Integrand size = 11, antiderivative size = 311 \[ \int \frac {1}{a-a \sin ^{12}(x)} \, dx=\frac {x}{6 \sqrt {2} a}+\frac {x}{2 \sqrt {3} a}+\frac {\arctan \left (\frac {1-2 \cos ^2(x)}{2+\sqrt {3}-2 \cos (x) \sin (x)}\right )}{4 \sqrt {3} a}-\frac {\arctan \left (\frac {1-2 \cos ^2(x)}{2+\sqrt {3}+2 \cos (x) \sin (x)}\right )}{4 \sqrt {3} a}+\frac {\arctan \left (\frac {\cos (x) \sin (x)}{1+\sqrt {2}+\sin ^2(x)}\right )}{6 \sqrt {2} a}-\frac {\sqrt {\frac {1}{3} \left (3+2 \sqrt {3}\right )} \arctan \left (2-\sqrt {3}-2 \sqrt {-3+2 \sqrt {3}} \tan (x)\right )}{12 a}+\frac {\sqrt {\frac {1}{3} \left (3+2 \sqrt {3}\right )} \arctan \left (2-\sqrt {3}+2 \sqrt {-3+2 \sqrt {3}} \tan (x)\right )}{12 a}+\frac {\text {arctanh}(\cos (x) \sin (x))}{12 a}+\frac {\sqrt {\frac {1}{3} \left (-3+2 \sqrt {3}\right )} \text {arctanh}\left (\frac {\sqrt {-3+2 \sqrt {3}} \tan (x)}{1+\sqrt {3} \tan ^2(x)}\right )}{12 a}+\frac {\tan (x)}{6 a} \] Output:
1/12*x*2^(1/2)/a+1/6*x*3^(1/2)/a+1/12*arctan((1-2*cos(x)^2)/(2+3^(1/2)-2*c os(x)*sin(x)))*3^(1/2)/a-1/12*arctan((1-2*cos(x)^2)/(2+3^(1/2)+2*cos(x)*si n(x)))*3^(1/2)/a+1/12*arctan(cos(x)*sin(x)/(1+2^(1/2)+sin(x)^2))*2^(1/2)/a +1/36*(9+6*3^(1/2))^(1/2)*arctan(-2+3^(1/2)+2*(-3+2*3^(1/2))^(1/2)*tan(x)) /a-1/36*(9+6*3^(1/2))^(1/2)*arctan(-2+3^(1/2)-2*(-3+2*3^(1/2))^(1/2)*tan(x ))/a+1/12*arctanh(cos(x)*sin(x))/a+1/36*(-9+6*3^(1/2))^(1/2)*arctanh((-3+2 *3^(1/2))^(1/2)*tan(x)/(1+3^(1/2)*tan(x)^2))/a+1/6*tan(x)/a
Result contains complex when optimal does not.
Time = 9.11 (sec) , antiderivative size = 361, normalized size of antiderivative = 1.16 \[ \int \frac {1}{a-a \sin ^{12}(x)} \, dx=\frac {12 \sqrt {2} \arctan \left (\sqrt {2} \tan (x)\right )-4 (-1)^{3/4} \sqrt [4]{3} \left (3 i+\sqrt {3}\right ) \arctan \left (\frac {1}{2} \sqrt [4]{-\frac {1}{3}} \left (-3 i+\sqrt {3}\right ) \tan (x)\right )+4 (-1)^{3/4} \sqrt [4]{3} \left (3+i \sqrt {3}\right ) \arctan \left (\frac {(-1)^{3/4} \left (3 i+\sqrt {3}\right ) \tan (x)}{2 \sqrt [4]{3}}\right )+24 \tan (x)+\frac {3 \left (i \sqrt {1-i \sqrt {3}} \left (3 i+\sqrt {3}\right ) \arctan \left (\frac {1}{2} \left (-i+\sqrt {3}\right ) \tan (x)\right )+\sqrt {1+i \sqrt {3}} \left (3+i \sqrt {3}\right ) \arctan \left (\frac {1}{2} \left (i+\sqrt {3}\right ) \tan (x)\right )\right ) (42+15 \cos (2 x)+6 \cos (4 x)+\cos (6 x)) \sec ^2(x)}{(-2+\sin (2 x)) (2+\sin (2 x)) \left (-3 i \sqrt {1-i \sqrt {3}}-3 i \sqrt {1+i \sqrt {3}}+\sqrt {3-3 i \sqrt {3}}-\sqrt {3+3 i \sqrt {3}}+2 \left (\sqrt {3-3 i \sqrt {3}}-\sqrt {3+3 i \sqrt {3}}\right ) \tan ^2(x)\right )}}{144 a} \] Input:
Integrate[(a - a*Sin[x]^12)^(-1),x]
Output:
(12*Sqrt[2]*ArcTan[Sqrt[2]*Tan[x]] - 4*(-1)^(3/4)*3^(1/4)*(3*I + Sqrt[3])* ArcTan[((-1/3)^(1/4)*(-3*I + Sqrt[3])*Tan[x])/2] + 4*(-1)^(3/4)*3^(1/4)*(3 + I*Sqrt[3])*ArcTan[((-1)^(3/4)*(3*I + Sqrt[3])*Tan[x])/(2*3^(1/4))] + 24 *Tan[x] + (3*(I*Sqrt[1 - I*Sqrt[3]]*(3*I + Sqrt[3])*ArcTan[((-I + Sqrt[3]) *Tan[x])/2] + Sqrt[1 + I*Sqrt[3]]*(3 + I*Sqrt[3])*ArcTan[((I + Sqrt[3])*Ta n[x])/2])*(42 + 15*Cos[2*x] + 6*Cos[4*x] + Cos[6*x])*Sec[x]^2)/((-2 + Sin[ 2*x])*(2 + Sin[2*x])*((-3*I)*Sqrt[1 - I*Sqrt[3]] - (3*I)*Sqrt[1 + I*Sqrt[3 ]] + Sqrt[3 - (3*I)*Sqrt[3]] - Sqrt[3 + (3*I)*Sqrt[3]] + 2*(Sqrt[3 - (3*I) *Sqrt[3]] - Sqrt[3 + (3*I)*Sqrt[3]])*Tan[x]^2)))/(144*a)
Time = 0.73 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.55, 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.
| \(\displaystyle \int \frac {1}{a-a \sin ^{12}(x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{a-a \sin (x)^{12}}dx\) |
| \(\Big \downarrow \) 3690 |
| \(\displaystyle \frac {\int \frac {1}{1-\sin ^2(x)}dx}{6 a}+\frac {\int \frac {1}{\sin ^2(x)+1}dx}{6 a}+\frac {\int \frac {1}{1-\sqrt [3]{-1} \sin ^2(x)}dx}{6 a}+\frac {\int \frac {1}{\sqrt [3]{-1} \sin ^2(x)+1}dx}{6 a}+\frac {\int \frac {1}{1-(-1)^{2/3} \sin ^2(x)}dx}{6 a}+\frac {\int \frac {1}{(-1)^{2/3} \sin ^2(x)+1}dx}{6 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {1}{1-\sin (x)^2}dx}{6 a}+\frac {\int \frac {1}{\sin (x)^2+1}dx}{6 a}+\frac {\int \frac {1}{1-\sqrt [3]{-1} \sin (x)^2}dx}{6 a}+\frac {\int \frac {1}{\sqrt [3]{-1} \sin (x)^2+1}dx}{6 a}+\frac {\int \frac {1}{1-(-1)^{2/3} \sin (x)^2}dx}{6 a}+\frac {\int \frac {1}{(-1)^{2/3} \sin (x)^2+1}dx}{6 a}\) |
| \(\Big \downarrow \) 3654 |
| \(\displaystyle \frac {\int \frac {1}{\sin (x)^2+1}dx}{6 a}+\frac {\int \frac {1}{1-\sqrt [3]{-1} \sin (x)^2}dx}{6 a}+\frac {\int \frac {1}{\sqrt [3]{-1} \sin (x)^2+1}dx}{6 a}+\frac {\int \frac {1}{1-(-1)^{2/3} \sin (x)^2}dx}{6 a}+\frac {\int \frac {1}{(-1)^{2/3} \sin (x)^2+1}dx}{6 a}+\frac {\int \sec ^2(x)dx}{6 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {1}{\sin (x)^2+1}dx}{6 a}+\frac {\int \frac {1}{1-\sqrt [3]{-1} \sin (x)^2}dx}{6 a}+\frac {\int \frac {1}{\sqrt [3]{-1} \sin (x)^2+1}dx}{6 a}+\frac {\int \frac {1}{1-(-1)^{2/3} \sin (x)^2}dx}{6 a}+\frac {\int \frac {1}{(-1)^{2/3} \sin (x)^2+1}dx}{6 a}+\frac {\int \csc \left (x+\frac {\pi }{2}\right )^2dx}{6 a}\) |
| \(\Big \downarrow \) 3660 |
| \(\displaystyle \frac {\int \frac {1}{2 \tan ^2(x)+1}d\tan (x)}{6 a}+\frac {\int \frac {1}{\left (1-\sqrt [3]{-1}\right ) \tan ^2(x)+1}d\tan (x)}{6 a}+\frac {\int \frac {1}{\left (1+\sqrt [3]{-1}\right ) \tan ^2(x)+1}d\tan (x)}{6 a}+\frac {\int \frac {1}{\left (1-(-1)^{2/3}\right ) \tan ^2(x)+1}d\tan (x)}{6 a}+\frac {\int \frac {1}{\left (1+(-1)^{2/3}\right ) \tan ^2(x)+1}d\tan (x)}{6 a}+\frac {\int \csc \left (x+\frac {\pi }{2}\right )^2dx}{6 a}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {\int \csc \left (x+\frac {\pi }{2}\right )^2dx}{6 a}+\frac {\arctan \left (\sqrt {2} \tan (x)\right )}{6 \sqrt {2} a}+\frac {\arctan \left (\sqrt {1-\sqrt [3]{-1}} \tan (x)\right )}{6 \sqrt {1-\sqrt [3]{-1}} a}+\frac {\arctan \left (\sqrt {1+\sqrt [3]{-1}} \tan (x)\right )}{6 \sqrt {1+\sqrt [3]{-1}} a}+\frac {\arctan \left (\sqrt {1-(-1)^{2/3}} \tan (x)\right )}{6 \sqrt {1-(-1)^{2/3}} a}+\frac {\arctan \left (\sqrt {1+(-1)^{2/3}} \tan (x)\right )}{6 \sqrt {1+(-1)^{2/3}} a}\) |
| \(\Big \downarrow \) 4254 |
| \(\displaystyle -\frac {\int 1d(-\tan (x))}{6 a}+\frac {\arctan \left (\sqrt {2} \tan (x)\right )}{6 \sqrt {2} a}+\frac {\arctan \left (\sqrt {1-\sqrt [3]{-1}} \tan (x)\right )}{6 \sqrt {1-\sqrt [3]{-1}} a}+\frac {\arctan \left (\sqrt {1+\sqrt [3]{-1}} \tan (x)\right )}{6 \sqrt {1+\sqrt [3]{-1}} a}+\frac {\arctan \left (\sqrt {1-(-1)^{2/3}} \tan (x)\right )}{6 \sqrt {1-(-1)^{2/3}} a}+\frac {\arctan \left (\sqrt {1+(-1)^{2/3}} \tan (x)\right )}{6 \sqrt {1+(-1)^{2/3}} a}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {\arctan \left (\sqrt {2} \tan (x)\right )}{6 \sqrt {2} a}+\frac {\arctan \left (\sqrt {1-\sqrt [3]{-1}} \tan (x)\right )}{6 \sqrt {1-\sqrt [3]{-1}} a}+\frac {\arctan \left (\sqrt {1+\sqrt [3]{-1}} \tan (x)\right )}{6 \sqrt {1+\sqrt [3]{-1}} a}+\frac {\arctan \left (\sqrt {1-(-1)^{2/3}} \tan (x)\right )}{6 \sqrt {1-(-1)^{2/3}} a}+\frac {\arctan \left (\sqrt {1+(-1)^{2/3}} \tan (x)\right )}{6 \sqrt {1+(-1)^{2/3}} a}+\frac {\tan (x)}{6 a}\) |
Input:
Int[(a - a*Sin[x]^12)^(-1),x]
Output:
ArcTan[Sqrt[2]*Tan[x]]/(6*Sqrt[2]*a) + ArcTan[Sqrt[1 - (-1)^(1/3)]*Tan[x]] /(6*Sqrt[1 - (-1)^(1/3)]*a) + ArcTan[Sqrt[1 + (-1)^(1/3)]*Tan[x]]/(6*Sqrt[ 1 + (-1)^(1/3)]*a) + ArcTan[Sqrt[1 - (-1)^(2/3)]*Tan[x]]/(6*Sqrt[1 - (-1)^ (2/3)]*a) + ArcTan[Sqrt[1 + (-1)^(2/3)]*Tan[x]]/(6*Sqrt[1 + (-1)^(2/3)]*a) + Tan[x]/(6*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]
Time = 36.85 (sec) , antiderivative size = 276, normalized size of antiderivative = 0.89
| method | result | size |
| default | \(\frac {\frac {\tan \left (x \right )}{6}+\frac {\ln \left (\tan \left (x \right )^{2}+\tan \left (x \right )+1\right )}{24}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 \tan \left (x \right )+1\right ) \sqrt {3}}{3}\right )}{12}+\frac {\sqrt {2}\, \arctan \left (\tan \left (x \right ) \sqrt {2}\right )}{12}-\frac {\ln \left (\tan \left (x \right )^{2}-\tan \left (x \right )+1\right )}{24}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 \tan \left (x \right )-1\right ) \sqrt {3}}{3}\right )}{12}+\frac {\sqrt {3}\, \left (-\frac {\sqrt {-3+2 \sqrt {3}}\, \ln \left (-\sqrt {-3+2 \sqrt {3}}\, \sqrt {3}\, \tan \left (x \right )+3 \tan \left (x \right )^{2}+\sqrt {3}\right )}{6}+\frac {2 \left (-\frac {\left (-3+2 \sqrt {3}\right ) \sqrt {3}}{6}+2\right ) \arctan \left (\frac {-\sqrt {3}\, \sqrt {-3+2 \sqrt {3}}+6 \tan \left (x \right )}{\sqrt {9+6 \sqrt {3}}}\right )}{\sqrt {9+6 \sqrt {3}}}\right )}{12}+\frac {\sqrt {3}\, \left (\frac {\sqrt {-3+2 \sqrt {3}}\, \ln \left (\sqrt {3}+\sqrt {-3+2 \sqrt {3}}\, \sqrt {3}\, \tan \left (x \right )+3 \tan \left (x \right )^{2}\right )}{6}+\frac {2 \left (-\frac {\left (-3+2 \sqrt {3}\right ) \sqrt {3}}{6}+2\right ) \arctan \left (\frac {\sqrt {3}\, \sqrt {-3+2 \sqrt {3}}+6 \tan \left (x \right )}{\sqrt {9+6 \sqrt {3}}}\right )}{\sqrt {9+6 \sqrt {3}}}\right )}{12}}{a}\) | \(276\) |
| risch | \(\frac {i}{3 a \left ({\mathrm e}^{2 i x}+1\right )}-\frac {\ln \left ({\mathrm e}^{2 i x}-i \sqrt {3}-2 i\right )}{24 a}+\frac {i \ln \left ({\mathrm e}^{2 i x}-i \sqrt {3}-2 i\right ) \sqrt {3}}{24 a}-\frac {\ln \left ({\mathrm e}^{2 i x}+i \sqrt {3}-2 i\right )}{24 a}-\frac {i \ln \left ({\mathrm e}^{2 i x}+i \sqrt {3}-2 i\right ) \sqrt {3}}{24 a}+\frac {i \sqrt {2}\, \ln \left ({\mathrm e}^{2 i x}-2 \sqrt {2}-3\right )}{24 a}-\frac {i \sqrt {2}\, \ln \left ({\mathrm e}^{2 i x}+2 \sqrt {2}-3\right )}{24 a}+\frac {\ln \left ({\mathrm e}^{2 i x}+i \sqrt {3}+2 i\right )}{24 a}+\frac {i \ln \left ({\mathrm e}^{2 i x}+i \sqrt {3}+2 i\right ) \sqrt {3}}{24 a}+\frac {\ln \left ({\mathrm e}^{2 i x}-i \sqrt {3}+2 i\right )}{24 a}-\frac {i \ln \left ({\mathrm e}^{2 i x}-i \sqrt {3}+2 i\right ) \sqrt {3}}{24 a}+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (62208 a^{4} \textit {\_Z}^{4}+432 a^{2} \textit {\_Z}^{2}+1\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 i x}-10368 i a^{3} \textit {\_R}^{3}+864 a^{2} \textit {\_R}^{2}+1\right )\right )\) | \(286\) |
Input:
int(1/(a-a*sin(x)^12),x,method=_RETURNVERBOSE)
Output:
1/a*(1/6*tan(x)+1/24*ln(tan(x)^2+tan(x)+1)+1/12*3^(1/2)*arctan(1/3*(2*tan( x)+1)*3^(1/2))+1/12*2^(1/2)*arctan(tan(x)*2^(1/2))-1/24*ln(tan(x)^2-tan(x) +1)+1/12*3^(1/2)*arctan(1/3*(2*tan(x)-1)*3^(1/2))+1/12*3^(1/2)*(-1/6*(-3+2 *3^(1/2))^(1/2)*ln(-(-3+2*3^(1/2))^(1/2)*3^(1/2)*tan(x)+3*tan(x)^2+3^(1/2) )+2*(-1/6*(-3+2*3^(1/2))*3^(1/2)+2)/(9+6*3^(1/2))^(1/2)*arctan((-3^(1/2)*( -3+2*3^(1/2))^(1/2)+6*tan(x))/(9+6*3^(1/2))^(1/2)))+1/12*3^(1/2)*(1/6*(-3+ 2*3^(1/2))^(1/2)*ln(3^(1/2)+(-3+2*3^(1/2))^(1/2)*3^(1/2)*tan(x)+3*tan(x)^2 )+2*(-1/6*(-3+2*3^(1/2))*3^(1/2)+2)/(9+6*3^(1/2))^(1/2)*arctan((3^(1/2)*(- 3+2*3^(1/2))^(1/2)+6*tan(x))/(9+6*3^(1/2))^(1/2))))
Leaf count of result is larger than twice the leaf count of optimal. 638 vs. \(2 (232) = 464\).
Time = 0.31 (sec) , antiderivative size = 638, normalized size of antiderivative = 2.05 \[ \int \frac {1}{a-a \sin ^{12}(x)} \, dx =\text {Too large to display} \] Input:
integrate(1/(a-a*sin(x)^12),x, algorithm="fricas")
Output:
-1/48*(2*sqrt(1/2)*a*sqrt(-(sqrt(1/3)*a^2*sqrt(-1/a^4) + 1)/a^2)*cos(x)*lo g(6*sqrt(1/2)*(sqrt(1/3)*a^3*sqrt(-1/a^4)*cos(x)*sin(x) + a*cos(x)*sin(x)) *sqrt(-(sqrt(1/3)*a^2*sqrt(-1/a^4) + 1)/a^2) + 3*sqrt(1/3)*(2*a^2*cos(x)^2 - a^2)*sqrt(-1/a^4) + 4*cos(x)^2 - 3) - 2*sqrt(1/2)*a*sqrt(-(sqrt(1/3)*a^ 2*sqrt(-1/a^4) + 1)/a^2)*cos(x)*log(-6*sqrt(1/2)*(sqrt(1/3)*a^3*sqrt(-1/a^ 4)*cos(x)*sin(x) + a*cos(x)*sin(x))*sqrt(-(sqrt(1/3)*a^2*sqrt(-1/a^4) + 1) /a^2) + 3*sqrt(1/3)*(2*a^2*cos(x)^2 - a^2)*sqrt(-1/a^4) + 4*cos(x)^2 - 3) + 2*sqrt(1/2)*a*sqrt((sqrt(1/3)*a^2*sqrt(-1/a^4) - 1)/a^2)*cos(x)*log(6*sq rt(1/2)*(sqrt(1/3)*a^3*sqrt(-1/a^4)*cos(x)*sin(x) - a*cos(x)*sin(x))*sqrt( (sqrt(1/3)*a^2*sqrt(-1/a^4) - 1)/a^2) + 3*sqrt(1/3)*(2*a^2*cos(x)^2 - a^2) *sqrt(-1/a^4) - 4*cos(x)^2 + 3) - 2*sqrt(1/2)*a*sqrt((sqrt(1/3)*a^2*sqrt(- 1/a^4) - 1)/a^2)*cos(x)*log(-6*sqrt(1/2)*(sqrt(1/3)*a^3*sqrt(-1/a^4)*cos(x )*sin(x) - a*cos(x)*sin(x))*sqrt((sqrt(1/3)*a^2*sqrt(-1/a^4) - 1)/a^2) + 3 *sqrt(1/3)*(2*a^2*cos(x)^2 - a^2)*sqrt(-1/a^4) - 4*cos(x)^2 + 3) - 2*sqrt( 3)*arctan(1/3*(4*sqrt(3)*cos(x)*sin(x) + sqrt(3))/(2*cos(x)^2 - 1))*cos(x) - 2*sqrt(3)*arctan(1/3*(4*sqrt(3)*cos(x)*sin(x) - sqrt(3))/(2*cos(x)^2 - 1))*cos(x) + 2*sqrt(2)*arctan(1/4*(3*sqrt(2)*cos(x)^2 - 2*sqrt(2))/(cos(x) *sin(x)))*cos(x) - cos(x)*log(-cos(x)^4 + cos(x)^2 + 2*cos(x)*sin(x) + 1) + cos(x)*log(-cos(x)^4 + cos(x)^2 - 2*cos(x)*sin(x) + 1) - 8*sin(x))/(a*co s(x))
Timed out. \[ \int \frac {1}{a-a \sin ^{12}(x)} \, dx=\text {Timed out} \] Input:
integrate(1/(a-a*sin(x)**12),x)
Output:
Timed out
\[ \int \frac {1}{a-a \sin ^{12}(x)} \, dx=\int { -\frac {1}{a \sin \left (x\right )^{12} - a} \,d x } \] Input:
integrate(1/(a-a*sin(x)^12),x, algorithm="maxima")
Output:
1/24*((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)) - 24*(a*cos(2*x)^2 + a*sin(2*x)^2 + 2*a*cos(2*x) + a)*inte grate(2/3*((cos(6*x) - 10*cos(4*x) + cos(2*x))*cos(8*x) + (110*cos(4*x) - 16*cos(2*x) + 1)*cos(6*x) - 8*cos(6*x)^2 + 10*(11*cos(2*x) - 1)*cos(4*x) - 300*cos(4*x)^2 - 8*cos(2*x)^2 + (sin(6*x) - 10*sin(4*x) + sin(2*x))*sin(8 *x) + 2*(55*sin(4*x) - 8*sin(2*x))*sin(6*x) - 8*sin(6*x)^2 - 300*sin(4*x)^ 2 + 110*sin(4*x)*sin(2*x) - 8*sin(2*x)^2 + cos(2*x))/(a*cos(8*x)^2 + 64*a* cos(6*x)^2 + 900*a*cos(4*x)^2 + 64*a*cos(2*x)^2 + a*sin(8*x)^2 + 64*a*sin( 6*x)^2 + 900*a*sin(4*x)^2 - 480*a*sin(4*x)*sin(2*x) + 64*a*sin(2*x)^2 - 2* (8*a*cos(6*x) - 30*a*cos(4*x) + 8*a*cos(2*x) - a)*cos(8*x) - 16*(30*a*cos( 4*x) - 8*a*cos(2*x) + a)*cos(6*x) - 60*(8*a*cos(2*x) - a)*cos(4*x) - 16*a* cos(2*x) - 4*(4*a*sin(6*x) - 15*a*sin(4*x) + 4*a*sin(2*x))*sin(8*x) - 32*( 15*a*sin(4*x) - 4*a*sin(2*x))*sin(6*x) + a), x) + 24*(a*cos(2*x)^2 + a*sin (2*x)^2 + 2*a*cos(2*x) + a)*integrate(1/6*((cos(3*x) + 4*cos(2*x) - cos...
Time = 0.50 (sec) , antiderivative size = 439, normalized size of antiderivative = 1.41 \[ \int \frac {1}{a-a \sin ^{12}(x)} \, dx =\text {Too large to display} \] Input:
integrate(1/(a-a*sin(x)^12),x, algorithm="giac")
Output:
1/12*sqrt(3)*(x + arctan(-(sqrt(3)*sin(2*x) + cos(2*x) - 2*sin(2*x) + 1)/( sqrt(3)*cos(2*x) + sqrt(3) - 2*cos(2*x) - sin(2*x) + 2)))/a + 1/12*sqrt(3) *(x + arctan(-(sqrt(3)*sin(2*x) - cos(2*x) - 2*sin(2*x) - 1)/(sqrt(3)*cos( 2*x) + sqrt(3) - 2*cos(2*x) + sin(2*x) + 2)))/a + 1/12*sqrt(2)*(x + arctan (-(sqrt(2)*sin(2*x) - 2*sin(2*x))/(sqrt(2)*cos(2*x) + sqrt(2) - 2*cos(2*x) + 2)))/a + 1/24*log(tan(x)^2 + tan(x) + 1)/a - 1/24*log(tan(x)^2 - tan(x) + 1)/a + 1/6*tan(x)/a + 1/72*(108^(1/4)*a^2 + 3*12^(1/4)*a*abs(a))*(pi*fl oor(x/pi + 1/2) - arctan(-3*(1/3)^(3/4)*((1/3)^(1/4)*(sqrt(6) - sqrt(2)) + 4*tan(x))/(sqrt(6) + sqrt(2))))/a^3 + 1/72*(108^(1/4)*a^2 + 3*12^(1/4)*a* abs(a))*(pi*floor(x/pi + 1/2) + arctan(-3*(1/3)^(3/4)*((1/3)^(1/4)*(sqrt(6 ) - sqrt(2)) - 4*tan(x))/(sqrt(6) + sqrt(2))))/a^3 - 1/144*(108^(1/4)*a^2 - 3*12^(1/4)*a*abs(a))*log(1/2*(sqrt(6)*(1/3)^(1/4) - sqrt(2)*(1/3)^(1/4)) *tan(x) + tan(x)^2 + sqrt(1/3))/a^3 + 1/144*(108^(1/4)*a^2 - 3*12^(1/4)*a* abs(a))*log(-1/2*(sqrt(6)*(1/3)^(1/4) - sqrt(2)*(1/3)^(1/4))*tan(x) + tan( x)^2 + sqrt(1/3))/a^3
Time = 37.93 (sec) , antiderivative size = 1615, normalized size of antiderivative = 5.19 \[ \int \frac {1}{a-a \sin ^{12}(x)} \, dx=\text {Too large to display} \] Input:
int(1/(a - a*sin(x)^12),x)
Output:
tan(x)/(6*a) + atan(a*tan(x)*(- (3^(1/2)*1i)/(864*a^2) - 1/(288*a^2))^(1/2 )*18i + 6*3^(1/2)*a*tan(x)*(- (3^(1/2)*1i)/(864*a^2) - 1/(288*a^2))^(1/2)) *(-(3^(1/2)*1i + 3)/(864*a^2))^(1/2)*2i + atan(a*tan(x)*((3^(1/2)*1i)/(864 *a^2) - 1/(288*a^2))^(1/2)*18i - 6*3^(1/2)*a*tan(x)*((3^(1/2)*1i)/(864*a^2 ) - 1/(288*a^2))^(1/2))*((3^(1/2)*1i - 3)/(864*a^2))^(1/2)*2i + (atan((((( 422*tan(x))/a^8 - ((3^(1/2)*1i - 1)*(13080/a^7 - ((3^(1/2)*1i - 1)*((26640 0*tan(x))/a^6 - ((3^(1/2)*1i - 1)*(3335040/a^5 - ((3^(1/2)*1i - 1)*((69672 960*tan(x))/a^4 - ((3^(1/2)*1i - 1)*(418037760/a^3 - ((3^(1/2)*1i - 1)*((8 778792960*tan(x))/a^2 - ((3^(1/2)*1i - 1)*(30098718720/a - (27088846848*ta n(x)*(3^(1/2)*1i - 1))/a))/(24*a)))/(24*a)))/(24*a)))/(24*a)))/(24*a)))/(2 4*a)))/(24*a))*(3^(1/2)*1i - 1)*1i)/(24*a) + (((422*tan(x))/a^8 + ((3^(1/2 )*1i - 1)*(13080/a^7 + ((3^(1/2)*1i - 1)*((266400*tan(x))/a^6 + ((3^(1/2)* 1i - 1)*(3335040/a^5 + ((3^(1/2)*1i - 1)*((69672960*tan(x))/a^4 + ((3^(1/2 )*1i - 1)*(418037760/a^3 + ((3^(1/2)*1i - 1)*((8778792960*tan(x))/a^2 + (( 3^(1/2)*1i - 1)*(30098718720/a + (27088846848*tan(x)*(3^(1/2)*1i - 1))/a)) /(24*a)))/(24*a)))/(24*a)))/(24*a)))/(24*a)))/(24*a)))/(24*a))*(3^(1/2)*1i - 1)*1i)/(24*a))/(42/a^9 - (((422*tan(x))/a^8 - ((3^(1/2)*1i - 1)*(13080/ a^7 - ((3^(1/2)*1i - 1)*((266400*tan(x))/a^6 - ((3^(1/2)*1i - 1)*(3335040/ a^5 - ((3^(1/2)*1i - 1)*((69672960*tan(x))/a^4 - ((3^(1/2)*1i - 1)*(418037 760/a^3 - ((3^(1/2)*1i - 1)*((8778792960*tan(x))/a^2 - ((3^(1/2)*1i - 1...
\[ \int \frac {1}{a-a \sin ^{12}(x)} \, dx=-\frac {\int \frac {1}{\sin \left (x \right )^{12}-1}d x}{a} \] Input:
int(1/(a-a*sin(x)^12),x)
Output:
( - int(1/(sin(x)**12 - 1),x))/a