Integrand size = 11, antiderivative size = 181 \[ \int \frac {1}{a-b \sin ^8(x)} \, dx=\frac {\arctan \left (\sqrt {1-i \sqrt [4]{\frac {b}{a}}} \tan (x)\right )}{4 a \sqrt {1-i \sqrt [4]{\frac {b}{a}}}}+\frac {\arctan \left (\sqrt {1+\sqrt [4]{\frac {b}{a}}} \tan (x)\right )}{4 a \sqrt {1+\sqrt [4]{\frac {b}{a}}}}+\frac {\text {arctanh}\left (\sqrt {-1-i \sqrt [4]{\frac {b}{a}}} \tan (x)\right )}{4 a \sqrt {-1-i \sqrt [4]{\frac {b}{a}}}}+\frac {\text {arctanh}\left (\sqrt {-1+\sqrt [4]{\frac {b}{a}}} \tan (x)\right )}{4 a \sqrt {-1+\sqrt [4]{\frac {b}{a}}}} \] Output:
1/4*arctan((1-I*(b/a)^(1/4))^(1/2)*tan(x))/a/(1-I*(b/a)^(1/4))^(1/2)+1/4*a rctan((1+(b/a)^(1/4))^(1/2)*tan(x))/a/(1+(b/a)^(1/4))^(1/2)+1/4*arctanh((- 1-I*(b/a)^(1/4))^(1/2)*tan(x))/a/(-1-I*(b/a)^(1/4))^(1/2)+1/4*arctanh((-1+ (b/a)^(1/4))^(1/2)*tan(x))/a/(-1+(b/a)^(1/4))^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 5.14 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.96 \[ \int \frac {1}{a-b \sin ^8(x)} \, dx=-8 \text {RootSum}\left [b-8 b \text {$\#$1}+28 b \text {$\#$1}^2-56 b \text {$\#$1}^3-256 a \text {$\#$1}^4+70 b \text {$\#$1}^4-56 b \text {$\#$1}^5+28 b \text {$\#$1}^6-8 b \text {$\#$1}^7+b \text {$\#$1}^8\&,\frac {2 \arctan \left (\frac {\sin (2 x)}{\cos (2 x)-\text {$\#$1}}\right ) \text {$\#$1}^3-i \log \left (1-2 \cos (2 x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^3}{-b+7 b \text {$\#$1}-21 b \text {$\#$1}^2-128 a \text {$\#$1}^3+35 b \text {$\#$1}^3-35 b \text {$\#$1}^4+21 b \text {$\#$1}^5-7 b \text {$\#$1}^6+b \text {$\#$1}^7}\&\right ] \] Input:
Integrate[(a - b*Sin[x]^8)^(-1),x]
Output:
-8*RootSum[b - 8*b*#1 + 28*b*#1^2 - 56*b*#1^3 - 256*a*#1^4 + 70*b*#1^4 - 5 6*b*#1^5 + 28*b*#1^6 - 8*b*#1^7 + b*#1^8 & , (2*ArcTan[Sin[2*x]/(Cos[2*x] - #1)]*#1^3 - I*Log[1 - 2*Cos[2*x]*#1 + #1^2]*#1^3)/(-b + 7*b*#1 - 21*b*#1 ^2 - 128*a*#1^3 + 35*b*#1^3 - 35*b*#1^4 + 21*b*#1^5 - 7*b*#1^6 + b*#1^7) & ]
Time = 0.50 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.18, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, 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.
| \(\displaystyle \int \frac {1}{a-b \sin ^8(x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{a-b \sin (x)^8}dx\) |
| \(\Big \downarrow \) 3690 |
| \(\displaystyle \frac {\int \frac {1}{1-\frac {\sqrt [4]{b} \sin ^2(x)}{\sqrt [4]{a}}}dx}{4 a}+\frac {\int \frac {1}{1-\frac {i \sqrt [4]{b} \sin ^2(x)}{\sqrt [4]{a}}}dx}{4 a}+\frac {\int \frac {1}{\frac {i \sqrt [4]{b} \sin ^2(x)}{\sqrt [4]{a}}+1}dx}{4 a}+\frac {\int \frac {1}{\frac {\sqrt [4]{b} \sin ^2(x)}{\sqrt [4]{a}}+1}dx}{4 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {1}{1-\frac {\sqrt [4]{b} \sin (x)^2}{\sqrt [4]{a}}}dx}{4 a}+\frac {\int \frac {1}{1-\frac {i \sqrt [4]{b} \sin (x)^2}{\sqrt [4]{a}}}dx}{4 a}+\frac {\int \frac {1}{\frac {i \sqrt [4]{b} \sin (x)^2}{\sqrt [4]{a}}+1}dx}{4 a}+\frac {\int \frac {1}{\frac {\sqrt [4]{b} \sin (x)^2}{\sqrt [4]{a}}+1}dx}{4 a}\) |
| \(\Big \downarrow \) 3660 |
| \(\displaystyle \frac {\int \frac {1}{\left (1-\frac {\sqrt [4]{b}}{\sqrt [4]{a}}\right ) \tan ^2(x)+1}d\tan (x)}{4 a}+\frac {\int \frac {1}{\left (1-\frac {i \sqrt [4]{b}}{\sqrt [4]{a}}\right ) \tan ^2(x)+1}d\tan (x)}{4 a}+\frac {\int \frac {1}{\left (\frac {i \sqrt [4]{b}}{\sqrt [4]{a}}+1\right ) \tan ^2(x)+1}d\tan (x)}{4 a}+\frac {\int \frac {1}{\left (\frac {\sqrt [4]{b}}{\sqrt [4]{a}}+1\right ) \tan ^2(x)+1}d\tan (x)}{4 a}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {\arctan \left (\frac {\sqrt {\sqrt [4]{a}-\sqrt [4]{b}} \tan (x)}{\sqrt [8]{a}}\right )}{4 a^{7/8} \sqrt {\sqrt [4]{a}-\sqrt [4]{b}}}+\frac {\arctan \left (\frac {\sqrt {\sqrt [4]{a}-i \sqrt [4]{b}} \tan (x)}{\sqrt [8]{a}}\right )}{4 a^{7/8} \sqrt {\sqrt [4]{a}-i \sqrt [4]{b}}}+\frac {\arctan \left (\frac {\sqrt {\sqrt [4]{a}+i \sqrt [4]{b}} \tan (x)}{\sqrt [8]{a}}\right )}{4 a^{7/8} \sqrt {\sqrt [4]{a}+i \sqrt [4]{b}}}+\frac {\arctan \left (\frac {\sqrt {\sqrt [4]{a}+\sqrt [4]{b}} \tan (x)}{\sqrt [8]{a}}\right )}{4 a^{7/8} \sqrt {\sqrt [4]{a}+\sqrt [4]{b}}}\) |
Input:
Int[(a - b*Sin[x]^8)^(-1),x]
Output:
ArcTan[(Sqrt[a^(1/4) - b^(1/4)]*Tan[x])/a^(1/8)]/(4*a^(7/8)*Sqrt[a^(1/4) - b^(1/4)]) + ArcTan[(Sqrt[a^(1/4) - I*b^(1/4)]*Tan[x])/a^(1/8)]/(4*a^(7/8) *Sqrt[a^(1/4) - I*b^(1/4)]) + ArcTan[(Sqrt[a^(1/4) + I*b^(1/4)]*Tan[x])/a^ (1/8)]/(4*a^(7/8)*Sqrt[a^(1/4) + I*b^(1/4)]) + ArcTan[(Sqrt[a^(1/4) + b^(1 /4)]*Tan[x])/a^(1/8)]/(4*a^(7/8)*Sqrt[a^(1/4) + b^(1/4)])
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 = 1.59 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.49
| method | result | size |
| default | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (a -b \right ) \textit {\_Z}^{8}+4 a \,\textit {\_Z}^{6}+6 a \,\textit {\_Z}^{4}+4 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (\textit {\_R}^{6}+3 \textit {\_R}^{4}+3 \textit {\_R}^{2}+1\right ) \ln \left (\tan \left (x \right )-\textit {\_R} \right )}{\textit {\_R}^{7} a -\textit {\_R}^{7} b +3 \textit {\_R}^{5} a +3 \textit {\_R}^{3} a +a \textit {\_R}}\right )}{8}\) | \(88\) |
| risch | \(\munderset {\textit {\_R} =\operatorname {RootOf}\left (1+\left (16777216 a^{8}-16777216 a^{7} b \right ) \textit {\_Z}^{8}+1048576 a^{6} \textit {\_Z}^{6}+24576 a^{4} \textit {\_Z}^{4}+256 a^{2} \textit {\_Z}^{2}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 i x}+\left (\frac {4194304 i a^{8}}{b}-4194304 i a^{7}\right ) \textit {\_R}^{7}+\left (-\frac {524288 a^{7}}{b}+524288 a^{6}\right ) \textit {\_R}^{6}+\left (\frac {196608 i a^{6}}{b}+65536 i a^{5}\right ) \textit {\_R}^{5}+\left (-\frac {24576 a^{5}}{b}-8192 a^{4}\right ) \textit {\_R}^{4}+\left (\frac {3072 i a^{4}}{b}-1024 i a^{3}\right ) \textit {\_R}^{3}+\left (-\frac {384 a^{3}}{b}+128 a^{2}\right ) \textit {\_R}^{2}+\left (\frac {16 i a^{2}}{b}+16 i a \right ) \textit {\_R} -\frac {2 a}{b}-1\right )\) | \(193\) |
Input:
int(1/(a-b*sin(x)^8),x,method=_RETURNVERBOSE)
Output:
1/8*sum((_R^6+3*_R^4+3*_R^2+1)/(_R^7*a-_R^7*b+3*_R^5*a+3*_R^3*a+_R*a)*ln(t an(x)-_R),_R=RootOf((a-b)*_Z^8+4*a*_Z^6+6*a*_Z^4+4*a*_Z^2+a))
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 643307 vs. \(2 (133) = 266\).
Time = 6.08 (sec) , antiderivative size = 643307, normalized size of antiderivative = 3554.18 \[ \int \frac {1}{a-b \sin ^8(x)} \, dx=\text {Too large to display} \] Input:
integrate(1/(a-b*sin(x)^8),x, algorithm="fricas")
Output:
Too large to include
\[ \int \frac {1}{a-b \sin ^8(x)} \, dx=\int \frac {1}{a - b \sin ^{8}{\left (x \right )}}\, dx \] Input:
integrate(1/(a-b*sin(x)**8),x)
Output:
Integral(1/(a - b*sin(x)**8), x)
\[ \int \frac {1}{a-b \sin ^8(x)} \, dx=\int { -\frac {1}{b \sin \left (x\right )^{8} - a} \,d x } \] Input:
integrate(1/(a-b*sin(x)^8),x, algorithm="maxima")
Output:
-integrate(1/(b*sin(x)^8 - a), x)
\[ \int \frac {1}{a-b \sin ^8(x)} \, dx=\int { -\frac {1}{b \sin \left (x\right )^{8} - a} \,d x } \] Input:
integrate(1/(a-b*sin(x)^8),x, algorithm="giac")
Output:
integrate(-1/(b*sin(x)^8 - a), x)
Time = 39.91 (sec) , antiderivative size = 818, normalized size of antiderivative = 4.52 \[ \int \frac {1}{a-b \sin ^8(x)} \, dx =\text {Too large to display} \] Input:
int(1/(a - b*sin(x)^8),x)
Output:
symsum(log(-2*b^5*(a - b)*(4*root(16777216*a^7*b*d^8 - 16777216*a^8*d^8 - 1048576*a^6*d^6 - 24576*a^4*d^4 - 256*a^2*d^2 - 1, d, k)*b*tan(x) - 43008* root(16777216*a^7*b*d^8 - 16777216*a^8*d^8 - 1048576*a^6*d^6 - 24576*a^4*d ^4 - 256*a^2*d^2 - 1, d, k)^4*a^4 - 786432*root(16777216*a^7*b*d^8 - 16777 216*a^8*d^8 - 1048576*a^6*d^6 - 24576*a^4*d^4 - 256*a^2*d^2 - 1, d, k)^6*a ^6 - 800*root(16777216*a^7*b*d^8 - 16777216*a^8*d^8 - 1048576*a^6*d^6 - 24 576*a^4*d^4 - 256*a^2*d^2 - 1, d, k)^2*a^2 - 6144*root(16777216*a^7*b*d^8 - 16777216*a^8*d^8 - 1048576*a^6*d^6 - 24576*a^4*d^4 - 256*a^2*d^2 - 1, d, k)^4*a^3*b + 786432*root(16777216*a^7*b*d^8 - 16777216*a^8*d^8 - 1048576* a^6*d^6 - 24576*a^4*d^4 - 256*a^2*d^2 - 1, d, k)^6*a^5*b + 9984*root(16777 216*a^7*b*d^8 - 16777216*a^8*d^8 - 1048576*a^6*d^6 - 24576*a^4*d^4 - 256*a ^2*d^2 - 1, d, k)^3*a^3*tan(x) + 557056*root(16777216*a^7*b*d^8 - 16777216 *a^8*d^8 - 1048576*a^6*d^6 - 24576*a^4*d^4 - 256*a^2*d^2 - 1, d, k)^5*a^5* tan(x) + 10485760*root(16777216*a^7*b*d^8 - 16777216*a^8*d^8 - 1048576*a^6 *d^6 - 24576*a^4*d^4 - 256*a^2*d^2 - 1, d, k)^7*a^7*tan(x) + 32*root(16777 216*a^7*b*d^8 - 16777216*a^8*d^8 - 1048576*a^6*d^6 - 24576*a^4*d^4 - 256*a ^2*d^2 - 1, d, k)^2*a*b + 60*root(16777216*a^7*b*d^8 - 16777216*a^8*d^8 - 1048576*a^6*d^6 - 24576*a^4*d^4 - 256*a^2*d^2 - 1, d, k)*a*tan(x) - 768*ro ot(16777216*a^7*b*d^8 - 16777216*a^8*d^8 - 1048576*a^6*d^6 - 24576*a^4*d^4 - 256*a^2*d^2 - 1, d, k)^3*a^2*b*tan(x) + 98304*root(16777216*a^7*b*d^...
\[ \int \frac {1}{a-b \sin ^8(x)} \, dx=-\left (\int \frac {1}{\sin \left (x \right )^{8} b -a}d x \right ) \] Input:
int(1/(a-b*sin(x)^8),x)
Output:
- int(1/(sin(x)**8*b - a),x)