\(\int \frac {\sin (100 x)+\sin (101 x)}{\cos (100 x)+\cos (101 x)} \, dx\) [430]

Optimal result

Integrand size = 21, antiderivative size = 11 \[ \int \frac {\sin (100 x)+\sin (101 x)}{\cos (100 x)+\cos (101 x)} \, dx=-\frac {2}{201} \log \left (\cos \left (\frac {201 x}{2}\right )\right ) \] Output:

-2/201*ln(cos(201/2*x))
 

Mathematica [F(-1)]

Timed out. \[ \int \frac {\sin (100 x)+\sin (101 x)}{\cos (100 x)+\cos (101 x)} \, dx=\text {\$Aborted} \] Input:

Integrate[(Sin[100*x] + Sin[101*x])/(Cos[100*x] + Cos[101*x]),x]
 

Output:

$Aborted
 

Rubi [F]

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.

\(\displaystyle \int \frac {\sin (100 x)+\sin (101 x)}{\cos (100 x)+\cos (101 x)} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\sin (100 x)+\sin (101 x)}{\cos (100 x)+\cos (101 x)}dx\)

\(\Big \downarrow \) 4901

\(\displaystyle \int \left (\frac {\sin (100 x)}{\cos (100 x)+\cos (101 x)}+\frac {\sin (101 x)}{\cos (100 x)+\cos (101 x)}\right )dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \left (\frac {\sin (100 x)}{\cos (100 x)+\cos (101 x)}+\frac {\sin (101 x)}{\cos (100 x)+\cos (101 x)}\right )dx\)

\(\Big \downarrow \) 4901

\(\displaystyle \int \left (\frac {\sin (100 x)}{\cos (100 x)+\cos (101 x)}+\frac {\sin (101 x)}{\cos (100 x)+\cos (101 x)}\right )dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \left (\frac {\sin (100 x)}{\cos (100 x)+\cos (101 x)}+\frac {\sin (101 x)}{\cos (100 x)+\cos (101 x)}\right )dx\)

\(\Big \downarrow \) 4901

\(\displaystyle \int \left (\frac {\sin (100 x)}{\cos (100 x)+\cos (101 x)}+\frac {\sin (101 x)}{\cos (100 x)+\cos (101 x)}\right )dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \left (\frac {\sin (100 x)}{\cos (100 x)+\cos (101 x)}+\frac {\sin (101 x)}{\cos (100 x)+\cos (101 x)}\right )dx\)

\(\Big \downarrow \) 4901

\(\displaystyle \int \left (\frac {\sin (100 x)}{\cos (100 x)+\cos (101 x)}+\frac {\sin (101 x)}{\cos (100 x)+\cos (101 x)}\right )dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \left (\frac {\sin (100 x)}{\cos (100 x)+\cos (101 x)}+\frac {\sin (101 x)}{\cos (100 x)+\cos (101 x)}\right )dx\)

\(\Big \downarrow \) 4901

\(\displaystyle \int \left (\frac {\sin (100 x)}{\cos (100 x)+\cos (101 x)}+\frac {\sin (101 x)}{\cos (100 x)+\cos (101 x)}\right )dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \left (\frac {\sin (100 x)}{\cos (100 x)+\cos (101 x)}+\frac {\sin (101 x)}{\cos (100 x)+\cos (101 x)}\right )dx\)

\(\Big \downarrow \) 4901

\(\displaystyle \int \left (\frac {\sin (100 x)}{\cos (100 x)+\cos (101 x)}+\frac {\sin (101 x)}{\cos (100 x)+\cos (101 x)}\right )dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \left (\frac {\sin (100 x)}{\cos (100 x)+\cos (101 x)}+\frac {\sin (101 x)}{\cos (100 x)+\cos (101 x)}\right )dx\)

\(\Big \downarrow \) 4901

\(\displaystyle \int \left (\frac {\sin (100 x)}{\cos (100 x)+\cos (101 x)}+\frac {\sin (101 x)}{\cos (100 x)+\cos (101 x)}\right )dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \left (\frac {\sin (100 x)}{\cos (100 x)+\cos (101 x)}+\frac {\sin (101 x)}{\cos (100 x)+\cos (101 x)}\right )dx\)

\(\Big \downarrow \) 4901

\(\displaystyle \int \left (\frac {\sin (100 x)}{\cos (100 x)+\cos (101 x)}+\frac {\sin (101 x)}{\cos (100 x)+\cos (101 x)}\right )dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \left (\frac {\sin (100 x)}{\cos (100 x)+\cos (101 x)}+\frac {\sin (101 x)}{\cos (100 x)+\cos (101 x)}\right )dx\)

\(\Big \downarrow \) 4901

\(\displaystyle \int \left (\frac {\sin (100 x)}{\cos (100 x)+\cos (101 x)}+\frac {\sin (101 x)}{\cos (100 x)+\cos (101 x)}\right )dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \left (\frac {\sin (100 x)}{\cos (100 x)+\cos (101 x)}+\frac {\sin (101 x)}{\cos (100 x)+\cos (101 x)}\right )dx\)

\(\Big \downarrow \) 4901

\(\displaystyle \int \left (\frac {\sin (100 x)}{\cos (100 x)+\cos (101 x)}+\frac {\sin (101 x)}{\cos (100 x)+\cos (101 x)}\right )dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \left (\frac {\sin (100 x)}{\cos (100 x)+\cos (101 x)}+\frac {\sin (101 x)}{\cos (100 x)+\cos (101 x)}\right )dx\)

\(\Big \downarrow \) 4901

\(\displaystyle \int \left (\frac {\sin (100 x)}{\cos (100 x)+\cos (101 x)}+\frac {\sin (101 x)}{\cos (100 x)+\cos (101 x)}\right )dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \left (\frac {\sin (100 x)}{\cos (100 x)+\cos (101 x)}+\frac {\sin (101 x)}{\cos (100 x)+\cos (101 x)}\right )dx\)

\(\Big \downarrow \) 4901

\(\displaystyle \int \left (\frac {\sin (100 x)}{\cos (100 x)+\cos (101 x)}+\frac {\sin (101 x)}{\cos (100 x)+\cos (101 x)}\right )dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \left (\frac {\sin (100 x)}{\cos (100 x)+\cos (101 x)}+\frac {\sin (101 x)}{\cos (100 x)+\cos (101 x)}\right )dx\)

\(\Big \downarrow \) 4901

\(\displaystyle \int \left (\frac {\sin (100 x)}{\cos (100 x)+\cos (101 x)}+\frac {\sin (101 x)}{\cos (100 x)+\cos (101 x)}\right )dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \left (\frac {\sin (100 x)}{\cos (100 x)+\cos (101 x)}+\frac {\sin (101 x)}{\cos (100 x)+\cos (101 x)}\right )dx\)

\(\Big \downarrow \) 4901

\(\displaystyle \int \left (\frac {\sin (100 x)}{\cos (100 x)+\cos (101 x)}+\frac {\sin (101 x)}{\cos (100 x)+\cos (101 x)}\right )dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \left (\frac {\sin (100 x)}{\cos (100 x)+\cos (101 x)}+\frac {\sin (101 x)}{\cos (100 x)+\cos (101 x)}\right )dx\)

\(\Big \downarrow \) 4901

\(\displaystyle \int \left (\frac {\sin (100 x)}{\cos (100 x)+\cos (101 x)}+\frac {\sin (101 x)}{\cos (100 x)+\cos (101 x)}\right )dx\)

Input:

Int[(Sin[100*x] + Sin[101*x])/(Cos[100*x] + Cos[101*x]),x]
 

Output:

$Aborted
 

Defintions of rubi rules used

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4901

Int[u_, x_Symbol] :> With[{v = ExpandTrig[u, x]}, Int[v, x] /; SumQ[v]] /; 
 !InertTrigFreeQ[u]
 

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.39 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.45

method	result	size
risch	\(i x -\frac {2 \ln \left ({\mathrm e}^{201 i x}+1\right )}{201}\)	\(16\)
parallelrisch	\(\ln \left (\left (\sec \left (50 x \right )^{2}\right )^{\frac {1}{201}}\right )+\ln \left (\left (\sec \left (\frac {101 x}{2}\right )^{2}\right )^{\frac {1}{201}}\right )+\ln \left (\frac {1}{\left (\tan \left (50 x \right ) \tan \left (\frac {101 x}{2}\right )-1\right )^{\frac {2}{201}}}\right )\)	\(34\)
default	\(\frac {\ln \left (1+\tan \left (50 x \right )^{2}\right )}{201}+\frac {\ln \left (1+\tan \left (\frac {101 x}{2}\right )^{2}\right )}{201}-\frac {2 \ln \left (\tan \left (50 x \right ) \tan \left (\frac {101 x}{2}\right )-1\right )}{201}\)	\(38\)

Input:

int((sin(100*x)+sin(101*x))/(cos(100*x)+cos(101*x)),x,method=_RETURNVERBOS 
E)
 

Output:

I*x-2/201*ln(exp(201*I*x)+1)
 

Fricas [F(-1)]

Timed out. \[ \int \frac {\sin (100 x)+\sin (101 x)}{\cos (100 x)+\cos (101 x)} \, dx=\text {Timed out} \] Input:

integrate((sin(100*x)+sin(101*x))/(cos(100*x)+cos(101*x)),x, algorithm="fr 
icas")
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\sin (100 x)+\sin (101 x)}{\cos (100 x)+\cos (101 x)} \, dx=\text {Timed out} \] Input:

integrate((sin(100*x)+sin(101*x))/(cos(100*x)+cos(101*x)),x)
                                                                                    
                                                                                    
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 63169 vs. \(2 (7) = 14\).

Time = 23.47 (sec) , antiderivative size = 63169, normalized size of antiderivative = 5742.64 \[ \int \frac {\sin (100 x)+\sin (101 x)}{\cos (100 x)+\cos (101 x)} \, dx=\text {Too large to display} \] Input:

integrate((sin(100*x)+sin(101*x))/(cos(100*x)+cos(101*x)),x, algorithm="ma 
xima")
 

Output:

-1/201*log(2*(cos(131*x) - cos(129*x) - cos(128*x) + cos(126*x) + cos(125* 
x) - cos(123*x) - cos(122*x) + cos(120*x) + cos(119*x) - cos(117*x) - cos( 
116*x) + cos(114*x) + cos(113*x) - cos(111*x) - cos(110*x) + cos(108*x) + 
cos(107*x) - cos(105*x) - cos(104*x) + cos(102*x) + cos(101*x) - cos(99*x) 
 - cos(98*x) + cos(96*x) + cos(95*x) - cos(93*x) - cos(92*x) + cos(90*x) + 
 cos(89*x) - cos(87*x) - cos(86*x) + cos(84*x) + cos(83*x) - cos(81*x) - c 
os(80*x) + cos(78*x) + cos(77*x) - cos(75*x) - cos(74*x) + cos(72*x) + cos 
(71*x) - cos(69*x) - cos(68*x) + cos(66*x) - cos(64*x) - cos(63*x) + cos(6 
1*x) + cos(60*x) - cos(58*x) - cos(57*x) + cos(55*x) + cos(54*x) - cos(52* 
x) - cos(51*x) + cos(49*x) + cos(48*x) - cos(46*x) - cos(45*x) + cos(43*x) 
 + cos(42*x) - cos(40*x) - cos(39*x) + cos(37*x) + cos(36*x) - cos(34*x) - 
 cos(33*x) + cos(31*x) + cos(30*x) - cos(28*x) - cos(27*x) + cos(25*x) + c 
os(24*x) - cos(22*x) - cos(21*x) + cos(19*x) + cos(18*x) - cos(16*x) - cos 
(15*x) + cos(13*x) + cos(12*x) - cos(10*x) - cos(9*x) + cos(7*x) + cos(6*x 
) - cos(4*x) - cos(3*x) + cos(x) + 1)*cos(132*x) + cos(132*x)^2 - 2*(cos(1 
29*x) + cos(128*x) - cos(126*x) - cos(125*x) + cos(123*x) + cos(122*x) - c 
os(120*x) - cos(119*x) + cos(117*x) + cos(116*x) - cos(114*x) - cos(113*x) 
 + cos(111*x) + cos(110*x) - cos(108*x) - cos(107*x) + cos(105*x) + cos(10 
4*x) - cos(102*x) - cos(101*x) + cos(99*x) + cos(98*x) - cos(96*x) - cos(9 
5*x) + cos(93*x) + cos(92*x) - cos(90*x) - cos(89*x) + cos(87*x) + cos(...
 

Giac [F]

\[ \int \frac {\sin (100 x)+\sin (101 x)}{\cos (100 x)+\cos (101 x)} \, dx=\int { \frac {\sin \left (101 \, x\right ) + \sin \left (100 \, x\right )}{\cos \left (101 \, x\right ) + \cos \left (100 \, x\right )} \,d x } \] Input:

integrate((sin(100*x)+sin(101*x))/(cos(100*x)+cos(101*x)),x, algorithm="gi 
ac")
 

Output:

integrate((sin(101*x) + sin(100*x))/(cos(101*x) + cos(100*x)), x)
 

Mupad [B] (verification not implemented)

Time = 57.00 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.36 \[ \int \frac {\sin (100 x)+\sin (101 x)}{\cos (100 x)+\cos (101 x)} \, dx=x\,1{}\mathrm {i}-\frac {2\,\ln \left ({\mathrm {e}}^{x\,201{}\mathrm {i}}+1\right )}{201} \] Input:

int((sin(100*x) + sin(101*x))/(cos(100*x) + cos(101*x)),x)
 

Output:

x*1i - (2*log(exp(x*201i) + 1))/201
 

Reduce [F]

\[ \int \frac {\sin (100 x)+\sin (101 x)}{\cos (100 x)+\cos (101 x)} \, dx=\frac {\left (\int \frac {\sin \left (100 x \right )}{\cos \left (101 x \right )+\cos \left (100 x \right )}d x \right )}{101}-\frac {\mathrm {log}\left (\cos \left (101 x \right )+\cos \left (100 x \right )\right )}{101} \] Input:

int((sin(100*x)+sin(101*x))/(cos(100*x)+cos(101*x)),x)
 

Output:

(int(sin(100*x)/(cos(101*x) + cos(100*x)),x) - log(cos(101*x) + cos(100*x) 
))/101