### 3.220 $$\int \text{csch}(6 x) \sinh (x) \, dx$$

Optimal. Leaf size=36 $\frac{1}{6} \tan ^{-1}(\sinh (x))+\frac{1}{6} \tan ^{-1}(2 \sinh (x))-\frac{\tan ^{-1}\left (\frac{2 \sinh (x)}{\sqrt{3}}\right )}{2 \sqrt{3}}$

[Out]

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

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Rubi [A]  time = 0.0454255, antiderivative size = 36, normalized size of antiderivative = 1., 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, 2057, 203} $\frac{1}{6} \tan ^{-1}(\sinh (x))+\frac{1}{6} \tan ^{-1}(2 \sinh (x))-\frac{\tan ^{-1}\left (\frac{2 \sinh (x)}{\sqrt{3}}\right )}{2 \sqrt{3}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Rule 12

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

Rule 2057

Int[(P_)^(p_), x_Symbol] :> With[{u = Factor[P /. x -> Sqrt[x]]}, Int[ExpandIntegrand[(u /. x -> x^2)^p, x], x
] /;  !SumQ[NonfreeFactors[u, x]]] /; PolyQ[P, x^2] && ILtQ[p, 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 \text{csch}(6 x) \sinh (x) \, dx &=\operatorname{Subst}\left (\int \frac{1}{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}{3+19 x^2+32 x^4+16 x^6} \, dx,x,\sinh (x)\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\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 )\\ &=\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}}\\ \end{align*}

Mathematica [A]  time = 0.0325924, size = 30, normalized size = 0.83 $\frac{1}{6} \left (\tan ^{-1}(\sinh (x))+\tan ^{-1}(2 \sinh (x))-\sqrt{3} \tan ^{-1}\left (\frac{2 \sinh (x)}{\sqrt{3}}\right )\right )$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [C]  time = 0.054, size = 92, normalized size = 2.6 \begin{align*}{\frac{i}{6}}\ln \left ({{\rm e}^{x}}+i \right ) -{\frac{i}{6}}\ln \left ({{\rm e}^{x}}-i \right ) +{\frac{i}{12}}\ln \left ({{\rm e}^{2\,x}}+i{{\rm e}^{x}}-1 \right ) -{\frac{i}{12}}\ln \left ({{\rm e}^{2\,x}}-i{{\rm e}^{x}}-1 \right ) +{\frac{i}{12}}\sqrt{3}\ln \left ({{\rm e}^{2\,x}}-i\sqrt{3}{{\rm e}^{x}}-1 \right ) -{\frac{i}{12}}\sqrt{3}\ln \left ({{\rm e}^{2\,x}}+i\sqrt{3}{{\rm e}^{x}}-1 \right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*I*ln(exp(x)+I)-1/6*I*ln(exp(x)-I)+1/12*I*ln(exp(2*x)+I*exp(x)-1)-1/12*I*ln(exp(2*x)-I*exp(x)-1)+1/12*I*3^(
1/2)*ln(exp(2*x)-I*3^(1/2)*exp(x)-1)-1/12*I*3^(1/2)*ln(exp(2*x)+I*3^(1/2)*exp(x)-1)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\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 ) + \int \frac{e^{\left (3 \, x\right )} + e^{x}}{6 \,{\left (e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )} + 1\right )}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 2.15969, size = 408, normalized size = 11.33 \begin{align*} -\frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3} \cosh \left (x\right ) + \frac{1}{3} \, \sqrt{3} \sinh \left (x\right )\right ) + \frac{1}{6} \, \sqrt{3} \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 ) - \frac{1}{6} \, \arctan \left (-\frac{\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) + \frac{1}{2} \, \arctan \left (\cosh \left (x\right ) + \sinh \left (x\right )\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sinh{\left (x \right )} \operatorname{csch}{\left (6 x \right )}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sinh(x)*csch(6*x), x)

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Giac [B]  time = 1.19096, size = 78, normalized size = 2.17 \begin{align*} \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 ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(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))