\(\int \coth (x) \sqrt {a+b \sinh (x)} \, dx\) [244]

Optimal result

Integrand size = 13, antiderivative size = 37 \[ \int \coth (x) \sqrt {a+b \sinh (x)} \, dx=-2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+b \sinh (x)}}{\sqrt {a}}\right )+2 \sqrt {a+b \sinh (x)} \]

[Out]

-2*arctanh((a+b*sinh(x))^(1/2)/a^(1/2))*a^(1/2)+2*(a+b*sinh(x))^(1/2)

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {2800, 52, 65, 213} \[ \int \coth (x) \sqrt {a+b \sinh (x)} \, dx=2 \sqrt {a+b \sinh (x)}-2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+b \sinh (x)}}{\sqrt {a}}\right ) \]

[In]

Int[Coth[x]*Sqrt[a + b*Sinh[x]],x]

[Out]

-2*Sqrt[a]*ArcTanh[Sqrt[a + b*Sinh[x]]/Sqrt[a]] + 2*Sqrt[a + b*Sinh[x]]

Rule 52

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] + Dist[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] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[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] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 213

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])

Rule 2800

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*tan[(e_.) + (f_.)*(x_)]^(p_.), x_Symbol] :> Dist[1/f, Subst[I
nt[(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]

Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {\sqrt {a+x}}{x} \, dx,x,b \sinh (x)\right ) \\ & = 2 \sqrt {a+b \sinh (x)}+a \text {Subst}\left (\int \frac {1}{x \sqrt {a+x}} \, dx,x,b \sinh (x)\right ) \\ & = 2 \sqrt {a+b \sinh (x)}+(2 a) \text {Subst}\left (\int \frac {1}{-a+x^2} \, dx,x,\sqrt {a+b \sinh (x)}\right ) \\ & = -2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+b \sinh (x)}}{\sqrt {a}}\right )+2 \sqrt {a+b \sinh (x)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00 \[ \int \coth (x) \sqrt {a+b \sinh (x)} \, dx=-2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+b \sinh (x)}}{\sqrt {a}}\right )+2 \sqrt {a+b \sinh (x)} \]

[In]

Integrate[Coth[x]*Sqrt[a + b*Sinh[x]],x]

[Out]

-2*Sqrt[a]*ArcTanh[Sqrt[a + b*Sinh[x]]/Sqrt[a]] + 2*Sqrt[a + b*Sinh[x]]

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 1.86 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.57

method	result	size
default	\(\operatorname {`\,int/indef0`\,}\left (\frac {\frac {a}{\sinh \left (x \right )}+b}{\sqrt {a +b \sinh \left (x \right )}}, \sinh \left (x \right )\right )\)	\(21\)

[In]

int(coth(x)*(a+b*sinh(x))^(1/2),x,method=_RETURNVERBOSE)

[Out]

`int/indef0`((1/sinh(x)*a+b)/(a+b*sinh(x))^(1/2),sinh(x))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 73 vs. \(2 (29) = 58\).

Time = 0.45 (sec) , antiderivative size = 356, normalized size of antiderivative = 9.62 \[ \int \coth (x) \sqrt {a+b \sinh (x)} \, dx=\left [\frac {1}{2} \, \sqrt {a} \log \left (-\frac {b^{2} \cosh \left (x\right )^{4} + b^{2} \sinh \left (x\right )^{4} + 16 \, a b \cosh \left (x\right )^{3} + 4 \, {\left (b^{2} \cosh \left (x\right ) + 4 \, a b\right )} \sinh \left (x\right )^{3} - 16 \, a b \cosh \left (x\right ) + 2 \, {\left (16 \, a^{2} - b^{2}\right )} \cosh \left (x\right )^{2} + 2 \, {\left (3 \, b^{2} \cosh \left (x\right )^{2} + 24 \, a b \cosh \left (x\right ) + 16 \, a^{2} - b^{2}\right )} \sinh \left (x\right )^{2} - 8 \, {\left (b \cosh \left (x\right )^{3} + b \sinh \left (x\right )^{3} + 4 \, a \cosh \left (x\right )^{2} + {\left (3 \, b \cosh \left (x\right ) + 4 \, a\right )} \sinh \left (x\right )^{2} - b \cosh \left (x\right ) + {\left (3 \, b \cosh \left (x\right )^{2} + 8 \, a \cosh \left (x\right ) - b\right )} \sinh \left (x\right )\right )} \sqrt {b \sinh \left (x\right ) + a} \sqrt {a} + b^{2} + 4 \, {\left (b^{2} \cosh \left (x\right )^{3} + 12 \, a b \cosh \left (x\right )^{2} - 4 \, a b + {\left (16 \, a^{2} - b^{2}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )}{\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + 2 \, {\left (3 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right )^{2} - 2 \, \cosh \left (x\right )^{2} + 4 \, {\left (\cosh \left (x\right )^{3} - \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1}\right ) + 2 \, \sqrt {b \sinh \left (x\right ) + a}, \sqrt {-a} \arctan \left (\frac {4 \, \sqrt {b \sinh \left (x\right ) + a} \sqrt {-a} {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )}}{b \cosh \left (x\right )^{2} + b \sinh \left (x\right )^{2} + 4 \, a \cosh \left (x\right ) + 2 \, {\left (b \cosh \left (x\right ) + 2 \, a\right )} \sinh \left (x\right ) - b}\right ) + 2 \, \sqrt {b \sinh \left (x\right ) + a}\right ] \]

[In]

integrate(coth(x)*(a+b*sinh(x))^(1/2),x, algorithm="fricas")

[Out]

[1/2*sqrt(a)*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*co
sh(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)) + 2*sqrt(b*sinh(x) + a), sqrt(-a)*arctan(4*sqrt(b*sinh(x) + a)*s
qrt(-a)*(cosh(x) + sinh(x))/(b*cosh(x)^2 + b*sinh(x)^2 + 4*a*cosh(x) + 2*(b*cosh(x) + 2*a)*sinh(x) - b)) + 2*s
qrt(b*sinh(x) + a)]

Sympy [F]

\[ \int \coth (x) \sqrt {a+b \sinh (x)} \, dx=\int \sqrt {a + b \sinh {\left (x \right )}} \coth {\left (x \right )}\, dx \]

[In]

integrate(coth(x)*(a+b*sinh(x))**(1/2),x)

[Out]

Integral(sqrt(a + b*sinh(x))*coth(x), x)

Maxima [F]

\[ \int \coth (x) \sqrt {a+b \sinh (x)} \, dx=\int { \sqrt {b \sinh \left (x\right ) + a} \coth \left (x\right ) \,d x } \]

[In]

integrate(coth(x)*(a+b*sinh(x))^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(b*sinh(x) + a)*coth(x), x)

Giac [F]

\[ \int \coth (x) \sqrt {a+b \sinh (x)} \, dx=\int { \sqrt {b \sinh \left (x\right ) + a} \coth \left (x\right ) \,d x } \]

[In]

integrate(coth(x)*(a+b*sinh(x))^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(b*sinh(x) + a)*coth(x), x)

Mupad [F(-1)]

Timed out. \[ \int \coth (x) \sqrt {a+b \sinh (x)} \, dx=\int \mathrm {coth}\left (x\right )\,\sqrt {a+b\,\mathrm {sinh}\left (x\right )} \,d x \]

[In]

int(coth(x)*(a + b*sinh(x))^(1/2),x)

[Out]

int(coth(x)*(a + b*sinh(x))^(1/2), x)