Integrand size = 10, antiderivative size = 187 \[ \int \frac {1}{1-\sin ^5(x)} \, dx=-\frac {2 \arctan \left (\frac {(-1)^{2/5}-\tan \left (\frac {x}{2}\right )}{\sqrt {1-(-1)^{4/5}}}\right )}{5 \sqrt {1-(-1)^{4/5}}}-\frac {2 \arctan \left (\frac {(-1)^{4/5}-\tan \left (\frac {x}{2}\right )}{\sqrt {1+(-1)^{3/5}}}\right )}{5 \sqrt {1+(-1)^{3/5}}}+\frac {2 \arctan \left (\frac {\sqrt [5]{-1}+\tan \left (\frac {x}{2}\right )}{\sqrt {1-(-1)^{2/5}}}\right )}{5 \sqrt {1-(-1)^{2/5}}}+\frac {2 \arctan \left (\frac {(-1)^{3/5}+\tan \left (\frac {x}{2}\right )}{\sqrt {1+\sqrt [5]{-1}}}\right )}{5 \sqrt {1+\sqrt [5]{-1}}}+\frac {\cos (x)}{5 (1-\sin (x))} \]
1/5*cos(x)/(1-sin(x))+2/5*arctan(((-1)^(3/5)+tan(1/2*x))/(1+(-1)^(1/5))^(1 /2))/(1+(-1)^(1/5))^(1/2)+2/5*arctan(((-1)^(1/5)+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 = 413, normalized size of antiderivative = 2.21 \[ \int \frac {1}{1-\sin ^5(x)} \, dx=\frac {1}{10} i \text {RootSum}\left [1-2 i \text {$\#$1}-8 \text {$\#$1}^2+14 i \text {$\#$1}^3+30 \text {$\#$1}^4-14 i \text {$\#$1}^5-8 \text {$\#$1}^6+2 i \text {$\#$1}^7+\text {$\#$1}^8\&,\frac {-2 \arctan \left (\frac {\sin (x)}{\cos (x)-\text {$\#$1}}\right )+i \log \left (1-2 \cos (x) \text {$\#$1}+\text {$\#$1}^2\right )+8 i \arctan \left (\frac {\sin (x)}{\cos (x)-\text {$\#$1}}\right ) \text {$\#$1}+4 \log \left (1-2 \cos (x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}+30 \arctan \left (\frac {\sin (x)}{\cos (x)-\text {$\#$1}}\right ) \text {$\#$1}^2-15 i \log \left (1-2 \cos (x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^2-80 i \arctan \left (\frac {\sin (x)}{\cos (x)-\text {$\#$1}}\right ) \text {$\#$1}^3-40 \log \left (1-2 \cos (x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^3-30 \arctan \left (\frac {\sin (x)}{\cos (x)-\text {$\#$1}}\right ) \text {$\#$1}^4+15 i \log \left (1-2 \cos (x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^4+8 i \arctan \left (\frac {\sin (x)}{\cos (x)-\text {$\#$1}}\right ) \text {$\#$1}^5+4 \log \left (1-2 \cos (x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^5+2 \arctan \left (\frac {\sin (x)}{\cos (x)-\text {$\#$1}}\right ) \text {$\#$1}^6-i \log \left (1-2 \cos (x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^6}{-i-8 \text {$\#$1}+21 i \text {$\#$1}^2+60 \text {$\#$1}^3-35 i \text {$\#$1}^4-24 \text {$\#$1}^5+7 i \text {$\#$1}^6+4 \text {$\#$1}^7}\&\right ]+\frac {2 \sin \left (\frac {x}{2}\right )}{5 \left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )} \]
(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*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]))
Time = 0.51 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.05, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, 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.
| \(\displaystyle \int \frac {1}{1-\sin ^5(x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{1-\sin (x)^5}dx\) |
| \(\Big \downarrow \) 3692 |
| \(\displaystyle \int \left (\frac {1}{5 \left (\sqrt [5]{-1} \sin (x)+1\right )}+\frac {1}{5 \left (1-(-1)^{2/5} \sin (x)\right )}+\frac {1}{5 \left ((-1)^{3/5} \sin (x)+1\right )}+\frac {1}{5 \left (1-(-1)^{4/5} \sin (x)\right )}+\frac {1}{5 (1-\sin (x))}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 \arctan \left (\frac {\tan \left (\frac {x}{2}\right )+\sqrt [5]{-1}}{\sqrt {1-(-1)^{2/5}}}\right )}{5 \sqrt {1-(-1)^{2/5}}}+\frac {2 \arctan \left (\frac {\tan \left (\frac {x}{2}\right )+(-1)^{3/5}}{\sqrt {1+\sqrt [5]{-1}}}\right )}{5 \sqrt {1+\sqrt [5]{-1}}}-\frac {2 \arctan \left (\frac {(-1)^{4/5} \left (\sqrt [5]{-1} \tan \left (\frac {x}{2}\right )+1\right )}{\sqrt {1+(-1)^{3/5}}}\right )}{5 \sqrt {1+(-1)^{3/5}}}-\frac {2 \arctan \left (\frac {(-1)^{2/5} \left ((-1)^{3/5} \tan \left (\frac {x}{2}\right )+1\right )}{\sqrt {1-(-1)^{4/5}}}\right )}{5 \sqrt {1-(-1)^{4/5}}}+\frac {\cos (x)}{5 (1-\sin (x))}\) |
(2*ArcTan[((-1)^(1/5) + Tan[x/2])/Sqrt[1 - (-1)^(2/5)]])/(5*Sqrt[1 - (-1)^ (2/5)]) + (2*ArcTan[((-1)^(3/5) + Tan[x/2])/Sqrt[1 + (-1)^(1/5)]])/(5*Sqrt [1 + (-1)^(1/5)]) - (2*ArcTan[((-1)^(4/5)*(1 + (-1)^(1/5)*Tan[x/2]))/Sqrt[ 1 + (-1)^(3/5)]])/(5*Sqrt[1 + (-1)^(3/5)]) - (2*ArcTan[((-1)^(2/5)*(1 + (- 1)^(3/5)*Tan[x/2]))/Sqrt[1 - (-1)^(4/5)]])/(5*Sqrt[1 - (-1)^(4/5)]) + Cos[ x]/(5*(1 - Sin[x]))
3.3.58.3.1 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 = 0.65 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.47
| 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+15625 0*_Z^6+6250*_Z^4+125*_Z^2+1))
Leaf count of result is larger than twice the leaf count of optimal. 858 vs. \(2 (127) = 254\).
Time = 0.39 (sec) , antiderivative size = 858, normalized size of antiderivative = 4.59 \[ \int \frac {1}{1-\sin ^5(x)} \, dx=\text {Too large to display} \]
-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(-...
\[ \int \frac {1}{1-\sin ^5(x)} \, dx=- \int \frac {1}{\sin ^{5}{\left (x \right )} - 1}\, dx \]
\[ \int \frac {1}{1-\sin ^5(x)} \, dx=\int { -\frac {1}{\sin \left (x\right )^{5} - 1} \,d x } \]
1/5*(5*(cos(x)^2 + sin(x)^2 - 2*sin(x) + 1)*integrate(2/5*((4*cos(6*x) - 4 0*cos(4*x) + 4*cos(2*x) + sin(7*x) - 15*sin(5*x) + 15*sin(3*x) - sin(x))*c os(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*(2 10*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*co s(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) + 1 5*cos(3*x) - cos(x) - 4*sin(6*x) + 40*sin(4*x) - 4*sin(2*x))*sin(8*x) - (1 6*cos(6*x) - 110*cos(4*x) + 16*cos(2*x) - 44*sin(5*x) + 44*sin(3*x) - 4*si n(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*si n(4*x) + 4*sin(2*x))*cos(7*x) - 4*cos(7*x)^2 + 16*(30*cos(4*x) - 8*cos(...
\[ \int \frac {1}{1-\sin ^5(x)} \, dx=\int { -\frac {1}{\sin \left (x\right )^{5} - 1} \,d x } \]
Time = 13.67 (sec) , antiderivative size = 3513, normalized size of antiderivative = 18.79 \[ \int \frac {1}{1-\sin ^5(x)} \, dx=\text {Too large to display} \]
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...