Integrand size = 29, antiderivative size = 180 \[ \int \frac {\text {csch}^2(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {a \arctan (\sinh (c+d x))}{2 \left (a^2+b^2\right ) d}-\frac {a \left (a^2+2 b^2\right ) \arctan (\sinh (c+d x))}{\left (a^2+b^2\right )^2 d}-\frac {\text {csch}(c+d x)}{a d}+\frac {b \left (a^2+2 b^2\right ) \log (\cosh (c+d x))}{\left (a^2+b^2\right )^2 d}-\frac {b \log (\sinh (c+d x))}{a^2 d}+\frac {b^5 \log (a+b \sinh (c+d x))}{a^2 \left (a^2+b^2\right )^2 d}-\frac {\text {sech}^2(c+d x) (b+a \sinh (c+d x))}{2 \left (a^2+b^2\right ) d} \] Output:
-1/2*a*arctan(sinh(d*x+c))/(a^2+b^2)/d-a*(a^2+2*b^2)*arctan(sinh(d*x+c))/( a^2+b^2)^2/d-csch(d*x+c)/a/d+b*(a^2+2*b^2)*ln(cosh(d*x+c))/(a^2+b^2)^2/d-b *ln(sinh(d*x+c))/a^2/d+b^5*ln(a+b*sinh(d*x+c))/a^2/(a^2+b^2)^2/d-1/2*sech( d*x+c)^2*(b+a*sinh(d*x+c))/(a^2+b^2)/d
Result contains complex when optimal does not.
Time = 0.70 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.26 \[ \int \frac {\text {csch}^2(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {\text {csch}(c+d x) (a+b \sinh (c+d x)) \left (\frac {a \arctan (\sinh (c+d x))}{a^2+b^2}+\frac {2 \text {csch}(c+d x)}{a}-\frac {(i a+b) \left (a^2+2 b^2\right ) \log (i-\sinh (c+d x))}{\left (a^2+b^2\right )^2}+\frac {2 b \log (\sinh (c+d x))}{a^2}+\frac {(i a-b) \left (a^2+2 b^2\right ) \log (i+\sinh (c+d x))}{\left (a^2+b^2\right )^2}-\frac {2 b^5 \log (a+b \sinh (c+d x))}{a^2 \left (a^2+b^2\right )^2}+\frac {b \text {sech}^2(c+d x)}{a^2+b^2}+\frac {a \text {sech}(c+d x) \tanh (c+d x)}{a^2+b^2}\right )}{2 d (b+a \text {csch}(c+d x))} \] Input:
Integrate[(Csch[c + d*x]^2*Sech[c + d*x]^3)/(a + b*Sinh[c + d*x]),x]
Output:
-1/2*(Csch[c + d*x]*(a + b*Sinh[c + d*x])*((a*ArcTan[Sinh[c + d*x]])/(a^2 + b^2) + (2*Csch[c + d*x])/a - ((I*a + b)*(a^2 + 2*b^2)*Log[I - Sinh[c + d *x]])/(a^2 + b^2)^2 + (2*b*Log[Sinh[c + d*x]])/a^2 + ((I*a - b)*(a^2 + 2*b ^2)*Log[I + Sinh[c + d*x]])/(a^2 + b^2)^2 - (2*b^5*Log[a + b*Sinh[c + d*x] ])/(a^2*(a^2 + b^2)^2) + (b*Sech[c + d*x]^2)/(a^2 + b^2) + (a*Sech[c + d*x ]*Tanh[c + d*x])/(a^2 + b^2)))/(d*(b + a*Csch[c + d*x]))
Time = 0.53 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.15, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {3042, 25, 3316, 25, 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.
| \(\displaystyle \int \frac {\text {csch}^2(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {1}{\sin (i c+i d x)^2 \cos (i c+i d x)^3 (a-i b \sin (i c+i d x))}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {1}{\cos (i c+i d x)^3 \sin (i c+i d x)^2 (a-i b \sin (i c+i d x))}dx\) |
| \(\Big \downarrow \) 3316 |
| \(\displaystyle -\frac {b^3 \int -\frac {\text {csch}^2(c+d x)}{(a+b \sinh (c+d x)) \left (\sinh ^2(c+d x) b^2+b^2\right )^2}d(b \sinh (c+d x))}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b^3 \int \frac {\text {csch}^2(c+d x)}{(a+b \sinh (c+d x)) \left (\sinh ^2(c+d x) b^2+b^2\right )^2}d(b \sinh (c+d x))}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b^5 \int \frac {\text {csch}^2(c+d x)}{b^2 (a+b \sinh (c+d x)) \left (\sinh ^2(c+d x) b^2+b^2\right )^2}d(b \sinh (c+d x))}{d}\) |
| \(\Big \downarrow \) 615 |
| \(\displaystyle \frac {b^5 \int \left (\frac {\text {csch}^2(c+d x)}{a b^6}-\frac {\text {csch}(c+d x)}{a^2 b^5}+\frac {1}{a^2 \left (a^2+b^2\right )^2 (a+b \sinh (c+d x))}-\frac {\left (a^2+2 b^2\right ) (a-b \sinh (c+d x))}{b^4 \left (a^2+b^2\right )^2 \left (\sinh ^2(c+d x) b^2+b^2\right )}+\frac {b \sinh (c+d x)-a}{b^2 \left (a^2+b^2\right ) \left (\sinh ^2(c+d x) b^2+b^2\right )^2}\right )d(b \sinh (c+d x))}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {b^5 \left (-\frac {a \left (a^2+2 b^2\right ) \arctan (\sinh (c+d x))}{b^5 \left (a^2+b^2\right )^2}-\frac {a \arctan (\sinh (c+d x))}{2 b^5 \left (a^2+b^2\right )}-\frac {\log (b \sinh (c+d x))}{a^2 b^4}+\frac {\log (a+b \sinh (c+d x))}{a^2 \left (a^2+b^2\right )^2}-\frac {a b \sinh (c+d x)+b^2}{2 b^4 \left (a^2+b^2\right ) \left (b^2 \sinh ^2(c+d x)+b^2\right )}+\frac {\left (a^2+2 b^2\right ) \log \left (b^2 \sinh ^2(c+d x)+b^2\right )}{2 b^4 \left (a^2+b^2\right )^2}-\frac {\text {csch}(c+d x)}{a b^5}\right )}{d}\) |
Input:
Int[(Csch[c + d*x]^2*Sech[c + d*x]^3)/(a + b*Sinh[c + d*x]),x]
Output:
(b^5*(-1/2*(a*ArcTan[Sinh[c + d*x]])/(b^5*(a^2 + b^2)) - (a*(a^2 + 2*b^2)* ArcTan[Sinh[c + d*x]])/(b^5*(a^2 + b^2)^2) - Csch[c + d*x]/(a*b^5) - Log[b *Sinh[c + d*x]]/(a^2*b^4) + Log[a + b*Sinh[c + d*x]]/(a^2*(a^2 + b^2)^2) + ((a^2 + 2*b^2)*Log[b^2 + b^2*Sinh[c + d*x]^2])/(2*b^4*(a^2 + b^2)^2) - (b ^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[(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 = 24.44 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.38
| method | result | size |
| derivativedivides | \(\frac {\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a}+\frac {b^{5} \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-a \right )}{\left (a^{2}+b^{2}\right )^{2} a^{2}}-\frac {2 \left (\frac {\left (-\frac {1}{2} a^{3}-\frac {1}{2} a \,b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (-a^{2} b -b^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\left (\frac {1}{2} a^{3}+\frac {1}{2} a \,b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2}}+\frac {\left (-2 a^{2} b -4 b^{3}\right ) \ln \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}{4}+\frac {\left (3 a^{3}+5 a \,b^{2}\right ) \arctan \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}\right )}{\left (a^{2}+b^{2}\right )^{2}}-\frac {1}{2 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {b \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{2}}}{d}\) | \(249\) |
| default | \(\frac {\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a}+\frac {b^{5} \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-a \right )}{\left (a^{2}+b^{2}\right )^{2} a^{2}}-\frac {2 \left (\frac {\left (-\frac {1}{2} a^{3}-\frac {1}{2} a \,b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (-a^{2} b -b^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\left (\frac {1}{2} a^{3}+\frac {1}{2} a \,b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2}}+\frac {\left (-2 a^{2} b -4 b^{3}\right ) \ln \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}{4}+\frac {\left (3 a^{3}+5 a \,b^{2}\right ) \arctan \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}\right )}{\left (a^{2}+b^{2}\right )^{2}}-\frac {1}{2 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {b \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{2}}}{d}\) | \(249\) |
| risch | \(-\frac {2 a^{2} b \,d^{2} x}{a^{4} d^{2}+2 a^{2} b^{2} d^{2}+b^{4} d^{2}}-\frac {2 a^{2} b d c}{a^{4} d^{2}+2 a^{2} b^{2} d^{2}+b^{4} d^{2}}-\frac {4 b^{3} d^{2} x}{a^{4} d^{2}+2 a^{2} b^{2} d^{2}+b^{4} d^{2}}-\frac {4 b^{3} d c}{a^{4} d^{2}+2 a^{2} b^{2} d^{2}+b^{4} d^{2}}-\frac {2 b^{5} x}{a^{2} \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {2 b^{5} c}{a^{2} d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {2 b x}{a^{2}}+\frac {2 b c}{a^{2} d}-\frac {{\mathrm e}^{d x +c} \left (3 \,{\mathrm e}^{4 d x +4 c} a^{2}+2 \,{\mathrm e}^{4 d x +4 c} b^{2}+2 \,{\mathrm e}^{3 d x +3 c} a b +2 \,{\mathrm e}^{2 d x +2 c} a^{2}+4 \,{\mathrm e}^{2 d x +2 c} b^{2}-2 b \,{\mathrm e}^{d x +c} a +3 a^{2}+2 b^{2}\right )}{d \left (a^{2}+b^{2}\right ) \left (1+{\mathrm e}^{2 d x +2 c}\right )^{2} a \left ({\mathrm e}^{2 d x +2 c}-1\right )}+\frac {3 i \ln \left ({\mathrm e}^{d x +c}-i\right ) a^{3}}{2 \left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) d}-\frac {5 i \ln \left ({\mathrm e}^{d x +c}+i\right ) a \,b^{2}}{2 \left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) d}+\frac {\ln \left ({\mathrm e}^{d x +c}-i\right ) a^{2} b}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) d}+\frac {2 \ln \left ({\mathrm e}^{d x +c}-i\right ) b^{3}}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) d}+\frac {5 i \ln \left ({\mathrm e}^{d x +c}-i\right ) a \,b^{2}}{2 \left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) d}-\frac {3 i \ln \left ({\mathrm e}^{d x +c}+i\right ) a^{3}}{2 \left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) d}+\frac {\ln \left ({\mathrm e}^{d x +c}+i\right ) a^{2} b}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) d}+\frac {2 \ln \left ({\mathrm e}^{d x +c}+i\right ) b^{3}}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) d}+\frac {b^{5} \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \,{\mathrm e}^{d x +c}}{b}-1\right )}{a^{2} d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {b \ln \left ({\mathrm e}^{2 d x +2 c}-1\right )}{a^{2} d}\) | \(717\) |
Input:
int(csch(d*x+c)^2*sech(d*x+c)^3/(a+b*sinh(d*x+c)),x,method=_RETURNVERBOSE)
Output:
1/d*(1/2/a*tanh(1/2*d*x+1/2*c)+b^5/(a^2+b^2)^2/a^2*ln(tanh(1/2*d*x+1/2*c)^ 2*a-2*b*tanh(1/2*d*x+1/2*c)-a)-2/(a^2+b^2)^2*(((-1/2*a^3-1/2*a*b^2)*tanh(1 /2*d*x+1/2*c)^3+(-a^2*b-b^3)*tanh(1/2*d*x+1/2*c)^2+(1/2*a^3+1/2*a*b^2)*tan h(1/2*d*x+1/2*c))/(1+tanh(1/2*d*x+1/2*c)^2)^2+1/4*(-2*a^2*b-4*b^3)*ln(1+ta nh(1/2*d*x+1/2*c)^2)+1/2*(3*a^3+5*a*b^2)*arctan(tanh(1/2*d*x+1/2*c)))-1/2/ a/tanh(1/2*d*x+1/2*c)-1/a^2*b*ln(tanh(1/2*d*x+1/2*c)))
Leaf count of result is larger than twice the leaf count of optimal. 2568 vs. \(2 (176) = 352\).
Time = 0.36 (sec) , antiderivative size = 2568, normalized size of antiderivative = 14.27 \[ \int \frac {\text {csch}^2(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \] Input:
integrate(csch(d*x+c)^2*sech(d*x+c)^3/(a+b*sinh(d*x+c)),x, algorithm="fric as")
Output:
-((3*a^5 + 5*a^3*b^2 + 2*a*b^4)*cosh(d*x + c)^5 + (3*a^5 + 5*a^3*b^2 + 2*a *b^4)*sinh(d*x + c)^5 + 2*(a^4*b + a^2*b^3)*cosh(d*x + c)^4 + (2*a^4*b + 2 *a^2*b^3 + 5*(3*a^5 + 5*a^3*b^2 + 2*a*b^4)*cosh(d*x + c))*sinh(d*x + c)^4 + 2*(a^5 + 3*a^3*b^2 + 2*a*b^4)*cosh(d*x + c)^3 + 2*(a^5 + 3*a^3*b^2 + 2*a *b^4 + 5*(3*a^5 + 5*a^3*b^2 + 2*a*b^4)*cosh(d*x + c)^2 + 4*(a^4*b + a^2*b^ 3)*cosh(d*x + c))*sinh(d*x + c)^3 - 2*(a^4*b + a^2*b^3)*cosh(d*x + c)^2 - 2*(a^4*b + a^2*b^3 - 5*(3*a^5 + 5*a^3*b^2 + 2*a*b^4)*cosh(d*x + c)^3 - 6*( a^4*b + a^2*b^3)*cosh(d*x + c)^2 - 3*(a^5 + 3*a^3*b^2 + 2*a*b^4)*cosh(d*x + c))*sinh(d*x + c)^2 + ((3*a^5 + 5*a^3*b^2)*cosh(d*x + c)^6 + 6*(3*a^5 + 5*a^3*b^2)*cosh(d*x + c)*sinh(d*x + c)^5 + (3*a^5 + 5*a^3*b^2)*sinh(d*x + c)^6 - 3*a^5 - 5*a^3*b^2 + (3*a^5 + 5*a^3*b^2)*cosh(d*x + c)^4 + (3*a^5 + 5*a^3*b^2 + 15*(3*a^5 + 5*a^3*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 4*(5 *(3*a^5 + 5*a^3*b^2)*cosh(d*x + c)^3 + (3*a^5 + 5*a^3*b^2)*cosh(d*x + c))* sinh(d*x + c)^3 - (3*a^5 + 5*a^3*b^2)*cosh(d*x + c)^2 - (3*a^5 + 5*a^3*b^2 - 15*(3*a^5 + 5*a^3*b^2)*cosh(d*x + c)^4 - 6*(3*a^5 + 5*a^3*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 2*(3*(3*a^5 + 5*a^3*b^2)*cosh(d*x + c)^5 + 2*(3 *a^5 + 5*a^3*b^2)*cosh(d*x + c)^3 - (3*a^5 + 5*a^3*b^2)*cosh(d*x + c))*sin h(d*x + c))*arctan(cosh(d*x + c) + sinh(d*x + c)) + (3*a^5 + 5*a^3*b^2 + 2 *a*b^4)*cosh(d*x + c) - (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*cosh(d*x + c...
Timed out. \[ \int \frac {\text {csch}^2(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \] Input:
integrate(csch(d*x+c)**2*sech(d*x+c)**3/(a+b*sinh(d*x+c)),x)
Output:
Timed out
Time = 0.13 (sec) , antiderivative size = 350, normalized size of antiderivative = 1.94 \[ \int \frac {\text {csch}^2(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {b^{5} \log \left (-2 \, a e^{\left (-d x - c\right )} + b e^{\left (-2 \, d x - 2 \, c\right )} - b\right )}{{\left (a^{6} + 2 \, a^{4} b^{2} + a^{2} b^{4}\right )} d} + \frac {{\left (3 \, a^{3} + 5 \, a b^{2}\right )} \arctan \left (e^{\left (-d x - c\right )}\right )}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d} + \frac {{\left (a^{2} b + 2 \, b^{3}\right )} \log \left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d} - \frac {2 \, a b e^{\left (-2 \, d x - 2 \, c\right )} - 2 \, a b e^{\left (-4 \, d x - 4 \, c\right )} + {\left (3 \, a^{2} + 2 \, b^{2}\right )} e^{\left (-d x - c\right )} + 2 \, {\left (a^{2} + 2 \, b^{2}\right )} e^{\left (-3 \, d x - 3 \, c\right )} + {\left (3 \, a^{2} + 2 \, b^{2}\right )} e^{\left (-5 \, d x - 5 \, c\right )}}{{\left (a^{3} + a b^{2} + {\left (a^{3} + a b^{2}\right )} e^{\left (-2 \, d x - 2 \, c\right )} - {\left (a^{3} + a b^{2}\right )} e^{\left (-4 \, d x - 4 \, c\right )} - {\left (a^{3} + a b^{2}\right )} e^{\left (-6 \, d x - 6 \, c\right )}\right )} d} - \frac {b \log \left (e^{\left (-d x - c\right )} + 1\right )}{a^{2} d} - \frac {b \log \left (e^{\left (-d x - c\right )} - 1\right )}{a^{2} d} \] Input:
integrate(csch(d*x+c)^2*sech(d*x+c)^3/(a+b*sinh(d*x+c)),x, algorithm="maxi ma")
Output:
b^5*log(-2*a*e^(-d*x - c) + b*e^(-2*d*x - 2*c) - b)/((a^6 + 2*a^4*b^2 + a^ 2*b^4)*d) + (3*a^3 + 5*a*b^2)*arctan(e^(-d*x - c))/((a^4 + 2*a^2*b^2 + b^4 )*d) + (a^2*b + 2*b^3)*log(e^(-2*d*x - 2*c) + 1)/((a^4 + 2*a^2*b^2 + b^4)* d) - (2*a*b*e^(-2*d*x - 2*c) - 2*a*b*e^(-4*d*x - 4*c) + (3*a^2 + 2*b^2)*e^ (-d*x - c) + 2*(a^2 + 2*b^2)*e^(-3*d*x - 3*c) + (3*a^2 + 2*b^2)*e^(-5*d*x - 5*c))/((a^3 + a*b^2 + (a^3 + a*b^2)*e^(-2*d*x - 2*c) - (a^3 + a*b^2)*e^( -4*d*x - 4*c) - (a^3 + a*b^2)*e^(-6*d*x - 6*c))*d) - b*log(e^(-d*x - c) + 1)/(a^2*d) - b*log(e^(-d*x - c) - 1)/(a^2*d)
Leaf count of result is larger than twice the leaf count of optimal. 458 vs. \(2 (176) = 352\).
Time = 0.13 (sec) , antiderivative size = 458, normalized size of antiderivative = 2.54 \[ \int \frac {\text {csch}^2(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {\frac {12 \, b^{6} \log \left ({\left | b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 2 \, a \right |}\right )}{a^{6} b + 2 \, a^{4} b^{3} + a^{2} b^{5}} - \frac {3 \, {\left (\pi + 2 \, \arctan \left (\frac {1}{2} \, {\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )} e^{\left (-d x - c\right )}\right )\right )} {\left (3 \, a^{3} + 5 \, a b^{2}\right )}}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {6 \, {\left (a^{2} b + 2 \, b^{3}\right )} \log \left ({\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} + 4\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} - \frac {12 \, b \log \left ({\left | e^{\left (d x + c\right )} - e^{\left (-d x - c\right )} \right |}\right )}{a^{2}} + \frac {4 \, {\left (b^{5} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} - 9 \, a^{5} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} - 15 \, a^{3} b^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} - 6 \, a b^{4} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} - 6 \, a^{4} b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} - 6 \, a^{2} b^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 4 \, b^{5} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} - 24 \, a^{5} - 48 \, a^{3} b^{2} - 24 \, a b^{4}\right )}}{{\left (a^{6} + 2 \, a^{4} b^{2} + a^{2} b^{4}\right )} {\left ({\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} + 4 \, e^{\left (d x + c\right )} - 4 \, e^{\left (-d x - c\right )}\right )}}}{12 \, d} \] Input:
integrate(csch(d*x+c)^2*sech(d*x+c)^3/(a+b*sinh(d*x+c)),x, algorithm="giac ")
Output:
1/12*(12*b^6*log(abs(b*(e^(d*x + c) - e^(-d*x - c)) + 2*a))/(a^6*b + 2*a^4 *b^3 + a^2*b^5) - 3*(pi + 2*arctan(1/2*(e^(2*d*x + 2*c) - 1)*e^(-d*x - c)) )*(3*a^3 + 5*a*b^2)/(a^4 + 2*a^2*b^2 + b^4) + 6*(a^2*b + 2*b^3)*log((e^(d* x + c) - e^(-d*x - c))^2 + 4)/(a^4 + 2*a^2*b^2 + b^4) - 12*b*log(abs(e^(d* x + c) - e^(-d*x - c)))/a^2 + 4*(b^5*(e^(d*x + c) - e^(-d*x - c))^3 - 9*a^ 5*(e^(d*x + c) - e^(-d*x - c))^2 - 15*a^3*b^2*(e^(d*x + c) - e^(-d*x - c)) ^2 - 6*a*b^4*(e^(d*x + c) - e^(-d*x - c))^2 - 6*a^4*b*(e^(d*x + c) - e^(-d *x - c)) - 6*a^2*b^3*(e^(d*x + c) - e^(-d*x - c)) + 4*b^5*(e^(d*x + c) - e ^(-d*x - c)) - 24*a^5 - 48*a^3*b^2 - 24*a*b^4)/((a^6 + 2*a^4*b^2 + a^2*b^4 )*((e^(d*x + c) - e^(-d*x - c))^3 + 4*e^(d*x + c) - 4*e^(-d*x - c))))/d
Time = 9.20 (sec) , antiderivative size = 398, normalized size of antiderivative = 2.21 \[ \int \frac {\text {csch}^2(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {2\,b}{d\,{\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}^2\,\left (a^2+b^2\right )}-\frac {b\,\ln \left ({\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}-1\right )}{a^2\,d}+\frac {2\,b\,\ln \left (1+{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,1{}\mathrm {i}\right )}{d\,{\left (-b+a\,1{}\mathrm {i}\right )}^2}-\frac {2\,b^3}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )\,{\left (a^2+b^2\right )}^2}-\frac {2\,{\mathrm {e}}^{c+d\,x}}{a\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )}+\frac {2\,b\,\ln \left ({\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c+1{}\mathrm {i}\right )}{d\,{\left (b+a\,1{}\mathrm {i}\right )}^2}-\frac {2\,a^2\,b}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )\,{\left (a^2+b^2\right )}^2}-\frac {a^3\,{\mathrm {e}}^{c+d\,x}}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )\,{\left (a^2+b^2\right )}^2}+\frac {b^5\,\ln \left (2\,a\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c-b+b\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}\right )}{a^2\,d\,{\left (a^2+b^2\right )}^2}+\frac {2\,a\,{\mathrm {e}}^{c+d\,x}}{d\,{\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}^2\,\left (a^2+b^2\right )}-\frac {a\,b^2\,{\mathrm {e}}^{c+d\,x}}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )\,{\left (a^2+b^2\right )}^2}-\frac {a\,\ln \left (1+{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,1{}\mathrm {i}\right )\,3{}\mathrm {i}}{2\,d\,{\left (-b+a\,1{}\mathrm {i}\right )}^2}+\frac {a\,\ln \left ({\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c+1{}\mathrm {i}\right )\,3{}\mathrm {i}}{2\,d\,{\left (b+a\,1{}\mathrm {i}\right )}^2} \] Input:
int(1/(cosh(c + d*x)^3*sinh(c + d*x)^2*(a + b*sinh(c + d*x))),x)
Output:
(2*b)/(d*(exp(2*c + 2*d*x) + 1)^2*(a^2 + b^2)) - (b*log(exp(2*c)*exp(2*d*x ) - 1))/(a^2*d) - (a*log(exp(d*x)*exp(c)*1i + 1)*3i)/(2*d*(a*1i - b)^2) + (2*b*log(exp(d*x)*exp(c)*1i + 1))/(d*(a*1i - b)^2) - (2*b^3)/(d*(exp(2*c + 2*d*x) + 1)*(a^2 + b^2)^2) - (2*exp(c + d*x))/(a*d*(exp(2*c + 2*d*x) - 1) ) + (a*log(exp(d*x)*exp(c) + 1i)*3i)/(2*d*(a*1i + b)^2) + (2*b*log(exp(d*x )*exp(c) + 1i))/(d*(a*1i + b)^2) - (2*a^2*b)/(d*(exp(2*c + 2*d*x) + 1)*(a^ 2 + b^2)^2) - (a^3*exp(c + d*x))/(d*(exp(2*c + 2*d*x) + 1)*(a^2 + b^2)^2) + (b^5*log(2*a*exp(d*x)*exp(c) - b + b*exp(2*c)*exp(2*d*x)))/(a^2*d*(a^2 + b^2)^2) + (2*a*exp(c + d*x))/(d*(exp(2*c + 2*d*x) + 1)^2*(a^2 + b^2)) - ( a*b^2*exp(c + d*x))/(d*(exp(2*c + 2*d*x) + 1)*(a^2 + b^2)^2)
Time = 0.20 (sec) , antiderivative size = 1469, normalized size of antiderivative = 8.16 \[ \int \frac {\text {csch}^2(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx =\text {Too large to display} \] Input:
int(csch(d*x+c)^2*sech(d*x+c)^3/(a+b*sinh(d*x+c)),x)
Output:
( - 3*e**(6*c + 6*d*x)*atan(e**(c + d*x))*a**5 - 5*e**(6*c + 6*d*x)*atan(e **(c + d*x))*a**3*b**2 - 3*e**(4*c + 4*d*x)*atan(e**(c + d*x))*a**5 - 5*e* *(4*c + 4*d*x)*atan(e**(c + d*x))*a**3*b**2 + 3*e**(2*c + 2*d*x)*atan(e**( c + d*x))*a**5 + 5*e**(2*c + 2*d*x)*atan(e**(c + d*x))*a**3*b**2 + 3*atan( e**(c + d*x))*a**5 + 5*atan(e**(c + d*x))*a**3*b**2 + e**(6*c + 6*d*x)*log (e**(2*c + 2*d*x) + 1)*a**4*b + 2*e**(6*c + 6*d*x)*log(e**(2*c + 2*d*x) + 1)*a**2*b**3 - 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**3 - e**(6*c + 6*d*x)*log(e**(c + d*x ) - 1)*b**5 - 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**3 - e**(6*c + 6*d*x)*log(e**(c + d*x) + 1)*b**5 + e**(6*c + 6*d*x)*log(e**(2*c + 2*d*x)*b + 2*e**(c + d*x)*a - b)*b**5 + 2*e**(6*c + 6*d*x)*a**4*b + 2*e**(6*c + 6*d*x)*a**2*b**3 - 3*e** (5*c + 5*d*x)*a**5 - 5*e**(5*c + 5*d*x)*a**3*b**2 - 2*e**(5*c + 5*d*x)*a*b **4 + e**(4*c + 4*d*x)*log(e**(2*c + 2*d*x) + 1)*a**4*b + 2*e**(4*c + 4*d* x)*log(e**(2*c + 2*d*x) + 1)*a**2*b**3 - 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**3 - e**(4* c + 4*d*x)*log(e**(c + d*x) - 1)*b**5 - 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**3 - e**(4*c + 4*d*x)*log(e**(c + d*x) + 1)*b**5 + e**(4*c + 4*d*x)*log(e**(2*c + 2*d* x)*b + 2*e**(c + d*x)*a - b)*b**5 - 2*e**(3*c + 3*d*x)*a**5 - 6*e**(3*c...