Integrand size = 11, antiderivative size = 167 \[ \int \frac {1}{a-a \sin ^6(x)} \, dx=-\frac {\sqrt {\frac {1}{3} \left (3+2 \sqrt {3}\right )} \arctan \left (2-\sqrt {3}-2 \sqrt {-3+2 \sqrt {3}} \tan (x)\right )}{6 a}+\frac {\sqrt {\frac {1}{3} \left (3+2 \sqrt {3}\right )} \arctan \left (2-\sqrt {3}+2 \sqrt {-3+2 \sqrt {3}} \tan (x)\right )}{6 a}+\frac {\sqrt {\frac {1}{3} \left (-3+2 \sqrt {3}\right )} \text {arctanh}\left (\frac {\sqrt {-3+2 \sqrt {3}} \tan (x)}{1+\sqrt {3} \tan ^2(x)}\right )}{6 a}+\frac {\tan (x)}{3 a} \] Output:
1/18*(9+6*3^(1/2))^(1/2)*arctan(-2+3^(1/2)+2*(-3+2*3^(1/2))^(1/2)*tan(x))/ a-1/18*(9+6*3^(1/2))^(1/2)*arctan(-2+3^(1/2)-2*(-3+2*3^(1/2))^(1/2)*tan(x) )/a+1/18*(-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+1/3*tan(x)/a
Result contains complex when optimal does not.
Time = 1.69 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.72 \[ \int \frac {1}{a-a \sin ^6(x)} \, dx=\frac {\cos (x) (15-8 \cos (2 x)+\cos (4 x)) \left (i \sqrt [4]{-3} \left (3 i+\sqrt {3}\right ) \arctan \left (\frac {1}{2} \sqrt [4]{-\frac {1}{3}} \left (-3 i+\sqrt {3}\right ) \tan (x)\right ) \cos (x)+\sqrt [4]{-3} \left (-3 i+\sqrt {3}\right ) \arctan \left (\frac {(-1)^{3/4} \left (3 i+\sqrt {3}\right ) \tan (x)}{2 \sqrt [4]{3}}\right ) \cos (x)-6 \sin (x)\right )}{144 a \left (-1+\sin ^6(x)\right )} \] Input:
Integrate[(a - a*Sin[x]^6)^(-1),x]
Output:
(Cos[x]*(15 - 8*Cos[2*x] + Cos[4*x])*(I*(-3)^(1/4)*(3*I + Sqrt[3])*ArcTan[ ((-1/3)^(1/4)*(-3*I + Sqrt[3])*Tan[x])/2]*Cos[x] + (-3)^(1/4)*(-3*I + Sqrt [3])*ArcTan[((-1)^(3/4)*(3*I + Sqrt[3])*Tan[x])/(2*3^(1/4))]*Cos[x] - 6*Si n[x]))/(144*a*(-1 + Sin[x]^6))
Time = 0.51 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.48, 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.
| \(\displaystyle \int \frac {1}{a-a \sin ^6(x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{a-a \sin (x)^6}dx\) |
| \(\Big \downarrow \) 3690 |
| \(\displaystyle \frac {\int \frac {1}{1-\sin ^2(x)}dx}{3 a}+\frac {\int \frac {1}{\sqrt [3]{-1} \sin ^2(x)+1}dx}{3 a}+\frac {\int \frac {1}{1-(-1)^{2/3} \sin ^2(x)}dx}{3 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {1}{1-\sin (x)^2}dx}{3 a}+\frac {\int \frac {1}{\sqrt [3]{-1} \sin (x)^2+1}dx}{3 a}+\frac {\int \frac {1}{1-(-1)^{2/3} \sin (x)^2}dx}{3 a}\) |
| \(\Big \downarrow \) 3654 |
| \(\displaystyle \frac {\int \frac {1}{\sqrt [3]{-1} \sin (x)^2+1}dx}{3 a}+\frac {\int \frac {1}{1-(-1)^{2/3} \sin (x)^2}dx}{3 a}+\frac {\int \sec ^2(x)dx}{3 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {1}{\sqrt [3]{-1} \sin (x)^2+1}dx}{3 a}+\frac {\int \frac {1}{1-(-1)^{2/3} \sin (x)^2}dx}{3 a}+\frac {\int \csc \left (x+\frac {\pi }{2}\right )^2dx}{3 a}\) |
| \(\Big \downarrow \) 3660 |
| \(\displaystyle \frac {\int \frac {1}{\left (1+\sqrt [3]{-1}\right ) \tan ^2(x)+1}d\tan (x)}{3 a}+\frac {\int \frac {1}{\left (1-(-1)^{2/3}\right ) \tan ^2(x)+1}d\tan (x)}{3 a}+\frac {\int \csc \left (x+\frac {\pi }{2}\right )^2dx}{3 a}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {\int \csc \left (x+\frac {\pi }{2}\right )^2dx}{3 a}+\frac {\arctan \left (\sqrt {1+\sqrt [3]{-1}} \tan (x)\right )}{3 \sqrt {1+\sqrt [3]{-1}} a}+\frac {\arctan \left (\sqrt {1-(-1)^{2/3}} \tan (x)\right )}{3 \sqrt {1-(-1)^{2/3}} a}\) |
| \(\Big \downarrow \) 4254 |
| \(\displaystyle -\frac {\int 1d(-\tan (x))}{3 a}+\frac {\arctan \left (\sqrt {1+\sqrt [3]{-1}} \tan (x)\right )}{3 \sqrt {1+\sqrt [3]{-1}} a}+\frac {\arctan \left (\sqrt {1-(-1)^{2/3}} \tan (x)\right )}{3 \sqrt {1-(-1)^{2/3}} a}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {\arctan \left (\sqrt {1+\sqrt [3]{-1}} \tan (x)\right )}{3 \sqrt {1+\sqrt [3]{-1}} a}+\frac {\arctan \left (\sqrt {1-(-1)^{2/3}} \tan (x)\right )}{3 \sqrt {1-(-1)^{2/3}} a}+\frac {\tan (x)}{3 a}\) |
Input:
Int[(a - a*Sin[x]^6)^(-1),x]
Output:
ArcTan[Sqrt[1 + (-1)^(1/3)]*Tan[x]]/(3*Sqrt[1 + (-1)^(1/3)]*a) + ArcTan[Sq rt[1 - (-1)^(2/3)]*Tan[x]]/(3*Sqrt[1 - (-1)^(2/3)]*a) + Tan[x]/(3*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 = 0.96 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.40
| method | result | size |
| risch | \(\frac {2 i}{3 a \left ({\mathrm e}^{2 i x}+1\right )}+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (3888 a^{4} \textit {\_Z}^{4}+108 a^{2} \textit {\_Z}^{2}+1\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 i x}-1296 i a^{3} \textit {\_R}^{3}+216 a^{2} \textit {\_R}^{2}+1\right )\right )\) | \(66\) |
| default | \(\frac {\frac {\tan \left (x \right )}{3}+\frac {\sqrt {3}\, \left (\frac {\sqrt {-3+2 \sqrt {3}}\, \ln \left (\sqrt {3}+\sqrt {-3+2 \sqrt {3}}\, \sqrt {3}\, \tan \left (x \right )+3 \tan \left (x \right )^{2}\right )}{6}+\frac {2 \left (-\frac {\left (-3+2 \sqrt {3}\right ) \sqrt {3}}{6}+2\right ) \arctan \left (\frac {\sqrt {3}\, \sqrt {-3+2 \sqrt {3}}+6 \tan \left (x \right )}{\sqrt {9+6 \sqrt {3}}}\right )}{\sqrt {9+6 \sqrt {3}}}\right )}{6}+\frac {\sqrt {3}\, \left (-\frac {\sqrt {-3+2 \sqrt {3}}\, \ln \left (-\sqrt {-3+2 \sqrt {3}}\, \sqrt {3}\, \tan \left (x \right )+3 \tan \left (x \right )^{2}+\sqrt {3}\right )}{6}+\frac {2 \left (-\frac {\left (-3+2 \sqrt {3}\right ) \sqrt {3}}{6}+2\right ) \arctan \left (\frac {-\sqrt {3}\, \sqrt {-3+2 \sqrt {3}}+6 \tan \left (x \right )}{\sqrt {9+6 \sqrt {3}}}\right )}{\sqrt {9+6 \sqrt {3}}}\right )}{6}}{a}\) | \(206\) |
Input:
int(1/(a-a*sin(x)^6),x,method=_RETURNVERBOSE)
Output:
2/3*I/a/(exp(2*I*x)+1)+sum(_R*ln(exp(2*I*x)-1296*I*a^3*_R^3+216*a^2*_R^2+1 ),_R=RootOf(3888*_Z^4*a^4+108*_Z^2*a^2+1))
Leaf count of result is larger than twice the leaf count of optimal. 490 vs. \(2 (113) = 226\).
Time = 0.12 (sec) , antiderivative size = 490, normalized size of antiderivative = 2.93 \[ \int \frac {1}{a-a \sin ^6(x)} \, dx =\text {Too large to display} \] Input:
integrate(1/(a-a*sin(x)^6),x, algorithm="fricas")
Output:
-1/12*(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))*s qrt(-(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) - sqrt(1/2)*a*sqrt(-(sqrt(1/3)*a^2*sq rt(-1/a^4) + 1)/a^2)*cos(x)*log(-6*sqrt(1/2)*(sqrt(1/3)*a^3*sqrt(-1/a^4)*c os(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) + sq rt(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) - 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) - 4*sin(x))/(a*co s(x))
Timed out. \[ \int \frac {1}{a-a \sin ^6(x)} \, dx=\text {Timed out} \] Input:
integrate(1/(a-a*sin(x)**6),x)
Output:
Timed out
\[ \int \frac {1}{a-a \sin ^6(x)} \, dx=\int { -\frac {1}{a \sin \left (x\right )^{6} - a} \,d x } \] Input:
integrate(1/(a-a*sin(x)^6),x, algorithm="maxima")
Output:
-1/3*(3*(a*cos(2*x)^2 + a*sin(2*x)^2 + 2*a*cos(2*x) + a)*integrate(4/3*((c os(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*s in(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*co s(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) - 2*sin(2*x))/(a*cos(2*x)^2 + a*sin(2*x )^2 + 2*a*cos(2*x) + a)
Time = 0.54 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.27 \[ \int \frac {1}{a-a \sin ^6(x)} \, dx=\frac {{\left (\pi \left \lfloor \frac {x}{\pi } + \frac {1}{2} \right \rfloor - \arctan \left (-\frac {3 \, \left (\frac {1}{3}\right )^{\frac {3}{4}} {\left (\left (\frac {1}{3}\right )^{\frac {1}{4}} {\left (\sqrt {6} - \sqrt {2}\right )} + 4 \, \tan \left (x\right )\right )}}{\sqrt {6} + \sqrt {2}}\right )\right )} \sqrt {6 \, \sqrt {3} + 9}}{18 \, a} + \frac {{\left (\pi \left \lfloor \frac {x}{\pi } + \frac {1}{2} \right \rfloor + \arctan \left (-\frac {3 \, \left (\frac {1}{3}\right )^{\frac {3}{4}} {\left (\left (\frac {1}{3}\right )^{\frac {1}{4}} {\left (\sqrt {6} - \sqrt {2}\right )} - 4 \, \tan \left (x\right )\right )}}{\sqrt {6} + \sqrt {2}}\right )\right )} \sqrt {6 \, \sqrt {3} + 9}}{18 \, a} + \frac {\sqrt {6 \, \sqrt {3} - 9} \log \left (\frac {1}{2} \, {\left (\sqrt {6} \left (\frac {1}{3}\right )^{\frac {1}{4}} - \sqrt {2} \left (\frac {1}{3}\right )^{\frac {1}{4}}\right )} \tan \left (x\right ) + \tan \left (x\right )^{2} + \sqrt {\frac {1}{3}}\right )}{36 \, a} - \frac {\sqrt {6 \, \sqrt {3} - 9} \log \left (-\frac {1}{2} \, {\left (\sqrt {6} \left (\frac {1}{3}\right )^{\frac {1}{4}} - \sqrt {2} \left (\frac {1}{3}\right )^{\frac {1}{4}}\right )} \tan \left (x\right ) + \tan \left (x\right )^{2} + \sqrt {\frac {1}{3}}\right )}{36 \, a} + \frac {\tan \left (x\right )}{3 \, a} \] Input:
integrate(1/(a-a*sin(x)^6),x, algorithm="giac")
Output:
1/18*(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))))*sqrt(6*sqrt(3) + 9)/a + 1/18*( 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))))*sqrt(6*sqrt(3) + 9)/a + 1/36*sqrt(6* sqrt(3) - 9)*log(1/2*(sqrt(6)*(1/3)^(1/4) - sqrt(2)*(1/3)^(1/4))*tan(x) + tan(x)^2 + sqrt(1/3))/a - 1/36*sqrt(6*sqrt(3) - 9)*log(-1/2*(sqrt(6)*(1/3) ^(1/4) - sqrt(2)*(1/3)^(1/4))*tan(x) + tan(x)^2 + sqrt(1/3))/a + 1/3*tan(x )/a
Time = 37.76 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.87 \[ \int \frac {1}{a-a \sin ^6(x)} \, dx=\frac {\mathrm {tan}\left (x\right )}{3\,a}+\frac {\mathrm {atan}\left (\frac {a\,\mathrm {tan}\left (x\right )\,\sqrt {-\frac {18}{a^2}-\frac {\sqrt {3}\,6{}\mathrm {i}}{a^2}}\,1{}\mathrm {i}}{4}+\frac {\sqrt {3}\,a\,\mathrm {tan}\left (x\right )\,\sqrt {-\frac {18}{a^2}-\frac {\sqrt {3}\,6{}\mathrm {i}}{a^2}}}{12}\right )\,\sqrt {-\frac {6\,\left (3+\sqrt {3}\,1{}\mathrm {i}\right )}{a^2}}\,1{}\mathrm {i}}{18}+\frac {\mathrm {atan}\left (\frac {a\,\mathrm {tan}\left (x\right )\,\sqrt {-\frac {18}{a^2}+\frac {\sqrt {3}\,6{}\mathrm {i}}{a^2}}\,1{}\mathrm {i}}{4}-\frac {\sqrt {3}\,a\,\mathrm {tan}\left (x\right )\,\sqrt {-\frac {18}{a^2}+\frac {\sqrt {3}\,6{}\mathrm {i}}{a^2}}}{12}\right )\,\sqrt {\frac {6\,\left (-3+\sqrt {3}\,1{}\mathrm {i}\right )}{a^2}}\,1{}\mathrm {i}}{18} \] Input:
int(1/(a - a*sin(x)^6),x)
Output:
tan(x)/(3*a) + (atan((a*tan(x)*(- (3^(1/2)*6i)/a^2 - 18/a^2)^(1/2)*1i)/4 + (3^(1/2)*a*tan(x)*(- (3^(1/2)*6i)/a^2 - 18/a^2)^(1/2))/12)*(-(6*(3^(1/2)* 1i + 3))/a^2)^(1/2)*1i)/18 + (atan((a*tan(x)*((3^(1/2)*6i)/a^2 - 18/a^2)^( 1/2)*1i)/4 - (3^(1/2)*a*tan(x)*((3^(1/2)*6i)/a^2 - 18/a^2)^(1/2))/12)*((6* (3^(1/2)*1i - 3))/a^2)^(1/2)*1i)/18
\[ \int \frac {1}{a-a \sin ^6(x)} \, dx=-\frac {\int \frac {1}{\sin \left (x \right )^{6}-1}d x}{a} \] Input:
int(1/(a-a*sin(x)^6),x)
Output:
( - int(1/(sin(x)**6 - 1),x))/a