Integrand size = 11, antiderivative size = 314 \[ \int \frac {1}{a-a \sin ^{16}(x)} \, dx=-\frac {\sqrt {1+\sqrt {2}} \arctan \left (\frac {\sqrt {-1+\sqrt {2}}-2 \tan (x)}{\sqrt {1+\sqrt {2}}}\right )}{16 a}+\frac {\arctan \left (\sqrt {2} \tan (x)\right )}{8 \sqrt {2} a}+\frac {\arctan \left (\sqrt {1+\sqrt [4]{-1}} \tan (x)\right )}{8 \sqrt {1+\sqrt [4]{-1}} a}+\frac {\arctan \left (\sqrt {1-(-1)^{3/4}} \tan (x)\right )}{8 \sqrt {1-(-1)^{3/4}} a}+\frac {\sqrt {1+\sqrt {2}} \arctan \left (\frac {\sqrt {-1+\sqrt {2}}+2 \tan (x)}{\sqrt {1+\sqrt {2}}}\right )}{16 a}+\frac {\text {arctanh}\left (\sqrt {-1+\sqrt [4]{-1}} \tan (x)\right )}{8 \sqrt {-1+\sqrt [4]{-1}} a}+\frac {\text {arctanh}\left (\sqrt {-1-(-1)^{3/4}} \tan (x)\right )}{8 \sqrt {-1-(-1)^{3/4}} a}+\frac {\sqrt {-1+\sqrt {2}} \text {arctanh}\left (\frac {\sqrt {2 \left (-1+\sqrt {2}\right )} \tan (x)}{1+\sqrt {2} \tan ^2(x)}\right )}{16 a}+\frac {\tan (x)}{8 a} \] Output:
-1/16*(1+2^(1/2))^(1/2)*arctan(((2^(1/2)-1)^(1/2)-2*tan(x))/(1+2^(1/2))^(1 /2))/a+1/16*arctan(tan(x)*2^(1/2))*2^(1/2)/a+1/8*arctan((1+(-1)^(1/4))^(1/ 2)*tan(x))/(1+(-1)^(1/4))^(1/2)/a+1/8*arctan((1-(-1)^(3/4))^(1/2)*tan(x))/ (1-(-1)^(3/4))^(1/2)/a+1/16*(1+2^(1/2))^(1/2)*arctan(((2^(1/2)-1)^(1/2)+2* tan(x))/(1+2^(1/2))^(1/2))/a+1/8*arctanh((-1+(-1)^(1/4))^(1/2)*tan(x))/(-1 +(-1)^(1/4))^(1/2)/a+1/8*arctanh((-1-(-1)^(3/4))^(1/2)*tan(x))/(-1-(-1)^(3 /4))^(1/2)/a+1/16*(2^(1/2)-1)^(1/2)*arctanh((-2+2*2^(1/2))^(1/2)*tan(x)/(1 +2^(1/2)*tan(x)^2))/a+1/8*tan(x)/a
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 5.51 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.66 \[ \int \frac {1}{a-a \sin ^{16}(x)} \, dx=\frac {\frac {2 \arctan \left (\sqrt {1-i} \tan (x)\right )}{\sqrt {1-i}}+\frac {2 \arctan \left (\sqrt {1+i} \tan (x)\right )}{\sqrt {1+i}}+\sqrt {2} \arctan \left (\sqrt {2} \tan (x)\right )+64 \text {RootSum}\left [1-8 \text {$\#$1}+28 \text {$\#$1}^2-56 \text {$\#$1}^3+326 \text {$\#$1}^4-56 \text {$\#$1}^5+28 \text {$\#$1}^6-8 \text {$\#$1}^7+\text {$\#$1}^8\&,\frac {2 \arctan \left (\frac {\sin (2 x)}{\cos (2 x)-\text {$\#$1}}\right ) \text {$\#$1}^3-i \log \left (1-2 \cos (2 x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^3}{-1+7 \text {$\#$1}-21 \text {$\#$1}^2+163 \text {$\#$1}^3-35 \text {$\#$1}^4+21 \text {$\#$1}^5-7 \text {$\#$1}^6+\text {$\#$1}^7}\&\right ]+2 \tan (x)}{16 a} \] Input:
Integrate[(a - a*Sin[x]^16)^(-1),x]
Output:
((2*ArcTan[Sqrt[1 - I]*Tan[x]])/Sqrt[1 - I] + (2*ArcTan[Sqrt[1 + I]*Tan[x] ])/Sqrt[1 + I] + Sqrt[2]*ArcTan[Sqrt[2]*Tan[x]] + 64*RootSum[1 - 8*#1 + 28 *#1^2 - 56*#1^3 + 326*#1^4 - 56*#1^5 + 28*#1^6 - 8*#1^7 + #1^8 & , (2*ArcT an[Sin[2*x]/(Cos[2*x] - #1)]*#1^3 - I*Log[1 - 2*Cos[2*x]*#1 + #1^2]*#1^3)/ (-1 + 7*#1 - 21*#1^2 + 163*#1^3 - 35*#1^4 + 21*#1^5 - 7*#1^6 + #1^7) & ] + 2*Tan[x])/(16*a)
Result contains complex when optimal does not.
Time = 0.86 (sec) , antiderivative size = 221, normalized size of antiderivative = 0.70, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.818, Rules used = {3042, 3690, 3042, 3654, 3042, 3660, 216, 4254, 24}
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.
| \(\displaystyle \int \frac {1}{a-a \sin ^{16}(x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{a-a \sin (x)^{16}}dx\) |
| \(\Big \downarrow \) 3690 |
| \(\displaystyle \frac {\int \frac {1}{1-\sin ^2(x)}dx}{8 a}+\frac {\int \frac {1}{1-i \sin ^2(x)}dx}{8 a}+\frac {\int \frac {1}{i \sin ^2(x)+1}dx}{8 a}+\frac {\int \frac {1}{\sin ^2(x)+1}dx}{8 a}+\frac {\int \frac {1}{1-\sqrt [4]{-1} \sin ^2(x)}dx}{8 a}+\frac {\int \frac {1}{\sqrt [4]{-1} \sin ^2(x)+1}dx}{8 a}+\frac {\int \frac {1}{1-(-1)^{3/4} \sin ^2(x)}dx}{8 a}+\frac {\int \frac {1}{(-1)^{3/4} \sin ^2(x)+1}dx}{8 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {1}{1-\sin (x)^2}dx}{8 a}+\frac {\int \frac {1}{1-i \sin (x)^2}dx}{8 a}+\frac {\int \frac {1}{i \sin (x)^2+1}dx}{8 a}+\frac {\int \frac {1}{\sin (x)^2+1}dx}{8 a}+\frac {\int \frac {1}{1-\sqrt [4]{-1} \sin (x)^2}dx}{8 a}+\frac {\int \frac {1}{\sqrt [4]{-1} \sin (x)^2+1}dx}{8 a}+\frac {\int \frac {1}{1-(-1)^{3/4} \sin (x)^2}dx}{8 a}+\frac {\int \frac {1}{(-1)^{3/4} \sin (x)^2+1}dx}{8 a}\) |
| \(\Big \downarrow \) 3654 |
| \(\displaystyle \frac {\int \frac {1}{1-i \sin (x)^2}dx}{8 a}+\frac {\int \frac {1}{i \sin (x)^2+1}dx}{8 a}+\frac {\int \frac {1}{\sin (x)^2+1}dx}{8 a}+\frac {\int \frac {1}{1-\sqrt [4]{-1} \sin (x)^2}dx}{8 a}+\frac {\int \frac {1}{\sqrt [4]{-1} \sin (x)^2+1}dx}{8 a}+\frac {\int \frac {1}{1-(-1)^{3/4} \sin (x)^2}dx}{8 a}+\frac {\int \frac {1}{(-1)^{3/4} \sin (x)^2+1}dx}{8 a}+\frac {\int \sec ^2(x)dx}{8 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {1}{1-i \sin (x)^2}dx}{8 a}+\frac {\int \frac {1}{i \sin (x)^2+1}dx}{8 a}+\frac {\int \frac {1}{\sin (x)^2+1}dx}{8 a}+\frac {\int \frac {1}{1-\sqrt [4]{-1} \sin (x)^2}dx}{8 a}+\frac {\int \frac {1}{\sqrt [4]{-1} \sin (x)^2+1}dx}{8 a}+\frac {\int \frac {1}{1-(-1)^{3/4} \sin (x)^2}dx}{8 a}+\frac {\int \frac {1}{(-1)^{3/4} \sin (x)^2+1}dx}{8 a}+\frac {\int \csc \left (x+\frac {\pi }{2}\right )^2dx}{8 a}\) |
| \(\Big \downarrow \) 3660 |
| \(\displaystyle \frac {\int \frac {1}{(1-i) \tan ^2(x)+1}d\tan (x)}{8 a}+\frac {\int \frac {1}{(1+i) \tan ^2(x)+1}d\tan (x)}{8 a}+\frac {\int \frac {1}{2 \tan ^2(x)+1}d\tan (x)}{8 a}+\frac {\int \frac {1}{\left (1-\sqrt [4]{-1}\right ) \tan ^2(x)+1}d\tan (x)}{8 a}+\frac {\int \frac {1}{\left (1+\sqrt [4]{-1}\right ) \tan ^2(x)+1}d\tan (x)}{8 a}+\frac {\int \frac {1}{\left (1-(-1)^{3/4}\right ) \tan ^2(x)+1}d\tan (x)}{8 a}+\frac {\int \frac {1}{\left (1+(-1)^{3/4}\right ) \tan ^2(x)+1}d\tan (x)}{8 a}+\frac {\int \csc \left (x+\frac {\pi }{2}\right )^2dx}{8 a}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {\int \csc \left (x+\frac {\pi }{2}\right )^2dx}{8 a}+\frac {\arctan \left (\sqrt {1-i} \tan (x)\right )}{8 \sqrt {1-i} a}+\frac {\arctan \left (\sqrt {1+i} \tan (x)\right )}{8 \sqrt {1+i} a}+\frac {\arctan \left (\sqrt {2} \tan (x)\right )}{8 \sqrt {2} a}+\frac {\arctan \left (\sqrt {1-\sqrt [4]{-1}} \tan (x)\right )}{8 \sqrt {1-\sqrt [4]{-1}} a}+\frac {\arctan \left (\sqrt {1+\sqrt [4]{-1}} \tan (x)\right )}{8 \sqrt {1+\sqrt [4]{-1}} a}+\frac {\arctan \left (\sqrt {1-(-1)^{3/4}} \tan (x)\right )}{8 \sqrt {1-(-1)^{3/4}} a}+\frac {\arctan \left (\sqrt {1+(-1)^{3/4}} \tan (x)\right )}{8 \sqrt {1+(-1)^{3/4}} a}\) |
| \(\Big \downarrow \) 4254 |
| \(\displaystyle -\frac {\int 1d(-\tan (x))}{8 a}+\frac {\arctan \left (\sqrt {1-i} \tan (x)\right )}{8 \sqrt {1-i} a}+\frac {\arctan \left (\sqrt {1+i} \tan (x)\right )}{8 \sqrt {1+i} a}+\frac {\arctan \left (\sqrt {2} \tan (x)\right )}{8 \sqrt {2} a}+\frac {\arctan \left (\sqrt {1-\sqrt [4]{-1}} \tan (x)\right )}{8 \sqrt {1-\sqrt [4]{-1}} a}+\frac {\arctan \left (\sqrt {1+\sqrt [4]{-1}} \tan (x)\right )}{8 \sqrt {1+\sqrt [4]{-1}} a}+\frac {\arctan \left (\sqrt {1-(-1)^{3/4}} \tan (x)\right )}{8 \sqrt {1-(-1)^{3/4}} a}+\frac {\arctan \left (\sqrt {1+(-1)^{3/4}} \tan (x)\right )}{8 \sqrt {1+(-1)^{3/4}} a}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {\arctan \left (\sqrt {1-i} \tan (x)\right )}{8 \sqrt {1-i} a}+\frac {\arctan \left (\sqrt {1+i} \tan (x)\right )}{8 \sqrt {1+i} a}+\frac {\arctan \left (\sqrt {2} \tan (x)\right )}{8 \sqrt {2} a}+\frac {\arctan \left (\sqrt {1-\sqrt [4]{-1}} \tan (x)\right )}{8 \sqrt {1-\sqrt [4]{-1}} a}+\frac {\arctan \left (\sqrt {1+\sqrt [4]{-1}} \tan (x)\right )}{8 \sqrt {1+\sqrt [4]{-1}} a}+\frac {\arctan \left (\sqrt {1-(-1)^{3/4}} \tan (x)\right )}{8 \sqrt {1-(-1)^{3/4}} a}+\frac {\arctan \left (\sqrt {1+(-1)^{3/4}} \tan (x)\right )}{8 \sqrt {1+(-1)^{3/4}} a}+\frac {\tan (x)}{8 a}\) |
Input:
Int[(a - a*Sin[x]^16)^(-1),x]
Output:
ArcTan[Sqrt[1 - I]*Tan[x]]/(8*Sqrt[1 - I]*a) + ArcTan[Sqrt[1 + I]*Tan[x]]/ (8*Sqrt[1 + I]*a) + ArcTan[Sqrt[2]*Tan[x]]/(8*Sqrt[2]*a) + ArcTan[Sqrt[1 - (-1)^(1/4)]*Tan[x]]/(8*Sqrt[1 - (-1)^(1/4)]*a) + ArcTan[Sqrt[1 + (-1)^(1/ 4)]*Tan[x]]/(8*Sqrt[1 + (-1)^(1/4)]*a) + ArcTan[Sqrt[1 - (-1)^(3/4)]*Tan[x ]]/(8*Sqrt[1 - (-1)^(3/4)]*a) + ArcTan[Sqrt[1 + (-1)^(3/4)]*Tan[x]]/(8*Sqr t[1 + (-1)^(3/4)]*a) + Tan[x]/(8*a)
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[(u_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Simp[ a^p Int[ActivateTrig[u*cos[e + f*x]^(2*p)], x], x] /; FreeQ[{a, b, e, f, p}, x] && EqQ[a + b, 0] && IntegerQ[p]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(-1), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[ff/f Subst[Int[1/(a + (a + b)*ff^2*x^ 2), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^(n_))^(-1), x_Symbol] :> Module[{ k}, Simp[2/(a*n) Sum[Int[1/(1 - Sin[e + f*x]^2/((-1)^(4*(k/n))*Rt[-a/b, n /2])), x], {k, 1, n/2}], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[n/2]
Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[Exp andIntegrand[(1 + x^2)^(n/2 - 1), x], x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 92.79 (sec) , antiderivative size = 280, normalized size of antiderivative = 0.89
| method | result | size |
| default | \(\frac {\frac {\tan \left (x \right )}{8}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (2 \textit {\_Z}^{8}+4 \textit {\_Z}^{6}+6 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right )}{\sum }\frac {\left (\textit {\_R}^{6}+3 \textit {\_R}^{4}+3 \textit {\_R}^{2}+1\right ) \ln \left (\tan \left (x \right )-\textit {\_R} \right )}{2 \textit {\_R}^{7}+3 \textit {\_R}^{5}+3 \textit {\_R}^{3}+\textit {\_R}}\right )}{16}+\frac {\sqrt {2}\, \arctan \left (\tan \left (x \right ) \sqrt {2}\right )}{16}+\frac {\sqrt {2}\, \left (-\frac {\sqrt {-2+2 \sqrt {2}}\, \ln \left (-\sqrt {-2+2 \sqrt {2}}\, \sqrt {2}\, \tan \left (x \right )+2 \tan \left (x \right )^{2}+\sqrt {2}\right )}{4}+\frac {\left (-\frac {\left (-2+2 \sqrt {2}\right ) \sqrt {2}}{4}+2\right ) \arctan \left (\frac {-\sqrt {2}\, \sqrt {-2+2 \sqrt {2}}+4 \tan \left (x \right )}{2 \sqrt {1+\sqrt {2}}}\right )}{\sqrt {1+\sqrt {2}}}\right )}{16}+\frac {\sqrt {2}\, \left (\frac {\sqrt {-2+2 \sqrt {2}}\, \ln \left (\sqrt {2}+\sqrt {-2+2 \sqrt {2}}\, \sqrt {2}\, \tan \left (x \right )+2 \tan \left (x \right )^{2}\right )}{4}+\frac {\left (-\frac {\left (-2+2 \sqrt {2}\right ) \sqrt {2}}{4}+2\right ) \arctan \left (\frac {\sqrt {2}\, \sqrt {-2+2 \sqrt {2}}+4 \tan \left (x \right )}{2 \sqrt {1+\sqrt {2}}}\right )}{\sqrt {1+\sqrt {2}}}\right )}{16}}{a}\) | \(280\) |
| risch | \(\frac {i}{4 a \left ({\mathrm e}^{2 i x}+1\right )}+\frac {\sqrt {-2+2 i}\, \ln \left ({\mathrm e}^{2 i x}+i \sqrt {-2+2 i}+\sqrt {-2+2 i}-1+2 i\right )}{32 a}-\frac {\sqrt {-2+2 i}\, \ln \left ({\mathrm e}^{2 i x}-i \sqrt {-2+2 i}-\sqrt {-2+2 i}-1+2 i\right )}{32 a}+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (131072 a^{4} \textit {\_Z}^{4}+\left (512 i a^{2}+512 a^{2}\right ) \textit {\_Z}^{2}+1+i\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 i x}+\left (-8192 i a^{3}+8192 a^{3}\right ) \textit {\_R}^{3}+\left (512 i a^{2}+512 a^{2}\right ) \textit {\_R}^{2}+\left (32 i a +32 a \right ) \textit {\_R} -1+2 i\right )\right )+\frac {i \sqrt {2}\, \ln \left ({\mathrm e}^{2 i x}-2 \sqrt {2}-3\right )}{32 a}-\frac {i \sqrt {2}\, \ln \left ({\mathrm e}^{2 i x}+2 \sqrt {2}-3\right )}{32 a}+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (131072 a^{4} \textit {\_Z}^{4}+\left (-512 i a^{2}+512 a^{2}\right ) \textit {\_Z}^{2}+1-i\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 i x}+\left (-8192 i a^{3}-8192 a^{3}\right ) \textit {\_R}^{3}+\left (-512 i a^{2}+512 a^{2}\right ) \textit {\_R}^{2}+\left (32 i a -32 a \right ) \textit {\_R} -1-2 i\right )\right )+\frac {\sqrt {-2-2 i}\, \ln \left ({\mathrm e}^{2 i x}+i \sqrt {-2-2 i}-\sqrt {-2-2 i}-1-2 i\right )}{32 a}-\frac {\sqrt {-2-2 i}\, \ln \left ({\mathrm e}^{2 i x}-i \sqrt {-2-2 i}+\sqrt {-2-2 i}-1-2 i\right )}{32 a}\) | \(369\) |
Input:
int(1/(a-a*sin(x)^16),x,method=_RETURNVERBOSE)
Output:
1/a*(1/8*tan(x)+1/16*sum((_R^6+3*_R^4+3*_R^2+1)/(2*_R^7+3*_R^5+3*_R^3+_R)* ln(tan(x)-_R),_R=RootOf(2*_Z^8+4*_Z^6+6*_Z^4+4*_Z^2+1))+1/16*2^(1/2)*arcta n(tan(x)*2^(1/2))+1/16*2^(1/2)*(-1/4*(-2+2*2^(1/2))^(1/2)*ln(-(-2+2*2^(1/2 ))^(1/2)*2^(1/2)*tan(x)+2*tan(x)^2+2^(1/2))+(-1/4*(-2+2*2^(1/2))*2^(1/2)+2 )/(1+2^(1/2))^(1/2)*arctan(1/2*(-2^(1/2)*(-2+2*2^(1/2))^(1/2)+4*tan(x))/(1 +2^(1/2))^(1/2)))+1/16*2^(1/2)*(1/4*(-2+2*2^(1/2))^(1/2)*ln(2^(1/2)+(-2+2* 2^(1/2))^(1/2)*2^(1/2)*tan(x)+2*tan(x)^2)+(-1/4*(-2+2*2^(1/2))*2^(1/2)+2)/ (1+2^(1/2))^(1/2)*arctan(1/2*(2^(1/2)*(-2+2*2^(1/2))^(1/2)+4*tan(x))/(1+2^ (1/2))^(1/2))))
Leaf count of result is larger than twice the leaf count of optimal. 2470 vs. \(2 (226) = 452\).
Time = 0.95 (sec) , antiderivative size = 2470, normalized size of antiderivative = 7.87 \[ \int \frac {1}{a-a \sin ^{16}(x)} \, dx=\text {Too large to display} \] Input:
integrate(1/(a-a*sin(x)^16),x, algorithm="fricas")
Output:
-1/32*(sqrt(1/2)*a*sqrt(-(a^2*sqrt(-(4*sqrt(1/2)*a^4*sqrt(a^(-8)) + 3)/a^4 ) + 1)/a^2)*cos(x)*log(1/2*sqrt(1/2)*(sqrt(1/2)*a^5*sqrt(a^(-8))*cos(x)*si n(x) - a*cos(x)*sin(x) + (sqrt(1/2)*a^7*sqrt(a^(-8))*cos(x)*sin(x) - a^3*c os(x)*sin(x))*sqrt(-(4*sqrt(1/2)*a^4*sqrt(a^(-8)) + 3)/a^4))*sqrt(-(a^2*sq rt(-(4*sqrt(1/2)*a^4*sqrt(a^(-8)) + 3)/a^4) + 1)/a^2) + 1/4*sqrt(1/2)*(2*a ^4*cos(x)^2 - a^4)*sqrt(a^(-8)) - 1/4*cos(x)^2 - 1/4*(2*a^2*cos(x)^2 - a^2 - sqrt(1/2)*(2*a^6*cos(x)^2 - a^6)*sqrt(a^(-8)))*sqrt(-(4*sqrt(1/2)*a^4*s qrt(a^(-8)) + 3)/a^4) + 1/4) - sqrt(1/2)*a*sqrt(-(a^2*sqrt(-(4*sqrt(1/2)*a ^4*sqrt(a^(-8)) + 3)/a^4) + 1)/a^2)*cos(x)*log(-1/2*sqrt(1/2)*(sqrt(1/2)*a ^5*sqrt(a^(-8))*cos(x)*sin(x) - a*cos(x)*sin(x) + (sqrt(1/2)*a^7*sqrt(a^(- 8))*cos(x)*sin(x) - a^3*cos(x)*sin(x))*sqrt(-(4*sqrt(1/2)*a^4*sqrt(a^(-8)) + 3)/a^4))*sqrt(-(a^2*sqrt(-(4*sqrt(1/2)*a^4*sqrt(a^(-8)) + 3)/a^4) + 1)/ a^2) + 1/4*sqrt(1/2)*(2*a^4*cos(x)^2 - a^4)*sqrt(a^(-8)) - 1/4*cos(x)^2 - 1/4*(2*a^2*cos(x)^2 - a^2 - sqrt(1/2)*(2*a^6*cos(x)^2 - a^6)*sqrt(a^(-8))) *sqrt(-(4*sqrt(1/2)*a^4*sqrt(a^(-8)) + 3)/a^4) + 1/4) - sqrt(1/2)*a*sqrt(( a^2*sqrt(-(4*sqrt(1/2)*a^4*sqrt(a^(-8)) + 3)/a^4) - 1)/a^2)*cos(x)*log(1/2 *sqrt(1/2)*(sqrt(1/2)*a^5*sqrt(a^(-8))*cos(x)*sin(x) - a*cos(x)*sin(x) - ( sqrt(1/2)*a^7*sqrt(a^(-8))*cos(x)*sin(x) - a^3*cos(x)*sin(x))*sqrt(-(4*sqr t(1/2)*a^4*sqrt(a^(-8)) + 3)/a^4))*sqrt((a^2*sqrt(-(4*sqrt(1/2)*a^4*sqrt(a ^(-8)) + 3)/a^4) - 1)/a^2) - 1/4*sqrt(1/2)*(2*a^4*cos(x)^2 - a^4)*sqrt(...
Timed out. \[ \int \frac {1}{a-a \sin ^{16}(x)} \, dx=\text {Timed out} \] Input:
integrate(1/(a-a*sin(x)**16),x)
Output:
Timed out
\[ \int \frac {1}{a-a \sin ^{16}(x)} \, dx=\int { -\frac {1}{a \sin \left (x\right )^{16} - a} \,d x } \] Input:
integrate(1/(a-a*sin(x)^16),x, algorithm="maxima")
Output:
1/32*((sqrt(2)*cos(2*x)^2 + sqrt(2)*sin(2*x)^2 + 2*sqrt(2)*cos(2*x) + sqrt (2))*arctan2(2*sqrt(2)*sin(x)/(2*(sqrt(2) + 1)*cos(x) + cos(x)^2 + sin(x)^ 2 + 2*sqrt(2) + 3), (cos(x)^2 + sin(x)^2 + 2*cos(x) - 1)/(2*(sqrt(2) + 1)* cos(x) + cos(x)^2 + sin(x)^2 + 2*sqrt(2) + 3)) - (sqrt(2)*cos(2*x)^2 + sqr t(2)*sin(2*x)^2 + 2*sqrt(2)*cos(2*x) + sqrt(2))*arctan2(2*sqrt(2)*sin(x)/( 2*(sqrt(2) - 1)*cos(x) + cos(x)^2 + sin(x)^2 - 2*sqrt(2) + 3), (cos(x)^2 + sin(x)^2 - 2*cos(x) - 1)/(2*(sqrt(2) - 1)*cos(x) + cos(x)^2 + sin(x)^2 - 2*sqrt(2) + 3)) + 4096*(a*cos(2*x)^2 + a*sin(2*x)^2 + 2*a*cos(2*x) + a)*in tegrate(-((56*cos(6*x) - 28*cos(4*x) + 8*cos(2*x) - 1)*cos(8*x) - cos(16*x )*cos(8*x) + 8*cos(14*x)*cos(8*x) - 28*cos(12*x)*cos(8*x) + 56*cos(10*x)*c os(8*x) - 326*cos(8*x)^2 + 4*(14*sin(6*x) - 7*sin(4*x) + 2*sin(2*x))*sin(8 *x) - sin(16*x)*sin(8*x) + 8*sin(14*x)*sin(8*x) - 28*sin(12*x)*sin(8*x) + 56*sin(10*x)*sin(8*x) - 326*sin(8*x)^2)/(a*cos(16*x)^2 + 64*a*cos(14*x)^2 + 784*a*cos(12*x)^2 + 3136*a*cos(10*x)^2 + 106276*a*cos(8*x)^2 + 3136*a*co s(6*x)^2 + 784*a*cos(4*x)^2 + 64*a*cos(2*x)^2 + a*sin(16*x)^2 + 64*a*sin(1 4*x)^2 + 784*a*sin(12*x)^2 + 3136*a*sin(10*x)^2 + 106276*a*sin(8*x)^2 + 31 36*a*sin(6*x)^2 + 784*a*sin(4*x)^2 - 448*a*sin(4*x)*sin(2*x) + 64*a*sin(2* x)^2 - 2*(8*a*cos(14*x) - 28*a*cos(12*x) + 56*a*cos(10*x) - 326*a*cos(8*x) + 56*a*cos(6*x) - 28*a*cos(4*x) + 8*a*cos(2*x) - a)*cos(16*x) - 16*(28*a* cos(12*x) - 56*a*cos(10*x) + 326*a*cos(8*x) - 56*a*cos(6*x) + 28*a*cos(...
Exception generated. \[ \int \frac {1}{a-a \sin ^{16}(x)} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(1/(a-a*sin(x)^16),x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to find common minimal polyn omial Error: Bad Argument Value
Time = 38.36 (sec) , antiderivative size = 961, normalized size of antiderivative = 3.06 \[ \int \frac {1}{a-a \sin ^{16}(x)} \, dx=\text {Too large to display} \] Input:
int(1/(a - a*sin(x)^16),x)
Output:
tan(x)/(8*a) + atan(a*tan(x)*((- 1/512 - 1i/512)/a^2)^(1/2)*(16 + 16i))*(( - 1/512 - 1i/512)/a^2)^(1/2)*2i - atan(a*tan(x)*((- 1/512 + 1i/512)/a^2)^( 1/2)*(16 - 16i))*((- 1/512 + 1i/512)/a^2)^(1/2)*2i - (atan((tan(x)*(- (2*( - 2*2^(1/2) - 3)^(1/2))/a^2 - 2/a^2)^(1/2)*17i)/(4*((7*2^(1/2))/(2*a) - 5/ a)) - (2^(1/2)*tan(x)*(- (2*(- 2*2^(1/2) - 3)^(1/2))/a^2 - 2/a^2)^(1/2)*3i )/((7*2^(1/2))/(2*a) - 5/a) - (tan(x)*(- (2*(- 2*2^(1/2) - 3)^(1/2))/a^2 - 2/a^2)^(1/2)*(- 2*2^(1/2) - 3)^(1/2)*17i)/(4*((7*2^(1/2))/(2*a) - 5/a)) + (2^(1/2)*tan(x)*(- (2*(- 2*2^(1/2) - 3)^(1/2))/a^2 - 2/a^2)^(1/2)*(- 2*2^ (1/2) - 3)^(1/2)*3i)/((7*2^(1/2))/(2*a) - 5/a))*(-(2*((- 2*2^(1/2) - 3)^(1 /2) + 1))/a^2)^(1/2)*1i)/16 - (atan((tan(x)*((2*(- 2*2^(1/2) - 3)^(1/2))/a ^2 - 2/a^2)^(1/2)*17i)/(4*((7*2^(1/2))/(2*a) - 5/a)) - (2^(1/2)*tan(x)*((2 *(- 2*2^(1/2) - 3)^(1/2))/a^2 - 2/a^2)^(1/2)*3i)/((7*2^(1/2))/(2*a) - 5/a) + (tan(x)*((2*(- 2*2^(1/2) - 3)^(1/2))/a^2 - 2/a^2)^(1/2)*(- 2*2^(1/2) - 3)^(1/2)*17i)/(4*((7*2^(1/2))/(2*a) - 5/a)) - (2^(1/2)*tan(x)*((2*(- 2*2^( 1/2) - 3)^(1/2))/a^2 - 2/a^2)^(1/2)*(- 2*2^(1/2) - 3)^(1/2)*3i)/((7*2^(1/2 ))/(2*a) - 5/a))*((2*((- 2*2^(1/2) - 3)^(1/2) - 1))/a^2)^(1/2)*1i)/16 + (a tan((tan(x)*(- (2*(2*2^(1/2) - 3)^(1/2))/a^2 - 2/a^2)^(1/2)*17i)/(4*((7*2^ (1/2))/(2*a) + 5/a)) + (2^(1/2)*tan(x)*(- (2*(2*2^(1/2) - 3)^(1/2))/a^2 - 2/a^2)^(1/2)*3i)/((7*2^(1/2))/(2*a) + 5/a) - (tan(x)*(- (2*(2*2^(1/2) - 3) ^(1/2))/a^2 - 2/a^2)^(1/2)*(2*2^(1/2) - 3)^(1/2)*17i)/(4*((7*2^(1/2))/(...
\[ \int \frac {1}{a-a \sin ^{16}(x)} \, dx=-\frac {\int \frac {1}{\sin \left (x \right )^{16}-1}d x}{a} \] Input:
int(1/(a-a*sin(x)^16),x)
Output:
( - int(1/(sin(x)**16 - 1),x))/a