\(\int \frac {\text {sech}^3(c+d x)}{a+b \sinh ^2(c+d x)} \, dx\) [289]

Optimal result

Integrand size = 23, antiderivative size = 92 \[ \int \frac {\text {sech}^3(c+d x)}{a+b \sinh ^2(c+d x)} \, dx=\frac {(a-3 b) \arctan (\sinh (c+d x))}{2 (a-b)^2 d}+\frac {b^{3/2} \arctan \left (\frac {\sqrt {b} \sinh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} (a-b)^2 d}+\frac {\text {sech}(c+d x) \tanh (c+d x)}{2 (a-b) d} \] Output:

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

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.99 \[ \int \frac {\text {sech}^3(c+d x)}{a+b \sinh ^2(c+d x)} \, dx=\frac {-2 b^{3/2} \arctan \left (\frac {\sqrt {a} \text {csch}(c+d x)}{\sqrt {b}}\right )+2 \sqrt {a} (a-3 b) \arctan \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )+\sqrt {a} (a-b) \text {sech}(c+d x) \tanh (c+d x)}{2 \sqrt {a} (a-b)^2 d} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.12, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {3042, 3669, 316, 25, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3669

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

\(\Big \downarrow \) 316

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 397

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

\(\Big \downarrow \) 216

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

\(\Big \downarrow \) 218

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

Input:

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

Output:

((((a - 3*b)*ArcTan[Sinh[c + d*x]])/(a - b) + (2*b^(3/2)*ArcTan[(Sqrt[b]*S 
inh[c + d*x])/Sqrt[a]])/(Sqrt[a]*(a - b)))/(2*(a - b)) + Sinh[c + d*x]/(2* 
(a - b)*(1 + Sinh[c + d*x]^2)))/d
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 [C] (verified)

Result contains complex when optimal does not.

Time = 135.32 (sec) , antiderivative size = 241, normalized size of antiderivative = 2.62

method	result	size
risch	\(\frac {{\mathrm e}^{d x +c} \left ({\mathrm e}^{2 d x +2 c}-1\right )}{d \left (a -b \right ) \left ({\mathrm e}^{2 d x +2 c}+1\right )^{2}}+\frac {i \ln \left ({\mathrm e}^{d x +c}+i\right ) a}{2 \left (a -b \right )^{2} d}-\frac {3 i \ln \left ({\mathrm e}^{d x +c}+i\right ) b}{2 \left (a -b \right )^{2} d}-\frac {i \ln \left ({\mathrm e}^{d x +c}-i\right ) a}{2 \left (a -b \right )^{2} d}+\frac {3 i \ln \left ({\mathrm e}^{d x +c}-i\right ) b}{2 \left (a -b \right )^{2} d}+\frac {\sqrt {-a b}\, b \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {-a b}\, {\mathrm e}^{d x +c}}{b}-1\right )}{2 a \left (a -b \right )^{2} d}-\frac {\sqrt {-a b}\, b \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {-a b}\, {\mathrm e}^{d x +c}}{b}-1\right )}{2 a \left (a -b \right )^{2} d}\)	\(241\)
derivativedivides	\(\frac {\frac {\frac {2 \left (\left (-\frac {a}{2}+\frac {b}{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (\frac {a}{2}-\frac {b}{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right )^{2}}+\left (a -3 b \right ) \arctan \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (a -b \right )^{2}}+\frac {2 b^{2} a \left (-\frac {\left (a +\sqrt {-b \left (a -b \right )}-b \right ) \operatorname {arctanh}\left (\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) a}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}+\frac {\left (-a +\sqrt {-b \left (a -b \right )}+b \right ) \arctan \left (\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) a}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}\right )}{\left (a -b \right )^{2}}}{d}\)	\(272\)
default	\(\frac {\frac {\frac {2 \left (\left (-\frac {a}{2}+\frac {b}{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (\frac {a}{2}-\frac {b}{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right )^{2}}+\left (a -3 b \right ) \arctan \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (a -b \right )^{2}}+\frac {2 b^{2} a \left (-\frac {\left (a +\sqrt {-b \left (a -b \right )}-b \right ) \operatorname {arctanh}\left (\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) a}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}+\frac {\left (-a +\sqrt {-b \left (a -b \right )}+b \right ) \arctan \left (\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) a}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}\right )}{\left (a -b \right )^{2}}}{d}\)	\(272\)

Input:

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

Output:

exp(d*x+c)*(exp(2*d*x+2*c)-1)/d/(a-b)/(exp(2*d*x+2*c)+1)^2+1/2*I/(a-b)^2/d 
*ln(exp(d*x+c)+I)*a-3/2*I/(a-b)^2/d*ln(exp(d*x+c)+I)*b-1/2*I/(a-b)^2/d*ln( 
exp(d*x+c)-I)*a+3/2*I/(a-b)^2/d*ln(exp(d*x+c)-I)*b+1/2/a*(-a*b)^(1/2)*b/(a 
-b)^2/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 
)*b/(a-b)^2/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. 772 vs. \(2 (80) = 160\).

Time = 0.15 (sec) , antiderivative size = 1644, normalized size of antiderivative = 17.87 \[ \int \frac {\text {sech}^3(c+d x)}{a+b \sinh ^2(c+d x)} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/2*(2*(a - b)*cosh(d*x + c)^3 + 6*(a - b)*cosh(d*x + c)*sinh(d*x + c)^2 
+ 2*(a - b)*sinh(d*x + c)^3 + (b*cosh(d*x + c)^4 + 4*b*cosh(d*x + c)*sinh( 
d*x + c)^3 + b*sinh(d*x + c)^4 + 2*b*cosh(d*x + c)^2 + 2*(3*b*cosh(d*x + c 
)^2 + b)*sinh(d*x + c)^2 + 4*(b*cosh(d*x + c)^3 + b*cosh(d*x + c))*sinh(d* 
x + c) + b)*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)*cos 
h(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)*sinh(d*x + c))*sqrt(-b/a) + b)/(b*cosh(d*x + c)^4 + 4*b*cosh(d*x + c)*s 
inh(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) + b)) + 2*((a - 3*b)*cosh(d*x + c)^4 + 4* 
(a - 3*b)*cosh(d*x + c)*sinh(d*x + c)^3 + (a - 3*b)*sinh(d*x + c)^4 + 2*(a 
 - 3*b)*cosh(d*x + c)^2 + 2*(3*(a - 3*b)*cosh(d*x + c)^2 + a - 3*b)*sinh(d 
*x + c)^2 + 4*((a - 3*b)*cosh(d*x + c)^3 + (a - 3*b)*cosh(d*x + c))*sinh(d 
*x + c) + a - 3*b)*arctan(cosh(d*x + c) + sinh(d*x + c)) - 2*(a - b)*cosh( 
d*x + c) + 2*(3*(a - b)*cosh(d*x + c)^2 - a + b)*sinh(d*x + c))/((a^2 - 2* 
a*b + b^2)*d*cosh(d*x + c)^4 + 4*(a^2 - 2*a*b + b^2)*d*cosh(d*x + c)*sinh( 
d*x + c)^3 + (a^2 - 2*a*b + b^2)*d*sinh(d*x + c)^4 + 2*(a^2 - 2*a*b + b...
 

Sympy [F]

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

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

Output:

Integral(sech(c + d*x)**3/(a + b*sinh(c + d*x)**2), x)
 

Maxima [F]

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

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

Output:

(a*e^c - 3*b*e^c)*arctan(e^(d*x + c))*e^(-c)/(a^2*d - 2*a*b*d + b^2*d) + ( 
e^(3*d*x + 3*c) - e^(d*x + c))/(a*d - b*d + (a*d*e^(4*c) - b*d*e^(4*c))*e^ 
(4*d*x) + 2*(a*d*e^(2*c) - b*d*e^(2*c))*e^(2*d*x)) + 8*integrate(1/4*(b^2* 
e^(3*d*x + 3*c) + b^2*e^(d*x + c))/(a^2*b - 2*a*b^2 + b^3 + (a^2*b*e^(4*c) 
 - 2*a*b^2*e^(4*c) + b^3*e^(4*c))*e^(4*d*x) + 2*(2*a^3*e^(2*c) - 5*a^2*b*e 
^(2*c) + 4*a*b^2*e^(2*c) - b^3*e^(2*c))*e^(2*d*x)), x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {\text {sech}^3(c+d x)}{a+b \sinh ^2(c+d x)} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(sech(d*x+c)^3/(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 = 6.90 (sec) , antiderivative size = 2797, normalized size of antiderivative = 30.40 \[ \int \frac {\text {sech}^3(c+d x)}{a+b \sinh ^2(c+d x)} \, dx=\text {Too large to display} \] Input:

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

Output:

exp(c + d*x)/((exp(2*c + 2*d*x) + 1)*(a*d - b*d)) - (2*exp(c + d*x))/((a*d 
 - b*d)*(2*exp(2*c + 2*d*x) + exp(4*c + 4*d*x) + 1)) + ((2*atan((b^2*exp(d 
*x)*exp(c)*(a*d^2*(a - b)^4)^(1/2))/(2*a*d*(a - b)^2*(b^3)^(1/2))) - 2*ata 
n((exp(d*x)*exp(c)*((64*(20*a^3*d*(b^3)^(5/2) - 232*a^6*d*(b^3)^(3/2) + 2* 
a^9*d*(b^3)^(1/2) + 2*a*b^5*d*(b^3)^(3/2) + 10*a^5*b*d*(b^3)^(3/2) - 22*a^ 
8*b*d*(b^3)^(1/2) - 10*a^2*b^4*d*(b^3)^(3/2) - 20*a^4*b^2*d*(b^3)^(3/2) - 
18*a^2*b^7*d*(b^3)^(1/2) + 102*a^3*b^6*d*(b^3)^(1/2) - 242*a^4*b^5*d*(b^3) 
^(1/2) + 310*a^5*b^4*d*(b^3)^(1/2) + 98*a^7*b^2*d*(b^3)^(1/2)))/(a*b^4*(a 
- b)^5*(a*b - a^2)*(a^2 - 2*a*b + b^2)*(a*d^2*(a - b)^4)^(1/2)*(3*a*b^2 - 
3*a^2*b + a^3 - b^3)*(9*a*b^2 - 6*a^2*b + a^3 - b^3)*(a^5*d^2 + a*b^4*d^2 
- 4*a^4*b*d^2 - 4*a^2*b^3*d^2 + 6*a^3*b^2*d^2)^(1/2)) - (32*(b^8*(a^5*d^2 
+ a*b^4*d^2 - 4*a^4*b*d^2 - 4*a^2*b^3*d^2 + 6*a^3*b^2*d^2)^(1/2) + 36*a^2* 
b^6*(a^5*d^2 + a*b^4*d^2 - 4*a^4*b*d^2 - 4*a^2*b^3*d^2 + 6*a^3*b^2*d^2)^(1 
/2) - 47*a^3*b^5*(a^5*d^2 + a*b^4*d^2 - 4*a^4*b*d^2 - 4*a^2*b^3*d^2 + 6*a^ 
3*b^2*d^2)^(1/2) + 30*a^4*b^4*(a^5*d^2 + a*b^4*d^2 - 4*a^4*b*d^2 - 4*a^2*b 
^3*d^2 + 6*a^3*b^2*d^2)^(1/2) - 9*a^5*b^3*(a^5*d^2 + a*b^4*d^2 - 4*a^4*b*d 
^2 - 4*a^2*b^3*d^2 + 6*a^3*b^2*d^2)^(1/2) + a^6*b^2*(a^5*d^2 + a*b^4*d^2 - 
 4*a^4*b*d^2 - 4*a^2*b^3*d^2 + 6*a^3*b^2*d^2)^(1/2) - 12*a*b^7*(a^5*d^2 + 
a*b^4*d^2 - 4*a^4*b*d^2 - 4*a^2*b^3*d^2 + 6*a^3*b^2*d^2)^(1/2)))/(a^2*b^2* 
d*(a - b)^7*(a*b - a^2)*(b^3)^(1/2)*(a^2 - 2*a*b + b^2)*(3*a*b^2 - 3*a^...
 

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 1396, normalized size of antiderivative = 15.17 \[ \int \frac {\text {sech}^3(c+d x)}{a+b \sinh ^2(c+d x)} \, dx =\text {Too large to display} \] Input:

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

Output:

(2*e**(4*c + 4*d*x)*atan(e**(c + d*x))*a**2 - 6*e**(4*c + 4*d*x)*atan(e**( 
c + d*x))*a*b + 4*e**(2*c + 2*d*x)*atan(e**(c + d*x))*a**2 - 12*e**(2*c + 
2*d*x)*atan(e**(c + d*x))*a*b + 2*atan(e**(c + d*x))*a**2 - 6*atan(e**(c + 
 d*x))*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))) - 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*sq 
rt(a)*sqrt(a - b) + 2*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))) + 2*e**(4*c + 4*d*x)*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 + 4*e**(2*c + 2*d*x)*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 + 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 - e**(4*c + 4*d*x 
)*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)) + e**(4*c + 
 4*d*x)*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)) - 2*e**( 
2*c + 2*d*x)*sqrt(b)*sqrt(a)*sqrt(a - b)*sqrt(2*sqrt(a)*sqrt(a - b) - 2...