\(\int \frac {\sinh (100 x)+\sinh (101 x)}{\cosh (100 x)+\cosh (101 x)} \, dx\) [438]

Optimal result

Integrand size = 21, antiderivative size = 11 \[ \int \frac {\sinh (100 x)+\sinh (101 x)}{\cosh (100 x)+\cosh (101 x)} \, dx=\frac {2}{201} \log \left (\cosh \left (\frac {201 x}{2}\right )\right ) \] Output:

2/201*ln(cosh(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 = 3.29 (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[Cosh[x/2]] + Log[1 - 2*Cosh[x]] + Log[1 - 2*Cosh[x] + 2*Co 
sh[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[2 
2*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[33*x]] + 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

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

\(\Big \downarrow \) 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 = 1.15 (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. 2905 vs. \(2 (7) = 14\).

Time = 0.16 (sec) , antiderivative size = 2905, normalized size of antiderivative = 264.09 \[ \int \frac {\sinh (100 x)+\sinh (101 x)}{\cosh (100 x)+\cosh (101 x)} \, dx=\text {Too large to display} \] Input:

integrate((sinh(100*x)+sinh(101*x))/(cosh(100*x)+cosh(101*x)),x, algorithm 
="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. 806 vs. \(2 (7) = 14\).

Time = 0.24 (sec) , antiderivative size = 806, normalized size of antiderivative = 73.27 \[ \int \frac {\sinh (100 x)+\sinh (101 x)}{\cosh (100 x)+\cosh (101 x)} \, dx=\text {Too large to display} \] Input:

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

Output:

x + 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^(-78*x) - e^(-80*x) - e^(-81*x) + e^(-83*x) + 
e^(-84*x) - e^(-86*x) - e^(-87*x) + e^(-89*x) + e^(-90*x) - e^(-92*x) - e^ 
(-93*x) + e^(-95*x) + e^(-96*x) - e^(-98*x) - e^(-99*x) + e^(-101*x) + e^( 
-102*x) - e^(-104*x) - e^(-105*x) + e^(-107*x) + e^(-108*x) - e^(-110*x) - 
 e^(-111*x) + e^(-113*x) + e^(-114*x) - e^(-116*x) - e^(-117*x) + e^(-119* 
x) + e^(-120*x) - e^(-122*x) - e^(-123*x) + e^(-125*x) + e^(-126*x) - e^(- 
128*x) - e^(-129*x) + e^(-131*x) + e^(-132*x) + 1) + 2/201*log(e^(-x) + 1) 
 + 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^(-...
 

Giac [A] (verification not implemented)

Time = 0.31 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.18 \[ \int \frac {\sinh (100 x)+\sinh (101 x)}{\cosh (100 x)+\cosh (101 x)} \, dx=-x + \frac {2}{201} \, \log \left (e^{\left (201 \, x\right )} + 1\right ) \] Input:

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

Output:

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

Mupad [B] (verification not implemented)

Time = 16.51 (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 = 0.17 (sec) , antiderivative size = 437, normalized size of antiderivative = 39.73 \[ \int \frac {\sinh (100 x)+\sinh (101 x)}{\cosh (100 x)+\cosh (101 x)} \, dx=\frac {\mathrm {log}\left (\cosh \left (101 x \right )+\cosh \left (100 x \right )\right )}{101}+\frac {\mathrm {log}\left (1+e^{198 x}-e^{195 x}+e^{192 x}-e^{189 x}+e^{186 x}-e^{183 x}+e^{180 x}-e^{177 x}+e^{174 x}-e^{171 x}+e^{168 x}-e^{165 x}+e^{162 x}-e^{159 x}+e^{156 x}-e^{153 x}+e^{150 x}-e^{147 x}+e^{144 x}-e^{141 x}+e^{138 x}-e^{135 x}+e^{132 x}-e^{129 x}+e^{126 x}-e^{123 x}+e^{120 x}-e^{117 x}+e^{114 x}-e^{111 x}+e^{108 x}-e^{105 x}+e^{102 x}-e^{99 x}+e^{96 x}-e^{93 x}+e^{90 x}-e^{87 x}+e^{84 x}-e^{81 x}+e^{78 x}-e^{75 x}+e^{72 x}-e^{69 x}+e^{66 x}-e^{63 x}+e^{60 x}-e^{57 x}+e^{54 x}-e^{51 x}+e^{48 x}-e^{45 x}+e^{42 x}-e^{39 x}+e^{36 x}-e^{33 x}+e^{30 x}-e^{27 x}+e^{24 x}-e^{21 x}+e^{18 x}-e^{15 x}+e^{12 x}-e^{9 x}+e^{6 x}-e^{3 x}\right )}{20301}+\frac {\mathrm {log}\left (e^{2 x}-e^{x}+1\right )}{20301}-\frac {200 \,\mathrm {log}\left (e^{x}+1\right )}{20301} \] 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