Integrand size = 13, antiderivative size = 24 \[ \int \frac {\coth (x)}{\sqrt {a+b \sinh (x)}} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {a+b \sinh (x)}}{\sqrt {a}}\right )}{\sqrt {a}} \] Output:
-2*arctanh((a+b*sinh(x))^(1/2)/a^(1/2))/a^(1/2)
Time = 0.01 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {\coth (x)}{\sqrt {a+b \sinh (x)}} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {a+b \sinh (x)}}{\sqrt {a}}\right )}{\sqrt {a}} \] Input:
Integrate[Coth[x]/Sqrt[a + b*Sinh[x]],x]
Output:
(-2*ArcTanh[Sqrt[a + b*Sinh[x]]/Sqrt[a]])/Sqrt[a]
Time = 0.25 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {3042, 26, 3200, 73, 220}
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.
\(\displaystyle \int \frac {\coth (x)}{\sqrt {a+b \sinh (x)}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {i}{\tan (i x) \sqrt {a-i b \sin (i x)}}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \int \frac {1}{\sqrt {a-i b \sin (i x)} \tan (i x)}dx\) |
\(\Big \downarrow \) 3200 |
\(\displaystyle \int \frac {\text {csch}(x)}{b \sqrt {a+b \sinh (x)}}d(b \sinh (x))\) |
\(\Big \downarrow \) 73 |
\(\displaystyle 2 \int \frac {1}{b^2 \sinh ^2(x)-a}d\sqrt {a+b \sinh (x)}\) |
\(\Big \downarrow \) 220 |
\(\displaystyle -\frac {2 \text {arctanh}\left (\frac {\sqrt {a+b \sinh (x)}}{\sqrt {a}}\right )}{\sqrt {a}}\) |
Input:
Int[Coth[x]/Sqrt[a + b*Sinh[x]],x]
Output:
(-2*ArcTanh[Sqrt[a + b*Sinh[x]]/Sqrt[a]])/Sqrt[a]
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] :> 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, 2]*Rt[b, 2])^(- 1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*tan[(e_.) + (f_.)*(x_)]^(p _.), x_Symbol] :> Simp[1/f Subst[Int[(x^p*(a + x)^m)/(b^2 - x^2)^((p + 1) /2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && NeQ[a^2 - b ^2, 0] && IntegerQ[(p + 1)/2]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.20 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.71
method | result | size |
default | \(\operatorname {`\,int/indef0`\,}\left (\frac {1}{\sinh \left (x \right ) \sqrt {a +b \sinh \left (x \right )}}, \sinh \left (x \right )\right )\) | \(17\) |
Input:
int(coth(x)/(a+b*sinh(x))^(1/2),x,method=_RETURNVERBOSE)
Output:
`int/indef0`(1/sinh(x)/(a+b*sinh(x))^(1/2),sinh(x))
Leaf count of result is larger than twice the leaf count of optimal. 99 vs. \(2 (18) = 36\).
Time = 0.12 (sec) , antiderivative size = 370, normalized size of antiderivative = 15.42 \[ \int \frac {\coth (x)}{\sqrt {a+b \sinh (x)}} \, dx=\left [\frac {\log \left (\frac {b^{2} \cosh \left (x\right )^{4} + b^{2} \sinh \left (x\right )^{4} + 16 \, a b \cosh \left (x\right )^{3} + 4 \, {\left (b^{2} \cosh \left (x\right ) + 4 \, a b\right )} \sinh \left (x\right )^{3} - 16 \, a b \cosh \left (x\right ) + 2 \, {\left (16 \, a^{2} - b^{2}\right )} \cosh \left (x\right )^{2} + 2 \, {\left (3 \, b^{2} \cosh \left (x\right )^{2} + 24 \, a b \cosh \left (x\right ) + 16 \, a^{2} - b^{2}\right )} \sinh \left (x\right )^{2} - 8 \, {\left (b \cosh \left (x\right )^{3} + b \sinh \left (x\right )^{3} + 4 \, a \cosh \left (x\right )^{2} + {\left (3 \, b \cosh \left (x\right ) + 4 \, a\right )} \sinh \left (x\right )^{2} - b \cosh \left (x\right ) + {\left (3 \, b \cosh \left (x\right )^{2} + 8 \, a \cosh \left (x\right ) - b\right )} \sinh \left (x\right )\right )} \sqrt {b \sinh \left (x\right ) + a} \sqrt {a} + b^{2} + 4 \, {\left (b^{2} \cosh \left (x\right )^{3} + 12 \, a b \cosh \left (x\right )^{2} - 4 \, a b + {\left (16 \, a^{2} - b^{2}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )}{\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + 2 \, {\left (3 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right )^{2} - 2 \, \cosh \left (x\right )^{2} + 4 \, {\left (\cosh \left (x\right )^{3} - \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1}\right )}{2 \, \sqrt {a}}, \frac {\sqrt {-a} \arctan \left (\frac {{\left (b \cosh \left (x\right )^{2} + b \sinh \left (x\right )^{2} + 4 \, a \cosh \left (x\right ) + 2 \, {\left (b \cosh \left (x\right ) + 2 \, a\right )} \sinh \left (x\right ) - b\right )} \sqrt {b \sinh \left (x\right ) + a} \sqrt {-a}}{2 \, {\left (a b \cosh \left (x\right )^{2} + a b \sinh \left (x\right )^{2} + 2 \, a^{2} \cosh \left (x\right ) - a b + 2 \, {\left (a b \cosh \left (x\right ) + a^{2}\right )} \sinh \left (x\right )\right )}}\right )}{a}\right ] \] Input:
integrate(coth(x)/(a+b*sinh(x))^(1/2),x, algorithm="fricas")
Output:
[1/2*log((b^2*cosh(x)^4 + b^2*sinh(x)^4 + 16*a*b*cosh(x)^3 + 4*(b^2*cosh(x ) + 4*a*b)*sinh(x)^3 - 16*a*b*cosh(x) + 2*(16*a^2 - b^2)*cosh(x)^2 + 2*(3* b^2*cosh(x)^2 + 24*a*b*cosh(x) + 16*a^2 - b^2)*sinh(x)^2 - 8*(b*cosh(x)^3 + b*sinh(x)^3 + 4*a*cosh(x)^2 + (3*b*cosh(x) + 4*a)*sinh(x)^2 - b*cosh(x) + (3*b*cosh(x)^2 + 8*a*cosh(x) - b)*sinh(x))*sqrt(b*sinh(x) + a)*sqrt(a) + b^2 + 4*(b^2*cosh(x)^3 + 12*a*b*cosh(x)^2 - 4*a*b + (16*a^2 - b^2)*cosh(x ))*sinh(x))/(cosh(x)^4 + 4*cosh(x)*sinh(x)^3 + sinh(x)^4 + 2*(3*cosh(x)^2 - 1)*sinh(x)^2 - 2*cosh(x)^2 + 4*(cosh(x)^3 - cosh(x))*sinh(x) + 1))/sqrt( a), sqrt(-a)*arctan(1/2*(b*cosh(x)^2 + b*sinh(x)^2 + 4*a*cosh(x) + 2*(b*co sh(x) + 2*a)*sinh(x) - b)*sqrt(b*sinh(x) + a)*sqrt(-a)/(a*b*cosh(x)^2 + a* b*sinh(x)^2 + 2*a^2*cosh(x) - a*b + 2*(a*b*cosh(x) + a^2)*sinh(x)))/a]
\[ \int \frac {\coth (x)}{\sqrt {a+b \sinh (x)}} \, dx=\int \frac {\coth {\left (x \right )}}{\sqrt {a + b \sinh {\left (x \right )}}}\, dx \] Input:
integrate(coth(x)/(a+b*sinh(x))**(1/2),x)
Output:
Integral(coth(x)/sqrt(a + b*sinh(x)), x)
\[ \int \frac {\coth (x)}{\sqrt {a+b \sinh (x)}} \, dx=\int { \frac {\coth \left (x\right )}{\sqrt {b \sinh \left (x\right ) + a}} \,d x } \] Input:
integrate(coth(x)/(a+b*sinh(x))^(1/2),x, algorithm="maxima")
Output:
integrate(coth(x)/sqrt(b*sinh(x) + a), x)
\[ \int \frac {\coth (x)}{\sqrt {a+b \sinh (x)}} \, dx=\int { \frac {\coth \left (x\right )}{\sqrt {b \sinh \left (x\right ) + a}} \,d x } \] Input:
integrate(coth(x)/(a+b*sinh(x))^(1/2),x, algorithm="giac")
Output:
integrate(coth(x)/sqrt(b*sinh(x) + a), x)
Timed out. \[ \int \frac {\coth (x)}{\sqrt {a+b \sinh (x)}} \, dx=\int \frac {\mathrm {coth}\left (x\right )}{\sqrt {a+b\,\mathrm {sinh}\left (x\right )}} \,d x \] Input:
int(coth(x)/(a + b*sinh(x))^(1/2),x)
Output:
int(coth(x)/(a + b*sinh(x))^(1/2), x)
\[ \int \frac {\coth (x)}{\sqrt {a+b \sinh (x)}} \, dx=\int \frac {\sqrt {\sinh \left (x \right ) b +a}\, \coth \left (x \right )}{\sinh \left (x \right ) b +a}d x \] Input:
int(coth(x)/(a+b*sinh(x))^(1/2),x)
Output:
int((sqrt(sinh(x)*b + a)*coth(x))/(sinh(x)*b + a),x)