Integrand size = 8, antiderivative size = 195 \[ \int \frac {1}{1+\sin ^5(x)} \, dx=\frac {2 \arctan \left (\frac {(-1)^{2/5}+\tan \left (\frac {x}{2}\right )}{\sqrt {1-(-1)^{4/5}}}\right )}{5 \sqrt {1-(-1)^{4/5}}}+\frac {2 \arctan \left (\frac {(-1)^{4/5}+\tan \left (\frac {x}{2}\right )}{\sqrt {1+(-1)^{3/5}}}\right )}{5 \sqrt {1+(-1)^{3/5}}}-\frac {2 \arctan \left (\frac {(-1)^{3/5} \left (1+(-1)^{2/5} \tan \left (\frac {x}{2}\right )\right )}{\sqrt {1+\sqrt [5]{-1}}}\right )}{5 \sqrt {1+\sqrt [5]{-1}}}-\frac {2 \arctan \left (\frac {\sqrt [5]{-1} \left (1+(-1)^{4/5} \tan \left (\frac {x}{2}\right )\right )}{\sqrt {1-(-1)^{2/5}}}\right )}{5 \sqrt {1-(-1)^{2/5}}}-\frac {\cos (x)}{5 (1+\sin (x))} \]
-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)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 5.09 (sec) , antiderivative size = 411, normalized size of antiderivative = 2.11 \[ \int \frac {1}{1+\sin ^5(x)} \, dx=-\frac {1}{10} i \text {RootSum}\left [1+2 i \text {$\#$1}-8 \text {$\#$1}^2-14 i \text {$\#$1}^3+30 \text {$\#$1}^4+14 i \text {$\#$1}^5-8 \text {$\#$1}^6-2 i \text {$\#$1}^7+\text {$\#$1}^8\&,\frac {-2 \arctan \left (\frac {\sin (x)}{\cos (x)-\text {$\#$1}}\right )+i \log \left (1-2 \cos (x) \text {$\#$1}+\text {$\#$1}^2\right )-8 i \arctan \left (\frac {\sin (x)}{\cos (x)-\text {$\#$1}}\right ) \text {$\#$1}-4 \log \left (1-2 \cos (x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}+30 \arctan \left (\frac {\sin (x)}{\cos (x)-\text {$\#$1}}\right ) \text {$\#$1}^2-15 i \log \left (1-2 \cos (x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^2+80 i \arctan \left (\frac {\sin (x)}{\cos (x)-\text {$\#$1}}\right ) \text {$\#$1}^3+40 \log \left (1-2 \cos (x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^3-30 \arctan \left (\frac {\sin (x)}{\cos (x)-\text {$\#$1}}\right ) \text {$\#$1}^4+15 i \log \left (1-2 \cos (x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^4-8 i \arctan \left (\frac {\sin (x)}{\cos (x)-\text {$\#$1}}\right ) \text {$\#$1}^5-4 \log \left (1-2 \cos (x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^5+2 \arctan \left (\frac {\sin (x)}{\cos (x)-\text {$\#$1}}\right ) \text {$\#$1}^6-i \log \left (1-2 \cos (x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^6}{i-8 \text {$\#$1}-21 i \text {$\#$1}^2+60 \text {$\#$1}^3+35 i \text {$\#$1}^4-24 \text {$\#$1}^5-7 i \text {$\#$1}^6+4 \text {$\#$1}^7}\&\right ]+\frac {2 \sin \left (\frac {x}{2}\right )}{5 \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )} \]
(-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*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]/(C os[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*S in[x/2])/(5*(Cos[x/2] + Sin[x/2]))
Time = 0.57 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {3042, 3692, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {1}{\sin ^5(x)+1} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\sin (x)^5+1}dx\) |
| \(\Big \downarrow \) 3692 |
| \(\displaystyle \int \left (-\frac {1}{5 \left (\sqrt [5]{-1} \sin (x)-1\right )}-\frac {1}{5 \left (-(-1)^{2/5} \sin (x)-1\right )}-\frac {1}{5 \left ((-1)^{3/5} \sin (x)-1\right )}-\frac {1}{5 \left (-(-1)^{4/5} \sin (x)-1\right )}-\frac {1}{5 (-\sin (x)-1)}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 \arctan \left (\frac {\tan \left (\frac {x}{2}\right )+(-1)^{2/5}}{\sqrt {1-(-1)^{4/5}}}\right )}{5 \sqrt {1-(-1)^{4/5}}}+\frac {2 \arctan \left (\frac {\tan \left (\frac {x}{2}\right )+(-1)^{4/5}}{\sqrt {1+(-1)^{3/5}}}\right )}{5 \sqrt {1+(-1)^{3/5}}}-\frac {2 \arctan \left (\frac {(-1)^{3/5} \left ((-1)^{2/5} \tan \left (\frac {x}{2}\right )+1\right )}{\sqrt {1+\sqrt [5]{-1}}}\right )}{5 \sqrt {1+\sqrt [5]{-1}}}-\frac {2 \arctan \left (\frac {\sqrt [5]{-1} \left ((-1)^{4/5} \tan \left (\frac {x}{2}\right )+1\right )}{\sqrt {1-(-1)^{2/5}}}\right )}{5 \sqrt {1-(-1)^{2/5}}}-\frac {\cos (x)}{5 (\sin (x)+1)}\) |
(2*ArcTan[((-1)^(2/5) + Tan[x/2])/Sqrt[1 - (-1)^(4/5)]])/(5*Sqrt[1 - (-1)^ (4/5)]) + (2*ArcTan[((-1)^(4/5) + Tan[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]))/Sqrt[ 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]))
3.3.55.3.1 Defintions of rubi rules used
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]))
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.68 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.45
| method | result | size |
| risch | \(-\frac {2}{5 \left ({\mathrm e}^{i x}+i\right )}+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (1953125 \textit {\_Z}^{8}+156250 \textit {\_Z}^{6}+6250 \textit {\_Z}^{4}+125 \textit {\_Z}^{2}+1\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{i x}+2343750 \textit {\_R}^{7}+234375 i \textit {\_R}^{6}+140625 \textit {\_R}^{5}+15625 i \textit {\_R}^{4}+4375 \textit {\_R}^{3}+500 i \textit {\_R}^{2}+50 \textit {\_R} +6 i\right )\right )\) | \(87\) |
| default | \(\frac {2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-2 \textit {\_Z}^{7}+8 \textit {\_Z}^{6}-14 \textit {\_Z}^{5}+30 \textit {\_Z}^{4}-14 \textit {\_Z}^{3}+8 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right )}{\sum }\frac {\left (2 \textit {\_R}^{6}-3 \textit {\_R}^{5}+10 \textit {\_R}^{4}-10 \textit {\_R}^{3}+10 \textit {\_R}^{2}-3 \textit {\_R} +2\right ) \ln \left (\tan \left (\frac {x}{2}\right )-\textit {\_R} \right )}{4 \textit {\_R}^{7}-7 \textit {\_R}^{6}+24 \textit {\_R}^{5}-35 \textit {\_R}^{4}+60 \textit {\_R}^{3}-21 \textit {\_R}^{2}+8 \textit {\_R} -1}\right )}{5}-\frac {2}{5 \left (\tan \left (\frac {x}{2}\right )+1\right )}\) | \(133\) |
-2/5/(exp(I*x)+I)+sum(_R*ln(exp(I*x)+2343750*_R^7+234375*I*_R^6+140625*_R^ 5+15625*I*_R^4+4375*_R^3+500*I*_R^2+50*_R+6*I),_R=RootOf(1953125*_Z^8+1562 50*_Z^6+6250*_Z^4+125*_Z^2+1))
Leaf count of result is larger than twice the leaf count of optimal. 848 vs. \(2 (133) = 266\).
Time = 0.41 (sec) , antiderivative size = 848, normalized size of antiderivative = 4.35 \[ \int \frac {1}{1+\sin ^5(x)} \, dx=\text {Too large to display} \]
-1/100*((sqrt(5)*cos(x) + sqrt(5)*sin(x) + sqrt(5))*sqrt(2*sqrt(5)*sqrt(2* sqrt(5) - 5) - 10)*log(-sqrt(2*sqrt(5)*sqrt(2*sqrt(5) - 5) - 10)*(3*sqrt(5 ) + 5)*sqrt(2*sqrt(5) - 5)*cos(x) + 5*sqrt(2*sqrt(5) - 5)*(sqrt(5) + 3)*si n(x) + 5*(sqrt(5) - 1)*sin(x) + 20) - (sqrt(5)*cos(x) + sqrt(5)*sin(x) + s qrt(5))*sqrt(2*sqrt(5)*sqrt(2*sqrt(5) - 5) - 10)*log(-sqrt(2*sqrt(5)*sqrt( 2*sqrt(5) - 5) - 10)*(3*sqrt(5) + 5)*sqrt(2*sqrt(5) - 5)*cos(x) - 5*sqrt(2 *sqrt(5) - 5)*(sqrt(5) + 3)*sin(x) - 5*(sqrt(5) - 1)*sin(x) - 20) + (sqrt( 5)*cos(x) + sqrt(5)*sin(x) + sqrt(5))*sqrt(-2*sqrt(5)*sqrt(2*sqrt(5) - 5) - 10)*log(-sqrt(-2*sqrt(5)*sqrt(2*sqrt(5) - 5) - 10)*(3*sqrt(5) + 5)*sqrt( 2*sqrt(5) - 5)*cos(x) + 5*sqrt(2*sqrt(5) - 5)*(sqrt(5) + 3)*sin(x) - 5*(sq rt(5) - 1)*sin(x) - 20) - (sqrt(5)*cos(x) + sqrt(5)*sin(x) + sqrt(5))*sqrt (-2*sqrt(5)*sqrt(2*sqrt(5) - 5) - 10)*log(-sqrt(-2*sqrt(5)*sqrt(2*sqrt(5) - 5) - 10)*(3*sqrt(5) + 5)*sqrt(2*sqrt(5) - 5)*cos(x) - 5*sqrt(2*sqrt(5) - 5)*(sqrt(5) + 3)*sin(x) + 5*(sqrt(5) - 1)*sin(x) + 20) - (sqrt(5)*cos(x) + sqrt(5)*sin(x) + sqrt(5))*sqrt(2*sqrt(5)*sqrt(-2*sqrt(5) - 5) - 10)*log( -sqrt(2*sqrt(5)*sqrt(-2*sqrt(5) - 5) - 10)*(3*sqrt(5) - 5)*sqrt(-2*sqrt(5) - 5)*cos(x) + 5*(sqrt(5) - 3)*sqrt(-2*sqrt(5) - 5)*sin(x) - 5*(sqrt(5) + 1)*sin(x) + 20) + (sqrt(5)*cos(x) + sqrt(5)*sin(x) + sqrt(5))*sqrt(2*sqrt( 5)*sqrt(-2*sqrt(5) - 5) - 10)*log(-sqrt(2*sqrt(5)*sqrt(-2*sqrt(5) - 5) - 1 0)*(3*sqrt(5) - 5)*sqrt(-2*sqrt(5) - 5)*cos(x) - 5*(sqrt(5) - 3)*sqrt(-...
\[ \int \frac {1}{1+\sin ^5(x)} \, dx=\int \frac {1}{\left (\sin {\left (x \right )} + 1\right ) \left (\sin ^{4}{\left (x \right )} - \sin ^{3}{\left (x \right )} + \sin ^{2}{\left (x \right )} - \sin {\left (x \right )} + 1\right )}\, dx \]
\[ \int \frac {1}{1+\sin ^5(x)} \, dx=\int { \frac {1}{\sin \left (x\right )^{5} + 1} \,d x } \]
-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*sin( 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*sin(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*sin(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))*cos(7*x) - 4*cos(7*x)^2 + 16*(30*cos(4*x) - 8*co...
\[ \int \frac {1}{1+\sin ^5(x)} \, dx=\int { \frac {1}{\sin \left (x\right )^{5} + 1} \,d x } \]
Time = 14.27 (sec) , antiderivative size = 3513, normalized size of antiderivative = 18.02 \[ \int \frac {1}{1+\sin ^5(x)} \, dx=\text {Too large to display} \]
2*atanh((989855744*((- (2*5^(1/2))/5 - 1)^(1/2)/50 - 1/50)^(1/2))/(5*((301 989888*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 + 1845493 76/25)) - (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 + (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 + 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 + (16 777216*5^(1/2))/5 - 16777216*(- (2*5^(1/2))/5 - 1)^(1/2) + (436207616*5...