Integrand size = 29, antiderivative size = 211 \[ \int \frac {\text {csch}^3(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {b \arctan (\sinh (c+d x))}{2 \left (a^2+b^2\right ) d}+\frac {b \left (a^2+2 b^2\right ) \arctan (\sinh (c+d x))}{\left (a^2+b^2\right )^2 d}+\frac {b \text {csch}(c+d x)}{a^2 d}-\frac {\text {csch}^2(c+d x)}{2 a d}+\frac {a \left (2 a^2+3 b^2\right ) \log (\cosh (c+d x))}{\left (a^2+b^2\right )^2 d}-\frac {\left (2 a^2-b^2\right ) \log (\sinh (c+d x))}{a^3 d}-\frac {b^6 \log (a+b \sinh (c+d x))}{a^3 \left (a^2+b^2\right )^2 d}-\frac {\text {sech}^2(c+d x) (a-b \sinh (c+d x))}{2 \left (a^2+b^2\right ) d} \] Output:
1/2*b*arctan(sinh(d*x+c))/(a^2+b^2)/d+b*(a^2+2*b^2)*arctan(sinh(d*x+c))/(a ^2+b^2)^2/d+b*csch(d*x+c)/a^2/d-1/2*csch(d*x+c)^2/a/d+a*(2*a^2+3*b^2)*ln(c osh(d*x+c))/(a^2+b^2)^2/d-(2*a^2-b^2)*ln(sinh(d*x+c))/a^3/d-b^6*ln(a+b*sin h(d*x+c))/a^3/(a^2+b^2)^2/d-1/2*sech(d*x+c)^2*(a-b*sinh(d*x+c))/(a^2+b^2)/ d
Result contains complex when optimal does not.
Time = 0.67 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.12 \[ \int \frac {\text {csch}^3(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {\frac {b \arctan (\sinh (c+d x))}{a^2+b^2}+\frac {2 b \text {csch}(c+d x)}{a^2}-\frac {\text {csch}^2(c+d x)}{a}+\frac {(a-i b) \left (2 a^2+i a b+2 b^2\right ) \log (i-\sinh (c+d x))}{\left (a^2+b^2\right )^2}-\frac {2 \left (2 a^2-b^2\right ) \log (\sinh (c+d x))}{a^3}+\frac {(a+i b) \left (2 a^2-i a b+2 b^2\right ) \log (i+\sinh (c+d x))}{\left (a^2+b^2\right )^2}-\frac {2 b^6 \log (a+b \sinh (c+d x))}{a^3 \left (a^2+b^2\right )^2}-\frac {a \text {sech}^2(c+d x)}{a^2+b^2}+\frac {b \text {sech}(c+d x) \tanh (c+d x)}{a^2+b^2}}{2 d} \] Input:
Integrate[(Csch[c + d*x]^3*Sech[c + d*x]^3)/(a + b*Sinh[c + d*x]),x]
Output:
((b*ArcTan[Sinh[c + d*x]])/(a^2 + b^2) + (2*b*Csch[c + d*x])/a^2 - Csch[c + d*x]^2/a + ((a - I*b)*(2*a^2 + I*a*b + 2*b^2)*Log[I - Sinh[c + d*x]])/(a ^2 + b^2)^2 - (2*(2*a^2 - b^2)*Log[Sinh[c + d*x]])/a^3 + ((a + I*b)*(2*a^2 - I*a*b + 2*b^2)*Log[I + Sinh[c + d*x]])/(a^2 + b^2)^2 - (2*b^6*Log[a + b *Sinh[c + d*x]])/(a^3*(a^2 + b^2)^2) - (a*Sech[c + d*x]^2)/(a^2 + b^2) + ( b*Sech[c + d*x]*Tanh[c + d*x])/(a^2 + b^2))/(2*d)
Time = 0.61 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.11, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {3042, 26, 3316, 26, 27, 615, 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.
| \(\displaystyle \int \frac {\text {csch}^3(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {i}{\sin (i c+i d x)^3 \cos (i c+i d x)^3 (a-i b \sin (i c+i d x))}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i \int \frac {1}{\cos (i c+i d x)^3 \sin (i c+i d x)^3 (a-i b \sin (i c+i d x))}dx\) |
| \(\Big \downarrow \) 3316 |
| \(\displaystyle -\frac {i b^3 \int \frac {i \text {csch}^3(c+d x)}{(a+b \sinh (c+d x)) \left (\sinh ^2(c+d x) b^2+b^2\right )^2}d(b \sinh (c+d x))}{d}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {b^3 \int \frac {\text {csch}^3(c+d x)}{(a+b \sinh (c+d x)) \left (\sinh ^2(c+d x) b^2+b^2\right )^2}d(b \sinh (c+d x))}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b^6 \int \frac {\text {csch}^3(c+d x)}{b^3 (a+b \sinh (c+d x)) \left (\sinh ^2(c+d x) b^2+b^2\right )^2}d(b \sinh (c+d x))}{d}\) |
| \(\Big \downarrow \) 615 |
| \(\displaystyle \frac {b^6 \int \left (\frac {\text {csch}^3(c+d x)}{a b^7}-\frac {\text {csch}^2(c+d x)}{a^2 b^6}+\frac {\left (b^2-2 a^2\right ) \text {csch}(c+d x)}{a^3 b^7}-\frac {1}{a^3 \left (a^2+b^2\right )^2 (a+b \sinh (c+d x))}+\frac {\left (a^2+2 b^2\right ) b^2+a \left (2 a^2+3 b^2\right ) \sinh (c+d x) b}{b^6 \left (a^2+b^2\right )^2 \left (\sinh ^2(c+d x) b^2+b^2\right )}+\frac {b^2+a \sinh (c+d x) b}{b^4 \left (a^2+b^2\right ) \left (\sinh ^2(c+d x) b^2+b^2\right )^2}\right )d(b \sinh (c+d x))}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {b^6 \left (\frac {\left (a^2+2 b^2\right ) \arctan (\sinh (c+d x))}{b^5 \left (a^2+b^2\right )^2}+\frac {\arctan (\sinh (c+d x))}{2 b^5 \left (a^2+b^2\right )}+\frac {\text {csch}(c+d x)}{a^2 b^5}+\frac {a \left (2 a^2+3 b^2\right ) \log \left (b^2 \sinh ^2(c+d x)+b^2\right )}{2 b^6 \left (a^2+b^2\right )^2}-\frac {a-b \sinh (c+d x)}{2 b^4 \left (a^2+b^2\right ) \left (b^2 \sinh ^2(c+d x)+b^2\right )}-\frac {\log (a+b \sinh (c+d x))}{a^3 \left (a^2+b^2\right )^2}-\frac {\left (2 a^2-b^2\right ) \log (b \sinh (c+d x))}{a^3 b^6}-\frac {\text {csch}^2(c+d x)}{2 a b^6}\right )}{d}\) |
Input:
Int[(Csch[c + d*x]^3*Sech[c + d*x]^3)/(a + b*Sinh[c + d*x]),x]
Output:
(b^6*(ArcTan[Sinh[c + d*x]]/(2*b^5*(a^2 + b^2)) + ((a^2 + 2*b^2)*ArcTan[Si nh[c + d*x]])/(b^5*(a^2 + b^2)^2) + Csch[c + d*x]/(a^2*b^5) - Csch[c + d*x ]^2/(2*a*b^6) - ((2*a^2 - b^2)*Log[b*Sinh[c + d*x]])/(a^3*b^6) - Log[a + b *Sinh[c + d*x]]/(a^3*(a^2 + b^2)^2) + (a*(2*a^2 + 3*b^2)*Log[b^2 + b^2*Sin h[c + d*x]^2])/(2*b^6*(a^2 + b^2)^2) - (a - b*Sinh[c + d*x])/(2*b^4*(a^2 + b^2)*(b^2 + b^2*Sinh[c + d*x]^2))))/d
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Int[ExpandIntegrand[(e*x)^m*(c + d*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && ILtQ[p, 0]
Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_ .)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Simp[1/(b^p* f) Subst[Int[(a + x)^m*(c + (d/b)*x)^n*(b^2 - x^2)^((p - 1)/2), x], x, b* Sin[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IntegerQ[(p - 1) /2] && NeQ[a^2 - b^2, 0]
Time = 47.01 (sec) , antiderivative size = 292, normalized size of antiderivative = 1.38
| method | result | size |
| derivativedivides | \(\frac {-\frac {\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a}{2}+2 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a^{2}}+\frac {\frac {2 \left (\left (-\frac {1}{2} a^{2} b -\frac {1}{2} b^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (a^{3}+a \,b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\left (\frac {1}{2} a^{2} b +\frac {1}{2} b^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2}}+\frac {\left (4 a^{3}+6 a \,b^{2}\right ) \ln \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}{2}+\left (3 a^{2} b +5 b^{3}\right ) \arctan \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (a^{2}+b^{2}\right )^{2}}-\frac {1}{8 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\left (-8 a^{2}+4 b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{3}}+\frac {b}{2 a^{2} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {b^{6} \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-a \right )}{\left (a^{2}+b^{2}\right )^{2} a^{3}}}{d}\) | \(292\) |
| default | \(\frac {-\frac {\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a}{2}+2 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a^{2}}+\frac {\frac {2 \left (\left (-\frac {1}{2} a^{2} b -\frac {1}{2} b^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (a^{3}+a \,b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\left (\frac {1}{2} a^{2} b +\frac {1}{2} b^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2}}+\frac {\left (4 a^{3}+6 a \,b^{2}\right ) \ln \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}{2}+\left (3 a^{2} b +5 b^{3}\right ) \arctan \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (a^{2}+b^{2}\right )^{2}}-\frac {1}{8 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\left (-8 a^{2}+4 b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{3}}+\frac {b}{2 a^{2} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {b^{6} \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-a \right )}{\left (a^{2}+b^{2}\right )^{2} a^{3}}}{d}\) | \(292\) |
| risch | \(-\frac {4 a^{3} d^{2} x}{a^{4} d^{2}+2 a^{2} b^{2} d^{2}+b^{4} d^{2}}-\frac {4 a^{3} d c}{a^{4} d^{2}+2 a^{2} b^{2} d^{2}+b^{4} d^{2}}-\frac {6 a \,b^{2} d^{2} x}{a^{4} d^{2}+2 a^{2} b^{2} d^{2}+b^{4} d^{2}}-\frac {6 a \,b^{2} d c}{a^{4} d^{2}+2 a^{2} b^{2} d^{2}+b^{4} d^{2}}+\frac {4 x}{a}+\frac {4 c}{d a}-\frac {2 b^{2} x}{a^{3}}-\frac {2 b^{2} c}{a^{3} d}+\frac {2 b^{6} x}{a^{3} \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {2 b^{6} c}{a^{3} d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {{\mathrm e}^{d x +c} \left (-3 b \,{\mathrm e}^{6 d x +6 c} a^{2}-2 \,{\mathrm e}^{6 d x +6 c} b^{3}+4 \,{\mathrm e}^{5 d x +5 c} a^{3}+2 \,{\mathrm e}^{5 d x +5 c} a \,b^{2}+a^{2} b \,{\mathrm e}^{4 d x +4 c}-2 \,{\mathrm e}^{4 d x +4 c} b^{3}+4 \,{\mathrm e}^{3 d x +3 c} a \,b^{2}-b \,{\mathrm e}^{2 d x +2 c} a^{2}+2 \,{\mathrm e}^{2 d x +2 c} b^{3}+4 \,{\mathrm e}^{d x +c} a^{3}+2 \,{\mathrm e}^{d x +c} a \,b^{2}+3 a^{2} b +2 b^{3}\right )}{d \left (a^{2}+b^{2}\right ) \left (1+{\mathrm e}^{2 d x +2 c}\right )^{2} a^{2} \left ({\mathrm e}^{2 d x +2 c}-1\right )^{2}}+\frac {3 i \ln \left ({\mathrm e}^{d x +c}+i\right ) a^{2} b}{2 \left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) d}-\frac {5 i \ln \left ({\mathrm e}^{d x +c}-i\right ) b^{3}}{2 \left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) d}+\frac {2 \ln \left ({\mathrm e}^{d x +c}+i\right ) a^{3}}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) d}+\frac {3 \ln \left ({\mathrm e}^{d x +c}+i\right ) a \,b^{2}}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) d}+\frac {5 i \ln \left ({\mathrm e}^{d x +c}+i\right ) b^{3}}{2 \left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) d}-\frac {3 i \ln \left ({\mathrm e}^{d x +c}-i\right ) a^{2} b}{2 \left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) d}+\frac {2 \ln \left ({\mathrm e}^{d x +c}-i\right ) a^{3}}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) d}+\frac {3 \ln \left ({\mathrm e}^{d x +c}-i\right ) a \,b^{2}}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) d}-\frac {2 \ln \left ({\mathrm e}^{2 d x +2 c}-1\right )}{a d}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}-1\right ) b^{2}}{a^{3} d}-\frac {b^{6} \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \,{\mathrm e}^{d x +c}}{b}-1\right )}{a^{3} d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}\) | \(835\) |
Input:
int(csch(d*x+c)^3*sech(d*x+c)^3/(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))+2/(a^2 +b^2)^2*(((-1/2*a^2*b-1/2*b^3)*tanh(1/2*d*x+1/2*c)^3+(a^3+a*b^2)*tanh(1/2* d*x+1/2*c)^2+(1/2*a^2*b+1/2*b^3)*tanh(1/2*d*x+1/2*c))/(1+tanh(1/2*d*x+1/2* c)^2)^2+1/4*(4*a^3+6*a*b^2)*ln(1+tanh(1/2*d*x+1/2*c)^2)+1/2*(3*a^2*b+5*b^3 )*arctan(tanh(1/2*d*x+1/2*c)))-1/8/a/tanh(1/2*d*x+1/2*c)^2+1/4/a^3*(-8*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)-b^6/(a^2+b^2 )^2/a^3*ln(tanh(1/2*d*x+1/2*c)^2*a-2*b*tanh(1/2*d*x+1/2*c)-a))
Leaf count of result is larger than twice the leaf count of optimal. 3148 vs. \(2 (206) = 412\).
Time = 0.48 (sec) , antiderivative size = 3148, normalized size of antiderivative = 14.92 \[ \int \frac {\text {csch}^3(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \] Input:
integrate(csch(d*x+c)^3*sech(d*x+c)^3/(a+b*sinh(d*x+c)),x, algorithm="fric as")
Output:
Too large to include
Timed out. \[ \int \frac {\text {csch}^3(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \] Input:
integrate(csch(d*x+c)**3*sech(d*x+c)**3/(a+b*sinh(d*x+c)),x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 418 vs. \(2 (206) = 412\).
Time = 0.13 (sec) , antiderivative size = 418, normalized size of antiderivative = 1.98 \[ \int \frac {\text {csch}^3(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {b^{6} \log \left (-2 \, a e^{\left (-d x - c\right )} + b e^{\left (-2 \, d x - 2 \, c\right )} - b\right )}{{\left (a^{7} + 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} d} - \frac {{\left (3 \, a^{2} b + 5 \, b^{3}\right )} \arctan \left (e^{\left (-d x - c\right )}\right )}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d} + \frac {{\left (2 \, a^{3} + 3 \, a b^{2}\right )} \log \left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d} - \frac {4 \, a b^{2} e^{\left (-4 \, d x - 4 \, c\right )} - {\left (3 \, a^{2} b + 2 \, b^{3}\right )} e^{\left (-d x - c\right )} + 2 \, {\left (2 \, a^{3} + a b^{2}\right )} e^{\left (-2 \, d x - 2 \, c\right )} + {\left (a^{2} b - 2 \, b^{3}\right )} e^{\left (-3 \, d x - 3 \, c\right )} - {\left (a^{2} b - 2 \, b^{3}\right )} e^{\left (-5 \, d x - 5 \, c\right )} + 2 \, {\left (2 \, a^{3} + a b^{2}\right )} e^{\left (-6 \, d x - 6 \, c\right )} + {\left (3 \, a^{2} b + 2 \, b^{3}\right )} e^{\left (-7 \, d x - 7 \, c\right )}}{{\left (a^{4} + a^{2} b^{2} - 2 \, {\left (a^{4} + a^{2} b^{2}\right )} e^{\left (-4 \, d x - 4 \, c\right )} + {\left (a^{4} + a^{2} b^{2}\right )} e^{\left (-8 \, d x - 8 \, c\right )}\right )} d} - \frac {{\left (2 \, a^{2} - b^{2}\right )} \log \left (e^{\left (-d x - c\right )} + 1\right )}{a^{3} d} - \frac {{\left (2 \, a^{2} - b^{2}\right )} \log \left (e^{\left (-d x - c\right )} - 1\right )}{a^{3} d} \] Input:
integrate(csch(d*x+c)^3*sech(d*x+c)^3/(a+b*sinh(d*x+c)),x, algorithm="maxi ma")
Output:
-b^6*log(-2*a*e^(-d*x - c) + b*e^(-2*d*x - 2*c) - b)/((a^7 + 2*a^5*b^2 + a ^3*b^4)*d) - (3*a^2*b + 5*b^3)*arctan(e^(-d*x - c))/((a^4 + 2*a^2*b^2 + b^ 4)*d) + (2*a^3 + 3*a*b^2)*log(e^(-2*d*x - 2*c) + 1)/((a^4 + 2*a^2*b^2 + b^ 4)*d) - (4*a*b^2*e^(-4*d*x - 4*c) - (3*a^2*b + 2*b^3)*e^(-d*x - c) + 2*(2* a^3 + a*b^2)*e^(-2*d*x - 2*c) + (a^2*b - 2*b^3)*e^(-3*d*x - 3*c) - (a^2*b - 2*b^3)*e^(-5*d*x - 5*c) + 2*(2*a^3 + a*b^2)*e^(-6*d*x - 6*c) + (3*a^2*b + 2*b^3)*e^(-7*d*x - 7*c))/((a^4 + a^2*b^2 - 2*(a^4 + a^2*b^2)*e^(-4*d*x - 4*c) + (a^4 + a^2*b^2)*e^(-8*d*x - 8*c))*d) - (2*a^2 - b^2)*log(e^(-d*x - c) + 1)/(a^3*d) - (2*a^2 - b^2)*log(e^(-d*x - c) - 1)/(a^3*d)
Leaf count of result is larger than twice the leaf count of optimal. 464 vs. \(2 (206) = 412\).
Time = 0.14 (sec) , antiderivative size = 464, normalized size of antiderivative = 2.20 \[ \int \frac {\text {csch}^3(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {\frac {4 \, b^{7} \log \left ({\left | b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 2 \, a \right |}\right )}{a^{7} b + 2 \, a^{5} b^{3} + a^{3} b^{5}} - \frac {{\left (\pi + 2 \, \arctan \left (\frac {1}{2} \, {\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )} e^{\left (-d x - c\right )}\right )\right )} {\left (3 \, a^{2} b + 5 \, b^{3}\right )}}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} - \frac {2 \, {\left (2 \, a^{3} + 3 \, a b^{2}\right )} \log \left ({\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} + 4\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {2 \, {\left (2 \, a^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} + 3 \, a b^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} - 2 \, a^{2} b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} - 2 \, b^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 12 \, a^{3} + 16 \, a b^{2}\right )}}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} {\left ({\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} + 4\right )}} + \frac {4 \, {\left (2 \, a^{2} - b^{2}\right )} \log \left ({\left | e^{\left (d x + c\right )} - e^{\left (-d x - c\right )} \right |}\right )}{a^{3}} - \frac {2 \, {\left (6 \, a^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} - 3 \, b^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} + 4 \, a b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} - 4 \, a^{2}\right )}}{a^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2}}}{4 \, d} \] Input:
integrate(csch(d*x+c)^3*sech(d*x+c)^3/(a+b*sinh(d*x+c)),x, algorithm="giac ")
Output:
-1/4*(4*b^7*log(abs(b*(e^(d*x + c) - e^(-d*x - c)) + 2*a))/(a^7*b + 2*a^5* b^3 + a^3*b^5) - (pi + 2*arctan(1/2*(e^(2*d*x + 2*c) - 1)*e^(-d*x - c)))*( 3*a^2*b + 5*b^3)/(a^4 + 2*a^2*b^2 + b^4) - 2*(2*a^3 + 3*a*b^2)*log((e^(d*x + c) - e^(-d*x - c))^2 + 4)/(a^4 + 2*a^2*b^2 + b^4) + 2*(2*a^3*(e^(d*x + c) - e^(-d*x - c))^2 + 3*a*b^2*(e^(d*x + c) - e^(-d*x - c))^2 - 2*a^2*b*(e ^(d*x + c) - e^(-d*x - c)) - 2*b^3*(e^(d*x + c) - e^(-d*x - c)) + 12*a^3 + 16*a*b^2)/((a^4 + 2*a^2*b^2 + b^4)*((e^(d*x + c) - e^(-d*x - c))^2 + 4)) + 4*(2*a^2 - b^2)*log(abs(e^(d*x + c) - e^(-d*x - c)))/a^3 - 2*(6*a^2*(e^( d*x + c) - e^(-d*x - c))^2 - 3*b^2*(e^(d*x + c) - e^(-d*x - c))^2 + 4*a*b* (e^(d*x + c) - e^(-d*x - c)) - 4*a^2)/(a^3*(e^(d*x + c) - e^(-d*x - c))^2) )/d
Time = 8.38 (sec) , antiderivative size = 554, normalized size of antiderivative = 2.63 \[ \int \frac {\text {csch}^3(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {\frac {4\,b^5}{a\,d\,\left (a^2\,b^3+b^5\right )}-\frac {4\,b^4\,{\mathrm {e}}^{3\,c+3\,d\,x}}{d\,\left (a^2\,b^3+b^5\right )}+\frac {4\,b^4\,{\mathrm {e}}^{c+d\,x}}{d\,\left (a^2\,b^3+b^5\right )}+\frac {4\,b^3\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (2\,a^2+b^2\right )}{a\,d\,\left (a^2\,b^3+b^5\right )}}{{\mathrm {e}}^{8\,c+8\,d\,x}-2\,{\mathrm {e}}^{4\,c+4\,d\,x}+1}-\frac {\frac {4\,\left (a^2\,b^5+b^7\right )}{a\,d\,\left (a^2\,b^3+b^5\right )\,\left (a^2+b^2\right )}+\frac {2\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (2\,a^4\,b^3+3\,a^2\,b^5+b^7\right )}{a\,d\,\left (a^2\,b^3+b^5\right )\,\left (a^2+b^2\right )}-\frac {{\mathrm {e}}^{3\,c+3\,d\,x}\,\left (3\,a^4\,b^4+5\,a^2\,b^6+2\,b^8\right )}{a^2\,d\,\left (a^2\,b^3+b^5\right )\,\left (a^2+b^2\right )}-\frac {b^4\,{\mathrm {e}}^{c+d\,x}\,\left (-a^4+a^2\,b^2+2\,b^4\right )}{a^2\,d\,\left (a^2\,b^3+b^5\right )\,\left (a^2+b^2\right )}}{{\mathrm {e}}^{4\,c+4\,d\,x}-1}+\frac {\ln \left (1+{\mathrm {e}}^{c+d\,x}\,1{}\mathrm {i}\right )\,\left (4\,a+b\,5{}\mathrm {i}\right )}{2\,\left (d\,a^2+2{}\mathrm {i}\,d\,a\,b-d\,b^2\right )}-\frac {b^6\,\ln \left (2\,a\,{\mathrm {e}}^{c+d\,x}-b+b\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{d\,a^7+2\,d\,a^5\,b^2+d\,a^3\,b^4}+\frac {\ln \left ({\mathrm {e}}^{c+d\,x}+1{}\mathrm {i}\right )\,\left (5\,b+a\,4{}\mathrm {i}\right )}{2\,\left (1{}\mathrm {i}\,d\,a^2+2\,d\,a\,b-1{}\mathrm {i}\,d\,b^2\right )}-\frac {\ln \left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )\,\left (2\,a^2-b^2\right )}{a^3\,d} \] Input:
int(1/(cosh(c + d*x)^3*sinh(c + d*x)^3*(a + b*sinh(c + d*x))),x)
Output:
(log(exp(c + d*x)*1i + 1)*(4*a + b*5i))/(2*(a^2*d - b^2*d + a*b*d*2i)) - ( (4*(b^7 + a^2*b^5))/(a*d*(b^5 + a^2*b^3)*(a^2 + b^2)) + (2*exp(2*c + 2*d*x )*(b^7 + 3*a^2*b^5 + 2*a^4*b^3))/(a*d*(b^5 + a^2*b^3)*(a^2 + b^2)) - (exp( 3*c + 3*d*x)*(2*b^8 + 5*a^2*b^6 + 3*a^4*b^4))/(a^2*d*(b^5 + a^2*b^3)*(a^2 + b^2)) - (b^4*exp(c + d*x)*(2*b^4 - a^4 + a^2*b^2))/(a^2*d*(b^5 + a^2*b^3 )*(a^2 + b^2)))/(exp(4*c + 4*d*x) - 1) - ((4*b^5)/(a*d*(b^5 + a^2*b^3)) - (4*b^4*exp(3*c + 3*d*x))/(d*(b^5 + a^2*b^3)) + (4*b^4*exp(c + d*x))/(d*(b^ 5 + a^2*b^3)) + (4*b^3*exp(2*c + 2*d*x)*(2*a^2 + b^2))/(a*d*(b^5 + a^2*b^3 )))/(exp(8*c + 8*d*x) - 2*exp(4*c + 4*d*x) + 1) - (b^6*log(2*a*exp(c + d*x ) - b + b*exp(2*c + 2*d*x)))/(a^7*d + a^3*b^4*d + 2*a^5*b^2*d) + (log(exp( c + d*x) + 1i)*(a*4i + 5*b))/(2*(a^2*d*1i - b^2*d*1i + 2*a*b*d)) - (log(ex p(2*c + 2*d*x) - 1)*(2*a^2 - b^2))/(a^3*d)
Time = 0.22 (sec) , antiderivative size = 1269, normalized size of antiderivative = 6.01 \[ \int \frac {\text {csch}^3(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx =\text {Too large to display} \] Input:
int(csch(d*x+c)^3*sech(d*x+c)^3/(a+b*sinh(d*x+c)),x)
Output:
(3*e**(8*c + 8*d*x)*atan(e**(c + d*x))*a**5*b + 5*e**(8*c + 8*d*x)*atan(e* *(c + d*x))*a**3*b**3 - 6*e**(4*c + 4*d*x)*atan(e**(c + d*x))*a**5*b - 10* e**(4*c + 4*d*x)*atan(e**(c + d*x))*a**3*b**3 + 3*atan(e**(c + d*x))*a**5* b + 5*atan(e**(c + d*x))*a**3*b**3 + 2*e**(8*c + 8*d*x)*log(e**(2*c + 2*d* x) + 1)*a**6 + 3*e**(8*c + 8*d*x)*log(e**(2*c + 2*d*x) + 1)*a**4*b**2 - 2* e**(8*c + 8*d*x)*log(e**(c + d*x) - 1)*a**6 - 3*e**(8*c + 8*d*x)*log(e**(c + d*x) - 1)*a**4*b**2 + e**(8*c + 8*d*x)*log(e**(c + d*x) - 1)*b**6 - 2*e **(8*c + 8*d*x)*log(e**(c + d*x) + 1)*a**6 - 3*e**(8*c + 8*d*x)*log(e**(c + d*x) + 1)*a**4*b**2 + e**(8*c + 8*d*x)*log(e**(c + d*x) + 1)*b**6 - e**( 8*c + 8*d*x)*log(e**(2*c + 2*d*x)*b + 2*e**(c + d*x)*a - b)*b**6 - 2*e**(8 *c + 8*d*x)*a**4*b**2 - 2*e**(8*c + 8*d*x)*a**2*b**4 + 3*e**(7*c + 7*d*x)* a**5*b + 5*e**(7*c + 7*d*x)*a**3*b**3 + 2*e**(7*c + 7*d*x)*a*b**5 - 4*e**( 6*c + 6*d*x)*a**6 - 6*e**(6*c + 6*d*x)*a**4*b**2 - 2*e**(6*c + 6*d*x)*a**2 *b**4 - e**(5*c + 5*d*x)*a**5*b + e**(5*c + 5*d*x)*a**3*b**3 + 2*e**(5*c + 5*d*x)*a*b**5 - 4*e**(4*c + 4*d*x)*log(e**(2*c + 2*d*x) + 1)*a**6 - 6*e** (4*c + 4*d*x)*log(e**(2*c + 2*d*x) + 1)*a**4*b**2 + 4*e**(4*c + 4*d*x)*log (e**(c + d*x) - 1)*a**6 + 6*e**(4*c + 4*d*x)*log(e**(c + d*x) - 1)*a**4*b* *2 - 2*e**(4*c + 4*d*x)*log(e**(c + d*x) - 1)*b**6 + 4*e**(4*c + 4*d*x)*lo g(e**(c + d*x) + 1)*a**6 + 6*e**(4*c + 4*d*x)*log(e**(c + d*x) + 1)*a**4*b **2 - 2*e**(4*c + 4*d*x)*log(e**(c + d*x) + 1)*b**6 + 2*e**(4*c + 4*d*x...