\(\int \frac {\tanh ^5(x)}{a+b \tanh (x)} \, dx\) [134]

Optimal result

Integrand size = 13, antiderivative size = 94 \[ \int \frac {\tanh ^5(x)}{a+b \tanh (x)} \, dx=-\frac {b x}{a^2-b^2}+\frac {a \log (\cosh (x))}{a^2-b^2}+\frac {a^5 \log (a+b \tanh (x))}{b^4 \left (a^2-b^2\right )}-\frac {\left (a^2+b^2\right ) \tanh (x)}{b^3}+\frac {a \tanh ^2(x)}{2 b^2}-\frac {\tanh ^3(x)}{3 b} \] Output:

-b*x/(a^2-b^2)+a*ln(cosh(x))/(a^2-b^2)+a^5*ln(a+b*tanh(x))/b^4/(a^2-b^2)-( 
a^2+b^2)*tanh(x)/b^3+1/2*a*tanh(x)^2/b^2-1/3*tanh(x)^3/b
 

Mathematica [A] (verified)

Time = 0.46 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.98 \[ \int \frac {\tanh ^5(x)}{a+b \tanh (x)} \, dx=\frac {1}{6} \left (-\frac {3 \log (1-\tanh (x))}{a+b}-\frac {3 \log (1+\tanh (x))}{a-b}+\frac {6 a^5 \log (a+b \tanh (x))}{b^4 \left (a^2-b^2\right )}-\frac {6 \left (a^2+b^2\right ) \tanh (x)}{b^3}+\frac {3 a \tanh ^2(x)}{b^2}-\frac {2 \tanh ^3(x)}{b}\right ) \] Input:

Integrate[Tanh[x]^5/(a + b*Tanh[x]),x]
 

Output:

((-3*Log[1 - Tanh[x]])/(a + b) - (3*Log[1 + Tanh[x]])/(a - b) + (6*a^5*Log 
[a + b*Tanh[x]])/(b^4*(a^2 - b^2)) - (6*(a^2 + b^2)*Tanh[x])/b^3 + (3*a*Ta 
nh[x]^2)/b^2 - (2*Tanh[x]^3)/b)/6
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 1.07 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.39, number of steps used = 21, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.538, Rules used = {3042, 26, 4049, 27, 3042, 25, 4130, 27, 3042, 26, 4131, 25, 3042, 4109, 26, 3042, 26, 3956, 4100, 16}

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.

\(\displaystyle \int \frac {\tanh ^5(x)}{a+b \tanh (x)} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int -\frac {i \tan (i x)^5}{a-i b \tan (i x)}dx\)

\(\Big \downarrow \) 26

\(\displaystyle -i \int \frac {\tan (i x)^5}{a-i b \tan (i x)}dx\)

\(\Big \downarrow \) 4049

\(\displaystyle -i \left (\frac {i \int \frac {3 \tanh ^2(x) \left (-a \tanh ^2(x)+b \tanh (x)+a\right )}{a+b \tanh (x)}dx}{3 b}-\frac {i \tanh ^3(x)}{3 b}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle -i \left (\frac {i \int \frac {\tanh ^2(x) \left (-a \tanh ^2(x)+b \tanh (x)+a\right )}{a+b \tanh (x)}dx}{b}-\frac {i \tanh ^3(x)}{3 b}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle -i \left (\frac {i \int -\frac {\tan (i x)^2 \left (a \tan (i x)^2-i b \tan (i x)+a\right )}{a-i b \tan (i x)}dx}{b}-\frac {i \tanh ^3(x)}{3 b}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle -i \left (-\frac {i \int \frac {\tan (i x)^2 \left (a \tan (i x)^2-i b \tan (i x)+a\right )}{a-i b \tan (i x)}dx}{b}-\frac {i \tanh ^3(x)}{3 b}\right )\)

\(\Big \downarrow \) 4130

\(\displaystyle -i \left (-\frac {i \left (-\frac {a \tanh ^2(x)}{2 b}+\frac {i \int -\frac {2 i \tanh (x) \left (a^2-\left (a^2+b^2\right ) \tanh ^2(x)\right )}{a+b \tanh (x)}dx}{2 b}\right )}{b}-\frac {i \tanh ^3(x)}{3 b}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle -i \left (-\frac {i \left (\frac {\int \frac {\tanh (x) \left (a^2-\left (a^2+b^2\right ) \tanh ^2(x)\right )}{a+b \tanh (x)}dx}{b}-\frac {a \tanh ^2(x)}{2 b}\right )}{b}-\frac {i \tanh ^3(x)}{3 b}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle -i \left (-\frac {i \left (-\frac {a \tanh ^2(x)}{2 b}+\frac {\int -\frac {i \tan (i x) \left (a^2+\left (a^2+b^2\right ) \tan (i x)^2\right )}{a-i b \tan (i x)}dx}{b}\right )}{b}-\frac {i \tanh ^3(x)}{3 b}\right )\)

\(\Big \downarrow \) 26

\(\displaystyle -i \left (-\frac {i \left (-\frac {a \tanh ^2(x)}{2 b}-\frac {i \int \frac {\tan (i x) \left (a^2+\left (a^2+b^2\right ) \tan (i x)^2\right )}{a-i b \tan (i x)}dx}{b}\right )}{b}-\frac {i \tanh ^3(x)}{3 b}\right )\)

\(\Big \downarrow \) 4131

\(\displaystyle -i \left (-\frac {i \left (-\frac {a \tanh ^2(x)}{2 b}-\frac {i \left (\frac {i \int -\frac {\tanh (x) b^3-a \left (a^2+b^2\right ) \tanh ^2(x)+a \left (a^2+b^2\right )}{a+b \tanh (x)}dx}{b}+\frac {i \left (a^2+b^2\right ) \tanh (x)}{b}\right )}{b}\right )}{b}-\frac {i \tanh ^3(x)}{3 b}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle -i \left (-\frac {i \left (-\frac {a \tanh ^2(x)}{2 b}-\frac {i \left (\frac {i \left (a^2+b^2\right ) \tanh (x)}{b}-\frac {i \int \frac {\tanh (x) b^3-a \left (a^2+b^2\right ) \tanh ^2(x)+a \left (a^2+b^2\right )}{a+b \tanh (x)}dx}{b}\right )}{b}\right )}{b}-\frac {i \tanh ^3(x)}{3 b}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle -i \left (-\frac {i \left (-\frac {a \tanh ^2(x)}{2 b}-\frac {i \left (\frac {i \left (a^2+b^2\right ) \tanh (x)}{b}-\frac {i \int \frac {-i \tan (i x) b^3+a \left (a^2+b^2\right ) \tan (i x)^2+a \left (a^2+b^2\right )}{a-i b \tan (i x)}dx}{b}\right )}{b}\right )}{b}-\frac {i \tanh ^3(x)}{3 b}\right )\)

\(\Big \downarrow \) 4109

\(\displaystyle -i \left (-\frac {i \left (-\frac {a \tanh ^2(x)}{2 b}-\frac {i \left (\frac {i \left (a^2+b^2\right ) \tanh (x)}{b}-\frac {i \left (-\frac {i a b^3 \int i \tanh (x)dx}{a^2-b^2}+\frac {a^5 \int \frac {1-\tanh ^2(x)}{a+b \tanh (x)}dx}{a^2-b^2}-\frac {b^4 x}{a^2-b^2}\right )}{b}\right )}{b}\right )}{b}-\frac {i \tanh ^3(x)}{3 b}\right )\)

\(\Big \downarrow \) 26

\(\displaystyle -i \left (-\frac {i \left (-\frac {a \tanh ^2(x)}{2 b}-\frac {i \left (\frac {i \left (a^2+b^2\right ) \tanh (x)}{b}-\frac {i \left (\frac {a b^3 \int \tanh (x)dx}{a^2-b^2}+\frac {a^5 \int \frac {1-\tanh ^2(x)}{a+b \tanh (x)}dx}{a^2-b^2}-\frac {b^4 x}{a^2-b^2}\right )}{b}\right )}{b}\right )}{b}-\frac {i \tanh ^3(x)}{3 b}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle -i \left (-\frac {i \left (-\frac {a \tanh ^2(x)}{2 b}-\frac {i \left (\frac {i \left (a^2+b^2\right ) \tanh (x)}{b}-\frac {i \left (\frac {a b^3 \int -i \tan (i x)dx}{a^2-b^2}+\frac {a^5 \int \frac {\tan (i x)^2+1}{a-i b \tan (i x)}dx}{a^2-b^2}-\frac {b^4 x}{a^2-b^2}\right )}{b}\right )}{b}\right )}{b}-\frac {i \tanh ^3(x)}{3 b}\right )\)

\(\Big \downarrow \) 26

\(\displaystyle -i \left (-\frac {i \left (-\frac {a \tanh ^2(x)}{2 b}-\frac {i \left (\frac {i \left (a^2+b^2\right ) \tanh (x)}{b}-\frac {i \left (-\frac {i a b^3 \int \tan (i x)dx}{a^2-b^2}+\frac {a^5 \int \frac {\tan (i x)^2+1}{a-i b \tan (i x)}dx}{a^2-b^2}-\frac {b^4 x}{a^2-b^2}\right )}{b}\right )}{b}\right )}{b}-\frac {i \tanh ^3(x)}{3 b}\right )\)

\(\Big \downarrow \) 3956

\(\displaystyle -i \left (-\frac {i \left (-\frac {a \tanh ^2(x)}{2 b}-\frac {i \left (\frac {i \left (a^2+b^2\right ) \tanh (x)}{b}-\frac {i \left (\frac {a^5 \int \frac {\tan (i x)^2+1}{a-i b \tan (i x)}dx}{a^2-b^2}-\frac {b^4 x}{a^2-b^2}+\frac {a b^3 \log (\cosh (x))}{a^2-b^2}\right )}{b}\right )}{b}\right )}{b}-\frac {i \tanh ^3(x)}{3 b}\right )\)

\(\Big \downarrow \) 4100

\(\displaystyle -i \left (-\frac {i \left (-\frac {a \tanh ^2(x)}{2 b}-\frac {i \left (\frac {i \left (a^2+b^2\right ) \tanh (x)}{b}-\frac {i \left (\frac {a^5 \int \frac {1}{a+b \tanh (x)}d(b \tanh (x))}{b \left (a^2-b^2\right )}-\frac {b^4 x}{a^2-b^2}+\frac {a b^3 \log (\cosh (x))}{a^2-b^2}\right )}{b}\right )}{b}\right )}{b}-\frac {i \tanh ^3(x)}{3 b}\right )\)

\(\Big \downarrow \) 16

\(\displaystyle -i \left (-\frac {i \left (-\frac {a \tanh ^2(x)}{2 b}-\frac {i \left (\frac {i \left (a^2+b^2\right ) \tanh (x)}{b}-\frac {i \left (-\frac {b^4 x}{a^2-b^2}+\frac {a b^3 \log (\cosh (x))}{a^2-b^2}+\frac {a^5 \log (a+b \tanh (x))}{b \left (a^2-b^2\right )}\right )}{b}\right )}{b}\right )}{b}-\frac {i \tanh ^3(x)}{3 b}\right )\)

Input:

Int[Tanh[x]^5/(a + b*Tanh[x]),x]
 

Output:

(-I)*(((-1/3*I)*Tanh[x]^3)/b - (I*(-1/2*(a*Tanh[x]^2)/b - (I*(((-I)*(-((b^ 
4*x)/(a^2 - b^2)) + (a*b^3*Log[Cosh[x]])/(a^2 - b^2) + (a^5*Log[a + b*Tanh 
[x]])/(b*(a^2 - b^2))))/b + (I*(a^2 + b^2)*Tanh[x])/b))/b))/b)
 

Defintions of rubi rules used

rule 16

Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + 
b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
 

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3956

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d 
*x], x]]/d, x] /; FreeQ[{c, d}, x]
 

rule 4049

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + 
 (f_.)*(x_)])^(n_), x_Symbol] :> Simp[b^2*(a + b*Tan[e + f*x])^(m - 2)*((c 
+ d*Tan[e + f*x])^(n + 1)/(d*f*(m + n - 1))), x] + Simp[1/(d*(m + n - 1)) 
 Int[(a + b*Tan[e + f*x])^(m - 3)*(c + d*Tan[e + f*x])^n*Simp[a^3*d*(m + n 
- 1) - b^2*(b*c*(m - 2) + a*d*(1 + n)) + b*d*(m + n - 1)*(3*a^2 - b^2)*Tan[ 
e + f*x] - b^2*(b*c*(m - 2) - a*d*(3*m + 2*n - 4))*Tan[e + f*x]^2, x], x], 
x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2 
, 0] && NeQ[c^2 + d^2, 0] && IntegerQ[2*m] && GtQ[m, 2] && (GeQ[n, -1] || I 
ntegerQ[m]) &&  !(IGtQ[n, 2] && ( !IntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])) 
)
 

rule 4100

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((A_) + (C_.)*tan[(e_.) + 
 (f_.)*(x_)]^2), x_Symbol] :> Simp[A/(b*f)   Subst[Int[(a + x)^m, x], x, b* 
Tan[e + f*x]], x] /; FreeQ[{a, b, e, f, A, C, m}, x] && EqQ[A, C]
 

rule 4109

Int[((A_) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2 
)/((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(a*A + b*B - a 
*C)*(x/(a^2 + b^2)), x] + (Simp[(A*b^2 - a*b*B + a^2*C)/(a^2 + b^2)   Int[( 
1 + Tan[e + f*x]^2)/(a + b*Tan[e + f*x]), x], x] - Simp[(A*b - a*B - b*C)/( 
a^2 + b^2)   Int[Tan[e + f*x], x], x]) /; FreeQ[{a, b, e, f, A, B, C}, x] & 
& NeQ[A*b^2 - a*b*B + a^2*C, 0] && NeQ[a^2 + b^2, 0] && NeQ[A*b - a*B - b*C 
, 0]
 

rule 4130

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*tan[(e_.) 
+ (f_.)*(x_)])^(n_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_. 
) + (f_.)*(x_)]^2), x_Symbol] :> Simp[C*(a + b*Tan[e + f*x])^m*((c + d*Tan[ 
e + f*x])^(n + 1)/(d*f*(m + n + 1))), x] + Simp[1/(d*(m + n + 1))   Int[(a 
+ b*Tan[e + f*x])^(m - 1)*(c + d*Tan[e + f*x])^n*Simp[a*A*d*(m + n + 1) - C 
*(b*c*m + a*d*(n + 1)) + d*(A*b + a*B - b*C)*(m + n + 1)*Tan[e + f*x] - (C* 
m*(b*c - a*d) - b*B*d*(m + n + 1))*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, 
b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && 
 NeQ[c^2 + d^2, 0] && GtQ[m, 0] &&  !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[ 
c, 0] && NeQ[a, 0])))
 

rule 4131

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*tan[(e_.) 
+ (f_.)*(x_)])^(n_)*((A_.) + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> 
 Simp[C*(a + b*Tan[e + f*x])^m*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(m + n + 
1))), x] + Simp[1/(d*(m + n + 1))   Int[(a + b*Tan[e + f*x])^(m - 1)*(c + d 
*Tan[e + f*x])^n*Simp[a*A*d*(m + n + 1) - C*(b*c*m + a*d*(n + 1)) + d*(A*b 
- b*C)*(m + n + 1)*Tan[e + f*x] - C*m*(b*c - a*d)*Tan[e + f*x]^2, x], x], x 
] /; FreeQ[{a, b, c, d, e, f, A, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 
+ b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 0] &&  !(IGtQ[n, 0] && ( !IntegerQ 
[m] || (EqQ[c, 0] && NeQ[a, 0])))
 

Maple [A] (verified)

Time = 0.23 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.02

method	result	size
derivativedivides	\(-\frac {\tanh \left (x \right )^{3}}{3 b}+\frac {a \tanh \left (x \right )^{2}}{2 b^{2}}-\frac {a^{2} \tanh \left (x \right )}{b^{3}}-\frac {\tanh \left (x \right )}{b}-\frac {\ln \left (\tanh \left (x \right )-1\right )}{2 b +2 a}+\frac {a^{5} \ln \left (a +b \tanh \left (x \right )\right )}{b^{4} \left (a +b \right ) \left (a -b \right )}-\frac {\ln \left (1+\tanh \left (x \right )\right )}{2 a -2 b}\)	\(96\)
default	\(-\frac {\tanh \left (x \right )^{3}}{3 b}+\frac {a \tanh \left (x \right )^{2}}{2 b^{2}}-\frac {a^{2} \tanh \left (x \right )}{b^{3}}-\frac {\tanh \left (x \right )}{b}-\frac {\ln \left (\tanh \left (x \right )-1\right )}{2 b +2 a}+\frac {a^{5} \ln \left (a +b \tanh \left (x \right )\right )}{b^{4} \left (a +b \right ) \left (a -b \right )}-\frac {\ln \left (1+\tanh \left (x \right )\right )}{2 a -2 b}\)	\(96\)
parallelrisch	\(\frac {-2 \tanh \left (x \right )^{3} a^{2} b^{3}+2 \tanh \left (x \right )^{3} b^{5}+3 \tanh \left (x \right )^{2} b^{2} a^{3}-3 \tanh \left (x \right )^{2} a \,b^{4}+6 a^{5} \ln \left (a +b \tanh \left (x \right )\right )-6 \ln \left (1-\tanh \left (x \right )\right ) a \,b^{4}-6 a \,b^{4} x -6 b^{5} x -6 \tanh \left (x \right ) b \,a^{4}+6 \tanh \left (x \right ) b^{5}}{6 b^{4} \left (a^{2}-b^{2}\right )}\)	\(114\)
risch	\(\frac {x}{a +b}+\frac {2 x \,a^{3}}{b^{4}}+\frac {2 a x}{b^{2}}-\frac {2 x \,a^{5}}{b^{4} \left (a^{2}-b^{2}\right )}+\frac {2 a^{2} {\mathrm e}^{4 x}-2 \,{\mathrm e}^{4 x} a b +4 b^{2} {\mathrm e}^{4 x}+4 \,{\mathrm e}^{2 x} a^{2}-2 \,{\mathrm e}^{2 x} a b +4 b^{2} {\mathrm e}^{2 x}+2 a^{2}+\frac {8 b^{2}}{3}}{b^{3} \left ({\mathrm e}^{2 x}+1\right )^{3}}-\frac {a^{3} \ln \left ({\mathrm e}^{2 x}+1\right )}{b^{4}}-\frac {a \ln \left ({\mathrm e}^{2 x}+1\right )}{b^{2}}+\frac {a^{5} \ln \left ({\mathrm e}^{2 x}+\frac {a -b}{a +b}\right )}{b^{4} \left (a^{2}-b^{2}\right )}\)	\(184\)

Input:

int(tanh(x)^5/(a+b*tanh(x)),x,method=_RETURNVERBOSE)
 

Output:

-1/3*tanh(x)^3/b+1/2*a*tanh(x)^2/b^2-1/b^3*a^2*tanh(x)-tanh(x)/b-1/(2*b+2* 
a)*ln(tanh(x)-1)+1/b^4*a^5/(a+b)/(a-b)*ln(a+b*tanh(x))-1/(2*a-2*b)*ln(1+ta 
nh(x))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1296 vs. \(2 (90) = 180\).

Time = 0.16 (sec) , antiderivative size = 1296, normalized size of antiderivative = 13.79 \[ \int \frac {\tanh ^5(x)}{a+b \tanh (x)} \, dx=\text {Too large to display} \] Input:

integrate(tanh(x)^5/(a+b*tanh(x)),x, algorithm="fricas")
 

Output:

-1/3*(3*(a*b^4 + b^5)*x*cosh(x)^6 + 18*(a*b^4 + b^5)*x*cosh(x)*sinh(x)^5 + 
 3*(a*b^4 + b^5)*x*sinh(x)^6 - 6*a^4*b - 2*a^2*b^3 + 8*b^5 - 3*(2*a^4*b - 
2*a^3*b^2 + 2*a^2*b^3 + 2*a*b^4 - 4*b^5 - 3*(a*b^4 + b^5)*x)*cosh(x)^4 - 3 
*(2*a^4*b - 2*a^3*b^2 + 2*a^2*b^3 + 2*a*b^4 - 4*b^5 - 15*(a*b^4 + b^5)*x*c 
osh(x)^2 - 3*(a*b^4 + b^5)*x)*sinh(x)^4 + 12*(5*(a*b^4 + b^5)*x*cosh(x)^3 
- (2*a^4*b - 2*a^3*b^2 + 2*a^2*b^3 + 2*a*b^4 - 4*b^5 - 3*(a*b^4 + b^5)*x)* 
cosh(x))*sinh(x)^3 - 3*(4*a^4*b - 2*a^3*b^2 + 2*a*b^4 - 4*b^5 - 3*(a*b^4 + 
 b^5)*x)*cosh(x)^2 + 3*(15*(a*b^4 + b^5)*x*cosh(x)^4 - 4*a^4*b + 2*a^3*b^2 
 - 2*a*b^4 + 4*b^5 - 6*(2*a^4*b - 2*a^3*b^2 + 2*a^2*b^3 + 2*a*b^4 - 4*b^5 
- 3*(a*b^4 + b^5)*x)*cosh(x)^2 + 3*(a*b^4 + b^5)*x)*sinh(x)^2 + 3*(a*b^4 + 
 b^5)*x - 3*(a^5*cosh(x)^6 + 6*a^5*cosh(x)*sinh(x)^5 + a^5*sinh(x)^6 + 3*a 
^5*cosh(x)^4 + 3*a^5*cosh(x)^2 + a^5 + 3*(5*a^5*cosh(x)^2 + a^5)*sinh(x)^4 
 + 4*(5*a^5*cosh(x)^3 + 3*a^5*cosh(x))*sinh(x)^3 + 3*(5*a^5*cosh(x)^4 + 6* 
a^5*cosh(x)^2 + a^5)*sinh(x)^2 + 6*(a^5*cosh(x)^5 + 2*a^5*cosh(x)^3 + a^5* 
cosh(x))*sinh(x))*log(2*(a*cosh(x) + b*sinh(x))/(cosh(x) - sinh(x))) + 3*( 
(a^5 - a*b^4)*cosh(x)^6 + 6*(a^5 - a*b^4)*cosh(x)*sinh(x)^5 + (a^5 - a*b^4 
)*sinh(x)^6 + a^5 - a*b^4 + 3*(a^5 - a*b^4)*cosh(x)^4 + 3*(a^5 - a*b^4 + 5 
*(a^5 - a*b^4)*cosh(x)^2)*sinh(x)^4 + 4*(5*(a^5 - a*b^4)*cosh(x)^3 + 3*(a^ 
5 - a*b^4)*cosh(x))*sinh(x)^3 + 3*(a^5 - a*b^4)*cosh(x)^2 + 3*(a^5 - a*b^4 
 + 5*(a^5 - a*b^4)*cosh(x)^4 + 6*(a^5 - a*b^4)*cosh(x)^2)*sinh(x)^2 + 6...
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 546 vs. \(2 (78) = 156\).

Time = 0.47 (sec) , antiderivative size = 546, normalized size of antiderivative = 5.81 \[ \int \frac {\tanh ^5(x)}{a+b \tanh (x)} \, dx =\text {Too large to display} \] Input:

integrate(tanh(x)**5/(a+b*tanh(x)),x)
 

Output:

Piecewise((zoo*(x - tanh(x)**3/3 - tanh(x)), Eq(a, 0) & Eq(b, 0)), ((x - l 
og(tanh(x) + 1) - tanh(x)**4/4 - tanh(x)**2/2)/a, Eq(b, 0)), (27*x*tanh(x) 
/(6*b*tanh(x) - 6*b) - 27*x/(6*b*tanh(x) - 6*b) - 12*log(tanh(x) + 1)*tanh 
(x)/(6*b*tanh(x) - 6*b) + 12*log(tanh(x) + 1)/(6*b*tanh(x) - 6*b) - 2*tanh 
(x)**4/(6*b*tanh(x) - 6*b) - tanh(x)**3/(6*b*tanh(x) - 6*b) - 9*tanh(x)**2 
/(6*b*tanh(x) - 6*b) + 15/(6*b*tanh(x) - 6*b), Eq(a, -b)), (3*x*tanh(x)/(6 
*b*tanh(x) + 6*b) + 3*x/(6*b*tanh(x) + 6*b) + 12*log(tanh(x) + 1)*tanh(x)/ 
(6*b*tanh(x) + 6*b) + 12*log(tanh(x) + 1)/(6*b*tanh(x) + 6*b) - 2*tanh(x)* 
*4/(6*b*tanh(x) + 6*b) + tanh(x)**3/(6*b*tanh(x) + 6*b) - 9*tanh(x)**2/(6* 
b*tanh(x) + 6*b) + 15/(6*b*tanh(x) + 6*b), Eq(a, b)), (6*a**5*log(a/b + ta 
nh(x))/(6*a**2*b**4 - 6*b**6) - 6*a**4*b*tanh(x)/(6*a**2*b**4 - 6*b**6) + 
3*a**3*b**2*tanh(x)**2/(6*a**2*b**4 - 6*b**6) - 2*a**2*b**3*tanh(x)**3/(6* 
a**2*b**4 - 6*b**6) + 6*a*b**4*x/(6*a**2*b**4 - 6*b**6) - 6*a*b**4*log(tan 
h(x) + 1)/(6*a**2*b**4 - 6*b**6) - 3*a*b**4*tanh(x)**2/(6*a**2*b**4 - 6*b* 
*6) - 6*b**5*x/(6*a**2*b**4 - 6*b**6) + 2*b**5*tanh(x)**3/(6*a**2*b**4 - 6 
*b**6) + 6*b**5*tanh(x)/(6*a**2*b**4 - 6*b**6), True))
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.60 \[ \int \frac {\tanh ^5(x)}{a+b \tanh (x)} \, dx=\frac {a^{5} \log \left (-{\left (a - b\right )} e^{\left (-2 \, x\right )} - a - b\right )}{a^{2} b^{4} - b^{6}} - \frac {2 \, {\left (3 \, a^{2} + 4 \, b^{2} + 3 \, {\left (2 \, a^{2} + a b + 2 \, b^{2}\right )} e^{\left (-2 \, x\right )} + 3 \, {\left (a^{2} + a b + 2 \, b^{2}\right )} e^{\left (-4 \, x\right )}\right )}}{3 \, {\left (3 \, b^{3} e^{\left (-2 \, x\right )} + 3 \, b^{3} e^{\left (-4 \, x\right )} + b^{3} e^{\left (-6 \, x\right )} + b^{3}\right )}} + \frac {x}{a + b} - \frac {{\left (a^{3} + a b^{2}\right )} \log \left (e^{\left (-2 \, x\right )} + 1\right )}{b^{4}} \] Input:

integrate(tanh(x)^5/(a+b*tanh(x)),x, algorithm="maxima")
 

Output:

a^5*log(-(a - b)*e^(-2*x) - a - b)/(a^2*b^4 - b^6) - 2/3*(3*a^2 + 4*b^2 + 
3*(2*a^2 + a*b + 2*b^2)*e^(-2*x) + 3*(a^2 + a*b + 2*b^2)*e^(-4*x))/(3*b^3* 
e^(-2*x) + 3*b^3*e^(-4*x) + b^3*e^(-6*x) + b^3) + x/(a + b) - (a^3 + a*b^2 
)*log(e^(-2*x) + 1)/b^4
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.51 \[ \int \frac {\tanh ^5(x)}{a+b \tanh (x)} \, dx=\frac {a^{5} \log \left ({\left | a e^{\left (2 \, x\right )} + b e^{\left (2 \, x\right )} + a - b \right |}\right )}{a^{2} b^{4} - b^{6}} - \frac {x}{a - b} - \frac {{\left (a^{3} + a b^{2}\right )} \log \left (e^{\left (2 \, x\right )} + 1\right )}{b^{4}} + \frac {2 \, {\left (3 \, a^{2} b + 4 \, b^{3} + 3 \, {\left (a^{2} b - a b^{2} + 2 \, b^{3}\right )} e^{\left (4 \, x\right )} + 3 \, {\left (2 \, a^{2} b - a b^{2} + 2 \, b^{3}\right )} e^{\left (2 \, x\right )}\right )}}{3 \, b^{4} {\left (e^{\left (2 \, x\right )} + 1\right )}^{3}} \] Input:

integrate(tanh(x)^5/(a+b*tanh(x)),x, algorithm="giac")
 

Output:

a^5*log(abs(a*e^(2*x) + b*e^(2*x) + a - b))/(a^2*b^4 - b^6) - x/(a - b) - 
(a^3 + a*b^2)*log(e^(2*x) + 1)/b^4 + 2/3*(3*a^2*b + 4*b^3 + 3*(a^2*b - a*b 
^2 + 2*b^3)*e^(4*x) + 3*(2*a^2*b - a*b^2 + 2*b^3)*e^(2*x))/(b^4*(e^(2*x) + 
 1)^3)
 

Mupad [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.90 \[ \int \frac {\tanh ^5(x)}{a+b \tanh (x)} \, dx=\frac {x}{a+b}-\frac {{\mathrm {tanh}\left (x\right )}^3}{3\,b}-\frac {a\,\ln \left (\mathrm {tanh}\left (x\right )+1\right )}{a^2-b^2}+\frac {a\,{\mathrm {tanh}\left (x\right )}^2}{2\,b^2}-\frac {\mathrm {tanh}\left (x\right )\,\left (a^2+b^2\right )}{b^3}+\frac {a^5\,\ln \left (a+b\,\mathrm {tanh}\left (x\right )\right )}{b^4\,\left (a^2-b^2\right )} \] Input:

int(tanh(x)^5/(a + b*tanh(x)),x)
 

Output:

x/(a + b) - tanh(x)^3/(3*b) - (a*log(tanh(x) + 1))/(a^2 - b^2) + (a*tanh(x 
)^2)/(2*b^2) - (tanh(x)*(a^2 + b^2))/b^3 + (a^5*log(a + b*tanh(x)))/(b^4*( 
a^2 - b^2))
 

Reduce [B] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.48 \[ \int \frac {\tanh ^5(x)}{a+b \tanh (x)} \, dx=\frac {6 \,\mathrm {log}\left (e^{2 x} a +e^{2 x} b +a -b \right ) a \,b^{4}+6 \,\mathrm {log}\left (\tanh \left (x \right ) b +a \right ) a^{5}-6 \,\mathrm {log}\left (\tanh \left (x \right ) b +a \right ) a \,b^{4}-2 \tanh \left (x \right )^{3} a^{2} b^{3}+2 \tanh \left (x \right )^{3} b^{5}+3 \tanh \left (x \right )^{2} a^{3} b^{2}-3 \tanh \left (x \right )^{2} a \,b^{4}-6 \tanh \left (x \right ) a^{4} b +6 \tanh \left (x \right ) b^{5}-6 a \,b^{4} x -6 b^{5} x}{6 b^{4} \left (a^{2}-b^{2}\right )} \] Input:

int(tanh(x)^5/(a+b*tanh(x)),x)
 

Output:

(6*log(e**(2*x)*a + e**(2*x)*b + a - b)*a*b**4 + 6*log(tanh(x)*b + a)*a**5 
 - 6*log(tanh(x)*b + a)*a*b**4 - 2*tanh(x)**3*a**2*b**3 + 2*tanh(x)**3*b** 
5 + 3*tanh(x)**2*a**3*b**2 - 3*tanh(x)**2*a*b**4 - 6*tanh(x)*a**4*b + 6*ta 
nh(x)*b**5 - 6*a*b**4*x - 6*b**5*x)/(6*b**4*(a**2 - b**2))