\(\int \frac {\text {sech}(c+d x)}{a+b \sinh ^2(c+d x)} \, dx\) [287]

Optimal result

Integrand size = 21, antiderivative size = 59 \[ \int \frac {\text {sech}(c+d x)}{a+b \sinh ^2(c+d x)} \, dx=\frac {\arctan (\sinh (c+d x))}{(a-b) d}-\frac {\sqrt {b} \arctan \left (\frac {\sqrt {b} \sinh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} (a-b) d} \] Output:

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

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.92 \[ \int \frac {\text {sech}(c+d x)}{a+b \sinh ^2(c+d x)} \, dx=\frac {\frac {\sqrt {b} \arctan \left (\frac {\sqrt {a} \text {csch}(c+d x)}{\sqrt {b}}\right )}{\sqrt {a}}+2 \arctan \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )}{a d-b d} \] Input:

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

Output:

((Sqrt[b]*ArcTan[(Sqrt[a]*Csch[c + d*x])/Sqrt[b]])/Sqrt[a] + 2*ArcTan[Tanh 
[(c + d*x)/2]])/(a*d - b*d)
 

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.97, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {3042, 3669, 303, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3669

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

\(\Big \downarrow \) 303

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

\(\Big \downarrow \) 216

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

\(\Big \downarrow \) 218

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

Input:

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

Output:

(ArcTan[Sinh[c + d*x]]/(a - b) - (Sqrt[b]*ArcTan[(Sqrt[b]*Sinh[c + d*x])/S 
qrt[a]])/(Sqrt[a]*(a - b)))/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 303

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

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 [C] (verified)

Result contains complex when optimal does not.

Time = 21.90 (sec) , antiderivative size = 148, normalized size of antiderivative = 2.51

method	result	size
risch	\(\frac {i \ln \left ({\mathrm e}^{d x +c}+i\right )}{\left (a -b \right ) d}-\frac {i \ln \left ({\mathrm e}^{d x +c}-i\right )}{\left (a -b \right ) d}+\frac {\sqrt {-a b}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {-a b}\, {\mathrm e}^{d x +c}}{b}-1\right )}{2 a \left (a -b \right ) d}-\frac {\sqrt {-a b}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {-a b}\, {\mathrm e}^{d x +c}}{b}-1\right )}{2 a \left (a -b \right ) d}\)	\(148\)
derivativedivides	\(\frac {-\frac {2 b a \left (-\frac {\left (a +\sqrt {-b \left (a -b \right )}-b \right ) \operatorname {arctanh}\left (\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) a}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}+\frac {\left (-a +\sqrt {-b \left (a -b \right )}+b \right ) \arctan \left (\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) a}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}\right )}{a -b}+\frac {2 \arctan \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a -b}}{d}\)	\(209\)
default	\(\frac {-\frac {2 b a \left (-\frac {\left (a +\sqrt {-b \left (a -b \right )}-b \right ) \operatorname {arctanh}\left (\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) a}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}+\frac {\left (-a +\sqrt {-b \left (a -b \right )}+b \right ) \arctan \left (\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) a}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}\right )}{a -b}+\frac {2 \arctan \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a -b}}{d}\)	\(209\)

Input:

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

Output:

I/(a-b)/d*ln(exp(d*x+c)+I)-I/(a-b)/d*ln(exp(d*x+c)-I)+1/2/a*(-a*b)^(1/2)/( 
a-b)/d*ln(exp(2*d*x+2*c)-2*(-a*b)^(1/2)/b*exp(d*x+c)-1)-1/2/a*(-a*b)^(1/2) 
/(a-b)/d*ln(exp(2*d*x+2*c)+2*(-a*b)^(1/2)/b*exp(d*x+c)-1)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 158 vs. \(2 (51) = 102\).

Time = 0.13 (sec) , antiderivative size = 511, normalized size of antiderivative = 8.66 \[ \int \frac {\text {sech}(c+d x)}{a+b \sinh ^2(c+d x)} \, dx=\left [-\frac {\sqrt {-\frac {b}{a}} \log \left (\frac {b \cosh \left (d x + c\right )^{4} + 4 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + b \sinh \left (d x + c\right )^{4} - 2 \, {\left (2 \, a + b\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, b \cosh \left (d x + c\right )^{2} - 2 \, a - b\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left (b \cosh \left (d x + c\right )^{3} - {\left (2 \, a + b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + 4 \, {\left (a \cosh \left (d x + c\right )^{3} + 3 \, a \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + a \sinh \left (d x + c\right )^{3} - a \cosh \left (d x + c\right ) + {\left (3 \, a \cosh \left (d x + c\right )^{2} - a\right )} \sinh \left (d x + c\right )\right )} \sqrt {-\frac {b}{a}} + b}{b \cosh \left (d x + c\right )^{4} + 4 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + b \sinh \left (d x + c\right )^{4} + 2 \, {\left (2 \, a - b\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, b \cosh \left (d x + c\right )^{2} + 2 \, a - b\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left (b \cosh \left (d x + c\right )^{3} + {\left (2 \, a - b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + b}\right ) - 4 \, \arctan \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right )}{2 \, {\left (a - b\right )} d}, -\frac {\sqrt {\frac {b}{a}} \arctan \left (\frac {1}{2} \, \sqrt {\frac {b}{a}} {\left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right )}\right ) + \sqrt {\frac {b}{a}} \arctan \left (\frac {{\left (b \cosh \left (d x + c\right )^{3} + 3 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + b \sinh \left (d x + c\right )^{3} + {\left (4 \, a - b\right )} \cosh \left (d x + c\right ) + {\left (3 \, b \cosh \left (d x + c\right )^{2} + 4 \, a - b\right )} \sinh \left (d x + c\right )\right )} \sqrt {\frac {b}{a}}}{2 \, b}\right ) - 2 \, \arctan \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right )}{{\left (a - b\right )} d}\right ] \] Input:

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

Output:

[-1/2*(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)*si 
nh(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)*c 
osh(d*x + c))*sinh(d*x + c) + b)) - 4*arctan(cosh(d*x + c) + sinh(d*x + c) 
))/((a - b)*d), -(sqrt(b/a)*arctan(1/2*sqrt(b/a)*(cosh(d*x + c) + sinh(d*x 
 + c))) + sqrt(b/a)*arctan(1/2*(b*cosh(d*x + c)^3 + 3*b*cosh(d*x + c)*sinh 
(d*x + c)^2 + b*sinh(d*x + c)^3 + (4*a - b)*cosh(d*x + c) + (3*b*cosh(d*x 
+ c)^2 + 4*a - b)*sinh(d*x + c))*sqrt(b/a)/b) - 2*arctan(cosh(d*x + c) + s 
inh(d*x + c)))/((a - b)*d)]
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

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

integrate(sech(d*x+c)/(a+b*sinh(d*x+c)^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 [B] (verification not implemented)

Time = 2.19 (sec) , antiderivative size = 648, normalized size of antiderivative = 10.98 \[ \int \frac {\text {sech}(c+d x)}{a+b \sinh ^2(c+d x)} \, dx=\frac {2\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (16\,a^2\,\sqrt {a^2\,d^2-2\,a\,b\,d^2+b^2\,d^2}+b^2\,\sqrt {a^2\,d^2-2\,a\,b\,d^2+b^2\,d^2}-8\,a\,b\,\sqrt {a^2\,d^2-2\,a\,b\,d^2+b^2\,d^2}\right )}{16\,d\,a^3-24\,d\,a^2\,b+9\,d\,a\,b^2-d\,b^3}\right )}{\sqrt {a^2\,d^2-2\,a\,b\,d^2+b^2\,d^2}}-\frac {\sqrt {b}\,\left (2\,\mathrm {atan}\left (\frac {\sqrt {b}\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {a\,d^2\,{\left (a-b\right )}^2}}{2\,a\,d\,\left (a-b\right )}\right )-2\,\mathrm {atan}\left (\frac {\left (a^3\,b^{5/2}\,\sqrt {a^3\,d^2-2\,a^2\,b\,d^2+a\,b^2\,d^2}-a^2\,b^{7/2}\,\sqrt {a^3\,d^2-2\,a^2\,b\,d^2+a\,b^2\,d^2}\right )\,\left ({\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (\frac {64\,\left (8\,d\,a^3-10\,d\,a^2\,b+2\,d\,a\,b^2\right )}{a\,b^3\,\left (a\,b-a^2\right )\,\sqrt {a\,d^2\,{\left (a-b\right )}^2}\,\sqrt {a^3\,d^2-2\,a^2\,b\,d^2+a\,b^2\,d^2}}+\frac {32\,\left (b^{3/2}\,\sqrt {a^3\,d^2-2\,a^2\,b\,d^2+a\,b^2\,d^2}-4\,a\,\sqrt {b}\,\sqrt {a^3\,d^2-2\,a^2\,b\,d^2+a\,b^2\,d^2}\right )}{a^2\,b^{5/2}\,d\,\left (a-b\right )\,\left (a\,b-a^2\right )\,\sqrt {a^3\,d^2-2\,a^2\,b\,d^2+a\,b^2\,d^2}}\right )-\frac {32\,{\mathrm {e}}^{3\,c}\,{\mathrm {e}}^{3\,d\,x}\,\left (b^{3/2}\,\sqrt {a^3\,d^2-2\,a^2\,b\,d^2+a\,b^2\,d^2}-4\,a\,\sqrt {b}\,\sqrt {a^3\,d^2-2\,a^2\,b\,d^2+a\,b^2\,d^2}\right )}{a^2\,b^{5/2}\,d\,\left (a-b\right )\,\left (a\,b-a^2\right )\,\sqrt {a^3\,d^2-2\,a^2\,b\,d^2+a\,b^2\,d^2}}\right )}{256\,a-64\,b}\right )\right )}{2\,\sqrt {a^3\,d^2-2\,a^2\,b\,d^2+a\,b^2\,d^2}} \] Input:

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

Output:

(2*atan((exp(d*x)*exp(c)*(16*a^2*(a^2*d^2 + b^2*d^2 - 2*a*b*d^2)^(1/2) + b 
^2*(a^2*d^2 + b^2*d^2 - 2*a*b*d^2)^(1/2) - 8*a*b*(a^2*d^2 + b^2*d^2 - 2*a* 
b*d^2)^(1/2)))/(16*a^3*d - b^3*d + 9*a*b^2*d - 24*a^2*b*d)))/(a^2*d^2 + b^ 
2*d^2 - 2*a*b*d^2)^(1/2) - (b^(1/2)*(2*atan((b^(1/2)*exp(d*x)*exp(c)*(a*d^ 
2*(a - b)^2)^(1/2))/(2*a*d*(a - b))) - 2*atan(((a^3*b^(5/2)*(a^3*d^2 + a*b 
^2*d^2 - 2*a^2*b*d^2)^(1/2) - a^2*b^(7/2)*(a^3*d^2 + a*b^2*d^2 - 2*a^2*b*d 
^2)^(1/2))*(exp(d*x)*exp(c)*((64*(8*a^3*d + 2*a*b^2*d - 10*a^2*b*d))/(a*b^ 
3*(a*b - a^2)*(a*d^2*(a - b)^2)^(1/2)*(a^3*d^2 + a*b^2*d^2 - 2*a^2*b*d^2)^ 
(1/2)) + (32*(b^(3/2)*(a^3*d^2 + a*b^2*d^2 - 2*a^2*b*d^2)^(1/2) - 4*a*b^(1 
/2)*(a^3*d^2 + a*b^2*d^2 - 2*a^2*b*d^2)^(1/2)))/(a^2*b^(5/2)*d*(a - b)*(a* 
b - a^2)*(a^3*d^2 + a*b^2*d^2 - 2*a^2*b*d^2)^(1/2))) - (32*exp(3*c)*exp(3* 
d*x)*(b^(3/2)*(a^3*d^2 + a*b^2*d^2 - 2*a^2*b*d^2)^(1/2) - 4*a*b^(1/2)*(a^3 
*d^2 + a*b^2*d^2 - 2*a^2*b*d^2)^(1/2)))/(a^2*b^(5/2)*d*(a - b)*(a*b - a^2) 
*(a^3*d^2 + a*b^2*d^2 - 2*a^2*b*d^2)^(1/2))))/(256*a - 64*b))))/(2*(a^3*d^ 
2 + a*b^2*d^2 - 2*a^2*b*d^2)^(1/2))
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 364, normalized size of antiderivative = 6.17 \[ \int \frac {\text {sech}(c+d x)}{a+b \sinh ^2(c+d x)} \, dx=\frac {4 \mathit {atan} \left (e^{d x +c}\right ) a b +2 \sqrt {b}\, \sqrt {a}\, \sqrt {a -b}\, \sqrt {2 \sqrt {a}\, \sqrt {a -b}+2 a -b}\, \mathit {atan} \left (\frac {e^{d x +c} b}{\sqrt {b}\, \sqrt {2 \sqrt {a}\, \sqrt {a -b}+2 a -b}}\right )-2 \sqrt {b}\, \sqrt {2 \sqrt {a}\, \sqrt {a -b}+2 a -b}\, \mathit {atan} \left (\frac {e^{d x +c} b}{\sqrt {b}\, \sqrt {2 \sqrt {a}\, \sqrt {a -b}+2 a -b}}\right ) a +\sqrt {b}\, \sqrt {a}\, \sqrt {a -b}\, \sqrt {2 \sqrt {a}\, \sqrt {a -b}-2 a +b}\, \mathrm {log}\left (-\sqrt {2 \sqrt {a}\, \sqrt {a -b}-2 a +b}+e^{d x +c} \sqrt {b}\right )-\sqrt {b}\, \sqrt {a}\, \sqrt {a -b}\, \sqrt {2 \sqrt {a}\, \sqrt {a -b}-2 a +b}\, \mathrm {log}\left (\sqrt {2 \sqrt {a}\, \sqrt {a -b}-2 a +b}+e^{d x +c} \sqrt {b}\right )+\sqrt {b}\, \sqrt {2 \sqrt {a}\, \sqrt {a -b}-2 a +b}\, \mathrm {log}\left (-\sqrt {2 \sqrt {a}\, \sqrt {a -b}-2 a +b}+e^{d x +c} \sqrt {b}\right ) a -\sqrt {b}\, \sqrt {2 \sqrt {a}\, \sqrt {a -b}-2 a +b}\, \mathrm {log}\left (\sqrt {2 \sqrt {a}\, \sqrt {a -b}-2 a +b}+e^{d x +c} \sqrt {b}\right ) a}{2 a b d \left (a -b \right )} \] Input:

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

Output:

(4*atan(e**(c + d*x))*a*b + 2*sqrt(b)*sqrt(a)*sqrt(a - 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))) - 2*sqrt(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 + sqrt 
(b)*sqrt(a)*sqrt(a - b)*sqrt(2*sqrt(a)*sqrt(a - b) - 2*a + b)*log( - sqrt( 
2*sqrt(a)*sqrt(a - b) - 2*a + b) + e**(c + d*x)*sqrt(b)) - sqrt(b)*sqrt(a) 
*sqrt(a - b)*sqrt(2*sqrt(a)*sqrt(a - b) - 2*a + b)*log(sqrt(2*sqrt(a)*sqrt 
(a - b) - 2*a + b) + e**(c + d*x)*sqrt(b)) + sqrt(b)*sqrt(2*sqrt(a)*sqrt(a 
 - b) - 2*a + b)*log( - sqrt(2*sqrt(a)*sqrt(a - b) - 2*a + b) + e**(c + d* 
x)*sqrt(b))*a - sqrt(b)*sqrt(2*sqrt(a)*sqrt(a - b) - 2*a + b)*log(sqrt(2*s 
qrt(a)*sqrt(a - b) - 2*a + b) + e**(c + d*x)*sqrt(b))*a)/(2*a*b*d*(a - b))