\(\int \frac {\text {csch}^3(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx\) [498]

Optimal result

Integrand size = 29, antiderivative size = 202 \[ \int \frac {\text {csch}^3(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {3 \text {arctanh}(\cosh (c+d x))}{2 a d}-\frac {b^2 \text {arctanh}(\cosh (c+d x))}{a^3 d}+\frac {2 b^5 \text {arctanh}\left (\frac {b-a \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right )^{3/2} d}+\frac {b \coth (c+d x)}{a^2 d}-\frac {\coth (c+d x) \text {csch}(c+d x)}{2 a d}-\frac {\text {sech}(c+d x)}{a d}+\frac {b^2 \text {sech}(c+d x)}{a^3 d}-\frac {b^3 \text {sech}(c+d x) (b+a \sinh (c+d x))}{a^3 \left (a^2+b^2\right ) d}+\frac {b \tanh (c+d x)}{a^2 d} \] Output:

3/2*arctanh(cosh(d*x+c))/a/d-b^2*arctanh(cosh(d*x+c))/a^3/d+2*b^5*arctanh( 
(b-a*tanh(1/2*d*x+1/2*c))/(a^2+b^2)^(1/2))/a^3/(a^2+b^2)^(3/2)/d+b*coth(d* 
x+c)/a^2/d-1/2*coth(d*x+c)*csch(d*x+c)/a/d-sech(d*x+c)/a/d+b^2*sech(d*x+c) 
/a^3/d-b^3*sech(d*x+c)*(b+a*sinh(d*x+c))/a^3/(a^2+b^2)/d+b*tanh(d*x+c)/a^2 
/d
 

Mathematica [A] (verified)

Time = 3.20 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.05 \[ \int \frac {\text {csch}^3(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {\frac {16 b^5 \arctan \left (\frac {b-a \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a^2-b^2}}\right )}{a^3 \left (-a^2-b^2\right )^{3/2}}+\frac {4 b \coth \left (\frac {1}{2} (c+d x)\right )}{a^2}-\frac {\text {csch}^2\left (\frac {1}{2} (c+d x)\right )}{a}+\frac {4 \left (3 a^2-2 b^2\right ) \log \left (\cosh \left (\frac {1}{2} (c+d x)\right )\right )}{a^3}+\frac {4 \left (-3 a^2+2 b^2\right ) \log \left (\sinh \left (\frac {1}{2} (c+d x)\right )\right )}{a^3}-\frac {\text {sech}^2\left (\frac {1}{2} (c+d x)\right )}{a}+\frac {8 \text {sech}(c+d x) (-a+b \sinh (c+d x))}{a^2+b^2}+\frac {4 b \tanh \left (\frac {1}{2} (c+d x)\right )}{a^2}}{8 d} \] Input:

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

Output:

((16*b^5*ArcTan[(b - a*Tanh[(c + d*x)/2])/Sqrt[-a^2 - b^2]])/(a^3*(-a^2 - 
b^2)^(3/2)) + (4*b*Coth[(c + d*x)/2])/a^2 - Csch[(c + d*x)/2]^2/a + (4*(3* 
a^2 - 2*b^2)*Log[Cosh[(c + d*x)/2]])/a^3 + (4*(-3*a^2 + 2*b^2)*Log[Sinh[(c 
 + d*x)/2]])/a^3 - Sech[(c + d*x)/2]^2/a + (8*Sech[c + d*x]*(-a + b*Sinh[c 
 + d*x]))/(a^2 + b^2) + (4*b*Tanh[(c + d*x)/2])/a^2)/(8*d)
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.66 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.14, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {3042, 26, 3377, 2009}

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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3377

\(\displaystyle -i \int \left (-\frac {i \text {sech}^2(c+d x) b^3}{a^3 (a+b \sinh (c+d x))}+\frac {i \text {csch}(c+d x) \text {sech}^2(c+d x) b^2}{a^3}-\frac {i \text {csch}^2(c+d x) \text {sech}^2(c+d x) b}{a^2}+\frac {i \text {csch}^3(c+d x) \text {sech}^2(c+d x)}{a}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -i \left (-\frac {i b^2 \text {arctanh}(\cosh (c+d x))}{a^3 d}+\frac {i b^2 \text {sech}(c+d x)}{a^3 d}+\frac {i b \tanh (c+d x)}{a^2 d}+\frac {i b \coth (c+d x)}{a^2 d}+\frac {2 i b^5 \text {arctanh}\left (\frac {b-a \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2+b^2}}\right )}{a^3 d \left (a^2+b^2\right )^{3/2}}-\frac {i b^3 \text {sech}(c+d x) (a \sinh (c+d x)+b)}{a^3 d \left (a^2+b^2\right )}+\frac {3 i \text {arctanh}(\cosh (c+d x))}{2 a d}-\frac {3 i \text {sech}(c+d x)}{2 a d}-\frac {i \text {csch}^2(c+d x) \text {sech}(c+d x)}{2 a d}\right )\)

Input:

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

Output:

(-I)*((((3*I)/2)*ArcTanh[Cosh[c + d*x]])/(a*d) - (I*b^2*ArcTanh[Cosh[c + d 
*x]])/(a^3*d) + ((2*I)*b^5*ArcTanh[(b - a*Tanh[(c + d*x)/2])/Sqrt[a^2 + b^ 
2]])/(a^3*(a^2 + b^2)^(3/2)*d) + (I*b*Coth[c + d*x])/(a^2*d) - (((3*I)/2)* 
Sech[c + d*x])/(a*d) + (I*b^2*Sech[c + d*x])/(a^3*d) - ((I/2)*Csch[c + d*x 
]^2*Sech[c + d*x])/(a*d) - (I*b^3*Sech[c + d*x]*(b + a*Sinh[c + d*x]))/(a^ 
3*(a^2 + b^2)*d) + (I*b*Tanh[c + d*x])/(a^2*d))
 

Defintions of rubi rules used

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

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

rule 3377

Int[((cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*sin[(e_.) + (f_.)*(x_)]^(n_))/((a 
_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Int[ExpandTrig[(g*cos[e + 
 f*x])^p, sin[e + f*x]^n/(a + b*sin[e + f*x]), x], x] /; FreeQ[{a, b, e, f, 
 g, p}, x] && NeQ[a^2 - b^2, 0] && IntegerQ[n] && (LtQ[n, 0] || IGtQ[p + 1/ 
2, 0])
 

Maple [A] (verified)

Time = 17.40 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.91

method	result	size
derivativedivides	\(\frac {\frac {\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a}{2}+2 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a^{2}}-\frac {1}{8 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\left (-6 a^{2}+4 b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{3}}+\frac {b}{2 a^{2} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {2 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2 a}{\left (a^{2}+b^{2}\right ) \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}-\frac {2 b^{5} \operatorname {arctanh}\left (\frac {2 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{a^{3} \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}}{d}\)	\(183\)
default	\(\frac {\frac {\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a}{2}+2 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a^{2}}-\frac {1}{8 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\left (-6 a^{2}+4 b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{3}}+\frac {b}{2 a^{2} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {2 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2 a}{\left (a^{2}+b^{2}\right ) \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}-\frac {2 b^{5} \operatorname {arctanh}\left (\frac {2 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{a^{3} \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}}{d}\)	\(183\)
risch	\(-\frac {3 \,{\mathrm e}^{d x +c} a^{3}+{\mathrm e}^{d x +c} a \,b^{2}+3 \,{\mathrm e}^{5 d x +5 c} a^{3}+{\mathrm e}^{5 d x +5 c} a \,b^{2}-2 \,{\mathrm e}^{4 d x +4 c} b^{3}-2 \,{\mathrm e}^{3 d x +3 c} a^{3}+2 \,{\mathrm e}^{3 d x +3 c} a \,b^{2}-4 b \,{\mathrm e}^{2 d x +2 c} a^{2}+4 a^{2} b +2 b^{3}}{d \left (a^{2}+b^{2}\right ) \left (1+{\mathrm e}^{2 d x +2 c}\right ) a^{2} \left ({\mathrm e}^{2 d x +2 c}-1\right )^{2}}+\frac {b^{5} \ln \left ({\mathrm e}^{d x +c}+\frac {a \left (a^{2}+b^{2}\right )^{\frac {3}{2}}+a^{4}+2 a^{2} b^{2}+b^{4}}{b \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}} d \,a^{3}}-\frac {b^{5} \ln \left ({\mathrm e}^{d x +c}+\frac {a \left (a^{2}+b^{2}\right )^{\frac {3}{2}}-a^{4}-2 a^{2} b^{2}-b^{4}}{b \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}} d \,a^{3}}+\frac {3 \ln \left ({\mathrm e}^{d x +c}+1\right )}{2 d a}-\frac {\ln \left ({\mathrm e}^{d x +c}+1\right ) b^{2}}{a^{3} d}-\frac {3 \ln \left ({\mathrm e}^{d x +c}-1\right )}{2 d a}+\frac {\ln \left ({\mathrm e}^{d x +c}-1\right ) b^{2}}{a^{3} d}\)	\(375\)

Input:

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

Output:

1/d*(1/4/a^2*(1/2*tanh(1/2*d*x+1/2*c)^2*a+2*b*tanh(1/2*d*x+1/2*c))-1/8/a/t 
anh(1/2*d*x+1/2*c)^2+1/4/a^3*(-6*a^2+4*b^2)*ln(tanh(1/2*d*x+1/2*c))+1/2*b/ 
a^2/tanh(1/2*d*x+1/2*c)+2/(a^2+b^2)*(b*tanh(1/2*d*x+1/2*c)-a)/(1+tanh(1/2* 
d*x+1/2*c)^2)-2/a^3*b^5/(a^2+b^2)^(3/2)*arctanh(1/2*(2*a*tanh(1/2*d*x+1/2* 
c)-2*b)/(a^2+b^2)^(1/2)))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2653 vs. \(2 (195) = 390\).

Time = 0.33 (sec) , antiderivative size = 2653, normalized size of antiderivative = 13.13 \[ \int \frac {\text {csch}^3(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \] Input:

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

Output:

-1/2*(8*a^5*b + 12*a^3*b^3 + 4*a*b^5 + 2*(3*a^6 + 4*a^4*b^2 + a^2*b^4)*cos 
h(d*x + c)^5 + 2*(3*a^6 + 4*a^4*b^2 + a^2*b^4)*sinh(d*x + c)^5 - 4*(a^3*b^ 
3 + a*b^5)*cosh(d*x + c)^4 - 2*(2*a^3*b^3 + 2*a*b^5 - 5*(3*a^6 + 4*a^4*b^2 
 + a^2*b^4)*cosh(d*x + c))*sinh(d*x + c)^4 - 4*(a^6 - a^2*b^4)*cosh(d*x + 
c)^3 - 4*(a^6 - a^2*b^4 - 5*(3*a^6 + 4*a^4*b^2 + a^2*b^4)*cosh(d*x + c)^2 
+ 4*(a^3*b^3 + a*b^5)*cosh(d*x + c))*sinh(d*x + c)^3 - 8*(a^5*b + a^3*b^3) 
*cosh(d*x + c)^2 - 4*(2*a^5*b + 2*a^3*b^3 - 5*(3*a^6 + 4*a^4*b^2 + a^2*b^4 
)*cosh(d*x + c)^3 + 6*(a^3*b^3 + a*b^5)*cosh(d*x + c)^2 + 3*(a^6 - a^2*b^4 
)*cosh(d*x + c))*sinh(d*x + c)^2 - 2*(b^5*cosh(d*x + c)^6 + 6*b^5*cosh(d*x 
 + c)*sinh(d*x + c)^5 + b^5*sinh(d*x + c)^6 - b^5*cosh(d*x + c)^4 - b^5*co 
sh(d*x + c)^2 + b^5 + (15*b^5*cosh(d*x + c)^2 - b^5)*sinh(d*x + c)^4 + 4*( 
5*b^5*cosh(d*x + c)^3 - b^5*cosh(d*x + c))*sinh(d*x + c)^3 + (15*b^5*cosh( 
d*x + c)^4 - 6*b^5*cosh(d*x + c)^2 - b^5)*sinh(d*x + c)^2 + 2*(3*b^5*cosh( 
d*x + c)^5 - 2*b^5*cosh(d*x + c)^3 - b^5*cosh(d*x + c))*sinh(d*x + c))*sqr 
t(a^2 + b^2)*log((b^2*cosh(d*x + c)^2 + b^2*sinh(d*x + c)^2 + 2*a*b*cosh(d 
*x + c) + 2*a^2 + b^2 + 2*(b^2*cosh(d*x + c) + a*b)*sinh(d*x + c) + 2*sqrt 
(a^2 + b^2)*(b*cosh(d*x + c) + b*sinh(d*x + c) + a))/(b*cosh(d*x + c)^2 + 
b*sinh(d*x + c)^2 + 2*a*cosh(d*x + c) + 2*(b*cosh(d*x + c) + a)*sinh(d*x + 
 c) - b)) + 2*(3*a^6 + 4*a^4*b^2 + a^2*b^4)*cosh(d*x + c) - ((3*a^6 + 4*a^ 
4*b^2 - a^2*b^4 - 2*b^6)*cosh(d*x + c)^6 + 6*(3*a^6 + 4*a^4*b^2 - a^2*b...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\text {csch}^3(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 334, normalized size of antiderivative = 1.65 \[ \int \frac {\text {csch}^3(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {b^{5} \log \left (\frac {b e^{\left (-d x - c\right )} - a - \sqrt {a^{2} + b^{2}}}{b e^{\left (-d x - c\right )} - a + \sqrt {a^{2} + b^{2}}}\right )}{{\left (a^{5} + a^{3} b^{2}\right )} \sqrt {a^{2} + b^{2}} d} - \frac {4 \, a^{2} b e^{\left (-2 \, d x - 2 \, c\right )} + 2 \, b^{3} e^{\left (-4 \, d x - 4 \, c\right )} - 4 \, a^{2} b - 2 \, b^{3} + {\left (3 \, a^{3} + a b^{2}\right )} e^{\left (-d x - c\right )} - 2 \, {\left (a^{3} - a b^{2}\right )} e^{\left (-3 \, d x - 3 \, c\right )} + {\left (3 \, a^{3} + a b^{2}\right )} e^{\left (-5 \, d x - 5 \, c\right )}}{{\left (a^{4} + a^{2} b^{2} - {\left (a^{4} + a^{2} b^{2}\right )} e^{\left (-2 \, d x - 2 \, c\right )} - {\left (a^{4} + a^{2} b^{2}\right )} e^{\left (-4 \, d x - 4 \, c\right )} + {\left (a^{4} + a^{2} b^{2}\right )} e^{\left (-6 \, d x - 6 \, c\right )}\right )} d} + \frac {{\left (3 \, a^{2} - 2 \, b^{2}\right )} \log \left (e^{\left (-d x - c\right )} + 1\right )}{2 \, a^{3} d} - \frac {{\left (3 \, a^{2} - 2 \, b^{2}\right )} \log \left (e^{\left (-d x - c\right )} - 1\right )}{2 \, a^{3} d} \] Input:

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

Output:

-b^5*log((b*e^(-d*x - c) - a - sqrt(a^2 + b^2))/(b*e^(-d*x - c) - a + sqrt 
(a^2 + b^2)))/((a^5 + a^3*b^2)*sqrt(a^2 + b^2)*d) - (4*a^2*b*e^(-2*d*x - 2 
*c) + 2*b^3*e^(-4*d*x - 4*c) - 4*a^2*b - 2*b^3 + (3*a^3 + a*b^2)*e^(-d*x - 
 c) - 2*(a^3 - a*b^2)*e^(-3*d*x - 3*c) + (3*a^3 + a*b^2)*e^(-5*d*x - 5*c)) 
/((a^4 + a^2*b^2 - (a^4 + a^2*b^2)*e^(-2*d*x - 2*c) - (a^4 + a^2*b^2)*e^(- 
4*d*x - 4*c) + (a^4 + a^2*b^2)*e^(-6*d*x - 6*c))*d) + 1/2*(3*a^2 - 2*b^2)* 
log(e^(-d*x - c) + 1)/(a^3*d) - 1/2*(3*a^2 - 2*b^2)*log(e^(-d*x - c) - 1)/ 
(a^3*d)
 

Giac [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.11 \[ \int \frac {\text {csch}^3(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {\frac {2 \, b^{5} \log \left (\frac {{\left | 2 \, b e^{\left (d x + c\right )} + 2 \, a - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, b e^{\left (d x + c\right )} + 2 \, a + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{{\left (a^{5} + a^{3} b^{2}\right )} \sqrt {a^{2} + b^{2}}} + \frac {4 \, {\left (a e^{\left (d x + c\right )} + b\right )}}{{\left (a^{2} + b^{2}\right )} {\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}} - \frac {{\left (3 \, a^{2} - 2 \, b^{2}\right )} \log \left (e^{\left (d x + c\right )} + 1\right )}{a^{3}} + \frac {{\left (3 \, a^{2} - 2 \, b^{2}\right )} \log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right )}{a^{3}} + \frac {2 \, {\left (a e^{\left (3 \, d x + 3 \, c\right )} - 2 \, b e^{\left (2 \, d x + 2 \, c\right )} + a e^{\left (d x + c\right )} + 2 \, b\right )}}{a^{2} {\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{2}}}{2 \, d} \] Input:

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

Output:

-1/2*(2*b^5*log(abs(2*b*e^(d*x + c) + 2*a - 2*sqrt(a^2 + b^2))/abs(2*b*e^( 
d*x + c) + 2*a + 2*sqrt(a^2 + b^2)))/((a^5 + a^3*b^2)*sqrt(a^2 + b^2)) + 4 
*(a*e^(d*x + c) + b)/((a^2 + b^2)*(e^(2*d*x + 2*c) + 1)) - (3*a^2 - 2*b^2) 
*log(e^(d*x + c) + 1)/a^3 + (3*a^2 - 2*b^2)*log(abs(e^(d*x + c) - 1))/a^3 
+ 2*(a*e^(3*d*x + 3*c) - 2*b*e^(2*d*x + 2*c) + a*e^(d*x + c) + 2*b)/(a^2*( 
e^(2*d*x + 2*c) - 1)^2))/d
 

Mupad [B] (verification not implemented)

Time = 4.65 (sec) , antiderivative size = 531, normalized size of antiderivative = 2.63 \[ \int \frac {\text {csch}^3(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {b^5\,\ln \left (2\,a^4\,b-4\,a^5\,{\mathrm {e}}^{c+d\,x}+b^5+3\,a^2\,b^3+4\,a^2\,{\mathrm {e}}^{c+d\,x}\,\sqrt {{\left (a^2+b^2\right )}^3}+b^2\,{\mathrm {e}}^{c+d\,x}\,\sqrt {{\left (a^2+b^2\right )}^3}-7\,a^3\,b^2\,{\mathrm {e}}^{c+d\,x}-2\,a\,b\,\sqrt {{\left (a^2+b^2\right )}^3}-3\,a\,b^4\,{\mathrm {e}}^{c+d\,x}\right )\,\sqrt {{\left (a^2+b^2\right )}^3}}{d\,a^9+3\,d\,a^7\,b^2+3\,d\,a^5\,b^4+d\,a^3\,b^6}-\frac {\frac {{\mathrm {e}}^{c+d\,x}}{a\,d}-\frac {2\,\left (a^2\,b+b^3\right )}{a^2\,d\,\left (a^2+b^2\right )}}{{\mathrm {e}}^{2\,c+2\,d\,x}-1}-\frac {\frac {2\,b}{d\,\left (a^2+b^2\right )}+\frac {2\,a\,{\mathrm {e}}^{c+d\,x}}{d\,\left (a^2+b^2\right )}}{{\mathrm {e}}^{2\,c+2\,d\,x}+1}-\frac {\ln \left ({\mathrm {e}}^{c+d\,x}-1\right )\,\left (3\,a^2-2\,b^2\right )}{2\,a^3\,d}+\frac {\ln \left ({\mathrm {e}}^{c+d\,x}+1\right )\,\left (3\,a^2-2\,b^2\right )}{2\,a^3\,d}-\frac {b^5\,\ln \left (4\,a^5\,{\mathrm {e}}^{c+d\,x}-2\,a^4\,b-b^5-3\,a^2\,b^3+4\,a^2\,{\mathrm {e}}^{c+d\,x}\,\sqrt {{\left (a^2+b^2\right )}^3}+b^2\,{\mathrm {e}}^{c+d\,x}\,\sqrt {{\left (a^2+b^2\right )}^3}+7\,a^3\,b^2\,{\mathrm {e}}^{c+d\,x}-2\,a\,b\,\sqrt {{\left (a^2+b^2\right )}^3}+3\,a\,b^4\,{\mathrm {e}}^{c+d\,x}\right )\,\sqrt {{\left (a^2+b^2\right )}^3}}{d\,a^9+3\,d\,a^7\,b^2+3\,d\,a^5\,b^4+d\,a^3\,b^6}-\frac {2\,{\mathrm {e}}^{c+d\,x}}{a\,d\,\left ({\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1\right )} \] Input:

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

Output:

(b^5*log(2*a^4*b - 4*a^5*exp(c + d*x) + b^5 + 3*a^2*b^3 + 4*a^2*exp(c + d* 
x)*((a^2 + b^2)^3)^(1/2) + b^2*exp(c + d*x)*((a^2 + b^2)^3)^(1/2) - 7*a^3* 
b^2*exp(c + d*x) - 2*a*b*((a^2 + b^2)^3)^(1/2) - 3*a*b^4*exp(c + d*x))*((a 
^2 + b^2)^3)^(1/2))/(a^9*d + a^3*b^6*d + 3*a^5*b^4*d + 3*a^7*b^2*d) - (exp 
(c + d*x)/(a*d) - (2*(a^2*b + b^3))/(a^2*d*(a^2 + b^2)))/(exp(2*c + 2*d*x) 
 - 1) - ((2*b)/(d*(a^2 + b^2)) + (2*a*exp(c + d*x))/(d*(a^2 + b^2)))/(exp( 
2*c + 2*d*x) + 1) - (log(exp(c + d*x) - 1)*(3*a^2 - 2*b^2))/(2*a^3*d) + (l 
og(exp(c + d*x) + 1)*(3*a^2 - 2*b^2))/(2*a^3*d) - (b^5*log(4*a^5*exp(c + d 
*x) - 2*a^4*b - b^5 - 3*a^2*b^3 + 4*a^2*exp(c + d*x)*((a^2 + b^2)^3)^(1/2) 
 + b^2*exp(c + d*x)*((a^2 + b^2)^3)^(1/2) + 7*a^3*b^2*exp(c + d*x) - 2*a*b 
*((a^2 + b^2)^3)^(1/2) + 3*a*b^4*exp(c + d*x))*((a^2 + b^2)^3)^(1/2))/(a^9 
*d + a^3*b^6*d + 3*a^5*b^4*d + 3*a^7*b^2*d) - (2*exp(c + d*x))/(a*d*(exp(4 
*c + 4*d*x) - 2*exp(2*c + 2*d*x) + 1))
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 1346, normalized size of antiderivative = 6.66 \[ \int \frac {\text {csch}^3(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 4*e**(6*c + 6*d*x)*sqrt(a**2 + b**2)*atan((e**(c + d*x)*b*i + a*i)/sqr 
t(a**2 + b**2))*b**5*i + 4*e**(4*c + 4*d*x)*sqrt(a**2 + b**2)*atan((e**(c 
+ d*x)*b*i + a*i)/sqrt(a**2 + b**2))*b**5*i + 4*e**(2*c + 2*d*x)*sqrt(a**2 
 + b**2)*atan((e**(c + d*x)*b*i + a*i)/sqrt(a**2 + b**2))*b**5*i - 4*sqrt( 
a**2 + b**2)*atan((e**(c + d*x)*b*i + a*i)/sqrt(a**2 + b**2))*b**5*i - 3*e 
**(6*c + 6*d*x)*log(e**(c + d*x) - 1)*a**6 - 4*e**(6*c + 6*d*x)*log(e**(c 
+ d*x) - 1)*a**4*b**2 + e**(6*c + 6*d*x)*log(e**(c + d*x) - 1)*a**2*b**4 + 
 2*e**(6*c + 6*d*x)*log(e**(c + d*x) - 1)*b**6 + 3*e**(6*c + 6*d*x)*log(e* 
*(c + d*x) + 1)*a**6 + 4*e**(6*c + 6*d*x)*log(e**(c + d*x) + 1)*a**4*b**2 
- e**(6*c + 6*d*x)*log(e**(c + d*x) + 1)*a**2*b**4 - 2*e**(6*c + 6*d*x)*lo 
g(e**(c + d*x) + 1)*b**6 + 4*e**(6*c + 6*d*x)*a**3*b**3 + 4*e**(6*c + 6*d* 
x)*a*b**5 - 6*e**(5*c + 5*d*x)*a**6 - 8*e**(5*c + 5*d*x)*a**4*b**2 - 2*e** 
(5*c + 5*d*x)*a**2*b**4 + 3*e**(4*c + 4*d*x)*log(e**(c + d*x) - 1)*a**6 + 
4*e**(4*c + 4*d*x)*log(e**(c + d*x) - 1)*a**4*b**2 - e**(4*c + 4*d*x)*log( 
e**(c + d*x) - 1)*a**2*b**4 - 2*e**(4*c + 4*d*x)*log(e**(c + d*x) - 1)*b** 
6 - 3*e**(4*c + 4*d*x)*log(e**(c + d*x) + 1)*a**6 - 4*e**(4*c + 4*d*x)*log 
(e**(c + d*x) + 1)*a**4*b**2 + e**(4*c + 4*d*x)*log(e**(c + d*x) + 1)*a**2 
*b**4 + 2*e**(4*c + 4*d*x)*log(e**(c + d*x) + 1)*b**6 + 4*e**(3*c + 3*d*x) 
*a**6 - 4*e**(3*c + 3*d*x)*a**2*b**4 + 3*e**(2*c + 2*d*x)*log(e**(c + d*x) 
 - 1)*a**6 + 4*e**(2*c + 2*d*x)*log(e**(c + d*x) - 1)*a**4*b**2 - e**(2...