Integrand size = 11, antiderivative size = 494 \[ \int \frac {1}{a-b \cos ^5(x)} \, dx=\frac {2 \arctan \left (\frac {\sqrt {\sqrt [5]{a}+\sqrt [5]{b}} \tan \left (\frac {x}{2}\right )}{\sqrt {\sqrt [5]{a}-\sqrt [5]{b}}}\right )}{5 a^{4/5} \sqrt {\sqrt [5]{a}-\sqrt [5]{b}} \sqrt {\sqrt [5]{a}+\sqrt [5]{b}}}+\frac {2 \arctan \left (\frac {\sqrt {\sqrt [5]{a}-\sqrt [5]{-1} \sqrt [5]{b}} \tan \left (\frac {x}{2}\right )}{\sqrt {\sqrt [5]{a}+\sqrt [5]{-1} \sqrt [5]{b}}}\right )}{5 a^{4/5} \sqrt {\sqrt [5]{a}-\sqrt [5]{-1} \sqrt [5]{b}} \sqrt {\sqrt [5]{a}+\sqrt [5]{-1} \sqrt [5]{b}}}+\frac {2 \arctan \left (\frac {\sqrt {\sqrt [5]{a}+(-1)^{2/5} \sqrt [5]{b}} \tan \left (\frac {x}{2}\right )}{\sqrt {\sqrt [5]{a}-(-1)^{2/5} \sqrt [5]{b}}}\right )}{5 a^{4/5} \sqrt {\sqrt [5]{a}-(-1)^{2/5} \sqrt [5]{b}} \sqrt {\sqrt [5]{a}+(-1)^{2/5} \sqrt [5]{b}}}+\frac {2 \arctan \left (\frac {\sqrt {\sqrt [5]{a}-(-1)^{3/5} \sqrt [5]{b}} \tan \left (\frac {x}{2}\right )}{\sqrt {\sqrt [5]{a}+(-1)^{3/5} \sqrt [5]{b}}}\right )}{5 a^{4/5} \sqrt {\sqrt [5]{a}-(-1)^{3/5} \sqrt [5]{b}} \sqrt {\sqrt [5]{a}+(-1)^{3/5} \sqrt [5]{b}}}+\frac {2 \arctan \left (\frac {\sqrt {\sqrt [5]{a}+(-1)^{4/5} \sqrt [5]{b}} \tan \left (\frac {x}{2}\right )}{\sqrt {\sqrt [5]{a}-(-1)^{4/5} \sqrt [5]{b}}}\right )}{5 a^{4/5} \sqrt {\sqrt [5]{a}-(-1)^{4/5} \sqrt [5]{b}} \sqrt {\sqrt [5]{a}+(-1)^{4/5} \sqrt [5]{b}}} \] Output:
2/5*arctan((a^(1/5)+b^(1/5))^(1/2)*tan(1/2*x)/(a^(1/5)-b^(1/5))^(1/2))/a^( 4/5)/(a^(1/5)-b^(1/5))^(1/2)/(a^(1/5)+b^(1/5))^(1/2)+2/5*arctan((a^(1/5)-( -1)^(1/5)*b^(1/5))^(1/2)*tan(1/2*x)/(a^(1/5)+(-1)^(1/5)*b^(1/5))^(1/2))/a^ (4/5)/(a^(1/5)-(-1)^(1/5)*b^(1/5))^(1/2)/(a^(1/5)+(-1)^(1/5)*b^(1/5))^(1/2 )+2/5*arctan((a^(1/5)+(-1)^(2/5)*b^(1/5))^(1/2)*tan(1/2*x)/(a^(1/5)-(-1)^( 2/5)*b^(1/5))^(1/2))/a^(4/5)/(a^(1/5)-(-1)^(2/5)*b^(1/5))^(1/2)/(a^(1/5)+( -1)^(2/5)*b^(1/5))^(1/2)+2/5*arctan((a^(1/5)-(-1)^(3/5)*b^(1/5))^(1/2)*tan (1/2*x)/(a^(1/5)+(-1)^(3/5)*b^(1/5))^(1/2))/a^(4/5)/(a^(1/5)-(-1)^(3/5)*b^ (1/5))^(1/2)/(a^(1/5)+(-1)^(3/5)*b^(1/5))^(1/2)+2/5*arctan((a^(1/5)+(-1)^( 4/5)*b^(1/5))^(1/2)*tan(1/2*x)/(a^(1/5)-(-1)^(4/5)*b^(1/5))^(1/2))/a^(4/5) /(a^(1/5)-(-1)^(4/5)*b^(1/5))^(1/2)/(a^(1/5)+(-1)^(4/5)*b^(1/5))^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 5.14 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.26 \[ \int \frac {1}{a-b \cos ^5(x)} \, dx=-\frac {8}{5} \text {RootSum}\left [b+5 b \text {$\#$1}^2+10 b \text {$\#$1}^4-32 a \text {$\#$1}^5+10 b \text {$\#$1}^6+5 b \text {$\#$1}^8+b \text {$\#$1}^{10}\&,\frac {2 \arctan \left (\frac {\sin (x)}{\cos (x)-\text {$\#$1}}\right ) \text {$\#$1}^3-i \log \left (1-2 \cos (x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^3}{b+4 b \text {$\#$1}^2-16 a \text {$\#$1}^3+6 b \text {$\#$1}^4+4 b \text {$\#$1}^6+b \text {$\#$1}^8}\&\right ] \] Input:
Integrate[(a - b*Cos[x]^5)^(-1),x]
Output:
(-8*RootSum[b + 5*b*#1^2 + 10*b*#1^4 - 32*a*#1^5 + 10*b*#1^6 + 5*b*#1^8 + b*#1^10 & , (2*ArcTan[Sin[x]/(Cos[x] - #1)]*#1^3 - I*Log[1 - 2*Cos[x]*#1 + #1^2]*#1^3)/(b + 4*b*#1^2 - 16*a*#1^3 + 6*b*#1^4 + 4*b*#1^6 + b*#1^8) & ] )/5
Time = 0.95 (sec) , antiderivative size = 494, 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.
| \(\displaystyle \int \frac {1}{a-b \cos ^5(x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{a-b \sin \left (x+\frac {\pi }{2}\right )^5}dx\) |
| \(\Big \downarrow \) 3692 |
| \(\displaystyle \int \left (\frac {1}{5 a^{4/5} \left (\sqrt [5]{a}-\sqrt [5]{b} \cos (x)\right )}+\frac {1}{5 a^{4/5} \left (\sqrt [5]{a}+\sqrt [5]{-1} \sqrt [5]{b} \cos (x)\right )}+\frac {1}{5 a^{4/5} \left (\sqrt [5]{a}-(-1)^{2/5} \sqrt [5]{b} \cos (x)\right )}+\frac {1}{5 a^{4/5} \left (\sqrt [5]{a}+(-1)^{3/5} \sqrt [5]{b} \cos (x)\right )}+\frac {1}{5 a^{4/5} \left (\sqrt [5]{a}-(-1)^{4/5} \sqrt [5]{b} \cos (x)\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 \arctan \left (\frac {\sqrt {\sqrt [5]{a}+\sqrt [5]{b}} \tan \left (\frac {x}{2}\right )}{\sqrt {\sqrt [5]{a}-\sqrt [5]{b}}}\right )}{5 a^{4/5} \sqrt {\sqrt [5]{a}-\sqrt [5]{b}} \sqrt {\sqrt [5]{a}+\sqrt [5]{b}}}+\frac {2 \arctan \left (\frac {\sqrt {\sqrt [5]{a}-\sqrt [5]{-1} \sqrt [5]{b}} \tan \left (\frac {x}{2}\right )}{\sqrt {\sqrt [5]{a}+\sqrt [5]{-1} \sqrt [5]{b}}}\right )}{5 a^{4/5} \sqrt {\sqrt [5]{a}-\sqrt [5]{-1} \sqrt [5]{b}} \sqrt {\sqrt [5]{a}+\sqrt [5]{-1} \sqrt [5]{b}}}+\frac {2 \arctan \left (\frac {\sqrt {\sqrt [5]{a}+(-1)^{2/5} \sqrt [5]{b}} \tan \left (\frac {x}{2}\right )}{\sqrt {\sqrt [5]{a}-(-1)^{2/5} \sqrt [5]{b}}}\right )}{5 a^{4/5} \sqrt {\sqrt [5]{a}-(-1)^{2/5} \sqrt [5]{b}} \sqrt {\sqrt [5]{a}+(-1)^{2/5} \sqrt [5]{b}}}+\frac {2 \arctan \left (\frac {\sqrt {\sqrt [5]{a}-(-1)^{3/5} \sqrt [5]{b}} \tan \left (\frac {x}{2}\right )}{\sqrt {\sqrt [5]{a}+(-1)^{3/5} \sqrt [5]{b}}}\right )}{5 a^{4/5} \sqrt {\sqrt [5]{a}-(-1)^{3/5} \sqrt [5]{b}} \sqrt {\sqrt [5]{a}+(-1)^{3/5} \sqrt [5]{b}}}+\frac {2 \arctan \left (\frac {\sqrt {\sqrt [5]{a}+(-1)^{4/5} \sqrt [5]{b}} \tan \left (\frac {x}{2}\right )}{\sqrt {\sqrt [5]{a}-(-1)^{4/5} \sqrt [5]{b}}}\right )}{5 a^{4/5} \sqrt {\sqrt [5]{a}-(-1)^{4/5} \sqrt [5]{b}} \sqrt {\sqrt [5]{a}+(-1)^{4/5} \sqrt [5]{b}}}\) |
Input:
Int[(a - b*Cos[x]^5)^(-1),x]
Output:
(2*ArcTan[(Sqrt[a^(1/5) + b^(1/5)]*Tan[x/2])/Sqrt[a^(1/5) - b^(1/5)]])/(5* a^(4/5)*Sqrt[a^(1/5) - b^(1/5)]*Sqrt[a^(1/5) + b^(1/5)]) + (2*ArcTan[(Sqrt [a^(1/5) - (-1)^(1/5)*b^(1/5)]*Tan[x/2])/Sqrt[a^(1/5) + (-1)^(1/5)*b^(1/5) ]])/(5*a^(4/5)*Sqrt[a^(1/5) - (-1)^(1/5)*b^(1/5)]*Sqrt[a^(1/5) + (-1)^(1/5 )*b^(1/5)]) + (2*ArcTan[(Sqrt[a^(1/5) + (-1)^(2/5)*b^(1/5)]*Tan[x/2])/Sqrt [a^(1/5) - (-1)^(2/5)*b^(1/5)]])/(5*a^(4/5)*Sqrt[a^(1/5) - (-1)^(2/5)*b^(1 /5)]*Sqrt[a^(1/5) + (-1)^(2/5)*b^(1/5)]) + (2*ArcTan[(Sqrt[a^(1/5) - (-1)^ (3/5)*b^(1/5)]*Tan[x/2])/Sqrt[a^(1/5) + (-1)^(3/5)*b^(1/5)]])/(5*a^(4/5)*S qrt[a^(1/5) - (-1)^(3/5)*b^(1/5)]*Sqrt[a^(1/5) + (-1)^(3/5)*b^(1/5)]) + (2 *ArcTan[(Sqrt[a^(1/5) + (-1)^(4/5)*b^(1/5)]*Tan[x/2])/Sqrt[a^(1/5) - (-1)^ (4/5)*b^(1/5)]])/(5*a^(4/5)*Sqrt[a^(1/5) - (-1)^(4/5)*b^(1/5)]*Sqrt[a^(1/5 ) + (-1)^(4/5)*b^(1/5)])
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 = 0.94 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.30
| method | result | size |
| default | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (a +b \right ) \textit {\_Z}^{10}+\left (5 a -5 b \right ) \textit {\_Z}^{8}+\left (10 a +10 b \right ) \textit {\_Z}^{6}+\left (10 a -10 b \right ) \textit {\_Z}^{4}+\left (5 a +5 b \right ) \textit {\_Z}^{2}+a -b \right )}{\sum }\frac {\left (\textit {\_R}^{8}+4 \textit {\_R}^{6}+6 \textit {\_R}^{4}+4 \textit {\_R}^{2}+1\right ) \ln \left (\tan \left (\frac {x}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{9} a +\textit {\_R}^{9} b +4 \textit {\_R}^{7} a -4 \textit {\_R}^{7} b +6 \textit {\_R}^{5} a +6 \textit {\_R}^{5} b +4 \textit {\_R}^{3} a -4 \textit {\_R}^{3} b +\textit {\_R} a +\textit {\_R} b}\right )}{5}\) | \(148\) |
| risch | \(\munderset {\textit {\_R} =\operatorname {RootOf}\left (1+\left (9765625 a^{10}-9765625 a^{8} b^{2}\right ) \textit {\_Z}^{10}+1953125 a^{8} \textit {\_Z}^{8}+156250 a^{6} \textit {\_Z}^{6}+6250 a^{4} \textit {\_Z}^{4}+125 a^{2} \textit {\_Z}^{2}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{i x}+\left (\frac {11718750 i a^{10}}{b}-11718750 i b \,a^{8}\right ) \textit {\_R}^{9}+\left (-\frac {1171875 a^{9}}{b}+1171875 a^{7} b \right ) \textit {\_R}^{8}+\left (\frac {2109375 i a^{8}}{b}+234375 i a^{6} b \right ) \textit {\_R}^{7}+\left (-\frac {218750 a^{7}}{b}-15625 a^{5} b \right ) \textit {\_R}^{6}+\left (\frac {143750 i a^{6}}{b}-3125 i b \,a^{4}\right ) \textit {\_R}^{5}-\frac {15625 a^{5} \textit {\_R}^{4}}{b}+\frac {4375 i a^{4} \textit {\_R}^{3}}{b}-\frac {500 a^{3} \textit {\_R}^{2}}{b}+\frac {50 i a^{2} \textit {\_R}}{b}-\frac {6 a}{b}\right )\) | \(217\) |
Input:
int(1/(a-b*cos(x)^5),x,method=_RETURNVERBOSE)
Output:
1/5*sum((_R^8+4*_R^6+6*_R^4+4*_R^2+1)/(_R^9*a+_R^9*b+4*_R^7*a-4*_R^7*b+6*_ R^5*a+6*_R^5*b+4*_R^3*a-4*_R^3*b+_R*a+_R*b)*ln(tan(1/2*x)-_R),_R=RootOf((a +b)*_Z^10+(5*a-5*b)*_Z^8+(10*a+10*b)*_Z^6+(10*a-10*b)*_Z^4+(5*a+5*b)*_Z^2+ a-b))
Exception generated. \[ \int \frac {1}{a-b \cos ^5(x)} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate(1/(a-b*cos(x)^5),x, algorithm="fricas")
Output:
Exception raised: RuntimeError >> no explicit roots found
\[ \int \frac {1}{a-b \cos ^5(x)} \, dx=\int \frac {1}{a - b \cos ^{5}{\left (x \right )}}\, dx \] Input:
integrate(1/(a-b*cos(x)**5),x)
Output:
Integral(1/(a - b*cos(x)**5), x)
\[ \int \frac {1}{a-b \cos ^5(x)} \, dx=\int { -\frac {1}{b \cos \left (x\right )^{5} - a} \,d x } \] Input:
integrate(1/(a-b*cos(x)^5),x, algorithm="maxima")
Output:
-integrate(1/(b*cos(x)^5 - a), x)
\[ \int \frac {1}{a-b \cos ^5(x)} \, dx=\int { -\frac {1}{b \cos \left (x\right )^{5} - a} \,d x } \] Input:
integrate(1/(a-b*cos(x)^5),x, algorithm="giac")
Output:
integrate(-1/(b*cos(x)^5 - a), x)
Time = 4.13 (sec) , antiderivative size = 1518, normalized size of antiderivative = 3.07 \[ \int \frac {1}{a-b \cos ^5(x)} \, dx=\text {Too large to display} \] Input:
int(1/(a - b*cos(x)^5),x)
Output:
symsum(log(-(10995116277760*b^7*(a + b)*(56*root(9765625*a^8*b^2*d^10 - 97 65625*a^10*d^10 - 1953125*a^8*d^8 - 156250*a^6*d^6 - 6250*a^4*d^4 - 125*a^ 2*d^2 - 1, d, k)*a - 7*cot(x/2) + root(9765625*a^8*b^2*d^10 - 9765625*a^10 *d^10 - 1953125*a^8*d^8 - 156250*a^6*d^6 - 6250*a^4*d^4 - 125*a^2*d^2 - 1, d, k)*b + 5800*root(9765625*a^8*b^2*d^10 - 9765625*a^10*d^10 - 1953125*a^ 8*d^8 - 156250*a^6*d^6 - 6250*a^4*d^4 - 125*a^2*d^2 - 1, d, k)^3*a^3 + 225 000*root(9765625*a^8*b^2*d^10 - 9765625*a^10*d^10 - 1953125*a^8*d^8 - 1562 50*a^6*d^6 - 6250*a^4*d^4 - 125*a^2*d^2 - 1, d, k)^5*a^5 + 3875000*root(97 65625*a^8*b^2*d^10 - 9765625*a^10*d^10 - 1953125*a^8*d^8 - 156250*a^6*d^6 - 6250*a^4*d^4 - 125*a^2*d^2 - 1, d, k)^7*a^7 + 25000000*root(9765625*a^8* b^2*d^10 - 9765625*a^10*d^10 - 1953125*a^8*d^8 - 156250*a^6*d^6 - 6250*a^4 *d^4 - 125*a^2*d^2 - 1, d, k)^9*a^9 - 735*root(9765625*a^8*b^2*d^10 - 9765 625*a^10*d^10 - 1953125*a^8*d^8 - 156250*a^6*d^6 - 6250*a^4*d^4 - 125*a^2* d^2 - 1, d, k)^2*a^2*cot(x/2) - 28875*root(9765625*a^8*b^2*d^10 - 9765625* a^10*d^10 - 1953125*a^8*d^8 - 156250*a^6*d^6 - 6250*a^4*d^4 - 125*a^2*d^2 - 1, d, k)^4*a^4*cot(x/2) - 503125*root(9765625*a^8*b^2*d^10 - 9765625*a^1 0*d^10 - 1953125*a^8*d^8 - 156250*a^6*d^6 - 6250*a^4*d^4 - 125*a^2*d^2 - 1 , d, k)^6*a^6*cot(x/2) - 3281250*root(9765625*a^8*b^2*d^10 - 9765625*a^10* d^10 - 1953125*a^8*d^8 - 156250*a^6*d^6 - 6250*a^4*d^4 - 125*a^2*d^2 - 1, d, k)^8*a^8*cot(x/2) + 800*root(9765625*a^8*b^2*d^10 - 9765625*a^10*d^1...
\[ \int \frac {1}{a-b \cos ^5(x)} \, dx=-\left (\int \frac {1}{\cos \left (x \right )^{5} b -a}d x \right ) \] Input:
int(1/(a-b*cos(x)^5),x)
Output:
- int(1/(cos(x)**5*b - a),x)