\(\int \frac {\sin (100 x)-\sin (101 x)}{\cos (100 x)+\cos (101 x)} \, dx\) [428]

Optimal result

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

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

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 9, normalized size of antiderivative = 1.00 \[ \int \frac {\sin (100 x)-\sin (101 x)}{\cos (100 x)+\cos (101 x)} \, dx=2 \log \left (\cos \left (\frac {x}{2}\right )\right ) \] Input:

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

Output:

2*Log[Cos[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 = 0.78 (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	\(-\ln \left (\sec \left (50 x \right )^{2}\right )-\ln \left (\sec \left (\frac {101 x}{2}\right )^{2}\right )+2 \ln \left (\tan \left (50 x \right ) \tan \left (\frac {101 x}{2}\right )+1\right )\)	\(34\)
default	\(-\ln \left (1+\tan \left (50 x \right )^{2}\right )-\ln \left (1+\tan \left (\frac {101 x}{2}\right )^{2}\right )+2 \ln \left (\tan \left (50 x \right ) \tan \left (\frac {101 x}{2}\right )+1\right )\)	\(38\)

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 = 21.13 (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]

\[ \int \frac {\sin (100 x)-\sin (101 x)}{\cos (100 x)+\cos (101 x)} \, dx=\int { -\frac {\sin \left (101 \, x\right ) - \sin \left (100 \, x\right )}{\cos \left (101 \, x\right ) + \cos \left (100 \, x\right )} \,d x } \] 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 = 0.11 (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 [F]

\[ \int \frac {\sin (100 x)-\sin (101 x)}{\cos (100 x)+\cos (101 x)} \, dx=\frac {201 \left (\int \frac {\sin \left (100 x \right )}{\cos \left (101 x \right )+\cos \left (100 x \right )}d x \right )}{101}+\frac {\mathrm {log}\left (\cos \left (101 x \right )+\cos \left (100 x \right )\right )}{101} \] Input:

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

Output:

(201*int(sin(100*x)/(cos(101*x) + cos(100*x)),x) + log(cos(101*x) + cos(10 
0*x)))/101