Integrand size = 21, antiderivative size = 156 \[ \int \frac {\text {csch}(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=-\frac {\text {arctanh}(\cosh (c+d x))}{a^3 d}+\frac {\sqrt {b} \left (15 a^2+20 a b+8 b^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \text {sech}(c+d x)}{\sqrt {a+b}}\right )}{8 a^3 (a+b)^{5/2} d}+\frac {b \text {sech}(c+d x)}{4 a (a+b) d \left (a+b-b \text {sech}^2(c+d x)\right )^2}+\frac {b (7 a+4 b) \text {sech}(c+d x)}{8 a^2 (a+b)^2 d \left (a+b-b \text {sech}^2(c+d x)\right )} \] Output:
-arctanh(cosh(d*x+c))/a^3/d+1/8*b^(1/2)*(15*a^2+20*a*b+8*b^2)*arctanh(b^(1 /2)*sech(d*x+c)/(a+b)^(1/2))/a^3/(a+b)^(5/2)/d+1/4*b*sech(d*x+c)/a/(a+b)/d /(a+b-b*sech(d*x+c)^2)^2+1/8*b*(7*a+4*b)*sech(d*x+c)/a^2/(a+b)^2/d/(a+b-b* sech(d*x+c)^2)
Result contains complex when optimal does not.
Time = 1.38 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.60 \[ \int \frac {\text {csch}(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\frac {\frac {i \sqrt {b} \left (15 a^2+20 a b+8 b^2\right ) \arctan \left (\frac {-i \sqrt {a+b}-\sqrt {a} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {b}}\right )}{(a+b)^{5/2}}+\frac {i \sqrt {b} \left (15 a^2+20 a b+8 b^2\right ) \arctan \left (\frac {-i \sqrt {a+b}+\sqrt {a} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {b}}\right )}{(a+b)^{5/2}}+\frac {8 a^2 b^2 \cosh (c+d x)}{(a+b)^2 (a-b+(a+b) \cosh (2 (c+d x)))^2}+\frac {2 a b (9 a+4 b) \cosh (c+d x)}{(a+b)^2 (a-b+(a+b) \cosh (2 (c+d x)))}-8 \log \left (\cosh \left (\frac {1}{2} (c+d x)\right )\right )+8 \log \left (\sinh \left (\frac {1}{2} (c+d x)\right )\right )}{8 a^3 d} \] Input:
Integrate[Csch[c + d*x]/(a + b*Tanh[c + d*x]^2)^3,x]
Output:
((I*Sqrt[b]*(15*a^2 + 20*a*b + 8*b^2)*ArcTan[((-I)*Sqrt[a + b] - Sqrt[a]*T anh[(c + d*x)/2])/Sqrt[b]])/(a + b)^(5/2) + (I*Sqrt[b]*(15*a^2 + 20*a*b + 8*b^2)*ArcTan[((-I)*Sqrt[a + b] + Sqrt[a]*Tanh[(c + d*x)/2])/Sqrt[b]])/(a + b)^(5/2) + (8*a^2*b^2*Cosh[c + d*x])/((a + b)^2*(a - b + (a + b)*Cosh[2* (c + d*x)])^2) + (2*a*b*(9*a + 4*b)*Cosh[c + d*x])/((a + b)^2*(a - b + (a + b)*Cosh[2*(c + d*x)])) - 8*Log[Cosh[(c + d*x)/2]] + 8*Log[Sinh[(c + d*x) /2]])/(8*a^3*d)
Time = 0.43 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.13, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.524, Rules used = {3042, 26, 4147, 25, 316, 25, 402, 25, 397, 219, 221}
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.
| \(\displaystyle \int \frac {\text {csch}(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {i}{\sin (i c+i d x) \left (a-b \tan (i c+i d x)^2\right )^3}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \int \frac {1}{\sin (i c+i d x) \left (a-b \tan (i c+i d x)^2\right )^3}dx\) |
| \(\Big \downarrow \) 4147 |
| \(\displaystyle \frac {\int -\frac {1}{\left (1-\text {sech}^2(c+d x)\right ) \left (-b \text {sech}^2(c+d x)+a+b\right )^3}d\text {sech}(c+d x)}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int \frac {1}{\left (1-\text {sech}^2(c+d x)\right ) \left (-b \text {sech}^2(c+d x)+a+b\right )^3}d\text {sech}(c+d x)}{d}\) |
| \(\Big \downarrow \) 316 |
| \(\displaystyle \frac {\frac {\int -\frac {3 b \text {sech}^2(c+d x)+4 a+b}{\left (1-\text {sech}^2(c+d x)\right ) \left (-b \text {sech}^2(c+d x)+a+b\right )^2}d\text {sech}(c+d x)}{4 a (a+b)}+\frac {b \text {sech}(c+d x)}{4 a (a+b) \left (a-b \text {sech}^2(c+d x)+b\right )^2}}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {b \text {sech}(c+d x)}{4 a (a+b) \left (a-b \text {sech}^2(c+d x)+b\right )^2}-\frac {\int \frac {3 b \text {sech}^2(c+d x)+4 a+b}{\left (1-\text {sech}^2(c+d x)\right ) \left (-b \text {sech}^2(c+d x)+a+b\right )^2}d\text {sech}(c+d x)}{4 a (a+b)}}{d}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {\frac {b \text {sech}(c+d x)}{4 a (a+b) \left (a-b \text {sech}^2(c+d x)+b\right )^2}-\frac {-\frac {\int -\frac {8 a^2+9 b a+4 b^2+b (7 a+4 b) \text {sech}^2(c+d x)}{\left (1-\text {sech}^2(c+d x)\right ) \left (-b \text {sech}^2(c+d x)+a+b\right )}d\text {sech}(c+d x)}{2 a (a+b)}-\frac {b (7 a+4 b) \text {sech}(c+d x)}{2 a (a+b) \left (a-b \text {sech}^2(c+d x)+b\right )}}{4 a (a+b)}}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {b \text {sech}(c+d x)}{4 a (a+b) \left (a-b \text {sech}^2(c+d x)+b\right )^2}-\frac {\frac {\int \frac {8 a^2+9 b a+4 b^2+b (7 a+4 b) \text {sech}^2(c+d x)}{\left (1-\text {sech}^2(c+d x)\right ) \left (-b \text {sech}^2(c+d x)+a+b\right )}d\text {sech}(c+d x)}{2 a (a+b)}-\frac {b (7 a+4 b) \text {sech}(c+d x)}{2 a (a+b) \left (a-b \text {sech}^2(c+d x)+b\right )}}{4 a (a+b)}}{d}\) |
| \(\Big \downarrow \) 397 |
| \(\displaystyle \frac {\frac {b \text {sech}(c+d x)}{4 a (a+b) \left (a-b \text {sech}^2(c+d x)+b\right )^2}-\frac {\frac {\frac {8 (a+b)^2 \int \frac {1}{1-\text {sech}^2(c+d x)}d\text {sech}(c+d x)}{a}-\frac {b \left (15 a^2+20 a b+8 b^2\right ) \int \frac {1}{-b \text {sech}^2(c+d x)+a+b}d\text {sech}(c+d x)}{a}}{2 a (a+b)}-\frac {b (7 a+4 b) \text {sech}(c+d x)}{2 a (a+b) \left (a-b \text {sech}^2(c+d x)+b\right )}}{4 a (a+b)}}{d}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {b \text {sech}(c+d x)}{4 a (a+b) \left (a-b \text {sech}^2(c+d x)+b\right )^2}-\frac {\frac {\frac {8 (a+b)^2 \text {arctanh}(\text {sech}(c+d x))}{a}-\frac {b \left (15 a^2+20 a b+8 b^2\right ) \int \frac {1}{-b \text {sech}^2(c+d x)+a+b}d\text {sech}(c+d x)}{a}}{2 a (a+b)}-\frac {b (7 a+4 b) \text {sech}(c+d x)}{2 a (a+b) \left (a-b \text {sech}^2(c+d x)+b\right )}}{4 a (a+b)}}{d}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {b \text {sech}(c+d x)}{4 a (a+b) \left (a-b \text {sech}^2(c+d x)+b\right )^2}-\frac {\frac {\frac {8 (a+b)^2 \text {arctanh}(\text {sech}(c+d x))}{a}-\frac {\sqrt {b} \left (15 a^2+20 a b+8 b^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \text {sech}(c+d x)}{\sqrt {a+b}}\right )}{a \sqrt {a+b}}}{2 a (a+b)}-\frac {b (7 a+4 b) \text {sech}(c+d x)}{2 a (a+b) \left (a-b \text {sech}^2(c+d x)+b\right )}}{4 a (a+b)}}{d}\) |
Input:
Int[Csch[c + d*x]/(a + b*Tanh[c + d*x]^2)^3,x]
Output:
((b*Sech[c + d*x])/(4*a*(a + b)*(a + b - b*Sech[c + d*x]^2)^2) - (((8*(a + b)^2*ArcTanh[Sech[c + d*x]])/a - (Sqrt[b]*(15*a^2 + 20*a*b + 8*b^2)*ArcTa nh[(Sqrt[b]*Sech[c + d*x])/Sqrt[a + b]])/(a*Sqrt[a + b]))/(2*a*(a + b)) - (b*(7*a + 4*b)*Sech[c + d*x])/(2*a*(a + b)*(a + b - b*Sech[c + d*x]^2)))/( 4*a*(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_) + (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[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
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]
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[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]^2)^ (p_.), x_Symbol] :> With[{ff = FreeFactors[Sec[e + f*x], x]}, Simp[1/(f*ff^ m) Subst[Int[(-1 + ff^2*x^2)^((m - 1)/2)*((a - b + b*ff^2*x^2)^p/x^(m + 1 )), x], x, Sec[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[( m - 1)/2]
Leaf count of result is larger than twice the leaf count of optimal. \(303\) vs. \(2(142)=284\).
Time = 5.88 (sec) , antiderivative size = 304, normalized size of antiderivative = 1.95
| method | result | size |
| derivativedivides | \(\frac {-\frac {2 b \left (\frac {-\frac {\left (9 a^{2}+28 a b +16 b^{2}\right ) a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{8 \left (a^{2}+2 a b +b^{2}\right )}-\frac {3 \left (9 a^{3}+30 a^{2} b +40 b^{2} a +16 b^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{8 \left (a^{2}+2 a b +b^{2}\right )}-\frac {a \left (27 a^{2}+68 a b +32 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{8 \left (a^{2}+2 a b +b^{2}\right )}-\frac {3 a^{2} \left (3 a +2 b \right )}{8 \left (a^{2}+2 a b +b^{2}\right )}}{\left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +4 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+a \right )^{2}}-\frac {\left (15 a^{2}+20 a b +8 b^{2}\right ) \operatorname {arctanh}\left (\frac {2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +2 a +4 b}{4 \sqrt {a b +b^{2}}}\right )}{16 \left (a^{2}+2 a b +b^{2}\right ) \sqrt {a b +b^{2}}}\right )}{a^{3}}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{3}}}{d}\) | \(304\) |
| default | \(\frac {-\frac {2 b \left (\frac {-\frac {\left (9 a^{2}+28 a b +16 b^{2}\right ) a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{8 \left (a^{2}+2 a b +b^{2}\right )}-\frac {3 \left (9 a^{3}+30 a^{2} b +40 b^{2} a +16 b^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{8 \left (a^{2}+2 a b +b^{2}\right )}-\frac {a \left (27 a^{2}+68 a b +32 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{8 \left (a^{2}+2 a b +b^{2}\right )}-\frac {3 a^{2} \left (3 a +2 b \right )}{8 \left (a^{2}+2 a b +b^{2}\right )}}{\left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +4 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+a \right )^{2}}-\frac {\left (15 a^{2}+20 a b +8 b^{2}\right ) \operatorname {arctanh}\left (\frac {2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +2 a +4 b}{4 \sqrt {a b +b^{2}}}\right )}{16 \left (a^{2}+2 a b +b^{2}\right ) \sqrt {a b +b^{2}}}\right )}{a^{3}}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{3}}}{d}\) | \(304\) |
| risch | \(\frac {\left (9 \,{\mathrm e}^{6 d x +6 c} a^{2}+13 \,{\mathrm e}^{6 d x +6 c} a b +4 \,{\mathrm e}^{6 d x +6 c} b^{2}+27 \,{\mathrm e}^{4 d x +4 c} a^{2}+11 \,{\mathrm e}^{4 d x +4 c} a b -4 \,{\mathrm e}^{4 d x +4 c} b^{2}+27 \,{\mathrm e}^{2 d x +2 c} a^{2}+11 \,{\mathrm e}^{2 d x +2 c} b a -4 b^{2} {\mathrm e}^{2 d x +2 c}+9 a^{2}+13 a b +4 b^{2}\right ) b \,{\mathrm e}^{d x +c}}{4 \left (a^{2}+2 a b +b^{2}\right ) \left ({\mathrm e}^{4 d x +4 c} a +b \,{\mathrm e}^{4 d x +4 c}+2 \,{\mathrm e}^{2 d x +2 c} a -2 \,{\mathrm e}^{2 d x +2 c} b +a +b \right )^{2} d \,a^{2}}-\frac {\ln \left ({\mathrm e}^{d x +c}+1\right )}{a^{3} d}+\frac {\ln \left ({\mathrm e}^{d x +c}-1\right )}{a^{3} d}+\frac {15 \sqrt {\left (a +b \right ) b}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {\left (a +b \right ) b}\, {\mathrm e}^{d x +c}}{a +b}+1\right )}{16 \left (a +b \right )^{3} d a}+\frac {5 \sqrt {\left (a +b \right ) b}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {\left (a +b \right ) b}\, {\mathrm e}^{d x +c}}{a +b}+1\right ) b}{4 \left (a +b \right )^{3} d \,a^{2}}+\frac {\sqrt {\left (a +b \right ) b}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {\left (a +b \right ) b}\, {\mathrm e}^{d x +c}}{a +b}+1\right ) b^{2}}{2 \left (a +b \right )^{3} d \,a^{3}}-\frac {15 \sqrt {\left (a +b \right ) b}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {\left (a +b \right ) b}\, {\mathrm e}^{d x +c}}{a +b}+1\right )}{16 \left (a +b \right )^{3} d a}-\frac {5 \sqrt {\left (a +b \right ) b}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {\left (a +b \right ) b}\, {\mathrm e}^{d x +c}}{a +b}+1\right ) b}{4 \left (a +b \right )^{3} d \,a^{2}}-\frac {\sqrt {\left (a +b \right ) b}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {\left (a +b \right ) b}\, {\mathrm e}^{d x +c}}{a +b}+1\right ) b^{2}}{2 \left (a +b \right )^{3} d \,a^{3}}\) | \(572\) |
Input:
int(csch(d*x+c)/(a+tanh(d*x+c)^2*b)^3,x,method=_RETURNVERBOSE)
Output:
1/d*(-2*b/a^3*((-1/8*(9*a^2+28*a*b+16*b^2)*a/(a^2+2*a*b+b^2)*tanh(1/2*d*x+ 1/2*c)^6-3/8*(9*a^3+30*a^2*b+40*a*b^2+16*b^3)/(a^2+2*a*b+b^2)*tanh(1/2*d*x +1/2*c)^4-1/8*a*(27*a^2+68*a*b+32*b^2)/(a^2+2*a*b+b^2)*tanh(1/2*d*x+1/2*c) ^2-3/8*a^2*(3*a+2*b)/(a^2+2*a*b+b^2))/(tanh(1/2*d*x+1/2*c)^4*a+2*tanh(1/2* d*x+1/2*c)^2*a+4*b*tanh(1/2*d*x+1/2*c)^2+a)^2-1/16*(15*a^2+20*a*b+8*b^2)/( a^2+2*a*b+b^2)/(a*b+b^2)^(1/2)*arctanh(1/4*(2*tanh(1/2*d*x+1/2*c)^2*a+2*a+ 4*b)/(a*b+b^2)^(1/2)))+1/a^3*ln(tanh(1/2*d*x+1/2*c)))
Leaf count of result is larger than twice the leaf count of optimal. 5783 vs. \(2 (148) = 296\).
Time = 0.30 (sec) , antiderivative size = 10716, normalized size of antiderivative = 68.69 \[ \int \frac {\text {csch}(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\text {Too large to display} \] Input:
integrate(csch(d*x+c)/(a+b*tanh(d*x+c)^2)^3,x, algorithm="fricas")
Output:
Too large to include
\[ \int \frac {\text {csch}(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\int \frac {\operatorname {csch}{\left (c + d x \right )}}{\left (a + b \tanh ^{2}{\left (c + d x \right )}\right )^{3}}\, dx \] Input:
integrate(csch(d*x+c)/(a+b*tanh(d*x+c)**2)**3,x)
Output:
Integral(csch(c + d*x)/(a + b*tanh(c + d*x)**2)**3, x)
\[ \int \frac {\text {csch}(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\int { \frac {\operatorname {csch}\left (d x + c\right )}{{\left (b \tanh \left (d x + c\right )^{2} + a\right )}^{3}} \,d x } \] Input:
integrate(csch(d*x+c)/(a+b*tanh(d*x+c)^2)^3,x, algorithm="maxima")
Output:
1/4*((9*a^2*b*e^(7*c) + 13*a*b^2*e^(7*c) + 4*b^3*e^(7*c))*e^(7*d*x) + (27* a^2*b*e^(5*c) + 11*a*b^2*e^(5*c) - 4*b^3*e^(5*c))*e^(5*d*x) + (27*a^2*b*e^ (3*c) + 11*a*b^2*e^(3*c) - 4*b^3*e^(3*c))*e^(3*d*x) + (9*a^2*b*e^c + 13*a* b^2*e^c + 4*b^3*e^c)*e^(d*x))/(a^6*d + 4*a^5*b*d + 6*a^4*b^2*d + 4*a^3*b^3 *d + a^2*b^4*d + (a^6*d*e^(8*c) + 4*a^5*b*d*e^(8*c) + 6*a^4*b^2*d*e^(8*c) + 4*a^3*b^3*d*e^(8*c) + a^2*b^4*d*e^(8*c))*e^(8*d*x) + 4*(a^6*d*e^(6*c) + 2*a^5*b*d*e^(6*c) - 2*a^3*b^3*d*e^(6*c) - a^2*b^4*d*e^(6*c))*e^(6*d*x) + 2 *(3*a^6*d*e^(4*c) + 4*a^5*b*d*e^(4*c) + 2*a^4*b^2*d*e^(4*c) + 4*a^3*b^3*d* e^(4*c) + 3*a^2*b^4*d*e^(4*c))*e^(4*d*x) + 4*(a^6*d*e^(2*c) + 2*a^5*b*d*e^ (2*c) - 2*a^3*b^3*d*e^(2*c) - a^2*b^4*d*e^(2*c))*e^(2*d*x)) - log((e^(d*x + c) + 1)*e^(-c))/(a^3*d) + log((e^(d*x + c) - 1)*e^(-c))/(a^3*d) - 2*inte grate(1/8*((15*a^2*b*e^(3*c) + 20*a*b^2*e^(3*c) + 8*b^3*e^(3*c))*e^(3*d*x) - (15*a^2*b*e^c + 20*a*b^2*e^c + 8*b^3*e^c)*e^(d*x))/(a^6 + 3*a^5*b + 3*a ^4*b^2 + a^3*b^3 + (a^6*e^(4*c) + 3*a^5*b*e^(4*c) + 3*a^4*b^2*e^(4*c) + a^ 3*b^3*e^(4*c))*e^(4*d*x) + 2*(a^6*e^(2*c) + a^5*b*e^(2*c) - a^4*b^2*e^(2*c ) - a^3*b^3*e^(2*c))*e^(2*d*x)), x)
Exception generated. \[ \int \frac {\text {csch}(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(csch(d*x+c)/(a+b*tanh(d*x+c)^2)^3,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. \[ \int \frac {\text {csch}(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\int \frac {1}{\mathrm {sinh}\left (c+d\,x\right )\,{\left (b\,{\mathrm {tanh}\left (c+d\,x\right )}^2+a\right )}^3} \,d x \] Input:
int(1/(sinh(c + d*x)*(a + b*tanh(c + d*x)^2)^3),x)
Output:
int(1/(sinh(c + d*x)*(a + b*tanh(c + d*x)^2)^3), x)
Time = 0.38 (sec) , antiderivative size = 4756, normalized size of antiderivative = 30.49 \[ \int \frac {\text {csch}(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx =\text {Too large to display} \] Input:
int(csch(d*x+c)/(a+b*tanh(d*x+c)^2)^3,x)
Output:
( - 15*e**(8*c + 8*d*x)*sqrt(b)*sqrt(a + b)*log(e**(2*c + 2*d*x)*sqrt(a + b) + sqrt(a + b) - 2*e**(c + d*x)*sqrt(b))*a**4 - 50*e**(8*c + 8*d*x)*sqrt (b)*sqrt(a + b)*log(e**(2*c + 2*d*x)*sqrt(a + b) + sqrt(a + b) - 2*e**(c + d*x)*sqrt(b))*a**3*b - 63*e**(8*c + 8*d*x)*sqrt(b)*sqrt(a + b)*log(e**(2* c + 2*d*x)*sqrt(a + b) + sqrt(a + b) - 2*e**(c + d*x)*sqrt(b))*a**2*b**2 - 36*e**(8*c + 8*d*x)*sqrt(b)*sqrt(a + b)*log(e**(2*c + 2*d*x)*sqrt(a + b) + sqrt(a + b) - 2*e**(c + d*x)*sqrt(b))*a*b**3 - 8*e**(8*c + 8*d*x)*sqrt(b )*sqrt(a + b)*log(e**(2*c + 2*d*x)*sqrt(a + b) + sqrt(a + b) - 2*e**(c + d *x)*sqrt(b))*b**4 + 15*e**(8*c + 8*d*x)*sqrt(b)*sqrt(a + b)*log(e**(2*c + 2*d*x)*sqrt(a + b) + sqrt(a + b) + 2*e**(c + d*x)*sqrt(b))*a**4 + 50*e**(8 *c + 8*d*x)*sqrt(b)*sqrt(a + b)*log(e**(2*c + 2*d*x)*sqrt(a + b) + sqrt(a + b) + 2*e**(c + d*x)*sqrt(b))*a**3*b + 63*e**(8*c + 8*d*x)*sqrt(b)*sqrt(a + b)*log(e**(2*c + 2*d*x)*sqrt(a + b) + sqrt(a + b) + 2*e**(c + d*x)*sqrt (b))*a**2*b**2 + 36*e**(8*c + 8*d*x)*sqrt(b)*sqrt(a + b)*log(e**(2*c + 2*d *x)*sqrt(a + b) + sqrt(a + b) + 2*e**(c + d*x)*sqrt(b))*a*b**3 + 8*e**(8*c + 8*d*x)*sqrt(b)*sqrt(a + b)*log(e**(2*c + 2*d*x)*sqrt(a + b) + sqrt(a + b) + 2*e**(c + d*x)*sqrt(b))*b**4 - 60*e**(6*c + 6*d*x)*sqrt(b)*sqrt(a + b )*log(e**(2*c + 2*d*x)*sqrt(a + b) + sqrt(a + b) - 2*e**(c + d*x)*sqrt(b)) *a**4 - 80*e**(6*c + 6*d*x)*sqrt(b)*sqrt(a + b)*log(e**(2*c + 2*d*x)*sqrt( a + b) + sqrt(a + b) - 2*e**(c + d*x)*sqrt(b))*a**3*b + 28*e**(6*c + 6*...