Integrand size = 11, antiderivative size = 52 \[ \int \frac {1}{(\cos (5 x)+\sin (4 x))^2} \, dx=-\frac {1}{16} \text {arctanh}(2 \cos (x) \sin (x))-\frac {\cot (x)}{32}+\frac {5 \tan (x)}{32}+\frac {\tan ^3(x)}{48}+\frac {\csc (x) \sec ^5(x)}{64 \left (1-\tan ^2(x)\right )} \] Output:
-1/16*arctanh(2*cos(x)*sin(x))-1/32*cot(x)+5/32*tan(x)+1/48*tan(x)^3+csc(x )*sec(x)^5/(64-64*tan(x)^2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.37 (sec) , antiderivative size = 345, normalized size of antiderivative = 6.63 \[ \int \frac {1}{(\cos (5 x)+\sin (4 x))^2} \, dx=\frac {1}{81} \left (8 \sqrt {3} \text {arctanh}\left (\frac {-2+\tan \left (\frac {x}{2}\right )}{\sqrt {3}}\right )+2 i \text {RootSum}\left [i+\text {$\#$1}^3-i \text {$\#$1}^6\&,\frac {22 \arctan \left (\frac {\sin (x)}{\cos (x)-\text {$\#$1}}\right )-11 i \log \left (1-2 \cos (x) \text {$\#$1}+\text {$\#$1}^2\right )-40 i \arctan \left (\frac {\sin (x)}{\cos (x)-\text {$\#$1}}\right ) \text {$\#$1}-20 \log \left (1-2 \cos (x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}-54 \arctan \left (\frac {\sin (x)}{\cos (x)-\text {$\#$1}}\right ) \text {$\#$1}^2+27 i \log \left (1-2 \cos (x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^2+40 i \arctan \left (\frac {\sin (x)}{\cos (x)-\text {$\#$1}}\right ) \text {$\#$1}^3+20 \log \left (1-2 \cos (x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^3+22 \arctan \left (\frac {\sin (x)}{\cos (x)-\text {$\#$1}}\right ) \text {$\#$1}^4-11 i \log \left (1-2 \cos (x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^4}{i \text {$\#$1}^2+2 \text {$\#$1}^5}\&\right ]+\frac {81 \sin \left (\frac {x}{2}\right )}{\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )}+\frac {\sin \left (\frac {x}{2}\right )}{\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )}+\frac {8 \cos (x)}{1-2 \sin (x)}+\frac {24 (4 \cos (x)-3 \cos (3 x)+5 \sin (2 x))}{1+2 \sin (3 x)}\right ) \] Input:
Integrate[(Cos[5*x] + Sin[4*x])^(-2),x]
Output:
(8*Sqrt[3]*ArcTanh[(-2 + Tan[x/2])/Sqrt[3]] + (2*I)*RootSum[I + #1^3 - I*# 1^6 & , (22*ArcTan[Sin[x]/(Cos[x] - #1)] - (11*I)*Log[1 - 2*Cos[x]*#1 + #1 ^2] - (40*I)*ArcTan[Sin[x]/(Cos[x] - #1)]*#1 - 20*Log[1 - 2*Cos[x]*#1 + #1 ^2]*#1 - 54*ArcTan[Sin[x]/(Cos[x] - #1)]*#1^2 + (27*I)*Log[1 - 2*Cos[x]*#1 + #1^2]*#1^2 + (40*I)*ArcTan[Sin[x]/(Cos[x] - #1)]*#1^3 + 20*Log[1 - 2*Co s[x]*#1 + #1^2]*#1^3 + 22*ArcTan[Sin[x]/(Cos[x] - #1)]*#1^4 - (11*I)*Log[1 - 2*Cos[x]*#1 + #1^2]*#1^4)/(I*#1^2 + 2*#1^5) & ] + (81*Sin[x/2])/(Cos[x/ 2] - Sin[x/2]) + Sin[x/2]/(Cos[x/2] + Sin[x/2]) + (8*Cos[x])/(1 - 2*Sin[x] ) + (24*(4*Cos[x] - 3*Cos[3*x] + 5*Sin[2*x]))/(1 + 2*Sin[3*x]))/81
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.
| \(\displaystyle \int \frac {1}{(\sin (4 x)+\cos (5 x))^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(\sin (4 x)+\cos (5 x))^2}dx\) |
| \(\Big \downarrow \) 4830 |
| \(\displaystyle 2 \int \frac {\left (\tan ^2\left (\frac {x}{2}\right )+1\right )^9}{\left (-\tan ^{10}\left (\frac {x}{2}\right )-8 \tan ^9\left (\frac {x}{2}\right )+45 \tan ^8\left (\frac {x}{2}\right )+48 \tan ^7\left (\frac {x}{2}\right )-210 \tan ^6\left (\frac {x}{2}\right )+210 \tan ^4\left (\frac {x}{2}\right )-48 \tan ^3\left (\frac {x}{2}\right )-45 \tan ^2\left (\frac {x}{2}\right )+8 \tan \left (\frac {x}{2}\right )+1\right )^2}d\tan \left (\frac {x}{2}\right )\) |
| \(\Big \downarrow \) 2462 |
| \(\displaystyle 2 \int \left (\frac {16 \tan \left (\frac {x}{2}\right )}{27 \left (\tan ^2\left (\frac {x}{2}\right )-4 \tan \left (\frac {x}{2}\right )+1\right )^2}+\frac {4}{81 \left (\tan ^2\left (\frac {x}{2}\right )-4 \tan \left (\frac {x}{2}\right )+1\right )}+\frac {4 \left (\tan ^4\left (\frac {x}{2}\right )-28 \tan ^3\left (\frac {x}{2}\right )+162 \tan ^2\left (\frac {x}{2}\right )-2140 \tan \left (\frac {x}{2}\right )+24577\right )}{9 \left (\tan ^6\left (\frac {x}{2}\right )+12 \tan ^5\left (\frac {x}{2}\right )+3 \tan ^4\left (\frac {x}{2}\right )-40 \tan ^3\left (\frac {x}{2}\right )+3 \tan ^2\left (\frac {x}{2}\right )+12 \tan \left (\frac {x}{2}\right )+1\right )}+\frac {1}{2 \left (\tan \left (\frac {x}{2}\right )-1\right )^2}+\frac {1}{162 \left (\tan \left (\frac {x}{2}\right )+1\right )^2}-\frac {256 \left (1461 \tan ^5\left (\frac {x}{2}\right )+828 \tan ^4\left (\frac {x}{2}\right )-5142 \tan ^3\left (\frac {x}{2}\right )+252 \tan ^2\left (\frac {x}{2}\right )+1525 \tan \left (\frac {x}{2}\right )+128\right )}{3 \left (\tan ^6\left (\frac {x}{2}\right )+12 \tan ^5\left (\frac {x}{2}\right )+3 \tan ^4\left (\frac {x}{2}\right )-40 \tan ^3\left (\frac {x}{2}\right )+3 \tan ^2\left (\frac {x}{2}\right )+12 \tan \left (\frac {x}{2}\right )+1\right )^2}\right )d\tan \left (\frac {x}{2}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 \left (\frac {2 \log \left (-\tan \left (\frac {x}{2}\right )-\sqrt {3}+2\right )}{27 \sqrt {3}}-\frac {2 \log \left (-\tan \left (\frac {x}{2}\right )+\sqrt {3}+2\right )}{27 \sqrt {3}}+\frac {715264}{3} \int \frac {1}{\left (\tan ^6\left (\frac {x}{2}\right )+12 \tan ^5\left (\frac {x}{2}\right )+3 \tan ^4\left (\frac {x}{2}\right )-40 \tan ^3\left (\frac {x}{2}\right )+3 \tan ^2\left (\frac {x}{2}\right )+12 \tan \left (\frac {x}{2}\right )+1\right )^2}d\tan \left (\frac {x}{2}\right )-\frac {16384}{3} \int \frac {\tan \left (\frac {x}{2}\right )}{\left (\tan ^6\left (\frac {x}{2}\right )+12 \tan ^5\left (\frac {x}{2}\right )+3 \tan ^4\left (\frac {x}{2}\right )-40 \tan ^3\left (\frac {x}{2}\right )+3 \tan ^2\left (\frac {x}{2}\right )+12 \tan \left (\frac {x}{2}\right )+1\right )^2}d\tan \left (\frac {x}{2}\right )-2514944 \int \frac {\tan ^2\left (\frac {x}{2}\right )}{\left (\tan ^6\left (\frac {x}{2}\right )+12 \tan ^5\left (\frac {x}{2}\right )+3 \tan ^4\left (\frac {x}{2}\right )-40 \tan ^3\left (\frac {x}{2}\right )+3 \tan ^2\left (\frac {x}{2}\right )+12 \tan \left (\frac {x}{2}\right )+1\right )^2}d\tan \left (\frac {x}{2}\right )+688128 \int \frac {\tan ^3\left (\frac {x}{2}\right )}{\left (\tan ^6\left (\frac {x}{2}\right )+12 \tan ^5\left (\frac {x}{2}\right )+3 \tan ^4\left (\frac {x}{2}\right )-40 \tan ^3\left (\frac {x}{2}\right )+3 \tan ^2\left (\frac {x}{2}\right )+12 \tan \left (\frac {x}{2}\right )+1\right )^2}d\tan \left (\frac {x}{2}\right )+1176064 \int \frac {\tan ^4\left (\frac {x}{2}\right )}{\left (\tan ^6\left (\frac {x}{2}\right )+12 \tan ^5\left (\frac {x}{2}\right )+3 \tan ^4\left (\frac {x}{2}\right )-40 \tan ^3\left (\frac {x}{2}\right )+3 \tan ^2\left (\frac {x}{2}\right )+12 \tan \left (\frac {x}{2}\right )+1\right )^2}d\tan \left (\frac {x}{2}\right )+\frac {98308}{9} \int \frac {1}{\tan ^6\left (\frac {x}{2}\right )+12 \tan ^5\left (\frac {x}{2}\right )+3 \tan ^4\left (\frac {x}{2}\right )-40 \tan ^3\left (\frac {x}{2}\right )+3 \tan ^2\left (\frac {x}{2}\right )+12 \tan \left (\frac {x}{2}\right )+1}d\tan \left (\frac {x}{2}\right )-\frac {8560}{9} \int \frac {\tan \left (\frac {x}{2}\right )}{\tan ^6\left (\frac {x}{2}\right )+12 \tan ^5\left (\frac {x}{2}\right )+3 \tan ^4\left (\frac {x}{2}\right )-40 \tan ^3\left (\frac {x}{2}\right )+3 \tan ^2\left (\frac {x}{2}\right )+12 \tan \left (\frac {x}{2}\right )+1}d\tan \left (\frac {x}{2}\right )+72 \int \frac {\tan ^2\left (\frac {x}{2}\right )}{\tan ^6\left (\frac {x}{2}\right )+12 \tan ^5\left (\frac {x}{2}\right )+3 \tan ^4\left (\frac {x}{2}\right )-40 \tan ^3\left (\frac {x}{2}\right )+3 \tan ^2\left (\frac {x}{2}\right )+12 \tan \left (\frac {x}{2}\right )+1}d\tan \left (\frac {x}{2}\right )-\frac {112}{9} \int \frac {\tan ^3\left (\frac {x}{2}\right )}{\tan ^6\left (\frac {x}{2}\right )+12 \tan ^5\left (\frac {x}{2}\right )+3 \tan ^4\left (\frac {x}{2}\right )-40 \tan ^3\left (\frac {x}{2}\right )+3 \tan ^2\left (\frac {x}{2}\right )+12 \tan \left (\frac {x}{2}\right )+1}d\tan \left (\frac {x}{2}\right )+\frac {4}{9} \int \frac {\tan ^4\left (\frac {x}{2}\right )}{\tan ^6\left (\frac {x}{2}\right )+12 \tan ^5\left (\frac {x}{2}\right )+3 \tan ^4\left (\frac {x}{2}\right )-40 \tan ^3\left (\frac {x}{2}\right )+3 \tan ^2\left (\frac {x}{2}\right )+12 \tan \left (\frac {x}{2}\right )+1}d\tan \left (\frac {x}{2}\right )+\frac {1}{2 \left (1-\tan \left (\frac {x}{2}\right )\right )}-\frac {1}{162 \left (\tan \left (\frac {x}{2}\right )+1\right )}+\frac {8 \left (1-2 \tan \left (\frac {x}{2}\right )\right )}{81 \left (\tan ^2\left (\frac {x}{2}\right )-4 \tan \left (\frac {x}{2}\right )+1\right )}+\frac {62336}{3 \left (\tan ^6\left (\frac {x}{2}\right )+12 \tan ^5\left (\frac {x}{2}\right )+3 \tan ^4\left (\frac {x}{2}\right )-40 \tan ^3\left (\frac {x}{2}\right )+3 \tan ^2\left (\frac {x}{2}\right )+12 \tan \left (\frac {x}{2}\right )+1\right )}\right )\) |
Input:
Int[(Cos[5*x] + Sin[4*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 = 2.47 (sec) , antiderivative size = 190, normalized size of antiderivative = 3.65
| method | result | size |
| risch | \(\frac {\frac {16 \,{\mathrm e}^{9 i x}}{9}-\frac {2 i {\mathrm e}^{8 i x}}{9}+\frac {2 \,{\mathrm e}^{7 i x}}{9}+\frac {2 i {\mathrm e}^{6 i x}}{9}-\frac {2 \,{\mathrm e}^{5 i x}}{9}-\frac {2 i {\mathrm e}^{4 i x}}{9}+\frac {2 \,{\mathrm e}^{3 i x}}{9}+\frac {2 i {\mathrm e}^{2 i x}}{9}-\frac {2 \,{\mathrm e}^{i x}}{9}+2 i}{-i {\mathrm e}^{9 i x}+{\mathrm e}^{10 i x}+i {\mathrm e}^{i x}+1}+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (94143178827 \textit {\_Z}^{6}-507606934032 \textit {\_Z}^{4}+171320832 \textit {\_Z}^{2}-4096\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{i x}-\frac {5965888110111 \textit {\_R}^{5}}{130316288}+\frac {12526595811 i \textit {\_R}^{4}}{32579072}+\frac {2010526865001 \textit {\_R}^{3}}{8144768}-\frac {2110877091 i \textit {\_R}^{2}}{1018096}-\frac {33777405 \textit {\_R}}{254524}+\frac {104321 i}{127262}\right )\right )+\frac {4 \sqrt {3}\, \ln \left ({\mathrm e}^{i x}-\frac {i}{2}-\frac {\sqrt {3}}{2}\right )}{81}-\frac {4 \sqrt {3}\, \ln \left ({\mathrm e}^{i x}-\frac {i}{2}+\frac {\sqrt {3}}{2}\right )}{81}\) | \(190\) |
| default | \(\frac {-\frac {64 \tan \left (\frac {x}{2}\right )^{5}}{27}-\frac {368 \tan \left (\frac {x}{2}\right )^{4}}{27}-\frac {320 \tan \left (\frac {x}{2}\right )^{3}}{27}+\frac {416 \tan \left (\frac {x}{2}\right )^{2}}{27}+\frac {256 \tan \left (\frac {x}{2}\right )}{27}+\frac {16}{27}}{\tan \left (\frac {x}{2}\right )^{6}+12 \tan \left (\frac {x}{2}\right )^{5}+3 \tan \left (\frac {x}{2}\right )^{4}-40 \tan \left (\frac {x}{2}\right )^{3}+3 \tan \left (\frac {x}{2}\right )^{2}+12 \tan \left (\frac {x}{2}\right )+1}+\frac {4 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}+12 \textit {\_Z}^{5}+3 \textit {\_Z}^{4}-40 \textit {\_Z}^{3}+3 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )}{\sum }\frac {\left (-5 \textit {\_R}^{4}-80 \textit {\_R}^{3}-186 \textit {\_R}^{2}-80 \textit {\_R} -5\right ) \ln \left (\tan \left (\frac {x}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5}+10 \textit {\_R}^{4}+2 \textit {\_R}^{3}-20 \textit {\_R}^{2}+\textit {\_R} +2}\right )}{81}+\frac {-\frac {32 \tan \left (\frac {x}{2}\right )}{81}+\frac {16}{81}}{\tan \left (\frac {x}{2}\right )^{2}-4 \tan \left (\frac {x}{2}\right )+1}+\frac {8 \sqrt {3}\, \operatorname {arctanh}\left (\frac {\left (2 \tan \left (\frac {x}{2}\right )-4\right ) \sqrt {3}}{6}\right )}{81}-\frac {1}{81 \left (\tan \left (\frac {x}{2}\right )+1\right )}-\frac {1}{\tan \left (\frac {x}{2}\right )-1}\) | \(244\) |
Input:
int(1/(cos(5*x)+sin(4*x))^2,x,method=_RETURNVERBOSE)
Output:
2/9*(8*exp(9*I*x)-I*exp(8*I*x)+exp(7*I*x)+I*exp(6*I*x)-exp(5*I*x)-I*exp(4* I*x)+exp(3*I*x)+I*exp(2*I*x)-exp(I*x)+9*I)/(-I*exp(9*I*x)+exp(10*I*x)+I*ex p(I*x)+1)+sum(_R*ln(exp(I*x)-5965888110111/130316288*_R^5+12526595811/3257 9072*I*_R^4+2010526865001/8144768*_R^3-2110877091/1018096*I*_R^2-33777405/ 254524*_R+104321/127262*I),_R=RootOf(94143178827*_Z^6-507606934032*_Z^4+17 1320832*_Z^2-4096))+4/81*3^(1/2)*ln(exp(I*x)-1/2*I-1/2*3^(1/2))-4/81*3^(1/ 2)*ln(exp(I*x)-1/2*I+1/2*3^(1/2))
Result contains complex when optimal does not.
Time = 1.00 (sec) , antiderivative size = 3679, normalized size of antiderivative = 70.75 \[ \int \frac {1}{(\cos (5 x)+\sin (4 x))^2} \, dx=\text {Too large to display} \] Input:
integrate(1/(cos(5*x)+sin(4*x))^2,x, algorithm="fricas")
Output:
Too large to include
\[ \int \frac {1}{(\cos (5 x)+\sin (4 x))^2} \, dx=\int \frac {1}{\left (\sin {\left (4 x \right )} + \cos {\left (5 x \right )}\right )^{2}}\, dx \] Input:
integrate(1/(cos(5*x)+sin(4*x))**2,x)
Output:
Integral((sin(4*x) + cos(5*x))**(-2), x)
Exception generated. \[ \int \frac {1}{(\cos (5 x)+\sin (4 x))^2} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate(1/(cos(5*x)+sin(4*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. \(2 (40) = 80\).
Time = 0.15 (sec) , antiderivative size = 238, normalized size of antiderivative = 4.58 \[ \int \frac {1}{(\cos (5 x)+\sin (4 x))^2} \, dx=-\frac {4}{81} \, \sqrt {3} \log \left (\frac {{\left | -2 \, \sqrt {3} + 2 \, \tan \left (\frac {1}{2} \, x\right ) - 4 \right |}}{{\left | 2 \, \sqrt {3} + 2 \, \tan \left (\frac {1}{2} \, x\right ) - 4 \right |}}\right ) - \frac {2 \, {\left (17 \, \tan \left (\frac {1}{2} \, x\right )^{9} + 80 \, \tan \left (\frac {1}{2} \, x\right )^{8} - 364 \, \tan \left (\frac {1}{2} \, x\right )^{7} - 712 \, \tan \left (\frac {1}{2} \, x\right )^{6} + 1094 \, \tan \left (\frac {1}{2} \, x\right )^{5} + 968 \, \tan \left (\frac {1}{2} \, x\right )^{4} - 748 \, \tan \left (\frac {1}{2} \, x\right )^{3} - 280 \, \tan \left (\frac {1}{2} \, x\right )^{2} + 81 \, \tan \left (\frac {1}{2} \, x\right ) + 8\right )}}{9 \, {\left (\tan \left (\frac {1}{2} \, x\right )^{10} + 8 \, \tan \left (\frac {1}{2} \, x\right )^{9} - 45 \, \tan \left (\frac {1}{2} \, x\right )^{8} - 48 \, \tan \left (\frac {1}{2} \, x\right )^{7} + 210 \, \tan \left (\frac {1}{2} \, x\right )^{6} - 210 \, \tan \left (\frac {1}{2} \, x\right )^{4} + 48 \, \tan \left (\frac {1}{2} \, x\right )^{3} + 45 \, \tan \left (\frac {1}{2} \, x\right )^{2} - 8 \, \tan \left (\frac {1}{2} \, x\right ) - 1\right )}} - 0.0176530836122963 \, \log \left (\tan \left (\frac {1}{2} \, x\right ) + 11.4300523028000\right ) - 0.00508872774952593 \, \log \left (\tan \left (\frac {1}{2} \, x\right ) + 2.14450692051000\right ) + 0.00508872774952593 \, \log \left (\tan \left (\frac {1}{2} \, x\right ) + 0.466307658155000\right ) + 0.0176530836122963 \, \log \left (\tan \left (\frac {1}{2} \, x\right ) + 0.0874886635259000\right ) + 2.32196543262222 \, \log \left (\tan \left (\frac {1}{2} \, x\right ) - 0.700207538210000\right ) - 2.32196543262222 \, \log \left (\tan \left (\frac {1}{2} \, x\right ) - 1.42814800674000\right ) \] Input:
integrate(1/(cos(5*x)+sin(4*x))^2,x, algorithm="giac")
Output:
-4/81*sqrt(3)*log(abs(-2*sqrt(3) + 2*tan(1/2*x) - 4)/abs(2*sqrt(3) + 2*tan (1/2*x) - 4)) - 2/9*(17*tan(1/2*x)^9 + 80*tan(1/2*x)^8 - 364*tan(1/2*x)^7 - 712*tan(1/2*x)^6 + 1094*tan(1/2*x)^5 + 968*tan(1/2*x)^4 - 748*tan(1/2*x) ^3 - 280*tan(1/2*x)^2 + 81*tan(1/2*x) + 8)/(tan(1/2*x)^10 + 8*tan(1/2*x)^9 - 45*tan(1/2*x)^8 - 48*tan(1/2*x)^7 + 210*tan(1/2*x)^6 - 210*tan(1/2*x)^4 + 48*tan(1/2*x)^3 + 45*tan(1/2*x)^2 - 8*tan(1/2*x) - 1) - 0.0176530836122 963*log(tan(1/2*x) + 11.4300523028000) - 0.00508872774952593*log(tan(1/2*x ) + 2.14450692051000) + 0.00508872774952593*log(tan(1/2*x) + 0.46630765815 5000) + 0.0176530836122963*log(tan(1/2*x) + 0.0874886635259000) + 2.321965 43262222*log(tan(1/2*x) - 0.700207538210000) - 2.32196543262222*log(tan(1/ 2*x) - 1.42814800674000)
Time = 22.90 (sec) , antiderivative size = 544, normalized size of antiderivative = 10.46 \[ \int \frac {1}{(\cos (5 x)+\sin (4 x))^2} \, dx=\text {Too large to display} \] Input:
int(1/(cos(5*x) + sin(4*x))^2,x)
Output:
(4*3^(1/2)*log(tan(x/2) + 3^(1/2) - 2))/81 + symsum(log((33577757957763849 25696*root(z^6 - (11792*z^4)/2187 + (8704*z^2)/4782969 - 4096/94143178827, z, k)^2*tan(x/2))/2187 - (25364273101350633472*tan(x/2))/531441 - (532577 677534325374976*root(z^6 - (11792*z^4)/2187 + (8704*z^2)/4782969 - 4096/94 143178827, z, k))/531441 + (6204370515848081702912*root(z^6 - (11792*z^4)/ 2187 + (8704*z^2)/4782969 - 4096/94143178827, z, k)^3*tan(x/2))/81 - 11417 12045134321811456*root(z^6 - (11792*z^4)/2187 + (8704*z^2)/4782969 - 4096/ 94143178827, z, k)^4*tan(x/2) - 21245848617185603223552*root(z^6 - (11792* z^4)/2187 + (8704*z^2)/4782969 - 4096/94143178827, z, k)^5*tan(x/2) + 1072 966004197394743296*root(z^6 - (11792*z^4)/2187 + (8704*z^2)/4782969 - 4096 /94143178827, z, k)^6*tan(x/2) + 4319334531957283356672*root(z^6 - (11792* z^4)/2187 + (8704*z^2)/4782969 - 4096/94143178827, z, k)^7*tan(x/2) + (185 737455832013996032*root(z^6 - (11792*z^4)/2187 + (8704*z^2)/4782969 - 4096 /94143178827, z, k)^2)/6561 + (625473427048096595968*root(z^6 - (11792*z^4 )/2187 + (8704*z^2)/4782969 - 4096/94143178827, z, k)^3)/81 + 248732118569 812230144*root(z^6 - (11792*z^4)/2187 + (8704*z^2)/4782969 - 4096/94143178 827, z, k)^4 + 1865156497001259466752*root(z^6 - (11792*z^4)/2187 + (8704* z^2)/4782969 - 4096/94143178827, z, k)^5 - 640363101472369410048*root(z^6 - (11792*z^4)/2187 + (8704*z^2)/4782969 - 4096/94143178827, z, k)^6 - 6170 47790279611908096*root(z^6 - (11792*z^4)/2187 + (8704*z^2)/4782969 - 40...
\[ \int \frac {1}{(\cos (5 x)+\sin (4 x))^2} \, dx=\int \frac {1}{\cos \left (5 x \right )^{2}+2 \cos \left (5 x \right ) \sin \left (4 x \right )+\sin \left (4 x \right )^{2}}d x \] Input:
int(1/(cos(5*x)+sin(4*x))^2,x)
Output:
int(1/(cos(5*x)**2 + 2*cos(5*x)*sin(4*x) + sin(4*x)**2),x)