\(\int \frac {\text {sech}^6(c+d x)}{(a+b \text {sech}^2(c+d x))^3} \, dx\) [100]

Optimal result

Integrand size = 23, antiderivative size = 140 \[ \int \frac {\text {sech}^6(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\frac {\left (3 a^2+8 a b+8 b^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{8 b^{5/2} (a+b)^{5/2} d}+\frac {a^2 \tanh (c+d x)}{4 b^2 (a+b) d \left (a+b-b \tanh ^2(c+d x)\right )^2}-\frac {a (5 a+8 b) \tanh (c+d x)}{8 b^2 (a+b)^2 d \left (a+b-b \tanh ^2(c+d x)\right )} \] Output:

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

Mathematica [A] (verified)

Time = 0.66 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.89 \[ \int \frac {\text {sech}^6(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\frac {\frac {\left (3 a^2+8 a b+8 b^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{(a+b)^{5/2}}-\frac {a \sqrt {b} \left (3 a^2+16 a b+16 b^2+3 a (a+2 b) \cosh (2 (c+d x))\right ) \sinh (2 (c+d x))}{(a+b)^2 (a+2 b+a \cosh (2 (c+d x)))^2}}{8 b^{5/2} d} \] Input:

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

Output:

(((3*a^2 + 8*a*b + 8*b^2)*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(a 
 + b)^(5/2) - (a*Sqrt[b]*(3*a^2 + 16*a*b + 16*b^2 + 3*a*(a + 2*b)*Cosh[2*( 
c + d*x)])*Sinh[2*(c + d*x)])/((a + b)^2*(a + 2*b + a*Cosh[2*(c + d*x)])^2 
))/(8*b^(5/2)*d)
 

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.11, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3042, 4634, 315, 25, 298, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4634

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

\(\Big \downarrow \) 315

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 298

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 25

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

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 298

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[(-( 
b*c - a*d))*x*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] - Simp[(a*d - b*c*( 
2*p + 3))/(2*a*b*(p + 1))   Int[(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, 
 c, d, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/2 + p, 0])
 

rule 315

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

rule 3042

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

rule 4634

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_) 
)^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[ff/f 
Subst[Int[(1 + ff^2*x^2)^(m/2 - 1)*ExpandToSum[a + b*(1 + ff^2*x^2)^(n/2), 
x]^p, x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ 
[m/2] && IntegerQ[n/2]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(327\) vs. \(2(126)=252\).

Time = 1.25 (sec) , antiderivative size = 328, normalized size of antiderivative = 2.34

method	result	size
derivativedivides	\(\frac {-\frac {2 \left (\frac {a \left (3 a +8 b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{8 \left (a +b \right ) b^{2}}+\frac {a \left (9 a^{2}+13 a b -8 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{8 \left (a +b \right )^{2} b^{2}}+\frac {a \left (9 a^{2}+13 a b -8 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{8 \left (a +b \right )^{2} b^{2}}+\frac {a \left (3 a +8 b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 \left (a +b \right ) b^{2}}\right )}{\left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} b +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a +b \right )^{2}}-\frac {\left (3 a^{2}+8 a b +8 b^{2}\right ) \left (-\frac {\ln \left (\sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}+\frac {\ln \left (\sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}\right )}{4 b^{2} \left (a^{2}+2 a b +b^{2}\right )}}{d}\)	\(328\)
default	\(\frac {-\frac {2 \left (\frac {a \left (3 a +8 b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{8 \left (a +b \right ) b^{2}}+\frac {a \left (9 a^{2}+13 a b -8 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{8 \left (a +b \right )^{2} b^{2}}+\frac {a \left (9 a^{2}+13 a b -8 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{8 \left (a +b \right )^{2} b^{2}}+\frac {a \left (3 a +8 b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 \left (a +b \right ) b^{2}}\right )}{\left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} b +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a +b \right )^{2}}-\frac {\left (3 a^{2}+8 a b +8 b^{2}\right ) \left (-\frac {\ln \left (\sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}+\frac {\ln \left (\sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}\right )}{4 b^{2} \left (a^{2}+2 a b +b^{2}\right )}}{d}\)	\(328\)
risch	\(\frac {3 a^{3} {\mathrm e}^{6 d x +6 c}+8 a^{2} b \,{\mathrm e}^{6 d x +6 c}+8 a \,b^{2} {\mathrm e}^{6 d x +6 c}+9 a^{3} {\mathrm e}^{4 d x +4 c}+42 a^{2} b \,{\mathrm e}^{4 d x +4 c}+72 a \,b^{2} {\mathrm e}^{4 d x +4 c}+48 b^{3} {\mathrm e}^{4 d x +4 c}+9 a^{3} {\mathrm e}^{2 d x +2 c}+40 a^{2} b \,{\mathrm e}^{2 d x +2 c}+40 a \,b^{2} {\mathrm e}^{2 d x +2 c}+3 a^{3}+6 a^{2} b}{4 d \,b^{2} \left (a +b \right )^{2} \left ({\mathrm e}^{4 d x +4 c} a +2 a \,{\mathrm e}^{2 d x +2 c}+4 b \,{\mathrm e}^{2 d x +2 c}+a \right )^{2}}+\frac {3 \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {a \sqrt {a b +b^{2}}+2 b \sqrt {a b +b^{2}}-2 a b -2 b^{2}}{a \sqrt {a b +b^{2}}}\right ) a^{2}}{16 \sqrt {a b +b^{2}}\, \left (a +b \right )^{2} d \,b^{2}}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {a \sqrt {a b +b^{2}}+2 b \sqrt {a b +b^{2}}-2 a b -2 b^{2}}{a \sqrt {a b +b^{2}}}\right ) a}{2 \sqrt {a b +b^{2}}\, \left (a +b \right )^{2} d b}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {a \sqrt {a b +b^{2}}+2 b \sqrt {a b +b^{2}}-2 a b -2 b^{2}}{a \sqrt {a b +b^{2}}}\right )}{2 \sqrt {a b +b^{2}}\, \left (a +b \right )^{2} d}-\frac {3 \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {a \sqrt {a b +b^{2}}+2 b \sqrt {a b +b^{2}}+2 a b +2 b^{2}}{a \sqrt {a b +b^{2}}}\right ) a^{2}}{16 \sqrt {a b +b^{2}}\, \left (a +b \right )^{2} d \,b^{2}}-\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {a \sqrt {a b +b^{2}}+2 b \sqrt {a b +b^{2}}+2 a b +2 b^{2}}{a \sqrt {a b +b^{2}}}\right ) a}{2 \sqrt {a b +b^{2}}\, \left (a +b \right )^{2} d b}-\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {a \sqrt {a b +b^{2}}+2 b \sqrt {a b +b^{2}}+2 a b +2 b^{2}}{a \sqrt {a b +b^{2}}}\right )}{2 \sqrt {a b +b^{2}}\, \left (a +b \right )^{2} d}\)	\(688\)

Input:

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

Output:

1/d*(-2*(1/8*a*(3*a+8*b)/(a+b)/b^2*tanh(1/2*d*x+1/2*c)^7+1/8*a*(9*a^2+13*a 
*b-8*b^2)/(a+b)^2/b^2*tanh(1/2*d*x+1/2*c)^5+1/8*a*(9*a^2+13*a*b-8*b^2)/(a+ 
b)^2/b^2*tanh(1/2*d*x+1/2*c)^3+1/8*a*(3*a+8*b)/(a+b)/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)^2-1/4*(3*a^2+8*a*b+8*b^2)/b^2/(a^2+2*a 
*b+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*t 
anh(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. 2823 vs. \(2 (132) = 264\).

Time = 0.32 (sec) , antiderivative size = 5887, normalized size of antiderivative = 42.05 \[ \int \frac {\text {sech}^6(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F]

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

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

Output:

Integral(sech(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. 395 vs. \(2 (132) = 264\).

Time = 0.27 (sec) , antiderivative size = 395, normalized size of antiderivative = 2.82 \[ \int \frac {\text {sech}^6(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=-\frac {{\left (3 \, a^{2} + 8 \, a b + 8 \, b^{2}\right )} \log \left (\frac {a e^{\left (-2 \, d x - 2 \, c\right )} + a + 2 \, b - 2 \, \sqrt {{\left (a + b\right )} b}}{a e^{\left (-2 \, d x - 2 \, c\right )} + a + 2 \, b + 2 \, \sqrt {{\left (a + b\right )} b}}\right )}{16 \, {\left (a^{2} b^{2} + 2 \, a b^{3} + b^{4}\right )} \sqrt {{\left (a + b\right )} b} d} - \frac {3 \, a^{3} + 6 \, a^{2} b + {\left (9 \, a^{3} + 40 \, a^{2} b + 40 \, a b^{2}\right )} e^{\left (-2 \, d x - 2 \, c\right )} + 3 \, {\left (3 \, a^{3} + 14 \, a^{2} b + 24 \, a b^{2} + 16 \, b^{3}\right )} e^{\left (-4 \, d x - 4 \, c\right )} + {\left (3 \, a^{3} + 8 \, a^{2} b + 8 \, a b^{2}\right )} e^{\left (-6 \, d x - 6 \, c\right )}}{4 \, {\left (a^{4} b^{2} + 2 \, a^{3} b^{3} + a^{2} b^{4} + 4 \, {\left (a^{4} b^{2} + 4 \, a^{3} b^{3} + 5 \, a^{2} b^{4} + 2 \, a b^{5}\right )} e^{\left (-2 \, d x - 2 \, c\right )} + 2 \, {\left (3 \, a^{4} b^{2} + 14 \, a^{3} b^{3} + 27 \, a^{2} b^{4} + 24 \, a b^{5} + 8 \, b^{6}\right )} e^{\left (-4 \, d x - 4 \, c\right )} + 4 \, {\left (a^{4} b^{2} + 4 \, a^{3} b^{3} + 5 \, a^{2} b^{4} + 2 \, a b^{5}\right )} e^{\left (-6 \, d x - 6 \, c\right )} + {\left (a^{4} b^{2} + 2 \, a^{3} b^{3} + a^{2} b^{4}\right )} e^{\left (-8 \, d x - 8 \, c\right )}\right )} d} \] Input:

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

Output:

-1/16*(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) - 1/4*(3*a^3 + 6*a^2*b + (9*a^3 + 40*a^ 
2*b + 40*a*b^2)*e^(-2*d*x - 2*c) + 3*(3*a^3 + 14*a^2*b + 24*a*b^2 + 16*b^3 
)*e^(-4*d*x - 4*c) + (3*a^3 + 8*a^2*b + 8*a*b^2)*e^(-6*d*x - 6*c))/((a^4*b 
^2 + 2*a^3*b^3 + a^2*b^4 + 4*(a^4*b^2 + 4*a^3*b^3 + 5*a^2*b^4 + 2*a*b^5)*e 
^(-2*d*x - 2*c) + 2*(3*a^4*b^2 + 14*a^3*b^3 + 27*a^2*b^4 + 24*a*b^5 + 8*b^ 
6)*e^(-4*d*x - 4*c) + 4*(a^4*b^2 + 4*a^3*b^3 + 5*a^2*b^4 + 2*a*b^5)*e^(-6* 
d*x - 6*c) + (a^4*b^2 + 2*a^3*b^3 + a^2*b^4)*e^(-8*d*x - 8*c))*d)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 302 vs. \(2 (132) = 264\).

Time = 0.35 (sec) , antiderivative size = 302, normalized size of antiderivative = 2.16 \[ \int \frac {\text {sech}^6(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\frac {\frac {{\left (3 \, a^{2} + 8 \, a b + 8 \, b^{2}\right )} \arctan \left (\frac {a e^{\left (2 \, d x + 2 \, c\right )} + a + 2 \, b}{2 \, \sqrt {-a b - b^{2}}}\right )}{{\left (a^{2} b^{2} + 2 \, a b^{3} + b^{4}\right )} \sqrt {-a b - b^{2}}} + \frac {2 \, {\left (3 \, a^{3} e^{\left (6 \, d x + 6 \, c\right )} + 8 \, a^{2} b e^{\left (6 \, d x + 6 \, c\right )} + 8 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 9 \, a^{3} e^{\left (4 \, d x + 4 \, c\right )} + 42 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} + 72 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 48 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 9 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} + 40 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} + 40 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 3 \, a^{3} + 6 \, a^{2} b\right )}}{{\left (a^{2} b^{2} + 2 \, a b^{3} + b^{4}\right )} {\left (a e^{\left (4 \, d x + 4 \, c\right )} + 2 \, a e^{\left (2 \, d x + 2 \, c\right )} + 4 \, b e^{\left (2 \, d x + 2 \, c\right )} + a\right )}^{2}}}{8 \, d} \] Input:

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

Output:

1/8*((3*a^2 + 8*a*b + 8*b^2)*arctan(1/2*(a*e^(2*d*x + 2*c) + a + 2*b)/sqrt 
(-a*b - b^2))/((a^2*b^2 + 2*a*b^3 + b^4)*sqrt(-a*b - b^2)) + 2*(3*a^3*e^(6 
*d*x + 6*c) + 8*a^2*b*e^(6*d*x + 6*c) + 8*a*b^2*e^(6*d*x + 6*c) + 9*a^3*e^ 
(4*d*x + 4*c) + 42*a^2*b*e^(4*d*x + 4*c) + 72*a*b^2*e^(4*d*x + 4*c) + 48*b 
^3*e^(4*d*x + 4*c) + 9*a^3*e^(2*d*x + 2*c) + 40*a^2*b*e^(2*d*x + 2*c) + 40 
*a*b^2*e^(2*d*x + 2*c) + 3*a^3 + 6*a^2*b)/((a^2*b^2 + 2*a*b^3 + b^4)*(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))/d
 

Mupad [F(-1)]

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

int(1/(cosh(c + d*x)^6*(a + b/cosh(c + d*x)^2)^3),x)
                                                                                    
                                                                                    
 

Output:

int(1/(cosh(c + d*x)^6*(a + b/cosh(c + d*x)^2)^3), x)
 

Reduce [B] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 4256, normalized size of antiderivative = 30.40 \[ \int \frac {\text {sech}^6(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx =\text {Too large to display} \] Input:

int(sech(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 + 14*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 + 24*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**3*b**2 + 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 + 14*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 + 24*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**3* 
b**2 + 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 - 14*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 - 24*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**3*b**2 
- 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)...