Integrand size = 23, antiderivative size = 196 \[ \int \frac {\text {csch}^3(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\frac {(a+6 b) \text {arctanh}(\cosh (c+d x))}{2 a^4 d}-\frac {\sqrt {b} \left (15 a^2+40 a b+24 b^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \text {sech}(c+d x)}{\sqrt {a+b}}\right )}{8 a^4 (a+b)^{3/2} d}-\frac {\coth (c+d x) \text {csch}(c+d x)}{2 a d \left (a+b-b \text {sech}^2(c+d x)\right )^2}-\frac {3 b \text {sech}(c+d x)}{4 a^2 d \left (a+b-b \text {sech}^2(c+d x)\right )^2}-\frac {b (11 a+12 b) \text {sech}(c+d x)}{8 a^3 (a+b) d \left (a+b-b \text {sech}^2(c+d x)\right )} \]
1/2*(a+6*b)*arctanh(cosh(d*x+c))/a^4/d-1/2*coth(d*x+c)*csch(d*x+c)/a/d/(a+ b-b*sech(d*x+c)^2)^2-3/4*b*sech(d*x+c)/a^2/d/(a+b-b*sech(d*x+c)^2)^2-1/8*b *(11*a+12*b)*sech(d*x+c)/a^3/(a+b)/d/(a+b-b*sech(d*x+c)^2)-1/8*(15*a^2+40* a*b+24*b^2)*arctanh(sech(d*x+c)*b^(1/2)/(a+b)^(1/2))*b^(1/2)/a^4/(a+b)^(3/ 2)/d
Result contains complex when optimal does not.
Time = 7.61 (sec) , antiderivative size = 401, normalized size of antiderivative = 2.05 \[ \int \frac {\text {csch}^3(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=-\frac {i \sqrt {b} \left (15 a^2+40 a b+24 b^2\right ) \arctan \left (\frac {\text {sech}\left (\frac {1}{2} (c+d x)\right ) \left (-i \sqrt {a+b} \cosh \left (\frac {1}{2} (c+d x)\right )-\sqrt {a} \sinh \left (\frac {1}{2} (c+d x)\right )\right )}{\sqrt {b}}\right )}{8 a^4 (a+b)^{3/2} d}-\frac {i \sqrt {b} \left (15 a^2+40 a b+24 b^2\right ) \arctan \left (\frac {\text {sech}\left (\frac {1}{2} (c+d x)\right ) \left (-i \sqrt {a+b} \cosh \left (\frac {1}{2} (c+d x)\right )+\sqrt {a} \sinh \left (\frac {1}{2} (c+d x)\right )\right )}{\sqrt {b}}\right )}{8 a^4 (a+b)^{3/2} d}-\frac {b^2 \cosh (c+d x)}{a^2 (a+b) d (a-b+a \cosh (2 (c+d x))+b \cosh (2 (c+d x)))^2}+\frac {-9 a b \cosh (c+d x)-8 b^2 \cosh (c+d x)}{4 a^3 (a+b) d (a-b+a \cosh (2 (c+d x))+b \cosh (2 (c+d x)))}-\frac {\text {csch}^2\left (\frac {1}{2} (c+d x)\right )}{8 a^3 d}+\frac {(a+6 b) \log \left (\cosh \left (\frac {1}{2} (c+d x)\right )\right )}{2 a^4 d}+\frac {(-a-6 b) \log \left (\sinh \left (\frac {1}{2} (c+d x)\right )\right )}{2 a^4 d}-\frac {\text {sech}^2\left (\frac {1}{2} (c+d x)\right )}{8 a^3 d} \]
((-1/8*I)*Sqrt[b]*(15*a^2 + 40*a*b + 24*b^2)*ArcTan[(Sech[(c + d*x)/2]*((- I)*Sqrt[a + b]*Cosh[(c + d*x)/2] - Sqrt[a]*Sinh[(c + d*x)/2]))/Sqrt[b]])/( a^4*(a + b)^(3/2)*d) - ((I/8)*Sqrt[b]*(15*a^2 + 40*a*b + 24*b^2)*ArcTan[(S ech[(c + d*x)/2]*((-I)*Sqrt[a + b]*Cosh[(c + d*x)/2] + Sqrt[a]*Sinh[(c + d *x)/2]))/Sqrt[b]])/(a^4*(a + b)^(3/2)*d) - (b^2*Cosh[c + d*x])/(a^2*(a + b )*d*(a - b + a*Cosh[2*(c + d*x)] + b*Cosh[2*(c + d*x)])^2) + (-9*a*b*Cosh[ c + d*x] - 8*b^2*Cosh[c + d*x])/(4*a^3*(a + b)*d*(a - b + a*Cosh[2*(c + d* x)] + b*Cosh[2*(c + d*x)])) - Csch[(c + d*x)/2]^2/(8*a^3*d) + ((a + 6*b)*L og[Cosh[(c + d*x)/2]])/(2*a^4*d) + ((-a - 6*b)*Log[Sinh[(c + d*x)/2]])/(2* a^4*d) - Sech[(c + d*x)/2]^2/(8*a^3*d)
Time = 0.44 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.13, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {3042, 26, 4147, 373, 402, 27, 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.
| \(\displaystyle \int \frac {\text {csch}^3(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {i}{\sin (i c+i d x)^3 \left (a-b \tan (i c+i d x)^2\right )^3}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i \int \frac {1}{\sin (i c+i d x)^3 \left (a-b \tan (i c+i d x)^2\right )^3}dx\) |
| \(\Big \downarrow \) 4147 |
| \(\displaystyle -\frac {\int \frac {\text {sech}^2(c+d x)}{\left (1-\text {sech}^2(c+d x)\right )^2 \left (-b \text {sech}^2(c+d x)+a+b\right )^3}d\text {sech}(c+d x)}{d}\) |
| \(\Big \downarrow \) 373 |
| \(\displaystyle -\frac {\frac {\text {sech}(c+d x)}{2 a \left (1-\text {sech}^2(c+d x)\right ) \left (a-b \text {sech}^2(c+d x)+b\right )^2}-\frac {\int \frac {5 b \text {sech}^2(c+d x)+a+b}{\left (1-\text {sech}^2(c+d x)\right ) \left (-b \text {sech}^2(c+d x)+a+b\right )^3}d\text {sech}(c+d x)}{2 a}}{d}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle -\frac {\frac {\text {sech}(c+d x)}{2 a \left (1-\text {sech}^2(c+d x)\right ) \left (a-b \text {sech}^2(c+d x)+b\right )^2}-\frac {-\frac {\int -\frac {2 (a+b) \left (9 b \text {sech}^2(c+d x)+2 a+3 b\right )}{\left (1-\text {sech}^2(c+d x)\right ) \left (-b \text {sech}^2(c+d x)+a+b\right )^2}d\text {sech}(c+d x)}{4 a (a+b)}-\frac {3 b \text {sech}(c+d x)}{2 a \left (a-b \text {sech}^2(c+d x)+b\right )^2}}{2 a}}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\text {sech}(c+d x)}{2 a \left (1-\text {sech}^2(c+d x)\right ) \left (a-b \text {sech}^2(c+d x)+b\right )^2}-\frac {\frac {\int \frac {9 b \text {sech}^2(c+d x)+2 a+3 b}{\left (1-\text {sech}^2(c+d x)\right ) \left (-b \text {sech}^2(c+d x)+a+b\right )^2}d\text {sech}(c+d x)}{2 a}-\frac {3 b \text {sech}(c+d x)}{2 a \left (a-b \text {sech}^2(c+d x)+b\right )^2}}{2 a}}{d}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle -\frac {\frac {\text {sech}(c+d x)}{2 a \left (1-\text {sech}^2(c+d x)\right ) \left (a-b \text {sech}^2(c+d x)+b\right )^2}-\frac {\frac {-\frac {\int -\frac {4 a^2+17 b a+12 b^2+b (11 a+12 b) \text {sech}^2(c+d x)}{\left (1-\text {sech}^2(c+d x)\right ) \left (-b \text {sech}^2(c+d x)+a+b\right )}d\text {sech}(c+d x)}{2 a (a+b)}-\frac {b (11 a+12 b) \text {sech}(c+d x)}{2 a (a+b) \left (a-b \text {sech}^2(c+d x)+b\right )}}{2 a}-\frac {3 b \text {sech}(c+d x)}{2 a \left (a-b \text {sech}^2(c+d x)+b\right )^2}}{2 a}}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {\text {sech}(c+d x)}{2 a \left (1-\text {sech}^2(c+d x)\right ) \left (a-b \text {sech}^2(c+d x)+b\right )^2}-\frac {\frac {\frac {\int \frac {4 a^2+17 b a+12 b^2+b (11 a+12 b) \text {sech}^2(c+d x)}{\left (1-\text {sech}^2(c+d x)\right ) \left (-b \text {sech}^2(c+d x)+a+b\right )}d\text {sech}(c+d x)}{2 a (a+b)}-\frac {b (11 a+12 b) \text {sech}(c+d x)}{2 a (a+b) \left (a-b \text {sech}^2(c+d x)+b\right )}}{2 a}-\frac {3 b \text {sech}(c+d x)}{2 a \left (a-b \text {sech}^2(c+d x)+b\right )^2}}{2 a}}{d}\) |
| \(\Big \downarrow \) 397 |
| \(\displaystyle -\frac {\frac {\text {sech}(c+d x)}{2 a \left (1-\text {sech}^2(c+d x)\right ) \left (a-b \text {sech}^2(c+d x)+b\right )^2}-\frac {\frac {\frac {\frac {4 (a+b) (a+6 b) \int \frac {1}{1-\text {sech}^2(c+d x)}d\text {sech}(c+d x)}{a}-\frac {b \left (15 a^2+40 a b+24 b^2\right ) \int \frac {1}{-b \text {sech}^2(c+d x)+a+b}d\text {sech}(c+d x)}{a}}{2 a (a+b)}-\frac {b (11 a+12 b) \text {sech}(c+d x)}{2 a (a+b) \left (a-b \text {sech}^2(c+d x)+b\right )}}{2 a}-\frac {3 b \text {sech}(c+d x)}{2 a \left (a-b \text {sech}^2(c+d x)+b\right )^2}}{2 a}}{d}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {\frac {\text {sech}(c+d x)}{2 a \left (1-\text {sech}^2(c+d x)\right ) \left (a-b \text {sech}^2(c+d x)+b\right )^2}-\frac {\frac {\frac {\frac {4 (a+b) (a+6 b) \text {arctanh}(\text {sech}(c+d x))}{a}-\frac {b \left (15 a^2+40 a b+24 b^2\right ) \int \frac {1}{-b \text {sech}^2(c+d x)+a+b}d\text {sech}(c+d x)}{a}}{2 a (a+b)}-\frac {b (11 a+12 b) \text {sech}(c+d x)}{2 a (a+b) \left (a-b \text {sech}^2(c+d x)+b\right )}}{2 a}-\frac {3 b \text {sech}(c+d x)}{2 a \left (a-b \text {sech}^2(c+d x)+b\right )^2}}{2 a}}{d}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {\frac {\text {sech}(c+d x)}{2 a \left (1-\text {sech}^2(c+d x)\right ) \left (a-b \text {sech}^2(c+d x)+b\right )^2}-\frac {\frac {\frac {\frac {4 (a+b) (a+6 b) \text {arctanh}(\text {sech}(c+d x))}{a}-\frac {\sqrt {b} \left (15 a^2+40 a b+24 b^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \text {sech}(c+d x)}{\sqrt {a+b}}\right )}{a \sqrt {a+b}}}{2 a (a+b)}-\frac {b (11 a+12 b) \text {sech}(c+d x)}{2 a (a+b) \left (a-b \text {sech}^2(c+d x)+b\right )}}{2 a}-\frac {3 b \text {sech}(c+d x)}{2 a \left (a-b \text {sech}^2(c+d x)+b\right )^2}}{2 a}}{d}\) |
-((Sech[c + d*x]/(2*a*(1 - Sech[c + d*x]^2)*(a + b - b*Sech[c + d*x]^2)^2) - ((-3*b*Sech[c + d*x])/(2*a*(a + b - b*Sech[c + d*x]^2)^2) + (((4*(a + b )*(a + 6*b)*ArcTanh[Sech[c + d*x]])/a - (Sqrt[b]*(15*a^2 + 40*a*b + 24*b^2 )*ArcTanh[(Sqrt[b]*Sech[c + d*x])/Sqrt[a + b]])/(a*Sqrt[a + b]))/(2*a*(a + b)) - (b*(11*a + 12*b)*Sech[c + d*x])/(2*a*(a + b)*(a + b - b*Sech[c + d* x]^2)))/(2*a))/(2*a))/d)
3.1.47.3.1 Defintions of rubi rules used
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[(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[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ ), x_Symbol] :> Simp[e*(e*x)^(m - 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(2*(b*c - a*d)*(p + 1))), x] - Simp[e^2/(2*(b*c - a*d)*(p + 1)) Int[(e *x)^(m - 2)*(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[c*(m - 1) + d*(m + 2*p + 2*q + 3)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && GtQ[m, 1] && LeQ[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[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]
Time = 7.89 (sec) , antiderivative size = 304, normalized size of antiderivative = 1.55
| method | result | size |
| derivativedivides | \(\frac {\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{8 a^{3}}+\frac {2 b \left (\frac {-\frac {\left (9 a^{2}+32 a b +24 b^{2}\right ) a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{8 \left (a +b \right )}-\frac {\left (27 a^{3}+102 a^{2} b +152 a \,b^{2}+80 b^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{8 \left (a +b \right )}-\frac {a \left (27 a^{2}+80 a b +56 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{8 \left (a +b \right )}-\frac {a^{2} \left (9 a +10 b \right )}{8 \left (a +b \right )}}{\left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +4 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a \right )^{2}}-\frac {\left (15 a^{2}+40 a b +24 b^{2}\right ) \operatorname {arctanh}\left (\frac {2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +2 a +4 b}{4 \sqrt {a b +b^{2}}}\right )}{16 \left (a +b \right ) \sqrt {a b +b^{2}}}\right )}{a^{4}}-\frac {1}{8 a^{3} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\left (-2 a -12 b \right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{4}}}{d}\) | \(304\) |
| default | \(\frac {\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{8 a^{3}}+\frac {2 b \left (\frac {-\frac {\left (9 a^{2}+32 a b +24 b^{2}\right ) a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{8 \left (a +b \right )}-\frac {\left (27 a^{3}+102 a^{2} b +152 a \,b^{2}+80 b^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{8 \left (a +b \right )}-\frac {a \left (27 a^{2}+80 a b +56 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{8 \left (a +b \right )}-\frac {a^{2} \left (9 a +10 b \right )}{8 \left (a +b \right )}}{\left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +4 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a \right )^{2}}-\frac {\left (15 a^{2}+40 a b +24 b^{2}\right ) \operatorname {arctanh}\left (\frac {2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +2 a +4 b}{4 \sqrt {a b +b^{2}}}\right )}{16 \left (a +b \right ) \sqrt {a b +b^{2}}}\right )}{a^{4}}-\frac {1}{8 a^{3} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\left (-2 a -12 b \right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{4}}}{d}\) | \(304\) |
| risch | \(-\frac {\left (4 a^{3} {\mathrm e}^{10 d x +10 c}+21 a^{2} b \,{\mathrm e}^{10 d x +10 c}+29 a \,b^{2} {\mathrm e}^{10 d x +10 c}+12 b^{3} {\mathrm e}^{10 d x +10 c}+20 a^{3} {\mathrm e}^{8 d x +8 c}+37 a^{2} b \,{\mathrm e}^{8 d x +8 c}-15 a \,b^{2} {\mathrm e}^{8 d x +8 c}-36 b^{3} {\mathrm e}^{8 d x +8 c}+40 a^{3} {\mathrm e}^{6 d x +6 c}+6 a^{2} b \,{\mathrm e}^{6 d x +6 c}-14 a \,b^{2} {\mathrm e}^{6 d x +6 c}+24 \,{\mathrm e}^{6 d x +6 c} b^{3}+40 a^{3} {\mathrm e}^{4 d x +4 c}+6 a^{2} b \,{\mathrm e}^{4 d x +4 c}-14 a \,b^{2} {\mathrm e}^{4 d x +4 c}+24 \,{\mathrm e}^{4 d x +4 c} b^{3}+20 a^{3} {\mathrm e}^{2 d x +2 c}+37 a^{2} b \,{\mathrm e}^{2 d x +2 c}-15 \,{\mathrm e}^{2 d x +2 c} a \,b^{2}-36 \,{\mathrm e}^{2 d x +2 c} b^{3}+4 a^{3}+21 a^{2} b +29 a \,b^{2}+12 b^{3}\right ) {\mathrm e}^{d x +c}}{4 \left (a +b \right ) \left (a \,{\mathrm e}^{4 d x +4 c}+b \,{\mathrm e}^{4 d x +4 c}+2 \,{\mathrm e}^{2 d x +2 c} a -2 b \,{\mathrm e}^{2 d x +2 c}+a +b \right )^{2} d \,a^{3} \left ({\mathrm e}^{2 d x +2 c}-1\right )^{2}}+\frac {\ln \left ({\mathrm e}^{d x +c}+1\right )}{2 a^{3} d}+\frac {3 \ln \left ({\mathrm e}^{d x +c}+1\right ) b}{d \,a^{4}}-\frac {\ln \left ({\mathrm e}^{d x +c}-1\right )}{2 a^{3} d}-\frac {3 \ln \left ({\mathrm e}^{d x +c}-1\right ) b}{d \,a^{4}}+\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 )^{2} d \,a^{2}}+\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}{2 \left (a +b \right )^{2} d \,a^{3}}+\frac {3 \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 )^{2} d \,a^{4}}-\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 )^{2} d \,a^{2}}-\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}{2 \left (a +b \right )^{2} d \,a^{3}}-\frac {3 \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 )^{2} d \,a^{4}}\) | \(788\) |
1/d*(1/8*tanh(1/2*d*x+1/2*c)^2/a^3+2*b/a^4*((-1/8*(9*a^2+32*a*b+24*b^2)*a/ (a+b)*tanh(1/2*d*x+1/2*c)^6-1/8*(27*a^3+102*a^2*b+152*a*b^2+80*b^3)/(a+b)* tanh(1/2*d*x+1/2*c)^4-1/8*a*(27*a^2+80*a*b+56*b^2)/(a+b)*tanh(1/2*d*x+1/2* c)^2-1/8*a^2*(9*a+10*b)/(a+b))/(tanh(1/2*d*x+1/2*c)^4*a+2*tanh(1/2*d*x+1/2 *c)^2*a+4*tanh(1/2*d*x+1/2*c)^2*b+a)^2-1/16*(15*a^2+40*a*b+24*b^2)/(a+b)/( 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)))-1/8/a^3/tanh(1/2*d*x+1/2*c)^2+1/4/a^4*(-2*a-12*b)*ln(tanh(1/2*d*x+1 /2*c)))
Leaf count of result is larger than twice the leaf count of optimal. 11576 vs. \(2 (187) = 374\).
Time = 0.56 (sec) , antiderivative size = 21301, normalized size of antiderivative = 108.68 \[ \int \frac {\text {csch}^3(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\text {Too large to display} \]
\[ \int \frac {\text {csch}^3(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\int \frac {\operatorname {csch}^{3}{\left (c + d x \right )}}{\left (a + b \tanh ^{2}{\left (c + d x \right )}\right )^{3}}\, dx \]
\[ \int \frac {\text {csch}^3(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\int { \frac {\operatorname {csch}\left (d x + c\right )^{3}}{{\left (b \tanh \left (d x + c\right )^{2} + a\right )}^{3}} \,d x } \]
-1/4*((4*a^3*e^(11*c) + 21*a^2*b*e^(11*c) + 29*a*b^2*e^(11*c) + 12*b^3*e^( 11*c))*e^(11*d*x) + (20*a^3*e^(9*c) + 37*a^2*b*e^(9*c) - 15*a*b^2*e^(9*c) - 36*b^3*e^(9*c))*e^(9*d*x) + 2*(20*a^3*e^(7*c) + 3*a^2*b*e^(7*c) - 7*a*b^ 2*e^(7*c) + 12*b^3*e^(7*c))*e^(7*d*x) + 2*(20*a^3*e^(5*c) + 3*a^2*b*e^(5*c ) - 7*a*b^2*e^(5*c) + 12*b^3*e^(5*c))*e^(5*d*x) + (20*a^3*e^(3*c) + 37*a^2 *b*e^(3*c) - 15*a*b^2*e^(3*c) - 36*b^3*e^(3*c))*e^(3*d*x) + (4*a^3*e^c + 2 1*a^2*b*e^c + 29*a*b^2*e^c + 12*b^3*e^c)*e^(d*x))/(a^6*d + 3*a^5*b*d + 3*a ^4*b^2*d + a^3*b^3*d + (a^6*d*e^(12*c) + 3*a^5*b*d*e^(12*c) + 3*a^4*b^2*d* e^(12*c) + a^3*b^3*d*e^(12*c))*e^(12*d*x) + 2*(a^6*d*e^(10*c) - a^5*b*d*e^ (10*c) - 5*a^4*b^2*d*e^(10*c) - 3*a^3*b^3*d*e^(10*c))*e^(10*d*x) - (a^6*d* e^(8*c) + 3*a^5*b*d*e^(8*c) - 13*a^4*b^2*d*e^(8*c) - 15*a^3*b^3*d*e^(8*c)) *e^(8*d*x) - 4*(a^6*d*e^(6*c) - a^5*b*d*e^(6*c) + 3*a^4*b^2*d*e^(6*c) + 5* a^3*b^3*d*e^(6*c))*e^(6*d*x) - (a^6*d*e^(4*c) + 3*a^5*b*d*e^(4*c) - 13*a^4 *b^2*d*e^(4*c) - 15*a^3*b^3*d*e^(4*c))*e^(4*d*x) + 2*(a^6*d*e^(2*c) - a^5* b*d*e^(2*c) - 5*a^4*b^2*d*e^(2*c) - 3*a^3*b^3*d*e^(2*c))*e^(2*d*x)) + 1/2* (a + 6*b)*log((e^(d*x + c) + 1)*e^(-c))/(a^4*d) - 1/2*(a + 6*b)*log((e^(d* x + c) - 1)*e^(-c))/(a^4*d) + 8*integrate(1/32*((15*a^2*b*e^(3*c) + 40*a*b ^2*e^(3*c) + 24*b^3*e^(3*c))*e^(3*d*x) - (15*a^2*b*e^c + 40*a*b^2*e^c + 24 *b^3*e^c)*e^(d*x))/(a^6 + 2*a^5*b + a^4*b^2 + (a^6*e^(4*c) + 2*a^5*b*e^(4* c) + a^4*b^2*e^(4*c))*e^(4*d*x) + 2*(a^6*e^(2*c) - a^4*b^2*e^(2*c))*e^(...
\[ \int \frac {\text {csch}^3(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\int { \frac {\operatorname {csch}\left (d x + c\right )^{3}}{{\left (b \tanh \left (d x + c\right )^{2} + a\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {\text {csch}^3(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\int \frac {1}{{\mathrm {sinh}\left (c+d\,x\right )}^3\,{\left (b\,{\mathrm {tanh}\left (c+d\,x\right )}^2+a\right )}^3} \,d x \]