\(\int \frac {\text {sech}(c+d x)}{(a+b \text {sech}^2(c+d x))^2} \, dx\) [86]

Optimal result

Integrand size = 21, antiderivative size = 82 \[ \int \frac {\text {sech}(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=\frac {(2 a+b) \arctan \left (\frac {\sqrt {a} \sinh (c+d x)}{\sqrt {a+b}}\right )}{2 a^{3/2} (a+b)^{3/2} d}-\frac {b \sinh (c+d x)}{2 a (a+b) d \left (a+b+a \sinh ^2(c+d x)\right )} \] Output:

1/2*(2*a+b)*arctan(a^(1/2)*sinh(d*x+c)/(a+b)^(1/2))/a^(3/2)/(a+b)^(3/2)/d- 
1/2*b*sinh(d*x+c)/a/(a+b)/d/(a+b+a*sinh(d*x+c)^2)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.24 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.51 \[ \int \frac {\text {sech}(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=\frac {\left (2 a^2+3 a b+b^2\right ) \arctan \left (\frac {\sqrt {a} \sinh (c+d x)}{\sqrt {a+b}}\right )-\sqrt {a} b \sqrt {a+b} \sinh (c+d x)+a (2 a+b) \arctan \left (\frac {\sqrt {a} \sinh (c+d x)}{\sqrt {a+b}}\right ) \sinh ^2(c+d x)}{a^{3/2} (a+b)^{3/2} d (a+2 b+a \cosh (2 (c+d x)))} \] Input:

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

Output:

((2*a^2 + 3*a*b + b^2)*ArcTan[(Sqrt[a]*Sinh[c + d*x])/Sqrt[a + b]] - Sqrt[ 
a]*b*Sqrt[a + b]*Sinh[c + d*x] + a*(2*a + b)*ArcTan[(Sqrt[a]*Sinh[c + d*x] 
)/Sqrt[a + b]]*Sinh[c + d*x]^2)/(a^(3/2)*(a + b)^(3/2)*d*(a + 2*b + a*Cosh 
[2*(c + d*x)]))
 

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.98, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3042, 4635, 298, 218}

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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4635

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

\(\Big \downarrow \) 298

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

\(\Big \downarrow \) 218

\(\displaystyle \frac {\frac {(2 a+b) \arctan \left (\frac {\sqrt {a} \sinh (c+d x)}{\sqrt {a+b}}\right )}{2 a^{3/2} (a+b)^{3/2}}-\frac {b \sinh (c+d x)}{2 a (a+b) \left (a \sinh ^2(c+d x)+a+b\right )}}{d}\)

Input:

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

Output:

(((2*a + b)*ArcTan[(Sqrt[a]*Sinh[c + d*x])/Sqrt[a + b]])/(2*a^(3/2)*(a + b 
)^(3/2)) - (b*Sinh[c + d*x])/(2*a*(a + b)*(a + b + a*Sinh[c + d*x]^2)))/d
 

Defintions of rubi rules used

rule 218

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R 
t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[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 4635

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(200\) vs. \(2(70)=140\).

Time = 0.53 (sec) , antiderivative size = 201, normalized size of antiderivative = 2.45

method	result	size
derivativedivides	\(\frac {\frac {\frac {b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{a \left (a +b \right )}-\frac {b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{a \left (a +b \right )}}{\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} b +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a +b}+\frac {\left (2 a +b \right ) \left (\frac {\arctan \left (\frac {2 \sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2 \sqrt {b}}{2 \sqrt {a}}\right )}{2 \sqrt {a +b}\, \sqrt {a}}+\frac {\arctan \left (\frac {2 \sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+2 \sqrt {b}}{2 \sqrt {a}}\right )}{2 \sqrt {a +b}\, \sqrt {a}}\right )}{\left (a +b \right ) a}}{d}\)	\(201\)
default	\(\frac {\frac {\frac {b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{a \left (a +b \right )}-\frac {b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{a \left (a +b \right )}}{\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} b +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a +b}+\frac {\left (2 a +b \right ) \left (\frac {\arctan \left (\frac {2 \sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2 \sqrt {b}}{2 \sqrt {a}}\right )}{2 \sqrt {a +b}\, \sqrt {a}}+\frac {\arctan \left (\frac {2 \sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+2 \sqrt {b}}{2 \sqrt {a}}\right )}{2 \sqrt {a +b}\, \sqrt {a}}\right )}{\left (a +b \right ) a}}{d}\)	\(201\)
risch	\(-\frac {{\mathrm e}^{d x +c} b \left ({\mathrm e}^{2 d x +2 c}-1\right )}{a d \left (a +b \right ) \left ({\mathrm e}^{4 d x +4 c} a +2 a \,{\mathrm e}^{2 d x +2 c}+4 b \,{\mathrm e}^{2 d x +2 c}+a \right )}-\frac {\ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \left (a +b \right ) {\mathrm e}^{d x +c}}{\sqrt {-a^{2}-a b}}-1\right )}{2 \sqrt {-a^{2}-a b}\, \left (a +b \right ) d}-\frac {b \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \left (a +b \right ) {\mathrm e}^{d x +c}}{\sqrt {-a^{2}-a b}}-1\right )}{4 \sqrt {-a^{2}-a b}\, \left (a +b \right ) d a}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \left (a +b \right ) {\mathrm e}^{d x +c}}{\sqrt {-a^{2}-a b}}-1\right )}{2 \sqrt {-a^{2}-a b}\, \left (a +b \right ) d}+\frac {b \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \left (a +b \right ) {\mathrm e}^{d x +c}}{\sqrt {-a^{2}-a b}}-1\right )}{4 \sqrt {-a^{2}-a b}\, \left (a +b \right ) d a}\)	\(308\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 932 vs. \(2 (70) = 140\).

Time = 0.25 (sec) , antiderivative size = 1856, normalized size of antiderivative = 22.63 \[ \int \frac {\text {sech}(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

[-1/4*(4*(a^2*b + a*b^2)*cosh(d*x + c)^3 + 12*(a^2*b + a*b^2)*cosh(d*x + c 
)*sinh(d*x + c)^2 + 4*(a^2*b + a*b^2)*sinh(d*x + c)^3 + ((2*a^2 + a*b)*cos 
h(d*x + c)^4 + 4*(2*a^2 + a*b)*cosh(d*x + c)*sinh(d*x + c)^3 + (2*a^2 + a* 
b)*sinh(d*x + c)^4 + 2*(2*a^2 + 5*a*b + 2*b^2)*cosh(d*x + c)^2 + 2*(3*(2*a 
^2 + a*b)*cosh(d*x + c)^2 + 2*a^2 + 5*a*b + 2*b^2)*sinh(d*x + c)^2 + 2*a^2 
 + a*b + 4*((2*a^2 + a*b)*cosh(d*x + c)^3 + (2*a^2 + 5*a*b + 2*b^2)*cosh(d 
*x + c))*sinh(d*x + c))*sqrt(-a^2 - a*b)*log((a*cosh(d*x + c)^4 + 4*a*cosh 
(d*x + c)*sinh(d*x + c)^3 + a*sinh(d*x + c)^4 - 2*(3*a + 2*b)*cosh(d*x + c 
)^2 + 2*(3*a*cosh(d*x + c)^2 - 3*a - 2*b)*sinh(d*x + c)^2 + 4*(a*cosh(d*x 
+ c)^3 - (3*a + 2*b)*cosh(d*x + c))*sinh(d*x + c) - 4*(cosh(d*x + c)^3 + 3 
*cosh(d*x + c)*sinh(d*x + c)^2 + sinh(d*x + c)^3 + (3*cosh(d*x + c)^2 - 1) 
*sinh(d*x + c) - cosh(d*x + c))*sqrt(-a^2 - a*b) + a)/(a*cosh(d*x + c)^4 + 
 4*a*cosh(d*x + c)*sinh(d*x + c)^3 + a*sinh(d*x + c)^4 + 2*(a + 2*b)*cosh( 
d*x + c)^2 + 2*(3*a*cosh(d*x + c)^2 + a + 2*b)*sinh(d*x + c)^2 + 4*(a*cosh 
(d*x + c)^3 + (a + 2*b)*cosh(d*x + c))*sinh(d*x + c) + a)) - 4*(a^2*b + a* 
b^2)*cosh(d*x + c) - 4*(a^2*b + a*b^2 - 3*(a^2*b + a*b^2)*cosh(d*x + c)^2) 
*sinh(d*x + c))/((a^5 + 2*a^4*b + a^3*b^2)*d*cosh(d*x + c)^4 + 4*(a^5 + 2* 
a^4*b + a^3*b^2)*d*cosh(d*x + c)*sinh(d*x + c)^3 + (a^5 + 2*a^4*b + a^3*b^ 
2)*d*sinh(d*x + c)^4 + 2*(a^5 + 4*a^4*b + 5*a^3*b^2 + 2*a^2*b^3)*d*cosh(d* 
x + c)^2 + 2*(3*(a^5 + 2*a^4*b + a^3*b^2)*d*cosh(d*x + c)^2 + (a^5 + 4*...
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

Exception generated. \[ \int \frac {\text {sech}(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Limit: Max order reached or unable 
to make series expansion Error: Bad Argument Value
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 3321, normalized size of antiderivative = 40.50 \[ \int \frac {\text {sech}(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 4*e**(4*c + 4*d*x)*sqrt(b)*sqrt(a)*sqrt(a + b)*sqrt(2*sqrt(b)*sqrt(a + 
 b) + a + 2*b)*atan((e**(c + d*x)*a)/(sqrt(a)*sqrt(2*sqrt(b)*sqrt(a + b) + 
 a + 2*b)))*a**2 - 2*e**(4*c + 4*d*x)*sqrt(b)*sqrt(a)*sqrt(a + b)*sqrt(2*s 
qrt(b)*sqrt(a + b) + a + 2*b)*atan((e**(c + d*x)*a)/(sqrt(a)*sqrt(2*sqrt(b 
)*sqrt(a + b) + a + 2*b)))*a*b - 8*e**(2*c + 2*d*x)*sqrt(b)*sqrt(a)*sqrt(a 
 + b)*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2*b)*atan((e**(c + d*x)*a)/(sqrt(a) 
*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2*b)))*a**2 - 20*e**(2*c + 2*d*x)*sqrt(b 
)*sqrt(a)*sqrt(a + b)*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2*b)*atan((e**(c + 
d*x)*a)/(sqrt(a)*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2*b)))*a*b - 8*e**(2*c + 
 2*d*x)*sqrt(b)*sqrt(a)*sqrt(a + b)*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2*b)* 
atan((e**(c + d*x)*a)/(sqrt(a)*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2*b)))*b** 
2 - 4*sqrt(b)*sqrt(a)*sqrt(a + b)*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2*b)*at 
an((e**(c + d*x)*a)/(sqrt(a)*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2*b)))*a**2 
- 2*sqrt(b)*sqrt(a)*sqrt(a + b)*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2*b)*atan 
((e**(c + d*x)*a)/(sqrt(a)*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2*b)))*a*b + 4 
*e**(4*c + 4*d*x)*sqrt(a)*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2*b)*atan((e**( 
c + d*x)*a)/(sqrt(a)*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2*b)))*a**3 + 6*e**( 
4*c + 4*d*x)*sqrt(a)*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2*b)*atan((e**(c + d 
*x)*a)/(sqrt(a)*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2*b)))*a**2*b + 2*e**(4*c 
 + 4*d*x)*sqrt(a)*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2*b)*atan((e**(c + d...