### 3.207 $$\int \coth (3 x) \sinh (x) \, dx$$

Optimal. Leaf size=20 $\sinh (x)-\frac{\tan ^{-1}\left (\frac{2 \sinh (x)}{\sqrt{3}}\right )}{\sqrt{3}}$

[Out]

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

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Rubi [A]  time = 0.0269765, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 7, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.286, Rules used = {388, 203} $\sinh (x)-\frac{\tan ^{-1}\left (\frac{2 \sinh (x)}{\sqrt{3}}\right )}{\sqrt{3}}$

Antiderivative was successfully veriﬁed.

[In]

Int[Coth[3*x]*Sinh[x],x]

[Out]

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

Rule 388

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(d*x*(a + b*x^n)^(p + 1))/(b*(n*
(p + 1) + 1)), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(b*(n*(p + 1) + 1)), Int[(a + b*x^n)^p, x], x] /; FreeQ[{
a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && NeQ[n*(p + 1) + 1, 0]

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])

Rubi steps

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

Mathematica [A]  time = 0.0157836, size = 20, normalized size = 1. $\sinh (x)-\frac{\tan ^{-1}\left (\frac{2 \sinh (x)}{\sqrt{3}}\right )}{\sqrt{3}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Coth[3*x]*Sinh[x],x]

[Out]

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

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Maple [B]  time = 0.049, size = 51, normalized size = 2.6 \begin{align*} -{\frac{\sqrt{3}}{3}\arctan \left ({\frac{\sqrt{3}}{3}\tanh \left ({\frac{x}{2}} \right ) } \right ) }- \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-1}-{\frac{\sqrt{3}}{3}\arctan \left ( \tanh \left ({\frac{x}{2}} \right ) \sqrt{3} \right ) }- \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-1} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(coth(3*x)*sinh(x),x)

[Out]

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

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Maxima [B]  time = 1.57482, size = 66, normalized size = 3.3 \begin{align*} \frac{1}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, e^{\left (-x\right )} + 1\right )}\right ) + \frac{1}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, e^{\left (-x\right )} - 1\right )}\right ) - \frac{1}{2} \, e^{\left (-x\right )} + \frac{1}{2} \, e^{x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(3*x)*sinh(x),x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 2.21664, size = 433, normalized size = 21.65 \begin{align*} -\frac{2 \,{\left (\sqrt{3} \cosh \left (x\right ) + \sqrt{3} \sinh \left (x\right )\right )} \arctan \left (\frac{1}{3} \, \sqrt{3} \cosh \left (x\right ) + \frac{1}{3} \, \sqrt{3} \sinh \left (x\right )\right ) - 2 \,{\left (\sqrt{3} \cosh \left (x\right ) + \sqrt{3} \sinh \left (x\right )\right )} \arctan \left (-\frac{\sqrt{3} \cosh \left (x\right )^{2} + 2 \, \sqrt{3} \cosh \left (x\right ) \sinh \left (x\right ) + \sqrt{3} \sinh \left (x\right )^{2} + 2 \, \sqrt{3}}{3 \,{\left (\cosh \left (x\right ) - \sinh \left (x\right )\right )}}\right ) - 3 \, \cosh \left (x\right )^{2} - 6 \, \cosh \left (x\right ) \sinh \left (x\right ) - 3 \, \sinh \left (x\right )^{2} + 3}{6 \,{\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(3*x)*sinh(x),x, algorithm="fricas")

[Out]

-1/6*(2*(sqrt(3)*cosh(x) + sqrt(3)*sinh(x))*arctan(1/3*sqrt(3)*cosh(x) + 1/3*sqrt(3)*sinh(x)) - 2*(sqrt(3)*cos
h(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*sqr
t(3))/(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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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sinh{\left (x \right )} \coth{\left (3 x \right )}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(3*x)*sinh(x),x)

[Out]

Integral(sinh(x)*coth(3*x), x)

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Giac [B]  time = 1.17834, size = 49, normalized size = 2.45 \begin{align*} -\frac{1}{6} \, \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}{2} \, e^{\left (-x\right )} + \frac{1}{2} \, e^{x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(3*x)*sinh(x),x, algorithm="giac")

[Out]

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