3.210 \(\int \coth (6 x) \sinh (x) \, dx\)

Optimal. Leaf size=38 \[ \sinh (x)-\frac {1}{6} \tan ^{-1}(\sinh (x))-\frac {1}{6} \tan ^{-1}(2 \sinh (x))-\frac {\tan ^{-1}\left (\frac {2 \sinh (x)}{\sqrt {3}}\right )}{2 \sqrt {3}} \]

[Out]

-1/6*arctan(sinh(x))-1/6*arctan(2*sinh(x))+sinh(x)-1/6*arctan(2/3*sinh(x)*3^(1/2))*3^(1/2)

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Rubi [A]  time = 0.08, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 3, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {12, 2073, 203} \[ \sinh (x)-\frac {1}{6} \tan ^{-1}(\sinh (x))-\frac {1}{6} \tan ^{-1}(2 \sinh (x))-\frac {\tan ^{-1}\left (\frac {2 \sinh (x)}{\sqrt {3}}\right )}{2 \sqrt {3}} \]

Antiderivative was successfully verified.

[In]

Int[Coth[6*x]*Sinh[x],x]

[Out]

-ArcTan[Sinh[x]]/6 - ArcTan[2*Sinh[x]]/6 - ArcTan[(2*Sinh[x])/Sqrt[3]]/(2*Sqrt[3]) + Sinh[x]

Rule 12

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

Rule 203

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

Rule 2073

Int[(P_)^(p_)*(Q_)^(q_.), x_Symbol] :> With[{PP = Factor[P /. x -> Sqrt[x]]}, Int[ExpandIntegrand[(PP /. x ->
x^2)^p*Q^q, x], x] /;  !SumQ[NonfreeFactors[PP, x]]] /; FreeQ[q, x] && PolyQ[P, x^2] && PolyQ[Q, x] && ILtQ[p,
0]

Rubi steps

\begin {align*} \int \coth (6 x) \sinh (x) \, dx &=\operatorname {Subst}\left (\int \frac {1+18 x^2+48 x^4+32 x^6}{2 \left (3+19 x^2+32 x^4+16 x^6\right )} \, dx,x,\sinh (x)\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1+18 x^2+48 x^4+32 x^6}{3+19 x^2+32 x^4+16 x^6} \, dx,x,\sinh (x)\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (2-\frac {1}{3 \left (1+x^2\right )}-\frac {2}{3 \left (1+4 x^2\right )}-\frac {2}{3+4 x^2}\right ) \, dx,x,\sinh (x)\right )\\ &=\sinh (x)-\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sinh (x)\right )-\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{1+4 x^2} \, dx,x,\sinh (x)\right )-\operatorname {Subst}\left (\int \frac {1}{3+4 x^2} \, dx,x,\sinh (x)\right )\\ &=-\frac {1}{6} \tan ^{-1}(\sinh (x))-\frac {1}{6} \tan ^{-1}(2 \sinh (x))-\frac {\tan ^{-1}\left (\frac {2 \sinh (x)}{\sqrt {3}}\right )}{2 \sqrt {3}}+\sinh (x)\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 38, normalized size = 1.00 \[ \sinh (x)-\frac {1}{6} \tan ^{-1}(\sinh (x))-\frac {1}{6} \tan ^{-1}(2 \sinh (x))-\frac {\tan ^{-1}\left (\frac {2 \sinh (x)}{\sqrt {3}}\right )}{2 \sqrt {3}} \]

Antiderivative was successfully verified.

[In]

Integrate[Coth[6*x]*Sinh[x],x]

[Out]

-1/6*ArcTan[Sinh[x]] - ArcTan[2*Sinh[x]]/6 - ArcTan[(2*Sinh[x])/Sqrt[3]]/(2*Sqrt[3]) + Sinh[x]

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fricas [B]  time = 0.46, size = 164, normalized size = 4.32 \[ -\frac {{\left (\sqrt {3} \cosh \relax (x) + \sqrt {3} \sinh \relax (x)\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} \cosh \relax (x) + \frac {1}{3} \, \sqrt {3} \sinh \relax (x)\right ) - {\left (\sqrt {3} \cosh \relax (x) + \sqrt {3} \sinh \relax (x)\right )} \arctan \left (-\frac {\sqrt {3} \cosh \relax (x)^{2} + 2 \, \sqrt {3} \cosh \relax (x) \sinh \relax (x) + \sqrt {3} \sinh \relax (x)^{2} + 2 \, \sqrt {3}}{3 \, {\left (\cosh \relax (x) - \sinh \relax (x)\right )}}\right ) - {\left (\cosh \relax (x) + \sinh \relax (x)\right )} \arctan \left (-\frac {\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2}}{\cosh \relax (x) - \sinh \relax (x)}\right ) + 3 \, {\left (\cosh \relax (x) + \sinh \relax (x)\right )} \arctan \left (\cosh \relax (x) + \sinh \relax (x)\right ) - 3 \, \cosh \relax (x)^{2} - 6 \, \cosh \relax (x) \sinh \relax (x) - 3 \, \sinh \relax (x)^{2} + 3}{6 \, {\left (\cosh \relax (x) + \sinh \relax (x)\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(6*x)*sinh(x),x, algorithm="fricas")

[Out]

-1/6*((sqrt(3)*cosh(x) + sqrt(3)*sinh(x))*arctan(1/3*sqrt(3)*cosh(x) + 1/3*sqrt(3)*sinh(x)) - (sqrt(3)*cosh(x)
+ sqrt(3)*sinh(x))*arctan(-1/3*(sqrt(3)*cosh(x)^2 + 2*sqrt(3)*cosh(x)*sinh(x) + sqrt(3)*sinh(x)^2 + 2*sqrt(3)
)/(cosh(x) - sinh(x))) - (cosh(x) + sinh(x))*arctan(-(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2)/(cosh(x) - si
nh(x))) + 3*(cosh(x) + sinh(x))*arctan(cosh(x) + sinh(x)) - 3*cosh(x)^2 - 6*cosh(x)*sinh(x) - 3*sinh(x)^2 + 3)
/(cosh(x) + sinh(x))

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giac [B]  time = 0.15, size = 68, normalized size = 1.79 \[ -\frac {1}{6} \, \pi - \frac {1}{12} \, \sqrt {3} {\left (\pi + 2 \, \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (e^{\left (2 \, x\right )} - 1\right )} e^{\left (-x\right )}\right )\right )} - \frac {1}{6} \, \arctan \left ({\left (e^{\left (2 \, x\right )} - 1\right )} e^{\left (-x\right )}\right ) - \frac {1}{6} \, \arctan \left (\frac {1}{2} \, {\left (e^{\left (2 \, x\right )} - 1\right )} e^{\left (-x\right )}\right ) - \frac {1}{2} \, e^{\left (-x\right )} + \frac {1}{2} \, e^{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(6*x)*sinh(x),x, algorithm="giac")

[Out]

-1/6*pi - 1/12*sqrt(3)*(pi + 2*arctan(1/3*sqrt(3)*(e^(2*x) - 1)*e^(-x))) - 1/6*arctan((e^(2*x) - 1)*e^(-x)) -
1/6*arctan(1/2*(e^(2*x) - 1)*e^(-x)) - 1/2*e^(-x) + 1/2*e^x

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maple [B]  time = 0.34, size = 172, normalized size = 4.53 \[ -\frac {1}{\tanh \left (\frac {x}{2}\right )+1}-\frac {\sqrt {3}\, \arctan \left (\frac {\tanh \left (\frac {x}{2}\right ) \sqrt {3}}{3}\right )}{6}-\frac {\sqrt {3}\, \arctan \left (\tanh \left (\frac {x}{2}\right ) \sqrt {3}\right )}{6}-\frac {\arctan \left (\tanh \left (\frac {x}{2}\right )\right )}{3}-\frac {\sqrt {3}\, \arctan \left (\frac {2 \tanh \left (\frac {x}{2}\right )}{4+2 \sqrt {3}}\right )}{3 \left (4+2 \sqrt {3}\right )}-\frac {2 \arctan \left (\frac {2 \tanh \left (\frac {x}{2}\right )}{4+2 \sqrt {3}}\right )}{3 \left (4+2 \sqrt {3}\right )}+\frac {\sqrt {3}\, \arctan \left (\frac {2 \tanh \left (\frac {x}{2}\right )}{4-2 \sqrt {3}}\right )}{12-6 \sqrt {3}}-\frac {2 \arctan \left (\frac {2 \tanh \left (\frac {x}{2}\right )}{4-2 \sqrt {3}}\right )}{3 \left (4-2 \sqrt {3}\right )}-\frac {1}{\tanh \left (\frac {x}{2}\right )-1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(6*x)*sinh(x),x)

[Out]

-1/(tanh(1/2*x)+1)-1/6*3^(1/2)*arctan(1/3*tanh(1/2*x)*3^(1/2))-1/6*3^(1/2)*arctan(tanh(1/2*x)*3^(1/2))-1/3*arc
tan(tanh(1/2*x))-1/3*3^(1/2)/(4+2*3^(1/2))*arctan(2*tanh(1/2*x)/(4+2*3^(1/2)))-2/3/(4+2*3^(1/2))*arctan(2*tanh
(1/2*x)/(4+2*3^(1/2)))+1/3*3^(1/2)/(4-2*3^(1/2))*arctan(2*tanh(1/2*x)/(4-2*3^(1/2)))-2/3/(4-2*3^(1/2))*arctan(
2*tanh(1/2*x)/(4-2*3^(1/2)))-1/(tanh(1/2*x)-1)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{2} \, {\left (e^{\left (2 \, x\right )} - 1\right )} e^{\left (-x\right )} - \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, e^{x} + 1\right )}\right ) - \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, e^{x} - 1\right )}\right ) - \frac {1}{3} \, \arctan \left (e^{x}\right ) - \frac {1}{2} \, \int \frac {e^{\left (3 \, x\right )} + e^{x}}{3 \, {\left (e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )} + 1\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(6*x)*sinh(x),x, algorithm="maxima")

[Out]

1/2*(e^(2*x) - 1)*e^(-x) - 1/6*sqrt(3)*arctan(1/3*sqrt(3)*(2*e^x + 1)) - 1/6*sqrt(3)*arctan(1/3*sqrt(3)*(2*e^x
- 1)) - 1/3*arctan(e^x) - 1/2*integrate(1/3*(e^(3*x) + e^x)/(e^(4*x) - e^(2*x) + 1), x)

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mupad [B]  time = 1.61, size = 56, normalized size = 1.47 \[ \frac {{\mathrm {e}}^x}{2}-\frac {\mathrm {atan}\left ({\mathrm {e}}^x\right )}{3}-\frac {{\mathrm {e}}^{-x}}{2}-\frac {\mathrm {atan}\left (36\,{\mathrm {e}}^{-x}\,\left (\frac {{\mathrm {e}}^{2\,x}}{36}-\frac {1}{36}\right )\right )}{6}-\frac {\sqrt {3}\,\mathrm {atan}\left (4\,\sqrt {3}\,{\mathrm {e}}^{-x}\,\left (\frac {{\mathrm {e}}^{2\,x}}{12}-\frac {1}{12}\right )\right )}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(6*x)*sinh(x),x)

[Out]

exp(x)/2 - atan(exp(x))/3 - exp(-x)/2 - atan(36*exp(-x)*(exp(2*x)/36 - 1/36))/6 - (3^(1/2)*atan(4*3^(1/2)*exp(
-x)*(exp(2*x)/12 - 1/12)))/6

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sinh {\relax (x )} \coth {\left (6 x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(6*x)*sinh(x),x)

[Out]

Integral(sinh(x)*coth(6*x), x)

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