Integrand size = 11, antiderivative size = 67 \[ \int \frac {1}{(a \coth (x)+b \text {csch}(x))^2} \, dx=\frac {x}{a^2}-\frac {2 b \arctan \left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{a^2 \sqrt {a-b} \sqrt {a+b}}-\frac {\sinh (x)}{a (b+a \cosh (x))} \]
x/a^2-sinh(x)/a/(b+a*cosh(x))-2*b*arctan((a-b)^(1/2)*tanh(1/2*x)/(a+b)^(1/ 2))/a^2/(a-b)^(1/2)/(a+b)^(1/2)
Time = 0.32 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.91 \[ \int \frac {1}{(a \coth (x)+b \text {csch}(x))^2} \, dx=\frac {x+\frac {2 b \arctan \left (\frac {(-a+b) \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}}-\frac {a \sinh (x)}{b+a \cosh (x)}}{a^2} \]
(x + (2*b*ArcTan[((-a + b)*Tanh[x/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] - (a*Sinh[x])/(b + a*Cosh[x]))/a^2
Time = 0.46 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.07, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.182, Rules used = {3042, 4892, 25, 25, 3042, 25, 3172, 25, 3042, 3214, 3042, 3138, 218}
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.
\(\displaystyle \int \frac {1}{(a \coth (x)+b \text {csch}(x))^2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{(i a \cot (i x)+i b \csc (i x))^2}dx\) |
\(\Big \downarrow \) 4892 |
\(\displaystyle \int -\frac {\sinh ^2(x)}{(i a \cosh (x)+i b)^2}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int -\frac {\sinh ^2(x)}{(b+a \cosh (x))^2}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \int \frac {\sinh ^2(x)}{(a \cosh (x)+b)^2}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -\frac {\cos \left (-\frac {\pi }{2}+i x\right )^2}{\left (b-a \sin \left (-\frac {\pi }{2}+i x\right )\right )^2}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int \frac {\cos \left (i x-\frac {\pi }{2}\right )^2}{\left (b-a \sin \left (i x-\frac {\pi }{2}\right )\right )^2}dx\) |
\(\Big \downarrow \) 3172 |
\(\displaystyle -\frac {\int -\frac {\cosh (x)}{b+a \cosh (x)}dx}{a}-\frac {\sinh (x)}{a (a \cosh (x)+b)}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\int \frac {\cosh (x)}{b+a \cosh (x)}dx}{a}-\frac {\sinh (x)}{a (a \cosh (x)+b)}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {\sinh (x)}{a (a \cosh (x)+b)}+\frac {\int \frac {\sin \left (i x+\frac {\pi }{2}\right )}{b+a \sin \left (i x+\frac {\pi }{2}\right )}dx}{a}\) |
\(\Big \downarrow \) 3214 |
\(\displaystyle \frac {\frac {x}{a}-\frac {b \int \frac {1}{b+a \cosh (x)}dx}{a}}{a}-\frac {\sinh (x)}{a (a \cosh (x)+b)}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {\sinh (x)}{a (a \cosh (x)+b)}+\frac {\frac {x}{a}-\frac {b \int \frac {1}{b+a \sin \left (i x+\frac {\pi }{2}\right )}dx}{a}}{a}\) |
\(\Big \downarrow \) 3138 |
\(\displaystyle \frac {\frac {x}{a}-\frac {2 b \int \frac {1}{(a-b) \tanh ^2\left (\frac {x}{2}\right )+a+b}d\tanh \left (\frac {x}{2}\right )}{a}}{a}-\frac {\sinh (x)}{a (a \cosh (x)+b)}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {\frac {x}{a}-\frac {2 b \arctan \left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{a \sqrt {a-b} \sqrt {a+b}}}{a}-\frac {\sinh (x)}{a (a \cosh (x)+b)}\) |
(x/a - (2*b*ArcTan[(Sqrt[a - b]*Tanh[x/2])/Sqrt[a + b]])/(a*Sqrt[a - b]*Sq rt[a + b]))/a - Sinh[x]/(a*(b + a*Cosh[x]))
3.7.50.3.1 Defintions of rubi rules used
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_.)*sin[Pi/2 + (c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{ e = FreeFactors[Tan[(c + d*x)/2], x]}, Simp[2*(e/d) Subst[Int[1/(a + b + (a - b)*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[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_), x_Symbol] :> Simp[g*(g*Cos[e + f*x])^(p - 1)*((a + b*Sin[e + f*x ])^(m + 1)/(b*f*(m + 1))), x] + Simp[g^2*((p - 1)/(b*(m + 1))) Int[(g*Cos [e + f*x])^(p - 2)*(a + b*Sin[e + f*x])^(m + 1)*Sin[e + f*x], x], x] /; Fre eQ[{a, b, e, f, g}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && GtQ[p, 1] && I ntegersQ[2*m, 2*p]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])/((c_.) + (d_.)*sin[(e_.) + (f_. )*(x_)]), x_Symbol] :> Simp[b*(x/d), x] - Simp[(b*c - a*d)/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[(cot[(c_.) + (d_.)*(x_)]^(n_.)*(a_.) + csc[(c_.) + (d_.)*(x_)]^(n_.)*(b _.))^(p_)*(u_.), x_Symbol] :> Int[ActivateTrig[u]*Csc[c + d*x]^(n*p)*(b + a *Cos[c + d*x]^n)^p, x] /; FreeQ[{a, b, c, d}, x] && IntegersQ[n, p]
Time = 2.02 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.43
method | result | size |
default | \(\frac {-\frac {2 \tanh \left (\frac {x}{2}\right ) a}{\tanh \left (\frac {x}{2}\right )^{2} a -\tanh \left (\frac {x}{2}\right )^{2} b +a +b}-\frac {2 b \arctan \left (\frac {\left (a -b \right ) \tanh \left (\frac {x}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}}{a^{2}}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{a^{2}}+\frac {\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{a^{2}}\) | \(96\) |
risch | \(\frac {x}{a^{2}}+\frac {2 \,{\mathrm e}^{x} b +2 a}{a^{2} \left (a \,{\mathrm e}^{2 x}+2 \,{\mathrm e}^{x} b +a \right )}-\frac {b \ln \left ({\mathrm e}^{x}+\frac {b \sqrt {-a^{2}+b^{2}}+a^{2}-b^{2}}{\sqrt {-a^{2}+b^{2}}\, a}\right )}{\sqrt {-a^{2}+b^{2}}\, a^{2}}+\frac {b \ln \left ({\mathrm e}^{x}+\frac {b \sqrt {-a^{2}+b^{2}}-a^{2}+b^{2}}{\sqrt {-a^{2}+b^{2}}\, a}\right )}{\sqrt {-a^{2}+b^{2}}\, a^{2}}\) | \(148\) |
2/a^2*(-tanh(1/2*x)*a/(tanh(1/2*x)^2*a-tanh(1/2*x)^2*b+a+b)-b/((a+b)*(a-b) )^(1/2)*arctan((a-b)*tanh(1/2*x)/((a+b)*(a-b))^(1/2)))-1/a^2*ln(tanh(1/2*x )-1)+1/a^2*ln(tanh(1/2*x)+1)
Leaf count of result is larger than twice the leaf count of optimal. 301 vs. \(2 (57) = 114\).
Time = 0.27 (sec) , antiderivative size = 682, normalized size of antiderivative = 10.18 \[ \int \frac {1}{(a \coth (x)+b \text {csch}(x))^2} \, dx=\left [\frac {{\left (a^{3} - a b^{2}\right )} x \cosh \left (x\right )^{2} + {\left (a^{3} - a b^{2}\right )} x \sinh \left (x\right )^{2} + 2 \, a^{3} - 2 \, a b^{2} - {\left (a b \cosh \left (x\right )^{2} + a b \sinh \left (x\right )^{2} + 2 \, b^{2} \cosh \left (x\right ) + a b + 2 \, {\left (a b \cosh \left (x\right ) + b^{2}\right )} \sinh \left (x\right )\right )} \sqrt {-a^{2} + b^{2}} \log \left (\frac {a^{2} \cosh \left (x\right )^{2} + a^{2} \sinh \left (x\right )^{2} + 2 \, a b \cosh \left (x\right ) - a^{2} + 2 \, b^{2} + 2 \, {\left (a^{2} \cosh \left (x\right ) + a b\right )} \sinh \left (x\right ) + 2 \, \sqrt {-a^{2} + b^{2}} {\left (a \cosh \left (x\right ) + a \sinh \left (x\right ) + b\right )}}{a \cosh \left (x\right )^{2} + a \sinh \left (x\right )^{2} + 2 \, b \cosh \left (x\right ) + 2 \, {\left (a \cosh \left (x\right ) + b\right )} \sinh \left (x\right ) + a}\right ) + {\left (a^{3} - a b^{2}\right )} x + 2 \, {\left (a^{2} b - b^{3} + {\left (a^{2} b - b^{3}\right )} x\right )} \cosh \left (x\right ) + 2 \, {\left (a^{2} b - b^{3} + {\left (a^{3} - a b^{2}\right )} x \cosh \left (x\right ) + {\left (a^{2} b - b^{3}\right )} x\right )} \sinh \left (x\right )}{a^{5} - a^{3} b^{2} + {\left (a^{5} - a^{3} b^{2}\right )} \cosh \left (x\right )^{2} + {\left (a^{5} - a^{3} b^{2}\right )} \sinh \left (x\right )^{2} + 2 \, {\left (a^{4} b - a^{2} b^{3}\right )} \cosh \left (x\right ) + 2 \, {\left (a^{4} b - a^{2} b^{3} + {\left (a^{5} - a^{3} b^{2}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )}, \frac {{\left (a^{3} - a b^{2}\right )} x \cosh \left (x\right )^{2} + {\left (a^{3} - a b^{2}\right )} x \sinh \left (x\right )^{2} + 2 \, a^{3} - 2 \, a b^{2} + 2 \, {\left (a b \cosh \left (x\right )^{2} + a b \sinh \left (x\right )^{2} + 2 \, b^{2} \cosh \left (x\right ) + a b + 2 \, {\left (a b \cosh \left (x\right ) + b^{2}\right )} \sinh \left (x\right )\right )} \sqrt {a^{2} - b^{2}} \arctan \left (-\frac {a \cosh \left (x\right ) + a \sinh \left (x\right ) + b}{\sqrt {a^{2} - b^{2}}}\right ) + {\left (a^{3} - a b^{2}\right )} x + 2 \, {\left (a^{2} b - b^{3} + {\left (a^{2} b - b^{3}\right )} x\right )} \cosh \left (x\right ) + 2 \, {\left (a^{2} b - b^{3} + {\left (a^{3} - a b^{2}\right )} x \cosh \left (x\right ) + {\left (a^{2} b - b^{3}\right )} x\right )} \sinh \left (x\right )}{a^{5} - a^{3} b^{2} + {\left (a^{5} - a^{3} b^{2}\right )} \cosh \left (x\right )^{2} + {\left (a^{5} - a^{3} b^{2}\right )} \sinh \left (x\right )^{2} + 2 \, {\left (a^{4} b - a^{2} b^{3}\right )} \cosh \left (x\right ) + 2 \, {\left (a^{4} b - a^{2} b^{3} + {\left (a^{5} - a^{3} b^{2}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )}\right ] \]
[((a^3 - a*b^2)*x*cosh(x)^2 + (a^3 - a*b^2)*x*sinh(x)^2 + 2*a^3 - 2*a*b^2 - (a*b*cosh(x)^2 + a*b*sinh(x)^2 + 2*b^2*cosh(x) + a*b + 2*(a*b*cosh(x) + b^2)*sinh(x))*sqrt(-a^2 + b^2)*log((a^2*cosh(x)^2 + a^2*sinh(x)^2 + 2*a*b* cosh(x) - a^2 + 2*b^2 + 2*(a^2*cosh(x) + a*b)*sinh(x) + 2*sqrt(-a^2 + b^2) *(a*cosh(x) + a*sinh(x) + b))/(a*cosh(x)^2 + a*sinh(x)^2 + 2*b*cosh(x) + 2 *(a*cosh(x) + b)*sinh(x) + a)) + (a^3 - a*b^2)*x + 2*(a^2*b - b^3 + (a^2*b - b^3)*x)*cosh(x) + 2*(a^2*b - b^3 + (a^3 - a*b^2)*x*cosh(x) + (a^2*b - b ^3)*x)*sinh(x))/(a^5 - a^3*b^2 + (a^5 - a^3*b^2)*cosh(x)^2 + (a^5 - a^3*b^ 2)*sinh(x)^2 + 2*(a^4*b - a^2*b^3)*cosh(x) + 2*(a^4*b - a^2*b^3 + (a^5 - a ^3*b^2)*cosh(x))*sinh(x)), ((a^3 - a*b^2)*x*cosh(x)^2 + (a^3 - a*b^2)*x*si nh(x)^2 + 2*a^3 - 2*a*b^2 + 2*(a*b*cosh(x)^2 + a*b*sinh(x)^2 + 2*b^2*cosh( x) + a*b + 2*(a*b*cosh(x) + b^2)*sinh(x))*sqrt(a^2 - b^2)*arctan(-(a*cosh( x) + a*sinh(x) + b)/sqrt(a^2 - b^2)) + (a^3 - a*b^2)*x + 2*(a^2*b - b^3 + (a^2*b - b^3)*x)*cosh(x) + 2*(a^2*b - b^3 + (a^3 - a*b^2)*x*cosh(x) + (a^2 *b - b^3)*x)*sinh(x))/(a^5 - a^3*b^2 + (a^5 - a^3*b^2)*cosh(x)^2 + (a^5 - a^3*b^2)*sinh(x)^2 + 2*(a^4*b - a^2*b^3)*cosh(x) + 2*(a^4*b - a^2*b^3 + (a ^5 - a^3*b^2)*cosh(x))*sinh(x))]
\[ \int \frac {1}{(a \coth (x)+b \text {csch}(x))^2} \, dx=\int \frac {1}{\left (a \coth {\left (x \right )} + b \operatorname {csch}{\left (x \right )}\right )^{2}}\, dx \]
Exception generated. \[ \int \frac {1}{(a \coth (x)+b \text {csch}(x))^2} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*b^2-4*a^2>0)', see `assume?` f or more de
Time = 0.26 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.01 \[ \int \frac {1}{(a \coth (x)+b \text {csch}(x))^2} \, dx=-\frac {2 \, b \arctan \left (\frac {a e^{x} + b}{\sqrt {a^{2} - b^{2}}}\right )}{\sqrt {a^{2} - b^{2}} a^{2}} + \frac {x}{a^{2}} + \frac {2 \, {\left (b e^{x} + a\right )}}{{\left (a e^{\left (2 \, x\right )} + 2 \, b e^{x} + a\right )} a^{2}} \]
-2*b*arctan((a*e^x + b)/sqrt(a^2 - b^2))/(sqrt(a^2 - b^2)*a^2) + x/a^2 + 2 *(b*e^x + a)/((a*e^(2*x) + 2*b*e^x + a)*a^2)
Time = 2.48 (sec) , antiderivative size = 139, normalized size of antiderivative = 2.07 \[ \int \frac {1}{(a \coth (x)+b \text {csch}(x))^2} \, dx=\frac {x}{a^2}+\frac {\frac {2}{a}+\frac {2\,b\,{\mathrm {e}}^x}{a^2}}{a+2\,b\,{\mathrm {e}}^x+a\,{\mathrm {e}}^{2\,x}}+\frac {b\,\ln \left (\frac {2\,b\,{\mathrm {e}}^x}{a^3}-\frac {2\,b\,\left (a+b\,{\mathrm {e}}^x\right )}{a^3\,\sqrt {a+b}\,\sqrt {b-a}}\right )}{a^2\,\sqrt {a+b}\,\sqrt {b-a}}-\frac {b\,\ln \left (\frac {2\,b\,{\mathrm {e}}^x}{a^3}+\frac {2\,b\,\left (a+b\,{\mathrm {e}}^x\right )}{a^3\,\sqrt {a+b}\,\sqrt {b-a}}\right )}{a^2\,\sqrt {a+b}\,\sqrt {b-a}} \]
x/a^2 + (2/a + (2*b*exp(x))/a^2)/(a + 2*b*exp(x) + a*exp(2*x)) + (b*log((2 *b*exp(x))/a^3 - (2*b*(a + b*exp(x)))/(a^3*(a + b)^(1/2)*(b - a)^(1/2))))/ (a^2*(a + b)^(1/2)*(b - a)^(1/2)) - (b*log((2*b*exp(x))/a^3 + (2*b*(a + b* exp(x)))/(a^3*(a + b)^(1/2)*(b - a)^(1/2))))/(a^2*(a + b)^(1/2)*(b - a)^(1 /2))