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