\(\int \frac {\text {sech}^4(c+d x)}{(a+b \text {sech}^2(c+d x))^3} \, dx\) [98]

Optimal result

Integrand size = 23, antiderivative size = 125 \[ \int \frac {\text {sech}^4(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\frac {(a+4 b) \text {arctanh}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{8 b^{3/2} (a+b)^{5/2} d}-\frac {a \tanh (c+d x)}{4 b (a+b) d \left (a+b-b \tanh ^2(c+d x)\right )^2}+\frac {(a+4 b) \tanh (c+d x)}{8 b (a+b)^2 d \left (a+b-b \tanh ^2(c+d x)\right )} \] Output:

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

Mathematica [A] (warning: unable to verify)

Time = 3.48 (sec) , antiderivative size = 250, normalized size of antiderivative = 2.00 \[ \int \frac {\text {sech}^4(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\frac {(a+2 b+a \cosh (2 (c+d x))) \text {sech}^6(c+d x) \left (\frac {(a+4 b) \text {arctanh}\left (\frac {\text {sech}(d x) (\cosh (2 c)-\sinh (2 c)) ((a+2 b) \sinh (d x)-a \sinh (2 c+d x))}{2 \sqrt {a+b} \sqrt {b (\cosh (c)-\sinh (c))^4}}\right ) (a+2 b+a \cosh (2 (c+d x)))^2 (\cosh (2 c)-\sinh (2 c))}{b \sqrt {a+b} \sqrt {b (\cosh (c)-\sinh (c))^4}}-\frac {4 (a+b) \text {sech}(2 c) ((a+2 b) \sinh (2 c)-a \sinh (2 d x))}{a}+\frac {(a+2 b+a \cosh (2 (c+d x))) \text {sech}(2 c) ((a+4 b) \sinh (2 c)-(a-2 b) \sinh (2 d x))}{b}\right )}{64 (a+b)^2 d \left (a+b \text {sech}^2(c+d x)\right )^3} \] Input:

Integrate[Sech[c + d*x]^4/(a + b*Sech[c + d*x]^2)^3,x]
 

Output:

((a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d*x]^6*(((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 
)])^2*(Cosh[2*c] - Sinh[2*c]))/(b*Sqrt[a + b]*Sqrt[b*(Cosh[c] - Sinh[c])^4 
]) - (4*(a + b)*Sech[2*c]*((a + 2*b)*Sinh[2*c] - a*Sinh[2*d*x]))/a + ((a + 
 2*b + a*Cosh[2*(c + d*x)])*Sech[2*c]*((a + 4*b)*Sinh[2*c] - (a - 2*b)*Sin 
h[2*d*x]))/b))/(64*(a + b)^2*d*(a + b*Sech[c + d*x]^2)^3)
 

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3042, 4634, 298, 215, 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.

\(\displaystyle \int \frac {\text {sech}^4(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\sec (i c+i d x)^4}{\left (a+b \sec (i c+i d x)^2\right )^3}dx\)

\(\Big \downarrow \) 4634

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

\(\Big \downarrow \) 298

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

\(\Big \downarrow \) 215

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

\(\Big \downarrow \) 221

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

Input:

Int[Sech[c + d*x]^4/(a + b*Sech[c + d*x]^2)^3,x]
 

Output:

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

Defintions of rubi rules used

rule 215

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) 
/(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1))   Int[(a + b*x^2)^(p + 1 
), x], x] /; FreeQ[{a, b}, x] && LtQ[p, -1] && (IntegerQ[4*p] || IntegerQ[6 
*p])
 

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 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. \(309\) vs. \(2(111)=222\).

Time = 1.25 (sec) , antiderivative size = 310, normalized size of antiderivative = 2.48

method	result	size
derivativedivides	\(\frac {-\frac {2 \left (\frac {\left (a -4 b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{8 b \left (a +b \right )}+\frac {\left (3 a^{2}-5 a b +4 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{8 \left (a +b \right )^{2} b}+\frac {\left (3 a^{2}-5 a b +4 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{8 \left (a +b \right )^{2} b}+\frac {\left (a -4 b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 b \left (a +b \right )}\right )}{\left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} b +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a +b \right )^{2}}-\frac {\left (a +4 b \right ) \left (-\frac {\ln \left (\sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}+\frac {\ln \left (\sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}\right )}{4 b \left (a^{2}+2 a b +b^{2}\right )}}{d}\)	\(310\)
default	\(\frac {-\frac {2 \left (\frac {\left (a -4 b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{8 b \left (a +b \right )}+\frac {\left (3 a^{2}-5 a b +4 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{8 \left (a +b \right )^{2} b}+\frac {\left (3 a^{2}-5 a b +4 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{8 \left (a +b \right )^{2} b}+\frac {\left (a -4 b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 b \left (a +b \right )}\right )}{\left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} b +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a +b \right )^{2}}-\frac {\left (a +4 b \right ) \left (-\frac {\ln \left (\sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}+\frac {\ln \left (\sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}\right )}{4 b \left (a^{2}+2 a b +b^{2}\right )}}{d}\)	\(310\)
risch	\(\frac {a^{3} {\mathrm e}^{6 d x +6 c}+4 a^{2} b \,{\mathrm e}^{6 d x +6 c}+3 a^{3} {\mathrm e}^{4 d x +4 c}+2 a^{2} b \,{\mathrm e}^{4 d x +4 c}-8 a \,b^{2} {\mathrm e}^{4 d x +4 c}-16 b^{3} {\mathrm e}^{4 d x +4 c}+3 a^{3} {\mathrm e}^{2 d x +2 c}-4 a^{2} b \,{\mathrm e}^{2 d x +2 c}-16 a \,b^{2} {\mathrm e}^{2 d x +2 c}+a^{3}-2 a^{2} b}{4 a d b \left (a +b \right )^{2} \left ({\mathrm e}^{4 d x +4 c} a +2 a \,{\mathrm e}^{2 d x +2 c}+4 b \,{\mathrm e}^{2 d x +2 c}+a \right )^{2}}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {a \sqrt {a b +b^{2}}+2 b \sqrt {a b +b^{2}}-2 a b -2 b^{2}}{a \sqrt {a b +b^{2}}}\right ) a}{16 \sqrt {a b +b^{2}}\, \left (a +b \right )^{2} d b}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {a \sqrt {a b +b^{2}}+2 b \sqrt {a b +b^{2}}-2 a b -2 b^{2}}{a \sqrt {a b +b^{2}}}\right )}{4 \sqrt {a b +b^{2}}\, \left (a +b \right )^{2} d}-\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {a \sqrt {a b +b^{2}}+2 b \sqrt {a b +b^{2}}+2 a b +2 b^{2}}{a \sqrt {a b +b^{2}}}\right ) a}{16 \sqrt {a b +b^{2}}\, \left (a +b \right )^{2} d b}-\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {a \sqrt {a b +b^{2}}+2 b \sqrt {a b +b^{2}}+2 a b +2 b^{2}}{a \sqrt {a b +b^{2}}}\right )}{4 \sqrt {a b +b^{2}}\, \left (a +b \right )^{2} d}\)	\(509\)

Input:

int(sech(d*x+c)^4/(a+b*sech(d*x+c)^2)^3,x,method=_RETURNVERBOSE)
 

Output:

1/d*(-2*(1/8*(a-4*b)/b/(a+b)*tanh(1/2*d*x+1/2*c)^7+1/8*(3*a^2-5*a*b+4*b^2) 
/(a+b)^2/b*tanh(1/2*d*x+1/2*c)^5+1/8*(3*a^2-5*a*b+4*b^2)/(a+b)^2/b*tanh(1/ 
2*d*x+1/2*c)^3+1/8*(a-4*b)/b/(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)^2-1/4*(a+4*b)/b/(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*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. 2603 vs. \(2 (117) = 234\).

Time = 0.49 (sec) , antiderivative size = 5447, normalized size of antiderivative = 43.58 \[ \int \frac {\text {sech}^4(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\text {Too large to display} \] Input:

integrate(sech(d*x+c)^4/(a+b*sech(d*x+c)^2)^3,x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F]

\[ \int \frac {\text {sech}^4(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\int \frac {\operatorname {sech}^{4}{\left (c + d x \right )}}{\left (a + b \operatorname {sech}^{2}{\left (c + d x \right )}\right )^{3}}\, dx \] Input:

integrate(sech(d*x+c)**4/(a+b*sech(d*x+c)**2)**3,x)
 

Output:

Integral(sech(c + d*x)**4/(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. 369 vs. \(2 (117) = 234\).

Time = 0.23 (sec) , antiderivative size = 369, normalized size of antiderivative = 2.95 \[ \int \frac {\text {sech}^4(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=-\frac {{\left (a + 4 \, b\right )} \log \left (\frac {a e^{\left (-2 \, d x - 2 \, c\right )} + a + 2 \, b - 2 \, \sqrt {{\left (a + b\right )} b}}{a e^{\left (-2 \, d x - 2 \, c\right )} + a + 2 \, b + 2 \, \sqrt {{\left (a + b\right )} b}}\right )}{16 \, {\left (a^{2} b + 2 \, a b^{2} + b^{3}\right )} \sqrt {{\left (a + b\right )} b} d} - \frac {a^{3} - 2 \, a^{2} b + {\left (3 \, a^{3} - 4 \, a^{2} b - 16 \, a b^{2}\right )} e^{\left (-2 \, d x - 2 \, c\right )} + {\left (3 \, a^{3} + 2 \, a^{2} b - 8 \, a b^{2} - 16 \, b^{3}\right )} e^{\left (-4 \, d x - 4 \, c\right )} + {\left (a^{3} + 4 \, a^{2} b\right )} e^{\left (-6 \, d x - 6 \, c\right )}}{4 \, {\left (a^{5} b + 2 \, a^{4} b^{2} + a^{3} b^{3} + 4 \, {\left (a^{5} b + 4 \, a^{4} b^{2} + 5 \, a^{3} b^{3} + 2 \, a^{2} b^{4}\right )} e^{\left (-2 \, d x - 2 \, c\right )} + 2 \, {\left (3 \, a^{5} b + 14 \, a^{4} b^{2} + 27 \, a^{3} b^{3} + 24 \, a^{2} b^{4} + 8 \, a b^{5}\right )} e^{\left (-4 \, d x - 4 \, c\right )} + 4 \, {\left (a^{5} b + 4 \, a^{4} b^{2} + 5 \, a^{3} b^{3} + 2 \, a^{2} b^{4}\right )} e^{\left (-6 \, d x - 6 \, c\right )} + {\left (a^{5} b + 2 \, a^{4} b^{2} + a^{3} b^{3}\right )} e^{\left (-8 \, d x - 8 \, c\right )}\right )} d} \] Input:

integrate(sech(d*x+c)^4/(a+b*sech(d*x+c)^2)^3,x, algorithm="maxima")
 

Output:

-1/16*(a + 4*b)*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*a*b^2 + b^3)* 
sqrt((a + b)*b)*d) - 1/4*(a^3 - 2*a^2*b + (3*a^3 - 4*a^2*b - 16*a*b^2)*e^( 
-2*d*x - 2*c) + (3*a^3 + 2*a^2*b - 8*a*b^2 - 16*b^3)*e^(-4*d*x - 4*c) + (a 
^3 + 4*a^2*b)*e^(-6*d*x - 6*c))/((a^5*b + 2*a^4*b^2 + a^3*b^3 + 4*(a^5*b + 
 4*a^4*b^2 + 5*a^3*b^3 + 2*a^2*b^4)*e^(-2*d*x - 2*c) + 2*(3*a^5*b + 14*a^4 
*b^2 + 27*a^3*b^3 + 24*a^2*b^4 + 8*a*b^5)*e^(-4*d*x - 4*c) + 4*(a^5*b + 4* 
a^4*b^2 + 5*a^3*b^3 + 2*a^2*b^4)*e^(-6*d*x - 6*c) + (a^5*b + 2*a^4*b^2 + a 
^3*b^3)*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. 274 vs. \(2 (117) = 234\).

Time = 0.37 (sec) , antiderivative size = 274, normalized size of antiderivative = 2.19 \[ \int \frac {\text {sech}^4(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\frac {\frac {{\left (a + 4 \, b\right )} \arctan \left (\frac {a e^{\left (2 \, d x + 2 \, c\right )} + a + 2 \, b}{2 \, \sqrt {-a b - b^{2}}}\right )}{{\left (a^{2} b + 2 \, a b^{2} + b^{3}\right )} \sqrt {-a b - b^{2}}} + \frac {2 \, {\left (a^{3} e^{\left (6 \, d x + 6 \, c\right )} + 4 \, a^{2} b e^{\left (6 \, d x + 6 \, c\right )} + 3 \, a^{3} e^{\left (4 \, d x + 4 \, c\right )} + 2 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} - 8 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} - 16 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 3 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} - 4 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} - 16 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} + a^{3} - 2 \, a^{2} b\right )}}{{\left (a^{3} b + 2 \, a^{2} b^{2} + a b^{3}\right )} {\left (a e^{\left (4 \, d x + 4 \, c\right )} + 2 \, a e^{\left (2 \, d x + 2 \, c\right )} + 4 \, b e^{\left (2 \, d x + 2 \, c\right )} + a\right )}^{2}}}{8 \, d} \] Input:

integrate(sech(d*x+c)^4/(a+b*sech(d*x+c)^2)^3,x, algorithm="giac")
 

Output:

1/8*((a + 4*b)*arctan(1/2*(a*e^(2*d*x + 2*c) + a + 2*b)/sqrt(-a*b - b^2))/ 
((a^2*b + 2*a*b^2 + b^3)*sqrt(-a*b - b^2)) + 2*(a^3*e^(6*d*x + 6*c) + 4*a^ 
2*b*e^(6*d*x + 6*c) + 3*a^3*e^(4*d*x + 4*c) + 2*a^2*b*e^(4*d*x + 4*c) - 8* 
a*b^2*e^(4*d*x + 4*c) - 16*b^3*e^(4*d*x + 4*c) + 3*a^3*e^(2*d*x + 2*c) - 4 
*a^2*b*e^(2*d*x + 2*c) - 16*a*b^2*e^(2*d*x + 2*c) + a^3 - 2*a^2*b)/((a^3*b 
 + 2*a^2*b^2 + a*b^3)*(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. \[ \int \frac {\text {sech}^4(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\int \frac {1}{{\mathrm {cosh}\left (c+d\,x\right )}^4\,{\left (a+\frac {b}{{\mathrm {cosh}\left (c+d\,x\right )}^2}\right )}^3} \,d x \] Input:

int(1/(cosh(c + d*x)^4*(a + b/cosh(c + d*x)^2)^3),x)
 

Output:

int(1/(cosh(c + d*x)^4*(a + b/cosh(c + d*x)^2)^3), x)
 

Reduce [B] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 3508, normalized size of antiderivative = 28.06 \[ \int \frac {\text {sech}^4(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx =\text {Too large to display} \] Input:

int(sech(d*x+c)^4/(a+b*sech(d*x+c)^2)^3,x)
 

Output:

(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 + 6*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 + 8*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 + e**(8*c + 8*d*x)*sq 
rt(b)*sqrt(a + b)*log(sqrt(2*sqrt(b)*sqrt(a + b) - a - 2*b) + e**(c + d*x) 
*sqrt(a))*a**5 + 6*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 + 8*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 - 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 - 6*e**(8*c + 8*d*x)*s 
qrt(b)*sqrt(a + b)*log(2*sqrt(b)*sqrt(a + b) + e**(2*c + 2*d*x)*a + a + 2* 
b)*a**4*b - 8*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 + 4*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)*sq 
rt(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 + 80*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)*sqrt(a + b)*log 
( - sqrt(2*sqrt(b)*sqrt(a + b) - a - 2*b) + e**(c + d*x)*sqrt(a))*a**2*...