Integrand size = 23, antiderivative size = 147 \[ \int \frac {\text {csch}^3(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=-\frac {(3 a-b) \sqrt {b} \arctan \left (\frac {\sqrt {a} \cosh (c+d x)}{\sqrt {b}}\right )}{2 \sqrt {a} (a+b)^3 d}+\frac {(a-3 b) \text {arctanh}(\cosh (c+d x))}{2 (a+b)^3 d}-\frac {(a-b) \cosh (c+d x)}{2 (a+b)^2 d \left (b+a \cosh ^2(c+d x)\right )}-\frac {\coth (c+d x) \text {csch}(c+d x)}{2 (a+b) d \left (b+a \cosh ^2(c+d x)\right )} \] Output:
-1/2*(3*a-b)*b^(1/2)*arctan(a^(1/2)*cosh(d*x+c)/b^(1/2))/a^(1/2)/(a+b)^3/d +1/2*(a-3*b)*arctanh(cosh(d*x+c))/(a+b)^3/d-1/2*(a-b)*cosh(d*x+c)/(a+b)^2/ d/(b+a*cosh(d*x+c)^2)-1/2*coth(d*x+c)*csch(d*x+c)/(a+b)/d/(b+a*cosh(d*x+c) ^2)
Result contains complex when optimal does not.
Time = 2.09 (sec) , antiderivative size = 462, normalized size of antiderivative = 3.14 \[ \int \frac {\text {csch}^3(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=\frac {(a+2 b+a \cosh (2 (c+d x))) \text {sech}^3(c+d x) \left (8 b (a+b)+\frac {4 \sqrt {b} (-3 a+b) \arctan \left (\frac {\left (\sqrt {a}-i \sqrt {a+b} \sqrt {(\cosh (c)-\sinh (c))^2}\right ) \sinh (c) \tanh \left (\frac {d x}{2}\right )+\cosh (c) \left (\sqrt {a}-i \sqrt {a+b} \sqrt {(\cosh (c)-\sinh (c))^2} \tanh \left (\frac {d x}{2}\right )\right )}{\sqrt {b}}\right ) (a+2 b+a \cosh (2 (c+d x))) \text {sech}(c+d x)}{\sqrt {a}}+\frac {4 \sqrt {b} (-3 a+b) \arctan \left (\frac {\left (\sqrt {a}+i \sqrt {a+b} \sqrt {(\cosh (c)-\sinh (c))^2}\right ) \sinh (c) \tanh \left (\frac {d x}{2}\right )+\cosh (c) \left (\sqrt {a}+i \sqrt {a+b} \sqrt {(\cosh (c)-\sinh (c))^2} \tanh \left (\frac {d x}{2}\right )\right )}{\sqrt {b}}\right ) (a+2 b+a \cosh (2 (c+d x))) \text {sech}(c+d x)}{\sqrt {a}}-(a+b) (a+2 b+a \cosh (2 (c+d x))) \text {csch}^2\left (\frac {1}{2} (c+d x)\right ) \text {sech}(c+d x)+4 (a-3 b) (a+2 b+a \cosh (2 (c+d x))) \log \left (\cosh \left (\frac {1}{2} (c+d x)\right )\right ) \text {sech}(c+d x)-4 (a-3 b) (a+2 b+a \cosh (2 (c+d x))) \log \left (\sinh \left (\frac {1}{2} (c+d x)\right )\right ) \text {sech}(c+d x)-(a+b) (a+2 b+a \cosh (2 (c+d x))) \text {sech}^2\left (\frac {1}{2} (c+d x)\right ) \text {sech}(c+d x)\right )}{32 (a+b)^3 d \left (a+b \text {sech}^2(c+d x)\right )^2} \] Input:
Integrate[Csch[c + d*x]^3/(a + b*Sech[c + d*x]^2)^2,x]
Output:
((a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d*x]^3*(8*b*(a + b) + (4*Sqrt[b] *(-3*a + b)*ArcTan[((Sqrt[a] - I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2])* Sinh[c]*Tanh[(d*x)/2] + Cosh[c]*(Sqrt[a] - I*Sqrt[a + b]*Sqrt[(Cosh[c] - S inh[c])^2]*Tanh[(d*x)/2]))/Sqrt[b]]*(a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d*x])/Sqrt[a] + (4*Sqrt[b]*(-3*a + b)*ArcTan[((Sqrt[a] + I*Sqrt[a + b]* Sqrt[(Cosh[c] - Sinh[c])^2])*Sinh[c]*Tanh[(d*x)/2] + Cosh[c]*(Sqrt[a] + I* Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2]*Tanh[(d*x)/2]))/Sqrt[b]]*(a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d*x])/Sqrt[a] - (a + b)*(a + 2*b + a*Cosh[ 2*(c + d*x)])*Csch[(c + d*x)/2]^2*Sech[c + d*x] + 4*(a - 3*b)*(a + 2*b + a *Cosh[2*(c + d*x)])*Log[Cosh[(c + d*x)/2]]*Sech[c + d*x] - 4*(a - 3*b)*(a + 2*b + a*Cosh[2*(c + d*x)])*Log[Sinh[(c + d*x)/2]]*Sech[c + d*x] - (a + b )*(a + 2*b + a*Cosh[2*(c + d*x)])*Sech[(c + d*x)/2]^2*Sech[c + d*x]))/(32* (a + b)^3*d*(a + b*Sech[c + d*x]^2)^2)
Time = 0.40 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.06, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {3042, 26, 4621, 372, 402, 27, 397, 218, 219}
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.
| \(\displaystyle \int \frac {\text {csch}^3(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {i}{\sin (i c+i d x)^3 \left (a+b \sec (i c+i d x)^2\right )^2}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i \int \frac {1}{\left (b \sec (i c+i d x)^2+a\right )^2 \sin (i c+i d x)^3}dx\) |
| \(\Big \downarrow \) 4621 |
| \(\displaystyle \frac {\int \frac {\cosh ^4(c+d x)}{\left (1-\cosh ^2(c+d x)\right )^2 \left (a \cosh ^2(c+d x)+b\right )^2}d\cosh (c+d x)}{d}\) |
| \(\Big \downarrow \) 372 |
| \(\displaystyle \frac {\frac {\cosh (c+d x)}{2 (a+b) \left (1-\cosh ^2(c+d x)\right ) \left (a \cosh ^2(c+d x)+b\right )}-\frac {\int \frac {b-(a-2 b) \cosh ^2(c+d x)}{\left (1-\cosh ^2(c+d x)\right ) \left (a \cosh ^2(c+d x)+b\right )^2}d\cosh (c+d x)}{2 (a+b)}}{d}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {\frac {\cosh (c+d x)}{2 (a+b) \left (1-\cosh ^2(c+d x)\right ) \left (a \cosh ^2(c+d x)+b\right )}-\frac {\frac {(a-b) \cosh (c+d x)}{(a+b) \left (a \cosh ^2(c+d x)+b\right )}-\frac {\int -\frac {2 b \left (2 b-(a-b) \cosh ^2(c+d x)\right )}{\left (1-\cosh ^2(c+d x)\right ) \left (a \cosh ^2(c+d x)+b\right )}d\cosh (c+d x)}{2 b (a+b)}}{2 (a+b)}}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\cosh (c+d x)}{2 (a+b) \left (1-\cosh ^2(c+d x)\right ) \left (a \cosh ^2(c+d x)+b\right )}-\frac {\frac {\int \frac {2 b-(a-b) \cosh ^2(c+d x)}{\left (1-\cosh ^2(c+d x)\right ) \left (a \cosh ^2(c+d x)+b\right )}d\cosh (c+d x)}{a+b}+\frac {(a-b) \cosh (c+d x)}{(a+b) \left (a \cosh ^2(c+d x)+b\right )}}{2 (a+b)}}{d}\) |
| \(\Big \downarrow \) 397 |
| \(\displaystyle \frac {\frac {\cosh (c+d x)}{2 (a+b) \left (1-\cosh ^2(c+d x)\right ) \left (a \cosh ^2(c+d x)+b\right )}-\frac {\frac {\frac {b (3 a-b) \int \frac {1}{a \cosh ^2(c+d x)+b}d\cosh (c+d x)}{a+b}-\frac {(a-3 b) \int \frac {1}{1-\cosh ^2(c+d x)}d\cosh (c+d x)}{a+b}}{a+b}+\frac {(a-b) \cosh (c+d x)}{(a+b) \left (a \cosh ^2(c+d x)+b\right )}}{2 (a+b)}}{d}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\frac {\cosh (c+d x)}{2 (a+b) \left (1-\cosh ^2(c+d x)\right ) \left (a \cosh ^2(c+d x)+b\right )}-\frac {\frac {\frac {\sqrt {b} (3 a-b) \arctan \left (\frac {\sqrt {a} \cosh (c+d x)}{\sqrt {b}}\right )}{\sqrt {a} (a+b)}-\frac {(a-3 b) \int \frac {1}{1-\cosh ^2(c+d x)}d\cosh (c+d x)}{a+b}}{a+b}+\frac {(a-b) \cosh (c+d x)}{(a+b) \left (a \cosh ^2(c+d x)+b\right )}}{2 (a+b)}}{d}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {\cosh (c+d x)}{2 (a+b) \left (1-\cosh ^2(c+d x)\right ) \left (a \cosh ^2(c+d x)+b\right )}-\frac {\frac {\frac {\sqrt {b} (3 a-b) \arctan \left (\frac {\sqrt {a} \cosh (c+d x)}{\sqrt {b}}\right )}{\sqrt {a} (a+b)}-\frac {(a-3 b) \text {arctanh}(\cosh (c+d x))}{a+b}}{a+b}+\frac {(a-b) \cosh (c+d x)}{(a+b) \left (a \cosh ^2(c+d x)+b\right )}}{2 (a+b)}}{d}\) |
Input:
Int[Csch[c + d*x]^3/(a + b*Sech[c + d*x]^2)^2,x]
Output:
(Cosh[c + d*x]/(2*(a + b)*(1 - Cosh[c + d*x]^2)*(b + a*Cosh[c + d*x]^2)) - ((((3*a - b)*Sqrt[b]*ArcTan[(Sqrt[a]*Cosh[c + d*x])/Sqrt[b]])/(Sqrt[a]*(a + b)) - ((a - 3*b)*ArcTanh[Cosh[c + d*x]])/(a + b))/(a + b) + ((a - b)*Co sh[c + d*x])/((a + b)*(b + a*Cosh[c + d*x]^2)))/(2*(a + b)))/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[((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]
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] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ ), x_Symbol] :> Simp[(-a)*e^3*(e*x)^(m - 3)*(a + b*x^2)^(p + 1)*((c + d*x^2 )^(q + 1)/(2*b*(b*c - a*d)*(p + 1))), x] + Simp[e^4/(2*b*(b*c - a*d)*(p + 1 )) Int[(e*x)^(m - 4)*(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[a*c*(m - 3) + (a*d*(m + 2*q - 1) + 2*b*c*(p + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && GtQ[m, 3] && IntBinomialQ[a , b, c, d, e, m, 2, p, q, x]
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]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*(x _)^2), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^ (q + 1)/(a*2*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*2*(b*c - a*d)*(p + 1)) Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[c*(b*e - a*f) + e*2*(b*c - a*d) *(p + 1) + d*(b*e - a*f)*(2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b , c, d, e, f, q}, x] && LtQ[p, -1]
Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_.)*sin[(e_.) + (f_.)*(x_ )]^(m_.), x_Symbol] :> With[{ff = FreeFactors[Cos[e + f*x], x]}, Simp[-ff/f Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*((b + a*(ff*x)^n)^p/(ff*x)^(n*p)), x], x, Cos[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[(m - 1)/ 2] && IntegerQ[n] && IntegerQ[p]
Time = 13.67 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.46
| method | result | size |
| derivativedivides | \(\frac {\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{8 a^{2}+16 a b +8 b^{2}}-\frac {2 b \left (\frac {\left (-\frac {a}{2}+\frac {b}{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-\frac {a}{2}-\frac {b}{2}}{\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} b +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a +b}+\frac {\left (3 a -b \right ) \arctan \left (\frac {2 \left (a +b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+2 a -2 b}{4 \sqrt {a b}}\right )}{4 \sqrt {a b}}\right )}{\left (a +b \right )^{3}}-\frac {1}{8 \left (a +b \right )^{2} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\left (-2 a +6 b \right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 \left (a +b \right )^{3}}}{d}\) | \(214\) |
| default | \(\frac {\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{8 a^{2}+16 a b +8 b^{2}}-\frac {2 b \left (\frac {\left (-\frac {a}{2}+\frac {b}{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-\frac {a}{2}-\frac {b}{2}}{\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} b +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a +b}+\frac {\left (3 a -b \right ) \arctan \left (\frac {2 \left (a +b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+2 a -2 b}{4 \sqrt {a b}}\right )}{4 \sqrt {a b}}\right )}{\left (a +b \right )^{3}}-\frac {1}{8 \left (a +b \right )^{2} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\left (-2 a +6 b \right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 \left (a +b \right )^{3}}}{d}\) | \(214\) |
| risch | \(-\frac {{\mathrm e}^{d x +c} \left ({\mathrm e}^{6 d x +6 c} a -{\mathrm e}^{6 d x +6 c} b +3 \,{\mathrm e}^{4 d x +4 c} a +5 \,{\mathrm e}^{4 d x +4 c} b +3 a \,{\mathrm e}^{2 d x +2 c}+5 b \,{\mathrm e}^{2 d x +2 c}+a -b \right )}{d \left (a +b \right )^{2} \left ({\mathrm e}^{4 d x +4 c} a +2 a \,{\mathrm e}^{2 d x +2 c}+4 b \,{\mathrm e}^{2 d x +2 c}+a \right ) \left ({\mathrm e}^{2 d x +2 c}-1\right )^{2}}+\frac {\ln \left ({\mathrm e}^{d x +c}+1\right ) a}{2 d \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}-\frac {3 \ln \left ({\mathrm e}^{d x +c}+1\right ) b}{2 d \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}-\frac {\ln \left ({\mathrm e}^{d x +c}-1\right ) a}{2 d \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}+\frac {3 \ln \left ({\mathrm e}^{d x +c}-1\right ) b}{2 d \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}+\frac {3 \sqrt {-a b}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {-a b}\, {\mathrm e}^{d x +c}}{a}+1\right )}{4 \left (a +b \right )^{3} d}-\frac {\sqrt {-a b}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {-a b}\, {\mathrm e}^{d x +c}}{a}+1\right ) b}{4 a \left (a +b \right )^{3} d}-\frac {3 \sqrt {-a b}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {-a b}\, {\mathrm e}^{d x +c}}{a}+1\right )}{4 \left (a +b \right )^{3} d}+\frac {\sqrt {-a b}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {-a b}\, {\mathrm e}^{d x +c}}{a}+1\right ) b}{4 a \left (a +b \right )^{3} d}\) | \(478\) |
Input:
int(csch(d*x+c)^3/(a+b*sech(d*x+c)^2)^2,x,method=_RETURNVERBOSE)
Output:
1/d*(1/8*tanh(1/2*d*x+1/2*c)^2/(a^2+2*a*b+b^2)-2*b/(a+b)^3*(((-1/2*a+1/2*b )*tanh(1/2*d*x+1/2*c)^2-1/2*a-1/2*b)/(tanh(1/2*d*x+1/2*c)^4*a+tanh(1/2*d*x +1/2*c)^4*b+2*tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+a+b)+1/4*( 3*a-b)/(a*b)^(1/2)*arctan(1/4*(2*(a+b)*tanh(1/2*d*x+1/2*c)^2+2*a-2*b)/(a*b )^(1/2)))-1/8/(a+b)^2/tanh(1/2*d*x+1/2*c)^2+1/4/(a+b)^3*(-2*a+6*b)*ln(tanh (1/2*d*x+1/2*c)))
Leaf count of result is larger than twice the leaf count of optimal. 3620 vs. \(2 (131) = 262\).
Time = 0.40 (sec) , antiderivative size = 6878, normalized size of antiderivative = 46.79 \[ \int \frac {\text {csch}^3(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=\text {Too large to display} \] Input:
integrate(csch(d*x+c)^3/(a+b*sech(d*x+c)^2)^2,x, algorithm="fricas")
Output:
Too large to include
\[ \int \frac {\text {csch}^3(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=\int \frac {\operatorname {csch}^{3}{\left (c + d x \right )}}{\left (a + b \operatorname {sech}^{2}{\left (c + d x \right )}\right )^{2}}\, dx \] Input:
integrate(csch(d*x+c)**3/(a+b*sech(d*x+c)**2)**2,x)
Output:
Integral(csch(c + d*x)**3/(a + b*sech(c + d*x)**2)**2, x)
\[ \int \frac {\text {csch}^3(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=\int { \frac {\operatorname {csch}\left (d x + c\right )^{3}}{{\left (b \operatorname {sech}\left (d x + c\right )^{2} + a\right )}^{2}} \,d x } \] Input:
integrate(csch(d*x+c)^3/(a+b*sech(d*x+c)^2)^2,x, algorithm="maxima")
Output:
1/2*(a - 3*b)*log((e^(d*x + c) + 1)*e^(-c))/(a^3*d + 3*a^2*b*d + 3*a*b^2*d + b^3*d) - 1/2*(a - 3*b)*log((e^(d*x + c) - 1)*e^(-c))/(a^3*d + 3*a^2*b*d + 3*a*b^2*d + b^3*d) - ((a*e^(7*c) - b*e^(7*c))*e^(7*d*x) + (3*a*e^(5*c) + 5*b*e^(5*c))*e^(5*d*x) + (3*a*e^(3*c) + 5*b*e^(3*c))*e^(3*d*x) + (a*e^c - b*e^c)*e^(d*x))/(a^3*d + 2*a^2*b*d + a*b^2*d + (a^3*d*e^(8*c) + 2*a^2*b* d*e^(8*c) + a*b^2*d*e^(8*c))*e^(8*d*x) + 4*(a^2*b*d*e^(6*c) + 2*a*b^2*d*e^ (6*c) + b^3*d*e^(6*c))*e^(6*d*x) - 2*(a^3*d*e^(4*c) + 6*a^2*b*d*e^(4*c) + 9*a*b^2*d*e^(4*c) + 4*b^3*d*e^(4*c))*e^(4*d*x) + 4*(a^2*b*d*e^(2*c) + 2*a* b^2*d*e^(2*c) + b^3*d*e^(2*c))*e^(2*d*x)) - 8*integrate(1/8*((3*a*b*e^(3*c ) - b^2*e^(3*c))*e^(3*d*x) - (3*a*b*e^c - b^2*e^c)*e^(d*x))/(a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3 + (a^4*e^(4*c) + 3*a^3*b*e^(4*c) + 3*a^2*b^2*e^(4*c) + a*b^3*e^(4*c))*e^(4*d*x) + 2*(a^4*e^(2*c) + 5*a^3*b*e^(2*c) + 9*a^2*b^2* e^(2*c) + 7*a*b^3*e^(2*c) + 2*b^4*e^(2*c))*e^(2*d*x)), x)
Exception generated. \[ \int \frac {\text {csch}^3(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(csch(d*x+c)^3/(a+b*sech(d*x+c)^2)^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
Timed out. \[ \int \frac {\text {csch}^3(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=\int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^4}{{\mathrm {sinh}\left (c+d\,x\right )}^3\,{\left (a\,{\mathrm {cosh}\left (c+d\,x\right )}^2+b\right )}^2} \,d x \] Input:
int(1/(sinh(c + d*x)^3*(a + b/cosh(c + d*x)^2)^2),x)
Output:
int(cosh(c + d*x)^4/(sinh(c + d*x)^3*(b + a*cosh(c + d*x)^2)^2), x)
Time = 0.78 (sec) , antiderivative size = 5156, normalized size of antiderivative = 35.07 \[ \int \frac {\text {csch}^3(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx =\text {Too large to display} \] Input:
int(csch(d*x+c)^3/(a+b*sech(d*x+c)^2)^2,x)
Output:
( - 6*e**(8*c + 8*d*x)*sqrt(b)*sqrt(a)*sqrt(a + b)*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2*b)*atan((e**(c + d*x)*a)/(sqrt(a)*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2*b)))*a**2 + 2*e**(8*c + 8*d*x)*sqrt(b)*sqrt(a)*sqrt(a + b)*sqrt(2*s qrt(b)*sqrt(a + b) + a + 2*b)*atan((e**(c + d*x)*a)/(sqrt(a)*sqrt(2*sqrt(b )*sqrt(a + b) + a + 2*b)))*a*b - 24*e**(6*c + 6*d*x)*sqrt(b)*sqrt(a)*sqrt( a + b)*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2*b)*atan((e**(c + d*x)*a)/(sqrt(a )*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2*b)))*a*b + 8*e**(6*c + 6*d*x)*sqrt(b) *sqrt(a)*sqrt(a + b)*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2*b)*atan((e**(c + d *x)*a)/(sqrt(a)*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2*b)))*b**2 + 12*e**(4*c + 4*d*x)*sqrt(b)*sqrt(a)*sqrt(a + b)*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2*b) *atan((e**(c + d*x)*a)/(sqrt(a)*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2*b)))*a* *2 + 44*e**(4*c + 4*d*x)*sqrt(b)*sqrt(a)*sqrt(a + b)*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2*b)*atan((e**(c + d*x)*a)/(sqrt(a)*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2*b)))*a*b - 16*e**(4*c + 4*d*x)*sqrt(b)*sqrt(a)*sqrt(a + b)*sqrt(2 *sqrt(b)*sqrt(a + b) + a + 2*b)*atan((e**(c + d*x)*a)/(sqrt(a)*sqrt(2*sqrt (b)*sqrt(a + b) + a + 2*b)))*b**2 - 24*e**(2*c + 2*d*x)*sqrt(b)*sqrt(a)*sq rt(a + b)*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2*b)*atan((e**(c + d*x)*a)/(sqr t(a)*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2*b)))*a*b + 8*e**(2*c + 2*d*x)*sqrt (b)*sqrt(a)*sqrt(a + b)*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2*b)*atan((e**(c + d*x)*a)/(sqrt(a)*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2*b)))*b**2 - 6*sqr...