\(\int \frac {\tanh ^6(c+d x)}{(a+b \text {sech}^2(c+d x))^3} \, dx\) [158]

Optimal result

Integrand size = 23, antiderivative size = 148 \[ \int \frac {\tanh ^6(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\frac {x}{a^3}-\frac {\sqrt {a+b} \left (3 a^2-4 a b+8 b^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{8 a^3 b^{5/2} d}-\frac {(a+b) \tanh ^3(c+d x)}{4 a b d \left (a+b-b \tanh ^2(c+d x)\right )^2}+\frac {(3 a-4 b) (a+b) \tanh (c+d x)}{8 a^2 b^2 d \left (a+b-b \tanh ^2(c+d x)\right )} \] Output:

x/a^3-1/8*(a+b)^(1/2)*(3*a^2-4*a*b+8*b^2)*arctanh(b^(1/2)*tanh(d*x+c)/(a+b 
)^(1/2))/a^3/b^(5/2)/d-1/4*(a+b)*tanh(d*x+c)^3/a/b/d/(a+b-b*tanh(d*x+c)^2) 
^2+1/8*(3*a-4*b)*(a+b)*tanh(d*x+c)/a^2/b^2/d/(a+b-b*tanh(d*x+c)^2)
                                                                                    
                                                                                    
 

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(515\) vs. \(2(148)=296\).

Time = 4.55 (sec) , antiderivative size = 515, normalized size of antiderivative = 3.48 \[ \int \frac {\tanh ^6(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\frac {(a+2 b+a \cosh (2 (c+d x))) \text {sech}^6(c+d x) \left (-\frac {2 \left (3 a^3-a^2 b+4 a b^2+8 b^3\right ) \text {arctanh}\left (\frac {\text {sech}(d x) (\cosh (2 c)-\sinh (2 c)) ((a+2 b) \sinh (d x)-a \sinh (2 c+d x))}{2 \sqrt {a+b} \sqrt {b (\cosh (c)-\sinh (c))^4}}\right ) (a+2 b+a \cosh (2 (c+d x)))^2 (\cosh (2 c)-\sinh (2 c))}{\sqrt {a+b} \sqrt {b (\cosh (c)-\sinh (c))^4}}+\text {sech}(2 c) \left (8 b^2 \left (3 a^2+8 a b+8 b^2\right ) d x \cosh (2 c)+16 a b^2 (a+2 b) d x \cosh (2 d x)+4 a^2 b^2 d x \cosh (2 (c+2 d x))+16 a^2 b^2 d x \cosh (4 c+2 d x)+32 a b^3 d x \cosh (4 c+2 d x)+4 a^2 b^2 d x \cosh (6 c+4 d x)-9 a^4 \sinh (2 c)-15 a^3 b \sinh (2 c)+18 a^2 b^2 \sinh (2 c)+72 a b^3 \sinh (2 c)+48 b^4 \sinh (2 c)+9 a^4 \sinh (2 d x)+13 a^3 b \sinh (2 d x)-28 a^2 b^2 \sinh (2 d x)-32 a b^3 \sinh (2 d x)+3 a^4 \sinh (2 (c+2 d x))-3 a^3 b \sinh (2 (c+2 d x))-6 a^2 b^2 \sinh (2 (c+2 d x))-3 a^4 \sinh (4 c+2 d x)+a^3 b \sinh (4 c+2 d x)+20 a^2 b^2 \sinh (4 c+2 d x)+16 a b^3 \sinh (4 c+2 d x)\right )\right )}{128 a^3 b^2 d \left (a+b \text {sech}^2(c+d x)\right )^3} \] Input:

Integrate[Tanh[c + d*x]^6/(a + b*Sech[c + d*x]^2)^3,x]
 

Output:

((a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d*x]^6*((-2*(3*a^3 - a^2*b + 4*a 
*b^2 + 8*b^3)*ArcTanh[(Sech[d*x]*(Cosh[2*c] - Sinh[2*c])*((a + 2*b)*Sinh[d 
*x] - a*Sinh[2*c + d*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cosh[c] - Sinh[c])^4])]*( 
a + 2*b + a*Cosh[2*(c + d*x)])^2*(Cosh[2*c] - Sinh[2*c]))/(Sqrt[a + b]*Sqr 
t[b*(Cosh[c] - Sinh[c])^4]) + Sech[2*c]*(8*b^2*(3*a^2 + 8*a*b + 8*b^2)*d*x 
*Cosh[2*c] + 16*a*b^2*(a + 2*b)*d*x*Cosh[2*d*x] + 4*a^2*b^2*d*x*Cosh[2*(c 
+ 2*d*x)] + 16*a^2*b^2*d*x*Cosh[4*c + 2*d*x] + 32*a*b^3*d*x*Cosh[4*c + 2*d 
*x] + 4*a^2*b^2*d*x*Cosh[6*c + 4*d*x] - 9*a^4*Sinh[2*c] - 15*a^3*b*Sinh[2* 
c] + 18*a^2*b^2*Sinh[2*c] + 72*a*b^3*Sinh[2*c] + 48*b^4*Sinh[2*c] + 9*a^4* 
Sinh[2*d*x] + 13*a^3*b*Sinh[2*d*x] - 28*a^2*b^2*Sinh[2*d*x] - 32*a*b^3*Sin 
h[2*d*x] + 3*a^4*Sinh[2*(c + 2*d*x)] - 3*a^3*b*Sinh[2*(c + 2*d*x)] - 6*a^2 
*b^2*Sinh[2*(c + 2*d*x)] - 3*a^4*Sinh[4*c + 2*d*x] + a^3*b*Sinh[4*c + 2*d* 
x] + 20*a^2*b^2*Sinh[4*c + 2*d*x] + 16*a*b^3*Sinh[4*c + 2*d*x])))/(128*a^3 
*b^2*d*(a + b*Sech[c + d*x]^2)^3)
 

Rubi [A] (verified)

Time = 0.56 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.17, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {3042, 25, 4629, 25, 2075, 372, 440, 397, 219, 221}

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.

\(\displaystyle \int \frac {\tanh ^6(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int -\frac {\tan (i c+i d x)^6}{\left (a+b \sec (i c+i d x)^2\right )^3}dx\)

\(\Big \downarrow \) 25

\(\displaystyle -\int \frac {\tan (i c+i d x)^6}{\left (b \sec (i c+i d x)^2+a\right )^3}dx\)

\(\Big \downarrow \) 4629

\(\displaystyle -\frac {\int -\frac {\tanh ^6(c+d x)}{\left (1-\tanh ^2(c+d x)\right ) \left (a+b \left (1-\tanh ^2(c+d x)\right )\right )^3}d\tanh (c+d x)}{d}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {\tanh ^6(c+d x)}{\left (1-\tanh ^2(c+d x)\right ) \left (a+b \left (1-\tanh ^2(c+d x)\right )\right )^3}d\tanh (c+d x)}{d}\)

\(\Big \downarrow \) 2075

\(\displaystyle \frac {\int \frac {\tanh ^6(c+d x)}{\left (1-\tanh ^2(c+d x)\right ) \left (-b \tanh ^2(c+d x)+a+b\right )^3}d\tanh (c+d x)}{d}\)

\(\Big \downarrow \) 372

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

\(\Big \downarrow \) 440

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

\(\Big \downarrow \) 397

\(\displaystyle -\frac {\frac {(a+b) \tanh ^3(c+d x)}{4 a b \left (a-b \tanh ^2(c+d x)+b\right )^2}-\frac {\frac {(3 a-4 b) (a+b) \tanh (c+d x)}{2 a b \left (a-b \tanh ^2(c+d x)+b\right )}-\frac {\frac {(a+b) \left (3 a^2-4 a b+8 b^2\right ) \int \frac {1}{-b \tanh ^2(c+d x)+a+b}d\tanh (c+d x)}{a}-\frac {8 b^2 \int \frac {1}{1-\tanh ^2(c+d x)}d\tanh (c+d x)}{a}}{2 a b}}{4 a b}}{d}\)

\(\Big \downarrow \) 219

\(\displaystyle -\frac {\frac {(a+b) \tanh ^3(c+d x)}{4 a b \left (a-b \tanh ^2(c+d x)+b\right )^2}-\frac {\frac {(3 a-4 b) (a+b) \tanh (c+d x)}{2 a b \left (a-b \tanh ^2(c+d x)+b\right )}-\frac {\frac {(a+b) \left (3 a^2-4 a b+8 b^2\right ) \int \frac {1}{-b \tanh ^2(c+d x)+a+b}d\tanh (c+d x)}{a}-\frac {8 b^2 \text {arctanh}(\tanh (c+d x))}{a}}{2 a b}}{4 a b}}{d}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {\frac {(a+b) \tanh ^3(c+d x)}{4 a b \left (a-b \tanh ^2(c+d x)+b\right )^2}-\frac {\frac {(3 a-4 b) (a+b) \tanh (c+d x)}{2 a b \left (a-b \tanh ^2(c+d x)+b\right )}-\frac {\frac {\sqrt {a+b} \left (3 a^2-4 a b+8 b^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{a \sqrt {b}}-\frac {8 b^2 \text {arctanh}(\tanh (c+d x))}{a}}{2 a b}}{4 a b}}{d}\)

Input:

Int[Tanh[c + d*x]^6/(a + b*Sech[c + d*x]^2)^3,x]
 

Output:

-((((a + b)*Tanh[c + d*x]^3)/(4*a*b*(a + b - b*Tanh[c + d*x]^2)^2) - (-1/2 
*((-8*b^2*ArcTanh[Tanh[c + d*x]])/a + (Sqrt[a + b]*(3*a^2 - 4*a*b + 8*b^2) 
*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(a*Sqrt[b]))/(a*b) + ((3*a 
- 4*b)*(a + b)*Tanh[c + d*x])/(2*a*b*(a + b - b*Tanh[c + d*x]^2)))/(4*a*b) 
)/d)
 

Defintions of rubi rules used

rule 25

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

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

rule 372

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ 
), x_Symbol] :> Simp[(-a)*e^3*(e*x)^(m - 3)*(a + b*x^2)^(p + 1)*((c + d*x^2 
)^(q + 1)/(2*b*(b*c - a*d)*(p + 1))), x] + Simp[e^4/(2*b*(b*c - a*d)*(p + 1 
))   Int[(e*x)^(m - 4)*(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[a*c*(m - 3) + 
 (a*d*(m + 2*q - 1) + 2*b*c*(p + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, 
e, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && GtQ[m, 3] && IntBinomialQ[a 
, b, c, d, e, m, 2, p, q, x]
 

rule 397

Int[((e_) + (f_.)*(x_)^2)/(((a_) + (b_.)*(x_)^2)*((c_) + (d_.)*(x_)^2)), x_ 
Symbol] :> Simp[(b*e - a*f)/(b*c - a*d)   Int[1/(a + b*x^2), x], x] - Simp[ 
(d*e - c*f)/(b*c - a*d)   Int[1/(c + d*x^2), x], x] /; FreeQ[{a, b, c, d, e 
, f}, x]
 

rule 440

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ 
)*((e_) + (f_.)*(x_)^2), x_Symbol] :> Simp[g*(b*e - a*f)*(g*x)^(m - 1)*(a + 
 b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(2*b*(b*c - a*d)*(p + 1))), x] - Simp[ 
g^2/(2*b*(b*c - a*d)*(p + 1))   Int[(g*x)^(m - 2)*(a + b*x^2)^(p + 1)*(c + 
d*x^2)^q*Simp[c*(b*e - a*f)*(m - 1) + (d*(b*e - a*f)*(m + 2*q + 1) - b*2*(c 
*f - d*e)*(p + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, q}, x] && 
 LtQ[p, -1] && GtQ[m, 1]
 

rule 2075

Int[(u_)^(p_.)*(v_)^(q_.)*((e_.)*(x_))^(m_.), x_Symbol] :> Int[(e*x)^m*Expa 
ndToSum[u, x]^p*ExpandToSum[v, x]^q, x] /; FreeQ[{e, m, p, q}, x] && Binomi 
alQ[{u, v}, x] && EqQ[BinomialDegree[u, x] - BinomialDegree[v, x], 0] &&  ! 
BinomialMatchQ[{u, v}, x]
 

rule 3042

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

rule 4629

Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_.)*((d_.)*tan[(e_.) + (f 
_.)*(x_)])^(m_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Sim 
p[ff/f   Subst[Int[(d*ff*x)^m*((a + b*(1 + ff^2*x^2)^(n/2))^p/(1 + ff^2*x^2 
)), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, d, e, f, m, p}, x] && Inte 
gerQ[n/2] && (IntegerQ[m/2] || EqQ[n, 2])
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(388\) vs. \(2(134)=268\).

Time = 185.05 (sec) , antiderivative size = 389, normalized size of antiderivative = 2.63

method	result	size
derivativedivides	\(\frac {\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a^{3}}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a^{3}}+\frac {\frac {2 \left (\frac {a \left (3 a^{3}+2 a^{2} b -5 a \,b^{2}-4 b^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{8 b^{2}}+\frac {\left (9 a^{3}-14 a^{2} b -19 a \,b^{2}+4 b^{3}\right ) a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{8 b^{2}}+\frac {\left (9 a^{3}-14 a^{2} b -19 a \,b^{2}+4 b^{3}\right ) a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{8 b^{2}}+\frac {a \left (3 a^{3}+2 a^{2} b -5 a \,b^{2}-4 b^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 b^{2}}\right )}{\left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} b +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a +b \right )^{2}}+\frac {\left (3 a^{3}-a^{2} b +4 a \,b^{2}+8 b^{3}\right ) \left (-\frac {\ln \left (\sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}+\frac {\ln \left (\sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}\right )}{4 b^{2}}}{a^{3}}}{d}\)	\(389\)
default	\(\frac {\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a^{3}}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a^{3}}+\frac {\frac {2 \left (\frac {a \left (3 a^{3}+2 a^{2} b -5 a \,b^{2}-4 b^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{8 b^{2}}+\frac {\left (9 a^{3}-14 a^{2} b -19 a \,b^{2}+4 b^{3}\right ) a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{8 b^{2}}+\frac {\left (9 a^{3}-14 a^{2} b -19 a \,b^{2}+4 b^{3}\right ) a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{8 b^{2}}+\frac {a \left (3 a^{3}+2 a^{2} b -5 a \,b^{2}-4 b^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 b^{2}}\right )}{\left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} b +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a +b \right )^{2}}+\frac {\left (3 a^{3}-a^{2} b +4 a \,b^{2}+8 b^{3}\right ) \left (-\frac {\ln \left (\sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}+\frac {\ln \left (\sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}\right )}{4 b^{2}}}{a^{3}}}{d}\)	\(389\)
risch	\(\frac {x}{a^{3}}-\frac {3 \,{\mathrm e}^{6 d x +6 c} a^{4}-a^{3} b \,{\mathrm e}^{6 d x +6 c}-20 a^{2} b^{2} {\mathrm e}^{6 d x +6 c}-16 a \,b^{3} {\mathrm e}^{6 d x +6 c}+9 a^{4} {\mathrm e}^{4 d x +4 c}+15 a^{3} b \,{\mathrm e}^{4 d x +4 c}-18 \,{\mathrm e}^{4 d x +4 c} a^{2} b^{2}-72 a \,b^{3} {\mathrm e}^{4 d x +4 c}-48 b^{4} {\mathrm e}^{4 d x +4 c}+9 a^{4} {\mathrm e}^{2 d x +2 c}+13 a^{3} b \,{\mathrm e}^{2 d x +2 c}-28 \,{\mathrm e}^{2 d x +2 c} a^{2} b^{2}-32 a \,b^{3} {\mathrm e}^{2 d x +2 c}+3 a^{4}-3 a^{3} b -6 a^{2} b^{2}}{4 a^{3} b^{2} d \left ({\mathrm e}^{4 d x +4 c} a +2 a \,{\mathrm e}^{2 d x +2 c}+4 b \,{\mathrm e}^{2 d x +2 c}+a \right )^{2}}+\frac {3 \sqrt {b \left (a +b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {a +2 \sqrt {b \left (a +b \right )}+2 b}{a}\right )}{16 b^{3} d a}-\frac {\sqrt {b \left (a +b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {a +2 \sqrt {b \left (a +b \right )}+2 b}{a}\right )}{4 b^{2} d \,a^{2}}+\frac {\sqrt {b \left (a +b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {a +2 \sqrt {b \left (a +b \right )}+2 b}{a}\right )}{2 b d \,a^{3}}-\frac {3 \sqrt {b \left (a +b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {-a +2 \sqrt {b \left (a +b \right )}-2 b}{a}\right )}{16 b^{3} d a}+\frac {\sqrt {b \left (a +b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {-a +2 \sqrt {b \left (a +b \right )}-2 b}{a}\right )}{4 b^{2} d \,a^{2}}-\frac {\sqrt {b \left (a +b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {-a +2 \sqrt {b \left (a +b \right )}-2 b}{a}\right )}{2 b d \,a^{3}}\)	\(565\)

Input:

int(tanh(d*x+c)^6/(a+b*sech(d*x+c)^2)^3,x,method=_RETURNVERBOSE)
 

Output:

1/d*(1/a^3*ln(tanh(1/2*d*x+1/2*c)+1)-1/a^3*ln(tanh(1/2*d*x+1/2*c)-1)+2/a^3 
*((1/8*a*(3*a^3+2*a^2*b-5*a*b^2-4*b^3)/b^2*tanh(1/2*d*x+1/2*c)^7+1/8*(9*a^ 
3-14*a^2*b-19*a*b^2+4*b^3)*a/b^2*tanh(1/2*d*x+1/2*c)^5+1/8*(9*a^3-14*a^2*b 
-19*a*b^2+4*b^3)*a/b^2*tanh(1/2*d*x+1/2*c)^3+1/8*a*(3*a^3+2*a^2*b-5*a*b^2- 
4*b^3)/b^2*tanh(1/2*d*x+1/2*c))/(tanh(1/2*d*x+1/2*c)^4*a+tanh(1/2*d*x+1/2* 
c)^4*b+2*tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+a+b)^2+1/8*(3*a 
^3-a^2*b+4*a*b^2+8*b^3)/b^2*(-1/4/b^(1/2)/(a+b)^(1/2)*ln((a+b)^(1/2)*tanh( 
1/2*d*x+1/2*c)^2+2*tanh(1/2*d*x+1/2*c)*b^(1/2)+(a+b)^(1/2))+1/4/b^(1/2)/(a 
+b)^(1/2)*ln((a+b)^(1/2)*tanh(1/2*d*x+1/2*c)^2-2*tanh(1/2*d*x+1/2*c)*b^(1/ 
2)+(a+b)^(1/2)))))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2604 vs. \(2 (140) = 280\).

Time = 0.35 (sec) , antiderivative size = 5463, normalized size of antiderivative = 36.91 \[ \int \frac {\tanh ^6(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\text {Too large to display} \] Input:

integrate(tanh(d*x+c)^6/(a+b*sech(d*x+c)^2)^3,x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F]

\[ \int \frac {\tanh ^6(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\int \frac {\tanh ^{6}{\left (c + d x \right )}}{\left (a + b \operatorname {sech}^{2}{\left (c + d x \right )}\right )^{3}}\, dx \] Input:

integrate(tanh(d*x+c)**6/(a+b*sech(d*x+c)**2)**3,x)
 

Output:

Integral(tanh(c + d*x)**6/(a + b*sech(c + d*x)**2)**3, x)
                                                                                    
                                                                                    
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3239 vs. \(2 (140) = 280\).

Time = 0.87 (sec) , antiderivative size = 3239, normalized size of antiderivative = 21.89 \[ \int \frac {\tanh ^6(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\text {Too large to display} \] Input:

integrate(tanh(d*x+c)^6/(a+b*sech(d*x+c)^2)^3,x, algorithm="maxima")
 

Output:

-45/1024*(a + 2*b)*a*log((a*e^(2*d*x + 2*c) + a + 2*b - 2*sqrt((a + b)*b)) 
/(a*e^(2*d*x + 2*c) + a + 2*b + 2*sqrt((a + b)*b)))/((a^2*b^2 + 2*a*b^3 + 
b^4)*sqrt((a + b)*b)*d) - 9/512*a^2*log((a*e^(2*d*x + 2*c) + a + 2*b - 2*s 
qrt((a + b)*b))/(a*e^(2*d*x + 2*c) + a + 2*b + 2*sqrt((a + b)*b)))/((a^2*b 
^2 + 2*a*b^3 + b^4)*sqrt((a + b)*b)*d) + 45/1024*(a + 2*b)*a*log((a*e^(-2* 
d*x - 2*c) + a + 2*b - 2*sqrt((a + b)*b))/(a*e^(-2*d*x - 2*c) + a + 2*b + 
2*sqrt((a + b)*b)))/((a^2*b^2 + 2*a*b^3 + b^4)*sqrt((a + b)*b)*d) + 9/512* 
a^2*log((a*e^(-2*d*x - 2*c) + a + 2*b - 2*sqrt((a + b)*b))/(a*e^(-2*d*x - 
2*c) + a + 2*b + 2*sqrt((a + b)*b)))/((a^2*b^2 + 2*a*b^3 + b^4)*sqrt((a + 
b)*b)*d) - 1/1024*(3*a^5 - 10*a^4*b + 80*a^3*b^2 + 480*a^2*b^3 + 640*a*b^4 
 + 256*b^5)*log((a*e^(2*d*x + 2*c) + a + 2*b - 2*sqrt((a + b)*b))/(a*e^(2* 
d*x + 2*c) + a + 2*b + 2*sqrt((a + b)*b)))/((a^5*b^2 + 2*a^4*b^3 + a^3*b^4 
)*sqrt((a + b)*b)*d) + 1/1024*(3*a^5 - 10*a^4*b + 80*a^3*b^2 + 480*a^2*b^3 
 + 640*a*b^4 + 256*b^5)*log((a*e^(-2*d*x - 2*c) + a + 2*b - 2*sqrt((a + b) 
*b))/(a*e^(-2*d*x - 2*c) + a + 2*b + 2*sqrt((a + b)*b)))/((a^5*b^2 + 2*a^4 
*b^3 + a^3*b^4)*sqrt((a + b)*b)*d) + 5/256*(3*a^2 + 8*a*b + 8*b^2)*log((a* 
e^(-2*d*x - 2*c) + a + 2*b - 2*sqrt((a + b)*b))/(a*e^(-2*d*x - 2*c) + a + 
2*b + 2*sqrt((a + b)*b)))/((a^2*b^2 + 2*a*b^3 + b^4)*sqrt((a + b)*b)*d) - 
1/256*(3*a^6 - 12*a^5*b - 204*a^4*b^2 - 384*a^3*b^3 - 192*a^2*b^4 + (3*a^6 
 - 10*a^5*b - 560*a^4*b^2 - 2080*a^3*b^3 - 2560*a^2*b^4 - 1024*a*b^5)*e...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 354 vs. \(2 (140) = 280\).

Time = 1.30 (sec) , antiderivative size = 354, normalized size of antiderivative = 2.39 \[ \int \frac {\tanh ^6(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\frac {\frac {8 \, {\left (d x + c\right )}}{a^{3}} - \frac {{\left (3 \, a^{3} - a^{2} b + 4 \, a b^{2} + 8 \, b^{3}\right )} \arctan \left (\frac {a e^{\left (2 \, d x + 2 \, c\right )} + a + 2 \, b}{2 \, \sqrt {-a b - b^{2}}}\right )}{\sqrt {-a b - b^{2}} a^{3} b^{2}} - \frac {2 \, {\left (3 \, a^{4} e^{\left (6 \, d x + 6 \, c\right )} - a^{3} b e^{\left (6 \, d x + 6 \, c\right )} - 20 \, a^{2} b^{2} e^{\left (6 \, d x + 6 \, c\right )} - 16 \, a b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 9 \, a^{4} e^{\left (4 \, d x + 4 \, c\right )} + 15 \, a^{3} b e^{\left (4 \, d x + 4 \, c\right )} - 18 \, a^{2} b^{2} e^{\left (4 \, d x + 4 \, c\right )} - 72 \, a b^{3} e^{\left (4 \, d x + 4 \, c\right )} - 48 \, b^{4} e^{\left (4 \, d x + 4 \, c\right )} + 9 \, a^{4} e^{\left (2 \, d x + 2 \, c\right )} + 13 \, a^{3} b e^{\left (2 \, d x + 2 \, c\right )} - 28 \, a^{2} b^{2} e^{\left (2 \, d x + 2 \, c\right )} - 32 \, a b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 3 \, a^{4} - 3 \, a^{3} b - 6 \, a^{2} b^{2}\right )}}{{\left (a e^{\left (4 \, d x + 4 \, c\right )} + 2 \, a e^{\left (2 \, d x + 2 \, c\right )} + 4 \, b e^{\left (2 \, d x + 2 \, c\right )} + a\right )}^{2} a^{3} b^{2}}}{8 \, d} \] Input:

integrate(tanh(d*x+c)^6/(a+b*sech(d*x+c)^2)^3,x, algorithm="giac")
 

Output:

1/8*(8*(d*x + c)/a^3 - (3*a^3 - a^2*b + 4*a*b^2 + 8*b^3)*arctan(1/2*(a*e^( 
2*d*x + 2*c) + a + 2*b)/sqrt(-a*b - b^2))/(sqrt(-a*b - b^2)*a^3*b^2) - 2*( 
3*a^4*e^(6*d*x + 6*c) - a^3*b*e^(6*d*x + 6*c) - 20*a^2*b^2*e^(6*d*x + 6*c) 
 - 16*a*b^3*e^(6*d*x + 6*c) + 9*a^4*e^(4*d*x + 4*c) + 15*a^3*b*e^(4*d*x + 
4*c) - 18*a^2*b^2*e^(4*d*x + 4*c) - 72*a*b^3*e^(4*d*x + 4*c) - 48*b^4*e^(4 
*d*x + 4*c) + 9*a^4*e^(2*d*x + 2*c) + 13*a^3*b*e^(2*d*x + 2*c) - 28*a^2*b^ 
2*e^(2*d*x + 2*c) - 32*a*b^3*e^(2*d*x + 2*c) + 3*a^4 - 3*a^3*b - 6*a^2*b^2 
)/((a*e^(4*d*x + 4*c) + 2*a*e^(2*d*x + 2*c) + 4*b*e^(2*d*x + 2*c) + a)^2*a 
^3*b^2))/d
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\tanh ^6(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\int \frac {{\left ({\mathrm {cosh}\left (c+d\,x\right )}^2-1\right )}^3}{{\left (a\,{\mathrm {cosh}\left (c+d\,x\right )}^2+b\right )}^3} \,d x \] Input:

int(tanh(c + d*x)^6/(a + b/cosh(c + d*x)^2)^3,x)
 

Output:

int((cosh(c + d*x)^2 - 1)^3/(b + a*cosh(c + d*x)^2)^3, x)
 

Reduce [B] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 3970, normalized size of antiderivative = 26.82 \[ \int \frac {\tanh ^6(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx =\text {Too large to display} \] Input:

int(tanh(d*x+c)^6/(a+b*sech(d*x+c)^2)^3,x)
 

Output:

( - 3*e**(8*c + 8*d*x)*sqrt(b)*sqrt(a + b)*log( - sqrt(2*sqrt(b)*sqrt(a + 
b) - a - 2*b) + e**(c + d*x)*sqrt(a))*a**5 - 2*e**(8*c + 8*d*x)*sqrt(b)*sq 
rt(a + b)*log( - sqrt(2*sqrt(b)*sqrt(a + b) - a - 2*b) + e**(c + d*x)*sqrt 
(a))*a**4*b - 16*e**(8*c + 8*d*x)*sqrt(b)*sqrt(a + b)*log( - sqrt(2*sqrt(b 
)*sqrt(a + b) - a - 2*b) + e**(c + d*x)*sqrt(a))*a**2*b**3 - 3*e**(8*c + 8 
*d*x)*sqrt(b)*sqrt(a + b)*log(sqrt(2*sqrt(b)*sqrt(a + b) - a - 2*b) + e**( 
c + d*x)*sqrt(a))*a**5 - 2*e**(8*c + 8*d*x)*sqrt(b)*sqrt(a + b)*log(sqrt(2 
*sqrt(b)*sqrt(a + b) - a - 2*b) + e**(c + d*x)*sqrt(a))*a**4*b - 16*e**(8* 
c + 8*d*x)*sqrt(b)*sqrt(a + b)*log(sqrt(2*sqrt(b)*sqrt(a + b) - a - 2*b) + 
 e**(c + d*x)*sqrt(a))*a**2*b**3 + 3*e**(8*c + 8*d*x)*sqrt(b)*sqrt(a + b)* 
log(2*sqrt(b)*sqrt(a + b) + e**(2*c + 2*d*x)*a + a + 2*b)*a**5 + 2*e**(8*c 
 + 8*d*x)*sqrt(b)*sqrt(a + b)*log(2*sqrt(b)*sqrt(a + b) + e**(2*c + 2*d*x) 
*a + a + 2*b)*a**4*b + 16*e**(8*c + 8*d*x)*sqrt(b)*sqrt(a + b)*log(2*sqrt( 
b)*sqrt(a + b) + e**(2*c + 2*d*x)*a + a + 2*b)*a**2*b**3 - 12*e**(6*c + 6* 
d*x)*sqrt(b)*sqrt(a + b)*log( - sqrt(2*sqrt(b)*sqrt(a + b) - a - 2*b) + e* 
*(c + d*x)*sqrt(a))*a**5 - 32*e**(6*c + 6*d*x)*sqrt(b)*sqrt(a + b)*log( - 
sqrt(2*sqrt(b)*sqrt(a + b) - a - 2*b) + e**(c + d*x)*sqrt(a))*a**4*b - 16* 
e**(6*c + 6*d*x)*sqrt(b)*sqrt(a + b)*log( - sqrt(2*sqrt(b)*sqrt(a + b) - a 
 - 2*b) + e**(c + d*x)*sqrt(a))*a**3*b**2 - 64*e**(6*c + 6*d*x)*sqrt(b)*sq 
rt(a + b)*log( - sqrt(2*sqrt(b)*sqrt(a + b) - a - 2*b) + e**(c + d*x)*s...