\(\int \frac {\sinh (100 x)-\sinh (101 x)}{\cosh (100 x)-\cosh (101 x)} \, dx\) [437]

Optimal result

Integrand size = 25, antiderivative size = 11 \[ \int \frac {\sinh (100 x)-\sinh (101 x)}{\cosh (100 x)-\cosh (101 x)} \, dx=\frac {2}{201} \log \left (\sinh \left (\frac {201 x}{2}\right )\right ) \] Output:

2/201*ln(sinh(201/2*x))
 

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.

Time = 3.35 (sec) , antiderivative size = 673, normalized size of antiderivative = 61.18 \[ \int \frac {\sinh (100 x)-\sinh (101 x)}{\cosh (100 x)-\cosh (101 x)} \, dx =\text {Too large to display} \] Input:

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

Output:

(2*(-66*x + Log[1 + 2*Cosh[x]] + Log[1 + 2*Cosh[x] + 2*Cosh[2*x] + 2*Cosh[ 
3*x] + 2*Cosh[4*x] + 2*Cosh[5*x] + 2*Cosh[6*x] + 2*Cosh[7*x] + 2*Cosh[8*x] 
 + 2*Cosh[9*x] + 2*Cosh[10*x] + 2*Cosh[11*x] + 2*Cosh[12*x] + 2*Cosh[13*x] 
 + 2*Cosh[14*x] + 2*Cosh[15*x] + 2*Cosh[16*x] + 2*Cosh[17*x] + 2*Cosh[18*x 
] + 2*Cosh[19*x] + 2*Cosh[20*x] + 2*Cosh[21*x] + 2*Cosh[22*x] + 2*Cosh[23* 
x] + 2*Cosh[24*x] + 2*Cosh[25*x] + 2*Cosh[26*x] + 2*Cosh[27*x] + 2*Cosh[28 
*x] + 2*Cosh[29*x] + 2*Cosh[30*x] + 2*Cosh[31*x] + 2*Cosh[32*x] + 2*Cosh[3 
3*x]] + Log[Sinh[x/2]] + RootSum[1 - #1 + #1^3 - #1^4 + #1^6 - #1^7 + #1^9 
 - #1^10 + #1^12 - #1^13 + #1^15 - #1^16 + #1^18 - #1^19 + #1^21 - #1^22 + 
 #1^24 - #1^25 + #1^27 - #1^28 + #1^30 - #1^31 + #1^33 - #1^34 + #1^36 - # 
1^37 + #1^39 - #1^40 + #1^42 - #1^43 + #1^45 - #1^46 + #1^48 - #1^49 + #1^ 
51 - #1^52 + #1^54 - #1^55 + #1^57 - #1^58 + #1^60 - #1^61 + #1^63 - #1^64 
 + #1^66 - #1^68 + #1^69 - #1^71 + #1^72 - #1^74 + #1^75 - #1^77 + #1^78 - 
 #1^80 + #1^81 - #1^83 + #1^84 - #1^86 + #1^87 - #1^89 + #1^90 - #1^92 + # 
1^93 - #1^95 + #1^96 - #1^98 + #1^99 - #1^101 + #1^102 - #1^104 + #1^105 - 
 #1^107 + #1^108 - #1^110 + #1^111 - #1^113 + #1^114 - #1^116 + #1^117 - # 
1^119 + #1^120 - #1^122 + #1^123 - #1^125 + #1^126 - #1^128 + #1^129 - #1^ 
131 + #1^132 & , Log[E^x - #1] & ]))/201
 

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 = 1.18 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.27

method	result	size
risch	\(-x +\frac {2 \ln \left ({\mathrm e}^{201 x}-1\right )}{201}\)	\(14\)

Input:

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

Output:

-x+2/201*ln(exp(201*x)-1)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2907 vs. \(2 (7) = 14\).

Time = 0.15 (sec) , antiderivative size = 2907, normalized size of antiderivative = 264.27 \[ \int \frac {\sinh (100 x)-\sinh (101 x)}{\cosh (100 x)-\cosh (101 x)} \, dx=\text {Too large to display} \] Input:

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

Output:

-x + 2/201*log((cosh(x)^101 + (101*cosh(x) - 1)*sinh(x)^100 + sinh(x)^101 
- cosh(x)^100 + 50*(101*cosh(x)^2 + 2*cosh(x))*sinh(x)^99 + 1650*(101*cosh 
(x)^3 - 3*cosh(x)^2)*sinh(x)^98 + 40425*(101*cosh(x)^4 + 4*cosh(x)^3)*sinh 
(x)^97 + 784245*(101*cosh(x)^5 - 5*cosh(x)^4)*sinh(x)^96 + 12547920*(101*c 
osh(x)^6 + 6*cosh(x)^5)*sinh(x)^95 + 170293200*(101*cosh(x)^7 - 7*cosh(x)^ 
6)*sinh(x)^94 + 2000945100*(101*cosh(x)^8 + 8*cosh(x)^7)*sinh(x)^93 + 2067 
6432700*(101*cosh(x)^9 - 9*cosh(x)^8)*sinh(x)^92 + 190223180840*(101*cosh( 
x)^10 + 10*cosh(x)^9)*sinh(x)^91 + 1573664496040*(101*cosh(x)^11 - 11*cosh 
(x)^10)*sinh(x)^90 + 11802483720300*(101*cosh(x)^12 + 12*cosh(x)^11)*sinh( 
x)^89 + 80801619315900*(101*cosh(x)^13 - 13*cosh(x)^12)*sinh(x)^88 + 50789 
5892842800*(101*cosh(x)^14 + 14*cosh(x)^13)*sinh(x)^87 + 2945796178488240* 
(101*cosh(x)^15 - 15*cosh(x)^14)*sinh(x)^86 + 15833654459374290*(101*cosh( 
x)^16 + 16*cosh(x)^15)*sinh(x)^85 + 79168272296871450*(101*cosh(x)^17 - 17 
*cosh(x)^16)*sinh(x)^84 + 369451937385400100*(101*cosh(x)^18 + 18*cosh(x)^ 
17)*sinh(x)^83 + 1613921621209905700*(101*cosh(x)^19 - 19*cosh(x)^18)*sinh 
(x)^82 + 6617078646960613370*(101*cosh(x)^20 + 20*cosh(x)^19)*sinh(x)^81 + 
 25523017638276651570*(101*cosh(x)^21 - 21*cosh(x)^20)*sinh(x)^80 + 928109 
73230096914800*(101*cosh(x)^22 + 22*cosh(x)^21)*sinh(x)^79 + 3187855167468 
54620400*(101*cosh(x)^23 - 23*cosh(x)^22)*sinh(x)^78 + 1036052929427277516 
300*(101*cosh(x)^24 + 24*cosh(x)^23)*sinh(x)^77 + 319104302263601475020...
 

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 [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 738 vs. \(2 (7) = 14\).

Time = 0.25 (sec) , antiderivative size = 738, normalized size of antiderivative = 67.09 \[ \int \frac {\sinh (100 x)-\sinh (101 x)}{\cosh (100 x)-\cosh (101 x)} \, dx=\text {Too large to display} \] Input:

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

Output:

x + 2/201*log(e^(-x) + e^(-2*x) + e^(-3*x) + e^(-4*x) + e^(-5*x) + e^(-6*x 
) + e^(-7*x) + e^(-8*x) + e^(-9*x) + e^(-10*x) + e^(-11*x) + e^(-12*x) + e 
^(-13*x) + e^(-14*x) + e^(-15*x) + e^(-16*x) + e^(-17*x) + e^(-18*x) + e^( 
-19*x) + e^(-20*x) + e^(-21*x) + e^(-22*x) + e^(-23*x) + e^(-24*x) + e^(-2 
5*x) + e^(-26*x) + e^(-27*x) + e^(-28*x) + e^(-29*x) + e^(-30*x) + e^(-31* 
x) + e^(-32*x) + e^(-33*x) + e^(-34*x) + e^(-35*x) + e^(-36*x) + e^(-37*x) 
 + e^(-38*x) + e^(-39*x) + e^(-40*x) + e^(-41*x) + e^(-42*x) + e^(-43*x) + 
 e^(-44*x) + e^(-45*x) + e^(-46*x) + e^(-47*x) + e^(-48*x) + e^(-49*x) + e 
^(-50*x) + e^(-51*x) + e^(-52*x) + e^(-53*x) + e^(-54*x) + e^(-55*x) + e^( 
-56*x) + e^(-57*x) + e^(-58*x) + e^(-59*x) + e^(-60*x) + e^(-61*x) + e^(-6 
2*x) + e^(-63*x) + e^(-64*x) + e^(-65*x) + e^(-66*x) + 1) + 2/201*log(e^(- 
x) + e^(-2*x) + 1) + 2/201*log(e^(-x) - 1) + 2/201*log(-e^(-x) + e^(-3*x) 
- e^(-4*x) + e^(-6*x) - e^(-7*x) + e^(-9*x) - e^(-10*x) + e^(-12*x) - e^(- 
13*x) + e^(-15*x) - e^(-16*x) + e^(-18*x) - e^(-19*x) + e^(-21*x) - e^(-22 
*x) + e^(-24*x) - e^(-25*x) + e^(-27*x) - e^(-28*x) + e^(-30*x) - e^(-31*x 
) + e^(-33*x) - e^(-34*x) + e^(-36*x) - e^(-37*x) + e^(-39*x) - e^(-40*x) 
+ e^(-42*x) - e^(-43*x) + e^(-45*x) - e^(-46*x) + e^(-48*x) - e^(-49*x) + 
e^(-51*x) - e^(-52*x) + e^(-54*x) - e^(-55*x) + e^(-57*x) - e^(-58*x) + e^ 
(-60*x) - e^(-61*x) + e^(-63*x) - e^(-64*x) + e^(-66*x) - e^(-68*x) + e^(- 
69*x) - e^(-71*x) + e^(-72*x) - e^(-74*x) + e^(-75*x) - e^(-77*x) + e^(...
 

Giac [A] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.27 \[ \int \frac {\sinh (100 x)-\sinh (101 x)}{\cosh (100 x)-\cosh (101 x)} \, dx=-x + \frac {2}{201} \, \log \left ({\left | e^{\left (201 \, x\right )} - 1 \right |}\right ) \] Input:

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

Output:

-x + 2/201*log(abs(e^(201*x) - 1))
 

Mupad [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.18 \[ \int \frac {\sinh (100 x)-\sinh (101 x)}{\cosh (100 x)-\cosh (101 x)} \, dx=\frac {2\,\ln \left ({\mathrm {e}}^{201\,x}-1\right )}{201}-x \] Input:

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

Output:

(2*log(exp(201*x) - 1))/201 - x
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 371, normalized size of antiderivative = 33.73 \[ \int \frac {\sinh (100 x)-\sinh (101 x)}{\cosh (100 x)-\cosh (101 x)} \, dx=\frac {\mathrm {log}\left (\cosh \left (101 x \right )-\cosh \left (100 x \right )\right )}{101}+\frac {\mathrm {log}\left (1+e^{198 x}+e^{195 x}+e^{192 x}+e^{189 x}+e^{186 x}+e^{183 x}+e^{180 x}+e^{177 x}+e^{174 x}+e^{171 x}+e^{168 x}+e^{165 x}+e^{162 x}+e^{159 x}+e^{156 x}+e^{153 x}+e^{150 x}+e^{147 x}+e^{144 x}+e^{141 x}+e^{138 x}+e^{135 x}+e^{132 x}+e^{129 x}+e^{126 x}+e^{123 x}+e^{120 x}+e^{117 x}+e^{114 x}+e^{111 x}+e^{108 x}+e^{105 x}+e^{102 x}+e^{99 x}+e^{96 x}+e^{93 x}+e^{90 x}+e^{87 x}+e^{84 x}+e^{81 x}+e^{78 x}+e^{75 x}+e^{72 x}+e^{69 x}+e^{66 x}+e^{63 x}+e^{60 x}+e^{57 x}+e^{54 x}+e^{51 x}+e^{48 x}+e^{45 x}+e^{42 x}+e^{39 x}+e^{36 x}+e^{33 x}+e^{30 x}+e^{27 x}+e^{24 x}+e^{21 x}+e^{18 x}+e^{15 x}+e^{12 x}+e^{9 x}+e^{6 x}+e^{3 x}\right )}{20301}+\frac {\mathrm {log}\left (e^{2 x}+e^{x}+1\right )}{20301}-\frac {200 \,\mathrm {log}\left (e^{x}-1\right )}{20301} \] Input:

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

Output:

(201*log(cosh(101*x) - cosh(100*x)) + log(e**(198*x) + e**(195*x) + e**(19 
2*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**(3 
3*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))/20301