∫ coth(x)∘ ________ a + b sinh3(x) dx [523]

Optimal. Leaf size=45 \[ -\frac {2}{3} \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b \sinh ^3(x)}}{\sqrt {a}}\right )+\frac {2}{3} \sqrt {a+b \sinh ^3(x)} \]

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Rubi [A]
time = 0.05, antiderivative size = 45, normalized size of antiderivative = 1.00, 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 = {3309, 272, 52, 65, 214} \begin {gather*} \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 ) \end {gather*}

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

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/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a

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]

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

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Mathematica [A]
time = 0.02, size = 45, normalized size = 1.00 \begin {gather*} -\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 {gather*}

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method	result	size

derivativedivides	\(-\frac {2 \arctanh \left (\frac {\sqrt {a +b \left (\sinh ^{3}\left (x \right )\right )}}{\sqrt {a}}\right ) \sqrt {a}}{3}+\frac {2 \sqrt {a +b \left (\sinh ^{3}\left (x \right )\right )}}{3}\)	\(34\)
default	\(-\frac {2 \arctanh \left (\frac {\sqrt {a +b \left (\sinh ^{3}\left (x \right )\right )}}{\sqrt {a}}\right ) \sqrt {a}}{3}+\frac {2 \sqrt {a +b \left (\sinh ^{3}\left (x \right )\right )}}{3}\)	\(34\)

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 275 vs. \(2 (33) = 66\).
time = 1.51, size = 1663, normalized size = 36.96 \begin {gather*} \text {Too large to display} \end {gather*}

[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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {a + b \sinh ^{3}{\left (x \right )}} \coth {\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*}

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \mathrm {coth}\left (x\right )\,\sqrt {b\,{\mathrm {sinh}\left (x\right )}^3+a} \,d x \end {gather*}

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3.6.23 \(\int \coth (x) \sqrt {a+b \sinh ^3(x)} \, dx\) [523]