\(\int \frac {\sinh (100 x)-\sinh (101 x)}{\cosh (100 x)+\cosh (101 x)} \, dx\) [436]

Optimal result

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

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

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 9, normalized size of antiderivative = 1.00 \[ \int \frac {\sinh (100 x)-\sinh (101 x)}{\cosh (100 x)+\cosh (101 x)} \, dx=-2 \log \left (\cosh \left (\frac {x}{2}\right )\right ) \] Input:

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

Output:

-2*Log[Cosh[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

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

\(\Big \downarrow \) 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

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

\(\Big \downarrow \) 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

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

\(\Big \downarrow \) 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

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

\(\Big \downarrow \) 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

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

\(\Big \downarrow \) 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

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

\(\Big \downarrow \) 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

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

\(\Big \downarrow \) 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

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

\(\Big \downarrow \) 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

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

\(\Big \downarrow \) 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

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

\(\Big \downarrow \) 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

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

\(\Big \downarrow \) 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

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

\(\Big \downarrow \) 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

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

\(\Big \downarrow \) 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

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

\(\Big \downarrow \) 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

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

\(\Big \downarrow \) 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 = 13.66 (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 = 0.23 (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 = 0.31 (sec) , antiderivative size = 9, normalized size of antiderivative = 1.00 \[ \int \frac {\sinh (100 x)-\sinh (101 x)}{\cosh (100 x)+\cosh (101 x)} \, dx=x - 2 \, \log \left (e^{x} + 1\right ) \] Input:

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

Output:

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

Mupad [B] (verification not implemented)

Time = 0.05 (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 = 0.17 (sec) , antiderivative size = 437, normalized size of antiderivative = 48.56 \[ \int \frac {\sinh (100 x)-\sinh (101 x)}{\cosh (100 x)+\cosh (101 x)} \, dx=-\frac {\mathrm {log}\left (\cosh \left (101 x \right )+\cosh \left (100 x \right )\right )}{101}+\frac {\mathrm {log}\left (1+e^{198 x}-e^{195 x}+e^{192 x}-e^{189 x}+e^{186 x}-e^{183 x}+e^{180 x}-e^{177 x}+e^{174 x}-e^{171 x}+e^{168 x}-e^{165 x}+e^{162 x}-e^{159 x}+e^{156 x}-e^{153 x}+e^{150 x}-e^{147 x}+e^{144 x}-e^{141 x}+e^{138 x}-e^{135 x}+e^{132 x}-e^{129 x}+e^{126 x}-e^{123 x}+e^{120 x}-e^{117 x}+e^{114 x}-e^{111 x}+e^{108 x}-e^{105 x}+e^{102 x}-e^{99 x}+e^{96 x}-e^{93 x}+e^{90 x}-e^{87 x}+e^{84 x}-e^{81 x}+e^{78 x}-e^{75 x}+e^{72 x}-e^{69 x}+e^{66 x}-e^{63 x}+e^{60 x}-e^{57 x}+e^{54 x}-e^{51 x}+e^{48 x}-e^{45 x}+e^{42 x}-e^{39 x}+e^{36 x}-e^{33 x}+e^{30 x}-e^{27 x}+e^{24 x}-e^{21 x}+e^{18 x}-e^{15 x}+e^{12 x}-e^{9 x}+e^{6 x}-e^{3 x}\right )}{101}+\frac {\mathrm {log}\left (e^{2 x}-e^{x}+1\right )}{101}-\frac {200 \,\mathrm {log}\left (e^{x}+1\right )}{101} \] 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