Integrand size = 11, antiderivative size = 115 \[ \int \frac {1}{a-a \sin ^3(x)} \, dx=\frac {2 \arctan \left (\frac {\sqrt [3]{-1}+\tan \left (\frac {x}{2}\right )}{\sqrt {1-(-1)^{2/3}}}\right )}{3 \sqrt {1-(-1)^{2/3}} a}-\frac {2 \arctan \left (\frac {(-1)^{2/3} \left (1+\sqrt [3]{-1} \tan \left (\frac {x}{2}\right )\right )}{\sqrt {1+\sqrt [3]{-1}}}\right )}{3 \sqrt {1+\sqrt [3]{-1}} a}+\frac {\cos (x)}{3 a (1-\sin (x))} \] Output:
2/3*arctan(((-1)^(1/3)+tan(1/2*x))/(1-(-1)^(2/3))^(1/2))/(1-(-1)^(2/3))^(1 /2)/a-2/3*arctan((-1)^(2/3)*(1+(-1)^(1/3)*tan(1/2*x))/(1+(-1)^(1/3))^(1/2) )/(1+(-1)^(1/3))^(1/2)/a+1/3*cos(x)/a/(1-sin(x))
Result contains complex when optimal does not.
Time = 5.89 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.28 \[ \int \frac {1}{a-a \sin ^3(x)} \, dx=\frac {i \sqrt {-18-6 i \sqrt {3}} \arctan \left (\frac {2+\left (1-i \sqrt {3}\right ) \tan \left (\frac {x}{2}\right )}{\sqrt {-6-2 i \sqrt {3}}}\right )-i \sqrt {6 i \left (3 i+\sqrt {3}\right )} \arctan \left (\frac {2+\left (1+i \sqrt {3}\right ) \tan \left (\frac {x}{2}\right )}{\sqrt {-6+2 i \sqrt {3}}}\right )+\frac {6 \sin \left (\frac {x}{2}\right )}{\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )}}{9 a} \] Input:
Integrate[(a - a*Sin[x]^3)^(-1),x]
Output:
(I*Sqrt[-18 - (6*I)*Sqrt[3]]*ArcTan[(2 + (1 - I*Sqrt[3])*Tan[x/2])/Sqrt[-6 - (2*I)*Sqrt[3]]] - I*Sqrt[(6*I)*(3*I + Sqrt[3])]*ArcTan[(2 + (1 + I*Sqrt [3])*Tan[x/2])/Sqrt[-6 + (2*I)*Sqrt[3]]] + (6*Sin[x/2])/(Cos[x/2] - Sin[x/ 2]))/(9*a)
Time = 0.43 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, 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.
| \(\displaystyle \int \frac {1}{a-a \sin ^3(x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{a-a \sin (x)^3}dx\) |
| \(\Big \downarrow \) 3692 |
| \(\displaystyle \int \left (\frac {1}{3 a \left (\sqrt [3]{-1} \sin (x)+1\right )}+\frac {1}{3 a \left (1-(-1)^{2/3} \sin (x)\right )}+\frac {1}{3 a (1-\sin (x))}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 \arctan \left (\frac {\tan \left (\frac {x}{2}\right )+\sqrt [3]{-1}}{\sqrt {1-(-1)^{2/3}}}\right )}{3 \sqrt {1-(-1)^{2/3}} a}-\frac {2 \arctan \left (\frac {(-1)^{2/3} \left (\sqrt [3]{-1} \tan \left (\frac {x}{2}\right )+1\right )}{\sqrt {1+\sqrt [3]{-1}}}\right )}{3 \sqrt {1+\sqrt [3]{-1}} a}+\frac {\cos (x)}{3 a (1-\sin (x))}\) |
Input:
Int[(a - a*Sin[x]^3)^(-1),x]
Output:
(2*ArcTan[((-1)^(1/3) + Tan[x/2])/Sqrt[1 - (-1)^(2/3)]])/(3*Sqrt[1 - (-1)^ (2/3)]*a) - (2*ArcTan[((-1)^(2/3)*(1 + (-1)^(1/3)*Tan[x/2]))/Sqrt[1 + (-1) ^(1/3)]])/(3*Sqrt[1 + (-1)^(1/3)]*a) + Cos[x]/(3*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 = 0.37 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.62
| method | result | size |
| risch | \(\frac {2}{3 \left ({\mathrm e}^{i x}-i\right ) a}+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (243 a^{4} \textit {\_Z}^{4}+27 a^{2} \textit {\_Z}^{2}+1\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{i x}+162 a^{3} \textit {\_R}^{3}+27 i a^{2} \textit {\_R}^{2}+9 a \textit {\_R} +2 i\right )\right )\) | \(71\) |
| default | \(\frac {\frac {2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}+2 \textit {\_Z}^{3}+6 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )}{\sum }\frac {\left (\textit {\_R}^{2}+\textit {\_R} +1\right ) \ln \left (\tan \left (\frac {x}{2}\right )-\textit {\_R} \right )}{2 \textit {\_R}^{3}+3 \textit {\_R}^{2}+6 \textit {\_R} +1}\right )}{3}-\frac {2}{3 \left (\tan \left (\frac {x}{2}\right )-1\right )}}{a}\) | \(74\) |
Input:
int(1/(a-a*sin(x)^3),x,method=_RETURNVERBOSE)
Output:
2/3/(exp(I*x)-I)/a+sum(_R*ln(exp(I*x)+162*a^3*_R^3+27*I*a^2*_R^2+9*a*_R+2* I),_R=RootOf(243*_Z^4*a^4+27*_Z^2*a^2+1))
Leaf count of result is larger than twice the leaf count of optimal. 439 vs. \(2 (81) = 162\).
Time = 0.10 (sec) , antiderivative size = 439, normalized size of antiderivative = 3.82 \[ \int \frac {1}{a-a \sin ^3(x)} \, dx=-\frac {\sqrt {2} {\left (a \cos \left (x\right ) - a \sin \left (x\right ) + a\right )} \sqrt {-\frac {\sqrt {\frac {1}{3}} a^{2} \sqrt {-\frac {1}{a^{4}}} + 1}{a^{2}}} \log \left (-3 \, \sqrt {2} \sqrt {\frac {1}{3}} a^{3} \sqrt {-\frac {\sqrt {\frac {1}{3}} a^{2} \sqrt {-\frac {1}{a^{4}}} + 1}{a^{2}}} \sqrt {-\frac {1}{a^{4}}} \cos \left (x\right ) + 3 \, \sqrt {\frac {1}{3}} a^{2} \sqrt {-\frac {1}{a^{4}}} \sin \left (x\right ) - \sin \left (x\right ) - 2\right ) - \sqrt {2} {\left (a \cos \left (x\right ) - a \sin \left (x\right ) + a\right )} \sqrt {-\frac {\sqrt {\frac {1}{3}} a^{2} \sqrt {-\frac {1}{a^{4}}} + 1}{a^{2}}} \log \left (-3 \, \sqrt {2} \sqrt {\frac {1}{3}} a^{3} \sqrt {-\frac {\sqrt {\frac {1}{3}} a^{2} \sqrt {-\frac {1}{a^{4}}} + 1}{a^{2}}} \sqrt {-\frac {1}{a^{4}}} \cos \left (x\right ) - 3 \, \sqrt {\frac {1}{3}} a^{2} \sqrt {-\frac {1}{a^{4}}} \sin \left (x\right ) + \sin \left (x\right ) + 2\right ) + \sqrt {2} {\left (a \cos \left (x\right ) - a \sin \left (x\right ) + a\right )} \sqrt {\frac {\sqrt {\frac {1}{3}} a^{2} \sqrt {-\frac {1}{a^{4}}} - 1}{a^{2}}} \log \left (-3 \, \sqrt {2} \sqrt {\frac {1}{3}} a^{3} \sqrt {\frac {\sqrt {\frac {1}{3}} a^{2} \sqrt {-\frac {1}{a^{4}}} - 1}{a^{2}}} \sqrt {-\frac {1}{a^{4}}} \cos \left (x\right ) + 3 \, \sqrt {\frac {1}{3}} a^{2} \sqrt {-\frac {1}{a^{4}}} \sin \left (x\right ) + \sin \left (x\right ) + 2\right ) - \sqrt {2} {\left (a \cos \left (x\right ) - a \sin \left (x\right ) + a\right )} \sqrt {\frac {\sqrt {\frac {1}{3}} a^{2} \sqrt {-\frac {1}{a^{4}}} - 1}{a^{2}}} \log \left (-3 \, \sqrt {2} \sqrt {\frac {1}{3}} a^{3} \sqrt {\frac {\sqrt {\frac {1}{3}} a^{2} \sqrt {-\frac {1}{a^{4}}} - 1}{a^{2}}} \sqrt {-\frac {1}{a^{4}}} \cos \left (x\right ) - 3 \, \sqrt {\frac {1}{3}} a^{2} \sqrt {-\frac {1}{a^{4}}} \sin \left (x\right ) - \sin \left (x\right ) - 2\right ) - 4 \, \cos \left (x\right ) - 4 \, \sin \left (x\right ) - 4}{12 \, {\left (a \cos \left (x\right ) - a \sin \left (x\right ) + a\right )}} \] Input:
integrate(1/(a-a*sin(x)^3),x, algorithm="fricas")
Output:
-1/12*(sqrt(2)*(a*cos(x) - a*sin(x) + a)*sqrt(-(sqrt(1/3)*a^2*sqrt(-1/a^4) + 1)/a^2)*log(-3*sqrt(2)*sqrt(1/3)*a^3*sqrt(-(sqrt(1/3)*a^2*sqrt(-1/a^4) + 1)/a^2)*sqrt(-1/a^4)*cos(x) + 3*sqrt(1/3)*a^2*sqrt(-1/a^4)*sin(x) - sin( x) - 2) - sqrt(2)*(a*cos(x) - a*sin(x) + a)*sqrt(-(sqrt(1/3)*a^2*sqrt(-1/a ^4) + 1)/a^2)*log(-3*sqrt(2)*sqrt(1/3)*a^3*sqrt(-(sqrt(1/3)*a^2*sqrt(-1/a^ 4) + 1)/a^2)*sqrt(-1/a^4)*cos(x) - 3*sqrt(1/3)*a^2*sqrt(-1/a^4)*sin(x) + s in(x) + 2) + sqrt(2)*(a*cos(x) - a*sin(x) + a)*sqrt((sqrt(1/3)*a^2*sqrt(-1 /a^4) - 1)/a^2)*log(-3*sqrt(2)*sqrt(1/3)*a^3*sqrt((sqrt(1/3)*a^2*sqrt(-1/a ^4) - 1)/a^2)*sqrt(-1/a^4)*cos(x) + 3*sqrt(1/3)*a^2*sqrt(-1/a^4)*sin(x) + sin(x) + 2) - sqrt(2)*(a*cos(x) - a*sin(x) + a)*sqrt((sqrt(1/3)*a^2*sqrt(- 1/a^4) - 1)/a^2)*log(-3*sqrt(2)*sqrt(1/3)*a^3*sqrt((sqrt(1/3)*a^2*sqrt(-1/ a^4) - 1)/a^2)*sqrt(-1/a^4)*cos(x) - 3*sqrt(1/3)*a^2*sqrt(-1/a^4)*sin(x) - sin(x) - 2) - 4*cos(x) - 4*sin(x) - 4)/(a*cos(x) - a*sin(x) + a)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 10462 vs. \(2 (105) = 210\).
Time = 19.69 (sec) , antiderivative size = 10462, normalized size of antiderivative = 90.97 \[ \int \frac {1}{a-a \sin ^3(x)} \, dx=\text {Too large to display} \] Input:
integrate(1/(a-a*sin(x)**3),x)
Output:
-406593*sqrt(6)*sqrt(-3 + sqrt(3)*I)*log(tan(x/2) + 1/2 + sqrt(2)*sqrt(-3 - sqrt(3)*I)/2 - sqrt(3)*I/2)*tan(x/2)/(1219779*sqrt(3)*a*sqrt(-3 - sqrt(3 )*I)*sqrt(-3 + sqrt(3)*I)*tan(x/2) - 4242534*I*a*sqrt(-3 - sqrt(3)*I)*sqrt (-3 + sqrt(3)*I)*tan(x/2) - 2698299*sqrt(6)*a*sqrt(-3 + sqrt(3)*I)*tan(x/2 ) - 4130865*sqrt(2)*I*a*sqrt(-3 + sqrt(3)*I)*tan(x/2) - 1219779*sqrt(3)*a* sqrt(-3 - sqrt(3)*I)*sqrt(-3 + sqrt(3)*I) + 4130865*sqrt(2)*I*a*sqrt(-3 + sqrt(3)*I) + 2698299*sqrt(6)*a*sqrt(-3 + sqrt(3)*I) + 4242534*I*a*sqrt(-3 - sqrt(3)*I)*sqrt(-3 + sqrt(3)*I)) - 238761*I*sqrt(-3 - sqrt(3)*I)*sqrt(-3 + sqrt(3)*I)*log(tan(x/2) + 1/2 + sqrt(2)*sqrt(-3 - sqrt(3)*I)/2 - sqrt(3 )*I/2)*tan(x/2)/(1219779*sqrt(3)*a*sqrt(-3 - sqrt(3)*I)*sqrt(-3 + sqrt(3)* I)*tan(x/2) - 4242534*I*a*sqrt(-3 - sqrt(3)*I)*sqrt(-3 + sqrt(3)*I)*tan(x/ 2) - 2698299*sqrt(6)*a*sqrt(-3 + sqrt(3)*I)*tan(x/2) - 4130865*sqrt(2)*I*a *sqrt(-3 + sqrt(3)*I)*tan(x/2) - 1219779*sqrt(3)*a*sqrt(-3 - sqrt(3)*I)*sq rt(-3 + sqrt(3)*I) + 4130865*sqrt(2)*I*a*sqrt(-3 + sqrt(3)*I) + 2698299*sq rt(6)*a*sqrt(-3 + sqrt(3)*I) + 4242534*I*a*sqrt(-3 - sqrt(3)*I)*sqrt(-3 + sqrt(3)*I)) - 679209*sqrt(3)*sqrt(-3 - sqrt(3)*I)*sqrt(-3 + sqrt(3)*I)*log (tan(x/2) + 1/2 + sqrt(2)*sqrt(-3 - sqrt(3)*I)/2 - sqrt(3)*I/2)*tan(x/2)/( 1219779*sqrt(3)*a*sqrt(-3 - sqrt(3)*I)*sqrt(-3 + sqrt(3)*I)*tan(x/2) - 424 2534*I*a*sqrt(-3 - sqrt(3)*I)*sqrt(-3 + sqrt(3)*I)*tan(x/2) - 2698299*sqrt (6)*a*sqrt(-3 + sqrt(3)*I)*tan(x/2) - 4130865*sqrt(2)*I*a*sqrt(-3 + sqr...
\[ \int \frac {1}{a-a \sin ^3(x)} \, dx=\int { -\frac {1}{a \sin \left (x\right )^{3} - a} \,d x } \] Input:
integrate(1/(a-a*sin(x)^3),x, algorithm="maxima")
Output:
1/3*(3*(a*cos(x)^2 + a*sin(x)^2 - 2*a*sin(x) + a)*integrate(-2/3*((4*cos(2 *x) + sin(3*x) - sin(x))*cos(4*x) + 2*(2*cos(x) + 7*sin(2*x))*cos(3*x) - 2 *cos(3*x)^2 + 2*(7*sin(x) + 2)*cos(2*x) - 24*cos(2*x)^2 - 2*cos(x)^2 - (co s(3*x) - cos(x) - 4*sin(2*x))*sin(4*x) - (14*cos(2*x) - 4*sin(x) - 1)*sin( 3*x) - 2*sin(3*x)^2 - 14*cos(x)*sin(2*x) - 24*sin(2*x)^2 - 2*sin(x)^2 - si n(x))/(a*cos(4*x)^2 + 4*a*cos(3*x)^2 + 36*a*cos(2*x)^2 + 4*a*cos(x)^2 + a* sin(4*x)^2 + 4*a*sin(3*x)^2 + 24*a*cos(x)*sin(2*x) + 36*a*sin(2*x)^2 + 4*a *sin(x)^2 - 2*(6*a*cos(2*x) + 2*a*sin(3*x) - 2*a*sin(x) - a)*cos(4*x) - 8* (a*cos(x) + 3*a*sin(2*x))*cos(3*x) - 12*(2*a*sin(x) + a)*cos(2*x) + 4*(a*c os(3*x) - a*cos(x) - 3*a*sin(2*x))*sin(4*x) + 4*(6*a*cos(2*x) - 2*a*sin(x) - a)*sin(3*x) + 4*a*sin(x) + a), x) + 2*cos(x))/(a*cos(x)^2 + a*sin(x)^2 - 2*a*sin(x) + a)
Leaf count of result is larger than twice the leaf count of optimal. 317 vs. \(2 (81) = 162\).
Time = 0.47 (sec) , antiderivative size = 317, normalized size of antiderivative = 2.76 \[ \int \frac {1}{a-a \sin ^3(x)} \, dx=-\frac {\sqrt {6 \, \sqrt {3} - 9} \log \left (196 \, {\left (2 \, \sqrt {3} \sqrt {6 \, \sqrt {3} - 9} + 3 \, \sqrt {3} + 3 \, \sqrt {6 \, \sqrt {3} - 9}\right )}^{2} + 196 \, {\left (\sqrt {3} \sqrt {6 \, \sqrt {3} - 9} + 6 \, \tan \left (\frac {1}{2} \, x\right ) + 3\right )}^{2}\right ) - \sqrt {6 \, \sqrt {3} - 9} \log \left (196 \, {\left (2 \, \sqrt {3} \sqrt {6 \, \sqrt {3} - 9} - 3 \, \sqrt {3} + 3 \, \sqrt {6 \, \sqrt {3} - 9}\right )}^{2} + 196 \, {\left (\sqrt {3} \sqrt {6 \, \sqrt {3} - 9} - 6 \, \tan \left (\frac {1}{2} \, x\right ) - 3\right )}^{2}\right ) - \frac {2 \, \sqrt {3} \sqrt {6 \, \sqrt {3} - 9} \arctan \left (\frac {3 \, {\left (\sqrt {2 \, \sqrt {3} - 3} + 2 \, \tan \left (\frac {1}{2} \, x\right ) + 1\right )}}{2 \, \sqrt {3} \sqrt {6 \, \sqrt {3} - 9} + 3 \, \sqrt {3} + 3 \, \sqrt {6 \, \sqrt {3} - 9}}\right )}{2 \, \sqrt {3} - 3} - \frac {2 \, \sqrt {3} \sqrt {6 \, \sqrt {3} - 9} \arctan \left (-\frac {3 \, {\left (\sqrt {2 \, \sqrt {3} - 3} - 2 \, \tan \left (\frac {1}{2} \, x\right ) - 1\right )}}{2 \, \sqrt {3} \sqrt {6 \, \sqrt {3} - 9} - 3 \, \sqrt {3} + 3 \, \sqrt {6 \, \sqrt {3} - 9}}\right )}{2 \, \sqrt {3} - 3}}{18 \, a} - \frac {2}{3 \, a {\left (\tan \left (\frac {1}{2} \, x\right ) - 1\right )}} \] Input:
integrate(1/(a-a*sin(x)^3),x, algorithm="giac")
Output:
-1/18*(sqrt(6*sqrt(3) - 9)*log(196*(2*sqrt(3)*sqrt(6*sqrt(3) - 9) + 3*sqrt (3) + 3*sqrt(6*sqrt(3) - 9))^2 + 196*(sqrt(3)*sqrt(6*sqrt(3) - 9) + 6*tan( 1/2*x) + 3)^2) - sqrt(6*sqrt(3) - 9)*log(196*(2*sqrt(3)*sqrt(6*sqrt(3) - 9 ) - 3*sqrt(3) + 3*sqrt(6*sqrt(3) - 9))^2 + 196*(sqrt(3)*sqrt(6*sqrt(3) - 9 ) - 6*tan(1/2*x) - 3)^2) - 2*sqrt(3)*sqrt(6*sqrt(3) - 9)*arctan(3*(sqrt(2* sqrt(3) - 3) + 2*tan(1/2*x) + 1)/(2*sqrt(3)*sqrt(6*sqrt(3) - 9) + 3*sqrt(3 ) + 3*sqrt(6*sqrt(3) - 9)))/(2*sqrt(3) - 3) - 2*sqrt(3)*sqrt(6*sqrt(3) - 9 )*arctan(-3*(sqrt(2*sqrt(3) - 3) - 2*tan(1/2*x) - 1)/(2*sqrt(3)*sqrt(6*sqr t(3) - 9) - 3*sqrt(3) + 3*sqrt(6*sqrt(3) - 9)))/(2*sqrt(3) - 3))/a - 2/3/( a*(tan(1/2*x) - 1))
Time = 37.52 (sec) , antiderivative size = 368, normalized size of antiderivative = 3.20 \[ \int \frac {1}{a-a \sin ^3(x)} \, dx=\frac {2}{3\,\left (a-a\,\mathrm {tan}\left (\frac {x}{2}\right )\right )}+2\,\mathrm {atanh}\left (\frac {5184\,a^3\,\sqrt {-\frac {1}{18\,a^2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{54\,a^2}}}{3456\,a^2\,\mathrm {tan}\left (\frac {x}{2}\right )+864\,a^2-\sqrt {3}\,a^2\,864{}\mathrm {i}}+\frac {2592\,a^3\,\mathrm {tan}\left (\frac {x}{2}\right )\,\sqrt {-\frac {1}{18\,a^2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{54\,a^2}}}{3456\,a^2\,\mathrm {tan}\left (\frac {x}{2}\right )+864\,a^2-\sqrt {3}\,a^2\,864{}\mathrm {i}}+\frac {\sqrt {3}\,a^3\,\mathrm {tan}\left (\frac {x}{2}\right )\,\sqrt {-\frac {1}{18\,a^2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{54\,a^2}}\,7776{}\mathrm {i}}{3456\,a^2\,\mathrm {tan}\left (\frac {x}{2}\right )+864\,a^2-\sqrt {3}\,a^2\,864{}\mathrm {i}}\right )\,\sqrt {-\frac {3+\sqrt {3}\,1{}\mathrm {i}}{54\,a^2}}+2\,\mathrm {atanh}\left (\frac {5184\,a^3\,\sqrt {-\frac {1}{18\,a^2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{54\,a^2}}}{3456\,a^2\,\mathrm {tan}\left (\frac {x}{2}\right )+864\,a^2+\sqrt {3}\,a^2\,864{}\mathrm {i}}+\frac {2592\,a^3\,\mathrm {tan}\left (\frac {x}{2}\right )\,\sqrt {-\frac {1}{18\,a^2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{54\,a^2}}}{3456\,a^2\,\mathrm {tan}\left (\frac {x}{2}\right )+864\,a^2+\sqrt {3}\,a^2\,864{}\mathrm {i}}-\frac {\sqrt {3}\,a^3\,\mathrm {tan}\left (\frac {x}{2}\right )\,\sqrt {-\frac {1}{18\,a^2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{54\,a^2}}\,7776{}\mathrm {i}}{3456\,a^2\,\mathrm {tan}\left (\frac {x}{2}\right )+864\,a^2+\sqrt {3}\,a^2\,864{}\mathrm {i}}\right )\,\sqrt {\frac {-3+\sqrt {3}\,1{}\mathrm {i}}{54\,a^2}} \] Input:
int(1/(a - a*sin(x)^3),x)
Output:
2/(3*(a - a*tan(x/2))) + 2*atanh((5184*a^3*(- (3^(1/2)*1i)/(54*a^2) - 1/(1 8*a^2))^(1/2))/(3456*a^2*tan(x/2) - 3^(1/2)*a^2*864i + 864*a^2) + (2592*a^ 3*tan(x/2)*(- (3^(1/2)*1i)/(54*a^2) - 1/(18*a^2))^(1/2))/(3456*a^2*tan(x/2 ) - 3^(1/2)*a^2*864i + 864*a^2) + (3^(1/2)*a^3*tan(x/2)*(- (3^(1/2)*1i)/(5 4*a^2) - 1/(18*a^2))^(1/2)*7776i)/(3456*a^2*tan(x/2) - 3^(1/2)*a^2*864i + 864*a^2))*(-(3^(1/2)*1i + 3)/(54*a^2))^(1/2) + 2*atanh((5184*a^3*((3^(1/2) *1i)/(54*a^2) - 1/(18*a^2))^(1/2))/(3456*a^2*tan(x/2) + 3^(1/2)*a^2*864i + 864*a^2) + (2592*a^3*tan(x/2)*((3^(1/2)*1i)/(54*a^2) - 1/(18*a^2))^(1/2)) /(3456*a^2*tan(x/2) + 3^(1/2)*a^2*864i + 864*a^2) - (3^(1/2)*a^3*tan(x/2)* ((3^(1/2)*1i)/(54*a^2) - 1/(18*a^2))^(1/2)*7776i)/(3456*a^2*tan(x/2) + 3^( 1/2)*a^2*864i + 864*a^2))*((3^(1/2)*1i - 3)/(54*a^2))^(1/2)
\[ \int \frac {1}{a-a \sin ^3(x)} \, dx=-\frac {\int \frac {1}{\sin \left (x \right )^{3}-1}d x}{a} \] Input:
int(1/(a-a*sin(x)^3),x)
Output:
( - int(1/(sin(x)**3 - 1),x))/a