3.6.22 \(\int \frac {\coth (x)}{\sqrt {a+b \sinh ^3(x)}} \, dx\) [522]

Optimal. Leaf size=28 \[ -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b \sinh ^3(x)}}{\sqrt {a}}\right )}{3 \sqrt {a}} \]

[Out]

-2/3*arctanh((a+b*sinh(x)^3)^(1/2)/a^(1/2))/a^(1/2)

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Rubi [A]
time = 0.05, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {3309, 272, 65, 214} \begin {gather*} -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b \sinh ^3(x)}}{\sqrt {a}}\right )}{3 \sqrt {a}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Coth[x]/Sqrt[a + b*Sinh[x]^3],x]

[Out]

(-2*ArcTanh[Sqrt[a + b*Sinh[x]^3]/Sqrt[a]])/(3*Sqrt[a])

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 214

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]

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[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]]

Rule 3309

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]}, Dist[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]

Rubi steps

\begin {align*} \int \frac {\coth (x)}{\sqrt {a+b \sinh ^3(x)}} \, dx &=\text {Subst}\left (\int \frac {1}{x \sqrt {a+b x^3}} \, dx,x,\sinh (x)\right )\\ &=\frac {1}{3} \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\sinh ^3(x)\right )\\ &=\frac {2 \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^3(x)}\right )}{3 b}\\ &=-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b \sinh ^3(x)}}{\sqrt {a}}\right )}{3 \sqrt {a}}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 28, normalized size = 1.00 \begin {gather*} -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b \sinh ^3(x)}}{\sqrt {a}}\right )}{3 \sqrt {a}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Coth[x]/Sqrt[a + b*Sinh[x]^3],x]

[Out]

(-2*ArcTanh[Sqrt[a + b*Sinh[x]^3]/Sqrt[a]])/(3*Sqrt[a])

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Maple [A]
time = 6.38, size = 21, normalized size = 0.75

method	result	size
derivativedivides	\(-\frac {2 \arctanh \left (\frac {\sqrt {a +b \left (\sinh ^{3}\left (x \right )\right )}}{\sqrt {a}}\right )}{3 \sqrt {a}}\)	\(21\)
default	\(-\frac {2 \arctanh \left (\frac {\sqrt {a +b \left (\sinh ^{3}\left (x \right )\right )}}{\sqrt {a}}\right )}{3 \sqrt {a}}\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(x)/(a+b*sinh(x)^3)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-2/3*arctanh((a+b*sinh(x)^3)^(1/2)/a^(1/2))/a^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x)/(a+b*sinh(x)^3)^(1/2),x, algorithm="maxima")

[Out]

integrate(coth(x)/sqrt(b*sinh(x)^3 + a), x)

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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x)/(a+b*sinh(x)^3)^(1/2),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   failed of mode Union(SparseUnivariatePol
ynomial(Expression(Integer)),failed) cannot be coerced to mode SparseUnivariatePolynomial(Expression(Integer))

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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 ^{3}{\left (x \right )}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x)/(a+b*sinh(x)**3)**(1/2),x)

[Out]

Integral(coth(x)/sqrt(a + b*sinh(x)**3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x)/(a+b*sinh(x)^3)^(1/2),x, algorithm="giac")

[Out]

integrate(coth(x)/sqrt(b*sinh(x)^3 + a), x)

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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 {b\,{\mathrm {sinh}\left (x\right )}^3+a}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(x)/(a + b*sinh(x)^3)^(1/2),x)

[Out]

int(coth(x)/(a + b*sinh(x)^3)^(1/2), x)

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