\(\int \frac {\sin (100 x)+\sin (101 x)}{\cos (100 x)-\cos (101 x)} \, dx\) [427]

Optimal result

Integrand size = 23, antiderivative size = 9 \[ \int \frac {\sin (100 x)+\sin (101 x)}{\cos (100 x)-\cos (101 x)} \, dx=2 \log \left (\sin \left (\frac {x}{2}\right )\right ) \] Output:

2*ln(sin(1/2*x))
 

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 9, normalized size of antiderivative = 1.00 \[ \int \frac {\sin (100 x)+\sin (101 x)}{\cos (100 x)-\cos (101 x)} \, dx=2 \log \left (\sin \left (\frac {x}{2}\right )\right ) \] Input:

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

Output:

2*Log[Sin[x/2]]
 

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 = 1.01 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.78

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

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.78 \[ \int \frac {\sin (100 x)+\sin (101 x)}{\cos (100 x)-\cos (101 x)} \, dx=\log \left (-\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) \] Input:

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

Output:

log(-1/2*cos(x) + 1/2)
 

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. 15 vs. \(2 (7) = 14\).

Time = 18.97 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.67 \[ \int \frac {\sin (100 x)+\sin (101 x)}{\cos (100 x)-\cos (101 x)} \, dx=\log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} - 2 \, \cos \left (x\right ) + 1\right ) \] Input:

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

Output:

log(cos(x)^2 + sin(x)^2 - 2*cos(x) + 1)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {\sin (100 x)+\sin (101 x)}{\cos (100 x)-\cos (101 x)} \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro 
unding error%%%{540174450688*i,[1,0,16]%%%}+%%%{1080348901376*i,[1,0,14]%% 
%}+%%%{94
 

Mupad [B] (verification not implemented)

Time = 15.72 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.67 \[ \int \frac {\sin (100 x)+\sin (101 x)}{\cos (100 x)-\cos (101 x)} \, dx=-x\,1{}\mathrm {i}+2\,\ln \left ({\mathrm {e}}^{x\,1{}\mathrm {i}}-1\right ) \] Input:

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

Output:

2*log(exp(x*1i) - 1) - x*1i
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 67, normalized size of antiderivative = 7.44 \[ \int \frac {\sin (100 x)+\sin (101 x)}{\cos (100 x)-\cos (101 x)} \, dx=\frac {\mathrm {log}\left (\cos \left (101 x \right )-\cos \left (100 x \right )\right )}{101}-\frac {100 \,\mathrm {log}\left (\tan \left (\frac {101 x}{2}\right )^{2}+1\right )}{101}-\frac {100 \,\mathrm {log}\left (\tan \left (50 x \right )^{2}+1\right )}{101}-\frac {\mathrm {log}\left (-\tan \left (\frac {101 x}{2}\right )-\tan \left (50 x \right )\right )}{101}+\frac {201 \,\mathrm {log}\left (\tan \left (\frac {101 x}{2}\right )-\tan \left (50 x \right )\right )}{101} \] Input:

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

Output:

(log(cos(101*x) - cos(100*x)) - 100*log(tan((101*x)/2)**2 + 1) - 100*log(t 
an(50*x)**2 + 1) - log( - tan((101*x)/2) - tan(50*x)) + 201*log(tan((101*x 
)/2) - tan(50*x)))/101