Integrand size = 21, antiderivative size = 110 \[ \int \frac {\text {csch}(c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^2} \, dx=-\frac {(3 a-2 b) \sqrt {b} \arctan \left (\frac {\sqrt {b} \cosh (c+d x)}{\sqrt {a-b}}\right )}{2 a^2 (a-b)^{3/2} d}-\frac {\text {arctanh}(\cosh (c+d x))}{a^2 d}-\frac {b \cosh (c+d x)}{2 a (a-b) d \left (a-b+b \cosh ^2(c+d x)\right )} \]
-arctanh(cosh(d*x+c))/a^2/d-1/2*b*cosh(d*x+c)/a/(a-b)/d/(a-b+b*cosh(d*x+c) ^2)-1/2*(3*a-2*b)*arctan(cosh(d*x+c)*b^(1/2)/(a-b)^(1/2))*b^(1/2)/a^2/(a-b )^(3/2)/d
Result contains complex when optimal does not.
Time = 1.71 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.72 \[ \int \frac {\text {csch}(c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^2} \, dx=\frac {\frac {\sqrt {b} (-3 a+2 b) \arctan \left (\frac {\sqrt {b}-i \sqrt {a} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a-b}}\right )}{(a-b)^{3/2}}+\frac {\sqrt {b} (-3 a+2 b) \arctan \left (\frac {\sqrt {b}+i \sqrt {a} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a-b}}\right )}{(a-b)^{3/2}}-\frac {2 a b \cosh (c+d x)}{(a-b) (2 a-b+b \cosh (2 (c+d x)))}-2 \log \left (\cosh \left (\frac {1}{2} (c+d x)\right )\right )+2 \log \left (\sinh \left (\frac {1}{2} (c+d x)\right )\right )}{2 a^2 d} \]
((Sqrt[b]*(-3*a + 2*b)*ArcTan[(Sqrt[b] - I*Sqrt[a]*Tanh[(c + d*x)/2])/Sqrt [a - b]])/(a - b)^(3/2) + (Sqrt[b]*(-3*a + 2*b)*ArcTan[(Sqrt[b] + I*Sqrt[a ]*Tanh[(c + d*x)/2])/Sqrt[a - b]])/(a - b)^(3/2) - (2*a*b*Cosh[c + d*x])/( (a - b)*(2*a - b + b*Cosh[2*(c + d*x)])) - 2*Log[Cosh[(c + d*x)/2]] + 2*Lo g[Sinh[(c + d*x)/2]])/(2*a^2*d)
Time = 0.31 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.12, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {3042, 26, 3665, 316, 25, 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.
| \(\displaystyle \int \frac {\text {csch}(c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {i}{\sin (i c+i d x) \left (a-b \sin (i c+i d x)^2\right )^2}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \int \frac {1}{\sin (i c+i d x) \left (a-b \sin (i c+i d x)^2\right )^2}dx\) |
| \(\Big \downarrow \) 3665 |
| \(\displaystyle -\frac {\int \frac {1}{\left (1-\cosh ^2(c+d x)\right ) \left (b \cosh ^2(c+d x)+a-b\right )^2}d\cosh (c+d x)}{d}\) |
| \(\Big \downarrow \) 316 |
| \(\displaystyle -\frac {\frac {b \cosh (c+d x)}{2 a (a-b) \left (a+b \cosh ^2(c+d x)-b\right )}-\frac {\int -\frac {-b \cosh ^2(c+d x)+2 a-b}{\left (1-\cosh ^2(c+d x)\right ) \left (b \cosh ^2(c+d x)+a-b\right )}d\cosh (c+d x)}{2 a (a-b)}}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {\int \frac {-b \cosh ^2(c+d x)+2 a-b}{\left (1-\cosh ^2(c+d x)\right ) \left (b \cosh ^2(c+d x)+a-b\right )}d\cosh (c+d x)}{2 a (a-b)}+\frac {b \cosh (c+d x)}{2 a (a-b) \left (a+b \cosh ^2(c+d x)-b\right )}}{d}\) |
| \(\Big \downarrow \) 397 |
| \(\displaystyle -\frac {\frac {\frac {2 (a-b) \int \frac {1}{1-\cosh ^2(c+d x)}d\cosh (c+d x)}{a}+\frac {b (3 a-2 b) \int \frac {1}{b \cosh ^2(c+d x)+a-b}d\cosh (c+d x)}{a}}{2 a (a-b)}+\frac {b \cosh (c+d x)}{2 a (a-b) \left (a+b \cosh ^2(c+d x)-b\right )}}{d}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle -\frac {\frac {\frac {2 (a-b) \int \frac {1}{1-\cosh ^2(c+d x)}d\cosh (c+d x)}{a}+\frac {\sqrt {b} (3 a-2 b) \arctan \left (\frac {\sqrt {b} \cosh (c+d x)}{\sqrt {a-b}}\right )}{a \sqrt {a-b}}}{2 a (a-b)}+\frac {b \cosh (c+d x)}{2 a (a-b) \left (a+b \cosh ^2(c+d x)-b\right )}}{d}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {\frac {\frac {\sqrt {b} (3 a-2 b) \arctan \left (\frac {\sqrt {b} \cosh (c+d x)}{\sqrt {a-b}}\right )}{a \sqrt {a-b}}+\frac {2 (a-b) \text {arctanh}(\cosh (c+d x))}{a}}{2 a (a-b)}+\frac {b \cosh (c+d x)}{2 a (a-b) \left (a+b \cosh ^2(c+d x)-b\right )}}{d}\) |
-(((((3*a - 2*b)*Sqrt[b]*ArcTan[(Sqrt[b]*Cosh[c + d*x])/Sqrt[a - b]])/(a*S qrt[a - b]) + (2*(a - b)*ArcTanh[Cosh[c + d*x]])/a)/(2*a*(a - b)) + (b*Cos h[c + d*x])/(2*a*(a - b)*(a - b + b*Cosh[c + d*x]^2)))/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_) + (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[((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[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^ (p_.), x_Symbol] :> With[{ff = FreeFactors[Cos[e + f*x], x]}, Simp[-ff/f Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b - b*ff^2*x^2)^p, x], x, Cos[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]
Time = 0.34 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.55
| method | result | size |
| derivativedivides | \(\frac {\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{2}}-\frac {4 b \left (\frac {-\frac {\left (a -2 b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{4 \left (a -b \right )}+\frac {a}{4 a -4 b}}{\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a -2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +4 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+a}+\frac {\left (3 a -2 b \right ) \arctan \left (\frac {2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 a +4 b}{4 \sqrt {a b -b^{2}}}\right )}{8 \left (a -b \right ) \sqrt {a b -b^{2}}}\right )}{a^{2}}}{d}\) | \(171\) |
| default | \(\frac {\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{2}}-\frac {4 b \left (\frac {-\frac {\left (a -2 b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{4 \left (a -b \right )}+\frac {a}{4 a -4 b}}{\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a -2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +4 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+a}+\frac {\left (3 a -2 b \right ) \arctan \left (\frac {2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 a +4 b}{4 \sqrt {a b -b^{2}}}\right )}{8 \left (a -b \right ) \sqrt {a b -b^{2}}}\right )}{a^{2}}}{d}\) | \(171\) |
| risch | \(-\frac {b \,{\mathrm e}^{d x +c} \left ({\mathrm e}^{2 d x +2 c}+1\right )}{a \left (a -b \right ) d \left (b \,{\mathrm e}^{4 d x +4 c}+4 a \,{\mathrm e}^{2 d x +2 c}-2 b \,{\mathrm e}^{2 d x +2 c}+b \right )}-\frac {\ln \left ({\mathrm e}^{d x +c}+1\right )}{a^{2} d}+\frac {\ln \left ({\mathrm e}^{d x +c}-1\right )}{a^{2} d}+\frac {3 \sqrt {-b \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {-b \left (a -b \right )}\, {\mathrm e}^{d x +c}}{b}+1\right )}{4 \left (a -b \right )^{2} d a}-\frac {\sqrt {-b \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {-b \left (a -b \right )}\, {\mathrm e}^{d x +c}}{b}+1\right ) b}{2 \left (a -b \right )^{2} d \,a^{2}}-\frac {3 \sqrt {-b \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {-b \left (a -b \right )}\, {\mathrm e}^{d x +c}}{b}+1\right )}{4 \left (a -b \right )^{2} d a}+\frac {\sqrt {-b \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {-b \left (a -b \right )}\, {\mathrm e}^{d x +c}}{b}+1\right ) b}{2 \left (a -b \right )^{2} d \,a^{2}}\) | \(341\) |
1/d*(1/a^2*ln(tanh(1/2*d*x+1/2*c))-4*b/a^2*((-1/4*(a-2*b)/(a-b)*tanh(1/2*d *x+1/2*c)^2+1/4*a/(a-b))/(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)+1/8*(3*a-2*b)/(a-b)/(a*b-b^2)^(1/2)*arctan( 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. 1256 vs. \(2 (98) = 196\).
Time = 0.35 (sec) , antiderivative size = 2529, normalized size of antiderivative = 22.99 \[ \int \frac {\text {csch}(c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^2} \, dx=\text {Too large to display} \]
[-1/4*(4*a*b*cosh(d*x + c)^3 + 12*a*b*cosh(d*x + c)*sinh(d*x + c)^2 + 4*a* b*sinh(d*x + c)^3 + 4*a*b*cosh(d*x + c) - ((3*a*b - 2*b^2)*cosh(d*x + c)^4 + 4*(3*a*b - 2*b^2)*cosh(d*x + c)*sinh(d*x + c)^3 + (3*a*b - 2*b^2)*sinh( d*x + c)^4 + 2*(6*a^2 - 7*a*b + 2*b^2)*cosh(d*x + c)^2 + 2*(3*(3*a*b - 2*b ^2)*cosh(d*x + c)^2 + 6*a^2 - 7*a*b + 2*b^2)*sinh(d*x + c)^2 + 3*a*b - 2*b ^2 + 4*((3*a*b - 2*b^2)*cosh(d*x + c)^3 + (6*a^2 - 7*a*b + 2*b^2)*cosh(d*x + c))*sinh(d*x + c))*sqrt(-b/(a - b))*log((b*cosh(d*x + c)^4 + 4*b*cosh(d *x + c)*sinh(d*x + c)^3 + b*sinh(d*x + c)^4 - 2*(2*a - 3*b)*cosh(d*x + c)^ 2 + 2*(3*b*cosh(d*x + c)^2 - 2*a + 3*b)*sinh(d*x + c)^2 + 4*(b*cosh(d*x + c)^3 - (2*a - 3*b)*cosh(d*x + c))*sinh(d*x + c) - 4*((a - b)*cosh(d*x + c) ^3 + 3*(a - b)*cosh(d*x + c)*sinh(d*x + c)^2 + (a - b)*sinh(d*x + c)^3 + ( a - b)*cosh(d*x + c) + (3*(a - b)*cosh(d*x + c)^2 + a - b)*sinh(d*x + c))* sqrt(-b/(a - b)) + b)/(b*cosh(d*x + c)^4 + 4*b*cosh(d*x + c)*sinh(d*x + c) ^3 + b*sinh(d*x + c)^4 + 2*(2*a - b)*cosh(d*x + c)^2 + 2*(3*b*cosh(d*x + c )^2 + 2*a - b)*sinh(d*x + c)^2 + 4*(b*cosh(d*x + c)^3 + (2*a - b)*cosh(d*x + c))*sinh(d*x + c) + b)) + 4*((a*b - b^2)*cosh(d*x + c)^4 + 4*(a*b - b^2 )*cosh(d*x + c)*sinh(d*x + c)^3 + (a*b - b^2)*sinh(d*x + c)^4 + 2*(2*a^2 - 3*a*b + b^2)*cosh(d*x + c)^2 + 2*(3*(a*b - b^2)*cosh(d*x + c)^2 + 2*a^2 - 3*a*b + b^2)*sinh(d*x + c)^2 + a*b - b^2 + 4*((a*b - b^2)*cosh(d*x + c)^3 + (2*a^2 - 3*a*b + b^2)*cosh(d*x + c))*sinh(d*x + c))*log(cosh(d*x + c...
Timed out. \[ \int \frac {\text {csch}(c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^2} \, dx=\text {Timed out} \]
\[ \int \frac {\text {csch}(c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^2} \, dx=\int { \frac {\operatorname {csch}\left (d x + c\right )}{{\left (b \sinh \left (d x + c\right )^{2} + a\right )}^{2}} \,d x } \]
-(b*e^(3*d*x + 3*c) + b*e^(d*x + c))/(a^2*b*d - a*b^2*d + (a^2*b*d*e^(4*c) - a*b^2*d*e^(4*c))*e^(4*d*x) + 2*(2*a^3*d*e^(2*c) - 3*a^2*b*d*e^(2*c) + a *b^2*d*e^(2*c))*e^(2*d*x)) - log((e^(d*x + c) + 1)*e^(-c))/(a^2*d) + log(( e^(d*x + c) - 1)*e^(-c))/(a^2*d) - 2*integrate(1/2*((3*a*b*e^(3*c) - 2*b^2 *e^(3*c))*e^(3*d*x) - (3*a*b*e^c - 2*b^2*e^c)*e^(d*x))/(a^3*b - a^2*b^2 + (a^3*b*e^(4*c) - a^2*b^2*e^(4*c))*e^(4*d*x) + 2*(2*a^4*e^(2*c) - 3*a^3*b*e ^(2*c) + a^2*b^2*e^(2*c))*e^(2*d*x)), x)
\[ \int \frac {\text {csch}(c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^2} \, dx=\int { \frac {\operatorname {csch}\left (d x + c\right )}{{\left (b \sinh \left (d x + c\right )^{2} + a\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {\text {csch}(c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^2} \, dx=\int \frac {1}{\mathrm {sinh}\left (c+d\,x\right )\,{\left (b\,{\mathrm {sinh}\left (c+d\,x\right )}^2+a\right )}^2} \,d x \]