3.6.21 \(\int \frac {\coth ^3(x)}{a+b \sinh ^3(x)} \, dx\) [521]

Optimal. Leaf size=152 \[ \frac {b^{2/3} \text {ArcTan}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sinh (x)}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{5/3}}-\frac {\text {csch}^2(x)}{2 a}+\frac {\log (\sinh (x))}{a}-\frac {b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sinh (x)\right )}{3 a^{5/3}}+\frac {b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sinh (x)+b^{2/3} \sinh ^2(x)\right )}{6 a^{5/3}}-\frac {\log \left (a+b \sinh ^3(x)\right )}{3 a} \]

[Out]

-1/2*csch(x)^2/a+ln(sinh(x))/a-1/3*b^(2/3)*ln(a^(1/3)+b^(1/3)*sinh(x))/a^(5/3)+1/6*b^(2/3)*ln(a^(2/3)-a^(1/3)*
b^(1/3)*sinh(x)+b^(2/3)*sinh(x)^2)/a^(5/3)-1/3*ln(a+b*sinh(x)^3)/a+1/3*b^(2/3)*arctan(1/3*(a^(1/3)-2*b^(1/3)*s
inh(x))/a^(1/3)*3^(1/2))/a^(5/3)*3^(1/2)

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Rubi [A]
time = 0.17, antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 11, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.733, Rules used = {3309, 1848, 1885, 12, 206, 31, 648, 631, 210, 642, 266} \begin {gather*} \frac {b^{2/3} \text {ArcTan}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sinh (x)}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{5/3}}+\frac {b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sinh (x)+b^{2/3} \sinh ^2(x)\right )}{6 a^{5/3}}-\frac {b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sinh (x)\right )}{3 a^{5/3}}-\frac {\log \left (a+b \sinh ^3(x)\right )}{3 a}-\frac {\text {csch}^2(x)}{2 a}+\frac {\log (\sinh (x))}{a} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Coth[x]^3/(a + b*Sinh[x]^3),x]

[Out]

(b^(2/3)*ArcTan[(a^(1/3) - 2*b^(1/3)*Sinh[x])/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*a^(5/3)) - Csch[x]^2/(2*a) + Log[Si
nh[x]]/a - (b^(2/3)*Log[a^(1/3) + b^(1/3)*Sinh[x]])/(3*a^(5/3)) + (b^(2/3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*Sinh[
x] + b^(2/3)*Sinh[x]^2])/(6*a^(5/3)) - Log[a + b*Sinh[x]^3]/(3*a)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 206

Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Dist[1/(3*Rt[a, 3]^2), Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Di
st[1/(3*Rt[a, 3]^2), Int[(2*Rt[a, 3] - Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2*x^2), x], x]
/; FreeQ[{a, b}, x]

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 266

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 648

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 1848

Int[((Pq_)*((c_.)*(x_))^(m_.))/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(Pq/(a + b*x
^n)), x], x] /; FreeQ[{a, b, c, m}, x] && PolyQ[Pq, x] && IntegerQ[n] &&  !IGtQ[m, 0]

Rule 1885

Int[(P2_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> With[{A = Coeff[P2, x, 0], B = Coeff[P2, x, 1], C = Coeff[P2, x,
2]}, Int[(A + B*x)/(a + b*x^3), x] + Dist[C, Int[x^2/(a + b*x^3), x], x] /; EqQ[a*B^3 - b*A^3, 0] ||  !Ration
alQ[a/b]] /; FreeQ[{a, b}, x] && PolyQ[P2, x, 2]

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 ^3(x)}{a+b \sinh ^3(x)} \, dx &=\text {Subst}\left (\int \frac {1+x^2}{x^3 \left (a+b x^3\right )} \, dx,x,\sinh (x)\right )\\ &=\text {Subst}\left (\int \left (\frac {1}{a x^3}+\frac {1}{a x}+\frac {-b-b x^2}{a \left (a+b x^3\right )}\right ) \, dx,x,\sinh (x)\right )\\ &=-\frac {\text {csch}^2(x)}{2 a}+\frac {\log (\sinh (x))}{a}+\frac {\text {Subst}\left (\int \frac {-b-b x^2}{a+b x^3} \, dx,x,\sinh (x)\right )}{a}\\ &=-\frac {\text {csch}^2(x)}{2 a}+\frac {\log (\sinh (x))}{a}-\frac {\text {Subst}\left (\int \frac {b}{a+b x^3} \, dx,x,\sinh (x)\right )}{a}-\frac {b \text {Subst}\left (\int \frac {x^2}{a+b x^3} \, dx,x,\sinh (x)\right )}{a}\\ &=-\frac {\text {csch}^2(x)}{2 a}+\frac {\log (\sinh (x))}{a}-\frac {\log \left (a+b \sinh ^3(x)\right )}{3 a}-\frac {b \text {Subst}\left (\int \frac {1}{a+b x^3} \, dx,x,\sinh (x)\right )}{a}\\ &=-\frac {\text {csch}^2(x)}{2 a}+\frac {\log (\sinh (x))}{a}-\frac {\log \left (a+b \sinh ^3(x)\right )}{3 a}-\frac {b \text {Subst}\left (\int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx,x,\sinh (x)\right )}{3 a^{5/3}}-\frac {b \text {Subst}\left (\int \frac {2 \sqrt [3]{a}-\sqrt [3]{b} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\sinh (x)\right )}{3 a^{5/3}}\\ &=-\frac {\text {csch}^2(x)}{2 a}+\frac {\log (\sinh (x))}{a}-\frac {b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sinh (x)\right )}{3 a^{5/3}}-\frac {\log \left (a+b \sinh ^3(x)\right )}{3 a}+\frac {b^{2/3} \text {Subst}\left (\int \frac {-\sqrt [3]{a} \sqrt [3]{b}+2 b^{2/3} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\sinh (x)\right )}{6 a^{5/3}}-\frac {b \text {Subst}\left (\int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\sinh (x)\right )}{2 a^{4/3}}\\ &=-\frac {\text {csch}^2(x)}{2 a}+\frac {\log (\sinh (x))}{a}-\frac {b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sinh (x)\right )}{3 a^{5/3}}+\frac {b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sinh (x)+b^{2/3} \sinh ^2(x)\right )}{6 a^{5/3}}-\frac {\log \left (a+b \sinh ^3(x)\right )}{3 a}-\frac {b^{2/3} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} \sinh (x)}{\sqrt [3]{a}}\right )}{a^{5/3}}\\ &=\frac {b^{2/3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} \sinh (x)}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt {3} a^{5/3}}-\frac {\text {csch}^2(x)}{2 a}+\frac {\log (\sinh (x))}{a}-\frac {b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sinh (x)\right )}{3 a^{5/3}}+\frac {b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sinh (x)+b^{2/3} \sinh ^2(x)\right )}{6 a^{5/3}}-\frac {\log \left (a+b \sinh ^3(x)\right )}{3 a}\\ \end {align*}

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Mathematica [A]
time = 0.25, size = 136, normalized size = 0.89 \begin {gather*} -\frac {\text {csch}^2(x)}{2 a}+\frac {\log (\sinh (x))}{a}-\frac {\left (a^{2/3}+(-1)^{2/3} b^{2/3}\right ) \log \left (-(-1)^{2/3} \sqrt [3]{a}-\sqrt [3]{b} \sinh (x)\right )+\left (a^{2/3}+b^{2/3}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sinh (x)\right )+\left (a^{2/3}-\sqrt [3]{-1} b^{2/3}\right ) \log \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} \sinh (x)\right )}{3 a^{5/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*Csch[x]^2/a + Log[Sinh[x]]/a - ((a^(2/3) + (-1)^(2/3)*b^(2/3))*Log[-((-1)^(2/3)*a^(1/3)) - b^(1/3)*Sinh[x
]] + (a^(2/3) + b^(2/3))*Log[a^(1/3) + b^(1/3)*Sinh[x]] + (a^(2/3) - (-1)^(1/3)*b^(2/3))*Log[a^(1/3) + (-1)^(2
/3)*b^(1/3)*Sinh[x]])/(3*a^(5/3))

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 1.02, size = 132, normalized size = 0.87

method	result	size
risch	\(-\frac {2 \,{\mathrm e}^{2 x}}{\left ({\mathrm e}^{2 x}-1\right )^{2} a}+\frac {\ln \left ({\mathrm e}^{2 x}-1\right )}{a}+\left (\munderset {\textit {\_R} =\RootOf \left (27 a^{5} \textit {\_Z}^{3}+27 a^{4} \textit {\_Z}^{2}+9 \textit {\_Z} \,a^{3}+a^{2}+b^{2}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 x}+\left (-\frac {6 a^{2} \textit {\_R}}{b}-\frac {2 a}{b}\right ) {\mathrm e}^{x}-1\right )\right )\)	\(91\)
default	\(-\frac {\tanh ^{2}\left (\frac {x}{2}\right )}{8 a}-\frac {1}{8 a \tanh \left (\frac {x}{2}\right )^{2}}+\frac {\ln \left (\tanh \left (\frac {x}{2}\right )\right )}{a}+\frac {\munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{6}-3 a \,\textit {\_Z}^{4}-8 b \,\textit {\_Z}^{3}+3 a \,\textit {\_Z}^{2}-a \right )}{\sum }\frac {\left (-\textit {\_R}^{5} a -\textit {\_R}^{4} b +2 \textit {\_R}^{3} a +4 \textit {\_R}^{2} b -\textit {\_R} a +b \right ) \ln \left (\tanh \left (\frac {x}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5} a -2 \textit {\_R}^{3} a -4 \textit {\_R}^{2} b +\textit {\_R} a}}{3 a}\)	\(132\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*tanh(1/2*x)^2/a-1/8/a/tanh(1/2*x)^2+1/a*ln(tanh(1/2*x))+1/3/a*sum((-_R^5*a-_R^4*b+2*_R^3*a+4*_R^2*b-_R*a+
b)/(_R^5*a-2*_R^3*a-4*_R^2*b+_R*a)*ln(tanh(1/2*x)-_R),_R=RootOf(_Z^6*a-3*_Z^4*a-8*_Z^3*b+3*_Z^2*a-a))

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

[Out]

2*b*(x/(a*b) - integrate((b*e^(5*x) - 3*b*e^(3*x) + 8*a*e^(2*x) + 3*b*e^x)*e^x/(b*e^(6*x) - 3*b*e^(4*x) + 8*a*
e^(3*x) + 3*b*e^(2*x) - b), x)/(a*b)) - 6*b*integrate(e^(4*x)/(b*e^(6*x) - 3*b*e^(4*x) + 8*a*e^(3*x) + 3*b*e^(
2*x) - b), x)/a - 2*(x*e^(4*x) - (2*x - 1)*e^(2*x) + x)/(a*e^(4*x) - 2*a*e^(2*x) + a) + log(e^x + 1)/a + log(e
^x - 1)/a + 8*integrate(e^(3*x)/(b*e^(6*x) - 3*b*e^(4*x) + 8*a*e^(3*x) + 3*b*e^(2*x) - b), x)

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Fricas [C] Result contains complex when optimal does not.
time = 1.19, size = 1115, normalized size = 7.34 \begin {gather*} -\frac {12 \, \sqrt {\frac {1}{3}} {\left (a e^{\left (4 \, x\right )} - 2 \, a e^{\left (2 \, x\right )} + a\right )} \sqrt {\frac {{\left (\left (\frac {1}{2}\right )^{\frac {1}{3}} {\left (i \, \sqrt {3} + 1\right )} {\left (\frac {1}{a^{3}} + \frac {b^{2}}{a^{5}} - \frac {a^{2} + b^{2}}{a^{5}}\right )}^{\frac {1}{3}} + \frac {2}{a}\right )}^{2} a^{2} - 4 \, {\left (\left (\frac {1}{2}\right )^{\frac {1}{3}} {\left (i \, \sqrt {3} + 1\right )} {\left (\frac {1}{a^{3}} + \frac {b^{2}}{a^{5}} - \frac {a^{2} + b^{2}}{a^{5}}\right )}^{\frac {1}{3}} + \frac {2}{a}\right )} a + 4}{a^{2}}} \arctan \left (-\frac {{\left (2 \, \sqrt {\frac {1}{3}} \sqrt {{\left (\left (\frac {1}{2}\right )^{\frac {1}{3}} {\left (i \, \sqrt {3} + 1\right )} {\left (\frac {1}{a^{3}} + \frac {b^{2}}{a^{5}} - \frac {a^{2} + b^{2}}{a^{5}}\right )}^{\frac {1}{3}} + \frac {2}{a}\right )}^{2} a^{4} e^{\left (2 \, x\right )} + b^{2} e^{\left (4 \, x\right )} + 2 \, a b e^{\left (3 \, x\right )} - 2 \, a b e^{x} - {\left (a^{2} b e^{\left (3 \, x\right )} + 4 \, a^{3} e^{\left (2 \, x\right )} - a^{2} b e^{x}\right )} {\left (\left (\frac {1}{2}\right )^{\frac {1}{3}} {\left (i \, \sqrt {3} + 1\right )} {\left (\frac {1}{a^{3}} + \frac {b^{2}}{a^{5}} - \frac {a^{2} + b^{2}}{a^{5}}\right )}^{\frac {1}{3}} + \frac {2}{a}\right )} + b^{2} + 2 \, {\left (2 \, a^{2} - b^{2}\right )} e^{\left (2 \, x\right )}} {\left ({\left (\left (\frac {1}{2}\right )^{\frac {1}{3}} {\left (i \, \sqrt {3} + 1\right )} {\left (\frac {1}{a^{3}} + \frac {b^{2}}{a^{5}} - \frac {a^{2} + b^{2}}{a^{5}}\right )}^{\frac {1}{3}} + \frac {2}{a}\right )} a^{3} - 2 \, a^{2}\right )} \sqrt {\frac {{\left (\left (\frac {1}{2}\right )^{\frac {1}{3}} {\left (i \, \sqrt {3} + 1\right )} {\left (\frac {1}{a^{3}} + \frac {b^{2}}{a^{5}} - \frac {a^{2} + b^{2}}{a^{5}}\right )}^{\frac {1}{3}} + \frac {2}{a}\right )}^{2} a^{2} - 4 \, {\left (\left (\frac {1}{2}\right )^{\frac {1}{3}} {\left (i \, \sqrt {3} + 1\right )} {\left (\frac {1}{a^{3}} + \frac {b^{2}}{a^{5}} - \frac {a^{2} + b^{2}}{a^{5}}\right )}^{\frac {1}{3}} + \frac {2}{a}\right )} a + 4}{a^{2}}} + \sqrt {\frac {1}{3}} {\left ({\left (\left (\frac {1}{2}\right )^{\frac {1}{3}} {\left (i \, \sqrt {3} + 1\right )} {\left (\frac {1}{a^{3}} + \frac {b^{2}}{a^{5}} - \frac {a^{2} + b^{2}}{a^{5}}\right )}^{\frac {1}{3}} + \frac {2}{a}\right )}^{2} a^{5} e^{x} + 4 \, a^{2} b e^{\left (2 \, x\right )} + 4 \, a^{3} e^{x} - 4 \, a^{2} b - 2 \, {\left (a^{3} b e^{\left (2 \, x\right )} + 2 \, a^{4} e^{x} - a^{3} b\right )} {\left (\left (\frac {1}{2}\right )^{\frac {1}{3}} {\left (i \, \sqrt {3} + 1\right )} {\left (\frac {1}{a^{3}} + \frac {b^{2}}{a^{5}} - \frac {a^{2} + b^{2}}{a^{5}}\right )}^{\frac {1}{3}} + \frac {2}{a}\right )}\right )} \sqrt {\frac {{\left (\left (\frac {1}{2}\right )^{\frac {1}{3}} {\left (i \, \sqrt {3} + 1\right )} {\left (\frac {1}{a^{3}} + \frac {b^{2}}{a^{5}} - \frac {a^{2} + b^{2}}{a^{5}}\right )}^{\frac {1}{3}} + \frac {2}{a}\right )}^{2} a^{2} - 4 \, {\left (\left (\frac {1}{2}\right )^{\frac {1}{3}} {\left (i \, \sqrt {3} + 1\right )} {\left (\frac {1}{a^{3}} + \frac {b^{2}}{a^{5}} - \frac {a^{2} + b^{2}}{a^{5}}\right )}^{\frac {1}{3}} + \frac {2}{a}\right )} a + 4}{a^{2}}}\right )} e^{\left (-x\right )}}{8 \, b^{2}}\right ) + 2 \, {\left (a e^{\left (4 \, x\right )} - 2 \, a e^{\left (2 \, x\right )} + a\right )} {\left (\left (\frac {1}{2}\right )^{\frac {1}{3}} {\left (i \, \sqrt {3} + 1\right )} {\left (\frac {1}{a^{3}} + \frac {b^{2}}{a^{5}} - \frac {a^{2} + b^{2}}{a^{5}}\right )}^{\frac {1}{3}} + \frac {2}{a}\right )} \log \left ({\left (\left (\frac {1}{2}\right )^{\frac {1}{3}} {\left (i \, \sqrt {3} + 1\right )} {\left (\frac {1}{a^{3}} + \frac {b^{2}}{a^{5}} - \frac {a^{2} + b^{2}}{a^{5}}\right )}^{\frac {1}{3}} + \frac {2}{a}\right )} a^{2} e^{x} + b e^{\left (2 \, x\right )} - 2 \, a e^{x} - b\right ) - {\left ({\left (a e^{\left (4 \, x\right )} - 2 \, a e^{\left (2 \, x\right )} + a\right )} {\left (\left (\frac {1}{2}\right )^{\frac {1}{3}} {\left (i \, \sqrt {3} + 1\right )} {\left (\frac {1}{a^{3}} + \frac {b^{2}}{a^{5}} - \frac {a^{2} + b^{2}}{a^{5}}\right )}^{\frac {1}{3}} + \frac {2}{a}\right )} - 6 \, e^{\left (4 \, x\right )} + 12 \, e^{\left (2 \, x\right )} - 6\right )} \log \left ({\left (\left (\frac {1}{2}\right )^{\frac {1}{3}} {\left (i \, \sqrt {3} + 1\right )} {\left (\frac {1}{a^{3}} + \frac {b^{2}}{a^{5}} - \frac {a^{2} + b^{2}}{a^{5}}\right )}^{\frac {1}{3}} + \frac {2}{a}\right )}^{2} a^{4} e^{\left (2 \, x\right )} + b^{2} e^{\left (4 \, x\right )} + 2 \, a b e^{\left (3 \, x\right )} - 2 \, a b e^{x} - {\left (a^{2} b e^{\left (3 \, x\right )} + 4 \, a^{3} e^{\left (2 \, x\right )} - a^{2} b e^{x}\right )} {\left (\left (\frac {1}{2}\right )^{\frac {1}{3}} {\left (i \, \sqrt {3} + 1\right )} {\left (\frac {1}{a^{3}} + \frac {b^{2}}{a^{5}} - \frac {a^{2} + b^{2}}{a^{5}}\right )}^{\frac {1}{3}} + \frac {2}{a}\right )} + b^{2} + 2 \, {\left (2 \, a^{2} - b^{2}\right )} e^{\left (2 \, x\right )}\right ) - 12 \, {\left (e^{\left (4 \, x\right )} - 2 \, e^{\left (2 \, x\right )} + 1\right )} \log \left (e^{\left (2 \, x\right )} - 1\right ) + 24 \, e^{\left (2 \, x\right )}}{12 \, {\left (a e^{\left (4 \, x\right )} - 2 \, a e^{\left (2 \, x\right )} + a\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*(12*sqrt(1/3)*(a*e^(4*x) - 2*a*e^(2*x) + a)*sqrt((((1/2)^(1/3)*(I*sqrt(3) + 1)*(1/a^3 + b^2/a^5 - (a^2 +
b^2)/a^5)^(1/3) + 2/a)^2*a^2 - 4*((1/2)^(1/3)*(I*sqrt(3) + 1)*(1/a^3 + b^2/a^5 - (a^2 + b^2)/a^5)^(1/3) + 2/a
)*a + 4)/a^2)*arctan(-1/8*(2*sqrt(1/3)*sqrt(((1/2)^(1/3)*(I*sqrt(3) + 1)*(1/a^3 + b^2/a^5 - (a^2 + b^2)/a^5)^(
1/3) + 2/a)^2*a^4*e^(2*x) + b^2*e^(4*x) + 2*a*b*e^(3*x) - 2*a*b*e^x - (a^2*b*e^(3*x) + 4*a^3*e^(2*x) - a^2*b*e
^x)*((1/2)^(1/3)*(I*sqrt(3) + 1)*(1/a^3 + b^2/a^5 - (a^2 + b^2)/a^5)^(1/3) + 2/a) + b^2 + 2*(2*a^2 - b^2)*e^(2
*x))*(((1/2)^(1/3)*(I*sqrt(3) + 1)*(1/a^3 + b^2/a^5 - (a^2 + b^2)/a^5)^(1/3) + 2/a)*a^3 - 2*a^2)*sqrt((((1/2)^
(1/3)*(I*sqrt(3) + 1)*(1/a^3 + b^2/a^5 - (a^2 + b^2)/a^5)^(1/3) + 2/a)^2*a^2 - 4*((1/2)^(1/3)*(I*sqrt(3) + 1)*
(1/a^3 + b^2/a^5 - (a^2 + b^2)/a^5)^(1/3) + 2/a)*a + 4)/a^2) + sqrt(1/3)*(((1/2)^(1/3)*(I*sqrt(3) + 1)*(1/a^3
+ b^2/a^5 - (a^2 + b^2)/a^5)^(1/3) + 2/a)^2*a^5*e^x + 4*a^2*b*e^(2*x) + 4*a^3*e^x - 4*a^2*b - 2*(a^3*b*e^(2*x)
+ 2*a^4*e^x - a^3*b)*((1/2)^(1/3)*(I*sqrt(3) + 1)*(1/a^3 + b^2/a^5 - (a^2 + b^2)/a^5)^(1/3) + 2/a))*sqrt((((1
/2)^(1/3)*(I*sqrt(3) + 1)*(1/a^3 + b^2/a^5 - (a^2 + b^2)/a^5)^(1/3) + 2/a)^2*a^2 - 4*((1/2)^(1/3)*(I*sqrt(3) +
1)*(1/a^3 + b^2/a^5 - (a^2 + b^2)/a^5)^(1/3) + 2/a)*a + 4)/a^2))*e^(-x)/b^2) + 2*(a*e^(4*x) - 2*a*e^(2*x) + a
)*((1/2)^(1/3)*(I*sqrt(3) + 1)*(1/a^3 + b^2/a^5 - (a^2 + b^2)/a^5)^(1/3) + 2/a)*log(((1/2)^(1/3)*(I*sqrt(3) +
1)*(1/a^3 + b^2/a^5 - (a^2 + b^2)/a^5)^(1/3) + 2/a)*a^2*e^x + b*e^(2*x) - 2*a*e^x - b) - ((a*e^(4*x) - 2*a*e^(
2*x) + a)*((1/2)^(1/3)*(I*sqrt(3) + 1)*(1/a^3 + b^2/a^5 - (a^2 + b^2)/a^5)^(1/3) + 2/a) - 6*e^(4*x) + 12*e^(2*
x) - 6)*log(((1/2)^(1/3)*(I*sqrt(3) + 1)*(1/a^3 + b^2/a^5 - (a^2 + b^2)/a^5)^(1/3) + 2/a)^2*a^4*e^(2*x) + b^2*
e^(4*x) + 2*a*b*e^(3*x) - 2*a*b*e^x - (a^2*b*e^(3*x) + 4*a^3*e^(2*x) - a^2*b*e^x)*((1/2)^(1/3)*(I*sqrt(3) + 1)
*(1/a^3 + b^2/a^5 - (a^2 + b^2)/a^5)^(1/3) + 2/a) + b^2 + 2*(2*a^2 - b^2)*e^(2*x)) - 12*(e^(4*x) - 2*e^(2*x) +
1)*log(e^(2*x) - 1) + 24*e^(2*x))/(a*e^(4*x) - 2*a*e^(2*x) + a)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\coth ^{3}{\left (x \right )}}{a + b \sinh ^{3}{\left (x \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.43, size = 209, normalized size = 1.38 \begin {gather*} \frac {b \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | -2 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}} - e^{\left (-x\right )} + e^{x} \right |}\right )}{3 \, a^{2}} - \frac {\log \left ({\left | -b {\left (e^{\left (-x\right )} - e^{x}\right )}^{3} + 8 \, a \right |}\right )}{3 \, a} + \frac {\log \left ({\left | -e^{\left (-x\right )} + e^{x} \right |}\right )}{a} - \frac {\sqrt {3} \left (-a b^{2}\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (\left (-\frac {a}{b}\right )^{\frac {1}{3}} - e^{\left (-x\right )} + e^{x}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, a^{2}} - \frac {\left (-a b^{2}\right )^{\frac {1}{3}} \log \left ({\left (e^{\left (-x\right )} - e^{x}\right )}^{2} - 2 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}} {\left (e^{\left (-x\right )} - e^{x}\right )} + 4 \, \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, a^{2}} - \frac {3 \, {\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 4}{2 \, a {\left (e^{\left (-x\right )} - e^{x}\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*b*(-a/b)^(1/3)*log(abs(-2*(-a/b)^(1/3) - e^(-x) + e^x))/a^2 - 1/3*log(abs(-b*(e^(-x) - e^x)^3 + 8*a))/a +
log(abs(-e^(-x) + e^x))/a - 1/3*sqrt(3)*(-a*b^2)^(1/3)*arctan(1/3*sqrt(3)*((-a/b)^(1/3) - e^(-x) + e^x)/(-a/b)
^(1/3))/a^2 - 1/6*(-a*b^2)^(1/3)*log((e^(-x) - e^x)^2 - 2*(-a/b)^(1/3)*(e^(-x) - e^x) + 4*(-a/b)^(2/3))/a^2 -
1/2*(3*(e^(-x) - e^x)^2 + 4)/(a*(e^(-x) - e^x)^2)

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Mupad [B]
time = 0.92, size = 1129, normalized size = 7.43 \begin {gather*} \frac {2}{a-a\,{\mathrm {e}}^{2\,x}}-\frac {2}{a-2\,a\,{\mathrm {e}}^{2\,x}+a\,{\mathrm {e}}^{4\,x}}+\left (\sum _{k=1}^3\ln \left (\frac {50331648\,a^6\,{\mathrm {e}}^{2\,x}+786432\,b^6\,{\mathrm {e}}^{2\,x}-\mathrm {root}\left (27\,a^5\,z^3+27\,a^4\,z^2+9\,a^3\,z+b^2+a^2,z,k\right )\,a^7\,452984832-50331648\,a^6-786432\,b^6-{\mathrm {root}\left (27\,a^5\,z^3+27\,a^4\,z^2+9\,a^3\,z+b^2+a^2,z,k\right )}^2\,a^8\,1358954496-{\mathrm {root}\left (27\,a^5\,z^3+27\,a^4\,z^2+9\,a^3\,z+b^2+a^2,z,k\right )}^3\,a^9\,1358954496-50593792\,a^2\,b^4-102498304\,a^4\,b^2+{\mathrm {root}\left (27\,a^5\,z^3+27\,a^4\,z^2+9\,a^3\,z+b^2+a^2,z,k\right )}^2\,a^8\,{\mathrm {e}}^{2\,x}\,1358954496+{\mathrm {root}\left (27\,a^5\,z^3+27\,a^4\,z^2+9\,a^3\,z+b^2+a^2,z,k\right )}^3\,a^9\,{\mathrm {e}}^{2\,x}\,1358954496+50593792\,a^2\,b^4\,{\mathrm {e}}^{2\,x}[phone]\,a^4\,b^2\,{\mathrm {e}}^{2\,x}-\mathrm {root}\left (27\,a^5\,z^3+27\,a^4\,z^2+9\,a^3\,z+b^2+a^2,z,k\right )\,a^3\,b^4\,7602176-\mathrm {root}\left (27\,a^5\,z^3+27\,a^4\,z^2+9\,a^3\,z+b^2+a^2,z,k\right )\,a^5\,b^2\,465305600+524288\,a\,b^5\,{\mathrm {e}}^x-{\mathrm {root}\left (27\,a^5\,z^3+27\,a^4\,z^2+9\,a^3\,z+b^2+a^2,z,k\right )}^2\,a^4\,b^4\,24379392-{\mathrm {root}\left (27\,a^5\,z^3+27\,a^4\,z^2+9\,a^3\,z+b^2+a^2,z,k\right )}^2\,a^6\,b^2\,1383333888-{\mathrm {root}\left (27\,a^5\,z^3+27\,a^4\,z^2+9\,a^3\,z+b^2+a^2,z,k\right )}^3\,a^5\,b^4\,18874368-{\mathrm {root}\left (27\,a^5\,z^3+27\,a^4\,z^2+9\,a^3\,z+b^2+a^2,z,k\right )}^3\,a^7\,b^2\,1370750976+\mathrm {root}\left (27\,a^5\,z^3+27\,a^4\,z^2+9\,a^3\,z+b^2+a^2,z,k\right )\,a^7\,{\mathrm {e}}^{2\,x}\,452984832+5242880\,a^3\,b^3\,{\mathrm {e}}^x-\mathrm {root}\left (27\,a^5\,z^3+27\,a^4\,z^2+9\,a^3\,z+b^2+a^2,z,k\right )\,a^2\,b^5\,{\mathrm {e}}^x\,524288+\mathrm {root}\left (27\,a^5\,z^3+27\,a^4\,z^2+9\,a^3\,z+b^2+a^2,z,k\right )\,a^4\,b^3\,{\mathrm {e}}^x\,8912896+\mathrm {root}\left (27\,a^5\,z^3+27\,a^4\,z^2+9\,a^3\,z+b^2+a^2,z,k\right )\,a^3\,b^4\,{\mathrm {e}}^{2\,x}\,7602176+\mathrm {root}\left (27\,a^5\,z^3+27\,a^4\,z^2+9\,a^3\,z+b^2+a^2,z,k\right )\,a^5\,b^2\,{\mathrm {e}}^{2\,x}\,465305600-{\mathrm {root}\left (27\,a^5\,z^3+27\,a^4\,z^2+9\,a^3\,z+b^2+a^2,z,k\right )}^3\,a^6\,b^3\,{\mathrm {e}}^x\,14155776+{\mathrm {root}\left (27\,a^5\,z^3+27\,a^4\,z^2+9\,a^3\,z+b^2+a^2,z,k\right )}^2\,a^4\,b^4\,{\mathrm {e}}^{2\,x}\,24379392+{\mathrm {root}\left (27\,a^5\,z^3+27\,a^4\,z^2+9\,a^3\,z+b^2+a^2,z,k\right )}^2\,a^6\,b^2\,{\mathrm {e}}^{2\,x}\,1383333888+{\mathrm {root}\left (27\,a^5\,z^3+27\,a^4\,z^2+9\,a^3\,z+b^2+a^2,z,k\right )}^3\,a^5\,b^4\,{\mathrm {e}}^{2\,x}\,18874368+{\mathrm {root}\left (27\,a^5\,z^3+27\,a^4\,z^2+9\,a^3\,z+b^2+a^2,z,k\right )}^3\,a^7\,b^2\,{\mathrm {e}}^{2\,x}\,1370750976}{a^6\,b^6}\right )\,\mathrm {root}\left (27\,a^5\,z^3+27\,a^4\,z^2+9\,a^3\,z+b^2+a^2,z,k\right )\right )+\frac {\ln \left (3221225472\,a^6\,{\mathrm {e}}^{2\,x}+786432\,b^6\,{\mathrm {e}}^{2\,x}-3221225472\,a^6-786432\,b^6-101449728\,a^2\,b^4-3321888768\,a^4\,b^2+101449728\,a^2\,b^4\,{\mathrm {e}}^{2\,x}[phone]\,a^4\,b^2\,{\mathrm {e}}^{2\,x}\right )}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(x)^3/(a + b*sinh(x)^3),x)

[Out]

2/(a - a*exp(2*x)) - 2/(a - 2*a*exp(2*x) + a*exp(4*x)) + symsum(log((50331648*a^6*exp(2*x) + 786432*b^6*exp(2*
x) - 452984832*root(27*a^5*z^3 + 27*a^4*z^2 + 9*a^3*z + b^2 + a^2, z, k)*a^7 - 50331648*a^6 - 786432*b^6 - 135
8954496*root(27*a^5*z^3 + 27*a^4*z^2 + 9*a^3*z + b^2 + a^2, z, k)^2*a^8 - 1358954496*root(27*a^5*z^3 + 27*a^4*
z^2 + 9*a^3*z + b^2 + a^2, z, k)^3*a^9 - 50593792*a^2*b^4 - 102498304*a^4*b^2 + 1358954496*root(27*a^5*z^3 + 2
7*a^4*z^2 + 9*a^3*z + b^2 + a^2, z, k)^2*a^8*exp(2*x) + 1358954496*root(27*a^5*z^3 + 27*a^4*z^2 + 9*a^3*z + b^
2 + a^2, z, k)^3*a^9*exp(2*x) + 50593792*a^2*b^4*exp(2*x) + 102498304*a^4*b^2*exp(2*x) - 7602176*root(27*a^5*z
^3 + 27*a^4*z^2 + 9*a^3*z + b^2 + a^2, z, k)*a^3*b^4 - 465305600*root(27*a^5*z^3 + 27*a^4*z^2 + 9*a^3*z + b^2
+ a^2, z, k)*a^5*b^2 + 524288*a*b^5*exp(x) - 24379392*root(27*a^5*z^3 + 27*a^4*z^2 + 9*a^3*z + b^2 + a^2, z, k
)^2*a^4*b^4 - 1383333888*root(27*a^5*z^3 + 27*a^4*z^2 + 9*a^3*z + b^2 + a^2, z, k)^2*a^6*b^2 - 18874368*root(2
7*a^5*z^3 + 27*a^4*z^2 + 9*a^3*z + b^2 + a^2, z, k)^3*a^5*b^4 - 1370750976*root(27*a^5*z^3 + 27*a^4*z^2 + 9*a^
3*z + b^2 + a^2, z, k)^3*a^7*b^2 + 452984832*root(27*a^5*z^3 + 27*a^4*z^2 + 9*a^3*z + b^2 + a^2, z, k)*a^7*exp
(2*x) + 5242880*a^3*b^3*exp(x) - 524288*root(27*a^5*z^3 + 27*a^4*z^2 + 9*a^3*z + b^2 + a^2, z, k)*a^2*b^5*exp(
x) + 8912896*root(27*a^5*z^3 + 27*a^4*z^2 + 9*a^3*z + b^2 + a^2, z, k)*a^4*b^3*exp(x) + 7602176*root(27*a^5*z^
3 + 27*a^4*z^2 + 9*a^3*z + b^2 + a^2, z, k)*a^3*b^4*exp(2*x) + 465305600*root(27*a^5*z^3 + 27*a^4*z^2 + 9*a^3*
z + b^2 + a^2, z, k)*a^5*b^2*exp(2*x) - 14155776*root(27*a^5*z^3 + 27*a^4*z^2 + 9*a^3*z + b^2 + a^2, z, k)^3*a
^6*b^3*exp(x) + 24379392*root(27*a^5*z^3 + 27*a^4*z^2 + 9*a^3*z + b^2 + a^2, z, k)^2*a^4*b^4*exp(2*x) + 138333
3888*root(27*a^5*z^3 + 27*a^4*z^2 + 9*a^3*z + b^2 + a^2, z, k)^2*a^6*b^2*exp(2*x) + 18874368*root(27*a^5*z^3 +
27*a^4*z^2 + 9*a^3*z + b^2 + a^2, z, k)^3*a^5*b^4*exp(2*x) + 1370750976*root(27*a^5*z^3 + 27*a^4*z^2 + 9*a^3*
z + b^2 + a^2, z, k)^3*a^7*b^2*exp(2*x))/(a^6*b^6))*root(27*a^5*z^3 + 27*a^4*z^2 + 9*a^3*z + b^2 + a^2, z, k),
k, 1, 3) + log(3221225472*a^6*exp(2*x) + 786432*b^6*exp(2*x) - 3221225472*a^6 - 786432*b^6 - 101449728*a^2*b^
4 - 3321888768*a^4*b^2 + 101449728*a^2*b^4*exp(2*x) + 3321888768*a^4*b^2*exp(2*x))/a

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