3.523 \(\int \coth (x) \sqrt{a+b \sinh ^3(x)} \, dx\)
Optimal. Leaf size=45 \[ \frac{2}{3} \sqrt{a+b \sinh ^3(x)}-\frac{2}{3} \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b \sinh ^3(x)}}{\sqrt{a}}\right ) \]
[Out]
(-2*Sqrt[a]*ArcTanh[Sqrt[a + b*Sinh[x]^3]/Sqrt[a]])/3 + (2*Sqrt[a + b*Sinh[x]^3])/3
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Rubi [A] time = 0.0822686, antiderivative size = 45, normalized size of antiderivative = 1.,
number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used =
{3230, 266, 50, 63, 208} \[ \frac{2}{3} \sqrt{a+b \sinh ^3(x)}-\frac{2}{3} \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b \sinh ^3(x)}}{\sqrt{a}}\right ) \]
Antiderivative was successfully verified.
[In]
Int[Coth[x]*Sqrt[a + b*Sinh[x]^3],x]
[Out]
(-2*Sqrt[a]*ArcTanh[Sqrt[a + b*Sinh[x]^3]/Sqrt[a]])/3 + (2*Sqrt[a + b*Sinh[x]^3])/3
Rule 3230
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]
Rule 266
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 50
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 63
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 208
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]
Rubi steps
\begin{align*} \int \coth (x) \sqrt{a+b \sinh ^3(x)} \, dx &=\operatorname{Subst}\left (\int \frac{\sqrt{a+b x^3}}{x} \, dx,x,\sinh (x)\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{\sqrt{a+b x}}{x} \, dx,x,\sinh ^3(x)\right )\\ &=\frac{2}{3} \sqrt{a+b \sinh ^3(x)}+\frac{1}{3} a \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,\sinh ^3(x)\right )\\ &=\frac{2}{3} \sqrt{a+b \sinh ^3(x)}+\frac{(2 a) \operatorname{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}{3} \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b \sinh ^3(x)}}{\sqrt{a}}\right )+\frac{2}{3} \sqrt{a+b \sinh ^3(x)}\\ \end{align*}
Mathematica [A] time = 0.0225346, size = 45, normalized size = 1. \[ \frac{2}{3} \sqrt{a+b \sinh ^3(x)}-\frac{2}{3} \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b \sinh ^3(x)}}{\sqrt{a}}\right ) \]
Antiderivative was successfully verified.
[In]
Integrate[Coth[x]*Sqrt[a + b*Sinh[x]^3],x]
[Out]
(-2*Sqrt[a]*ArcTanh[Sqrt[a + b*Sinh[x]^3]/Sqrt[a]])/3 + (2*Sqrt[a + b*Sinh[x]^3])/3
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Maple [A] time = 0.042, size = 34, normalized size = 0.8 \begin{align*} -{\frac{2}{3}{\it Artanh} \left ({\sqrt{a+b \left ( \sinh \left ( x \right ) \right ) ^{3}}{\frac{1}{\sqrt{a}}}} \right ) \sqrt{a}}+{\frac{2}{3}\sqrt{a+b \left ( \sinh \left ( x \right ) \right ) ^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(coth(x)*(a+b*sinh(x)^3)^(1/2),x)
[Out]
-2/3*arctanh((a+b*sinh(x)^3)^(1/2)/a^(1/2))*a^(1/2)+2/3*(a+b*sinh(x)^3)^(1/2)
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{b \sinh \left (x\right )^{3} + a} \coth \left (x\right )\,{d x} \end{align*}
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(sqrt(b*sinh(x)^3 + a)*coth(x), x)
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Fricas [B] time = 15.5511, size = 5184, normalized size = 115.2 \begin{align*} \text{result too large to display} \end{align*}
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]
[1/6*(sqrt(a)*(cosh(x) + sinh(x))*log(-(b^2*cosh(x)^12 + 12*b^2*cosh(x)*sinh(x)^11 + b^2*sinh(x)^12 - 6*b^2*co
sh(x)^10 + 64*a*b*cosh(x)^9 + 6*(11*b^2*cosh(x)^2 - b^2)*sinh(x)^10 + 15*b^2*cosh(x)^8 + 4*(55*b^2*cosh(x)^3 -
15*b^2*cosh(x) + 16*a*b)*sinh(x)^9 - 192*a*b*cosh(x)^7 + 3*(165*b^2*cosh(x)^4 - 90*b^2*cosh(x)^2 + 192*a*b*co
sh(x) + 5*b^2)*sinh(x)^8 + 24*(33*b^2*cosh(x)^5 - 30*b^2*cosh(x)^3 + 96*a*b*cosh(x)^2 + 5*b^2*cosh(x) - 8*a*b)
*sinh(x)^7 + 192*a*b*cosh(x)^5 + 4*(128*a^2 - 5*b^2)*cosh(x)^6 + 4*(231*b^2*cosh(x)^6 - 315*b^2*cosh(x)^4 + 13
44*a*b*cosh(x)^3 + 105*b^2*cosh(x)^2 - 336*a*b*cosh(x) + 128*a^2 - 5*b^2)*sinh(x)^6 + 15*b^2*cosh(x)^4 + 24*(3
3*b^2*cosh(x)^7 - 63*b^2*cosh(x)^5 + 336*a*b*cosh(x)^4 + 35*b^2*cosh(x)^3 - 168*a*b*cosh(x)^2 + 8*a*b + (128*a
^2 - 5*b^2)*cosh(x))*sinh(x)^5 - 64*a*b*cosh(x)^3 + 3*(165*b^2*cosh(x)^8 - 420*b^2*cosh(x)^6 + 2688*a*b*cosh(x
)^5 + 350*b^2*cosh(x)^4 - 2240*a*b*cosh(x)^3 + 320*a*b*cosh(x) + 20*(128*a^2 - 5*b^2)*cosh(x)^2 + 5*b^2)*sinh(
x)^4 - 6*b^2*cosh(x)^2 + 4*(55*b^2*cosh(x)^9 - 180*b^2*cosh(x)^7 + 1344*a*b*cosh(x)^6 + 210*b^2*cosh(x)^5 - 16
80*a*b*cosh(x)^4 + 480*a*b*cosh(x)^2 + 20*(128*a^2 - 5*b^2)*cosh(x)^3 + 15*b^2*cosh(x) - 16*a*b)*sinh(x)^3 + 6
*(11*b^2*cosh(x)^10 - 45*b^2*cosh(x)^8 + 384*a*b*cosh(x)^7 + 70*b^2*cosh(x)^6 - 672*a*b*cosh(x)^5 + 320*a*b*co
sh(x)^3 + 10*(128*a^2 - 5*b^2)*cosh(x)^4 + 15*b^2*cosh(x)^2 - 32*a*b*cosh(x) - b^2)*sinh(x)^2 + b^2 - 16*(b*co
sh(x)^8 + 8*b*cosh(x)*sinh(x)^7 + b*sinh(x)^8 - 3*b*cosh(x)^6 + (28*b*cosh(x)^2 - 3*b)*sinh(x)^6 + 16*a*cosh(x
)^5 + 2*(28*b*cosh(x)^3 - 9*b*cosh(x) + 8*a)*sinh(x)^5 + 3*b*cosh(x)^4 + (70*b*cosh(x)^4 - 45*b*cosh(x)^2 + 80
*a*cosh(x) + 3*b)*sinh(x)^4 + 4*(14*b*cosh(x)^5 - 15*b*cosh(x)^3 + 40*a*cosh(x)^2 + 3*b*cosh(x))*sinh(x)^3 - b
*cosh(x)^2 + (28*b*cosh(x)^6 - 45*b*cosh(x)^4 + 160*a*cosh(x)^3 + 18*b*cosh(x)^2 - b)*sinh(x)^2 + 2*(4*b*cosh(
x)^7 - 9*b*cosh(x)^5 + 40*a*cosh(x)^4 + 6*b*cosh(x)^3 - b*cosh(x))*sinh(x))*sqrt(a)*sqrt((b*sinh(x)^3 + 3*(b*c
osh(x)^2 - b)*sinh(x) + 4*a)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2)) + 12*(b^2*cosh(x)^11 - 5*b^2*cosh(x)
^9 + 48*a*b*cosh(x)^8 + 10*b^2*cosh(x)^7 - 112*a*b*cosh(x)^6 + 80*a*b*cosh(x)^4 + 2*(128*a^2 - 5*b^2)*cosh(x)^
5 + 5*b^2*cosh(x)^3 - 16*a*b*cosh(x)^2 - b^2*cosh(x))*sinh(x))/(cosh(x)^12 + 12*cosh(x)*sinh(x)^11 + sinh(x)^1
2 + 6*(11*cosh(x)^2 - 1)*sinh(x)^10 - 6*cosh(x)^10 + 20*(11*cosh(x)^3 - 3*cosh(x))*sinh(x)^9 + 15*(33*cosh(x)^
4 - 18*cosh(x)^2 + 1)*sinh(x)^8 + 15*cosh(x)^8 + 24*(33*cosh(x)^5 - 30*cosh(x)^3 + 5*cosh(x))*sinh(x)^7 + 4*(2
31*cosh(x)^6 - 315*cosh(x)^4 + 105*cosh(x)^2 - 5)*sinh(x)^6 - 20*cosh(x)^6 + 24*(33*cosh(x)^7 - 63*cosh(x)^5 +
35*cosh(x)^3 - 5*cosh(x))*sinh(x)^5 + 15*(33*cosh(x)^8 - 84*cosh(x)^6 + 70*cosh(x)^4 - 20*cosh(x)^2 + 1)*sinh
(x)^4 + 15*cosh(x)^4 + 20*(11*cosh(x)^9 - 36*cosh(x)^7 + 42*cosh(x)^5 - 20*cosh(x)^3 + 3*cosh(x))*sinh(x)^3 +
6*(11*cosh(x)^10 - 45*cosh(x)^8 + 70*cosh(x)^6 - 50*cosh(x)^4 + 15*cosh(x)^2 - 1)*sinh(x)^2 - 6*cosh(x)^2 + 12
*(cosh(x)^11 - 5*cosh(x)^9 + 10*cosh(x)^7 - 10*cosh(x)^5 + 5*cosh(x)^3 - cosh(x))*sinh(x) + 1)) + 2*sqrt((b*si
nh(x)^3 + 3*(b*cosh(x)^2 - b)*sinh(x) + 4*a)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2)))/(cosh(x) + sinh(x))
, 1/3*(sqrt(-a)*(cosh(x) + sinh(x))*arctan(8*(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2)*sqrt(-a)*sqrt((b*sinh
(x)^3 + 3*(b*cosh(x)^2 - b)*sinh(x) + 4*a)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2))/(b*cosh(x)^6 + 6*b*cos
h(x)*sinh(x)^5 + b*sinh(x)^6 - 3*b*cosh(x)^4 + 3*(5*b*cosh(x)^2 - b)*sinh(x)^4 + 16*a*cosh(x)^3 + 4*(5*b*cosh(
x)^3 - 3*b*cosh(x) + 4*a)*sinh(x)^3 + 3*b*cosh(x)^2 + 3*(5*b*cosh(x)^4 - 6*b*cosh(x)^2 + 16*a*cosh(x) + b)*sin
h(x)^2 + 6*(b*cosh(x)^5 - 2*b*cosh(x)^3 + 8*a*cosh(x)^2 + b*cosh(x))*sinh(x) - b)) + sqrt((b*sinh(x)^3 + 3*(b*
cosh(x)^2 - b)*sinh(x) + 4*a)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2)))/(cosh(x) + sinh(x))]
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a + b \sinh ^{3}{\left (x \right )}} \coth{\left (x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(coth(x)*(a+b*sinh(x)**3)**(1/2),x)
[Out]
Integral(sqrt(a + b*sinh(x)**3)*coth(x), x)
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{b \sinh \left (x\right )^{3} + a} \coth \left (x\right )\,{d x} \end{align*}
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(sqrt(b*sinh(x)^3 + a)*coth(x), x)