Integrand size = 23, antiderivative size = 141 \[ \int \frac {\text {csch}^3(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx=\frac {(a+4 b) \text {arctanh}(\cosh (c+d x))}{2 a^3 d}-\frac {\sqrt {b} (3 a+4 b) \text {arctanh}\left (\frac {\sqrt {b} \text {sech}(c+d x)}{\sqrt {a+b}}\right )}{2 a^3 \sqrt {a+b} d}-\frac {\coth (c+d x) \text {csch}(c+d x)}{2 a d \left (a+b-b \text {sech}^2(c+d x)\right )}-\frac {b \text {sech}(c+d x)}{a^2 d \left (a+b-b \text {sech}^2(c+d x)\right )} \]
1/2*(a+4*b)*arctanh(cosh(d*x+c))/a^3/d-1/2*coth(d*x+c)*csch(d*x+c)/a/d/(a+ b-b*sech(d*x+c)^2)-b*sech(d*x+c)/a^2/d/(a+b-b*sech(d*x+c)^2)-1/2*(3*a+4*b) *arctanh(sech(d*x+c)*b^(1/2)/(a+b)^(1/2))*b^(1/2)/a^3/d/(a+b)^(1/2)
Result contains complex when optimal does not.
Time = 5.85 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.57 \[ \int \frac {\text {csch}^3(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx=-\frac {\frac {4 i \sqrt {b} (3 a+4 b) \arctan \left (\frac {-i \sqrt {a+b}-\sqrt {a} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {b}}\right )}{\sqrt {a+b}}+\frac {4 i \sqrt {b} (3 a+4 b) \arctan \left (\frac {-i \sqrt {a+b}+\sqrt {a} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {b}}\right )}{\sqrt {a+b}}+\frac {8 a b \cosh (c+d x)}{a-b+(a+b) \cosh (2 (c+d x))}+a \text {csch}^2\left (\frac {1}{2} (c+d x)\right )-4 (a+4 b) \log \left (\cosh \left (\frac {1}{2} (c+d x)\right )\right )+4 (a+4 b) \log \left (\sinh \left (\frac {1}{2} (c+d x)\right )\right )+a \text {sech}^2\left (\frac {1}{2} (c+d x)\right )}{8 a^3 d} \]
-1/8*(((4*I)*Sqrt[b]*(3*a + 4*b)*ArcTan[((-I)*Sqrt[a + b] - Sqrt[a]*Tanh[( c + d*x)/2])/Sqrt[b]])/Sqrt[a + b] + ((4*I)*Sqrt[b]*(3*a + 4*b)*ArcTan[((- I)*Sqrt[a + b] + Sqrt[a]*Tanh[(c + d*x)/2])/Sqrt[b]])/Sqrt[a + b] + (8*a*b *Cosh[c + d*x])/(a - b + (a + b)*Cosh[2*(c + d*x)]) + a*Csch[(c + d*x)/2]^ 2 - 4*(a + 4*b)*Log[Cosh[(c + d*x)/2]] + 4*(a + 4*b)*Log[Sinh[(c + d*x)/2] ] + a*Sech[(c + d*x)/2]^2)/(a^3*d)
Time = 0.35 (sec) , antiderivative size = 150, 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, 4147, 373, 402, 27, 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.
| \(\displaystyle \int \frac {\text {csch}^3(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {i}{\sin (i c+i d x)^3 \left (a-b \tan (i c+i d x)^2\right )^2}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i \int \frac {1}{\sin (i c+i d x)^3 \left (a-b \tan (i c+i d x)^2\right )^2}dx\) |
| \(\Big \downarrow \) 4147 |
| \(\displaystyle -\frac {\int \frac {\text {sech}^2(c+d x)}{\left (1-\text {sech}^2(c+d x)\right )^2 \left (-b \text {sech}^2(c+d x)+a+b\right )^2}d\text {sech}(c+d x)}{d}\) |
| \(\Big \downarrow \) 373 |
| \(\displaystyle -\frac {\frac {\text {sech}(c+d x)}{2 a \left (1-\text {sech}^2(c+d x)\right ) \left (a-b \text {sech}^2(c+d x)+b\right )}-\frac {\int \frac {3 b \text {sech}^2(c+d x)+a+b}{\left (1-\text {sech}^2(c+d x)\right ) \left (-b \text {sech}^2(c+d x)+a+b\right )^2}d\text {sech}(c+d x)}{2 a}}{d}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle -\frac {\frac {\text {sech}(c+d x)}{2 a \left (1-\text {sech}^2(c+d x)\right ) \left (a-b \text {sech}^2(c+d x)+b\right )}-\frac {-\frac {\int -\frac {2 (a+b) \left (2 b \text {sech}^2(c+d x)+a+2 b\right )}{\left (1-\text {sech}^2(c+d x)\right ) \left (-b \text {sech}^2(c+d x)+a+b\right )}d\text {sech}(c+d x)}{2 a (a+b)}-\frac {2 b \text {sech}(c+d x)}{a \left (a-b \text {sech}^2(c+d x)+b\right )}}{2 a}}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\text {sech}(c+d x)}{2 a \left (1-\text {sech}^2(c+d x)\right ) \left (a-b \text {sech}^2(c+d x)+b\right )}-\frac {\frac {\int \frac {2 b \text {sech}^2(c+d x)+a+2 b}{\left (1-\text {sech}^2(c+d x)\right ) \left (-b \text {sech}^2(c+d x)+a+b\right )}d\text {sech}(c+d x)}{a}-\frac {2 b \text {sech}(c+d x)}{a \left (a-b \text {sech}^2(c+d x)+b\right )}}{2 a}}{d}\) |
| \(\Big \downarrow \) 397 |
| \(\displaystyle -\frac {\frac {\text {sech}(c+d x)}{2 a \left (1-\text {sech}^2(c+d x)\right ) \left (a-b \text {sech}^2(c+d x)+b\right )}-\frac {\frac {\frac {(a+4 b) \int \frac {1}{1-\text {sech}^2(c+d x)}d\text {sech}(c+d x)}{a}-\frac {b (3 a+4 b) \int \frac {1}{-b \text {sech}^2(c+d x)+a+b}d\text {sech}(c+d x)}{a}}{a}-\frac {2 b \text {sech}(c+d x)}{a \left (a-b \text {sech}^2(c+d x)+b\right )}}{2 a}}{d}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {\frac {\text {sech}(c+d x)}{2 a \left (1-\text {sech}^2(c+d x)\right ) \left (a-b \text {sech}^2(c+d x)+b\right )}-\frac {\frac {\frac {(a+4 b) \text {arctanh}(\text {sech}(c+d x))}{a}-\frac {b (3 a+4 b) \int \frac {1}{-b \text {sech}^2(c+d x)+a+b}d\text {sech}(c+d x)}{a}}{a}-\frac {2 b \text {sech}(c+d x)}{a \left (a-b \text {sech}^2(c+d x)+b\right )}}{2 a}}{d}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {\frac {\text {sech}(c+d x)}{2 a \left (1-\text {sech}^2(c+d x)\right ) \left (a-b \text {sech}^2(c+d x)+b\right )}-\frac {\frac {\frac {(a+4 b) \text {arctanh}(\text {sech}(c+d x))}{a}-\frac {\sqrt {b} (3 a+4 b) \text {arctanh}\left (\frac {\sqrt {b} \text {sech}(c+d x)}{\sqrt {a+b}}\right )}{a \sqrt {a+b}}}{a}-\frac {2 b \text {sech}(c+d x)}{a \left (a-b \text {sech}^2(c+d x)+b\right )}}{2 a}}{d}\) |
-((Sech[c + d*x]/(2*a*(1 - Sech[c + d*x]^2)*(a + b - b*Sech[c + d*x]^2)) - ((((a + 4*b)*ArcTanh[Sech[c + d*x]])/a - (Sqrt[b]*(3*a + 4*b)*ArcTanh[(Sq rt[b]*Sech[c + d*x])/Sqrt[a + b]])/(a*Sqrt[a + b]))/a - (2*b*Sech[c + d*x] )/(a*(a + b - b*Sech[c + d*x]^2)))/(2*a))/d)
3.1.39.3.1 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 = 2.45 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.33
| method | result | size |
| derivativedivides | \(\frac {\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{8 a^{2}}-\frac {1}{8 a^{2} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\left (-2 a -8 b \right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{3}}+\frac {2 b \left (\frac {\left (-\frac {a}{2}-b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-\frac {a}{2}}{\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +4 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a}-\frac {\left (3 a +4 b \right ) \operatorname {arctanh}\left (\frac {2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +2 a +4 b}{4 \sqrt {a b +b^{2}}}\right )}{4 \sqrt {a b +b^{2}}}\right )}{a^{3}}}{d}\) | \(187\) |
| default | \(\frac {\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{8 a^{2}}-\frac {1}{8 a^{2} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\left (-2 a -8 b \right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{3}}+\frac {2 b \left (\frac {\left (-\frac {a}{2}-b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-\frac {a}{2}}{\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +4 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a}-\frac {\left (3 a +4 b \right ) \operatorname {arctanh}\left (\frac {2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +2 a +4 b}{4 \sqrt {a b +b^{2}}}\right )}{4 \sqrt {a b +b^{2}}}\right )}{a^{3}}}{d}\) | \(187\) |
| risch | \(-\frac {{\mathrm e}^{d x +c} \left (a \,{\mathrm e}^{6 d x +6 c}+2 b \,{\mathrm e}^{6 d x +6 c}+3 a \,{\mathrm e}^{4 d x +4 c}-2 b \,{\mathrm e}^{4 d x +4 c}+3 \,{\mathrm e}^{2 d x +2 c} a -2 b \,{\mathrm e}^{2 d x +2 c}+a +2 b \right )}{d \,a^{2} \left (a \,{\mathrm e}^{4 d x +4 c}+b \,{\mathrm e}^{4 d x +4 c}+2 \,{\mathrm e}^{2 d x +2 c} a -2 b \,{\mathrm e}^{2 d x +2 c}+a +b \right ) \left ({\mathrm e}^{2 d x +2 c}-1\right )^{2}}+\frac {\ln \left ({\mathrm e}^{d x +c}+1\right )}{2 a^{2} d}+\frac {2 \ln \left ({\mathrm e}^{d x +c}+1\right ) b}{d \,a^{3}}-\frac {\ln \left ({\mathrm e}^{d x +c}-1\right )}{2 a^{2} d}-\frac {2 \ln \left ({\mathrm e}^{d x +c}-1\right ) b}{d \,a^{3}}+\frac {3 \sqrt {\left (a +b \right ) b}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {\left (a +b \right ) b}\, {\mathrm e}^{d x +c}}{a +b}+1\right )}{4 \left (a +b \right ) d \,a^{2}}+\frac {\sqrt {\left (a +b \right ) b}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {\left (a +b \right ) b}\, {\mathrm e}^{d x +c}}{a +b}+1\right ) b}{\left (a +b \right ) d \,a^{3}}-\frac {3 \sqrt {\left (a +b \right ) b}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {\left (a +b \right ) b}\, {\mathrm e}^{d x +c}}{a +b}+1\right )}{4 \left (a +b \right ) d \,a^{2}}-\frac {\sqrt {\left (a +b \right ) b}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {\left (a +b \right ) b}\, {\mathrm e}^{d x +c}}{a +b}+1\right ) b}{\left (a +b \right ) d \,a^{3}}\) | \(435\) |
1/d*(1/8*tanh(1/2*d*x+1/2*c)^2/a^2-1/8/a^2/tanh(1/2*d*x+1/2*c)^2+1/4/a^3*( -2*a-8*b)*ln(tanh(1/2*d*x+1/2*c))+2*b/a^3*(((-1/2*a-b)*tanh(1/2*d*x+1/2*c) ^2-1/2*a)/(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)-1/4*(3*a+4*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))))
Leaf count of result is larger than twice the leaf count of optimal. 3368 vs. \(2 (132) = 264\).
Time = 0.35 (sec) , antiderivative size = 6335, normalized size of antiderivative = 44.93 \[ \int \frac {\text {csch}^3(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx=\text {Too large to display} \]
\[ \int \frac {\text {csch}^3(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx=\int \frac {\operatorname {csch}^{3}{\left (c + d x \right )}}{\left (a + b \tanh ^{2}{\left (c + d x \right )}\right )^{2}}\, dx \]
\[ \int \frac {\text {csch}^3(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx=\int { \frac {\operatorname {csch}\left (d x + c\right )^{3}}{{\left (b \tanh \left (d x + c\right )^{2} + a\right )}^{2}} \,d x } \]
((a*e^(7*c) + 2*b*e^(7*c))*e^(7*d*x) + (3*a*e^(5*c) - 2*b*e^(5*c))*e^(5*d* x) + (3*a*e^(3*c) - 2*b*e^(3*c))*e^(3*d*x) + (a*e^c + 2*b*e^c)*e^(d*x))/(4 *a^2*b*d*e^(6*d*x + 6*c) + 4*a^2*b*d*e^(2*d*x + 2*c) - a^3*d - a^2*b*d - ( a^3*d*e^(8*c) + a^2*b*d*e^(8*c))*e^(8*d*x) + 2*(a^3*d*e^(4*c) - 3*a^2*b*d* e^(4*c))*e^(4*d*x)) + 1/2*(a + 4*b)*log((e^(d*x + c) + 1)*e^(-c))/(a^3*d) - 1/2*(a + 4*b)*log((e^(d*x + c) - 1)*e^(-c))/(a^3*d) + 8*integrate(1/8*(( 3*a*b*e^(3*c) + 4*b^2*e^(3*c))*e^(3*d*x) - (3*a*b*e^c + 4*b^2*e^c)*e^(d*x) )/(a^4 + a^3*b + (a^4*e^(4*c) + a^3*b*e^(4*c))*e^(4*d*x) + 2*(a^4*e^(2*c) - a^3*b*e^(2*c))*e^(2*d*x)), x)
\[ \int \frac {\text {csch}^3(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx=\int { \frac {\operatorname {csch}\left (d x + c\right )^{3}}{{\left (b \tanh \left (d x + c\right )^{2} + a\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {\text {csch}^3(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx=\int \frac {1}{{\mathrm {sinh}\left (c+d\,x\right )}^3\,{\left (b\,{\mathrm {tanh}\left (c+d\,x\right )}^2+a\right )}^2} \,d x \]