Integrand size = 12, antiderivative size = 54 \[ \int \frac {1}{a+b \text {csch}(c+d x)} \, dx=\frac {x}{a}+\frac {2 b \text {arctanh}\left (\frac {a-b \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d} \] Output:
x/a+2*b*arctanh((a-b*tanh(1/2*d*x+1/2*c))/(a^2+b^2)^(1/2))/a/(a^2+b^2)^(1/ 2)/d
Time = 0.19 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.19 \[ \int \frac {1}{a+b \text {csch}(c+d x)} \, dx=\frac {\frac {c}{d}+x-\frac {2 b \arctan \left (\frac {a-b \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a^2-b^2}}\right )}{\sqrt {-a^2-b^2} d}}{a} \] Input:
Integrate[(a + b*Csch[c + d*x])^(-1),x]
Output:
(c/d + x - (2*b*ArcTan[(a - b*Tanh[(c + d*x)/2])/Sqrt[-a^2 - b^2]])/(Sqrt[ -a^2 - b^2]*d))/a
Time = 0.31 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.98, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3042, 4270, 3042, 3139, 1083, 217}
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.
\(\displaystyle \int \frac {1}{a+b \text {csch}(c+d x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{a+i b \csc (i c+i d x)}dx\) |
\(\Big \downarrow \) 4270 |
\(\displaystyle \frac {x}{a}-\frac {\int \frac {1}{\frac {a \sinh (c+d x)}{b}+1}dx}{a}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {x}{a}-\frac {\int \frac {1}{1-\frac {i a \sin (i c+i d x)}{b}}dx}{a}\) |
\(\Big \downarrow \) 3139 |
\(\displaystyle \frac {x}{a}+\frac {2 i \int \frac {1}{-\tanh ^2\left (\frac {1}{2} (c+d x)\right )+\frac {2 a \tanh \left (\frac {1}{2} (c+d x)\right )}{b}+1}d\left (i \tanh \left (\frac {1}{2} (c+d x)\right )\right )}{a d}\) |
\(\Big \downarrow \) 1083 |
\(\displaystyle \frac {x}{a}-\frac {4 i \int \frac {1}{\tanh ^2\left (\frac {1}{2} (c+d x)\right )-4 \left (\frac {a^2}{b^2}+1\right )}d\left (2 i \tanh \left (\frac {1}{2} (c+d x)\right )-\frac {2 i a}{b}\right )}{a d}\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {x}{a}-\frac {2 b \text {arctanh}\left (\frac {b \tanh \left (\frac {1}{2} (c+d x)\right )}{2 \sqrt {a^2+b^2}}\right )}{a d \sqrt {a^2+b^2}}\) |
Input:
Int[(a + b*Csch[c + d*x])^(-1),x]
Output:
x/a - (2*b*ArcTanh[(b*Tanh[(c + d*x)/2])/(2*Sqrt[a^2 + b^2])])/(a*Sqrt[a^2 + b^2]*d)
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = Fre eFactors[Tan[(c + d*x)/2], x]}, Simp[2*(e/d) Subst[Int[1/(a + 2*b*e*x + a *e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] && NeQ [a^2 - b^2, 0]
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(-1), x_Symbol] :> Simp[x/a, x] - Simp[1/a Int[1/(1 + (a/b)*Sin[c + d*x]), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0]
Time = 0.21 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.52
method | result | size |
derivativedivides | \(\frac {\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a}+\frac {2 b \,\operatorname {arctanh}\left (\frac {-2 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+2 a}{2 \sqrt {a^{2}+b^{2}}}\right )}{a \sqrt {a^{2}+b^{2}}}}{d}\) | \(82\) |
default | \(\frac {\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a}+\frac {2 b \,\operatorname {arctanh}\left (\frac {-2 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+2 a}{2 \sqrt {a^{2}+b^{2}}}\right )}{a \sqrt {a^{2}+b^{2}}}}{d}\) | \(82\) |
risch | \(\frac {x}{a}+\frac {b \ln \left ({\mathrm e}^{d x +c}+\frac {b \sqrt {a^{2}+b^{2}}+a^{2}+b^{2}}{\sqrt {a^{2}+b^{2}}\, a}\right )}{\sqrt {a^{2}+b^{2}}\, d a}-\frac {b \ln \left ({\mathrm e}^{d x +c}+\frac {b \sqrt {a^{2}+b^{2}}-a^{2}-b^{2}}{\sqrt {a^{2}+b^{2}}\, a}\right )}{\sqrt {a^{2}+b^{2}}\, d a}\) | \(124\) |
Input:
int(1/(a+csch(d*x+c)*b),x,method=_RETURNVERBOSE)
Output:
1/d*(1/a*ln(tanh(1/2*d*x+1/2*c)+1)-1/a*ln(tanh(1/2*d*x+1/2*c)-1)+2/a*b/(a^ 2+b^2)^(1/2)*arctanh(1/2*(-2*b*tanh(1/2*d*x+1/2*c)+2*a)/(a^2+b^2)^(1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 186 vs. \(2 (51) = 102\).
Time = 0.10 (sec) , antiderivative size = 186, normalized size of antiderivative = 3.44 \[ \int \frac {1}{a+b \text {csch}(c+d x)} \, dx=\frac {{\left (a^{2} + b^{2}\right )} d x + \sqrt {a^{2} + b^{2}} b \log \left (\frac {a^{2} \cosh \left (d x + c\right )^{2} + a^{2} \sinh \left (d x + c\right )^{2} + 2 \, a b \cosh \left (d x + c\right ) + a^{2} + 2 \, b^{2} + 2 \, {\left (a^{2} \cosh \left (d x + c\right ) + a b\right )} \sinh \left (d x + c\right ) + 2 \, \sqrt {a^{2} + b^{2}} {\left (a \cosh \left (d x + c\right ) + a \sinh \left (d x + c\right ) + b\right )}}{a \cosh \left (d x + c\right )^{2} + a \sinh \left (d x + c\right )^{2} + 2 \, b \cosh \left (d x + c\right ) + 2 \, {\left (a \cosh \left (d x + c\right ) + b\right )} \sinh \left (d x + c\right ) - a}\right )}{{\left (a^{3} + a b^{2}\right )} d} \] Input:
integrate(1/(a+b*csch(d*x+c)),x, algorithm="fricas")
Output:
((a^2 + b^2)*d*x + sqrt(a^2 + b^2)*b*log((a^2*cosh(d*x + c)^2 + a^2*sinh(d *x + c)^2 + 2*a*b*cosh(d*x + c) + a^2 + 2*b^2 + 2*(a^2*cosh(d*x + c) + a*b )*sinh(d*x + c) + 2*sqrt(a^2 + b^2)*(a*cosh(d*x + c) + a*sinh(d*x + c) + b ))/(a*cosh(d*x + c)^2 + a*sinh(d*x + c)^2 + 2*b*cosh(d*x + c) + 2*(a*cosh( d*x + c) + b)*sinh(d*x + c) - a)))/((a^3 + a*b^2)*d)
\[ \int \frac {1}{a+b \text {csch}(c+d x)} \, dx=\int \frac {1}{a + b \operatorname {csch}{\left (c + d x \right )}}\, dx \] Input:
integrate(1/(a+b*csch(d*x+c)),x)
Output:
Integral(1/(a + b*csch(c + d*x)), x)
Time = 0.12 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.57 \[ \int \frac {1}{a+b \text {csch}(c+d x)} \, dx=-\frac {b \log \left (\frac {a e^{\left (-d x - c\right )} - b - \sqrt {a^{2} + b^{2}}}{a e^{\left (-d x - c\right )} - b + \sqrt {a^{2} + b^{2}}}\right )}{\sqrt {a^{2} + b^{2}} a d} + \frac {d x + c}{a d} \] Input:
integrate(1/(a+b*csch(d*x+c)),x, algorithm="maxima")
Output:
-b*log((a*e^(-d*x - c) - b - sqrt(a^2 + b^2))/(a*e^(-d*x - c) - b + sqrt(a ^2 + b^2)))/(sqrt(a^2 + b^2)*a*d) + (d*x + c)/(a*d)
Time = 0.13 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.56 \[ \int \frac {1}{a+b \text {csch}(c+d x)} \, dx=-\frac {\frac {b \log \left (\frac {{\left | 2 \, a e^{\left (d x + c\right )} + 2 \, b - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, a e^{\left (d x + c\right )} + 2 \, b + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{\sqrt {a^{2} + b^{2}} a} - \frac {d x + c}{a}}{d} \] Input:
integrate(1/(a+b*csch(d*x+c)),x, algorithm="giac")
Output:
-(b*log(abs(2*a*e^(d*x + c) + 2*b - 2*sqrt(a^2 + b^2))/abs(2*a*e^(d*x + c) + 2*b + 2*sqrt(a^2 + b^2)))/(sqrt(a^2 + b^2)*a) - (d*x + c)/a)/d
Time = 0.30 (sec) , antiderivative size = 121, normalized size of antiderivative = 2.24 \[ \int \frac {1}{a+b \text {csch}(c+d x)} \, dx=\frac {x}{a}-\frac {b\,\ln \left (\frac {2\,b\,{\mathrm {e}}^{c+d\,x}}{a^2}-\frac {2\,b\,\left (a-b\,{\mathrm {e}}^{c+d\,x}\right )}{a^2\,\sqrt {a^2+b^2}}\right )}{a\,d\,\sqrt {a^2+b^2}}+\frac {b\,\ln \left (\frac {2\,b\,{\mathrm {e}}^{c+d\,x}}{a^2}+\frac {2\,b\,\left (a-b\,{\mathrm {e}}^{c+d\,x}\right )}{a^2\,\sqrt {a^2+b^2}}\right )}{a\,d\,\sqrt {a^2+b^2}} \] Input:
int(1/(a + b/sinh(c + d*x)),x)
Output:
x/a - (b*log((2*b*exp(c + d*x))/a^2 - (2*b*(a - b*exp(c + d*x)))/(a^2*(a^2 + b^2)^(1/2))))/(a*d*(a^2 + b^2)^(1/2)) + (b*log((2*b*exp(c + d*x))/a^2 + (2*b*(a - b*exp(c + d*x)))/(a^2*(a^2 + b^2)^(1/2))))/(a*d*(a^2 + b^2)^(1/ 2))
Time = 0.25 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.24 \[ \int \frac {1}{a+b \text {csch}(c+d x)} \, dx=\frac {-2 \sqrt {a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{d x +c} a i +b i}{\sqrt {a^{2}+b^{2}}}\right ) b i +a^{2} d x +b^{2} d x}{a d \left (a^{2}+b^{2}\right )} \] Input:
int(1/(a+b*csch(d*x+c)),x)
Output:
( - 2*sqrt(a**2 + b**2)*atan((e**(c + d*x)*a*i + b*i)/sqrt(a**2 + b**2))*b *i + a**2*d*x + b**2*d*x)/(a*d*(a**2 + b**2))