Integrand size = 10, antiderivative size = 210 \[ \int \frac {1}{a+a \sin ^5(x)} \, dx=\frac {2 \arctan \left (\frac {(-1)^{2/5}+\tan \left (\frac {x}{2}\right )}{\sqrt {1-(-1)^{4/5}}}\right )}{5 \sqrt {1-(-1)^{4/5}} a}+\frac {2 \arctan \left (\frac {(-1)^{4/5}+\tan \left (\frac {x}{2}\right )}{\sqrt {1+(-1)^{3/5}}}\right )}{5 \sqrt {1+(-1)^{3/5}} a}-\frac {2 \arctan \left (\frac {(-1)^{3/5} \left (1+(-1)^{2/5} \tan \left (\frac {x}{2}\right )\right )}{\sqrt {1+\sqrt [5]{-1}}}\right )}{5 \sqrt {1+\sqrt [5]{-1}} a}-\frac {2 \arctan \left (\frac {\sqrt [5]{-1} \left (1+(-1)^{4/5} \tan \left (\frac {x}{2}\right )\right )}{\sqrt {1-(-1)^{2/5}}}\right )}{5 \sqrt {1-(-1)^{2/5}} a}-\frac {\cos (x)}{5 a (1+\sin (x))} \] Output:
2/5*arctan(((-1)^(2/5)+tan(1/2*x))/(1-(-1)^(4/5))^(1/2))/(1-(-1)^(4/5))^(1 /2)/a+2/5*arctan(((-1)^(4/5)+tan(1/2*x))/(1+(-1)^(3/5))^(1/2))/(1+(-1)^(3/ 5))^(1/2)/a-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)/a-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)/a-1/5*cos(x)/a/(1+sin(x))
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 5.14 (sec) , antiderivative size = 414, normalized size of antiderivative = 1.97 \[ \int \frac {1}{a+a \sin ^5(x)} \, dx=\frac {-i \text {RootSum}\left [1+2 i \text {$\#$1}-8 \text {$\#$1}^2-14 i \text {$\#$1}^3+30 \text {$\#$1}^4+14 i \text {$\#$1}^5-8 \text {$\#$1}^6-2 i \text {$\#$1}^7+\text {$\#$1}^8\&,\frac {-2 \arctan \left (\frac {\sin (x)}{\cos (x)-\text {$\#$1}}\right )+i \log \left (1-2 \cos (x) \text {$\#$1}+\text {$\#$1}^2\right )-8 i \arctan \left (\frac {\sin (x)}{\cos (x)-\text {$\#$1}}\right ) \text {$\#$1}-4 \log \left (1-2 \cos (x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}+30 \arctan \left (\frac {\sin (x)}{\cos (x)-\text {$\#$1}}\right ) \text {$\#$1}^2-15 i \log \left (1-2 \cos (x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^2+80 i \arctan \left (\frac {\sin (x)}{\cos (x)-\text {$\#$1}}\right ) \text {$\#$1}^3+40 \log \left (1-2 \cos (x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^3-30 \arctan \left (\frac {\sin (x)}{\cos (x)-\text {$\#$1}}\right ) \text {$\#$1}^4+15 i \log \left (1-2 \cos (x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^4-8 i \arctan \left (\frac {\sin (x)}{\cos (x)-\text {$\#$1}}\right ) \text {$\#$1}^5-4 \log \left (1-2 \cos (x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^5+2 \arctan \left (\frac {\sin (x)}{\cos (x)-\text {$\#$1}}\right ) \text {$\#$1}^6-i \log \left (1-2 \cos (x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^6}{i-8 \text {$\#$1}-21 i \text {$\#$1}^2+60 \text {$\#$1}^3+35 i \text {$\#$1}^4-24 \text {$\#$1}^5-7 i \text {$\#$1}^6+4 \text {$\#$1}^7}\&\right ]+\frac {4 \sin \left (\frac {x}{2}\right )}{\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )}}{10 a} \] Input:
Integrate[(a + a*Sin[x]^5)^(-1),x]
Output:
((-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)*L og[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) & ] + (4*Sin[x /2])/(Cos[x/2] + Sin[x/2]))/(10*a)
Time = 0.55 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, 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.
| \(\displaystyle \int \frac {1}{a \sin ^5(x)+a} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{a \sin (x)^5+a}dx\) |
| \(\Big \downarrow \) 3692 |
| \(\displaystyle \int \left (-\frac {1}{5 a \left (\sqrt [5]{-1} \sin (x)-1\right )}-\frac {1}{5 a \left (-(-1)^{2/5} \sin (x)-1\right )}-\frac {1}{5 a \left ((-1)^{3/5} \sin (x)-1\right )}-\frac {1}{5 a \left (-(-1)^{4/5} \sin (x)-1\right )}-\frac {1}{5 a (-\sin (x)-1)}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 \arctan \left (\frac {\tan \left (\frac {x}{2}\right )+(-1)^{2/5}}{\sqrt {1-(-1)^{4/5}}}\right )}{5 \sqrt {1-(-1)^{4/5}} a}+\frac {2 \arctan \left (\frac {\tan \left (\frac {x}{2}\right )+(-1)^{4/5}}{\sqrt {1+(-1)^{3/5}}}\right )}{5 \sqrt {1+(-1)^{3/5}} a}-\frac {2 \arctan \left (\frac {(-1)^{3/5} \left ((-1)^{2/5} \tan \left (\frac {x}{2}\right )+1\right )}{\sqrt {1+\sqrt [5]{-1}}}\right )}{5 \sqrt {1+\sqrt [5]{-1}} a}-\frac {2 \arctan \left (\frac {\sqrt [5]{-1} \left ((-1)^{4/5} \tan \left (\frac {x}{2}\right )+1\right )}{\sqrt {1-(-1)^{2/5}}}\right )}{5 \sqrt {1-(-1)^{2/5}} a}-\frac {\cos (x)}{5 a (\sin (x)+1)}\) |
Input:
Int[(a + a*Sin[x]^5)^(-1),x]
Output:
(2*ArcTan[((-1)^(2/5) + Tan[x/2])/Sqrt[1 - (-1)^(4/5)]])/(5*Sqrt[1 - (-1)^ (4/5)]*a) + (2*ArcTan[((-1)^(4/5) + Tan[x/2])/Sqrt[1 + (-1)^(3/5)]])/(5*Sq rt[1 + (-1)^(3/5)]*a) - (2*ArcTan[((-1)^(3/5)*(1 + (-1)^(2/5)*Tan[x/2]))/S qrt[1 + (-1)^(1/5)]])/(5*Sqrt[1 + (-1)^(1/5)]*a) - (2*ArcTan[((-1)^(1/5)*( 1 + (-1)^(4/5)*Tan[x/2]))/Sqrt[1 - (-1)^(2/5)]])/(5*Sqrt[1 - (-1)^(2/5)]*a ) - Cos[x]/(5*a*(1 + Sin[x]))
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 = 0.80 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.58
| method | result | size |
| risch | \(-\frac {2}{5 \left ({\mathrm e}^{i x}+i\right ) a}+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (1953125 a^{8} \textit {\_Z}^{8}+156250 a^{6} \textit {\_Z}^{6}+6250 a^{4} \textit {\_Z}^{4}+125 a^{2} \textit {\_Z}^{2}+1\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{i x}+2343750 a^{7} \textit {\_R}^{7}+234375 i a^{6} \textit {\_R}^{6}+140625 a^{5} \textit {\_R}^{5}+15625 i a^{4} \textit {\_R}^{4}+4375 a^{3} \textit {\_R}^{3}+500 i a^{2} \textit {\_R}^{2}+50 a \textit {\_R} +6 i\right )\right )\) | \(121\) |
| default | \(\frac {\frac {2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-2 \textit {\_Z}^{7}+8 \textit {\_Z}^{6}-14 \textit {\_Z}^{5}+30 \textit {\_Z}^{4}-14 \textit {\_Z}^{3}+8 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right )}{\sum }\frac {\left (2 \textit {\_R}^{6}-3 \textit {\_R}^{5}+10 \textit {\_R}^{4}-10 \textit {\_R}^{3}+10 \textit {\_R}^{2}-3 \textit {\_R} +2\right ) \ln \left (\tan \left (\frac {x}{2}\right )-\textit {\_R} \right )}{4 \textit {\_R}^{7}-7 \textit {\_R}^{6}+24 \textit {\_R}^{5}-35 \textit {\_R}^{4}+60 \textit {\_R}^{3}-21 \textit {\_R}^{2}+8 \textit {\_R} -1}\right )}{5}-\frac {2}{5 \left (\tan \left (\frac {x}{2}\right )+1\right )}}{a}\) | \(138\) |
Input:
int(1/(a+a*sin(x)^5),x,method=_RETURNVERBOSE)
Output:
-2/5/(exp(I*x)+I)/a+sum(_R*ln(exp(I*x)+2343750*a^7*_R^7+234375*I*a^6*_R^6+ 140625*a^5*_R^5+15625*I*a^4*_R^4+4375*a^3*_R^3+500*I*a^2*_R^2+50*a*_R+6*I) ,_R=RootOf(1953125*_Z^8*a^8+156250*_Z^6*a^6+6250*_Z^4*a^4+125*_Z^2*a^2+1))
Leaf count of result is larger than twice the leaf count of optimal. 1612 vs. \(2 (148) = 296\).
Time = 0.25 (sec) , antiderivative size = 1612, normalized size of antiderivative = 7.68 \[ \int \frac {1}{a+a \sin ^5(x)} \, dx=\text {Too large to display} \] Input:
integrate(1/(a+a*sin(x)^5),x, algorithm="fricas")
Output:
1/20*(sqrt(2)*(a*cos(x) + a*sin(x) + a)*sqrt(-(a^2*sqrt(-(2*sqrt(1/5)*a^4* sqrt(a^(-8)) + 1)/a^4) + 1)/a^2)*log(5*sqrt(1/5)*a^4*sqrt(a^(-8))*sin(x) - 5*sqrt(2)*(3*sqrt(1/5)*a^7*sqrt(a^(-8))*cos(x) - a^3*cos(x))*sqrt(-(a^2*s qrt(-(2*sqrt(1/5)*a^4*sqrt(a^(-8)) + 1)/a^4) + 1)/a^2)*sqrt(-(2*sqrt(1/5)* a^4*sqrt(a^(-8)) + 1)/a^4) - 5*(3*sqrt(1/5)*a^6*sqrt(a^(-8))*sin(x) - a^2* sin(x))*sqrt(-(2*sqrt(1/5)*a^4*sqrt(a^(-8)) + 1)/a^4) + sin(x) - 4) - sqrt (2)*(a*cos(x) + a*sin(x) + a)*sqrt((a^2*sqrt(-(2*sqrt(1/5)*a^4*sqrt(a^(-8) ) + 1)/a^4) - 1)/a^2)*log(5*sqrt(1/5)*a^4*sqrt(a^(-8))*sin(x) - 5*sqrt(2)* (3*sqrt(1/5)*a^7*sqrt(a^(-8))*cos(x) - a^3*cos(x))*sqrt((a^2*sqrt(-(2*sqrt (1/5)*a^4*sqrt(a^(-8)) + 1)/a^4) - 1)/a^2)*sqrt(-(2*sqrt(1/5)*a^4*sqrt(a^( -8)) + 1)/a^4) + 5*(3*sqrt(1/5)*a^6*sqrt(a^(-8))*sin(x) - a^2*sin(x))*sqrt (-(2*sqrt(1/5)*a^4*sqrt(a^(-8)) + 1)/a^4) + sin(x) - 4) + sqrt(2)*(a*cos(x ) + a*sin(x) + a)*sqrt(-(a^2*sqrt((2*sqrt(1/5)*a^4*sqrt(a^(-8)) - 1)/a^4) + 1)/a^2)*log(5*sqrt(1/5)*a^4*sqrt(a^(-8))*sin(x) - 5*sqrt(2)*(3*sqrt(1/5) *a^7*sqrt(a^(-8))*cos(x) + a^3*cos(x))*sqrt(-(a^2*sqrt((2*sqrt(1/5)*a^4*sq rt(a^(-8)) - 1)/a^4) + 1)/a^2)*sqrt((2*sqrt(1/5)*a^4*sqrt(a^(-8)) - 1)/a^4 ) - 5*(3*sqrt(1/5)*a^6*sqrt(a^(-8))*sin(x) + a^2*sin(x))*sqrt((2*sqrt(1/5) *a^4*sqrt(a^(-8)) - 1)/a^4) - sin(x) + 4) - sqrt(2)*(a*cos(x) + a*sin(x) + a)*sqrt((a^2*sqrt((2*sqrt(1/5)*a^4*sqrt(a^(-8)) - 1)/a^4) - 1)/a^2)*log(5 *sqrt(1/5)*a^4*sqrt(a^(-8))*sin(x) - 5*sqrt(2)*(3*sqrt(1/5)*a^7*sqrt(a^...
\[ \int \frac {1}{a+a \sin ^5(x)} \, dx=\frac {\int \frac {1}{\sin ^{5}{\left (x \right )} + 1}\, dx}{a} \] Input:
integrate(1/(a+a*sin(x)**5),x)
Output:
Integral(1/(sin(x)**5 + 1), x)/a
\[ \int \frac {1}{a+a \sin ^5(x)} \, dx=\int { \frac {1}{a \sin \left (x\right )^{5} + a} \,d x } \] Input:
integrate(1/(a+a*sin(x)^5),x, algorithm="maxima")
Output:
-1/5*(5*(a*cos(x)^2 + a*sin(x)^2 + 2*a*sin(x) + a)*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) + si n(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) - 21 0*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))/(a*cos(8*x)^2 + 4*a*cos( 7*x)^2 + 64*a*cos(6*x)^2 + 196*a*cos(5*x)^2 + 900*a*cos(4*x)^2 + 196*a*cos (3*x)^2 + 64*a*cos(2*x)^2 + 4*a*cos(x)^2 + a*sin(8*x)^2 + 4*a*sin(7*x)^2 + 64*a*sin(6*x)^2 + 196*a*sin(5*x)^2 + 900*a*sin(4*x)^2 + 196*a*sin(3*x)...
Leaf count of result is larger than twice the leaf count of optimal. 2033 vs. \(2 (148) = 296\).
Time = 2.45 (sec) , antiderivative size = 2033, normalized size of antiderivative = 9.68 \[ \int \frac {1}{a+a \sin ^5(x)} \, dx=\text {Too large to display} \] Input:
integrate(1/(a+a*sin(x)^5),x, algorithm="giac")
Output:
1/50*(2*sqrt(1/5)*sqrt(2*sqrt(5) + 5)*sqrt(5*sqrt(10*sqrt(5) + 50) - 25)*( arctan(1/2) + arctan(1/2*(24209596193492233425*sqrt(5)*(sqrt(5) + 5) - 496 45000851686087842*sqrt(5)*sqrt(10*sqrt(5) + 50)*sqrt(5*sqrt(10*sqrt(5) + 5 0) - 25) + 248225004258430439210*sqrt(10*sqrt(5) + 50)*sqrt(5*sqrt(10*sqrt (5) + 50) - 25) - 2603298023551765559225*sqrt(5) + 9929000170337217568400* tan(1/2*x) - 2603298023551765559225)/(1474476248065267148200*sqrt(5)*(sqrt (5) + 5) + 148935002555058263526*sqrt(5)*sqrt(10*sqrt(5) + 50)*sqrt(5*sqrt (10*sqrt(5) + 50) - 25) + 124112502129215219605*sqrt(5)*sqrt(10*sqrt(5) + 50) + 744675012775291317630*sqrt(5)*sqrt(5*sqrt(10*sqrt(5) + 50) - 25) - 2 48225004258430439210*sqrt(10*sqrt(5) + 50)*sqrt(5*sqrt(10*sqrt(5) + 50) - 25) - 7372381240326335741000*sqrt(5) - 620562510646076098025*sqrt(10*sqrt( 5) + 50) - 1241125021292152196050*sqrt(5*sqrt(10*sqrt(5) + 50) - 25) - 737 2381240326335741000)))/(2*sqrt(1/10)*sqrt(sqrt(5) + 5) - 1) + 2*sqrt(1/5)* sqrt(2*sqrt(5) + 5)*sqrt(5*sqrt(10*sqrt(5) + 50) - 25)*arctan(1/2*(5878490 76675773567575*sqrt(5)*(sqrt(5) + 5) + 49645000851686087842*sqrt(5)*sqrt(1 0*sqrt(5) + 50)*sqrt(5*sqrt(10*sqrt(5) + 50) - 25) - 248225004258430439210 *sqrt(10*sqrt(5) + 50)*sqrt(5*sqrt(10*sqrt(5) + 50) - 25) - 54214954259631 72229975*sqrt(5) + 9929000170337217568400*tan(1/2*x) - 5421495425963172229 975)/(1701866802206171210550*sqrt(5)*(sqrt(5) + 5) + 148935002555058263526 *sqrt(5)*sqrt(10*sqrt(5) + 50)*sqrt(5*sqrt(10*sqrt(5) + 50) - 25) - 124...
Time = 37.31 (sec) , antiderivative size = 4652, normalized size of antiderivative = 22.15 \[ \int \frac {1}{a+a \sin ^5(x)} \, dx=\text {Too large to display} \] Input:
int(1/(a + a*sin(x)^5),x)
Output:
2*atanh((241664000000000000*a^7*((- (2*5^(1/2))/5 - 1)^(1/2)/(50*a^2) - 1/ (50*a^2))^(1/2))/(73728000000000000*a^6*tan(x/2) + 4096000000000000*5^(1/2 )*a^6 - 20480000000000000*a^6*(- (2*5^(1/2))/5 - 1)^(1/2) + 90112000000000 00*a^6 + 23265280000000000*5^(1/2)*a^6*tan(x/2) + 63897600000000000*a^6*ta n(x/2)*(- (2*5^(1/2))/5 - 1)^(1/2) - 22118400000000000*5^(1/2)*a^6*(- (2*5 ^(1/2))/5 - 1)^(1/2) + 21299200000000000*5^(1/2)*a^6*tan(x/2)*(- (2*5^(1/2 ))/5 - 1)^(1/2)) - (495616000000000000*a^7*tan(x/2)*((- (2*5^(1/2))/5 - 1) ^(1/2)/(50*a^2) - 1/(50*a^2))^(1/2))/(73728000000000000*a^6*tan(x/2) + 409 6000000000000*5^(1/2)*a^6 - 20480000000000000*a^6*(- (2*5^(1/2))/5 - 1)^(1 /2) + 9011200000000000*a^6 + 23265280000000000*5^(1/2)*a^6*tan(x/2) + 6389 7600000000000*a^6*tan(x/2)*(- (2*5^(1/2))/5 - 1)^(1/2) - 22118400000000000 *5^(1/2)*a^6*(- (2*5^(1/2))/5 - 1)^(1/2) + 21299200000000000*5^(1/2)*a^6*t an(x/2)*(- (2*5^(1/2))/5 - 1)^(1/2)) + (79462400000000000*5^(1/2)*a^7*((- (2*5^(1/2))/5 - 1)^(1/2)/(50*a^2) - 1/(50*a^2))^(1/2))/(73728000000000000* a^6*tan(x/2) + 4096000000000000*5^(1/2)*a^6 - 20480000000000000*a^6*(- (2* 5^(1/2))/5 - 1)^(1/2) + 9011200000000000*a^6 + 23265280000000000*5^(1/2)*a ^6*tan(x/2) + 63897600000000000*a^6*tan(x/2)*(- (2*5^(1/2))/5 - 1)^(1/2) - 22118400000000000*5^(1/2)*a^6*(- (2*5^(1/2))/5 - 1)^(1/2) + 2129920000000 0000*5^(1/2)*a^6*tan(x/2)*(- (2*5^(1/2))/5 - 1)^(1/2)) + (1351680000000000 00*a^7*((- (2*5^(1/2))/5 - 1)^(1/2)/(50*a^2) - 1/(50*a^2))^(1/2)*(- (2*...
\[ \int \frac {1}{a+a \sin ^5(x)} \, dx=\frac {\int \frac {1}{\sin \left (x \right )^{5}+1}d x}{a} \] Input:
int(1/(a+a*sin(x)^5),x)
Output:
int(1/(sin(x)**5 + 1),x)/a