\(\int \coth ^5(c+d x) (a+b \tanh ^2(c+d x))^2 \, dx\) [153]

Optimal result

Integrand size = 23, antiderivative size = 72 \[ \int \coth ^5(c+d x) \left (a+b \tanh ^2(c+d x)\right )^2 \, dx=-\frac {a (a+2 b) \coth ^2(c+d x)}{2 d}-\frac {a^2 \coth ^4(c+d x)}{4 d}+\frac {(a+b)^2 \log (\cosh (c+d x))}{d}+\frac {(a+b)^2 \log (\tanh (c+d x))}{d} \] Output:

-1/2*a*(a+2*b)*coth(d*x+c)^2/d-1/4*a^2*coth(d*x+c)^4/d+(a+b)^2*ln(cosh(d*x 
+c))/d+(a+b)^2*ln(tanh(d*x+c))/d
 

Mathematica [A] (verified)

Time = 0.25 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.67 \[ \int \coth ^5(c+d x) \left (a+b \tanh ^2(c+d x)\right )^2 \, dx=-\frac {4 a (a+b) \text {csch}^2(c+d x)+a^2 \text {csch}^4(c+d x)-4 (a+b)^2 \log (\sinh (c+d x))}{4 d} \] Input:

Integrate[Coth[c + d*x]^5*(a + b*Tanh[c + d*x]^2)^2,x]
 

Output:

-1/4*(4*a*(a + b)*Csch[c + d*x]^2 + a^2*Csch[c + d*x]^4 - 4*(a + b)^2*Log[ 
Sinh[c + d*x]])/d
 

Rubi [A] (warning: unable to verify)

Time = 0.32 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {3042, 26, 4153, 26, 354, 99, 2009}

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.

\(\displaystyle \int \coth ^5(c+d x) \left (a+b \tanh ^2(c+d x)\right )^2 \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {i \left (a-b \tan (i c+i d x)^2\right )^2}{\tan (i c+i d x)^5}dx\)

\(\Big \downarrow \) 26

\(\displaystyle i \int \frac {\left (a-b \tan (i c+i d x)^2\right )^2}{\tan (i c+i d x)^5}dx\)

\(\Big \downarrow \) 4153

\(\displaystyle \frac {i \int -\frac {i \coth ^5(c+d x) \left (b \tanh ^2(c+d x)+a\right )^2}{1-\tanh ^2(c+d x)}d\tanh (c+d x)}{d}\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {\int \frac {\coth ^5(c+d x) \left (b \tanh ^2(c+d x)+a\right )^2}{1-\tanh ^2(c+d x)}d\tanh (c+d x)}{d}\)

\(\Big \downarrow \) 354

\(\displaystyle \frac {\int \frac {\coth ^3(c+d x) \left (b \tanh ^2(c+d x)+a\right )^2}{1-\tanh ^2(c+d x)}d\tanh ^2(c+d x)}{2 d}\)

\(\Big \downarrow \) 99

\(\displaystyle \frac {\int \left (a^2 \coth ^3(c+d x)+a (a+2 b) \coth ^2(c+d x)+(a+b)^2 \coth (c+d x)-\frac {(a+b)^2}{\tanh ^2(c+d x)-1}\right )d\tanh ^2(c+d x)}{2 d}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {-\frac {1}{2} a^2 \coth ^2(c+d x)-a (a+2 b) \coth (c+d x)+(a+b)^2 \log \left (\tanh ^2(c+d x)\right )-(a+b)^2 \log \left (1-\tanh ^2(c+d x)\right )}{2 d}\)

Input:

Int[Coth[c + d*x]^5*(a + b*Tanh[c + d*x]^2)^2,x]
 

Output:

(-(a*(a + 2*b)*Coth[c + d*x]) - (a^2*Coth[c + d*x]^2)/2 + (a + b)^2*Log[Ta 
nh[c + d*x]^2] - (a + b)^2*Log[1 - Tanh[c + d*x]^2])/(2*d)
 

Defintions of rubi rules used

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 99

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], 
 x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] && (IntegerQ[p] | 
| (GtQ[m, 0] && GeQ[n, -1]))
 

rule 354

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_S 
ymbol] :> Simp[1/2   Subst[Int[x^((m - 1)/2)*(a + b*x)^p*(c + d*x)^q, x], x 
, x^2], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IntegerQ 
[(m - 1)/2]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

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

rule 4153

Int[((d_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*((c_.)*tan[(e_.) + 
(f_.)*(x_)])^(n_))^(p_.), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], 
 x]}, Simp[c*(ff/f)   Subst[Int[(d*ff*(x/c))^m*((a + b*(ff*x)^n)^p/(c^2 + f 
f^2*x^2)), x], x, c*(Tan[e + f*x]/ff)], x]] /; FreeQ[{a, b, c, d, e, f, m, 
n, p}, x] && (IGtQ[p, 0] || EqQ[n, 2] || EqQ[n, 4] || (IntegerQ[p] && Ratio 
nalQ[n]))
 

Maple [A] (verified)

Time = 0.18 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.07

method	result	size
parallelrisch	\(\frac {-4 \left (a +b \right )^{2} \ln \left (1-\tanh \left (d x +c \right )\right )+4 \left (a +b \right )^{2} \ln \left (\tanh \left (d x +c \right )\right )-\coth \left (d x +c \right )^{4} a^{2}-2 a \coth \left (d x +c \right )^{2} \left (a +2 b \right )-4 d x \left (a +b \right )^{2}}{4 d}\)	\(77\)
derivativedivides	\(-\frac {\left (-a^{2}-2 a b -b^{2}\right ) \ln \left (\tanh \left (d x +c \right )\right )+\frac {a^{2}}{4 \tanh \left (d x +c \right )^{4}}+\frac {a \left (a +2 b \right )}{2 \tanh \left (d x +c \right )^{2}}+\left (\frac {1}{2} a^{2}+a b +\frac {1}{2} b^{2}\right ) \ln \left (1+\tanh \left (d x +c \right )\right )+\left (\frac {1}{2} a^{2}+a b +\frac {1}{2} b^{2}\right ) \ln \left (-1+\tanh \left (d x +c \right )\right )}{d}\)	\(107\)
default	\(-\frac {\left (-a^{2}-2 a b -b^{2}\right ) \ln \left (\tanh \left (d x +c \right )\right )+\frac {a^{2}}{4 \tanh \left (d x +c \right )^{4}}+\frac {a \left (a +2 b \right )}{2 \tanh \left (d x +c \right )^{2}}+\left (\frac {1}{2} a^{2}+a b +\frac {1}{2} b^{2}\right ) \ln \left (1+\tanh \left (d x +c \right )\right )+\left (\frac {1}{2} a^{2}+a b +\frac {1}{2} b^{2}\right ) \ln \left (-1+\tanh \left (d x +c \right )\right )}{d}\)	\(107\)
risch	\(-a^{2} x -2 a b x -b^{2} x -\frac {2 a^{2} c}{d}-\frac {4 a b c}{d}-\frac {2 b^{2} c}{d}-\frac {4 a \,{\mathrm e}^{2 d x +2 c} \left (a \,{\mathrm e}^{4 d x +4 c}+b \,{\mathrm e}^{4 d x +4 c}-a \,{\mathrm e}^{2 d x +2 c}-2 b \,{\mathrm e}^{2 d x +2 c}+a +b \right )}{d \left ({\mathrm e}^{2 d x +2 c}-1\right )^{4}}+\frac {a^{2} \ln \left ({\mathrm e}^{2 d x +2 c}-1\right )}{d}+\frac {2 a \ln \left ({\mathrm e}^{2 d x +2 c}-1\right ) b}{d}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}-1\right ) b^{2}}{d}\)	\(179\)

Input:

int(coth(d*x+c)^5*(a+b*tanh(d*x+c)^2)^2,x,method=_RETURNVERBOSE)
 

Output:

1/4*(-4*(a+b)^2*ln(1-tanh(d*x+c))+4*(a+b)^2*ln(tanh(d*x+c))-coth(d*x+c)^4* 
a^2-2*a*coth(d*x+c)^2*(a+2*b)-4*d*x*(a+b)^2)/d
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1649 vs. \(2 (68) = 136\).

Time = 0.11 (sec) , antiderivative size = 1649, normalized size of antiderivative = 22.90 \[ \int \coth ^5(c+d x) \left (a+b \tanh ^2(c+d x)\right )^2 \, dx=\text {Too large to display} \] Input:

integrate(coth(d*x+c)^5*(a+b*tanh(d*x+c)^2)^2,x, algorithm="fricas")
 

Output:

-((a^2 + 2*a*b + b^2)*d*x*cosh(d*x + c)^8 + 8*(a^2 + 2*a*b + b^2)*d*x*cosh 
(d*x + c)*sinh(d*x + c)^7 + (a^2 + 2*a*b + b^2)*d*x*sinh(d*x + c)^8 - 4*(( 
a^2 + 2*a*b + b^2)*d*x - a^2 - a*b)*cosh(d*x + c)^6 + 4*(7*(a^2 + 2*a*b + 
b^2)*d*x*cosh(d*x + c)^2 - (a^2 + 2*a*b + b^2)*d*x + a^2 + a*b)*sinh(d*x + 
 c)^6 + 8*(7*(a^2 + 2*a*b + b^2)*d*x*cosh(d*x + c)^3 - 3*((a^2 + 2*a*b + b 
^2)*d*x - a^2 - a*b)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(3*(a^2 + 2*a*b + 
b^2)*d*x - 2*a^2 - 4*a*b)*cosh(d*x + c)^4 + 2*(35*(a^2 + 2*a*b + b^2)*d*x* 
cosh(d*x + c)^4 + 3*(a^2 + 2*a*b + b^2)*d*x - 30*((a^2 + 2*a*b + b^2)*d*x 
- a^2 - a*b)*cosh(d*x + c)^2 - 2*a^2 - 4*a*b)*sinh(d*x + c)^4 + 8*(7*(a^2 
+ 2*a*b + b^2)*d*x*cosh(d*x + c)^5 - 10*((a^2 + 2*a*b + b^2)*d*x - a^2 - a 
*b)*cosh(d*x + c)^3 + (3*(a^2 + 2*a*b + b^2)*d*x - 2*a^2 - 4*a*b)*cosh(d*x 
 + c))*sinh(d*x + c)^3 + (a^2 + 2*a*b + b^2)*d*x - 4*((a^2 + 2*a*b + b^2)* 
d*x - a^2 - a*b)*cosh(d*x + c)^2 + 4*(7*(a^2 + 2*a*b + b^2)*d*x*cosh(d*x + 
 c)^6 - 15*((a^2 + 2*a*b + b^2)*d*x - a^2 - a*b)*cosh(d*x + c)^4 - (a^2 + 
2*a*b + b^2)*d*x + 3*(3*(a^2 + 2*a*b + b^2)*d*x - 2*a^2 - 4*a*b)*cosh(d*x 
+ c)^2 + a^2 + a*b)*sinh(d*x + c)^2 - ((a^2 + 2*a*b + b^2)*cosh(d*x + c)^8 
 + 8*(a^2 + 2*a*b + b^2)*cosh(d*x + c)*sinh(d*x + c)^7 + (a^2 + 2*a*b + b^ 
2)*sinh(d*x + c)^8 - 4*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^6 + 4*(7*(a^2 + 2 
*a*b + b^2)*cosh(d*x + c)^2 - a^2 - 2*a*b - b^2)*sinh(d*x + c)^6 + 8*(7*(a 
^2 + 2*a*b + b^2)*cosh(d*x + c)^3 - 3*(a^2 + 2*a*b + b^2)*cosh(d*x + c)...
 

Sympy [F]

\[ \int \coth ^5(c+d x) \left (a+b \tanh ^2(c+d x)\right )^2 \, dx=\int \left (a + b \tanh ^{2}{\left (c + d x \right )}\right )^{2} \coth ^{5}{\left (c + d x \right )}\, dx \] Input:

integrate(coth(d*x+c)**5*(a+b*tanh(d*x+c)**2)**2,x)
 

Output:

Integral((a + b*tanh(c + d*x)**2)**2*coth(c + d*x)**5, x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 236 vs. \(2 (68) = 136\).

Time = 0.04 (sec) , antiderivative size = 236, normalized size of antiderivative = 3.28 \[ \int \coth ^5(c+d x) \left (a+b \tanh ^2(c+d x)\right )^2 \, dx=a^{2} {\left (x + \frac {c}{d} + \frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{d} + \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{d} + \frac {4 \, {\left (e^{\left (-2 \, d x - 2 \, c\right )} - e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )}\right )}}{d {\left (4 \, e^{\left (-2 \, d x - 2 \, c\right )} - 6 \, e^{\left (-4 \, d x - 4 \, c\right )} + 4 \, e^{\left (-6 \, d x - 6 \, c\right )} - e^{\left (-8 \, d x - 8 \, c\right )} - 1\right )}}\right )} + 2 \, a b {\left (x + \frac {c}{d} + \frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{d} + \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{d} + \frac {2 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} - e^{\left (-4 \, d x - 4 \, c\right )} - 1\right )}}\right )} + \frac {b^{2} \log \left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}{d} \] Input:

integrate(coth(d*x+c)^5*(a+b*tanh(d*x+c)^2)^2,x, algorithm="maxima")
                                                                                    
                                                                                    
 

Output:

a^2*(x + c/d + log(e^(-d*x - c) + 1)/d + log(e^(-d*x - c) - 1)/d + 4*(e^(- 
2*d*x - 2*c) - e^(-4*d*x - 4*c) + e^(-6*d*x - 6*c))/(d*(4*e^(-2*d*x - 2*c) 
 - 6*e^(-4*d*x - 4*c) + 4*e^(-6*d*x - 6*c) - e^(-8*d*x - 8*c) - 1))) + 2*a 
*b*(x + c/d + log(e^(-d*x - c) + 1)/d + log(e^(-d*x - c) - 1)/d + 2*e^(-2* 
d*x - 2*c)/(d*(2*e^(-2*d*x - 2*c) - e^(-4*d*x - 4*c) - 1))) + b^2*log(e^(d 
*x + c) - e^(-d*x - c))/d
 

Giac [A] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.64 \[ \int \coth ^5(c+d x) \left (a+b \tanh ^2(c+d x)\right )^2 \, dx=-\frac {{\left (a^{2} + 2 \, a b + b^{2}\right )} {\left (d x + c\right )} - {\left (a^{2} + 2 \, a b + b^{2}\right )} \log \left ({\left | e^{\left (2 \, d x + 2 \, c\right )} - 1 \right |}\right ) + \frac {4 \, {\left ({\left (a^{2} + a b\right )} e^{\left (6 \, d x + 6 \, c\right )} - {\left (a^{2} + 2 \, a b\right )} e^{\left (4 \, d x + 4 \, c\right )} + {\left (a^{2} + a b\right )} e^{\left (2 \, d x + 2 \, c\right )}\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{4}}}{d} \] Input:

integrate(coth(d*x+c)^5*(a+b*tanh(d*x+c)^2)^2,x, algorithm="giac")
 

Output:

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

Mupad [B] (verification not implemented)

Time = 2.50 (sec) , antiderivative size = 197, normalized size of antiderivative = 2.74 \[ \int \coth ^5(c+d x) \left (a+b \tanh ^2(c+d x)\right )^2 \, dx=\frac {\ln \left ({\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}-1\right )\,\left (a^2+2\,a\,b+b^2\right )}{d}-x\,{\left (a+b\right )}^2-\frac {4\,\left (2\,a^2+b\,a\right )}{d\,\left ({\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}-\frac {8\,a^2}{d\,\left (3\,{\mathrm {e}}^{2\,c+2\,d\,x}-3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}-1\right )}-\frac {4\,a^2}{d\,\left (6\,{\mathrm {e}}^{4\,c+4\,d\,x}-4\,{\mathrm {e}}^{2\,c+2\,d\,x}-4\,{\mathrm {e}}^{6\,c+6\,d\,x}+{\mathrm {e}}^{8\,c+8\,d\,x}+1\right )}-\frac {4\,\left (a^2+b\,a\right )}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )} \] Input:

int(coth(c + d*x)^5*(a + b*tanh(c + d*x)^2)^2,x)
 

Output:

(log(exp(2*c)*exp(2*d*x) - 1)*(2*a*b + a^2 + b^2))/d - x*(a + b)^2 - (4*(a 
*b + 2*a^2))/(d*(exp(4*c + 4*d*x) - 2*exp(2*c + 2*d*x) + 1)) - (8*a^2)/(d* 
(3*exp(2*c + 2*d*x) - 3*exp(4*c + 4*d*x) + exp(6*c + 6*d*x) - 1)) - (4*a^2 
)/(d*(6*exp(4*c + 4*d*x) - 4*exp(2*c + 2*d*x) - 4*exp(6*c + 6*d*x) + exp(8 
*c + 8*d*x) + 1)) - (4*(a*b + a^2))/(d*(exp(2*c + 2*d*x) - 1))
 

Reduce [B] (verification not implemented)

Time = 0.34 (sec) , antiderivative size = 1014, normalized size of antiderivative = 14.08 \[ \int \coth ^5(c+d x) \left (a+b \tanh ^2(c+d x)\right )^2 \, dx =\text {Too large to display} \] Input:

int(coth(d*x+c)^5*(a+b*tanh(d*x+c)^2)^2,x)
 

Output:

(e**(8*c + 8*d*x)*log(e**(c + d*x) - 1)*a**2 + 2*e**(8*c + 8*d*x)*log(e**( 
c + d*x) - 1)*a*b + e**(8*c + 8*d*x)*log(e**(c + d*x) - 1)*b**2 + e**(8*c 
+ 8*d*x)*log(e**(c + d*x) + 1)*a**2 + 2*e**(8*c + 8*d*x)*log(e**(c + d*x) 
+ 1)*a*b + e**(8*c + 8*d*x)*log(e**(c + d*x) + 1)*b**2 - e**(8*c + 8*d*x)* 
a**2*d*x - e**(8*c + 8*d*x)*a**2 - 2*e**(8*c + 8*d*x)*a*b*d*x - e**(8*c + 
8*d*x)*a*b - e**(8*c + 8*d*x)*b**2*d*x - 4*e**(6*c + 6*d*x)*log(e**(c + d* 
x) - 1)*a**2 - 8*e**(6*c + 6*d*x)*log(e**(c + d*x) - 1)*a*b - 4*e**(6*c + 
6*d*x)*log(e**(c + d*x) - 1)*b**2 - 4*e**(6*c + 6*d*x)*log(e**(c + d*x) + 
1)*a**2 - 8*e**(6*c + 6*d*x)*log(e**(c + d*x) + 1)*a*b - 4*e**(6*c + 6*d*x 
)*log(e**(c + d*x) + 1)*b**2 + 4*e**(6*c + 6*d*x)*a**2*d*x + 8*e**(6*c + 6 
*d*x)*a*b*d*x + 4*e**(6*c + 6*d*x)*b**2*d*x + 6*e**(4*c + 4*d*x)*log(e**(c 
 + d*x) - 1)*a**2 + 12*e**(4*c + 4*d*x)*log(e**(c + d*x) - 1)*a*b + 6*e**( 
4*c + 4*d*x)*log(e**(c + d*x) - 1)*b**2 + 6*e**(4*c + 4*d*x)*log(e**(c + d 
*x) + 1)*a**2 + 12*e**(4*c + 4*d*x)*log(e**(c + d*x) + 1)*a*b + 6*e**(4*c 
+ 4*d*x)*log(e**(c + d*x) + 1)*b**2 - 6*e**(4*c + 4*d*x)*a**2*d*x - 2*e**( 
4*c + 4*d*x)*a**2 - 12*e**(4*c + 4*d*x)*a*b*d*x + 2*e**(4*c + 4*d*x)*a*b - 
 6*e**(4*c + 4*d*x)*b**2*d*x - 4*e**(2*c + 2*d*x)*log(e**(c + d*x) - 1)*a* 
*2 - 8*e**(2*c + 2*d*x)*log(e**(c + d*x) - 1)*a*b - 4*e**(2*c + 2*d*x)*log 
(e**(c + d*x) - 1)*b**2 - 4*e**(2*c + 2*d*x)*log(e**(c + d*x) + 1)*a**2 - 
8*e**(2*c + 2*d*x)*log(e**(c + d*x) + 1)*a*b - 4*e**(2*c + 2*d*x)*log(e...