Integrand size = 15, antiderivative size = 47 \[ \int \coth (x) \sqrt {a+b \sinh ^n(x)} \, dx=-\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+b \sinh ^n(x)}}{\sqrt {a}}\right )}{n}+\frac {2 \sqrt {a+b \sinh ^n(x)}}{n} \]
Time = 0.02 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.96 \[ \int \coth (x) \sqrt {a+b \sinh ^n(x)} \, dx=\frac {-2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+b \sinh ^n(x)}}{\sqrt {a}}\right )+2 \sqrt {a+b \sinh ^n(x)}}{n} \]
Time = 0.27 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.96, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {3042, 26, 3709, 798, 60, 73, 221}
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.
\(\displaystyle \int \coth (x) \sqrt {a+b \sinh ^n(x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {i \sqrt {a+b (-i \sin (i x))^n}}{\tan (i x)}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \int \frac {\sqrt {b (-i \sin (i x))^n+a}}{\tan (i x)}dx\) |
\(\Big \downarrow \) 3709 |
\(\displaystyle \int \text {csch}(x) \sqrt {a+b \sinh ^n(x)}d\sinh (x)\) |
\(\Big \downarrow \) 798 |
\(\displaystyle \frac {\int \text {csch}(x) \sqrt {b \sinh ^n(x)+a}d\sinh ^n(x)}{n}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {a \int \frac {\text {csch}(x)}{\sqrt {b \sinh ^n(x)+a}}d\sinh ^n(x)+2 \sqrt {a+b \sinh ^n(x)}}{n}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {\frac {2 a \int \frac {1}{\frac {\sinh ^{2 n}(x)}{b}-\frac {a}{b}}d\sqrt {b \sinh ^n(x)+a}}{b}+2 \sqrt {a+b \sinh ^n(x)}}{n}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {2 \sqrt {a+b \sinh ^n(x)}-2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+b \sinh ^n(x)}}{\sqrt {a}}\right )}{n}\) |
3.6.25.3.1 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_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[1/n Subst [Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
Int[((a_) + (b_.)*((c_.)*sin[(e_.) + (f_.)*(x_)])^(n_))^(p_.)*tan[(e_.) + ( f_.)*(x_)]^(m_.), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Si mp[ff^(m + 1)/f Subst[Int[x^m*((a + b*(c*ff*x)^n)^p/(1 - ff^2*x^2)^((m + 1)/2)), x], x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, c, e, f, n, p}, x] && ILtQ[(m - 1)/2, 0]
Time = 0.21 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.81
method | result | size |
derivativedivides | \(\frac {2 \sqrt {a +b \sinh \left (x \right )^{n}}-2 \sqrt {a}\, \operatorname {arctanh}\left (\frac {\sqrt {a +b \sinh \left (x \right )^{n}}}{\sqrt {a}}\right )}{n}\) | \(38\) |
default | \(\frac {2 \sqrt {a +b \sinh \left (x \right )^{n}}-2 \sqrt {a}\, \operatorname {arctanh}\left (\frac {\sqrt {a +b \sinh \left (x \right )^{n}}}{\sqrt {a}}\right )}{n}\) | \(38\) |
Time = 0.26 (sec) , antiderivative size = 156, normalized size of antiderivative = 3.32 \[ \int \coth (x) \sqrt {a+b \sinh ^n(x)} \, dx=\left [\frac {\sqrt {a} \log \left (\frac {b \cosh \left (n \log \left (\sinh \left (x\right )\right )\right ) + b \sinh \left (n \log \left (\sinh \left (x\right )\right )\right ) - 2 \, \sqrt {b \cosh \left (n \log \left (\sinh \left (x\right )\right )\right ) + b \sinh \left (n \log \left (\sinh \left (x\right )\right )\right ) + a} \sqrt {a} + 2 \, a}{\cosh \left (n \log \left (\sinh \left (x\right )\right )\right ) + \sinh \left (n \log \left (\sinh \left (x\right )\right )\right )}\right ) + 2 \, \sqrt {b \cosh \left (n \log \left (\sinh \left (x\right )\right )\right ) + b \sinh \left (n \log \left (\sinh \left (x\right )\right )\right ) + a}}{n}, \frac {2 \, {\left (\sqrt {-a} \arctan \left (\frac {\sqrt {b \cosh \left (n \log \left (\sinh \left (x\right )\right )\right ) + b \sinh \left (n \log \left (\sinh \left (x\right )\right )\right ) + a} \sqrt {-a}}{a}\right ) + \sqrt {b \cosh \left (n \log \left (\sinh \left (x\right )\right )\right ) + b \sinh \left (n \log \left (\sinh \left (x\right )\right )\right ) + a}\right )}}{n}\right ] \]
[(sqrt(a)*log((b*cosh(n*log(sinh(x))) + b*sinh(n*log(sinh(x))) - 2*sqrt(b* cosh(n*log(sinh(x))) + b*sinh(n*log(sinh(x))) + a)*sqrt(a) + 2*a)/(cosh(n* log(sinh(x))) + sinh(n*log(sinh(x))))) + 2*sqrt(b*cosh(n*log(sinh(x))) + b *sinh(n*log(sinh(x))) + a))/n, 2*(sqrt(-a)*arctan(sqrt(b*cosh(n*log(sinh(x ))) + b*sinh(n*log(sinh(x))) + a)*sqrt(-a)/a) + sqrt(b*cosh(n*log(sinh(x)) ) + b*sinh(n*log(sinh(x))) + a))/n]
\[ \int \coth (x) \sqrt {a+b \sinh ^n(x)} \, dx=\int \sqrt {a + b \sinh ^{n}{\left (x \right )}} \coth {\left (x \right )}\, dx \]
\[ \int \coth (x) \sqrt {a+b \sinh ^n(x)} \, dx=\int { \sqrt {b \sinh \left (x\right )^{n} + a} \coth \left (x\right ) \,d x } \]
\[ \int \coth (x) \sqrt {a+b \sinh ^n(x)} \, dx=\int { \sqrt {b \sinh \left (x\right )^{n} + a} \coth \left (x\right ) \,d x } \]
Timed out. \[ \int \coth (x) \sqrt {a+b \sinh ^n(x)} \, dx=\int \mathrm {coth}\left (x\right )\,\sqrt {a+b\,{\mathrm {sinh}\left (x\right )}^n} \,d x \]