Integrand size = 11, antiderivative size = 50 \[ \int \frac {\text {csch}(x)}{a+b \sinh (x)} \, dx=-\frac {\text {arctanh}(\cosh (x))}{a}+\frac {2 b \text {arctanh}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2}} \] Output:
-arctanh(cosh(x))/a+2*b*arctanh((b-a*tanh(1/2*x))/(a^2+b^2)^(1/2))/a/(a^2+ b^2)^(1/2)
Time = 0.28 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.34 \[ \int \frac {\text {csch}(x)}{a+b \sinh (x)} \, dx=\frac {-\frac {2 b \arctan \left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {-a^2-b^2}}\right )}{\sqrt {-a^2-b^2}}-\log \left (\cosh \left (\frac {x}{2}\right )\right )+\log \left (\sinh \left (\frac {x}{2}\right )\right )}{a} \] Input:
Integrate[Csch[x]/(a + b*Sinh[x]),x]
Output:
((-2*b*ArcTan[(b - a*Tanh[x/2])/Sqrt[-a^2 - b^2]])/Sqrt[-a^2 - b^2] - Log[ Cosh[x/2]] + Log[Sinh[x/2]])/a
Result contains complex when optimal does not.
Time = 0.38 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.26, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.909, Rules used = {3042, 26, 3226, 26, 3042, 26, 3139, 1083, 219, 4257}
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.
\(\displaystyle \int \frac {\text {csch}(x)}{a+b \sinh (x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {i}{\sin (i x) (a-i b \sin (i x))}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \int \frac {1}{\sin (i x) (a-i b \sin (i x))}dx\) |
\(\Big \downarrow \) 3226 |
\(\displaystyle i \left (\frac {i b \int \frac {1}{a+b \sinh (x)}dx}{a}+\frac {\int -i \text {csch}(x)dx}{a}\right )\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \left (\frac {i b \int \frac {1}{a+b \sinh (x)}dx}{a}-\frac {i \int \text {csch}(x)dx}{a}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle i \left (\frac {i b \int \frac {1}{a-i b \sin (i x)}dx}{a}-\frac {i \int i \csc (i x)dx}{a}\right )\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \left (\frac {i b \int \frac {1}{a-i b \sin (i x)}dx}{a}+\frac {\int \csc (i x)dx}{a}\right )\) |
\(\Big \downarrow \) 3139 |
\(\displaystyle i \left (\frac {2 i b \int \frac {1}{-a \tanh ^2\left (\frac {x}{2}\right )+2 b \tanh \left (\frac {x}{2}\right )+a}d\tanh \left (\frac {x}{2}\right )}{a}+\frac {\int \csc (i x)dx}{a}\right )\) |
\(\Big \downarrow \) 1083 |
\(\displaystyle i \left (\frac {\int \csc (i x)dx}{a}-\frac {4 i b \int \frac {1}{4 \left (a^2+b^2\right )-\left (2 b-2 a \tanh \left (\frac {x}{2}\right )\right )^2}d\left (2 b-2 a \tanh \left (\frac {x}{2}\right )\right )}{a}\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle i \left (\frac {\int \csc (i x)dx}{a}-\frac {2 i b \text {arctanh}\left (\frac {2 b-2 a \tanh \left (\frac {x}{2}\right )}{2 \sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2}}\right )\) |
\(\Big \downarrow \) 4257 |
\(\displaystyle i \left (\frac {i \text {arctanh}(\cosh (x))}{a}-\frac {2 i b \text {arctanh}\left (\frac {2 b-2 a \tanh \left (\frac {x}{2}\right )}{2 \sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2}}\right )\) |
Input:
Int[Csch[x]/(a + b*Sinh[x]),x]
Output:
I*((I*ArcTanh[Cosh[x]])/a - ((2*I)*b*ArcTanh[(2*b - 2*a*Tanh[x/2])/(2*Sqrt [a^2 + b^2])])/(a*Sqrt[a^2 + b^2]))
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[(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_) + (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[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + ( f_.)*(x_)])), x_Symbol] :> Simp[b/(b*c - a*d) Int[1/(a + b*Sin[e + f*x]), x], x] - Simp[d/(b*c - a*d) Int[1/(c + d*Sin[e + f*x]), x], x] /; FreeQ[ {a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Time = 0.16 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.98
method | result | size |
default | \(-\frac {2 b \,\operatorname {arctanh}\left (\frac {2 a \tanh \left (\frac {x}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{a \sqrt {a^{2}+b^{2}}}+\frac {\ln \left (\tanh \left (\frac {x}{2}\right )\right )}{a}\) | \(49\) |
risch | \(\frac {b \ln \left ({\mathrm e}^{x}+\frac {a \sqrt {a^{2}+b^{2}}+a^{2}+b^{2}}{\sqrt {a^{2}+b^{2}}\, b}\right )}{\sqrt {a^{2}+b^{2}}\, a}-\frac {b \ln \left ({\mathrm e}^{x}+\frac {a \sqrt {a^{2}+b^{2}}-a^{2}-b^{2}}{\sqrt {a^{2}+b^{2}}\, b}\right )}{\sqrt {a^{2}+b^{2}}\, a}+\frac {\ln \left ({\mathrm e}^{x}-1\right )}{a}-\frac {\ln \left ({\mathrm e}^{x}+1\right )}{a}\) | \(124\) |
Input:
int(csch(x)/(a+b*sinh(x)),x,method=_RETURNVERBOSE)
Output:
-2*b/a/(a^2+b^2)^(1/2)*arctanh(1/2*(2*a*tanh(1/2*x)-2*b)/(a^2+b^2)^(1/2))+ 1/a*ln(tanh(1/2*x))
Leaf count of result is larger than twice the leaf count of optimal. 156 vs. \(2 (46) = 92\).
Time = 0.11 (sec) , antiderivative size = 156, normalized size of antiderivative = 3.12 \[ \int \frac {\text {csch}(x)}{a+b \sinh (x)} \, dx=\frac {\sqrt {a^{2} + b^{2}} b \log \left (\frac {b^{2} \cosh \left (x\right )^{2} + b^{2} \sinh \left (x\right )^{2} + 2 \, a b \cosh \left (x\right ) + 2 \, a^{2} + b^{2} + 2 \, {\left (b^{2} \cosh \left (x\right ) + a b\right )} \sinh \left (x\right ) + 2 \, \sqrt {a^{2} + b^{2}} {\left (b \cosh \left (x\right ) + b \sinh \left (x\right ) + a\right )}}{b \cosh \left (x\right )^{2} + b \sinh \left (x\right )^{2} + 2 \, a \cosh \left (x\right ) + 2 \, {\left (b \cosh \left (x\right ) + a\right )} \sinh \left (x\right ) - b}\right ) - {\left (a^{2} + b^{2}\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) + {\left (a^{2} + b^{2}\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) - 1\right )}{a^{3} + a b^{2}} \] Input:
integrate(csch(x)/(a+b*sinh(x)),x, algorithm="fricas")
Output:
(sqrt(a^2 + b^2)*b*log((b^2*cosh(x)^2 + b^2*sinh(x)^2 + 2*a*b*cosh(x) + 2* a^2 + b^2 + 2*(b^2*cosh(x) + a*b)*sinh(x) + 2*sqrt(a^2 + b^2)*(b*cosh(x) + b*sinh(x) + a))/(b*cosh(x)^2 + b*sinh(x)^2 + 2*a*cosh(x) + 2*(b*cosh(x) + a)*sinh(x) - b)) - (a^2 + b^2)*log(cosh(x) + sinh(x) + 1) + (a^2 + b^2)*l og(cosh(x) + sinh(x) - 1))/(a^3 + a*b^2)
\[ \int \frac {\text {csch}(x)}{a+b \sinh (x)} \, dx=\int \frac {\operatorname {csch}{\left (x \right )}}{a + b \sinh {\left (x \right )}}\, dx \] Input:
integrate(csch(x)/(a+b*sinh(x)),x)
Output:
Integral(csch(x)/(a + b*sinh(x)), x)
Time = 0.13 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.66 \[ \int \frac {\text {csch}(x)}{a+b \sinh (x)} \, dx=-\frac {b \log \left (\frac {b e^{\left (-x\right )} - a - \sqrt {a^{2} + b^{2}}}{b e^{\left (-x\right )} - a + \sqrt {a^{2} + b^{2}}}\right )}{\sqrt {a^{2} + b^{2}} a} - \frac {\log \left (e^{\left (-x\right )} + 1\right )}{a} + \frac {\log \left (e^{\left (-x\right )} - 1\right )}{a} \] Input:
integrate(csch(x)/(a+b*sinh(x)),x, algorithm="maxima")
Output:
-b*log((b*e^(-x) - a - sqrt(a^2 + b^2))/(b*e^(-x) - a + sqrt(a^2 + b^2)))/ (sqrt(a^2 + b^2)*a) - log(e^(-x) + 1)/a + log(e^(-x) - 1)/a
Time = 0.13 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.64 \[ \int \frac {\text {csch}(x)}{a+b \sinh (x)} \, dx=-\frac {b \log \left (\frac {{\left | 2 \, b e^{x} + 2 \, a - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, b e^{x} + 2 \, a + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{\sqrt {a^{2} + b^{2}} a} - \frac {\log \left (e^{x} + 1\right )}{a} + \frac {\log \left ({\left | e^{x} - 1 \right |}\right )}{a} \] Input:
integrate(csch(x)/(a+b*sinh(x)),x, algorithm="giac")
Output:
-b*log(abs(2*b*e^x + 2*a - 2*sqrt(a^2 + b^2))/abs(2*b*e^x + 2*a + 2*sqrt(a ^2 + b^2)))/(sqrt(a^2 + b^2)*a) - log(e^x + 1)/a + log(abs(e^x - 1))/a
Time = 2.01 (sec) , antiderivative size = 287, normalized size of antiderivative = 5.74 \[ \int \frac {\text {csch}(x)}{a+b \sinh (x)} \, dx=\frac {\ln \left (32\,a-32\,a\,{\mathrm {e}}^x\right )}{a}-\frac {\ln \left (32\,a+32\,a\,{\mathrm {e}}^x\right )}{a}+\frac {b\,\ln \left (128\,a^5\,{\mathrm {e}}^x-64\,a^2\,b^3-64\,a^4\,b-128\,a^4\,{\mathrm {e}}^x\,\sqrt {a^2+b^2}+32\,a\,b^4\,{\mathrm {e}}^x+160\,a^3\,b^2\,{\mathrm {e}}^x+32\,a\,b^3\,\sqrt {a^2+b^2}+64\,a^3\,b\,\sqrt {a^2+b^2}-96\,a^2\,b^2\,{\mathrm {e}}^x\,\sqrt {a^2+b^2}\right )\,\sqrt {a^2+b^2}}{a^3+a\,b^2}-\frac {b\,\ln \left (64\,a^4\,b+64\,a^2\,b^3-128\,a^5\,{\mathrm {e}}^x-128\,a^4\,{\mathrm {e}}^x\,\sqrt {a^2+b^2}-32\,a\,b^4\,{\mathrm {e}}^x-160\,a^3\,b^2\,{\mathrm {e}}^x+32\,a\,b^3\,\sqrt {a^2+b^2}+64\,a^3\,b\,\sqrt {a^2+b^2}-96\,a^2\,b^2\,{\mathrm {e}}^x\,\sqrt {a^2+b^2}\right )\,\sqrt {a^2+b^2}}{a^3+a\,b^2} \] Input:
int(1/(sinh(x)*(a + b*sinh(x))),x)
Output:
log(32*a - 32*a*exp(x))/a - log(32*a + 32*a*exp(x))/a + (b*log(128*a^5*exp (x) - 64*a^2*b^3 - 64*a^4*b - 128*a^4*exp(x)*(a^2 + b^2)^(1/2) + 32*a*b^4* exp(x) + 160*a^3*b^2*exp(x) + 32*a*b^3*(a^2 + b^2)^(1/2) + 64*a^3*b*(a^2 + b^2)^(1/2) - 96*a^2*b^2*exp(x)*(a^2 + b^2)^(1/2))*(a^2 + b^2)^(1/2))/(a*b ^2 + a^3) - (b*log(64*a^4*b + 64*a^2*b^3 - 128*a^5*exp(x) - 128*a^4*exp(x) *(a^2 + b^2)^(1/2) - 32*a*b^4*exp(x) - 160*a^3*b^2*exp(x) + 32*a*b^3*(a^2 + b^2)^(1/2) + 64*a^3*b*(a^2 + b^2)^(1/2) - 96*a^2*b^2*exp(x)*(a^2 + b^2)^ (1/2))*(a^2 + b^2)^(1/2))/(a*b^2 + a^3)
Time = 0.17 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.80 \[ \int \frac {\text {csch}(x)}{a+b \sinh (x)} \, dx=\frac {-2 \sqrt {a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{x} b i +a i}{\sqrt {a^{2}+b^{2}}}\right ) b i +\mathrm {log}\left (e^{x}-1\right ) a^{2}+\mathrm {log}\left (e^{x}-1\right ) b^{2}-\mathrm {log}\left (e^{x}+1\right ) a^{2}-\mathrm {log}\left (e^{x}+1\right ) b^{2}}{a \left (a^{2}+b^{2}\right )} \] Input:
int(csch(x)/(a+b*sinh(x)),x)
Output:
( - 2*sqrt(a**2 + b**2)*atan((e**x*b*i + a*i)/sqrt(a**2 + b**2))*b*i + log (e**x - 1)*a**2 + log(e**x - 1)*b**2 - log(e**x + 1)*a**2 - log(e**x + 1)* b**2)/(a*(a**2 + b**2))