∫ coth(x)_____∘ a + b sinh(x) dx [245]

Optimal. Leaf size=24 \[ -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b \sinh (x)}}{\sqrt {a}}\right )}{\sqrt {a}} \]

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Rubi [A]
time = 0.04, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2800, 65, 213} \begin {gather*} -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b \sinh (x)}}{\sqrt {a}}\right )}{\sqrt {a}} \end {gather*}

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,

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] :> 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

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

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Mathematica [A]
time = 0.01, size = 24, normalized size = 1.00 \begin {gather*} -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b \sinh (x)}}{\sqrt {a}}\right )}{\sqrt {a}} \end {gather*}

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method	result	size

derivativedivides	\(-\frac {2 \arctanh \left (\frac {\sqrt {a +b \sinh \left (x \right )}}{\sqrt {a}}\right )}{\sqrt {a}}\)	\(19\)
default	\(-\frac {2 \arctanh \left (\frac {\sqrt {a +b \sinh \left (x \right )}}{\sqrt {a}}\right )}{\sqrt {a}}\)	\(19\)

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 99 vs. \(2 (18) = 36\).
time = 0.46, size = 370, normalized size = 15.42 \begin {gather*} \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 ] \end {gather*}

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

[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)*co

sh(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*cosh(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) -

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\coth {\left (x \right )}}{\sqrt {a + b \sinh {\left (x \right )}}}\, dx \end {gather*}

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {\mathrm {coth}\left (x\right )}{\sqrt {a+b\,\mathrm {sinh}\left (x\right )}} \,d x \end {gather*}

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3.3.45 \(\int \frac {\coth (x)}{\sqrt {a+b \sinh (x)}} \, dx\) [245]