\(\int \sqrt {a+b \text {csch}^2(c+d x)} \, dx\) [11]

Optimal result

Integrand size = 16, antiderivative size = 84 \[ \int \sqrt {a+b \text {csch}^2(c+d x)} \, dx=\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \coth (c+d x)}{\sqrt {a-b+b \coth ^2(c+d x)}}\right )}{d}-\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \coth (c+d x)}{\sqrt {a-b+b \coth ^2(c+d x)}}\right )}{d} \]

[Out]

arctanh(coth(d*x+c)*a^(1/2)/(a-b+b*coth(d*x+c)^2)^(1/2))*a^(1/2)/d-arctanh(coth(d*x+c)*b^(1/2)/(a-b+b*coth(d*x
+c)^2)^(1/2))*b^(1/2)/d

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {4213, 399, 223, 212, 385} \[ \int \sqrt {a+b \text {csch}^2(c+d x)} \, dx=\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \coth (c+d x)}{\sqrt {a+b \coth ^2(c+d x)-b}}\right )}{d}-\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \coth (c+d x)}{\sqrt {a+b \coth ^2(c+d x)-b}}\right )}{d} \]

[In]

Int[Sqrt[a + b*Csch[c + d*x]^2],x]

[Out]

(Sqrt[a]*ArcTanh[(Sqrt[a]*Coth[c + d*x])/Sqrt[a - b + b*Coth[c + d*x]^2]])/d - (Sqrt[b]*ArcTanh[(Sqrt[b]*Coth[
c + d*x])/Sqrt[a - b + b*Coth[c + d*x]^2]])/d

Rule 212

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

Rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 385

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]

Rule 399

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Dist[b/d, Int[(a + b*x^n)^(p - 1), x]
, x] - Dist[(b*c - a*d)/d, Int[(a + b*x^n)^(p - 1)/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b*c
- a*d, 0] && EqQ[n*(p - 1) + 1, 0] && IntegerQ[n]

Rule 4213

Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist
[ff/f, Subst[Int[(a + b + b*ff^2*x^2)^p/(1 + ff^2*x^2), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p},
x] && NeQ[a + b, 0] && NeQ[p, -1]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\sqrt {a-b+b x^2}}{1-x^2} \, dx,x,\coth (c+d x)\right )}{d} \\ & = \frac {a \text {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \sqrt {a-b+b x^2}} \, dx,x,\coth (c+d x)\right )}{d}-\frac {b \text {Subst}\left (\int \frac {1}{\sqrt {a-b+b x^2}} \, dx,x,\coth (c+d x)\right )}{d} \\ & = \frac {a \text {Subst}\left (\int \frac {1}{1-a x^2} \, dx,x,\frac {\coth (c+d x)}{\sqrt {a-b+b \coth ^2(c+d x)}}\right )}{d}-\frac {b \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\coth (c+d x)}{\sqrt {a-b+b \coth ^2(c+d x)}}\right )}{d} \\ & = \frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \coth (c+d x)}{\sqrt {a-b+b \coth ^2(c+d x)}}\right )}{d}-\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \coth (c+d x)}{\sqrt {a-b+b \coth ^2(c+d x)}}\right )}{d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.70 \[ \int \sqrt {a+b \text {csch}^2(c+d x)} \, dx=\frac {\sqrt {a+b \text {csch}^2(c+d x)} \left (-\sqrt {b} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {b} \cosh (c+d x)}{\sqrt {-a+2 b+a \cosh (2 (c+d x))}}\right )+\sqrt {a} \log \left (\sqrt {2} \sqrt {a} \cosh (c+d x)+\sqrt {-a+2 b+a \cosh (2 (c+d x))}\right )\right ) \sinh (c+d x)}{d \sqrt {-\frac {a}{2}+b+\frac {1}{2} a \cosh (2 (c+d x))}} \]

[In]

Integrate[Sqrt[a + b*Csch[c + d*x]^2],x]

[Out]

(Sqrt[a + b*Csch[c + d*x]^2]*(-(Sqrt[b]*ArcTanh[(Sqrt[2]*Sqrt[b]*Cosh[c + d*x])/Sqrt[-a + 2*b + a*Cosh[2*(c +
d*x)]]]) + Sqrt[a]*Log[Sqrt[2]*Sqrt[a]*Cosh[c + d*x] + Sqrt[-a + 2*b + a*Cosh[2*(c + d*x)]]])*Sinh[c + d*x])/(
d*Sqrt[-1/2*a + b + (a*Cosh[2*(c + d*x)])/2])

Maple [F]

\[\int \sqrt {a +b \operatorname {csch}\left (d x +c \right )^{2}}d x\]

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 709 vs. \(2 (72) = 144\).

Time = 0.37 (sec) , antiderivative size = 4389, normalized size of antiderivative = 52.25 \[ \int \sqrt {a+b \text {csch}^2(c+d x)} \, dx=\text {Too large to display} \]

[In]

integrate((a+b*csch(d*x+c)^2)^(1/2),x, algorithm="fricas")

[Out]

[1/4*(sqrt(a)*log((a*b^2*cosh(d*x + c)^8 + 8*a*b^2*cosh(d*x + c)*sinh(d*x + c)^7 + a*b^2*sinh(d*x + c)^8 + 2*(
a*b^2 + b^3)*cosh(d*x + c)^6 + 2*(14*a*b^2*cosh(d*x + c)^2 + a*b^2 + b^3)*sinh(d*x + c)^6 + 4*(14*a*b^2*cosh(d
*x + c)^3 + 3*(a*b^2 + b^3)*cosh(d*x + c))*sinh(d*x + c)^5 + (a^3 - 4*a^2*b + 9*a*b^2)*cosh(d*x + c)^4 + (70*a
*b^2*cosh(d*x + c)^4 + a^3 - 4*a^2*b + 9*a*b^2 + 30*(a*b^2 + b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 4*(14*a*b
^2*cosh(d*x + c)^5 + 10*(a*b^2 + b^3)*cosh(d*x + c)^3 + (a^3 - 4*a^2*b + 9*a*b^2)*cosh(d*x + c))*sinh(d*x + c)
^3 + a^3 - 2*(a^3 - 3*a^2*b)*cosh(d*x + c)^2 + 2*(14*a*b^2*cosh(d*x + c)^6 + 15*(a*b^2 + b^3)*cosh(d*x + c)^4
- a^3 + 3*a^2*b + 3*(a^3 - 4*a^2*b + 9*a*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + sqrt(2)*(b^2*cosh(d*x + c)^6
+ 6*b^2*cosh(d*x + c)*sinh(d*x + c)^5 + b^2*sinh(d*x + c)^6 + 3*b^2*cosh(d*x + c)^4 + 3*(5*b^2*cosh(d*x + c)^2
+ b^2)*sinh(d*x + c)^4 + 4*(5*b^2*cosh(d*x + c)^3 + 3*b^2*cosh(d*x + c))*sinh(d*x + c)^3 - (a^2 - 4*a*b)*cosh
(d*x + c)^2 + (15*b^2*cosh(d*x + c)^4 + 18*b^2*cosh(d*x + c)^2 - a^2 + 4*a*b)*sinh(d*x + c)^2 + a^2 + 2*(3*b^2
*cosh(d*x + c)^5 + 6*b^2*cosh(d*x + c)^3 - (a^2 - 4*a*b)*cosh(d*x + c))*sinh(d*x + c))*sqrt(a)*sqrt((a*cosh(d*
x + c)^2 + a*sinh(d*x + c)^2 - a + 2*b)/(cosh(d*x + c)^2 - 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2)) +
4*(2*a*b^2*cosh(d*x + c)^7 + 3*(a*b^2 + b^3)*cosh(d*x + c)^5 + (a^3 - 4*a^2*b + 9*a*b^2)*cosh(d*x + c)^3 - (a
^3 - 3*a^2*b)*cosh(d*x + c))*sinh(d*x + c))/(cosh(d*x + c)^6 + 6*cosh(d*x + c)^5*sinh(d*x + c) + 15*cosh(d*x +
c)^4*sinh(d*x + c)^2 + 20*cosh(d*x + c)^3*sinh(d*x + c)^3 + 15*cosh(d*x + c)^2*sinh(d*x + c)^4 + 6*cosh(d*x +
c)*sinh(d*x + c)^5 + sinh(d*x + c)^6)) + 2*sqrt(b)*log(((a + b)*cosh(d*x + c)^4 + 4*(a + b)*cosh(d*x + c)*sin
h(d*x + c)^3 + (a + b)*sinh(d*x + c)^4 - 2*(a - 3*b)*cosh(d*x + c)^2 + 2*(3*(a + b)*cosh(d*x + c)^2 - a + 3*b)
*sinh(d*x + c)^2 - 2*sqrt(2)*(cosh(d*x + c)^2 + 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2 + 1)*sqrt(b)*s
qrt((a*cosh(d*x + c)^2 + a*sinh(d*x + c)^2 - a + 2*b)/(cosh(d*x + c)^2 - 2*cosh(d*x + c)*sinh(d*x + c) + sinh(
d*x + c)^2)) + 4*((a + b)*cosh(d*x + c)^3 - (a - 3*b)*cosh(d*x + c))*sinh(d*x + c) + a + b)/(cosh(d*x + c)^4 +
4*cosh(d*x + c)*sinh(d*x + c)^3 + sinh(d*x + c)^4 + 2*(3*cosh(d*x + c)^2 - 1)*sinh(d*x + c)^2 - 2*cosh(d*x +
c)^2 + 4*(cosh(d*x + c)^3 - cosh(d*x + c))*sinh(d*x + c) + 1)) + sqrt(a)*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*(a - b)*cosh(d*x + c)^2 + 2*(3*a*cosh(d*x + c)^2 - a + b)*sinh(
d*x + c)^2 + sqrt(2)*(cosh(d*x + c)^2 + 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2 - 1)*sqrt(a)*sqrt((a*c
osh(d*x + c)^2 + a*sinh(d*x + c)^2 - a + 2*b)/(cosh(d*x + c)^2 - 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)
^2)) + 4*(a*cosh(d*x + c)^3 - (a - b)*cosh(d*x + c))*sinh(d*x + c) + a)/(cosh(d*x + c)^2 + 2*cosh(d*x + c)*sin
h(d*x + c) + sinh(d*x + c)^2)))/d, 1/4*(4*sqrt(-b)*arctan(sqrt(2)*(cosh(d*x + c)^2 + 2*cosh(d*x + c)*sinh(d*x
+ c) + sinh(d*x + c)^2 + 1)*sqrt(-b)*sqrt((a*cosh(d*x + c)^2 + a*sinh(d*x + c)^2 - a + 2*b)/(cosh(d*x + c)^2 -
2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2))/(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)) + sqrt(a)*log((a*b^2*cosh(d*x + c)^8 + 8*a*b^2*cosh(
d*x + c)*sinh(d*x + c)^7 + a*b^2*sinh(d*x + c)^8 + 2*(a*b^2 + b^3)*cosh(d*x + c)^6 + 2*(14*a*b^2*cosh(d*x + c)
^2 + a*b^2 + b^3)*sinh(d*x + c)^6 + 4*(14*a*b^2*cosh(d*x + c)^3 + 3*(a*b^2 + b^3)*cosh(d*x + c))*sinh(d*x + c)
^5 + (a^3 - 4*a^2*b + 9*a*b^2)*cosh(d*x + c)^4 + (70*a*b^2*cosh(d*x + c)^4 + a^3 - 4*a^2*b + 9*a*b^2 + 30*(a*b
^2 + b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 4*(14*a*b^2*cosh(d*x + c)^5 + 10*(a*b^2 + b^3)*cosh(d*x + c)^3 +
(a^3 - 4*a^2*b + 9*a*b^2)*cosh(d*x + c))*sinh(d*x + c)^3 + a^3 - 2*(a^3 - 3*a^2*b)*cosh(d*x + c)^2 + 2*(14*a*b
^2*cosh(d*x + c)^6 + 15*(a*b^2 + b^3)*cosh(d*x + c)^4 - a^3 + 3*a^2*b + 3*(a^3 - 4*a^2*b + 9*a*b^2)*cosh(d*x +
c)^2)*sinh(d*x + c)^2 + sqrt(2)*(b^2*cosh(d*x + c)^6 + 6*b^2*cosh(d*x + c)*sinh(d*x + c)^5 + b^2*sinh(d*x + c
)^6 + 3*b^2*cosh(d*x + c)^4 + 3*(5*b^2*cosh(d*x + c)^2 + b^2)*sinh(d*x + c)^4 + 4*(5*b^2*cosh(d*x + c)^3 + 3*b
^2*cosh(d*x + c))*sinh(d*x + c)^3 - (a^2 - 4*a*b)*cosh(d*x + c)^2 + (15*b^2*cosh(d*x + c)^4 + 18*b^2*cosh(d*x
+ c)^2 - a^2 + 4*a*b)*sinh(d*x + c)^2 + a^2 + 2*(3*b^2*cosh(d*x + c)^5 + 6*b^2*cosh(d*x + c)^3 - (a^2 - 4*a*b)
*cosh(d*x + c))*sinh(d*x + c))*sqrt(a)*sqrt((a*cosh(d*x + c)^2 + a*sinh(d*x + c)^2 - a + 2*b)/(cosh(d*x + c)^2
- 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2)) + 4*(2*a*b^2*cosh(d*x + c)^7 + 3*(a*b^2 + b^3)*cosh(d*x +
c)^5 + (a^3 - 4*a^2*b + 9*a*b^2)*cosh(d*x + c)^3 - (a^3 - 3*a^2*b)*cosh(d*x + c))*sinh(d*x + c))/(cosh(d*x +
c)^6 + 6*cosh(d*x + c)^5*sinh(d*x + c) + 15*cosh(d*x + c)^4*sinh(d*x + c)^2 + 20*cosh(d*x + c)^3*sinh(d*x + c)
^3 + 15*cosh(d*x + c)^2*sinh(d*x + c)^4 + 6*cosh(d*x + c)*sinh(d*x + c)^5 + sinh(d*x + c)^6)) + sqrt(a)*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*(a - b)*cosh(d*x + c)^2 + 2*(3*a
*cosh(d*x + c)^2 - a + b)*sinh(d*x + c)^2 + sqrt(2)*(cosh(d*x + c)^2 + 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*
x + c)^2 - 1)*sqrt(a)*sqrt((a*cosh(d*x + c)^2 + a*sinh(d*x + c)^2 - a + 2*b)/(cosh(d*x + c)^2 - 2*cosh(d*x + c
)*sinh(d*x + c) + sinh(d*x + c)^2)) + 4*(a*cosh(d*x + c)^3 - (a - b)*cosh(d*x + c))*sinh(d*x + c) + a)/(cosh(d
*x + c)^2 + 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2)))/d, -1/2*(sqrt(-a)*arctan(sqrt(2)*(b*cosh(d*x +
c)^2 + 2*b*cosh(d*x + c)*sinh(d*x + c) + b*sinh(d*x + c)^2 + a)*sqrt(-a)*sqrt((a*cosh(d*x + c)^2 + a*sinh(d*x
+ c)^2 - a + 2*b)/(cosh(d*x + c)^2 - 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2))/(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 - (a^2 - 3*a*b)*cosh(d*x + c)^2 + (6*a*b*cosh(d*x +
c)^2 - a^2 + 3*a*b)*sinh(d*x + c)^2 + a^2 + 2*(2*a*b*cosh(d*x + c)^3 - (a^2 - 3*a*b)*cosh(d*x + c))*sinh(d*x +
c))) + sqrt(-a)*arctan(sqrt(2)*(cosh(d*x + c)^2 + 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2 - 1)*sqrt(-
a)*sqrt((a*cosh(d*x + c)^2 + a*sinh(d*x + c)^2 - a + 2*b)/(cosh(d*x + c)^2 - 2*cosh(d*x + c)*sinh(d*x + c) + s
inh(d*x + c)^2))/(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)) - sqrt(b)*log(((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*(a - 3*b)*cosh(d*x + c)^2 + 2*(3*(a + b)*cosh(d*x + c)^2 - a + 3*b)*sinh(d*x + c)^2 -
2*sqrt(2)*(cosh(d*x + c)^2 + 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2 + 1)*sqrt(b)*sqrt((a*cosh(d*x +
c)^2 + a*sinh(d*x + c)^2 - a + 2*b)/(cosh(d*x + c)^2 - 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2)) + 4*(
(a + b)*cosh(d*x + c)^3 - (a - 3*b)*cosh(d*x + c))*sinh(d*x + c) + a + b)/(cosh(d*x + c)^4 + 4*cosh(d*x + c)*s
inh(d*x + c)^3 + sinh(d*x + c)^4 + 2*(3*cosh(d*x + c)^2 - 1)*sinh(d*x + c)^2 - 2*cosh(d*x + c)^2 + 4*(cosh(d*x
+ c)^3 - cosh(d*x + c))*sinh(d*x + c) + 1)))/d, -1/2*(sqrt(-a)*arctan(sqrt(2)*(b*cosh(d*x + c)^2 + 2*b*cosh(d
*x + c)*sinh(d*x + c) + b*sinh(d*x + c)^2 + a)*sqrt(-a)*sqrt((a*cosh(d*x + c)^2 + a*sinh(d*x + c)^2 - a + 2*b)
/(cosh(d*x + c)^2 - 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2))/(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 - (a^2 - 3*a*b)*cosh(d*x + c)^2 + (6*a*b*cosh(d*x + c)^2 - a^2 + 3*a*
b)*sinh(d*x + c)^2 + a^2 + 2*(2*a*b*cosh(d*x + c)^3 - (a^2 - 3*a*b)*cosh(d*x + c))*sinh(d*x + c))) + sqrt(-a)*
arctan(sqrt(2)*(cosh(d*x + c)^2 + 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2 - 1)*sqrt(-a)*sqrt((a*cosh(d
*x + c)^2 + a*sinh(d*x + c)^2 - a + 2*b)/(cosh(d*x + c)^2 - 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2))/
(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)) - 2*sqrt(-b)*arctan(sqrt(2)*(cosh(d*x + c)^2 + 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2 + 1)*sqr
t(-b)*sqrt((a*cosh(d*x + c)^2 + a*sinh(d*x + c)^2 - a + 2*b)/(cosh(d*x + c)^2 - 2*cosh(d*x + c)*sinh(d*x + c)
+ sinh(d*x + c)^2))/(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)*c
osh(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)))/d]

Sympy [F]

\[ \int \sqrt {a+b \text {csch}^2(c+d x)} \, dx=\int \sqrt {a + b \operatorname {csch}^{2}{\left (c + d x \right )}}\, dx \]

[In]

integrate((a+b*csch(d*x+c)**2)**(1/2),x)

[Out]

Integral(sqrt(a + b*csch(c + d*x)**2), x)

Maxima [F]

\[ \int \sqrt {a+b \text {csch}^2(c+d x)} \, dx=\int { \sqrt {b \operatorname {csch}\left (d x + c\right )^{2} + a} \,d x } \]

[In]

integrate((a+b*csch(d*x+c)^2)^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(b*csch(d*x + c)^2 + a), x)

Giac [F(-2)]

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

[In]

integrate((a+b*csch(d*x+c)^2)^(1/2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Error: Bad Argument Type

Mupad [F(-1)]

Timed out. \[ \int \sqrt {a+b \text {csch}^2(c+d x)} \, dx=\int \sqrt {a+\frac {b}{{\mathrm {sinh}\left (c+d\,x\right )}^2}} \,d x \]

[In]

int((a + b/sinh(c + d*x)^2)^(1/2),x)

[Out]

int((a + b/sinh(c + d*x)^2)^(1/2), x)