Integrand size = 11, antiderivative size = 32 \[ \int \frac {1}{\cos (5 x)+\sin (2 x)} \, dx=-\frac {1}{6} \text {arctanh}(\cos (x))-\frac {2}{3} \text {arctanh}(2 \cos (x))+\frac {\text {arctanh}\left (\sqrt {2} \cos (x)\right )}{\sqrt {2}} \] Output:
-1/6*arctanh(cos(x))-2/3*arctanh(2*cos(x))+1/2*arctanh(cos(x)*2^(1/2))*2^( 1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.17 (sec) , antiderivative size = 116, normalized size of antiderivative = 3.62 \[ \int \frac {1}{\cos (5 x)+\sin (2 x)} \, dx=\frac {1}{21} \left (-12 \log \left (\sec ^2\left (\frac {x}{2}\right )\right )-3 \log \left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )+7 \log \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )-14 \log (1-2 \sin (x))+24 \text {RootSum}\left [1-4 \text {$\#$1}-32 \text {$\#$1}^2+64 \text {$\#$1}^3\&,\log \left (-\sec ^2\left (\frac {x}{2}\right ) \left (-1-4 \sin (x)-8 \sin (x) \text {$\#$1}+32 \sin (x) \text {$\#$1}^2\right )\right ) \text {$\#$1}\&\right ]\right ) \] Input:
Integrate[(Cos[5*x] + Sin[2*x])^(-1),x]
Output:
(-12*Log[Sec[x/2]^2] - 3*Log[Cos[x/2] - Sin[x/2]] + 7*Log[Cos[x/2] + Sin[x /2]] - 14*Log[1 - 2*Sin[x]] + 24*RootSum[1 - 4*#1 - 32*#1^2 + 64*#1^3 & , Log[-(Sec[x/2]^2*(-1 - 4*Sin[x] - 8*Sin[x]*#1 + 32*Sin[x]*#1^2))]*#1 & ])/ 21
Result contains complex when optimal does not.
Time = 4.29 (sec) , antiderivative size = 890, normalized size of antiderivative = 27.81, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {3042, 4829, 2462, 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.
| \(\displaystyle \int \frac {1}{\sin (2 x)+\cos (5 x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\sin (2 x)+\cos (5 x)}dx\) |
| \(\Big \downarrow \) 4829 |
| \(\displaystyle \int \frac {1}{\left (1-\sin ^2(x)\right ) \left (16 \sin ^4(x)-12 \sin ^2(x)+2 \sin (x)+1\right )}d\sin (x)\) |
| \(\Big \downarrow \) 2462 |
| \(\displaystyle \int \left (\frac {4 \left (8 \sin ^2(x)+12 \sin (x)+1\right )}{7 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )}-\frac {1}{14 (\sin (x)-1)}+\frac {1}{6 (\sin (x)+1)}-\frac {4}{3 (2 \sin (x)-1)}\right )d\sin (x)\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {6\ 2^{2/3} \sqrt [3]{7} \left (1+3 i \sqrt {3}\right ) \left (2^{2/3} \sqrt [3]{7}-\left (1+3 i \sqrt {3}\right )^{2/3}\right ) \text {arctanh}\left (\frac {4 i \left (1+3 i \sqrt {3}\right )^{2/3} (6 \sin (x)+1)+\sqrt [3]{14} \left (\sqrt [3]{2} \left (i-3 \sqrt {3}\right )+2 i \sqrt [3]{7+21 i \sqrt {3}}\right )}{2 \sqrt {3 \left (-\sqrt [3]{2} 7^{2/3} \left (13-3 i \sqrt {3}\right )+7 \sqrt [3]{7} \left (2+6 i \sqrt {3}\right )^{2/3}-\sqrt [3]{1+3 i \sqrt {3}} \left (14+42 i \sqrt {3}\right )\right )}}\right )}{\sqrt {-\sqrt [3]{2} 7^{2/3} \left (13-3 i \sqrt {3}\right )+7 \sqrt [3]{7} \left (2+6 i \sqrt {3}\right )^{2/3}-\sqrt [3]{1+3 i \sqrt {3}} \left (14+42 i \sqrt {3}\right )} \left (14 \left (1+3 i \sqrt {3}\right )^{2/3}+\sqrt [3]{14} \left (14 \sqrt [3]{2}+\sqrt [3]{7} \left (1+3 i \sqrt {3}\right )^{4/3}\right )\right )}-\frac {2}{3} \log (1-2 \sin (x))-\frac {1}{14} \log (1-\sin (x))+\frac {1}{6} \log (\sin (x)+1)-\frac {2 \sqrt [3]{1+3 i \sqrt {3}} \left (2 \sqrt [3]{2} 7^{2/3}+\sqrt [3]{1+3 i \sqrt {3}}+\sqrt [3]{7} \left (2+6 i \sqrt {3}\right )^{2/3}\right ) \log \left (24 \left (1+3 i \sqrt {3}\right )^{2/3} \sin ^2(x)+4\ 7^{2/3} \sqrt [3]{2 \left (1+3 i \sqrt {3}\right )} \sin (x)+8 \left (1+3 i \sqrt {3}\right )^{2/3} \sin (x)+6 i 2^{2/3} \sqrt {3} \sqrt [3]{7} \sin (x)+2\ 2^{2/3} \sqrt [3]{7} \sin (x)+\sqrt [3]{14} \left (\sqrt [3]{2} \left (5+i \sqrt {3}\right )+\left (1+i \sqrt {3}\right ) \sqrt [3]{7+21 i \sqrt {3}}\right )-4 \left (1+3 i \sqrt {3}\right )^{2/3}\right )}{3 \left (14 \left (1+3 i \sqrt {3}\right )^{2/3}+\sqrt [3]{14} \left (14 \sqrt [3]{2}+\sqrt [3]{7} \left (1+3 i \sqrt {3}\right )^{4/3}\right )\right )}+\frac {4}{21} \log \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )+\frac {2 \left (i-3 \sqrt {3}\right ) \left (2 \sqrt [3]{2} 7^{2/3}+\sqrt [3]{1+3 i \sqrt {3}}+\sqrt [3]{7} \left (2+6 i \sqrt {3}\right )^{2/3}\right ) \log \left (\sqrt [3]{14} \left (2 \sqrt [3]{7}+\sqrt [3]{2} \left (1+3 i \sqrt {3}\right )^{2/3}\right )-2 \sqrt [3]{1+3 i \sqrt {3}} (6 \sin (x)+1)\right )}{3 \left (7 \left (i-3 \sqrt {3}\right ) \sqrt [3]{1+3 i \sqrt {3}}+7 i \sqrt [3]{7} \left (2+6 i \sqrt {3}\right )^{2/3}-\sqrt [3]{2} 7^{2/3} \left (13 i+3 \sqrt {3}\right )\right )}\) |
Input:
Int[(Cos[5*x] + Sin[2*x])^(-1),x]
Output:
(6*2^(2/3)*7^(1/3)*(1 + (3*I)*Sqrt[3])*(2^(2/3)*7^(1/3) - (1 + (3*I)*Sqrt[ 3])^(2/3))*ArcTanh[(14^(1/3)*(2^(1/3)*(I - 3*Sqrt[3]) + (2*I)*(7 + (21*I)* Sqrt[3])^(1/3)) + (4*I)*(1 + (3*I)*Sqrt[3])^(2/3)*(1 + 6*Sin[x]))/(2*Sqrt[ 3*(-(2^(1/3)*7^(2/3)*(13 - (3*I)*Sqrt[3])) + 7*7^(1/3)*(2 + (6*I)*Sqrt[3]) ^(2/3) - (1 + (3*I)*Sqrt[3])^(1/3)*(14 + (42*I)*Sqrt[3]))])])/(Sqrt[-(2^(1 /3)*7^(2/3)*(13 - (3*I)*Sqrt[3])) + 7*7^(1/3)*(2 + (6*I)*Sqrt[3])^(2/3) - (1 + (3*I)*Sqrt[3])^(1/3)*(14 + (42*I)*Sqrt[3])]*(14*(1 + (3*I)*Sqrt[3])^( 2/3) + 14^(1/3)*(14*2^(1/3) + 7^(1/3)*(1 + (3*I)*Sqrt[3])^(4/3)))) - (2*Lo g[1 - 2*Sin[x]])/3 - Log[1 - Sin[x]]/14 + Log[1 + Sin[x]]/6 - (2*(1 + (3*I )*Sqrt[3])^(1/3)*(2*2^(1/3)*7^(2/3) + (1 + (3*I)*Sqrt[3])^(1/3) + 7^(1/3)* (2 + (6*I)*Sqrt[3])^(2/3))*Log[-4*(1 + (3*I)*Sqrt[3])^(2/3) + 14^(1/3)*(2^ (1/3)*(5 + I*Sqrt[3]) + (1 + I*Sqrt[3])*(7 + (21*I)*Sqrt[3])^(1/3)) + 2*2^ (2/3)*7^(1/3)*Sin[x] + (6*I)*2^(2/3)*Sqrt[3]*7^(1/3)*Sin[x] + 8*(1 + (3*I) *Sqrt[3])^(2/3)*Sin[x] + 4*7^(2/3)*(2*(1 + (3*I)*Sqrt[3]))^(1/3)*Sin[x] + 24*(1 + (3*I)*Sqrt[3])^(2/3)*Sin[x]^2])/(3*(14*(1 + (3*I)*Sqrt[3])^(2/3) + 14^(1/3)*(14*2^(1/3) + 7^(1/3)*(1 + (3*I)*Sqrt[3])^(4/3)))) + (4*Log[-1 - 4*Sin[x] + 4*Sin[x]^2 + 8*Sin[x]^3])/21 + (2*(I - 3*Sqrt[3])*(2*2^(1/3)*7 ^(2/3) + (1 + (3*I)*Sqrt[3])^(1/3) + 7^(1/3)*(2 + (6*I)*Sqrt[3])^(2/3))*Lo g[14^(1/3)*(2*7^(1/3) + 2^(1/3)*(1 + (3*I)*Sqrt[3])^(2/3)) - 2*(1 + (3*I)* Sqrt[3])^(1/3)*(1 + 6*Sin[x])])/(3*(7*(I - 3*Sqrt[3])*(1 + (3*I)*Sqrt[3...
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px]}, Int[ExpandIntegr and[u*Qx^p, x], x] /; !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && GtQ [Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0 ] && RationalFunctionQ[u, x]
Int[(cos[(n_.)*((c_.) + (d_.)*(x_))]*(b_.) + (a_.)*sin[(m_.)*((c_.) + (d_.) *(x_))])^(p_), x_Symbol] :> Simp[1/d Subst[Int[Simplify[TrigExpand[a*Sin[ m*ArcSin[x]] + b*Cos[n*ArcSin[x]]]]^p/Sqrt[1 - x^2], x], x, Sin[c + d*x]], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[(p - 1)/2, 0] && IntegerQ[m/2] && Inte gerQ[(n - 1)/2]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.74 (sec) , antiderivative size = 76, normalized size of antiderivative = 2.38
| method | result | size |
| default | \(-\frac {\ln \left (\sin \left (x \right )-1\right )}{14}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (8 \textit {\_Z}^{3}+4 \textit {\_Z}^{2}-4 \textit {\_Z} -1\right )}{\sum }\frac {\left (8 \textit {\_R}^{2}+12 \textit {\_R} +1\right ) \ln \left (\sin \left (x \right )-\textit {\_R} \right )}{6 \textit {\_R}^{2}+2 \textit {\_R} -1}\right )}{7}-\frac {2 \ln \left (2 \sin \left (x \right )-1\right )}{3}+\frac {\ln \left (1+\sin \left (x \right )\right )}{6}\) | \(76\) |
| risch | \(-\frac {\ln \left ({\mathrm e}^{i x}-i\right )}{7}+\frac {\ln \left ({\mathrm e}^{i x}+i\right )}{3}+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (343 \textit {\_Z}^{3}-196 \textit {\_Z}^{2}-28 \textit {\_Z} +8\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 i x}+\left (-\frac {7}{2} i \textit {\_R} +i\right ) {\mathrm e}^{i x}-1\right )\right )-\frac {2 \ln \left (-i {\mathrm e}^{i x}+{\mathrm e}^{2 i x}-1\right )}{3}\) | \(84\) |
Input:
int(1/(cos(5*x)+sin(2*x)),x,method=_RETURNVERBOSE)
Output:
-1/14*ln(sin(x)-1)+1/7*sum((8*_R^2+12*_R+1)/(6*_R^2+2*_R-1)*ln(sin(x)-_R), _R=RootOf(8*_Z^3+4*_Z^2-4*_Z-1))-2/3*ln(2*sin(x)-1)+1/6*ln(1+sin(x))
Result contains complex when optimal does not.
Time = 0.69 (sec) , antiderivative size = 607, normalized size of antiderivative = 18.97 \[ \int \frac {1}{\cos (5 x)+\sin (2 x)} \, dx=\text {Too large to display} \] Input:
integrate(1/(cos(5*x)+sin(2*x)),x, algorithm="fricas")
Output:
1/252*(63*(4/441*I*sqrt(3) + 4/1323)^(1/3)*(-I*sqrt(3) + 1) + 4*(I*sqrt(3) + 1)/(4/441*I*sqrt(3) + 4/1323)^(1/3) + 18*sqrt(-1/108*(63*(4/441*I*sqrt( 3) + 4/1323)^(1/3)*(-I*sqrt(3) + 1) + 4*(I*sqrt(3) + 1)/(4/441*I*sqrt(3) + 4/1323)^(1/3) - 24)^2 - 28*(4/441*I*sqrt(3) + 4/1323)^(1/3)*(-I*sqrt(3) + 1) - 16/9*(I*sqrt(3) + 1)/(4/441*I*sqrt(3) + 4/1323)^(1/3) + 128/3) + 48) *log(7/2*(4/441*I*sqrt(3) + 4/1323)^(1/3)*(-I*sqrt(3) + 1) + 2/9*(I*sqrt(3 ) + 1)/(4/441*I*sqrt(3) + 4/1323)^(1/3) + sqrt(-1/108*(63*(4/441*I*sqrt(3) + 4/1323)^(1/3)*(-I*sqrt(3) + 1) + 4*(I*sqrt(3) + 1)/(4/441*I*sqrt(3) + 4 /1323)^(1/3) - 24)^2 - 28*(4/441*I*sqrt(3) + 4/1323)^(1/3)*(-I*sqrt(3) + 1 ) - 16/9*(I*sqrt(3) + 1)/(4/441*I*sqrt(3) + 4/1323)^(1/3) + 128/3) - 8*sin (x) - 4/3) + 1/252*(63*(4/441*I*sqrt(3) + 4/1323)^(1/3)*(-I*sqrt(3) + 1) + 4*(I*sqrt(3) + 1)/(4/441*I*sqrt(3) + 4/1323)^(1/3) - 18*sqrt(-1/108*(63*( 4/441*I*sqrt(3) + 4/1323)^(1/3)*(-I*sqrt(3) + 1) + 4*(I*sqrt(3) + 1)/(4/44 1*I*sqrt(3) + 4/1323)^(1/3) - 24)^2 - 28*(4/441*I*sqrt(3) + 4/1323)^(1/3)* (-I*sqrt(3) + 1) - 16/9*(I*sqrt(3) + 1)/(4/441*I*sqrt(3) + 4/1323)^(1/3) + 128/3) + 48)*log(-7/2*(4/441*I*sqrt(3) + 4/1323)^(1/3)*(-I*sqrt(3) + 1) - 2/9*(I*sqrt(3) + 1)/(4/441*I*sqrt(3) + 4/1323)^(1/3) + sqrt(-1/108*(63*(4 /441*I*sqrt(3) + 4/1323)^(1/3)*(-I*sqrt(3) + 1) + 4*(I*sqrt(3) + 1)/(4/441 *I*sqrt(3) + 4/1323)^(1/3) - 24)^2 - 28*(4/441*I*sqrt(3) + 4/1323)^(1/3)*( -I*sqrt(3) + 1) - 16/9*(I*sqrt(3) + 1)/(4/441*I*sqrt(3) + 4/1323)^(1/3)...
\[ \int \frac {1}{\cos (5 x)+\sin (2 x)} \, dx=\int \frac {1}{\sin {\left (2 x \right )} + \cos {\left (5 x \right )}}\, dx \] Input:
integrate(1/(cos(5*x)+sin(2*x)),x)
Output:
Integral(1/(sin(2*x) + cos(5*x)), x)
\[ \int \frac {1}{\cos (5 x)+\sin (2 x)} \, dx=\int { \frac {1}{\cos \left (5 \, x\right ) + \sin \left (2 \, x\right )} \,d x } \] Input:
integrate(1/(cos(5*x)+sin(2*x)),x, algorithm="maxima")
Output:
-4/21*integrate(((cos(2*x)*sin(8*x) - cos(2*x)*sin(6*x) + cos(2*x)*sin(4*x ) - cos(8*x)*sin(2*x) + cos(6*x)*sin(2*x) - cos(4*x)*sin(2*x) - sin(2*x))* cos(1/2*arctan2(sin(2*x), cos(2*x)))^2 + (cos(2*x)*sin(8*x) - cos(2*x)*sin (6*x) + cos(2*x)*sin(4*x) - cos(8*x)*sin(2*x) + cos(6*x)*sin(2*x) - cos(4* x)*sin(2*x) - sin(2*x))*sin(1/2*arctan2(sin(2*x), cos(2*x)))^2 + (cos(6*x) *cos(2*x) - cos(4*x)*cos(2*x) - (cos(2*x)*sin(8*x) - cos(2*x)*sin(6*x) + c os(2*x)*sin(4*x) - cos(8*x)*sin(2*x) + cos(6*x)*sin(2*x) - cos(4*x)*sin(2* x) - sin(2*x))*cos(1/2*arctan2(sin(2*x), cos(2*x))) + sin(6*x)*sin(2*x) - sin(4*x)*sin(2*x) - (cos(8*x)*cos(2*x) - cos(6*x)*cos(2*x) + cos(4*x)*cos( 2*x) - cos(2*x)^2 + sin(8*x)*sin(2*x) - sin(6*x)*sin(2*x) + sin(4*x)*sin(2 *x) - sin(2*x)^2 + cos(2*x))*sin(1/2*arctan2(sin(2*x), cos(2*x))))*cos(7/2 *arctan2(sin(2*x), cos(2*x))) - (cos(6*x)*cos(2*x) - cos(4*x)*cos(2*x) - ( cos(2*x)*sin(8*x) - cos(2*x)*sin(6*x) + cos(2*x)*sin(4*x) - cos(8*x)*sin(2 *x) + cos(6*x)*sin(2*x) - cos(4*x)*sin(2*x) - sin(2*x))*cos(1/2*arctan2(si n(2*x), cos(2*x))) + sin(6*x)*sin(2*x) - sin(4*x)*sin(2*x) - (cos(8*x)*cos (2*x) - cos(6*x)*cos(2*x) + cos(4*x)*cos(2*x) - cos(2*x)^2 + sin(8*x)*sin( 2*x) - sin(6*x)*sin(2*x) + sin(4*x)*sin(2*x) - sin(2*x)^2 + cos(2*x))*sin( 1/2*arctan2(sin(2*x), cos(2*x))))*cos(5/2*arctan2(sin(2*x), cos(2*x))) + ( cos(6*x)*cos(2*x) - cos(4*x)*cos(2*x) - (cos(2*x)*sin(8*x) - cos(2*x)*sin( 6*x) + cos(2*x)*sin(4*x) - cos(8*x)*sin(2*x) + cos(6*x)*sin(2*x) - cos(...
Exception generated. \[ \int \frac {1}{\cos (5 x)+\sin (2 x)} \, dx=\text {Exception raised: NotImplementedError} \] Input:
integrate(1/(cos(5*x)+sin(2*x)),x, algorithm="giac")
Output:
Exception raised: NotImplementedError >> unable to parse Giac output: 2*(- 1/28*ln(-sin(sageVARx)+1)+1/12*ln(sin(sageVARx)+1)-1/3*ln(abs(2*sin(sageVA Rx)-1))+((-7/32768*rootof([[-3,0,35840,0,-66060288],[1,0,-14336,0,51380224 ,0,-52613349376]])+
Time = 23.35 (sec) , antiderivative size = 582, normalized size of antiderivative = 18.19 \[ \int \frac {1}{\cos (5 x)+\sin (2 x)} \, dx=\text {Too large to display} \] Input:
int(1/(cos(5*x) + sin(2*x)),x)
Output:
log(tan(x/2) + 1)/3 - log(tan(x/2) - 1)/7 - (2*log(tan(x/2)^2 - 4*tan(x/2) + 1))/3 + symsum(log((140737488355328*(11902752*root(z^3 - (4*z^2)/7 - (4 *z)/49 + 8/343, z, k)*cos(x) - 728000*cos(x) - 2080*sin(x) - 14590784*root (z^3 - (4*z^2)/7 - (4*z)/49 + 8/343, z, k) + 29331680*root(z^3 - (4*z^2)/7 - (4*z)/49 + 8/343, z, k)*sin(x) - 549239760*root(z^3 - (4*z^2)/7 - (4*z) /49 + 8/343, z, k)^2 + 108864280*root(z^3 - (4*z^2)/7 - (4*z)/49 + 8/343, z, k)^3 + 28013641068*root(z^3 - (4*z^2)/7 - (4*z)/49 + 8/343, z, k)^4 + 1 6847690972*root(z^3 - (4*z^2)/7 - (4*z)/49 + 8/343, z, k)^5 - 321476748285 *root(z^3 - (4*z^2)/7 - (4*z)/49 + 8/343, z, k)^6 - 28330551480*root(z^3 - (4*z^2)/7 - (4*z)/49 + 8/343, z, k)^7 + 602671372485*root(z^3 - (4*z^2)/7 - (4*z)/49 + 8/343, z, k)^8 + 592615840*root(z^3 - (4*z^2)/7 - (4*z)/49 + 8/343, z, k)^2*cos(x) + 152859056*root(z^3 - (4*z^2)/7 - (4*z)/49 + 8/343 , z, k)^3*cos(x) - 32417181972*root(z^3 - (4*z^2)/7 - (4*z)/49 + 8/343, z, k)^4*cos(x) - 21798153132*root(z^3 - (4*z^2)/7 - (4*z)/49 + 8/343, z, k)^ 5*cos(x) + 390994442041*root(z^3 - (4*z^2)/7 - (4*z)/49 + 8/343, z, k)^6*c os(x) + 35652852368*root(z^3 - (4*z^2)/7 - (4*z)/49 + 8/343, z, k)^7*cos(x ) - 741082462953*root(z^3 - (4*z^2)/7 - (4*z)/49 + 8/343, z, k)^8*cos(x) + 54174840*root(z^3 - (4*z^2)/7 - (4*z)/49 + 8/343, z, k)^2*sin(x) - 174956 6588*root(z^3 - (4*z^2)/7 - (4*z)/49 + 8/343, z, k)^3*sin(x) - 1411646320* root(z^3 - (4*z^2)/7 - (4*z)/49 + 8/343, z, k)^4*sin(x) + 21036147076*r...
\[ \int \frac {1}{\cos (5 x)+\sin (2 x)} \, dx=\int \frac {1}{\cos \left (5 x \right )+\sin \left (2 x \right )}d x \] Input:
int(1/(cos(5*x)+sin(2*x)),x)
Output:
int(1/(cos(5*x) + sin(2*x)),x)