Integrand size = 21, antiderivative size = 60 \[ \int \frac {\text {csch}(c+d x)}{a+b \sinh ^2(c+d x)} \, dx=-\frac {\sqrt {b} \arctan \left (\frac {\sqrt {b} \cosh (c+d x)}{\sqrt {a-b}}\right )}{a \sqrt {a-b} d}-\frac {\text {arctanh}(\cosh (c+d x))}{a d} \]
Result contains complex when optimal does not.
Time = 1.18 (sec) , antiderivative size = 135, normalized size of antiderivative = 2.25 \[ \int \frac {\text {csch}(c+d x)}{a+b \sinh ^2(c+d x)} \, dx=-\frac {\frac {\sqrt {b} \arctan \left (\frac {\sqrt {b}-i \sqrt {a} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a-b}}\right )}{\sqrt {a-b}}+\frac {\sqrt {b} \arctan \left (\frac {\sqrt {b}+i \sqrt {a} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a-b}}\right )}{\sqrt {a-b}}+\log \left (\cosh \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\sinh \left (\frac {1}{2} (c+d x)\right )\right )}{a d} \]
-(((Sqrt[b]*ArcTan[(Sqrt[b] - I*Sqrt[a]*Tanh[(c + d*x)/2])/Sqrt[a - b]])/S qrt[a - b] + (Sqrt[b]*ArcTan[(Sqrt[b] + I*Sqrt[a]*Tanh[(c + d*x)/2])/Sqrt[ a - b]])/Sqrt[a - b] + Log[Cosh[(c + d*x)/2]] - Log[Sinh[(c + d*x)/2]])/(a *d))
Time = 0.25 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.95, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3042, 26, 3665, 303, 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.
| \(\displaystyle \int \frac {\text {csch}(c+d x)}{a+b \sinh ^2(c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {i}{\sin (i c+i d x) \left (a-b \sin (i c+i d x)^2\right )}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \int \frac {1}{\sin (i c+i d x) \left (a-b \sin (i c+i d x)^2\right )}dx\) |
| \(\Big \downarrow \) 3665 |
| \(\displaystyle -\frac {\int \frac {1}{\left (1-\cosh ^2(c+d x)\right ) \left (b \cosh ^2(c+d x)+a-b\right )}d\cosh (c+d x)}{d}\) |
| \(\Big \downarrow \) 303 |
| \(\displaystyle -\frac {\frac {b \int \frac {1}{b \cosh ^2(c+d x)+a-b}d\cosh (c+d x)}{a}+\frac {\int \frac {1}{1-\cosh ^2(c+d x)}d\cosh (c+d x)}{a}}{d}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle -\frac {\frac {\int \frac {1}{1-\cosh ^2(c+d x)}d\cosh (c+d x)}{a}+\frac {\sqrt {b} \arctan \left (\frac {\sqrt {b} \cosh (c+d x)}{\sqrt {a-b}}\right )}{a \sqrt {a-b}}}{d}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {\frac {\sqrt {b} \arctan \left (\frac {\sqrt {b} \cosh (c+d x)}{\sqrt {a-b}}\right )}{a \sqrt {a-b}}+\frac {\text {arctanh}(\cosh (c+d x))}{a}}{d}\) |
-(((Sqrt[b]*ArcTan[(Sqrt[b]*Cosh[c + d*x])/Sqrt[a - b]])/(a*Sqrt[a - b]) + ArcTanh[Cosh[c + d*x]]/a)/d)
3.1.36.3.1 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[1/(((a_) + (b_.)*(x_)^2)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Simp[b/(b *c - a*d) Int[1/(a + b*x^2), x], x] - Simp[d/(b*c - a*d) Int[1/(c + d*x ^2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
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 = 0.15 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.20
| method | result | size |
| derivativedivides | \(\frac {\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {b \arctan \left (\frac {2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 a +4 b}{4 \sqrt {a b -b^{2}}}\right )}{a \sqrt {a b -b^{2}}}}{d}\) | \(72\) |
| default | \(\frac {\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {b \arctan \left (\frac {2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 a +4 b}{4 \sqrt {a b -b^{2}}}\right )}{a \sqrt {a b -b^{2}}}}{d}\) | \(72\) |
| risch | \(\frac {\ln \left ({\mathrm e}^{d x +c}-1\right )}{a d}-\frac {\ln \left ({\mathrm e}^{d x +c}+1\right )}{a d}+\frac {\sqrt {-b \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {-b \left (a -b \right )}\, {\mathrm e}^{d x +c}}{b}+1\right )}{2 \left (a -b \right ) d a}-\frac {\sqrt {-b \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {-b \left (a -b \right )}\, {\mathrm e}^{d x +c}}{b}+1\right )}{2 \left (a -b \right ) d a}\) | \(151\) |
1/d*(1/a*ln(tanh(1/2*d*x+1/2*c))-b/a/(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. 187 vs. \(2 (52) = 104\).
Time = 0.31 (sec) , antiderivative size = 586, normalized size of antiderivative = 9.77 \[ \int \frac {\text {csch}(c+d x)}{a+b \sinh ^2(c+d x)} \, dx=\left [\frac {\sqrt {-\frac {b}{a - b}} \log \left (\frac {b \cosh \left (d x + c\right )^{4} + 4 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + b \sinh \left (d x + c\right )^{4} - 2 \, {\left (2 \, a - 3 \, b\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, b \cosh \left (d x + c\right )^{2} - 2 \, a + 3 \, b\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left (b \cosh \left (d x + c\right )^{3} - {\left (2 \, a - 3 \, b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) - 4 \, {\left ({\left (a - b\right )} \cosh \left (d x + c\right )^{3} + 3 \, {\left (a - b\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + {\left (a - b\right )} \sinh \left (d x + c\right )^{3} + {\left (a - b\right )} \cosh \left (d x + c\right ) + {\left (3 \, {\left (a - b\right )} \cosh \left (d x + c\right )^{2} + a - b\right )} \sinh \left (d x + c\right )\right )} \sqrt {-\frac {b}{a - b}} + b}{b \cosh \left (d x + c\right )^{4} + 4 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + b \sinh \left (d x + c\right )^{4} + 2 \, {\left (2 \, a - b\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, b \cosh \left (d x + c\right )^{2} + 2 \, a - b\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left (b \cosh \left (d x + c\right )^{3} + {\left (2 \, a - b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + b}\right ) - 2 \, \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) + 1\right ) + 2 \, \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) - 1\right )}{2 \, a d}, -\frac {\sqrt {\frac {b}{a - b}} \arctan \left (\frac {1}{2} \, \sqrt {\frac {b}{a - b}} {\left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right )}\right ) - \sqrt {\frac {b}{a - b}} \arctan \left (\frac {{\left (b \cosh \left (d x + c\right )^{3} + 3 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + b \sinh \left (d x + c\right )^{3} + {\left (4 \, a - 3 \, b\right )} \cosh \left (d x + c\right ) + {\left (3 \, b \cosh \left (d x + c\right )^{2} + 4 \, a - 3 \, b\right )} \sinh \left (d x + c\right )\right )} \sqrt {\frac {b}{a - b}}}{2 \, b}\right ) + \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) + 1\right ) - \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) - 1\right )}{a d}\right ] \]
[1/2*(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)*c osh(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)*s inh(d*x + c)^2 + 4*(b*cosh(d*x + c)^3 + (2*a - b)*cosh(d*x + c))*sinh(d*x + c) + b)) - 2*log(cosh(d*x + c) + sinh(d*x + c) + 1) + 2*log(cosh(d*x + c ) + sinh(d*x + c) - 1))/(a*d), -(sqrt(b/(a - b))*arctan(1/2*sqrt(b/(a - b) )*(cosh(d*x + c) + sinh(d*x + c))) - sqrt(b/(a - b))*arctan(1/2*(b*cosh(d* x + c)^3 + 3*b*cosh(d*x + c)*sinh(d*x + c)^2 + b*sinh(d*x + c)^3 + (4*a - 3*b)*cosh(d*x + c) + (3*b*cosh(d*x + c)^2 + 4*a - 3*b)*sinh(d*x + c))*sqrt (b/(a - b))/b) + log(cosh(d*x + c) + sinh(d*x + c) + 1) - log(cosh(d*x + c ) + sinh(d*x + c) - 1))/(a*d)]
\[ \int \frac {\text {csch}(c+d x)}{a+b \sinh ^2(c+d x)} \, dx=\int \frac {\operatorname {csch}{\left (c + d x \right )}}{a + b \sinh ^{2}{\left (c + d x \right )}}\, dx \]
\[ \int \frac {\text {csch}(c+d x)}{a+b \sinh ^2(c+d x)} \, dx=\int { \frac {\operatorname {csch}\left (d x + c\right )}{b \sinh \left (d x + c\right )^{2} + a} \,d x } \]
-log((e^(d*x + c) + 1)*e^(-c))/(a*d) + log((e^(d*x + c) - 1)*e^(-c))/(a*d) - 2*integrate((b*e^(3*d*x + 3*c) - b*e^(d*x + c))/(a*b*e^(4*d*x + 4*c) + a*b + 2*(2*a^2*e^(2*c) - a*b*e^(2*c))*e^(2*d*x)), x)
\[ \int \frac {\text {csch}(c+d x)}{a+b \sinh ^2(c+d x)} \, dx=\int { \frac {\operatorname {csch}\left (d x + c\right )}{b \sinh \left (d x + c\right )^{2} + a} \,d x } \]
Time = 1.82 (sec) , antiderivative size = 323, normalized size of antiderivative = 5.38 \[ \int \frac {\text {csch}(c+d x)}{a+b \sinh ^2(c+d x)} \, dx=-\frac {2\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (16\,a^2\,\sqrt {-a^2\,d^2}+9\,b^2\,\sqrt {-a^2\,d^2}-24\,a\,b\,\sqrt {-a^2\,d^2}\right )}{16\,d\,a^3-24\,d\,a^2\,b+9\,d\,a\,b^2}\right )}{\sqrt {-a^2\,d^2}}-\frac {\sqrt {b}\,\left (2\,\mathrm {atan}\left (\frac {\sqrt {b}\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {a^2\,d^2\,\left (a-b\right )}}{2\,a\,d\,\left (a-b\right )}\right )+2\,\mathrm {atan}\left (\frac {4\,a^4\,d^2\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c+4\,a^2\,b^2\,d^2\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c+b\,{\mathrm {e}}^{3\,c}\,{\mathrm {e}}^{3\,d\,x}\,\sqrt {a^3\,d^2-a^2\,b\,d^2}\,\sqrt {a^2\,d^2\,\left (a-b\right )}-8\,a^3\,b\,d^2\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c+b\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {a^3\,d^2-a^2\,b\,d^2}\,\sqrt {a^2\,d^2\,\left (a-b\right )}}{\sqrt {b}\,d\,\left (2\,a\,b-2\,a^2\right )\,\sqrt {a^2\,d^2\,\left (a-b\right )}}\right )\right )}{2\,\sqrt {a^3\,d^2-a^2\,b\,d^2}} \]
- (2*atan((exp(d*x)*exp(c)*(16*a^2*(-a^2*d^2)^(1/2) + 9*b^2*(-a^2*d^2)^(1/ 2) - 24*a*b*(-a^2*d^2)^(1/2)))/(16*a^3*d + 9*a*b^2*d - 24*a^2*b*d)))/(-a^2 *d^2)^(1/2) - (b^(1/2)*(2*atan((b^(1/2)*exp(d*x)*exp(c)*(a^2*d^2*(a - b))^ (1/2))/(2*a*d*(a - b))) + 2*atan((4*a^4*d^2*exp(d*x)*exp(c) + 4*a^2*b^2*d^ 2*exp(d*x)*exp(c) + b*exp(3*c)*exp(3*d*x)*(a^3*d^2 - a^2*b*d^2)^(1/2)*(a^2 *d^2*(a - b))^(1/2) - 8*a^3*b*d^2*exp(d*x)*exp(c) + b*exp(d*x)*exp(c)*(a^3 *d^2 - a^2*b*d^2)^(1/2)*(a^2*d^2*(a - b))^(1/2))/(b^(1/2)*d*(2*a*b - 2*a^2 )*(a^2*d^2*(a - b))^(1/2)))))/(2*(a^3*d^2 - a^2*b*d^2)^(1/2))