Integrand size = 11, antiderivative size = 90 \[ \int \frac {1}{(\cos (5 x)+\sin (2 x))^2} \, dx=-\frac {3}{4} \text {arctanh}(2 \cos (x) \sin (x))-\frac {\cot (x)}{36}-\frac {4 \log \left (\sqrt {3} \cos (x)-\sin (x)\right )}{3 \sqrt {3}}+\frac {4 \log \left (\sqrt {3} \cos (x)+\sin (x)\right )}{3 \sqrt {3}}+\frac {\tan (x) \left (43-25 \tan ^2(x)\right )}{18 \left (3-4 \tan ^2(x)+\tan ^4(x)\right )} \] Output:
-3/4*arctanh(2*cos(x)*sin(x))-1/36*cot(x)-4/9*ln(3^(1/2)*cos(x)-sin(x))*3^ (1/2)+4/9*ln(3^(1/2)*cos(x)+sin(x))*3^(1/2)+tan(x)*(43-25*tan(x)^2)/(54-72 *tan(x)^2+18*tan(x)^4)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.38 (sec) , antiderivative size = 401, normalized size of antiderivative = 4.46 \[ \int \frac {1}{(\cos (5 x)+\sin (2 x))^2} \, dx=\frac {1}{441} \left (2744 \sqrt {3} \text {arctanh}\left (\frac {-2+\tan \left (\frac {x}{2}\right )}{\sqrt {3}}\right )-378 i \text {RootSum}\left [i+\text {$\#$1}-i \text {$\#$1}^2-\text {$\#$1}^3+i \text {$\#$1}^4+\text {$\#$1}^5-i \text {$\#$1}^6\&,\frac {10 \arctan \left (\frac {\sin (x)}{\cos (x)-\text {$\#$1}}\right )-5 i \log \left (1-2 \cos (x) \text {$\#$1}+\text {$\#$1}^2\right )-22 i \arctan \left (\frac {\sin (x)}{\cos (x)-\text {$\#$1}}\right ) \text {$\#$1}-11 \log \left (1-2 \cos (x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}-28 \arctan \left (\frac {\sin (x)}{\cos (x)-\text {$\#$1}}\right ) \text {$\#$1}^2+14 i \log \left (1-2 \cos (x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^2+22 i \arctan \left (\frac {\sin (x)}{\cos (x)-\text {$\#$1}}\right ) \text {$\#$1}^3+11 \log \left (1-2 \cos (x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^3+10 \arctan \left (\frac {\sin (x)}{\cos (x)-\text {$\#$1}}\right ) \text {$\#$1}^4-5 i \log \left (1-2 \cos (x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^4}{i+2 \text {$\#$1}-3 i \text {$\#$1}^2-4 \text {$\#$1}^3+5 i \text {$\#$1}^4+6 \text {$\#$1}^5}\&\right ]+\frac {9 \sin \left (\frac {x}{2}\right )}{\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )}+\frac {49 \sin \left (\frac {x}{2}\right )}{\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )}+\frac {392 \cos (x)}{1-2 \sin (x)}+\frac {72 (9 \cos (x)-6 \cos (3 x)+11 \sin (2 x))}{-1+2 \cos (2 x)-2 \sin (x)+2 \sin (3 x)}\right ) \] Input:
Integrate[(Cos[5*x] + Sin[2*x])^(-2),x]
Output:
(2744*Sqrt[3]*ArcTanh[(-2 + Tan[x/2])/Sqrt[3]] - (378*I)*RootSum[I + #1 - I*#1^2 - #1^3 + I*#1^4 + #1^5 - I*#1^6 & , (10*ArcTan[Sin[x]/(Cos[x] - #1) ] - (5*I)*Log[1 - 2*Cos[x]*#1 + #1^2] - (22*I)*ArcTan[Sin[x]/(Cos[x] - #1) ]*#1 - 11*Log[1 - 2*Cos[x]*#1 + #1^2]*#1 - 28*ArcTan[Sin[x]/(Cos[x] - #1)] *#1^2 + (14*I)*Log[1 - 2*Cos[x]*#1 + #1^2]*#1^2 + (22*I)*ArcTan[Sin[x]/(Co s[x] - #1)]*#1^3 + 11*Log[1 - 2*Cos[x]*#1 + #1^2]*#1^3 + 10*ArcTan[Sin[x]/ (Cos[x] - #1)]*#1^4 - (5*I)*Log[1 - 2*Cos[x]*#1 + #1^2]*#1^4)/(I + 2*#1 - (3*I)*#1^2 - 4*#1^3 + (5*I)*#1^4 + 6*#1^5) & ] + (9*Sin[x/2])/(Cos[x/2] - Sin[x/2]) + (49*Sin[x/2])/(Cos[x/2] + Sin[x/2]) + (392*Cos[x])/(1 - 2*Sin[ x]) + (72*(9*Cos[x] - 6*Cos[3*x] + 11*Sin[2*x]))/(-1 + 2*Cos[2*x] - 2*Sin[ x] + 2*Sin[3*x]))/441
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.
| \(\displaystyle \int \frac {1}{(\sin (2 x)+\cos (5 x))^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(\sin (2 x)+\cos (5 x))^2}dx\) |
| \(\Big \downarrow \) 4830 |
| \(\displaystyle 2 \int \frac {\left (\tan ^2\left (\frac {x}{2}\right )+1\right )^9}{\left (-\tan ^{10}\left (\frac {x}{2}\right )-4 \tan ^9\left (\frac {x}{2}\right )+45 \tan ^8\left (\frac {x}{2}\right )-8 \tan ^7\left (\frac {x}{2}\right )-210 \tan ^6\left (\frac {x}{2}\right )+210 \tan ^4\left (\frac {x}{2}\right )+8 \tan ^3\left (\frac {x}{2}\right )-45 \tan ^2\left (\frac {x}{2}\right )+4 \tan \left (\frac {x}{2}\right )+1\right )^2}d\tan \left (\frac {x}{2}\right )\) |
| \(\Big \downarrow \) 2462 |
| \(\displaystyle 2 \int \left (\frac {16 \tan \left (\frac {x}{2}\right )}{3 \left (\tan ^2\left (\frac {x}{2}\right )-4 \tan \left (\frac {x}{2}\right )+1\right )^2}-\frac {68}{9 \left (\tan ^2\left (\frac {x}{2}\right )-4 \tan \left (\frac {x}{2}\right )+1\right )}+\frac {32 \left (13 \tan ^4\left (\frac {x}{2}\right )+130 \tan ^3\left (\frac {x}{2}\right )+372 \tan ^2\left (\frac {x}{2}\right )-206 \tan \left (\frac {x}{2}\right )+5613\right )}{49 \left (\tan ^6\left (\frac {x}{2}\right )+8 \tan ^5\left (\frac {x}{2}\right )-13 \tan ^4\left (\frac {x}{2}\right )-48 \tan ^3\left (\frac {x}{2}\right )-13 \tan ^2\left (\frac {x}{2}\right )+8 \tan \left (\frac {x}{2}\right )+1\right )}+\frac {1}{98 \left (\tan \left (\frac {x}{2}\right )-1\right )^2}+\frac {1}{18 \left (\tan \left (\frac {x}{2}\right )+1\right )^2}-\frac {512 \left (373 \tan ^5\left (\frac {x}{2}\right )-532 \tan ^4\left (\frac {x}{2}\right )-2358 \tan ^3\left (\frac {x}{2}\right )-674 \tan ^2\left (\frac {x}{2}\right )+397 \tan \left (\frac {x}{2}\right )+50\right )}{7 \left (\tan ^6\left (\frac {x}{2}\right )+8 \tan ^5\left (\frac {x}{2}\right )-13 \tan ^4\left (\frac {x}{2}\right )-48 \tan ^3\left (\frac {x}{2}\right )-13 \tan ^2\left (\frac {x}{2}\right )+8 \tan \left (\frac {x}{2}\right )+1\right )^2}\right )d\tan \left (\frac {x}{2}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 \left (\frac {14 \log \left (-\tan \left (\frac {x}{2}\right )-\sqrt {3}+2\right )}{3 \sqrt {3}}-\frac {14 \log \left (-\tan \left (\frac {x}{2}\right )+\sqrt {3}+2\right )}{3 \sqrt {3}}+\frac {687104}{21} \int \frac {1}{\left (\tan ^6\left (\frac {x}{2}\right )+8 \tan ^5\left (\frac {x}{2}\right )-13 \tan ^4\left (\frac {x}{2}\right )-48 \tan ^3\left (\frac {x}{2}\right )-13 \tan ^2\left (\frac {x}{2}\right )+8 \tan \left (\frac {x}{2}\right )+1\right )^2}d\tan \left (\frac {x}{2}\right )-\frac {3092480}{21} \int \frac {\tan \left (\frac {x}{2}\right )}{\left (\tan ^6\left (\frac {x}{2}\right )+8 \tan ^5\left (\frac {x}{2}\right )-13 \tan ^4\left (\frac {x}{2}\right )-48 \tan ^3\left (\frac {x}{2}\right )-13 \tan ^2\left (\frac {x}{2}\right )+8 \tan \left (\frac {x}{2}\right )+1\right )^2}d\tan \left (\frac {x}{2}\right )-\frac {4238336}{7} \int \frac {\tan ^2\left (\frac {x}{2}\right )}{\left (\tan ^6\left (\frac {x}{2}\right )+8 \tan ^5\left (\frac {x}{2}\right )-13 \tan ^4\left (\frac {x}{2}\right )-48 \tan ^3\left (\frac {x}{2}\right )-13 \tan ^2\left (\frac {x}{2}\right )+8 \tan \left (\frac {x}{2}\right )+1\right )^2}d\tan \left (\frac {x}{2}\right )-\frac {1343488}{21} \int \frac {\tan ^3\left (\frac {x}{2}\right )}{\left (\tan ^6\left (\frac {x}{2}\right )+8 \tan ^5\left (\frac {x}{2}\right )-13 \tan ^4\left (\frac {x}{2}\right )-48 \tan ^3\left (\frac {x}{2}\right )-13 \tan ^2\left (\frac {x}{2}\right )+8 \tan \left (\frac {x}{2}\right )+1\right )^2}d\tan \left (\frac {x}{2}\right )+\frac {4636672}{21} \int \frac {\tan ^4\left (\frac {x}{2}\right )}{\left (\tan ^6\left (\frac {x}{2}\right )+8 \tan ^5\left (\frac {x}{2}\right )-13 \tan ^4\left (\frac {x}{2}\right )-48 \tan ^3\left (\frac {x}{2}\right )-13 \tan ^2\left (\frac {x}{2}\right )+8 \tan \left (\frac {x}{2}\right )+1\right )^2}d\tan \left (\frac {x}{2}\right )+\frac {179616}{49} \int \frac {1}{\tan ^6\left (\frac {x}{2}\right )+8 \tan ^5\left (\frac {x}{2}\right )-13 \tan ^4\left (\frac {x}{2}\right )-48 \tan ^3\left (\frac {x}{2}\right )-13 \tan ^2\left (\frac {x}{2}\right )+8 \tan \left (\frac {x}{2}\right )+1}d\tan \left (\frac {x}{2}\right )-\frac {6592}{49} \int \frac {\tan \left (\frac {x}{2}\right )}{\tan ^6\left (\frac {x}{2}\right )+8 \tan ^5\left (\frac {x}{2}\right )-13 \tan ^4\left (\frac {x}{2}\right )-48 \tan ^3\left (\frac {x}{2}\right )-13 \tan ^2\left (\frac {x}{2}\right )+8 \tan \left (\frac {x}{2}\right )+1}d\tan \left (\frac {x}{2}\right )+\frac {11904}{49} \int \frac {\tan ^2\left (\frac {x}{2}\right )}{\tan ^6\left (\frac {x}{2}\right )+8 \tan ^5\left (\frac {x}{2}\right )-13 \tan ^4\left (\frac {x}{2}\right )-48 \tan ^3\left (\frac {x}{2}\right )-13 \tan ^2\left (\frac {x}{2}\right )+8 \tan \left (\frac {x}{2}\right )+1}d\tan \left (\frac {x}{2}\right )+\frac {4160}{49} \int \frac {\tan ^3\left (\frac {x}{2}\right )}{\tan ^6\left (\frac {x}{2}\right )+8 \tan ^5\left (\frac {x}{2}\right )-13 \tan ^4\left (\frac {x}{2}\right )-48 \tan ^3\left (\frac {x}{2}\right )-13 \tan ^2\left (\frac {x}{2}\right )+8 \tan \left (\frac {x}{2}\right )+1}d\tan \left (\frac {x}{2}\right )+\frac {416}{49} \int \frac {\tan ^4\left (\frac {x}{2}\right )}{\tan ^6\left (\frac {x}{2}\right )+8 \tan ^5\left (\frac {x}{2}\right )-13 \tan ^4\left (\frac {x}{2}\right )-48 \tan ^3\left (\frac {x}{2}\right )-13 \tan ^2\left (\frac {x}{2}\right )+8 \tan \left (\frac {x}{2}\right )+1}d\tan \left (\frac {x}{2}\right )+\frac {1}{98 \left (1-\tan \left (\frac {x}{2}\right )\right )}-\frac {1}{18 \left (\tan \left (\frac {x}{2}\right )+1\right )}+\frac {8 \left (1-2 \tan \left (\frac {x}{2}\right )\right )}{9 \left (\tan ^2\left (\frac {x}{2}\right )-4 \tan \left (\frac {x}{2}\right )+1\right )}+\frac {95488}{21 \left (\tan ^6\left (\frac {x}{2}\right )+8 \tan ^5\left (\frac {x}{2}\right )-13 \tan ^4\left (\frac {x}{2}\right )-48 \tan ^3\left (\frac {x}{2}\right )-13 \tan ^2\left (\frac {x}{2}\right )+8 \tan \left (\frac {x}{2}\right )+1\right )}\right )\) |
Input:
Int[(Cos[5*x] + Sin[2*x])^(-2),x]
Output:
$Aborted
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[2/d Subst[Int[Simplify[TrigExpand[a*Sin[ 2*m*ArcTan[x]] + b*Cos[2*n*ArcTan[x]]]]^p/(1 + x^2), x], x, Tan[(1/2)*(c + d*x)]], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[p, 0] && IntegerQ[m] && Intege rQ[n]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 2.28 (sec) , antiderivative size = 194, normalized size of antiderivative = 2.16
| method | result | size |
| risch | \(\frac {\frac {16 \,{\mathrm e}^{9 i x}}{21}-\frac {10 i {\mathrm e}^{8 i x}}{21}+\frac {12 \,{\mathrm e}^{7 i x}}{7}+\frac {2 i {\mathrm e}^{6 i x}}{7}-\frac {2 \,{\mathrm e}^{5 i x}}{7}-\frac {2 i {\mathrm e}^{4 i x}}{7}-\frac {2 \,{\mathrm e}^{3 i x}}{7}+\frac {10 i {\mathrm e}^{2 i x}}{21}+\frac {16 \,{\mathrm e}^{i x}}{21}+2 i}{{\mathrm e}^{10 i x}-i {\mathrm e}^{7 i x}+i {\mathrm e}^{3 i x}+1}+\frac {28 \sqrt {3}\, \ln \left ({\mathrm e}^{i x}-\frac {i}{2}-\frac {\sqrt {3}}{2}\right )}{9}-\frac {28 \sqrt {3}\, \ln \left ({\mathrm e}^{i x}-\frac {i}{2}+\frac {\sqrt {3}}{2}\right )}{9}+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (1977326743 \textit {\_Z}^{6}-58109194080 \textit {\_Z}^{4}+2339995392 \textit {\_Z}^{2}-2985984\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{i x}-\frac {93499307419 \textit {\_R}^{5}}{3560454144}-\frac {547656095 i \textit {\_R}^{4}}{890113536}+\frac {794954293 \textit {\_R}^{3}}{1030224}+\frac {55720007 i \textit {\_R}^{2}}{3090672}-\frac {7175315 \textit {\_R}}{257556}+\frac {8531 i}{42926}\right )\right )\) | \(194\) |
| default | \(-\frac {1}{9 \left (\tan \left (\frac {x}{2}\right )+1\right )}-\frac {1}{49 \left (\tan \left (\frac {x}{2}\right )-1\right )}+\frac {-\frac {160 \tan \left (\frac {x}{2}\right )^{5}}{49}-\frac {1104 \tan \left (\frac {x}{2}\right )^{4}}{49}-\frac {1152 \tan \left (\frac {x}{2}\right )^{3}}{49}+\frac {480 \tan \left (\frac {x}{2}\right )^{2}}{49}+\frac {544 \tan \left (\frac {x}{2}\right )}{49}+\frac {48}{49}}{\tan \left (\frac {x}{2}\right )^{6}+8 \tan \left (\frac {x}{2}\right )^{5}-13 \tan \left (\frac {x}{2}\right )^{4}-48 \tan \left (\frac {x}{2}\right )^{3}-13 \tan \left (\frac {x}{2}\right )^{2}+8 \tan \left (\frac {x}{2}\right )+1}+\frac {48 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}+8 \textit {\_Z}^{5}-13 \textit {\_Z}^{4}-48 \textit {\_Z}^{3}-13 \textit {\_Z}^{2}+8 \textit {\_Z} +1\right )}{\sum }\frac {\left (\textit {\_R}^{4}+11 \textit {\_R}^{3}+22 \textit {\_R}^{2}+11 \textit {\_R} +1\right ) \ln \left (\tan \left (\frac {x}{2}\right )-\textit {\_R} \right )}{3 \textit {\_R}^{5}+20 \textit {\_R}^{4}-26 \textit {\_R}^{3}-72 \textit {\_R}^{2}-13 \textit {\_R} +4}\right )}{7}-\frac {8 \left (4 \tan \left (\frac {x}{2}\right )-2\right )}{9 \left (\tan \left (\frac {x}{2}\right )^{2}-4 \tan \left (\frac {x}{2}\right )+1\right )}+\frac {56 \sqrt {3}\, \operatorname {arctanh}\left (\frac {\left (2 \tan \left (\frac {x}{2}\right )-4\right ) \sqrt {3}}{6}\right )}{9}\) | \(246\) |
Input:
int(1/(cos(5*x)+sin(2*x))^2,x,method=_RETURNVERBOSE)
Output:
2/21*(8*exp(9*I*x)-5*I*exp(8*I*x)+18*exp(7*I*x)+3*I*exp(6*I*x)-3*exp(5*I*x )-3*I*exp(4*I*x)-3*exp(3*I*x)+5*I*exp(2*I*x)+8*exp(I*x)+21*I)/(exp(10*I*x) -I*exp(7*I*x)+I*exp(3*I*x)+1)+28/9*3^(1/2)*ln(exp(I*x)-1/2*I-1/2*3^(1/2))- 28/9*3^(1/2)*ln(exp(I*x)-1/2*I+1/2*3^(1/2))+sum(_R*ln(exp(I*x)-93499307419 /3560454144*_R^5-547656095/890113536*I*_R^4+794954293/1030224*_R^3+5572000 7/3090672*I*_R^2-7175315/257556*_R+8531/42926*I),_R=RootOf(1977326743*_Z^6 -58109194080*_Z^4+2339995392*_Z^2-2985984))
Result contains complex when optimal does not.
Time = 1.05 (sec) , antiderivative size = 3630, normalized size of antiderivative = 40.33 \[ \int \frac {1}{(\cos (5 x)+\sin (2 x))^2} \, dx=\text {Too large to display} \] Input:
integrate(1/(cos(5*x)+sin(2*x))^2,x, algorithm="fricas")
Output:
Too large to include
\[ \int \frac {1}{(\cos (5 x)+\sin (2 x))^2} \, dx=\int \frac {1}{\left (\sin {\left (2 x \right )} + \cos {\left (5 x \right )}\right )^{2}}\, dx \] Input:
integrate(1/(cos(5*x)+sin(2*x))**2,x)
Output:
Integral((sin(2*x) + cos(5*x))**(-2), x)
Exception generated. \[ \int \frac {1}{(\cos (5 x)+\sin (2 x))^2} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate(1/(cos(5*x)+sin(2*x))^2,x, algorithm="maxima")
Output:
Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is un defined.
Leaf count of result is larger than twice the leaf count of optimal. 238 vs. \(2 (72) = 144\).
Time = 0.14 (sec) , antiderivative size = 238, normalized size of antiderivative = 2.64 \[ \int \frac {1}{(\cos (5 x)+\sin (2 x))^2} \, dx=-\frac {28}{9} \, \sqrt {3} \log \left (\frac {{\left | -2 \, \sqrt {3} + 2 \, \tan \left (\frac {1}{2} \, x\right ) - 4 \right |}}{{\left | 2 \, \sqrt {3} + 2 \, \tan \left (\frac {1}{2} \, x\right ) - 4 \right |}}\right ) - \frac {2 \, {\left (73 \, \tan \left (\frac {1}{2} \, x\right )^{9} + 384 \, \tan \left (\frac {1}{2} \, x\right )^{8} - 1436 \, \tan \left (\frac {1}{2} \, x\right )^{7} - 2724 \, \tan \left (\frac {1}{2} \, x\right )^{6} + 2470 \, \tan \left (\frac {1}{2} \, x\right )^{5} + 3156 \, \tan \left (\frac {1}{2} \, x\right )^{4} - 1212 \, \tan \left (\frac {1}{2} \, x\right )^{3} - 876 \, \tan \left (\frac {1}{2} \, x\right )^{2} + 185 \, \tan \left (\frac {1}{2} \, x\right ) + 28\right )}}{21 \, {\left (\tan \left (\frac {1}{2} \, x\right )^{10} + 4 \, \tan \left (\frac {1}{2} \, x\right )^{9} - 45 \, \tan \left (\frac {1}{2} \, x\right )^{8} + 8 \, \tan \left (\frac {1}{2} \, x\right )^{7} + 210 \, \tan \left (\frac {1}{2} \, x\right )^{6} - 210 \, \tan \left (\frac {1}{2} \, x\right )^{4} - 8 \, \tan \left (\frac {1}{2} \, x\right )^{3} + 45 \, \tan \left (\frac {1}{2} \, x\right )^{2} - 4 \, \tan \left (\frac {1}{2} \, x\right ) - 1\right )}} - 0.0363219532408571 \, \log \left (\tan \left (\frac {1}{2} \, x\right ) + 8.87524547497000\right ) + 0.197492344812286 \, \log \left (\tan \left (\frac {1}{2} \, x\right ) + 1.59149088298000\right ) - 0.197492344812286 \, \log \left (\tan \left (\frac {1}{2} \, x\right ) + 0.628341645367000\right ) + 0.0363219532408571 \, \log \left (\tan \left (\frac {1}{2} \, x\right ) + 0.112672939900000\right ) - 5.41732707074286 \, \log \left (\tan \left (\frac {1}{2} \, x\right ) - 0.349915133947000\right ) + 5.41732707074286 \, \log \left (\tan \left (\frac {1}{2} \, x\right ) - 2.85783580927000\right ) \] Input:
integrate(1/(cos(5*x)+sin(2*x))^2,x, algorithm="giac")
Output:
-28/9*sqrt(3)*log(abs(-2*sqrt(3) + 2*tan(1/2*x) - 4)/abs(2*sqrt(3) + 2*tan (1/2*x) - 4)) - 2/21*(73*tan(1/2*x)^9 + 384*tan(1/2*x)^8 - 1436*tan(1/2*x) ^7 - 2724*tan(1/2*x)^6 + 2470*tan(1/2*x)^5 + 3156*tan(1/2*x)^4 - 1212*tan( 1/2*x)^3 - 876*tan(1/2*x)^2 + 185*tan(1/2*x) + 28)/(tan(1/2*x)^10 + 4*tan( 1/2*x)^9 - 45*tan(1/2*x)^8 + 8*tan(1/2*x)^7 + 210*tan(1/2*x)^6 - 210*tan(1 /2*x)^4 - 8*tan(1/2*x)^3 + 45*tan(1/2*x)^2 - 4*tan(1/2*x) - 1) - 0.0363219 532408571*log(tan(1/2*x) + 8.87524547497000) + 0.197492344812286*log(tan(1 /2*x) + 1.59149088298000) - 0.197492344812286*log(tan(1/2*x) + 0.628341645 367000) + 0.0363219532408571*log(tan(1/2*x) + 0.112672939900000) - 5.41732 707074286*log(tan(1/2*x) - 0.349915133947000) + 5.41732707074286*log(tan(1 /2*x) - 2.85783580927000)
Time = 21.24 (sec) , antiderivative size = 544, normalized size of antiderivative = 6.04 \[ \int \frac {1}{(\cos (5 x)+\sin (2 x))^2} \, dx=\text {Too large to display} \] Input:
int(1/(cos(5*x) + sin(2*x))^2,x)
Output:
(28*3^(1/2)*log(tan(x/2) + 3^(1/2) - 2))/9 + symsum(log((16267598490940865 595310080*root(z^6 - (1440*z^4)/49 + (974592*z^2)/823543 - 2985984/1977326 743, z, k)^2*tan(x/2))/16807 - (415508298574289222762496*tan(x/2))/16807 - (8951559695274229792505856*root(z^6 - (1440*z^4)/49 + (974592*z^2)/823543 - 2985984/1977326743, z, k))/117649 + (44675218075436881973084160*root(z^ 6 - (1440*z^4)/49 + (974592*z^2)/823543 - 2985984/1977326743, z, k)^3*tan( x/2))/2401 - (2003907688503534405287936*root(z^6 - (1440*z^4)/49 + (974592 *z^2)/823543 - 2985984/1977326743, z, k)^4*tan(x/2))/147 - (93995352480177 163599872*root(z^6 - (1440*z^4)/49 + (974592*z^2)/823543 - 2985984/1977326 743, z, k)^5*tan(x/2))/9 + 456834590888836464640*root(z^6 - (1440*z^4)/49 + (974592*z^2)/823543 - 2985984/1977326743, z, k)^6*tan(x/2) + 33550309573 5661887488*root(z^6 - (1440*z^4)/49 + (974592*z^2)/823543 - 2985984/197732 6743, z, k)^7*tan(x/2) - (2430569047972215231873024*root(z^6 - (1440*z^4)/ 49 + (974592*z^2)/823543 - 2985984/1977326743, z, k)^2)/16807 + (596916683 2256521743630336*root(z^6 - (1440*z^4)/49 + (974592*z^2)/823543 - 2985984/ 1977326743, z, k)^3)/2401 + (319229402799190778576896*root(z^6 - (1440*z^4 )/49 + (974592*z^2)/823543 - 2985984/1977326743, z, k)^4)/147 + (230348635 99711839846400*root(z^6 - (1440*z^4)/49 + (974592*z^2)/823543 - 2985984/19 77326743, z, k)^5)/9 - (214614677380201971712*root(z^6 - (1440*z^4)/49 + ( 974592*z^2)/823543 - 2985984/1977326743, z, k)^6)/3 - 90517831318333554...
\[ \int \frac {1}{(\cos (5 x)+\sin (2 x))^2} \, dx=\int \frac {1}{\cos \left (5 x \right )^{2}+2 \cos \left (5 x \right ) \sin \left (2 x \right )+\sin \left (2 x \right )^{2}}d x \] Input:
int(1/(cos(5*x)+sin(2*x))^2,x)
Output:
int(1/(cos(5*x)**2 + 2*cos(5*x)*sin(2*x) + sin(2*x)**2),x)