\(\int \frac {\sinh (100 x)+\sinh (101 x)}{\cosh (100 x)-\cosh (101 x)} \, dx\) [435]

Optimal result

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

-2*ln(sinh(1/2*x))
 

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 9, normalized size of antiderivative = 1.00 \[ \int \frac {\sinh (100 x)+\sinh (101 x)}{\cosh (100 x)-\cosh (101 x)} \, dx=-2 \log \left (\sinh \left (\frac {x}{2}\right )\right ) \] Input:

Integrate[(Sinh[100*x] + Sinh[101*x])/(Cosh[100*x] - Cosh[101*x]),x]
 

Output:

-2*Log[Sinh[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.

\(\displaystyle \int \frac {\sinh (100 x)+\sinh (101 x)}{\cosh (100 x)-\cosh (101 x)} \, dx\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4901

\(\displaystyle \int \left (\frac {\sinh (100 x)}{\cosh (100 x)-\cosh (101 x)}+\frac {\sinh (101 x)}{\cosh (100 x)-\cosh (101 x)}\right )dx\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4901

\(\displaystyle \int \left (\frac {\sinh (100 x)}{\cosh (100 x)-\cosh (101 x)}+\frac {\sinh (101 x)}{\cosh (100 x)-\cosh (101 x)}\right )dx\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4901

\(\displaystyle \int \left (\frac {\sinh (100 x)}{\cosh (100 x)-\cosh (101 x)}+\frac {\sinh (101 x)}{\cosh (100 x)-\cosh (101 x)}\right )dx\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4901

\(\displaystyle \int \left (\frac {\sinh (100 x)}{\cosh (100 x)-\cosh (101 x)}+\frac {\sinh (101 x)}{\cosh (100 x)-\cosh (101 x)}\right )dx\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4901

\(\displaystyle \int \left (\frac {\sinh (100 x)}{\cosh (100 x)-\cosh (101 x)}+\frac {\sinh (101 x)}{\cosh (100 x)-\cosh (101 x)}\right )dx\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4901

\(\displaystyle \int \left (\frac {\sinh (100 x)}{\cosh (100 x)-\cosh (101 x)}+\frac {\sinh (101 x)}{\cosh (100 x)-\cosh (101 x)}\right )dx\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4901

\(\displaystyle \int \left (\frac {\sinh (100 x)}{\cosh (100 x)-\cosh (101 x)}+\frac {\sinh (101 x)}{\cosh (100 x)-\cosh (101 x)}\right )dx\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4901

\(\displaystyle \int \left (\frac {\sinh (100 x)}{\cosh (100 x)-\cosh (101 x)}+\frac {\sinh (101 x)}{\cosh (100 x)-\cosh (101 x)}\right )dx\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4901

\(\displaystyle \int \left (\frac {\sinh (100 x)}{\cosh (100 x)-\cosh (101 x)}+\frac {\sinh (101 x)}{\cosh (100 x)-\cosh (101 x)}\right )dx\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4901

\(\displaystyle \int \left (\frac {\sinh (100 x)}{\cosh (100 x)-\cosh (101 x)}+\frac {\sinh (101 x)}{\cosh (100 x)-\cosh (101 x)}\right )dx\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4901

\(\displaystyle \int \left (\frac {\sinh (100 x)}{\cosh (100 x)-\cosh (101 x)}+\frac {\sinh (101 x)}{\cosh (100 x)-\cosh (101 x)}\right )dx\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4901

\(\displaystyle \int \left (\frac {\sinh (100 x)}{\cosh (100 x)-\cosh (101 x)}+\frac {\sinh (101 x)}{\cosh (100 x)-\cosh (101 x)}\right )dx\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4901

\(\displaystyle \int \left (\frac {\sinh (100 x)}{\cosh (100 x)-\cosh (101 x)}+\frac {\sinh (101 x)}{\cosh (100 x)-\cosh (101 x)}\right )dx\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4901

\(\displaystyle \int \left (\frac {\sinh (100 x)}{\cosh (100 x)-\cosh (101 x)}+\frac {\sinh (101 x)}{\cosh (100 x)-\cosh (101 x)}\right )dx\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4901

\(\displaystyle \int \left (\frac {\sinh (100 x)}{\cosh (100 x)-\cosh (101 x)}+\frac {\sinh (101 x)}{\cosh (100 x)-\cosh (101 x)}\right )dx\)

Input:

Int[(Sinh[100*x] + Sinh[101*x])/(Cosh[100*x] - Cosh[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 [A] (verified)

Time = 14.60 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.11

method	result	size
risch	\(x -2 \ln \left ({\mathrm e}^{x}-1\right )\)	\(10\)

Input:

int((sinh(100*x)+sinh(101*x))/(cosh(100*x)-cosh(101*x)),x,method=_RETURNVE 
RBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.22 \[ \int \frac {\sinh (100 x)+\sinh (101 x)}{\cosh (100 x)-\cosh (101 x)} \, dx=x - 2 \, \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) - 1\right ) \] Input:

integrate((sinh(100*x)+sinh(101*x))/(cosh(100*x)-cosh(101*x)),x, algorithm 
="fricas")
 

Output:

x - 2*log(cosh(x) + sinh(x) - 1)
 

Sympy [F]

\[ \int \frac {\sinh (100 x)+\sinh (101 x)}{\cosh (100 x)-\cosh (101 x)} \, dx=\int \frac {\sinh {\left (100 x \right )} + \sinh {\left (101 x \right )}}{\cosh {\left (100 x \right )} - \cosh {\left (101 x \right )}}\, dx \] Input:

integrate((sinh(100*x)+sinh(101*x))/(cosh(100*x)-cosh(101*x)),x)
 

Output:

Integral((sinh(100*x) + sinh(101*x))/(cosh(100*x) - cosh(101*x)), x)
 

Maxima [A] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.44 \[ \int \frac {\sinh (100 x)+\sinh (101 x)}{\cosh (100 x)-\cosh (101 x)} \, dx=-x - 2 \, \log \left (e^{\left (-x\right )} - 1\right ) \] Input:

integrate((sinh(100*x)+sinh(101*x))/(cosh(100*x)-cosh(101*x)),x, algorithm 
="maxima")
 

Output:

-x - 2*log(e^(-x) - 1)
 

Giac [A] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.11 \[ \int \frac {\sinh (100 x)+\sinh (101 x)}{\cosh (100 x)-\cosh (101 x)} \, dx=x - 2 \, \log \left ({\left | e^{x} - 1 \right |}\right ) \] Input:

integrate((sinh(100*x)+sinh(101*x))/(cosh(100*x)-cosh(101*x)),x, algorithm 
="giac")
 

Output:

x - 2*log(abs(e^x - 1))
 

Mupad [B] (verification not implemented)

Time = 16.99 (sec) , antiderivative size = 9, normalized size of antiderivative = 1.00 \[ \int \frac {\sinh (100 x)+\sinh (101 x)}{\cosh (100 x)-\cosh (101 x)} \, dx=x-2\,\ln \left ({\mathrm {e}}^x-1\right ) \] Input:

int((sinh(100*x) + sinh(101*x))/(cosh(100*x) - cosh(101*x)),x)
 

Output:

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

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 371, normalized size of antiderivative = 41.22 \[ \int \frac {\sinh (100 x)+\sinh (101 x)}{\cosh (100 x)-\cosh (101 x)} \, dx=-\frac {\mathrm {log}\left (\cosh \left (101 x \right )-\cosh \left (100 x \right )\right )}{101}+\frac {\mathrm {log}\left (1+e^{198 x}+e^{195 x}+e^{192 x}+e^{189 x}+e^{186 x}+e^{183 x}+e^{180 x}+e^{177 x}+e^{174 x}+e^{171 x}+e^{168 x}+e^{165 x}+e^{162 x}+e^{159 x}+e^{156 x}+e^{153 x}+e^{150 x}+e^{147 x}+e^{144 x}+e^{141 x}+e^{138 x}+e^{135 x}+e^{132 x}+e^{129 x}+e^{126 x}+e^{123 x}+e^{120 x}+e^{117 x}+e^{114 x}+e^{111 x}+e^{108 x}+e^{105 x}+e^{102 x}+e^{99 x}+e^{96 x}+e^{93 x}+e^{90 x}+e^{87 x}+e^{84 x}+e^{81 x}+e^{78 x}+e^{75 x}+e^{72 x}+e^{69 x}+e^{66 x}+e^{63 x}+e^{60 x}+e^{57 x}+e^{54 x}+e^{51 x}+e^{48 x}+e^{45 x}+e^{42 x}+e^{39 x}+e^{36 x}+e^{33 x}+e^{30 x}+e^{27 x}+e^{24 x}+e^{21 x}+e^{18 x}+e^{15 x}+e^{12 x}+e^{9 x}+e^{6 x}+e^{3 x}\right )}{101}+\frac {\mathrm {log}\left (e^{2 x}+e^{x}+1\right )}{101}-\frac {200 \,\mathrm {log}\left (e^{x}-1\right )}{101} \] Input:

int((sinh(100*x)+sinh(101*x))/(cosh(100*x)-cosh(101*x)),x)
 

Output:

( - log(cosh(101*x) - cosh(100*x)) + log(e**(198*x) + e**(195*x) + e**(192 
*x) + e**(189*x) + e**(186*x) + e**(183*x) + e**(180*x) + e**(177*x) + e** 
(174*x) + e**(171*x) + e**(168*x) + e**(165*x) + e**(162*x) + e**(159*x) + 
 e**(156*x) + e**(153*x) + e**(150*x) + e**(147*x) + e**(144*x) + e**(141* 
x) + e**(138*x) + e**(135*x) + e**(132*x) + e**(129*x) + e**(126*x) + e**( 
123*x) + e**(120*x) + e**(117*x) + e**(114*x) + e**(111*x) + e**(108*x) + 
e**(105*x) + e**(102*x) + e**(99*x) + e**(96*x) + e**(93*x) + e**(90*x) + 
e**(87*x) + e**(84*x) + e**(81*x) + e**(78*x) + e**(75*x) + e**(72*x) + e* 
*(69*x) + e**(66*x) + e**(63*x) + e**(60*x) + e**(57*x) + e**(54*x) + e**( 
51*x) + e**(48*x) + e**(45*x) + e**(42*x) + e**(39*x) + e**(36*x) + e**(33 
*x) + e**(30*x) + e**(27*x) + e**(24*x) + e**(21*x) + e**(18*x) + e**(15*x 
) + e**(12*x) + e**(9*x) + e**(6*x) + e**(3*x) + 1) + log(e**(2*x) + e**x 
+ 1) - 200*log(e**x - 1))/101