Integrand size = 23, antiderivative size = 92 \[ \int \frac {\text {csch}^2(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=\frac {3 \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{2 (a+b)^{5/2} d}-\frac {3 \coth (c+d x)}{2 (a+b)^2 d}+\frac {\coth (c+d x)}{2 (a+b) d \left (a+b-b \tanh ^2(c+d x)\right )} \]
-3/2*coth(d*x+c)/(a+b)^2/d+3/2*arctanh(b^(1/2)*tanh(d*x+c)/(a+b)^(1/2))*b^ (1/2)/(a+b)^(5/2)/d+1/2*coth(d*x+c)/(a+b)/d/(a+b-b*tanh(d*x+c)^2)
Leaf count is larger than twice the leaf count of optimal. \(220\) vs. \(2(92)=184\).
Time = 3.87 (sec) , antiderivative size = 220, normalized size of antiderivative = 2.39 \[ \int \frac {\text {csch}^2(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=\frac {(a+2 b+a \cosh (2 (c+d x))) \text {sech}^4(c+d x) \left (\frac {3 b \text {arctanh}\left (\frac {\text {sech}(d x) (\cosh (2 c)-\sinh (2 c)) ((a+2 b) \sinh (d x)-a \sinh (2 c+d x))}{2 \sqrt {a+b} \sqrt {b (\cosh (c)-\sinh (c))^4}}\right ) (a+2 b+a \cosh (2 (c+d x))) (\cosh (2 c)-\sinh (2 c))}{\sqrt {a+b} \sqrt {b (\cosh (c)-\sinh (c))^4}}+2 (a+2 b+a \cosh (2 (c+d x))) \text {csch}(c) \text {csch}(c+d x) \sinh (d x)+b \text {sech}(2 c) \sinh (2 d x)-\frac {b (a+2 b) \tanh (2 c)}{a}\right )}{8 (a+b)^2 d \left (a+b \text {sech}^2(c+d x)\right )^2} \]
((a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d*x]^4*((3*b*ArcTanh[(Sech[d*x]* (Cosh[2*c] - Sinh[2*c])*((a + 2*b)*Sinh[d*x] - a*Sinh[2*c + d*x]))/(2*Sqrt [a + b]*Sqrt[b*(Cosh[c] - Sinh[c])^4])]*(a + 2*b + a*Cosh[2*(c + d*x)])*(C osh[2*c] - Sinh[2*c]))/(Sqrt[a + b]*Sqrt[b*(Cosh[c] - Sinh[c])^4]) + 2*(a + 2*b + a*Cosh[2*(c + d*x)])*Csch[c]*Csch[c + d*x]*Sinh[d*x] + b*Sech[2*c] *Sinh[2*d*x] - (b*(a + 2*b)*Tanh[2*c])/a))/(8*(a + b)^2*d*(a + b*Sech[c + d*x]^2)^2)
Time = 0.28 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3042, 25, 4620, 253, 264, 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.
| \(\displaystyle \int \frac {\text {csch}^2(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {1}{\sin (i c+i d x)^2 \left (a+b \sec (i c+i d x)^2\right )^2}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {1}{\left (b \sec (i c+i d x)^2+a\right )^2 \sin (i c+i d x)^2}dx\) |
| \(\Big \downarrow \) 4620 |
| \(\displaystyle \frac {\int \frac {\coth ^2(c+d x)}{\left (-b \tanh ^2(c+d x)+a+b\right )^2}d\tanh (c+d x)}{d}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {\frac {3 \int \frac {\coth ^2(c+d x)}{-b \tanh ^2(c+d x)+a+b}d\tanh (c+d x)}{2 (a+b)}+\frac {\coth (c+d x)}{2 (a+b) \left (a-b \tanh ^2(c+d x)+b\right )}}{d}\) |
| \(\Big \downarrow \) 264 |
| \(\displaystyle \frac {\frac {3 \left (\frac {b \int \frac {1}{-b \tanh ^2(c+d x)+a+b}d\tanh (c+d x)}{a+b}-\frac {\coth (c+d x)}{a+b}\right )}{2 (a+b)}+\frac {\coth (c+d x)}{2 (a+b) \left (a-b \tanh ^2(c+d x)+b\right )}}{d}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {3 \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{(a+b)^{3/2}}-\frac {\coth (c+d x)}{a+b}\right )}{2 (a+b)}+\frac {\coth (c+d x)}{2 (a+b) \left (a-b \tanh ^2(c+d x)+b\right )}}{d}\) |
((3*((Sqrt[b]*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(a + b)^(3/2) - Coth[c + d*x]/(a + b)))/(2*(a + b)) + Coth[c + d*x]/(2*(a + b)*(a + b - b*Tanh[c + d*x]^2)))/d
3.1.38.3.1 Defintions of rubi rules used
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[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(c*x )^(m + 1))*((a + b*x^2)^(p + 1)/(2*a*c*(p + 1))), x] + Simp[(m + 2*p + 3)/( 2*a*(p + 1)) Int[(c*x)^m*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, m }, x] && LtQ[p, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^( m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + 2*p + 3)/(a*c ^2*(m + 1))) Int[(c*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, p }, x] && LtQ[m, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_.)*sin[(e_.) + (f_.)*(x_ )]^(m_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[ff^(m + 1)/f Subst[Int[x^m*(ExpandToSum[a + b*(1 + ff^2*x^2)^(n/2), x]^p/(1 + f f^2*x^2)^(m/2 + 1)), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[m/2] && IntegerQ[n/2]
Leaf count of result is larger than twice the leaf count of optimal. \(237\) vs. \(2(78)=156\).
Time = 25.41 (sec) , antiderivative size = 238, normalized size of antiderivative = 2.59
| method | result | size |
| derivativedivides | \(\frac {-\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a^{2}+2 a b +b^{2}\right )}-\frac {1}{2 \left (a +b \right )^{2} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {4 b \left (\frac {-\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{4}-\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{4}}{\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} b +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a +b}+\frac {3 \ln \left (\sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{16 \sqrt {b}\, \sqrt {a +b}}-\frac {3 \ln \left (\sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{16 \sqrt {b}\, \sqrt {a +b}}\right )}{\left (a +b \right )^{2}}}{d}\) | \(238\) |
| default | \(\frac {-\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a^{2}+2 a b +b^{2}\right )}-\frac {1}{2 \left (a +b \right )^{2} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {4 b \left (\frac {-\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{4}-\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{4}}{\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} b +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a +b}+\frac {3 \ln \left (\sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{16 \sqrt {b}\, \sqrt {a +b}}-\frac {3 \ln \left (\sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{16 \sqrt {b}\, \sqrt {a +b}}\right )}{\left (a +b \right )^{2}}}{d}\) | \(238\) |
| risch | \(-\frac {2 a^{2} {\mathrm e}^{4 d x +4 c}+a b \,{\mathrm e}^{4 d x +4 c}+2 \,{\mathrm e}^{4 d x +4 c} b^{2}+4 a^{2} {\mathrm e}^{2 d x +2 c}+8 a b \,{\mathrm e}^{2 d x +2 c}-2 \,{\mathrm e}^{2 d x +2 c} b^{2}+2 a^{2}-a b}{d \left (a +b \right )^{2} \left ({\mathrm e}^{2 d x +2 c}-1\right ) a \left (a \,{\mathrm e}^{4 d x +4 c}+2 \,{\mathrm e}^{2 d x +2 c} a +4 b \,{\mathrm e}^{2 d x +2 c}+a \right )}+\frac {3 \sqrt {\left (a +b \right ) b}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {\left (a +b \right ) b}-a -2 b}{a}\right )}{4 \left (a +b \right )^{3} d}-\frac {3 \sqrt {\left (a +b \right ) b}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {\left (a +b \right ) b}+a +2 b}{a}\right )}{4 \left (a +b \right )^{3} d}\) | \(253\) |
1/d*(-1/2/(a^2+2*a*b+b^2)*tanh(1/2*d*x+1/2*c)-1/2/(a+b)^2/tanh(1/2*d*x+1/2 *c)-4*b/(a+b)^2*((-1/4*tanh(1/2*d*x+1/2*c)^3-1/4*tanh(1/2*d*x+1/2*c))/(tan h(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*t anh(1/2*d*x+1/2*c)^2*b+a+b)+3/16/b^(1/2)/(a+b)^(1/2)*ln((a+b)^(1/2)*tanh(1 /2*d*x+1/2*c)^2-2*tanh(1/2*d*x+1/2*c)*b^(1/2)+(a+b)^(1/2))-3/16/b^(1/2)/(a +b)^(1/2)*ln((a+b)^(1/2)*tanh(1/2*d*x+1/2*c)^2+2*tanh(1/2*d*x+1/2*c)*b^(1/ 2)+(a+b)^(1/2))))
Leaf count of result is larger than twice the leaf count of optimal. 1065 vs. \(2 (81) = 162\).
Time = 0.32 (sec) , antiderivative size = 2407, normalized size of antiderivative = 26.16 \[ \int \frac {\text {csch}^2(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=\text {Too large to display} \]
[-1/4*(4*(2*a^2 + a*b + 2*b^2)*cosh(d*x + c)^4 + 16*(2*a^2 + a*b + 2*b^2)* cosh(d*x + c)*sinh(d*x + c)^3 + 4*(2*a^2 + a*b + 2*b^2)*sinh(d*x + c)^4 + 8*(2*a^2 + 4*a*b - b^2)*cosh(d*x + c)^2 + 8*(3*(2*a^2 + a*b + 2*b^2)*cosh( d*x + c)^2 + 2*a^2 + 4*a*b - b^2)*sinh(d*x + c)^2 - 3*(a^2*cosh(d*x + c)^6 + 6*a^2*cosh(d*x + c)*sinh(d*x + c)^5 + a^2*sinh(d*x + c)^6 + (a^2 + 4*a* b)*cosh(d*x + c)^4 + (15*a^2*cosh(d*x + c)^2 + a^2 + 4*a*b)*sinh(d*x + c)^ 4 + 4*(5*a^2*cosh(d*x + c)^3 + (a^2 + 4*a*b)*cosh(d*x + c))*sinh(d*x + c)^ 3 - (a^2 + 4*a*b)*cosh(d*x + c)^2 + (15*a^2*cosh(d*x + c)^4 + 6*(a^2 + 4*a *b)*cosh(d*x + c)^2 - a^2 - 4*a*b)*sinh(d*x + c)^2 - a^2 + 2*(3*a^2*cosh(d *x + c)^5 + 2*(a^2 + 4*a*b)*cosh(d*x + c)^3 - (a^2 + 4*a*b)*cosh(d*x + c)) *sinh(d*x + c))*sqrt(b/(a + b))*log((a^2*cosh(d*x + c)^4 + 4*a^2*cosh(d*x + c)*sinh(d*x + c)^3 + a^2*sinh(d*x + c)^4 + 2*(a^2 + 2*a*b)*cosh(d*x + c) ^2 + 2*(3*a^2*cosh(d*x + c)^2 + a^2 + 2*a*b)*sinh(d*x + c)^2 + a^2 + 8*a*b + 8*b^2 + 4*(a^2*cosh(d*x + c)^3 + (a^2 + 2*a*b)*cosh(d*x + c))*sinh(d*x + c) - 4*((a^2 + a*b)*cosh(d*x + c)^2 + 2*(a^2 + a*b)*cosh(d*x + c)*sinh(d *x + c) + (a^2 + a*b)*sinh(d*x + c)^2 + a^2 + 3*a*b + 2*b^2)*sqrt(b/(a + b )))/(a*cosh(d*x + c)^4 + 4*a*cosh(d*x + c)*sinh(d*x + c)^3 + a*sinh(d*x + c)^4 + 2*(a + 2*b)*cosh(d*x + c)^2 + 2*(3*a*cosh(d*x + c)^2 + a + 2*b)*sin h(d*x + c)^2 + 4*(a*cosh(d*x + c)^3 + (a + 2*b)*cosh(d*x + c))*sinh(d*x + c) + a)) + 8*a^2 - 4*a*b + 16*((2*a^2 + a*b + 2*b^2)*cosh(d*x + c)^3 + ...
\[ \int \frac {\text {csch}^2(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=\int \frac {\operatorname {csch}^{2}{\left (c + d x \right )}}{\left (a + b \operatorname {sech}^{2}{\left (c + d x \right )}\right )^{2}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 262 vs. \(2 (81) = 162\).
Time = 0.30 (sec) , antiderivative size = 262, normalized size of antiderivative = 2.85 \[ \int \frac {\text {csch}^2(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=-\frac {3 \, b \log \left (\frac {a e^{\left (-2 \, d x - 2 \, c\right )} + a + 2 \, b - 2 \, \sqrt {{\left (a + b\right )} b}}{a e^{\left (-2 \, d x - 2 \, c\right )} + a + 2 \, b + 2 \, \sqrt {{\left (a + b\right )} b}}\right )}{4 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \sqrt {{\left (a + b\right )} b} d} - \frac {2 \, a^{2} - a b + 2 \, {\left (2 \, a^{2} + 4 \, a b - b^{2}\right )} e^{\left (-2 \, d x - 2 \, c\right )} + {\left (2 \, a^{2} + a b + 2 \, b^{2}\right )} e^{\left (-4 \, d x - 4 \, c\right )}}{{\left (a^{4} + 2 \, a^{3} b + a^{2} b^{2} + {\left (a^{4} + 6 \, a^{3} b + 9 \, a^{2} b^{2} + 4 \, a b^{3}\right )} e^{\left (-2 \, d x - 2 \, c\right )} - {\left (a^{4} + 6 \, a^{3} b + 9 \, a^{2} b^{2} + 4 \, a b^{3}\right )} e^{\left (-4 \, d x - 4 \, c\right )} - {\left (a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right )} e^{\left (-6 \, d x - 6 \, c\right )}\right )} d} \]
-3/4*b*log((a*e^(-2*d*x - 2*c) + a + 2*b - 2*sqrt((a + b)*b))/(a*e^(-2*d*x - 2*c) + a + 2*b + 2*sqrt((a + b)*b)))/((a^2 + 2*a*b + b^2)*sqrt((a + b)* b)*d) - (2*a^2 - a*b + 2*(2*a^2 + 4*a*b - b^2)*e^(-2*d*x - 2*c) + (2*a^2 + a*b + 2*b^2)*e^(-4*d*x - 4*c))/((a^4 + 2*a^3*b + a^2*b^2 + (a^4 + 6*a^3*b + 9*a^2*b^2 + 4*a*b^3)*e^(-2*d*x - 2*c) - (a^4 + 6*a^3*b + 9*a^2*b^2 + 4* a*b^3)*e^(-4*d*x - 4*c) - (a^4 + 2*a^3*b + a^2*b^2)*e^(-6*d*x - 6*c))*d)
\[ \int \frac {\text {csch}^2(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=\int { \frac {\operatorname {csch}\left (d x + c\right )^{2}}{{\left (b \operatorname {sech}\left (d x + c\right )^{2} + a\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {\text {csch}^2(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=\int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^4}{{\mathrm {sinh}\left (c+d\,x\right )}^2\,{\left (a\,{\mathrm {cosh}\left (c+d\,x\right )}^2+b\right )}^2} \,d x \]