\(\int \frac {\text {sech}(c+d x)}{(a+b \sinh ^2(c+d x))^2} \, dx\) [299]

Optimal result

Integrand size = 21, antiderivative size = 106 \[ \int \frac {\text {sech}(c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^2} \, dx=\frac {\arctan (\sinh (c+d x))}{(a-b)^2 d}-\frac {(3 a-b) \sqrt {b} \arctan \left (\frac {\sqrt {b} \sinh (c+d x)}{\sqrt {a}}\right )}{2 a^{3/2} (a-b)^2 d}-\frac {b \sinh (c+d x)}{2 a (a-b) d \left (a+b \sinh ^2(c+d x)\right )} \] Output:

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

Mathematica [A] (verified)

Time = 0.31 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.64 \[ \int \frac {\text {sech}(c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^2} \, dx=\frac {(2 a-b) \left (-\sqrt {b} (-3 a+b) \arctan \left (\frac {\sqrt {a} \text {csch}(c+d x)}{\sqrt {b}}\right )+4 a^{3/2} \arctan \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )\right )+\left (-b^{3/2} (-3 a+b) \arctan \left (\frac {\sqrt {a} \text {csch}(c+d x)}{\sqrt {b}}\right )+4 a^{3/2} b \arctan \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )\right ) \cosh (2 (c+d x))-2 \sqrt {a} (a-b) b \sinh (c+d x)}{2 a^{3/2} (a-b)^2 d (2 a-b+b \cosh (2 (c+d x)))} \] Input:

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

Output:

((2*a - b)*(-(Sqrt[b]*(-3*a + b)*ArcTan[(Sqrt[a]*Csch[c + d*x])/Sqrt[b]]) 
+ 4*a^(3/2)*ArcTan[Tanh[(c + d*x)/2]]) + (-(b^(3/2)*(-3*a + b)*ArcTan[(Sqr 
t[a]*Csch[c + d*x])/Sqrt[b]]) + 4*a^(3/2)*b*ArcTan[Tanh[(c + d*x)/2]])*Cos 
h[2*(c + d*x)] - 2*Sqrt[a]*(a - b)*b*Sinh[c + d*x])/(2*a^(3/2)*(a - b)^2*d 
*(2*a - b + b*Cosh[2*(c + d*x)]))
 

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.09, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3042, 3669, 316, 397, 216, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3669

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

\(\Big \downarrow \) 316

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

\(\Big \downarrow \) 397

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

\(\Big \downarrow \) 216

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

\(\Big \downarrow \) 218

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 216

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A 
rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a 
, 0] || GtQ[b, 0])
 

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 316

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim 
p[(-b)*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(2*a*(p + 1)*(b*c - a*d)) 
), x] + Simp[1/(2*a*(p + 1)*(b*c - a*d))   Int[(a + b*x^2)^(p + 1)*(c + d*x 
^2)^q*Simp[b*c + 2*(p + 1)*(b*c - a*d) + d*b*(2*(p + q + 2) + 1)*x^2, x], x 
], x] /; FreeQ[{a, b, c, d, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] &&  ! 
( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomialQ[a, b, c, d, 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 3042

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

rule 3669

Int[cos[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^ 
(p_.), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Simp[ff/f   S 
ubst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b*ff^2*x^2)^p, x], x, Sin[e + f*x] 
/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(304\) vs. \(2(94)=188\).

Time = 0.24 (sec) , antiderivative size = 305, normalized size of antiderivative = 2.88

\[\frac {-\frac {2 b \left (\frac {-\frac {\left (a -b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{2 a}+\frac {\left (a -b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a}}{\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a -2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +4 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+a}+\frac {\left (3 a -b \right ) \left (-\frac {\left (a +\sqrt {-b \left (a -b \right )}-b \right ) \operatorname {arctanh}\left (\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) a}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}+\frac {\left (-a +\sqrt {-b \left (a -b \right )}+b \right ) \arctan \left (\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) a}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}\right )}{2}\right )}{\left (a -b \right )^{2}}+\frac {2 \arctan \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (a -b \right )^{2}}}{d}\]

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1068 vs. \(2 (94) = 188\).

Time = 0.15 (sec) , antiderivative size = 2143, normalized size of antiderivative = 20.22 \[ \int \frac {\text {sech}(c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

[-1/4*(4*(a*b - b^2)*cosh(d*x + c)^3 + 12*(a*b - b^2)*cosh(d*x + c)*sinh(d 
*x + c)^2 + 4*(a*b - b^2)*sinh(d*x + c)^3 + ((3*a*b - b^2)*cosh(d*x + c)^4 
 + 4*(3*a*b - b^2)*cosh(d*x + c)*sinh(d*x + c)^3 + (3*a*b - b^2)*sinh(d*x 
+ c)^4 + 2*(6*a^2 - 5*a*b + b^2)*cosh(d*x + c)^2 + 2*(3*(3*a*b - b^2)*cosh 
(d*x + c)^2 + 6*a^2 - 5*a*b + b^2)*sinh(d*x + c)^2 + 3*a*b - b^2 + 4*((3*a 
*b - b^2)*cosh(d*x + c)^3 + (6*a^2 - 5*a*b + b^2)*cosh(d*x + c))*sinh(d*x 
+ c))*sqrt(-b/a)*log((b*cosh(d*x + c)^4 + 4*b*cosh(d*x + c)*sinh(d*x + c)^ 
3 + b*sinh(d*x + c)^4 - 2*(2*a + b)*cosh(d*x + c)^2 + 2*(3*b*cosh(d*x + c) 
^2 - 2*a - b)*sinh(d*x + c)^2 + 4*(b*cosh(d*x + c)^3 - (2*a + b)*cosh(d*x 
+ c))*sinh(d*x + c) + 4*(a*cosh(d*x + c)^3 + 3*a*cosh(d*x + c)*sinh(d*x + 
c)^2 + a*sinh(d*x + c)^3 - a*cosh(d*x + c) + (3*a*cosh(d*x + c)^2 - a)*sin 
h(d*x + c))*sqrt(-b/a) + b)/(b*cosh(d*x + c)^4 + 4*b*cosh(d*x + c)*sinh(d* 
x + c)^3 + b*sinh(d*x + c)^4 + 2*(2*a - b)*cosh(d*x + c)^2 + 2*(3*b*cosh(d 
*x + c)^2 + 2*a - b)*sinh(d*x + c)^2 + 4*(b*cosh(d*x + c)^3 + (2*a - b)*co 
sh(d*x + c))*sinh(d*x + c) + b)) - 8*(a*b*cosh(d*x + c)^4 + 4*a*b*cosh(d*x 
 + c)*sinh(d*x + c)^3 + a*b*sinh(d*x + c)^4 + 2*(2*a^2 - a*b)*cosh(d*x + c 
)^2 + 2*(3*a*b*cosh(d*x + c)^2 + 2*a^2 - a*b)*sinh(d*x + c)^2 + a*b + 4*(a 
*b*cosh(d*x + c)^3 + (2*a^2 - a*b)*cosh(d*x + c))*sinh(d*x + c))*arctan(co 
sh(d*x + c) + sinh(d*x + c)) - 4*(a*b - b^2)*cosh(d*x + c) + 4*(3*(a*b - b 
^2)*cosh(d*x + c)^2 - a*b + b^2)*sinh(d*x + c))/((a^3*b - 2*a^2*b^2 + a...
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

-(b*e^(3*d*x + 3*c) - b*e^(d*x + c))/(a^2*b*d - a*b^2*d + (a^2*b*d*e^(4*c) 
 - a*b^2*d*e^(4*c))*e^(4*d*x) + 2*(2*a^3*d*e^(2*c) - 3*a^2*b*d*e^(2*c) + a 
*b^2*d*e^(2*c))*e^(2*d*x)) + 2*arctan(e^(d*x + c))/(a^2*d - 2*a*b*d + b^2* 
d) - 2*integrate(1/2*((3*a*b*e^(3*c) - b^2*e^(3*c))*e^(3*d*x) + (3*a*b*e^c 
 - b^2*e^c)*e^(d*x))/(a^3*b - 2*a^2*b^2 + a*b^3 + (a^3*b*e^(4*c) - 2*a^2*b 
^2*e^(4*c) + a*b^3*e^(4*c))*e^(4*d*x) + 2*(2*a^4*e^(2*c) - 5*a^3*b*e^(2*c) 
 + 4*a^2*b^2*e^(2*c) - a*b^3*e^(2*c))*e^(2*d*x)), x)
                                                                                    
                                                                                    
 

Giac [F(-2)]

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

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 3045, normalized size of antiderivative = 28.73 \[ \int \frac {\text {sech}(c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

(8*e**(4*c + 4*d*x)*atan(e**(c + d*x))*a**2*b**2 + 32*e**(2*c + 2*d*x)*ata 
n(e**(c + d*x))*a**3*b - 16*e**(2*c + 2*d*x)*atan(e**(c + d*x))*a**2*b**2 
+ 8*atan(e**(c + d*x))*a**2*b**2 + 6*e**(4*c + 4*d*x)*sqrt(b)*sqrt(a)*sqrt 
(a - b)*sqrt(2*sqrt(a)*sqrt(a - b) + 2*a - b)*atan((e**(c + d*x)*b)/(sqrt( 
b)*sqrt(2*sqrt(a)*sqrt(a - b) + 2*a - b)))*a*b - 2*e**(4*c + 4*d*x)*sqrt(b 
)*sqrt(a)*sqrt(a - b)*sqrt(2*sqrt(a)*sqrt(a - b) + 2*a - b)*atan((e**(c + 
d*x)*b)/(sqrt(b)*sqrt(2*sqrt(a)*sqrt(a - b) + 2*a - b)))*b**2 + 24*e**(2*c 
 + 2*d*x)*sqrt(b)*sqrt(a)*sqrt(a - b)*sqrt(2*sqrt(a)*sqrt(a - b) + 2*a - b 
)*atan((e**(c + d*x)*b)/(sqrt(b)*sqrt(2*sqrt(a)*sqrt(a - b) + 2*a - b)))*a 
**2 - 20*e**(2*c + 2*d*x)*sqrt(b)*sqrt(a)*sqrt(a - b)*sqrt(2*sqrt(a)*sqrt( 
a - b) + 2*a - b)*atan((e**(c + d*x)*b)/(sqrt(b)*sqrt(2*sqrt(a)*sqrt(a - b 
) + 2*a - b)))*a*b + 4*e**(2*c + 2*d*x)*sqrt(b)*sqrt(a)*sqrt(a - b)*sqrt(2 
*sqrt(a)*sqrt(a - b) + 2*a - b)*atan((e**(c + d*x)*b)/(sqrt(b)*sqrt(2*sqrt 
(a)*sqrt(a - b) + 2*a - b)))*b**2 + 6*sqrt(b)*sqrt(a)*sqrt(a - b)*sqrt(2*s 
qrt(a)*sqrt(a - b) + 2*a - b)*atan((e**(c + d*x)*b)/(sqrt(b)*sqrt(2*sqrt(a 
)*sqrt(a - b) + 2*a - b)))*a*b - 2*sqrt(b)*sqrt(a)*sqrt(a - b)*sqrt(2*sqrt 
(a)*sqrt(a - b) + 2*a - b)*atan((e**(c + d*x)*b)/(sqrt(b)*sqrt(2*sqrt(a)*s 
qrt(a - b) + 2*a - b)))*b**2 - 6*e**(4*c + 4*d*x)*sqrt(b)*sqrt(2*sqrt(a)*s 
qrt(a - b) + 2*a - b)*atan((e**(c + d*x)*b)/(sqrt(b)*sqrt(2*sqrt(a)*sqrt(a 
 - b) + 2*a - b)))*a**2*b + 2*e**(4*c + 4*d*x)*sqrt(b)*sqrt(2*sqrt(a)*s...