\(\int \frac {\text {sech}^6(c+d x)}{(a+b \text {sech}^2(c+d x))^2} \, dx\) [91]

Optimal result

Integrand size = 23, antiderivative size = 101 \[ \int \frac {\text {sech}^6(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=-\frac {a (3 a+4 b) \text {arctanh}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{2 b^{5/2} (a+b)^{3/2} d}+\frac {\tanh (c+d x)}{b^2 d}+\frac {a^2 \tanh (c+d x)}{2 b^2 (a+b) d \left (a+b-b \tanh ^2(c+d x)\right )} \] Output:

-1/2*a*(3*a+4*b)*arctanh(b^(1/2)*tanh(d*x+c)/(a+b)^(1/2))/b^(5/2)/(a+b)^(3 
/2)/d+tanh(d*x+c)/b^2/d+1/2*a^2*tanh(d*x+c)/b^2/(a+b)/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. \(229\) vs. \(2(101)=202\).

Time = 3.75 (sec) , antiderivative size = 229, normalized size of antiderivative = 2.27 \[ \int \frac {\text {sech}^6(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=\frac {(a+2 b+a \cosh (2 (c+d x))) \text {sech}^4(c+d x) \left (-\frac {a (3 a+4 b) \text {arctanh}\left (\frac {\text {sech}(d x) (\cosh (2 c)-\sinh (2 c)) ((a+2 b) \sinh (d x)-a \sinh (2 c+d x))}{2 \sqrt {a+b} \sqrt {b (\cosh (c)-\sinh (c))^4}}\right ) (a+2 b+a \cosh (2 (c+d x))) (\cosh (2 c)-\sinh (2 c))}{(a+b)^{3/2} \sqrt {b (\cosh (c)-\sinh (c))^4}}+2 (a+2 b+a \cosh (2 (c+d x))) \text {sech}(c) \text {sech}(c+d x) \sinh (d x)+\frac {a (a \text {sech}(2 c) \sinh (2 d x)-(a+2 b) \tanh (2 c))}{a+b}\right )}{8 b^2 d \left (a+b \text {sech}^2(c+d x)\right )^2} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.95, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3042, 4634, 300, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4634

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

\(\Big \downarrow \) 300

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\frac {a^2 \tanh (c+d x)}{2 b^2 (a+b) \left (a-b \tanh ^2(c+d x)+b\right )}-\frac {a (3 a+4 b) \text {arctanh}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{2 b^{5/2} (a+b)^{3/2}}+\frac {\tanh (c+d x)}{b^2}}{d}\)

Input:

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

Output:

(-1/2*(a*(3*a + 4*b)*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(b^(5/2 
)*(a + b)^(3/2)) + Tanh[c + d*x]/b^2 + (a^2*Tanh[c + d*x])/(2*b^2*(a + b)* 
(a + b - b*Tanh[c + d*x]^2)))/d
 

Defintions of rubi rules used

rule 300

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Int 
[PolynomialDivide[(a + b*x^2)^p, (c + d*x^2)^(-q), x], x] /; FreeQ[{a, b, c 
, d}, x] && NeQ[b*c - a*d, 0] && IGtQ[p, 0] && ILtQ[q, 0] && GeQ[p, -q]
 

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 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. \(249\) vs. \(2(89)=178\).

Time = 1.96 (sec) , antiderivative size = 250, normalized size of antiderivative = 2.48

method	result	size
derivativedivides	\(\frac {\frac {2 a \left (\frac {\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} a}{2 a +2 b}+\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a +2 b}}{\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} b +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a +b}+\frac {\left (3 a +4 b \right ) \left (-\frac {\ln \left (\sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}+\frac {\ln \left (\sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}\right )}{2 a +2 b}\right )}{b^{2}}+\frac {2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{b^{2} \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}}{d}\)	\(250\)
default	\(\frac {\frac {2 a \left (\frac {\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} a}{2 a +2 b}+\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a +2 b}}{\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} b +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a +b}+\frac {\left (3 a +4 b \right ) \left (-\frac {\ln \left (\sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}+\frac {\ln \left (\sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}\right )}{2 a +2 b}\right )}{b^{2}}+\frac {2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{b^{2} \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}}{d}\)	\(250\)
risch	\(-\frac {3 a^{2} {\mathrm e}^{4 d x +4 c}+4 a b \,{\mathrm e}^{4 d x +4 c}+6 a^{2} {\mathrm e}^{2 d x +2 c}+14 a b \,{\mathrm e}^{2 d x +2 c}+8 b^{2} {\mathrm e}^{2 d x +2 c}+3 a^{2}+2 a b}{b^{2} \left (a +b \right ) d \left ({\mathrm e}^{4 d x +4 c} a +2 a \,{\mathrm e}^{2 d x +2 c}+4 b \,{\mathrm e}^{2 d x +2 c}+a \right ) \left ({\mathrm e}^{2 d x +2 c}+1\right )}+\frac {3 a^{2} \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {a \sqrt {a b +b^{2}}+2 b \sqrt {a b +b^{2}}+2 a b +2 b^{2}}{a \sqrt {a b +b^{2}}}\right )}{4 \sqrt {a b +b^{2}}\, \left (a +b \right ) d \,b^{2}}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {a \sqrt {a b +b^{2}}+2 b \sqrt {a b +b^{2}}+2 a b +2 b^{2}}{a \sqrt {a b +b^{2}}}\right ) a}{\sqrt {a b +b^{2}}\, \left (a +b \right ) d b}-\frac {3 a^{2} \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {a \sqrt {a b +b^{2}}+2 b \sqrt {a b +b^{2}}-2 a b -2 b^{2}}{a \sqrt {a b +b^{2}}}\right )}{4 \sqrt {a b +b^{2}}\, \left (a +b \right ) d \,b^{2}}-\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {a \sqrt {a b +b^{2}}+2 b \sqrt {a b +b^{2}}-2 a b -2 b^{2}}{a \sqrt {a b +b^{2}}}\right ) a}{\sqrt {a b +b^{2}}\, \left (a +b \right ) d b}\)	\(468\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1358 vs. \(2 (92) = 184\).

Time = 0.24 (sec) , antiderivative size = 2958, normalized size of antiderivative = 29.29 \[ \int \frac {\text {sech}^6(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

[-1/4*(4*(3*a^3*b + 7*a^2*b^2 + 4*a*b^3)*cosh(d*x + c)^4 + 16*(3*a^3*b + 7 
*a^2*b^2 + 4*a*b^3)*cosh(d*x + c)*sinh(d*x + c)^3 + 4*(3*a^3*b + 7*a^2*b^2 
 + 4*a*b^3)*sinh(d*x + c)^4 + 12*a^3*b + 20*a^2*b^2 + 8*a*b^3 + 8*(3*a^3*b 
 + 10*a^2*b^2 + 11*a*b^3 + 4*b^4)*cosh(d*x + c)^2 + 8*(3*a^3*b + 10*a^2*b^ 
2 + 11*a*b^3 + 4*b^4 + 3*(3*a^3*b + 7*a^2*b^2 + 4*a*b^3)*cosh(d*x + c)^2)* 
sinh(d*x + c)^2 - ((3*a^3 + 4*a^2*b)*cosh(d*x + c)^6 + 6*(3*a^3 + 4*a^2*b) 
*cosh(d*x + c)*sinh(d*x + c)^5 + (3*a^3 + 4*a^2*b)*sinh(d*x + c)^6 + (9*a^ 
3 + 24*a^2*b + 16*a*b^2)*cosh(d*x + c)^4 + (9*a^3 + 24*a^2*b + 16*a*b^2 + 
15*(3*a^3 + 4*a^2*b)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 4*(5*(3*a^3 + 4*a^ 
2*b)*cosh(d*x + c)^3 + (9*a^3 + 24*a^2*b + 16*a*b^2)*cosh(d*x + c))*sinh(d 
*x + c)^3 + 3*a^3 + 4*a^2*b + (9*a^3 + 24*a^2*b + 16*a*b^2)*cosh(d*x + c)^ 
2 + (15*(3*a^3 + 4*a^2*b)*cosh(d*x + c)^4 + 9*a^3 + 24*a^2*b + 16*a*b^2 + 
6*(9*a^3 + 24*a^2*b + 16*a*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 2*(3*(3 
*a^3 + 4*a^2*b)*cosh(d*x + c)^5 + 2*(9*a^3 + 24*a^2*b + 16*a*b^2)*cosh(d*x 
 + c)^3 + (9*a^3 + 24*a^2*b + 16*a*b^2)*cosh(d*x + c))*sinh(d*x + c))*sqrt 
(a*b + b^2)*log((a^2*cosh(d*x + c)^4 + 4*a^2*cosh(d*x + c)*sinh(d*x + c)^3 
 + a^2*sinh(d*x + c)^4 + 2*(a^2 + 2*a*b)*cosh(d*x + c)^2 + 2*(3*a^2*cosh(d 
*x + c)^2 + a^2 + 2*a*b)*sinh(d*x + c)^2 + a^2 + 8*a*b + 8*b^2 + 4*(a^2*co 
sh(d*x + c)^3 + (a^2 + 2*a*b)*cosh(d*x + c))*sinh(d*x + c) + 4*(a*cosh(d*x 
 + c)^2 + 2*a*cosh(d*x + c)*sinh(d*x + c) + a*sinh(d*x + c)^2 + a + 2*b...
 

Sympy [F]

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

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

Output:

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 244 vs. \(2 (92) = 184\).

Time = 0.20 (sec) , antiderivative size = 244, normalized size of antiderivative = 2.42 \[ \int \frac {\text {sech}^6(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=\frac {{\left (3 \, a + 4 \, b\right )} a \log \left (\frac {a e^{\left (-2 \, d x - 2 \, c\right )} + a + 2 \, b - 2 \, \sqrt {{\left (a + b\right )} b}}{a e^{\left (-2 \, d x - 2 \, c\right )} + a + 2 \, b + 2 \, \sqrt {{\left (a + b\right )} b}}\right )}{4 \, {\left (a b^{2} + b^{3}\right )} \sqrt {{\left (a + b\right )} b} d} + \frac {3 \, a^{2} + 2 \, a b + 2 \, {\left (3 \, a^{2} + 7 \, a b + 4 \, b^{2}\right )} e^{\left (-2 \, d x - 2 \, c\right )} + {\left (3 \, a^{2} + 4 \, a b\right )} e^{\left (-4 \, d x - 4 \, c\right )}}{{\left (a^{2} b^{2} + a b^{3} + {\left (3 \, a^{2} b^{2} + 7 \, a b^{3} + 4 \, b^{4}\right )} e^{\left (-2 \, d x - 2 \, c\right )} + {\left (3 \, a^{2} b^{2} + 7 \, a b^{3} + 4 \, b^{4}\right )} e^{\left (-4 \, d x - 4 \, c\right )} + {\left (a^{2} b^{2} + a b^{3}\right )} e^{\left (-6 \, d x - 6 \, c\right )}\right )} d} \] Input:

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

Output:

1/4*(3*a + 4*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*b^2 + b^3)*sqrt((a 
+ b)*b)*d) + (3*a^2 + 2*a*b + 2*(3*a^2 + 7*a*b + 4*b^2)*e^(-2*d*x - 2*c) + 
 (3*a^2 + 4*a*b)*e^(-4*d*x - 4*c))/((a^2*b^2 + a*b^3 + (3*a^2*b^2 + 7*a*b^ 
3 + 4*b^4)*e^(-2*d*x - 2*c) + (3*a^2*b^2 + 7*a*b^3 + 4*b^4)*e^(-4*d*x - 4* 
c) + (a^2*b^2 + a*b^3)*e^(-6*d*x - 6*c))*d)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 225 vs. \(2 (92) = 184\).

Time = 0.26 (sec) , antiderivative size = 225, normalized size of antiderivative = 2.23 \[ \int \frac {\text {sech}^6(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=-\frac {\frac {{\left (3 \, a^{2} + 4 \, a b\right )} \arctan \left (\frac {a e^{\left (2 \, d x + 2 \, c\right )} + a + 2 \, b}{2 \, \sqrt {-a b - b^{2}}}\right )}{{\left (a b^{2} + b^{3}\right )} \sqrt {-a b - b^{2}}} + \frac {2 \, {\left (3 \, a^{2} e^{\left (4 \, d x + 4 \, c\right )} + 4 \, a b e^{\left (4 \, d x + 4 \, c\right )} + 6 \, a^{2} e^{\left (2 \, d x + 2 \, c\right )} + 14 \, a b e^{\left (2 \, d x + 2 \, c\right )} + 8 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 3 \, a^{2} + 2 \, a b\right )}}{{\left (a b^{2} + b^{3}\right )} {\left (a e^{\left (6 \, d x + 6 \, c\right )} + 3 \, a e^{\left (4 \, d x + 4 \, c\right )} + 4 \, b e^{\left (4 \, d x + 4 \, c\right )} + 3 \, a e^{\left (2 \, d x + 2 \, c\right )} + 4 \, b e^{\left (2 \, d x + 2 \, c\right )} + a\right )}}}{2 \, d} \] Input:

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

Output:

-1/2*((3*a^2 + 4*a*b)*arctan(1/2*(a*e^(2*d*x + 2*c) + a + 2*b)/sqrt(-a*b - 
 b^2))/((a*b^2 + b^3)*sqrt(-a*b - b^2)) + 2*(3*a^2*e^(4*d*x + 4*c) + 4*a*b 
*e^(4*d*x + 4*c) + 6*a^2*e^(2*d*x + 2*c) + 14*a*b*e^(2*d*x + 2*c) + 8*b^2* 
e^(2*d*x + 2*c) + 3*a^2 + 2*a*b)/((a*b^2 + b^3)*(a*e^(6*d*x + 6*c) + 3*a*e 
^(4*d*x + 4*c) + 4*b*e^(4*d*x + 4*c) + 3*a*e^(2*d*x + 2*c) + 4*b*e^(2*d*x 
+ 2*c) + a)))/d
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.31 (sec) , antiderivative size = 1742, normalized size of antiderivative = 17.25 \[ \int \frac {\text {sech}^6(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 3*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 - 4*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)*sqrt 
(a))*a**2*b - 3*e**(6*c + 6*d*x)*sqrt(b)*sqrt(a + b)*log(sqrt(2*sqrt(b)*sq 
rt(a + b) - a - 2*b) + e**(c + d*x)*sqrt(a))*a**3 - 4*e**(6*c + 6*d*x)*sqr 
t(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*e**(6*c + 6*d*x)*sqrt(b)*sqrt(a + b)*log(2*sqrt(b)*sqr 
t(a + b) + e**(2*c + 2*d*x)*a + a + 2*b)*a**3 + 4*e**(6*c + 6*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 - 9*e**(4*c + 4*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 - 24*e**(4*c + 4*d*x)*sqrt(b) 
*sqrt(a + b)*log( - sqrt(2*sqrt(b)*sqrt(a + b) - a - 2*b) + e**(c + d*x)*s 
qrt(a))*a**2*b - 16*e**(4*c + 4*d*x)*sqrt(b)*sqrt(a + b)*log( - sqrt(2*sqr 
t(b)*sqrt(a + b) - a - 2*b) + e**(c + d*x)*sqrt(a))*a*b**2 - 9*e**(4*c + 4 
*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 - 24*e**(4*c + 4*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 - 16*e**(4 
*c + 4*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*b**2 + 9*e**(4*c + 4*d*x)*sqrt(b)*sqrt(a + b)*lo 
g(2*sqrt(b)*sqrt(a + b) + e**(2*c + 2*d*x)*a + a + 2*b)*a**3 + 24*e**(4...