\(\int \frac {1}{a+a \sin ^6(x)} \, dx\) [66]

Optimal result

Integrand size = 10, antiderivative size = 142 \[ \int \frac {1}{a+a \sin ^6(x)} \, dx=\frac {x}{3 \sqrt {2} a}+\frac {x}{\sqrt {3} a}+\frac {\arctan \left (\frac {1-2 \cos ^2(x)}{2+\sqrt {3}-2 \cos (x) \sin (x)}\right )}{2 \sqrt {3} a}-\frac {\arctan \left (\frac {1-2 \cos ^2(x)}{2+\sqrt {3}+2 \cos (x) \sin (x)}\right )}{2 \sqrt {3} a}+\frac {\arctan \left (\frac {\cos (x) \sin (x)}{1+\sqrt {2}+\sin ^2(x)}\right )}{3 \sqrt {2} a}+\frac {\text {arctanh}(\cos (x) \sin (x))}{6 a} \] Output:

1/6*x*2^(1/2)/a+1/3*x*3^(1/2)/a+1/6*arctan((1-2*cos(x)^2)/(2+3^(1/2)-2*cos 
(x)*sin(x)))*3^(1/2)/a-1/6*arctan((1-2*cos(x)^2)/(2+3^(1/2)+2*cos(x)*sin(x 
)))*3^(1/2)/a+1/6*arctan(cos(x)*sin(x)/(1+2^(1/2)+sin(x)^2))*2^(1/2)/a+1/6 
*arctanh(cos(x)*sin(x))/a
 

Mathematica [A] (verified)

Time = 5.14 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.58 \[ \int \frac {1}{a+a \sin ^6(x)} \, dx=\frac {-2 \sqrt {3} \arctan \left (\frac {1-2 \tan (x)}{\sqrt {3}}\right )+2 \sqrt {2} \arctan \left (\sqrt {2} \tan (x)\right )+2 \sqrt {3} \arctan \left (\frac {1+2 \tan (x)}{\sqrt {3}}\right )-\log (2-\sin (2 x))+\log (2+\sin (2 x))}{12 a} \] Input:

Integrate[(a + a*Sin[x]^6)^(-1),x]
 

Output:

(-2*Sqrt[3]*ArcTan[(1 - 2*Tan[x])/Sqrt[3]] + 2*Sqrt[2]*ArcTan[Sqrt[2]*Tan[ 
x]] + 2*Sqrt[3]*ArcTan[(1 + 2*Tan[x])/Sqrt[3]] - Log[2 - Sin[2*x]] + Log[2 
 + Sin[2*x]])/(12*a)
 

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.65, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3042, 3690, 3042, 3660, 216}

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.

\(\displaystyle \int \frac {1}{a \sin ^6(x)+a} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{a \sin (x)^6+a}dx\)

\(\Big \downarrow \) 3690

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3660

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

\(\Big \downarrow \) 216

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

Input:

Int[(a + a*Sin[x]^6)^(-1),x]
 

Output:

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

Defintions of rubi rules used

rule 216

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])
 

rule 3042

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

rule 3660

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]
 

rule 3690

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]
 

Maple [A] (verified)

Time = 1.86 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.54

method	result	size
default	\(\frac {\frac {\ln \left (\tan \left (x \right )^{2}+\tan \left (x \right )+1\right )}{12}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 \tan \left (x \right )+1\right ) \sqrt {3}}{3}\right )}{6}+\frac {\sqrt {2}\, \arctan \left (\tan \left (x \right ) \sqrt {2}\right )}{6}-\frac {\ln \left (\tan \left (x \right )^{2}-\tan \left (x \right )+1\right )}{12}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 \tan \left (x \right )-1\right ) \sqrt {3}}{3}\right )}{6}}{a}\)	\(76\)
risch	\(\frac {i \sqrt {2}\, \ln \left ({\mathrm e}^{2 i x}-2 \sqrt {2}-3\right )}{12 a}-\frac {i \sqrt {2}\, \ln \left ({\mathrm e}^{2 i x}+2 \sqrt {2}-3\right )}{12 a}-\frac {\ln \left ({\mathrm e}^{2 i x}-i \sqrt {3}-2 i\right )}{12 a}+\frac {i \ln \left ({\mathrm e}^{2 i x}-i \sqrt {3}-2 i\right ) \sqrt {3}}{12 a}-\frac {\ln \left ({\mathrm e}^{2 i x}+i \sqrt {3}-2 i\right )}{12 a}-\frac {i \ln \left ({\mathrm e}^{2 i x}+i \sqrt {3}-2 i\right ) \sqrt {3}}{12 a}+\frac {\ln \left ({\mathrm e}^{2 i x}+i \sqrt {3}+2 i\right )}{12 a}+\frac {i \ln \left ({\mathrm e}^{2 i x}+i \sqrt {3}+2 i\right ) \sqrt {3}}{12 a}+\frac {\ln \left ({\mathrm e}^{2 i x}-i \sqrt {3}+2 i\right )}{12 a}-\frac {i \ln \left ({\mathrm e}^{2 i x}-i \sqrt {3}+2 i\right ) \sqrt {3}}{12 a}\)	\(222\)

Input:

int(1/(a+a*sin(x)^6),x,method=_RETURNVERBOSE)
 

Output:

1/a*(1/12*ln(tan(x)^2+tan(x)+1)+1/6*3^(1/2)*arctan(1/3*(2*tan(x)+1)*3^(1/2 
))+1/6*2^(1/2)*arctan(tan(x)*2^(1/2))-1/12*ln(tan(x)^2-tan(x)+1)+1/6*3^(1/ 
2)*arctan(1/3*(2*tan(x)-1)*3^(1/2)))
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.99 \[ \int \frac {1}{a+a \sin ^6(x)} \, dx=\frac {2 \, \sqrt {3} \arctan \left (\frac {4 \, \sqrt {3} \cos \left (x\right ) \sin \left (x\right ) + \sqrt {3}}{3 \, {\left (2 \, \cos \left (x\right )^{2} - 1\right )}}\right ) + 2 \, \sqrt {3} \arctan \left (\frac {4 \, \sqrt {3} \cos \left (x\right ) \sin \left (x\right ) - \sqrt {3}}{3 \, {\left (2 \, \cos \left (x\right )^{2} - 1\right )}}\right ) - 2 \, \sqrt {2} \arctan \left (\frac {3 \, \sqrt {2} \cos \left (x\right )^{2} - 2 \, \sqrt {2}}{4 \, \cos \left (x\right ) \sin \left (x\right )}\right ) + \log \left (-\cos \left (x\right )^{4} + \cos \left (x\right )^{2} + 2 \, \cos \left (x\right ) \sin \left (x\right ) + 1\right ) - \log \left (-\cos \left (x\right )^{4} + \cos \left (x\right )^{2} - 2 \, \cos \left (x\right ) \sin \left (x\right ) + 1\right )}{24 \, a} \] Input:

integrate(1/(a+a*sin(x)^6),x, algorithm="fricas")
 

Output:

1/24*(2*sqrt(3)*arctan(1/3*(4*sqrt(3)*cos(x)*sin(x) + sqrt(3))/(2*cos(x)^2 
 - 1)) + 2*sqrt(3)*arctan(1/3*(4*sqrt(3)*cos(x)*sin(x) - sqrt(3))/(2*cos(x 
)^2 - 1)) - 2*sqrt(2)*arctan(1/4*(3*sqrt(2)*cos(x)^2 - 2*sqrt(2))/(cos(x)* 
sin(x))) + log(-cos(x)^4 + cos(x)^2 + 2*cos(x)*sin(x) + 1) - log(-cos(x)^4 
 + cos(x)^2 - 2*cos(x)*sin(x) + 1))/a
 

Sympy [F(-1)]

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
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.61 \[ \int \frac {1}{a+a \sin ^6(x)} \, dx=\frac {\sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \tan \left (x\right ) + 1\right )}\right )}{6 \, a} + \frac {\sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \tan \left (x\right ) - 1\right )}\right )}{6 \, a} + \frac {\sqrt {2} \arctan \left (\sqrt {2} \tan \left (x\right )\right )}{6 \, a} + \frac {\log \left (\tan \left (x\right )^{2} + \tan \left (x\right ) + 1\right )}{12 \, a} - \frac {\log \left (\tan \left (x\right )^{2} - \tan \left (x\right ) + 1\right )}{12 \, a} \] Input:

integrate(1/(a+a*sin(x)^6),x, algorithm="maxima")
 

Output:

1/6*sqrt(3)*arctan(1/3*sqrt(3)*(2*tan(x) + 1))/a + 1/6*sqrt(3)*arctan(1/3* 
sqrt(3)*(2*tan(x) - 1))/a + 1/6*sqrt(2)*arctan(sqrt(2)*tan(x))/a + 1/12*lo 
g(tan(x)^2 + tan(x) + 1)/a - 1/12*log(tan(x)^2 - tan(x) + 1)/a
 

Giac [A] (verification not implemented)

Time = 0.43 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.41 \[ \int \frac {1}{a+a \sin ^6(x)} \, dx=\frac {\sqrt {3} {\left (x + \arctan \left (-\frac {\sqrt {3} \sin \left (2 \, x\right ) + \cos \left (2 \, x\right ) - 2 \, \sin \left (2 \, x\right ) + 1}{\sqrt {3} \cos \left (2 \, x\right ) + \sqrt {3} - 2 \, \cos \left (2 \, x\right ) - \sin \left (2 \, x\right ) + 2}\right )\right )}}{6 \, a} + \frac {\sqrt {3} {\left (x + \arctan \left (-\frac {\sqrt {3} \sin \left (2 \, x\right ) - \cos \left (2 \, x\right ) - 2 \, \sin \left (2 \, x\right ) - 1}{\sqrt {3} \cos \left (2 \, x\right ) + \sqrt {3} - 2 \, \cos \left (2 \, x\right ) + \sin \left (2 \, x\right ) + 2}\right )\right )}}{6 \, a} + \frac {\sqrt {2} {\left (x + \arctan \left (-\frac {\sqrt {2} \sin \left (2 \, x\right ) - 2 \, \sin \left (2 \, x\right )}{\sqrt {2} \cos \left (2 \, x\right ) + \sqrt {2} - 2 \, \cos \left (2 \, x\right ) + 2}\right )\right )}}{6 \, a} + \frac {\log \left (\tan \left (x\right )^{2} + \tan \left (x\right ) + 1\right )}{12 \, a} - \frac {\log \left (\tan \left (x\right )^{2} - \tan \left (x\right ) + 1\right )}{12 \, a} \] Input:

integrate(1/(a+a*sin(x)^6),x, algorithm="giac")
 

Output:

1/6*sqrt(3)*(x + arctan(-(sqrt(3)*sin(2*x) + cos(2*x) - 2*sin(2*x) + 1)/(s 
qrt(3)*cos(2*x) + sqrt(3) - 2*cos(2*x) - sin(2*x) + 2)))/a + 1/6*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 + 1/6*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 + 1/12*log(tan(x)^2 + tan(x) + 1)/a - 1/12*log(tan(x)^2 - tan(x) + 
1)/a
 

Mupad [B] (verification not implemented)

Time = 38.05 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.51 \[ \int \frac {1}{a+a \sin ^6(x)} \, dx=\frac {\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,\mathrm {tan}\left (x\right )\right )}{6\,a}+\frac {\mathrm {atan}\left (-\frac {\sqrt {3}\,\mathrm {tan}\left (x\right )}{2}+\frac {\mathrm {tan}\left (x\right )\,1{}\mathrm {i}}{2}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{6\,a}-\frac {\mathrm {atan}\left (\frac {\sqrt {3}\,\mathrm {tan}\left (x\right )}{2}+\frac {\mathrm {tan}\left (x\right )\,1{}\mathrm {i}}{2}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{6\,a} \] Input:

int(1/(a + a*sin(x)^6),x)
 

Output:

(2^(1/2)*atan(2^(1/2)*tan(x)))/(6*a) + (atan((tan(x)*1i)/2 - (3^(1/2)*tan( 
x))/2)*(3^(1/2)*1i - 1)*1i)/(6*a) - (atan((tan(x)*1i)/2 + (3^(1/2)*tan(x)) 
/2)*(3^(1/2)*1i + 1)*1i)/(6*a)
 

Reduce [F]

\[ \int \frac {1}{a+a \sin ^6(x)} \, dx=\frac {\int \frac {1}{\sin \left (x \right )^{6}+1}d x}{a} \] Input:

int(1/(a+a*sin(x)^6),x)
 

Output:

int(1/(sin(x)**6 + 1),x)/a