Integrand size = 11, antiderivative size = 421 \[ \int \frac {1}{a-a \sin ^{10}(x)} \, dx=\frac {\arctan \left (\frac {\sqrt [5]{-1}+\tan \left (\frac {x}{2}\right )}{\sqrt {1-(-1)^{2/5}}}\right )}{5 \sqrt {1-(-1)^{2/5}} a}+\frac {\arctan \left (\frac {(-1)^{2/5}+\tan \left (\frac {x}{2}\right )}{\sqrt {1-(-1)^{4/5}}}\right )}{5 \sqrt {1-(-1)^{4/5}} a}+\frac {\arctan \left (\frac {(-1)^{3/5}+\tan \left (\frac {x}{2}\right )}{\sqrt {1+\sqrt [5]{-1}}}\right )}{5 \sqrt {1+\sqrt [5]{-1}} a}+\frac {\arctan \left (\frac {(-1)^{4/5}+\tan \left (\frac {x}{2}\right )}{\sqrt {1+(-1)^{3/5}}}\right )}{5 \sqrt {1+(-1)^{3/5}} a}-\frac {\arctan \left (\frac {(-1)^{4/5} \left (1+\sqrt [5]{-1} \tan \left (\frac {x}{2}\right )\right )}{\sqrt {1+(-1)^{3/5}}}\right )}{5 \sqrt {1+(-1)^{3/5}} a}-\frac {\arctan \left (\frac {(-1)^{3/5} \left (1+(-1)^{2/5} \tan \left (\frac {x}{2}\right )\right )}{\sqrt {1+\sqrt [5]{-1}}}\right )}{5 \sqrt {1+\sqrt [5]{-1}} a}-\frac {\arctan \left (\frac {(-1)^{2/5} \left (1+(-1)^{3/5} \tan \left (\frac {x}{2}\right )\right )}{\sqrt {1-(-1)^{4/5}}}\right )}{5 \sqrt {1-(-1)^{4/5}} a}-\frac {\arctan \left (\frac {\sqrt [5]{-1} \left (1+(-1)^{4/5} \tan \left (\frac {x}{2}\right )\right )}{\sqrt {1-(-1)^{2/5}}}\right )}{5 \sqrt {1-(-1)^{2/5}} a}+\frac {\cos (x)}{10 a (1-\sin (x))}-\frac {\cos (x)}{10 a (1+\sin (x))} \] Output:
1/5*arctan(((-1)^(1/5)+tan(1/2*x))/(1-(-1)^(2/5))^(1/2))/(1-(-1)^(2/5))^(1 /2)/a+1/5*arctan(((-1)^(2/5)+tan(1/2*x))/(1-(-1)^(4/5))^(1/2))/(1-(-1)^(4/ 5))^(1/2)/a+1/5*arctan(((-1)^(3/5)+tan(1/2*x))/(1+(-1)^(1/5))^(1/2))/(1+(- 1)^(1/5))^(1/2)/a+1/5*arctan(((-1)^(4/5)+tan(1/2*x))/(1+(-1)^(3/5))^(1/2)) /(1+(-1)^(3/5))^(1/2)/a-1/5*arctan((-1)^(4/5)*(1+(-1)^(1/5)*tan(1/2*x))/(1 +(-1)^(3/5))^(1/2))/(1+(-1)^(3/5))^(1/2)/a-1/5*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-1/5*arctan(( -1)^(2/5)*(1+(-1)^(3/5)*tan(1/2*x))/(1-(-1)^(4/5))^(1/2))/(1-(-1)^(4/5))^( 1/2)/a-1/5*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+1/10*cos(x)/a/(1-sin(x))-1/10*cos(x)/a/(1+sin(x) )
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 5.11 (sec) , antiderivative size = 417, normalized size of antiderivative = 0.99 \[ \int \frac {1}{a-a \sin ^{10}(x)} \, dx=-\frac {\text {RootSum}\left [1-12 \text {$\#$1}+68 \text {$\#$1}^2-244 \text {$\#$1}^3+630 \text {$\#$1}^4-244 \text {$\#$1}^5+68 \text {$\#$1}^6-12 \text {$\#$1}^7+\text {$\#$1}^8\&,\frac {2 \arctan \left (\frac {\sin (2 x)}{\cos (2 x)-\text {$\#$1}}\right )-i \log \left (1-2 \cos (2 x) \text {$\#$1}+\text {$\#$1}^2\right )-28 \arctan \left (\frac {\sin (2 x)}{\cos (2 x)-\text {$\#$1}}\right ) \text {$\#$1}+14 i \log \left (1-2 \cos (2 x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}+190 \arctan \left (\frac {\sin (2 x)}{\cos (2 x)-\text {$\#$1}}\right ) \text {$\#$1}^2-95 i \log \left (1-2 \cos (2 x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^2-840 \arctan \left (\frac {\sin (2 x)}{\cos (2 x)-\text {$\#$1}}\right ) \text {$\#$1}^3+420 i \log \left (1-2 \cos (2 x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^3+190 \arctan \left (\frac {\sin (2 x)}{\cos (2 x)-\text {$\#$1}}\right ) \text {$\#$1}^4-95 i \log \left (1-2 \cos (2 x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^4-28 \arctan \left (\frac {\sin (2 x)}{\cos (2 x)-\text {$\#$1}}\right ) \text {$\#$1}^5+14 i \log \left (1-2 \cos (2 x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^5+2 \arctan \left (\frac {\sin (2 x)}{\cos (2 x)-\text {$\#$1}}\right ) \text {$\#$1}^6-i \log \left (1-2 \cos (2 x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^6}{-3+34 \text {$\#$1}-183 \text {$\#$1}^2+630 \text {$\#$1}^3-305 \text {$\#$1}^4+102 \text {$\#$1}^5-21 \text {$\#$1}^6+2 \text {$\#$1}^7}\&\right ]-4 \tan (x)}{20 a} \] Input:
Integrate[(a - a*Sin[x]^10)^(-1),x]
Output:
-1/20*(RootSum[1 - 12*#1 + 68*#1^2 - 244*#1^3 + 630*#1^4 - 244*#1^5 + 68*# 1^6 - 12*#1^7 + #1^8 & , (2*ArcTan[Sin[2*x]/(Cos[2*x] - #1)] - I*Log[1 - 2 *Cos[2*x]*#1 + #1^2] - 28*ArcTan[Sin[2*x]/(Cos[2*x] - #1)]*#1 + (14*I)*Log [1 - 2*Cos[2*x]*#1 + #1^2]*#1 + 190*ArcTan[Sin[2*x]/(Cos[2*x] - #1)]*#1^2 - (95*I)*Log[1 - 2*Cos[2*x]*#1 + #1^2]*#1^2 - 840*ArcTan[Sin[2*x]/(Cos[2*x ] - #1)]*#1^3 + (420*I)*Log[1 - 2*Cos[2*x]*#1 + #1^2]*#1^3 + 190*ArcTan[Si n[2*x]/(Cos[2*x] - #1)]*#1^4 - (95*I)*Log[1 - 2*Cos[2*x]*#1 + #1^2]*#1^4 - 28*ArcTan[Sin[2*x]/(Cos[2*x] - #1)]*#1^5 + (14*I)*Log[1 - 2*Cos[2*x]*#1 + #1^2]*#1^5 + 2*ArcTan[Sin[2*x]/(Cos[2*x] - #1)]*#1^6 - I*Log[1 - 2*Cos[2* x]*#1 + #1^2]*#1^6)/(-3 + 34*#1 - 183*#1^2 + 630*#1^3 - 305*#1^4 + 102*#1^ 5 - 21*#1^6 + 2*#1^7) & ] - 4*Tan[x])/a
Time = 0.71 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.36, 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.
| \(\displaystyle \int \frac {1}{a-a \sin ^{10}(x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{a-a \sin (x)^{10}}dx\) |
| \(\Big \downarrow \) 3690 |
| \(\displaystyle \frac {\int \frac {1}{1-\sin ^2(x)}dx}{5 a}+\frac {\int \frac {1}{\sqrt [5]{-1} \sin ^2(x)+1}dx}{5 a}+\frac {\int \frac {1}{1-(-1)^{2/5} \sin ^2(x)}dx}{5 a}+\frac {\int \frac {1}{(-1)^{3/5} \sin ^2(x)+1}dx}{5 a}+\frac {\int \frac {1}{1-(-1)^{4/5} \sin ^2(x)}dx}{5 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {1}{1-\sin (x)^2}dx}{5 a}+\frac {\int \frac {1}{\sqrt [5]{-1} \sin (x)^2+1}dx}{5 a}+\frac {\int \frac {1}{1-(-1)^{2/5} \sin (x)^2}dx}{5 a}+\frac {\int \frac {1}{(-1)^{3/5} \sin (x)^2+1}dx}{5 a}+\frac {\int \frac {1}{1-(-1)^{4/5} \sin (x)^2}dx}{5 a}\) |
| \(\Big \downarrow \) 3654 |
| \(\displaystyle \frac {\int \frac {1}{\sqrt [5]{-1} \sin (x)^2+1}dx}{5 a}+\frac {\int \frac {1}{1-(-1)^{2/5} \sin (x)^2}dx}{5 a}+\frac {\int \frac {1}{(-1)^{3/5} \sin (x)^2+1}dx}{5 a}+\frac {\int \frac {1}{1-(-1)^{4/5} \sin (x)^2}dx}{5 a}+\frac {\int \sec ^2(x)dx}{5 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {1}{\sqrt [5]{-1} \sin (x)^2+1}dx}{5 a}+\frac {\int \frac {1}{1-(-1)^{2/5} \sin (x)^2}dx}{5 a}+\frac {\int \frac {1}{(-1)^{3/5} \sin (x)^2+1}dx}{5 a}+\frac {\int \frac {1}{1-(-1)^{4/5} \sin (x)^2}dx}{5 a}+\frac {\int \csc \left (x+\frac {\pi }{2}\right )^2dx}{5 a}\) |
| \(\Big \downarrow \) 3660 |
| \(\displaystyle \frac {\int \frac {1}{\left (1+\sqrt [5]{-1}\right ) \tan ^2(x)+1}d\tan (x)}{5 a}+\frac {\int \frac {1}{\left (1-(-1)^{2/5}\right ) \tan ^2(x)+1}d\tan (x)}{5 a}+\frac {\int \frac {1}{\left (1+(-1)^{3/5}\right ) \tan ^2(x)+1}d\tan (x)}{5 a}+\frac {\int \frac {1}{\left (1-(-1)^{4/5}\right ) \tan ^2(x)+1}d\tan (x)}{5 a}+\frac {\int \csc \left (x+\frac {\pi }{2}\right )^2dx}{5 a}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {\int \csc \left (x+\frac {\pi }{2}\right )^2dx}{5 a}+\frac {\arctan \left (\sqrt {1+\sqrt [5]{-1}} \tan (x)\right )}{5 \sqrt {1+\sqrt [5]{-1}} a}+\frac {\arctan \left (\sqrt {1-(-1)^{2/5}} \tan (x)\right )}{5 \sqrt {1-(-1)^{2/5}} a}+\frac {\arctan \left (\sqrt {1+(-1)^{3/5}} \tan (x)\right )}{5 \sqrt {1+(-1)^{3/5}} a}+\frac {\arctan \left (\sqrt {1-(-1)^{4/5}} \tan (x)\right )}{5 \sqrt {1-(-1)^{4/5}} a}\) |
| \(\Big \downarrow \) 4254 |
| \(\displaystyle -\frac {\int 1d(-\tan (x))}{5 a}+\frac {\arctan \left (\sqrt {1+\sqrt [5]{-1}} \tan (x)\right )}{5 \sqrt {1+\sqrt [5]{-1}} a}+\frac {\arctan \left (\sqrt {1-(-1)^{2/5}} \tan (x)\right )}{5 \sqrt {1-(-1)^{2/5}} a}+\frac {\arctan \left (\sqrt {1+(-1)^{3/5}} \tan (x)\right )}{5 \sqrt {1+(-1)^{3/5}} a}+\frac {\arctan \left (\sqrt {1-(-1)^{4/5}} \tan (x)\right )}{5 \sqrt {1-(-1)^{4/5}} a}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {\arctan \left (\sqrt {1+\sqrt [5]{-1}} \tan (x)\right )}{5 \sqrt {1+\sqrt [5]{-1}} a}+\frac {\arctan \left (\sqrt {1-(-1)^{2/5}} \tan (x)\right )}{5 \sqrt {1-(-1)^{2/5}} a}+\frac {\arctan \left (\sqrt {1+(-1)^{3/5}} \tan (x)\right )}{5 \sqrt {1+(-1)^{3/5}} a}+\frac {\arctan \left (\sqrt {1-(-1)^{4/5}} \tan (x)\right )}{5 \sqrt {1-(-1)^{4/5}} a}+\frac {\tan (x)}{5 a}\) |
Input:
Int[(a - a*Sin[x]^10)^(-1),x]
Output:
ArcTan[Sqrt[1 + (-1)^(1/5)]*Tan[x]]/(5*Sqrt[1 + (-1)^(1/5)]*a) + ArcTan[Sq rt[1 - (-1)^(2/5)]*Tan[x]]/(5*Sqrt[1 - (-1)^(2/5)]*a) + ArcTan[Sqrt[1 + (- 1)^(3/5)]*Tan[x]]/(5*Sqrt[1 + (-1)^(3/5)]*a) + ArcTan[Sqrt[1 - (-1)^(4/5)] *Tan[x]]/(5*Sqrt[1 - (-1)^(4/5)]*a) + Tan[x]/(5*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 = 3.67 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.19
| method | result | size |
| default | \(\frac {\frac {\tan \left (x \right )}{5}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (5 \textit {\_Z}^{8}+10 \textit {\_Z}^{6}+10 \textit {\_Z}^{4}+5 \textit {\_Z}^{2}+1\right )}{\sum }\frac {\left (10 \textit {\_R}^{6}+20 \textit {\_R}^{4}+15 \textit {\_R}^{2}+4\right ) \ln \left (\tan \left (x \right )-\textit {\_R} \right )}{4 \textit {\_R}^{7}+6 \textit {\_R}^{5}+4 \textit {\_R}^{3}+\textit {\_R}}\right )}{50}}{a}\) | \(82\) |
| risch | \(\frac {2 i}{5 a \left ({\mathrm e}^{2 i x}+1\right )}+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (500000000 a^{8} \textit {\_Z}^{8}+10000000 a^{6} \textit {\_Z}^{6}+100000 a^{4} \textit {\_Z}^{4}+500 a^{2} \textit {\_Z}^{2}+1\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 i x}-100000000 i a^{7} \textit {\_R}^{7}+10000000 a^{6} \textit {\_R}^{6}-1000000 i a^{5} \textit {\_R}^{5}+100000 a^{4} \textit {\_R}^{4}-10000 i a^{3} \textit {\_R}^{3}+1000 a^{2} \textit {\_R}^{2}+1\right )\right )\) | \(116\) |
Input:
int(1/(a-a*sin(x)^10),x,method=_RETURNVERBOSE)
Output:
1/a*(1/5*tan(x)+1/50*sum((10*_R^6+20*_R^4+15*_R^2+4)/(4*_R^7+6*_R^5+4*_R^3 +_R)*ln(tan(x)-_R),_R=RootOf(5*_Z^8+10*_Z^6+10*_Z^4+5*_Z^2+1)))
Leaf count of result is larger than twice the leaf count of optimal. 2010 vs. \(2 (295) = 590\).
Time = 0.39 (sec) , antiderivative size = 2010, normalized size of antiderivative = 4.77 \[ \int \frac {1}{a-a \sin ^{10}(x)} \, dx=\text {Too large to display} \] Input:
integrate(1/(a-a*sin(x)^10),x, algorithm="fricas")
Output:
-1/20*(sqrt(1/2)*a*sqrt(-(a^2*sqrt(-(2*sqrt(1/5)*a^4*sqrt(a^(-8)) + 1)/a^4 ) + 1)/a^2)*cos(x)*log(10*sqrt(1/2)*(sqrt(1/5)*a^5*sqrt(a^(-8))*cos(x)*sin (x) - a*cos(x)*sin(x) + (sqrt(1/5)*a^7*sqrt(a^(-8))*cos(x)*sin(x) - a^3*co s(x)*sin(x))*sqrt(-(2*sqrt(1/5)*a^4*sqrt(a^(-8)) + 1)/a^4))*sqrt(-(a^2*sqr t(-(2*sqrt(1/5)*a^4*sqrt(a^(-8)) + 1)/a^4) + 1)/a^2) + 5*sqrt(1/5)*(2*a^4* cos(x)^2 - a^4)*sqrt(a^(-8)) - 6*cos(x)^2 - 5*(2*a^2*cos(x)^2 - a^2 - sqrt (1/5)*(2*a^6*cos(x)^2 - a^6)*sqrt(a^(-8)))*sqrt(-(2*sqrt(1/5)*a^4*sqrt(a^( -8)) + 1)/a^4) + 5) - sqrt(1/2)*a*sqrt(-(a^2*sqrt(-(2*sqrt(1/5)*a^4*sqrt(a ^(-8)) + 1)/a^4) + 1)/a^2)*cos(x)*log(-10*sqrt(1/2)*(sqrt(1/5)*a^5*sqrt(a^ (-8))*cos(x)*sin(x) - a*cos(x)*sin(x) + (sqrt(1/5)*a^7*sqrt(a^(-8))*cos(x) *sin(x) - a^3*cos(x)*sin(x))*sqrt(-(2*sqrt(1/5)*a^4*sqrt(a^(-8)) + 1)/a^4) )*sqrt(-(a^2*sqrt(-(2*sqrt(1/5)*a^4*sqrt(a^(-8)) + 1)/a^4) + 1)/a^2) + 5*s qrt(1/5)*(2*a^4*cos(x)^2 - a^4)*sqrt(a^(-8)) - 6*cos(x)^2 - 5*(2*a^2*cos(x )^2 - a^2 - sqrt(1/5)*(2*a^6*cos(x)^2 - a^6)*sqrt(a^(-8)))*sqrt(-(2*sqrt(1 /5)*a^4*sqrt(a^(-8)) + 1)/a^4) + 5) - sqrt(1/2)*a*sqrt((a^2*sqrt(-(2*sqrt( 1/5)*a^4*sqrt(a^(-8)) + 1)/a^4) - 1)/a^2)*cos(x)*log(10*sqrt(1/2)*(sqrt(1/ 5)*a^5*sqrt(a^(-8))*cos(x)*sin(x) - a*cos(x)*sin(x) - (sqrt(1/5)*a^7*sqrt( a^(-8))*cos(x)*sin(x) - a^3*cos(x)*sin(x))*sqrt(-(2*sqrt(1/5)*a^4*sqrt(a^( -8)) + 1)/a^4))*sqrt((a^2*sqrt(-(2*sqrt(1/5)*a^4*sqrt(a^(-8)) + 1)/a^4) - 1)/a^2) - 5*sqrt(1/5)*(2*a^4*cos(x)^2 - a^4)*sqrt(a^(-8)) + 6*cos(x)^2 ...
\[ \int \frac {1}{a-a \sin ^{10}(x)} \, dx=- \frac {\int \frac {1}{\sin ^{10}{\left (x \right )} - 1}\, dx}{a} \] Input:
integrate(1/(a-a*sin(x)**10),x)
Output:
-Integral(1/(sin(x)**10 - 1), x)/a
\[ \int \frac {1}{a-a \sin ^{10}(x)} \, dx=\int { -\frac {1}{a \sin \left (x\right )^{10} - a} \,d x } \] Input:
integrate(1/(a-a*sin(x)^10),x, algorithm="maxima")
Output:
-1/5*(5*(a*cos(2*x)^2 + a*sin(2*x)^2 + 2*a*cos(2*x) + a)*integrate(4/5*((c os(14*x) - 14*cos(12*x) + 95*cos(10*x) - 420*cos(8*x) + 95*cos(6*x) - 14*c os(4*x) + cos(2*x))*cos(16*x) + (236*cos(12*x) - 1384*cos(10*x) + 5670*cos (8*x) - 1384*cos(6*x) + 236*cos(4*x) - 24*cos(2*x) + 1)*cos(14*x) - 12*cos (14*x)^2 + 2*(4938*cos(10*x) - 18690*cos(8*x) + 4938*cos(6*x) - 952*cos(4* x) + 118*cos(2*x) - 7)*cos(12*x) - 952*cos(12*x)^2 + (162330*cos(8*x) - 46 360*cos(6*x) + 9876*cos(4*x) - 1384*cos(2*x) + 95)*cos(10*x) - 23180*cos(1 0*x)^2 + 210*(773*cos(6*x) - 178*cos(4*x) + 27*cos(2*x) - 2)*cos(8*x) - 26 4600*cos(8*x)^2 + (9876*cos(4*x) - 1384*cos(2*x) + 95)*cos(6*x) - 23180*co s(6*x)^2 + 2*(118*cos(2*x) - 7)*cos(4*x) - 952*cos(4*x)^2 - 12*cos(2*x)^2 + (sin(14*x) - 14*sin(12*x) + 95*sin(10*x) - 420*sin(8*x) + 95*sin(6*x) - 14*sin(4*x) + sin(2*x))*sin(16*x) + 2*(118*sin(12*x) - 692*sin(10*x) + 283 5*sin(8*x) - 692*sin(6*x) + 118*sin(4*x) - 12*sin(2*x))*sin(14*x) - 12*sin (14*x)^2 + 4*(2469*sin(10*x) - 9345*sin(8*x) + 2469*sin(6*x) - 476*sin(4*x ) + 59*sin(2*x))*sin(12*x) - 952*sin(12*x)^2 + 2*(81165*sin(8*x) - 23180*s in(6*x) + 4938*sin(4*x) - 692*sin(2*x))*sin(10*x) - 23180*sin(10*x)^2 + 21 0*(773*sin(6*x) - 178*sin(4*x) + 27*sin(2*x))*sin(8*x) - 264600*sin(8*x)^2 + 4*(2469*sin(4*x) - 346*sin(2*x))*sin(6*x) - 23180*sin(6*x)^2 - 952*sin( 4*x)^2 + 236*sin(4*x)*sin(2*x) - 12*sin(2*x)^2 + cos(2*x))/(a*cos(16*x)^2 + 144*a*cos(14*x)^2 + 4624*a*cos(12*x)^2 + 59536*a*cos(10*x)^2 + 396900...
Exception generated. \[ \int \frac {1}{a-a \sin ^{10}(x)} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(1/(a-a*sin(x)^10),x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Degree mismatch inside factorisatio n over extensionUnable to transpose Error: Bad Argument ValueDegree mismat ch inside
Time = 37.65 (sec) , antiderivative size = 1176, normalized size of antiderivative = 2.79 \[ \int \frac {1}{a-a \sin ^{10}(x)} \, dx=\text {Too large to display} \] Input:
int(1/(a - a*sin(x)^10),x)
Output:
tan(x)/(5*a) + atan((tan(x)*(- (- (2*5^(1/2))/5 - 1)^(1/2)/(200*a^2) - 1/( 200*a^2))^(1/2)*5i)/((4*(- (2*5^(1/2))/5 - 1)^(1/2))/a - (2*5^(1/2)*(- (2* 5^(1/2))/5 - 1)^(1/2))/a) - (5^(1/2)*tan(x)*(- (- (2*5^(1/2))/5 - 1)^(1/2) /(200*a^2) - 1/(200*a^2))^(1/2)*5i)/((4*(- (2*5^(1/2))/5 - 1)^(1/2))/a - ( 2*5^(1/2)*(- (2*5^(1/2))/5 - 1)^(1/2))/a) + (tan(x)*(- (- (2*5^(1/2))/5 - 1)^(1/2)/(200*a^2) - 1/(200*a^2))^(1/2)*(- (2*5^(1/2))/5 - 1)^(1/2)*75i)/( (4*(- (2*5^(1/2))/5 - 1)^(1/2))/a - (2*5^(1/2)*(- (2*5^(1/2))/5 - 1)^(1/2) )/a) - (5^(1/2)*tan(x)*(- (- (2*5^(1/2))/5 - 1)^(1/2)/(200*a^2) - 1/(200*a ^2))^(1/2)*(- (2*5^(1/2))/5 - 1)^(1/2)*35i)/((4*(- (2*5^(1/2))/5 - 1)^(1/2 ))/a - (2*5^(1/2)*(- (2*5^(1/2))/5 - 1)^(1/2))/a))*(-((- (2*5^(1/2))/5 - 1 )^(1/2) + 1)/(200*a^2))^(1/2)*2i - atan((tan(x)*((- (2*5^(1/2))/5 - 1)^(1/ 2)/(200*a^2) - 1/(200*a^2))^(1/2)*5i)/((4*(- (2*5^(1/2))/5 - 1)^(1/2))/a - (2*5^(1/2)*(- (2*5^(1/2))/5 - 1)^(1/2))/a) - (5^(1/2)*tan(x)*((- (2*5^(1/ 2))/5 - 1)^(1/2)/(200*a^2) - 1/(200*a^2))^(1/2)*5i)/((4*(- (2*5^(1/2))/5 - 1)^(1/2))/a - (2*5^(1/2)*(- (2*5^(1/2))/5 - 1)^(1/2))/a) - (tan(x)*((- (2 *5^(1/2))/5 - 1)^(1/2)/(200*a^2) - 1/(200*a^2))^(1/2)*(- (2*5^(1/2))/5 - 1 )^(1/2)*75i)/((4*(- (2*5^(1/2))/5 - 1)^(1/2))/a - (2*5^(1/2)*(- (2*5^(1/2) )/5 - 1)^(1/2))/a) + (5^(1/2)*tan(x)*((- (2*5^(1/2))/5 - 1)^(1/2)/(200*a^2 ) - 1/(200*a^2))^(1/2)*(- (2*5^(1/2))/5 - 1)^(1/2)*35i)/((4*(- (2*5^(1/2)) /5 - 1)^(1/2))/a - (2*5^(1/2)*(- (2*5^(1/2))/5 - 1)^(1/2))/a))*(((- (2*...
\[ \int \frac {1}{a-a \sin ^{10}(x)} \, dx=-\frac {\int \frac {1}{\sin \left (x \right )^{10}-1}d x}{a} \] Input:
int(1/(a-a*sin(x)^10),x)
Output:
( - int(1/(sin(x)**10 - 1),x))/a