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]

(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]))

________________________________________________________________________________________

Rubi [A]  time = 0.382604, antiderivative size = 195, normalized size of antiderivative = 1., 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 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]))

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 618

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

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

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 = 0.159132, size = 411, normalized size = 2.11 \[ \frac{2 \sin \left (\frac{x}{2}\right )}{5 \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )}-\frac{1}{10} i \text{RootSum}\left [\text{$\#$1}^8-2 i \text{$\#$1}^7-8 \text{$\#$1}^6+14 i \text{$\#$1}^5+30 \text{$\#$1}^4-14 i \text{$\#$1}^3-8 \text{$\#$1}^2+2 i \text{$\#$1}+1\& ,\frac{-i \text{$\#$1}^6 \log \left (\text{$\#$1}^2-2 \text{$\#$1} \cos (x)+1\right )-4 \text{$\#$1}^5 \log \left (\text{$\#$1}^2-2 \text{$\#$1} \cos (x)+1\right )+15 i \text{$\#$1}^4 \log \left (\text{$\#$1}^2-2 \text{$\#$1} \cos (x)+1\right )+40 \text{$\#$1}^3 \log \left (\text{$\#$1}^2-2 \text{$\#$1} \cos (x)+1\right )-15 i \text{$\#$1}^2 \log \left (\text{$\#$1}^2-2 \text{$\#$1} \cos (x)+1\right )-4 \text{$\#$1} \log \left (\text{$\#$1}^2-2 \text{$\#$1} \cos (x)+1\right )+i \log \left (\text{$\#$1}^2-2 \text{$\#$1} \cos (x)+1\right )+2 \text{$\#$1}^6 \tan ^{-1}\left (\frac{\sin (x)}{\cos (x)-\text{$\#$1}}\right )-8 i \text{$\#$1}^5 \tan ^{-1}\left (\frac{\sin (x)}{\cos (x)-\text{$\#$1}}\right )-30 \text{$\#$1}^4 \tan ^{-1}\left (\frac{\sin (x)}{\cos (x)-\text{$\#$1}}\right )+80 i \text{$\#$1}^3 \tan ^{-1}\left (\frac{\sin (x)}{\cos (x)-\text{$\#$1}}\right )+30 \text{$\#$1}^2 \tan ^{-1}\left (\frac{\sin (x)}{\cos (x)-\text{$\#$1}}\right )-8 i \text{$\#$1} \tan ^{-1}\left (\frac{\sin (x)}{\cos (x)-\text{$\#$1}}\right )-2 \tan ^{-1}\left (\frac{\sin (x)}{\cos (x)-\text{$\#$1}}\right )}{4 \text{$\#$1}^7-7 i \text{$\#$1}^6-24 \text{$\#$1}^5+35 i \text{$\#$1}^4+60 \text{$\#$1}^3-21 i \text{$\#$1}^2-8 \text{$\#$1}+i}\& \right ] \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + Sin[x]^5)^(-1),x]

[Out]

(-I/10)*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*Lo
g[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*Co
s[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]
))

________________________________________________________________________________________

Maple [C]  time = 0.079, size = 133, normalized size = 0.7 \begin{align*} -{\frac{2}{5} \left ( \tan \left ({\frac{x}{2}} \right ) +1 \right ) ^{-1}}+{\frac{2}{5}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{8}-2\,{{\it \_Z}}^{7}+8\,{{\it \_Z}}^{6}-14\,{{\it \_Z}}^{5}+30\,{{\it \_Z}}^{4}-14\,{{\it \_Z}}^{3}+8\,{{\it \_Z}}^{2}-2\,{\it \_Z}+1 \right ) }{\frac{2\,{{\it \_R}}^{6}-3\,{{\it \_R}}^{5}+10\,{{\it \_R}}^{4}-10\,{{\it \_R}}^{3}+10\,{{\it \_R}}^{2}-3\,{\it \_R}+2}{4\,{{\it \_R}}^{7}-7\,{{\it \_R}}^{6}+24\,{{\it \_R}}^{5}-35\,{{\it \_R}}^{4}+60\,{{\it \_R}}^{3}-21\,{{\it \_R}}^{2}+8\,{\it \_R}-1}\ln \left ( \tan \left ({\frac{x}{2}} \right ) -{\it \_R} \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1+sin(x)^5),x)

[Out]

-2/5/(tan(1/2*x)+1)+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))

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}

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)

________________________________________________________________________________________

Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+sin(x)^5),x, algorithm="fricas")

[Out]

Timed out

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (\sin{\left (x \right )} + 1\right ) \left (\sin ^{4}{\left (x \right )} - \sin ^{3}{\left (x \right )} + \sin ^{2}{\left (x \right )} - \sin{\left (x \right )} + 1\right )}\, dx \end{align*}

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)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sin \left (x\right )^{5} + 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+sin(x)^5),x, algorithm="giac")

[Out]

integrate(1/(sin(x)^5 + 1), x)