3.2.63 \(\int \coth ^3(c+d x) (a+b \tanh ^2(c+d x))^3 \, dx\) [163]

3.2.63.1 Optimal result

Integrand size = 23, antiderivative size = 72 \[ \int \coth ^3(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=-\frac {a^3 \coth ^2(c+d x)}{2 d}+\frac {(a+b)^3 \log (\cosh (c+d x))}{d}+\frac {a^2 (a+3 b) \log (\tanh (c+d x))}{d}-\frac {b^3 \tanh ^2(c+d x)}{2 d} \]

output

-1/2*a^3*coth(d*x+c)^2/d+(a+b)^3*ln(cosh(d*x+c))/d+a^2*(a+3*b)*ln(tanh(d*x 
+c))/d-1/2*b^3*tanh(d*x+c)^2/d
 

3.2.63.2 Mathematica [A] (verified)

Time = 0.49 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.88 \[ \int \coth ^3(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=-\frac {a^3 \coth ^2(c+d x)-2 (a+b)^3 \log (\cosh (c+d x))-2 a^2 (a+3 b) \log (\tanh (c+d x))+b^3 \tanh ^2(c+d x)}{2 d} \]

input

Integrate[Coth[c + d*x]^3*(a + b*Tanh[c + d*x]^2)^3,x]
 

output

-1/2*(a^3*Coth[c + d*x]^2 - 2*(a + b)^3*Log[Cosh[c + d*x]] - 2*a^2*(a + 3* 
b)*Log[Tanh[c + d*x]] + b^3*Tanh[c + d*x]^2)/d
 

3.2.63.3 Rubi [A] (warning: unable to verify)

Time = 0.29 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.97, 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.

\(\displaystyle \int \coth ^3(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 4153

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 354

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

\(\Big \downarrow \) 99

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

\(\Big \downarrow \) 2009

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

input

Int[Coth[c + d*x]^3*(a + b*Tanh[c + d*x]^2)^3,x]
 

output

(-(a^3*Coth[c + d*x]) + a^2*(a + 3*b)*Log[Tanh[c + d*x]^2] - (a + b)^3*Log 
[1 - Tanh[c + d*x]^2] - b^3*Tanh[c + d*x]^2)/(2*d)
 

3.2.63.3.1 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]))
 

3.2.63.4 Maple [A] (verified)

Time = 0.25 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.07

method	result	size
parallelrisch	\(\frac {-2 \left (a +b \right )^{3} \ln \left (1-\tanh \left (d x +c \right )\right )+2 a^{2} \left (a +3 b \right ) \ln \left (\tanh \left (d x +c \right )\right )-\coth \left (d x +c \right )^{2} a^{3}-b^{3} \tanh \left (d x +c \right )^{2}-2 d x \left (a +b \right )^{3}}{2 d}\)	\(77\)
derivativedivides	\(-\frac {\frac {b^{3} \tanh \left (d x +c \right )^{2}}{2}+\left (\frac {1}{2} a^{3}+\frac {3}{2} a^{2} b +\frac {3}{2} a \,b^{2}+\frac {1}{2} b^{3}\right ) \ln \left (\tanh \left (d x +c \right )+1\right )+\frac {a^{3}}{2 \tanh \left (d x +c \right )^{2}}-a^{2} \left (a +3 b \right ) \ln \left (\tanh \left (d x +c \right )\right )+\left (\frac {1}{2} a^{3}+\frac {3}{2} a^{2} b +\frac {3}{2} a \,b^{2}+\frac {1}{2} b^{3}\right ) \ln \left (\tanh \left (d x +c \right )-1\right )}{d}\)	\(116\)
default	\(-\frac {\frac {b^{3} \tanh \left (d x +c \right )^{2}}{2}+\left (\frac {1}{2} a^{3}+\frac {3}{2} a^{2} b +\frac {3}{2} a \,b^{2}+\frac {1}{2} b^{3}\right ) \ln \left (\tanh \left (d x +c \right )+1\right )+\frac {a^{3}}{2 \tanh \left (d x +c \right )^{2}}-a^{2} \left (a +3 b \right ) \ln \left (\tanh \left (d x +c \right )\right )+\left (\frac {1}{2} a^{3}+\frac {3}{2} a^{2} b +\frac {3}{2} a \,b^{2}+\frac {1}{2} b^{3}\right ) \ln \left (\tanh \left (d x +c \right )-1\right )}{d}\)	\(116\)
risch	\(-a^{3} x -3 b \,a^{2} x -3 a \,b^{2} x -b^{3} x -\frac {6 a \,b^{2} c}{d}-\frac {2 b^{3} c}{d}-\frac {2 a^{3} c}{d}-\frac {6 b c \,a^{2}}{d}-\frac {2 \,{\mathrm e}^{2 d x +2 c} \left (a^{3} {\mathrm e}^{4 d x +4 c}-{\mathrm e}^{4 d x +4 c} b^{3}+2 a^{3} {\mathrm e}^{2 d x +2 c}+2 \,{\mathrm e}^{2 d x +2 c} b^{3}+a^{3}-b^{3}\right )}{d \left ({\mathrm e}^{2 d x +2 c}+1\right )^{2} \left ({\mathrm e}^{2 d x +2 c}-1\right )^{2}}+\frac {3 \ln \left ({\mathrm e}^{2 d x +2 c}+1\right ) a \,b^{2}}{d}+\frac {b^{3} \ln \left ({\mathrm e}^{2 d x +2 c}+1\right )}{d}+\frac {a^{3} \ln \left ({\mathrm e}^{2 d x +2 c}-1\right )}{d}+\frac {3 a^{2} \ln \left ({\mathrm e}^{2 d x +2 c}-1\right ) b}{d}\)	\(250\)

input

int(coth(d*x+c)^3*(a+b*tanh(d*x+c)^2)^3,x,method=_RETURNVERBOSE)
 

output

1/2*(-2*(a+b)^3*ln(1-tanh(d*x+c))+2*a^2*(a+3*b)*ln(tanh(d*x+c))-coth(d*x+c 
)^2*a^3-b^3*tanh(d*x+c)^2-2*d*x*(a+b)^3)/d
 

3.2.63.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1686 vs. \(2 (68) = 136\).

Time = 0.27 (sec) , antiderivative size = 1686, normalized size of antiderivative = 23.42 \[ \int \coth ^3(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=\text {Too large to display} \]

input

integrate(coth(d*x+c)^3*(a+b*tanh(d*x+c)^2)^3,x, algorithm="fricas")
 

output

-((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*x*cosh(d*x + c)^8 + 8*(a^3 + 3*a^2*b + 
 3*a*b^2 + b^3)*d*x*cosh(d*x + c)*sinh(d*x + c)^7 + (a^3 + 3*a^2*b + 3*a*b 
^2 + b^3)*d*x*sinh(d*x + c)^8 + 2*(a^3 - b^3)*cosh(d*x + c)^6 + 2*(14*(a^3 
 + 3*a^2*b + 3*a*b^2 + b^3)*d*x*cosh(d*x + c)^2 + a^3 - b^3)*sinh(d*x + c) 
^6 + 4*(14*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*x*cosh(d*x + c)^3 + 3*(a^3 - 
b^3)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(2*a^3 + 2*b^3 - (a^3 + 3*a^2*b + 
3*a*b^2 + b^3)*d*x)*cosh(d*x + c)^4 + 2*(35*(a^3 + 3*a^2*b + 3*a*b^2 + b^3 
)*d*x*cosh(d*x + c)^4 + 2*a^3 + 2*b^3 - (a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d* 
x + 15*(a^3 - b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 8*(7*(a^3 + 3*a^2*b 
+ 3*a*b^2 + b^3)*d*x*cosh(d*x + c)^5 + 5*(a^3 - b^3)*cosh(d*x + c)^3 + (2* 
a^3 + 2*b^3 - (a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*x)*cosh(d*x + c))*sinh(d*x 
 + c)^3 + (a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*x + 2*(a^3 - b^3)*cosh(d*x + c 
)^2 + 2*(14*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*x*cosh(d*x + c)^6 + 15*(a^3 
- b^3)*cosh(d*x + c)^4 + a^3 - b^3 + 6*(2*a^3 + 2*b^3 - (a^3 + 3*a^2*b + 3 
*a*b^2 + b^3)*d*x)*cosh(d*x + c)^2)*sinh(d*x + c)^2 - ((3*a*b^2 + b^3)*cos 
h(d*x + c)^8 + 56*(3*a*b^2 + b^3)*cosh(d*x + c)^3*sinh(d*x + c)^5 + 28*(3* 
a*b^2 + b^3)*cosh(d*x + c)^2*sinh(d*x + c)^6 + 8*(3*a*b^2 + b^3)*cosh(d*x 
+ c)*sinh(d*x + c)^7 + (3*a*b^2 + b^3)*sinh(d*x + c)^8 - 2*(3*a*b^2 + b^3) 
*cosh(d*x + c)^4 + 2*(35*(3*a*b^2 + b^3)*cosh(d*x + c)^4 - 3*a*b^2 - b^3)* 
sinh(d*x + c)^4 + 8*(7*(3*a*b^2 + b^3)*cosh(d*x + c)^5 - (3*a*b^2 + b^3...
 

3.2.63.6 Sympy [F]

\[ \int \coth ^3(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=\int \left (a + b \tanh ^{2}{\left (c + d x \right )}\right )^{3} \coth ^{3}{\left (c + d x \right )}\, dx \]

input

integrate(coth(d*x+c)**3*(a+b*tanh(d*x+c)**2)**3,x)
 

output

Integral((a + b*tanh(c + d*x)**2)**3*coth(c + d*x)**3, x)
 

3.2.63.7 Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 203 vs. \(2 (68) = 136\).

Time = 0.29 (sec) , antiderivative size = 203, normalized size of antiderivative = 2.82 \[ \int \coth ^3(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=a^{3} {\left (x + \frac {c}{d} + \frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{d} + \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{d} + \frac {2 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} - e^{\left (-4 \, d x - 4 \, c\right )} - 1\right )}}\right )} + b^{3} {\left (x + \frac {c}{d} + \frac {\log \left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}{d} + \frac {2 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} + e^{\left (-4 \, d x - 4 \, c\right )} + 1\right )}}\right )} + \frac {3 \, a b^{2} \log \left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}{d} + \frac {3 \, a^{2} b \log \left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}{d} \]

input

integrate(coth(d*x+c)^3*(a+b*tanh(d*x+c)^2)^3,x, algorithm="maxima")
 

output

a^3*(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^3*(x + c/ 
d + log(e^(-2*d*x - 2*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))) + 3*a*b^2*log(e^(d*x + c) + e^(-d*x - c))/d + 
3*a^2*b*log(e^(d*x + c) - e^(-d*x - c))/d
 

3.2.63.8 Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 274 vs. \(2 (68) = 136\).

Time = 0.47 (sec) , antiderivative size = 274, normalized size of antiderivative = 3.81 \[ \int \coth ^3(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=\frac {2 \, {\left (3 \, a b^{2} + b^{3}\right )} \log \left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )} + 2\right ) + 2 \, {\left (a^{3} + 3 \, a^{2} b\right )} \log \left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )} - 2\right ) - \frac {a^{3} {\left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )}\right )}^{2} + 3 \, a^{2} b {\left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )}\right )}^{2} + 3 \, a b^{2} {\left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )}\right )}^{2} + b^{3} {\left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )}\right )}^{2} + 8 \, a^{3} {\left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )}\right )} - 8 \, b^{3} {\left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )}\right )} + 12 \, a^{3} - 12 \, a^{2} b - 12 \, a b^{2} + 12 \, b^{3}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )}\right )}^{2} - 4}}{4 \, d} \]

input

integrate(coth(d*x+c)^3*(a+b*tanh(d*x+c)^2)^3,x, algorithm="giac")
 

output

1/4*(2*(3*a*b^2 + b^3)*log(e^(2*d*x + 2*c) + e^(-2*d*x - 2*c) + 2) + 2*(a^ 
3 + 3*a^2*b)*log(e^(2*d*x + 2*c) + e^(-2*d*x - 2*c) - 2) - (a^3*(e^(2*d*x 
+ 2*c) + e^(-2*d*x - 2*c))^2 + 3*a^2*b*(e^(2*d*x + 2*c) + e^(-2*d*x - 2*c) 
)^2 + 3*a*b^2*(e^(2*d*x + 2*c) + e^(-2*d*x - 2*c))^2 + b^3*(e^(2*d*x + 2*c 
) + e^(-2*d*x - 2*c))^2 + 8*a^3*(e^(2*d*x + 2*c) + e^(-2*d*x - 2*c)) - 8*b 
^3*(e^(2*d*x + 2*c) + e^(-2*d*x - 2*c)) + 12*a^3 - 12*a^2*b - 12*a*b^2 + 1 
2*b^3)/((e^(2*d*x + 2*c) + e^(-2*d*x - 2*c))^2 - 4))/d
 

3.2.63.9 Mupad [B] (verification not implemented)

Time = 2.91 (sec) , antiderivative size = 327, normalized size of antiderivative = 4.54 \[ \int \coth ^3(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=\frac {\ln \left ({\mathrm {e}}^{4\,c+4\,d\,x}-1\right )\,\left (d\,a^3+3\,d\,a^2\,b+3\,d\,a\,b^2+d\,b^3\right )}{2\,d^2}-\frac {\frac {4\,\left (a^3+b^3\right )}{d}+\frac {2\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (a^3-b^3\right )}{d}}{{\mathrm {e}}^{4\,c+4\,d\,x}-1}-\frac {\frac {4\,\left (a^3+b^3\right )}{d}+\frac {4\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (a^3-b^3\right )}{d}}{{\mathrm {e}}^{8\,c+8\,d\,x}-2\,{\mathrm {e}}^{4\,c+4\,d\,x}+1}-\frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}\,\left (a^3\,\sqrt {-d^2}-b^3\,\sqrt {-d^2}-3\,a\,b^2\,\sqrt {-d^2}+3\,a^2\,b\,\sqrt {-d^2}\right )}{d\,\sqrt {a^6+6\,a^5\,b+3\,a^4\,b^2-20\,a^3\,b^3+3\,a^2\,b^4+6\,a\,b^5+b^6}}\right )\,\sqrt {a^6+6\,a^5\,b+3\,a^4\,b^2-20\,a^3\,b^3+3\,a^2\,b^4+6\,a\,b^5+b^6}}{\sqrt {-d^2}}-x\,{\left (a+b\right )}^3 \]

input

int(coth(c + d*x)^3*(a + b*tanh(c + d*x)^2)^3,x)
 

output

(log(exp(4*c + 4*d*x) - 1)*(a^3*d + b^3*d + 3*a*b^2*d + 3*a^2*b*d))/(2*d^2 
) - ((4*(a^3 + b^3))/d + (2*exp(2*c + 2*d*x)*(a^3 - b^3))/d)/(exp(4*c + 4* 
d*x) - 1) - ((4*(a^3 + b^3))/d + (4*exp(2*c + 2*d*x)*(a^3 - b^3))/d)/(exp( 
8*c + 8*d*x) - 2*exp(4*c + 4*d*x) + 1) - (atan((exp(2*c)*exp(2*d*x)*(a^3*( 
-d^2)^(1/2) - b^3*(-d^2)^(1/2) - 3*a*b^2*(-d^2)^(1/2) + 3*a^2*b*(-d^2)^(1/ 
2)))/(d*(6*a*b^5 + 6*a^5*b + a^6 + b^6 + 3*a^2*b^4 - 20*a^3*b^3 + 3*a^4*b^ 
2)^(1/2)))*(6*a*b^5 + 6*a^5*b + a^6 + b^6 + 3*a^2*b^4 - 20*a^3*b^3 + 3*a^4 
*b^2)^(1/2))/(-d^2)^(1/2) - x*(a + b)^3