Integrand size = 10, antiderivative size = 179 \[ \int \frac {1}{a+b \sin ^6(x)} \, dx=\frac {\arctan \left (\frac {\sqrt {\sqrt [3]{a}+\sqrt [3]{b}} \tan (x)}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt {\sqrt [3]{a}+\sqrt [3]{b}}}+\frac {\text {arctanh}\left (\frac {\sqrt {-\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b}} \tan (x)}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt {-\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b}}}+\frac {\text {arctanh}\left (\frac {\sqrt {-\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b}} \tan (x)}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt {-\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b}}} \] Output:
1/3*arctan((a^(1/3)+b^(1/3))^(1/2)*tan(x)/a^(1/6))/a^(5/6)/(a^(1/3)+b^(1/3 ))^(1/2)+1/3*arctanh((-a^(1/3)+(-1)^(1/3)*b^(1/3))^(1/2)*tan(x)/a^(1/6))/a ^(5/6)/(-a^(1/3)+(-1)^(1/3)*b^(1/3))^(1/2)+1/3*arctanh((-a^(1/3)-(-1)^(2/3 )*b^(1/3))^(1/2)*tan(x)/a^(1/6))/a^(5/6)/(-a^(1/3)-(-1)^(2/3)*b^(1/3))^(1/ 2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 5.08 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.83 \[ \int \frac {1}{a+b \sin ^6(x)} \, dx=-\frac {8}{3} \text {RootSum}\left [b-6 b \text {$\#$1}+15 b \text {$\#$1}^2-64 a \text {$\#$1}^3-20 b \text {$\#$1}^3+15 b \text {$\#$1}^4-6 b \text {$\#$1}^5+b \text {$\#$1}^6\&,\frac {2 \arctan \left (\frac {\sin (2 x)}{\cos (2 x)-\text {$\#$1}}\right ) \text {$\#$1}^2-i \log \left (1-2 \cos (2 x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^2}{-b+5 b \text {$\#$1}-32 a \text {$\#$1}^2-10 b \text {$\#$1}^2+10 b \text {$\#$1}^3-5 b \text {$\#$1}^4+b \text {$\#$1}^5}\&\right ] \] Input:
Integrate[(a + b*Sin[x]^6)^(-1),x]
Output:
(-8*RootSum[b - 6*b*#1 + 15*b*#1^2 - 64*a*#1^3 - 20*b*#1^3 + 15*b*#1^4 - 6 *b*#1^5 + b*#1^6 & , (2*ArcTan[Sin[2*x]/(Cos[2*x] - #1)]*#1^2 - I*Log[1 - 2*Cos[2*x]*#1 + #1^2]*#1^2)/(-b + 5*b*#1 - 32*a*#1^2 - 10*b*#1^2 + 10*b*#1 ^3 - 5*b*#1^4 + b*#1^5) & ])/3
Time = 0.50 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.96, 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.
| \(\displaystyle \int \frac {1}{a+b \sin ^6(x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{a+b \sin (x)^6}dx\) |
| \(\Big \downarrow \) 3690 |
| \(\displaystyle \frac {\int \frac {1}{\frac {\sqrt [3]{b} \sin ^2(x)}{\sqrt [3]{a}}+1}dx}{3 a}+\frac {\int \frac {1}{1-\frac {\sqrt [3]{-1} \sqrt [3]{b} \sin ^2(x)}{\sqrt [3]{a}}}dx}{3 a}+\frac {\int \frac {1}{\frac {(-1)^{2/3} \sqrt [3]{b} \sin ^2(x)}{\sqrt [3]{a}}+1}dx}{3 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {1}{\frac {\sqrt [3]{b} \sin (x)^2}{\sqrt [3]{a}}+1}dx}{3 a}+\frac {\int \frac {1}{1-\frac {\sqrt [3]{-1} \sqrt [3]{b} \sin (x)^2}{\sqrt [3]{a}}}dx}{3 a}+\frac {\int \frac {1}{\frac {(-1)^{2/3} \sqrt [3]{b} \sin (x)^2}{\sqrt [3]{a}}+1}dx}{3 a}\) |
| \(\Big \downarrow \) 3660 |
| \(\displaystyle \frac {\int \frac {1}{\left (\frac {\sqrt [3]{b}}{\sqrt [3]{a}}+1\right ) \tan ^2(x)+1}d\tan (x)}{3 a}+\frac {\int \frac {1}{\left (1-\frac {\sqrt [3]{-1} \sqrt [3]{b}}{\sqrt [3]{a}}\right ) \tan ^2(x)+1}d\tan (x)}{3 a}+\frac {\int \frac {1}{\left (\frac {(-1)^{2/3} \sqrt [3]{b}}{\sqrt [3]{a}}+1\right ) \tan ^2(x)+1}d\tan (x)}{3 a}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {\arctan \left (\frac {\sqrt {\sqrt [3]{a}+\sqrt [3]{b}} \tan (x)}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt {\sqrt [3]{a}+\sqrt [3]{b}}}+\frac {\arctan \left (\frac {\sqrt {\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b}} \tan (x)}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt {\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b}}}+\frac {\arctan \left (\frac {\sqrt {\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b}} \tan (x)}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt {\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b}}}\) |
Input:
Int[(a + b*Sin[x]^6)^(-1),x]
Output:
ArcTan[(Sqrt[a^(1/3) + b^(1/3)]*Tan[x])/a^(1/6)]/(3*a^(5/6)*Sqrt[a^(1/3) + b^(1/3)]) + ArcTan[(Sqrt[a^(1/3) - (-1)^(1/3)*b^(1/3)]*Tan[x])/a^(1/6)]/( 3*a^(5/6)*Sqrt[a^(1/3) - (-1)^(1/3)*b^(1/3)]) + ArcTan[(Sqrt[a^(1/3) + (-1 )^(2/3)*b^(1/3)]*Tan[x])/a^(1/6)]/(3*a^(5/6)*Sqrt[a^(1/3) + (-1)^(2/3)*b^( 1/3)])
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[((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]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.48 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.38
| method | result | size |
| default | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (a +b \right ) \textit {\_Z}^{6}+3 a \,\textit {\_Z}^{4}+3 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (\textit {\_R}^{4}+2 \textit {\_R}^{2}+1\right ) \ln \left (\tan \left (x \right )-\textit {\_R} \right )}{\textit {\_R}^{5} a +\textit {\_R}^{5} b +2 \textit {\_R}^{3} a +a \textit {\_R}}\right )}{6}\) | \(68\) |
| risch | \(\munderset {\textit {\_R} =\operatorname {RootOf}\left (1+\left (46656 a^{6}+46656 a^{5} b \right ) \textit {\_Z}^{6}+3888 a^{4} \textit {\_Z}^{4}+108 a^{2} \textit {\_Z}^{2}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 i x}+\left (\frac {15552 i a^{6}}{b}+15552 i a^{5}\right ) \textit {\_R}^{5}+\left (-\frac {2592 a^{5}}{b}-2592 a^{4}\right ) \textit {\_R}^{4}+\left (\frac {864 i a^{4}}{b}-432 i a^{3}\right ) \textit {\_R}^{3}+\left (-\frac {144 a^{3}}{b}+72 a^{2}\right ) \textit {\_R}^{2}+\left (\frac {12 i a^{2}}{b}+12 i a \right ) \textit {\_R} -\frac {2 a}{b}-1\right )\) | \(147\) |
Input:
int(1/(a+b*sin(x)^6),x,method=_RETURNVERBOSE)
Output:
1/6*sum((_R^4+2*_R^2+1)/(_R^5*a+_R^5*b+2*_R^3*a+_R*a)*ln(tan(x)-_R),_R=Roo tOf((a+b)*_Z^6+3*a*_Z^4+3*a*_Z^2+a))
Result contains complex when optimal does not.
Time = 1.52 (sec) , antiderivative size = 15501, normalized size of antiderivative = 86.60 \[ \int \frac {1}{a+b \sin ^6(x)} \, dx=\text {Too large to display} \] Input:
integrate(1/(a+b*sin(x)^6),x, algorithm="fricas")
Output:
Too large to include
\[ \int \frac {1}{a+b \sin ^6(x)} \, dx=\int \frac {1}{a + b \sin ^{6}{\left (x \right )}}\, dx \] Input:
integrate(1/(a+b*sin(x)**6),x)
Output:
Integral(1/(a + b*sin(x)**6), x)
\[ \int \frac {1}{a+b \sin ^6(x)} \, dx=\int { \frac {1}{b \sin \left (x\right )^{6} + a} \,d x } \] Input:
integrate(1/(a+b*sin(x)^6),x, algorithm="maxima")
Output:
integrate(1/(b*sin(x)^6 + a), x)
\[ \int \frac {1}{a+b \sin ^6(x)} \, dx=\int { \frac {1}{b \sin \left (x\right )^{6} + a} \,d x } \] Input:
integrate(1/(a+b*sin(x)^6),x, algorithm="giac")
Output:
integrate(1/(b*sin(x)^6 + a), x)
Time = 40.27 (sec) , antiderivative size = 513, normalized size of antiderivative = 2.87 \[ \int \frac {1}{a+b \sin ^6(x)} \, dx=\sum _{k=1}^6\ln \left (-\frac {b^3\,\left (a+b\right )\,\left (-\mathrm {cot}\left (x\right )+\mathrm {root}\left (46656\,a^5\,b\,d^6+46656\,a^6\,d^6+3888\,a^4\,d^4+108\,a^2\,d^2+1,d,k\right )\,a\,8+\mathrm {root}\left (46656\,a^5\,b\,d^6+46656\,a^6\,d^6+3888\,a^4\,d^4+108\,a^2\,d^2+1,d,k\right )\,b\,2+{\mathrm {root}\left (46656\,a^5\,b\,d^6+46656\,a^6\,d^6+3888\,a^4\,d^4+108\,a^2\,d^2+1,d,k\right )}^3\,a^3\,504+{\mathrm {root}\left (46656\,a^5\,b\,d^6+46656\,a^6\,d^6+3888\,a^4\,d^4+108\,a^2\,d^2+1,d,k\right )}^5\,a^5\,7776-{\mathrm {root}\left (46656\,a^5\,b\,d^6+46656\,a^6\,d^6+3888\,a^4\,d^4+108\,a^2\,d^2+1,d,k\right )}^3\,a^2\,b\,144+{\mathrm {root}\left (46656\,a^5\,b\,d^6+46656\,a^6\,d^6+3888\,a^4\,d^4+108\,a^2\,d^2+1,d,k\right )}^5\,a^4\,b\,7776-{\mathrm {root}\left (46656\,a^5\,b\,d^6+46656\,a^6\,d^6+3888\,a^4\,d^4+108\,a^2\,d^2+1,d,k\right )}^2\,a^2\,\mathrm {cot}\left (x\right )\,60-{\mathrm {root}\left (46656\,a^5\,b\,d^6+46656\,a^6\,d^6+3888\,a^4\,d^4+108\,a^2\,d^2+1,d,k\right )}^4\,a^4\,\mathrm {cot}\left (x\right )\,864-{\mathrm {root}\left (46656\,a^5\,b\,d^6+46656\,a^6\,d^6+3888\,a^4\,d^4+108\,a^2\,d^2+1,d,k\right )}^4\,a^3\,b\,\mathrm {cot}\left (x\right )\,864+{\mathrm {root}\left (46656\,a^5\,b\,d^6+46656\,a^6\,d^6+3888\,a^4\,d^4+108\,a^2\,d^2+1,d,k\right )}^2\,a\,b\,\mathrm {cot}\left (x\right )\,12\right )\,3}{\mathrm {cot}\left (x\right )}\right )\,\mathrm {root}\left (46656\,a^5\,b\,d^6+46656\,a^6\,d^6+3888\,a^4\,d^4+108\,a^2\,d^2+1,d,k\right ) \] Input:
int(1/(a + b*sin(x)^6),x)
Output:
symsum(log(-(3*b^3*(a + b)*(8*root(46656*a^5*b*d^6 + 46656*a^6*d^6 + 3888* a^4*d^4 + 108*a^2*d^2 + 1, d, k)*a - cot(x) + 2*root(46656*a^5*b*d^6 + 466 56*a^6*d^6 + 3888*a^4*d^4 + 108*a^2*d^2 + 1, d, k)*b + 504*root(46656*a^5* b*d^6 + 46656*a^6*d^6 + 3888*a^4*d^4 + 108*a^2*d^2 + 1, d, k)^3*a^3 + 7776 *root(46656*a^5*b*d^6 + 46656*a^6*d^6 + 3888*a^4*d^4 + 108*a^2*d^2 + 1, d, k)^5*a^5 - 144*root(46656*a^5*b*d^6 + 46656*a^6*d^6 + 3888*a^4*d^4 + 108* a^2*d^2 + 1, d, k)^3*a^2*b + 7776*root(46656*a^5*b*d^6 + 46656*a^6*d^6 + 3 888*a^4*d^4 + 108*a^2*d^2 + 1, d, k)^5*a^4*b - 60*root(46656*a^5*b*d^6 + 4 6656*a^6*d^6 + 3888*a^4*d^4 + 108*a^2*d^2 + 1, d, k)^2*a^2*cot(x) - 864*ro ot(46656*a^5*b*d^6 + 46656*a^6*d^6 + 3888*a^4*d^4 + 108*a^2*d^2 + 1, d, k) ^4*a^4*cot(x) - 864*root(46656*a^5*b*d^6 + 46656*a^6*d^6 + 3888*a^4*d^4 + 108*a^2*d^2 + 1, d, k)^4*a^3*b*cot(x) + 12*root(46656*a^5*b*d^6 + 46656*a^ 6*d^6 + 3888*a^4*d^4 + 108*a^2*d^2 + 1, d, k)^2*a*b*cot(x)))/cot(x))*root( 46656*a^5*b*d^6 + 46656*a^6*d^6 + 3888*a^4*d^4 + 108*a^2*d^2 + 1, d, k), k , 1, 6)
\[ \int \frac {1}{a+b \sin ^6(x)} \, dx=\int \frac {1}{\sin \left (x \right )^{6} b +a}d x \] Input:
int(1/(a+b*sin(x)^6),x)
Output:
int(1/(sin(x)**6*b + a),x)