\(\int \frac {1}{a+a \sin ^3(x)} \, dx\) [69]

Optimal result

Integrand size = 10, antiderivative size = 113 \[ \int \frac {1}{a+a \sin ^3(x)} \, dx=\frac {2 \arctan \left (\frac {(-1)^{2/3}+\tan \left (\frac {x}{2}\right )}{\sqrt {1+\sqrt [3]{-1}}}\right )}{3 \sqrt {1+\sqrt [3]{-1}} a}-\frac {2 \arctan \left (\frac {\sqrt [3]{-1} \left (1+(-1)^{2/3} \tan \left (\frac {x}{2}\right )\right )}{\sqrt {1-(-1)^{2/3}}}\right )}{3 \sqrt {1-(-1)^{2/3}} a}-\frac {\cos (x)}{3 a (1+\sin (x))} \] Output:

2/3*arctan(((-1)^(2/3)+tan(1/2*x))/(1+(-1)^(1/3))^(1/2))/(1+(-1)^(1/3))^(1 
/2)/a-2/3*arctan((-1)^(1/3)*(1+(-1)^(2/3)*tan(1/2*x))/(1-(-1)^(2/3))^(1/2) 
)/(1-(-1)^(2/3))^(1/2)/a-1/3*cos(x)/a/(1+sin(x))
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 5.89 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.29 \[ \int \frac {1}{a+a \sin ^3(x)} \, dx=\frac {i \sqrt {6 i \left (3 i+\sqrt {3}\right )} \arctan \left (\frac {2+\left (-1-i \sqrt {3}\right ) \tan \left (\frac {x}{2}\right )}{\sqrt {-6+2 i \sqrt {3}}}\right )-i \sqrt {-18-6 i \sqrt {3}} \arctan \left (\frac {2+i \left (i+\sqrt {3}\right ) \tan \left (\frac {x}{2}\right )}{\sqrt {-6-2 i \sqrt {3}}}\right )+\frac {6 \sin \left (\frac {x}{2}\right )}{\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )}}{9 a} \] Input:

Integrate[(a + a*Sin[x]^3)^(-1),x]
 

Output:

(I*Sqrt[(6*I)*(3*I + Sqrt[3])]*ArcTan[(2 + (-1 - I*Sqrt[3])*Tan[x/2])/Sqrt 
[-6 + (2*I)*Sqrt[3]]] - I*Sqrt[-18 - (6*I)*Sqrt[3]]*ArcTan[(2 + I*(I + Sqr 
t[3])*Tan[x/2])/Sqrt[-6 - (2*I)*Sqrt[3]]] + (6*Sin[x/2])/(Cos[x/2] + Sin[x 
/2]))/(9*a)
 

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, 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.

\(\displaystyle \int \frac {1}{a \sin ^3(x)+a} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{a \sin (x)^3+a}dx\)

\(\Big \downarrow \) 3692

\(\displaystyle \int \left (-\frac {1}{3 a \left (\sqrt [3]{-1} \sin (x)-1\right )}-\frac {1}{3 a \left (-(-1)^{2/3} \sin (x)-1\right )}-\frac {1}{3 a (-\sin (x)-1)}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {2 \arctan \left (\frac {\tan \left (\frac {x}{2}\right )+(-1)^{2/3}}{\sqrt {1+\sqrt [3]{-1}}}\right )}{3 \sqrt {1+\sqrt [3]{-1}} a}-\frac {2 \arctan \left (\frac {\sqrt [3]{-1} \left ((-1)^{2/3} \tan \left (\frac {x}{2}\right )+1\right )}{\sqrt {1-(-1)^{2/3}}}\right )}{3 \sqrt {1-(-1)^{2/3}} a}-\frac {\cos (x)}{3 a (\sin (x)+1)}\)

Input:

Int[(a + a*Sin[x]^3)^(-1),x]
 

Output:

(2*ArcTan[((-1)^(2/3) + Tan[x/2])/Sqrt[1 + (-1)^(1/3)]])/(3*Sqrt[1 + (-1)^ 
(1/3)]*a) - (2*ArcTan[((-1)^(1/3)*(1 + (-1)^(2/3)*Tan[x/2]))/Sqrt[1 - (-1) 
^(2/3)]])/(3*Sqrt[1 - (-1)^(2/3)]*a) - Cos[x]/(3*a*(1 + Sin[x]))
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3692

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]))
 

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.34 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.63

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	\(\frac {\frac {2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}-2 \textit {\_Z}^{3}+6 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right )}{\sum }\frac {\left (\textit {\_R}^{2}-\textit {\_R} +1\right ) \ln \left (\tan \left (\frac {x}{2}\right )-\textit {\_R} \right )}{2 \textit {\_R}^{3}-3 \textit {\_R}^{2}+6 \textit {\_R} -1}\right )}{3}-\frac {2}{3 \left (\tan \left (\frac {x}{2}\right )+1\right )}}{a}\)	\(76\)

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))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 434 vs. \(2 (81) = 162\).

Time = 0.10 (sec) , antiderivative size = 434, normalized size of antiderivative = 3.84 \[ \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)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 18.62 (sec) , antiderivative size = 10462, normalized size of antiderivative = 92.58 \[ \int \frac {1}{a+a \sin ^3(x)} \, dx=\text {Too large to display} \] 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) + 4130865*sqrt(2)*I*a*sqrt(-3 + sqrt(3)*I)*tan(x 
/2) + 2698299*sqrt(6)*a*sqrt(-3 + sqrt(3)*I)*tan(x/2) + 1219779*sqrt(3)*a* 
sqrt(-3 - sqrt(3)*I)*sqrt(-3 + sqrt(3)*I) - 4242534*I*a*sqrt(-3 - sqrt(3)* 
I)*sqrt(-3 + sqrt(3)*I) + 4130865*sqrt(2)*I*a*sqrt(-3 + sqrt(3)*I) + 26982 
99*sqrt(6)*a*sqrt(-3 + sqrt(3)*I)) + 679209*sqrt(3)*sqrt(-3 - sqrt(3)*I)*s 
qrt(-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 + sq 
rt(3)*I)*tan(x/2) - 4242534*I*a*sqrt(-3 - sqrt(3)*I)*sqrt(-3 + sqrt(3)*I)* 
tan(x/2) + 4130865*sqrt(2)*I*a*sqrt(-3 + sqrt(3)*I)*tan(x/2) + 2698299*sqr 
t(6)*a*sqrt(-3 + sqrt(3)*I)*tan(x/2) + 1219779*sqrt(3)*a*sqrt(-3 - sqrt(3) 
*I)*sqrt(-3 + sqrt(3)*I) - 4242534*I*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)) + 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) - 424 
2534*I*a*sqrt(-3 - sqrt(3)*I)*sqrt(-3 + sqrt(3)*I)*tan(x/2) + 4130865*sqrt 
(2)*I*a*sqrt(-3 + sqrt(3)*I)*tan(x/2) + 2698299*sqrt(6)*a*sqrt(-3 + sqr...
 

Maxima [F]

\[ \int \frac {1}{a+a \sin ^3(x)} \, dx=\int { \frac {1}{a \sin \left (x\right )^{3} + a} \,d x } \] 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)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 317 vs. \(2 (81) = 162\).

Time = 0.43 (sec) , antiderivative size = 317, normalized size of antiderivative = 2.81 \[ \int \frac {1}{a+a \sin ^3(x)} \, dx=\frac {\sqrt {6 \, \sqrt {3} - 9} \log \left (196 \, {\left (2 \, \sqrt {3} \sqrt {6 \, \sqrt {3} - 9} + 3 \, \sqrt {3} + 3 \, \sqrt {6 \, \sqrt {3} - 9}\right )}^{2} + 196 \, {\left (\sqrt {3} \sqrt {6 \, \sqrt {3} - 9} - 6 \, \tan \left (\frac {1}{2} \, x\right ) + 3\right )}^{2}\right ) - \sqrt {6 \, \sqrt {3} - 9} \log \left (196 \, {\left (2 \, \sqrt {3} \sqrt {6 \, \sqrt {3} - 9} - 3 \, \sqrt {3} + 3 \, \sqrt {6 \, \sqrt {3} - 9}\right )}^{2} + 196 \, {\left (\sqrt {3} \sqrt {6 \, \sqrt {3} - 9} + 6 \, \tan \left (\frac {1}{2} \, x\right ) - 3\right )}^{2}\right ) + \frac {2 \, \sqrt {3} \sqrt {6 \, \sqrt {3} - 9} \arctan \left (\frac {3 \, {\left (\sqrt {2 \, \sqrt {3} - 3} + 2 \, \tan \left (\frac {1}{2} \, x\right ) - 1\right )}}{2 \, \sqrt {3} \sqrt {6 \, \sqrt {3} - 9} - 3 \, \sqrt {3} + 3 \, \sqrt {6 \, \sqrt {3} - 9}}\right )}{2 \, \sqrt {3} - 3} + \frac {2 \, \sqrt {3} \sqrt {6 \, \sqrt {3} - 9} \arctan \left (-\frac {3 \, {\left (\sqrt {2 \, \sqrt {3} - 3} - 2 \, \tan \left (\frac {1}{2} \, x\right ) + 1\right )}}{2 \, \sqrt {3} \sqrt {6 \, \sqrt {3} - 9} + 3 \, \sqrt {3} + 3 \, \sqrt {6 \, \sqrt {3} - 9}}\right )}{2 \, \sqrt {3} - 3}}{18 \, a} - \frac {2}{3 \, a {\left (\tan \left (\frac {1}{2} \, x\right ) + 1\right )}} \] 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*s 
qrt(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*sqrt 
(3) - 9) + 3*sqrt(3) + 3*sqrt(6*sqrt(3) - 9)))/(2*sqrt(3) - 3))/a - 2/3/(a 
*(tan(1/2*x) + 1))
 

Mupad [B] (verification not implemented)

Time = 37.30 (sec) , antiderivative size = 368, normalized size of antiderivative = 3.26 \[ \int \frac {1}{a+a \sin ^3(x)} \, dx=-\frac {2}{3\,\left (a+a\,\mathrm {tan}\left (\frac {x}{2}\right )\right )}-2\,\mathrm {atanh}\left (-\frac {5184\,a^3\,\sqrt {-\frac {1}{18\,a^2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{54\,a^2}}}{3456\,a^2\,\mathrm {tan}\left (\frac {x}{2}\right )-864\,a^2+\sqrt {3}\,a^2\,864{}\mathrm {i}}+\frac {2592\,a^3\,\mathrm {tan}\left (\frac {x}{2}\right )\,\sqrt {-\frac {1}{18\,a^2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{54\,a^2}}}{3456\,a^2\,\mathrm {tan}\left (\frac {x}{2}\right )-864\,a^2+\sqrt {3}\,a^2\,864{}\mathrm {i}}+\frac {\sqrt {3}\,a^3\,\mathrm {tan}\left (\frac {x}{2}\right )\,\sqrt {-\frac {1}{18\,a^2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{54\,a^2}}\,7776{}\mathrm {i}}{3456\,a^2\,\mathrm {tan}\left (\frac {x}{2}\right )-864\,a^2+\sqrt {3}\,a^2\,864{}\mathrm {i}}\right )\,\sqrt {-\frac {3+\sqrt {3}\,1{}\mathrm {i}}{54\,a^2}}-2\,\mathrm {atanh}\left (\frac {5184\,a^3\,\sqrt {-\frac {1}{18\,a^2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{54\,a^2}}}{864\,a^2-3456\,a^2\,\mathrm {tan}\left (\frac {x}{2}\right )+\sqrt {3}\,a^2\,864{}\mathrm {i}}-\frac {2592\,a^3\,\mathrm {tan}\left (\frac {x}{2}\right )\,\sqrt {-\frac {1}{18\,a^2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{54\,a^2}}}{864\,a^2-3456\,a^2\,\mathrm {tan}\left (\frac {x}{2}\right )+\sqrt {3}\,a^2\,864{}\mathrm {i}}+\frac {\sqrt {3}\,a^3\,\mathrm {tan}\left (\frac {x}{2}\right )\,\sqrt {-\frac {1}{18\,a^2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{54\,a^2}}\,7776{}\mathrm {i}}{864\,a^2-3456\,a^2\,\mathrm {tan}\left (\frac {x}{2}\right )+\sqrt {3}\,a^2\,864{}\mathrm {i}}\right )\,\sqrt {\frac {-3+\sqrt {3}\,1{}\mathrm {i}}{54\,a^2}} \] Input:

int(1/(a + a*sin(x)^3),x)
 

Output:

- 2/(3*(a + a*tan(x/2))) - 2*atanh((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) 
 - (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) + (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) - 2*atanh((5184*a^3*((3^(1/ 
2)*1i)/(54*a^2) - 1/(18*a^2))^(1/2))/(3^(1/2)*a^2*864i - 3456*a^2*tan(x/2) 
 + 864*a^2) - (2592*a^3*tan(x/2)*((3^(1/2)*1i)/(54*a^2) - 1/(18*a^2))^(1/2 
))/(3^(1/2)*a^2*864i - 3456*a^2*tan(x/2) + 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)/(3^(1/2)*a^2*864i - 34 
56*a^2*tan(x/2) + 864*a^2))*((3^(1/2)*1i - 3)/(54*a^2))^(1/2)
 

Reduce [F]

\[ \int \frac {1}{a+a \sin ^3(x)} \, dx=\frac {\int \frac {1}{\sin \left (x \right )^{3}+1}d x}{a} \] Input:

int(1/(a+a*sin(x)^3),x)
 

Output:

int(1/(sin(x)**3 + 1),x)/a