(1/2))/25 - (452984832*5^(1/2)*((2*5^(1/2))/5 - 1)^(1/2))/25 - (16777
216*5^(1/2))/5 + 16777216*((2*5^(1/2))/5 - 1)^(1/2) + (436207616*5^(1/2)*tan(x/2)*((2*5^(1/2))/5 - 1)^(1/2))/2
5 + 184549376/25)) + (5083496448*5^(1/2)*tan(x/2)*(- ((2*5^(1/2))/5 - 1)^(1/2)/50 - 1/50)^(1/2))/(25*((3019898
88*tan(x/2))/5 - (2382364672*5^(1/2)*tan(x/2))/125 - (1308622848*tan(x/2)*((2*5^(1/2))/5 - 1)^(1/2))/25 - (452
984832*5^(1/2)*((2*5^(1/2))/5 - 1)^(1/2))/25 - (16777216*5^(1/2))/5 + 16777216*((2*5^(1/2))/5 - 1)^(1/2) + (43
6207616*5^(1/2)*tan(x/2)*((2*5^(1/2))/5 - 1)^(1/2))/25 + 184549376/25)) + (553648128*tan(x/2)*((2*5^(1/2))/5 -
1)^(1/2)*(- ((2*5^(1/2))/5 - 1)^(1/2)/50 - 1/50)^(1/2))/(5*((301989888*tan(x/2))/5 - (2382364672*5^(1/2)*tan(
x/2))/125 - (1308622848*tan(x/2)*((2*5^(1/2))/5 - 1)^(1/2))/25 - (452984832*5^(1/2)*((2*5^(1/2))/5 - 1)^(1/2))
/25 - (16777216*5^(1/2))/5 + 16777216*((2*5^(1/2))/5 - 1)^(1/2) + (436207616*5^(1/2)*tan(x/2)*((2*5^(1/2))/5 -
1)^(1/2))/25 + 184549376/25)) + (553648128*5^(1/2)*tan(x/2)*((2*5^(1/2))/5 - 1)^(1/2)*(- ((2*5^(1/2))/5 - 1)^
(1/2)/50 - 1/50)^(1/2))/(5*((301989888*tan(x/2))/5 - (2382364672*5^(1/2)*tan(x/2))/125 - (1308622848*tan(x/2)*
((2*5^(1/2))/5 - 1)^(1/2))/25 - (452984832*5^(1/2)*((2*5^(1/2))/5 - 1)^(1/2))/25 - (16777216*5^(1/2))/5 + 1677
7216*((2*5^(1/2))/5 - 1)^(1/2) + (436207616*5^(1/2)*tan(x/2)*((2*5^(1/2))/5 - 1)^(1/2))/25 + 184549376/25)))*(
- ((2*5^(1/2))/5 - 1)^(1/2)/50 - 1/50)^(1/2) - 2*atanh((2030043136*tan(x/2)*(((2*5^(1/2))/5 - 1)^(1/2)/50 - 1/
50)^(1/2))/(5*((301989888*tan(x/2))/5 - (2382364672*5^(1/2)*tan(x/2))/125 + (1308622848*tan(x/2)*((2*5^(1/2))/
5 - 1)^(1/2))/25 + (452984832*5^(1/2)*((2*5^(1/2))/5 - 1)^(1/2))/25 - (16777216*5^(1/2))/5 - 16777216*((2*5^(1
/2))/5 - 1)^(1/2) - (436207616*5^(1/2)*tan(x/2)*((2*5^(1/2))/5 - 1)^(1/2))/25 + 184549376/25)) - (989855744*((
(2*5^(1/2))/5 - 1)^(1/2)/50 - 1/50)^(1/2))/(5*((301989888*tan(x/2))/5 - (2382364672*5^(1/2)*tan(x/2))/125 + (1
308622848*tan(x/2)*((2*5^(1/2))/5 - 1)^(1/2))/25 + (452984832*5^(1/2)*((2*5^(1/2))/5 - 1)^(1/2))/25 - (1677721
6*5^(1/2))/5 - 16777216*((2*5^(1/2))/5 - 1)^(1/2) - (436207616*5^(1/2)*tan(x/2)*((2*5^(1/2))/5 - 1)^(1/2))/25
+ 184549376/25)) + (1627389952*5^(1/2)*(((2*5^(1/2))/5 - 1)^(1/2)/50 - 1/50)^(1/2))/(25*((301989888*tan(x/2))/
5 - (2382364672*5^(1/2)*tan(x/2))/125 + (1308622848*tan(x/2)*((2*5^(1/2))/5 - 1)^(1/2))/25 + (452984832*5^(1/2
)*((2*5^(1/2))/5 - 1)^(1/2))/25 - (16777216*5^(1/2))/5 - 16777216*((2*5^(1/2))/5 - 1)^(1/2) - (436207616*5^(1/
2)*tan(x/2)*((2*5^(1/2))/5 - 1)^(1/2))/25 + 184549376/25)) - (553648128*((2*5^(1/2))/5 - 1)^(1/2)*(((2*5^(1/2)
)/5 - 1)^(1/2)/50 - 1/50)^(1/2))/(5*((301989888*tan(x/2))/5 - (2382364672*5^(1/2)*tan(x/2))/125 + (1308622848*
tan(x/2)*((2*5^(1/2))/5 - 1)^(1/2))/25 + (452984832*5^(1/2)*((2*5^(1/2))/5 - 1)^(1/2))/25 - (16777216*5^(1/2))
/5 - 16777216*((2*5^(1/2))/5 - 1)^(1/2) - (436207616*5^(1/2)*tan(x/2)*((2*5^(1/2))/5 - 1)^(1/2))/25 + 18454937
6/25)) + (184549376*5^(1/2)*((2*5^(1/2))/5 - 1)^(1/2)*(((2*5^(1/2))/5 - 1)^(1/2)/50 - 1/50)^(1/2))/(5*((301989
888*tan(x/2))/5 - (2382364672*5^(1/2)*tan(x/2))/125 + (1308622848*tan(x/2)*((2*5^(1/2))/5 - 1)^(1/2))/25 + (45
2984832*5^(1/2)*((2*5^(1/2))/5 - 1)^(1/2))/25 - (16777216*5^(1/2))/5 - 16777216*((2*5^(1/2))/5 - 1)^(1/2) - (4
36207616*5^(1/2)*tan(x/2)*((2*5^(1/2))/5 - 1)^(1/2))/25 + 184549376/25)) - (5083496448*5^(1/2)*tan(x/2)*(((2*5
^(1/2))/5 - 1)^(1/2)/50 - 1/50)^(1/2))/(25*((301989888*tan(x/2))/5 - (2382364672*5^(1/2)*tan(x/2))/125 + (1308
622848*tan(x/2)*((2*5^(1/2))/5 - 1)^(1/2))/25 + (452984832*5^(1/2)*((2*5^(1/2))/5 - 1)^(1/2))/25 - (16777216*5
^(1/2))/5 - 16777216*((2*5^(1/2))/5 - 1)^(1/2) - (436207616*5^(1/2)*tan(x/2)*((2*5^(1/2))/5 - 1)^(1/2))/25 + 1
84549376/25)) + (553648128*tan(x/2)*((2*5^(1/2))/5 - 1)^(1/2)*(((2*5^(1/2))/5 - 1)^(1/2)/50 - 1/50)^(1/2))/(5*
((301989888*tan(x/2))/5 - (2382364672*5^(1/2)*tan(x/2))/125 + (1308622848*tan(x/2)*((2*5^(1/2))/5 - 1)^(1/2))/
25 + (452984832*5^(1/2)*((2*5^(1/2))/5 - 1)^(1/2))/25 - (16777216*5^(1/2))/5 - 16777216*((2*5^(1/2))/5 - 1)^(1
/2) - (436207616*5^(1/2)*tan(x/2)*((2*5^(1/2))/5 - 1)^(1/2))/25 + 184549376/25)) + (553648128*5^(1/2)*tan(x/2)
*((2*5^(1/2))/5 - 1)^(1/2)*(((2*5^(1/2))/5 - 1)^(1/2)/50 - 1/50)^(1/2))/(5*((301989888*tan(x/2))/5 - (23823646
72*5^(1/2)*tan(x/2))/125 + (1308622848*tan(x/2)*((2*5^(1/2))/5 - 1)^(1/2))/25 + (452984832*5^(1/2)*((2*5^(1/2)
)/5 - 1)^(1/2))/25 - (16777216*5^(1/2))/5 - 16777216*((2*5^(1/2))/5 - 1)^(1/2) - (436207616*5^(1/2)*tan(x/2)*(
(2*5^(1/2))/5 - 1)^(1/2))/25 + 184549376/25)))*(((2*5^(1/2))/5 - 1)^(1/2)/50 - 1/50)^(1/2)
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (\sin {\relax (x )} + 1\right ) \left (\sin ^{4}{\relax (x )} - \sin ^{3}{\relax (x )} + \sin ^{2}{\relax (x )} - \sin {\relax (x )} + 1\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(1+sin(x)**5),x)
[Out]
Integral(1/((sin(x) + 1)*(sin(x)**4 - sin(x)**3 + sin(x)**2 - sin(x) + 1)), x)
________________________________________________________________________________________