### 3.249 $$\int \cosh (x) \text{csch}(6 x) \, dx$$

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

[Out]

-ArcTanh[Cosh[x]]/6 - ArcTanh[2*Cosh[x]]/6 + ArcTanh[(2*Cosh[x])/Sqrt[3]]/(2*Sqrt[3])

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Rubi [A]  time = 0.05212, 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, 207} $-\frac{1}{6} \tanh ^{-1}(\cosh (x))-\frac{1}{6} \tanh ^{-1}(2 \cosh (x))+\frac{\tanh ^{-1}\left (\frac{2 \cosh (x)}{\sqrt{3}}\right )}{2 \sqrt{3}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-ArcTanh[Cosh[x]]/6 - ArcTanh[2*Cosh[x]]/6 + ArcTanh[(2*Cosh[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 207

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

Rubi steps

\begin{align*} \int \cosh (x) \text{csch}(6 x) \, dx &=-\operatorname{Subst}\left (\int \frac{1}{2 \left (3-19 x^2+32 x^4-16 x^6\right )} \, dx,x,\cosh (x)\right )\\ &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{3-19 x^2+32 x^4-16 x^6} \, dx,x,\cosh (x)\right )\right )\\ &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \left (-\frac{1}{3 \left (-1+x^2\right )}+\frac{2}{-3+4 x^2}-\frac{2}{3 \left (-1+4 x^2\right )}\right ) \, dx,x,\cosh (x)\right )\right )\\ &=\frac{1}{6} \operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,\cosh (x)\right )+\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{-1+4 x^2} \, dx,x,\cosh (x)\right )-\operatorname{Subst}\left (\int \frac{1}{-3+4 x^2} \, dx,x,\cosh (x)\right )\\ &=-\frac{1}{6} \tanh ^{-1}(\cosh (x))-\frac{1}{6} \tanh ^{-1}(2 \cosh (x))+\frac{\tanh ^{-1}\left (\frac{2 \cosh (x)}{\sqrt{3}}\right )}{2 \sqrt{3}}\\ \end{align*}

Mathematica [C]  time = 0.0615614, size = 91, normalized size = 2.53 $\frac{1}{12} \left (2 \sqrt{3} \tanh ^{-1}\left (\frac{2-i \tanh \left (\frac{x}{2}\right )}{\sqrt{3}}\right )+2 \sqrt{3} \tanh ^{-1}\left (\frac{2+i \tanh \left (\frac{x}{2}\right )}{\sqrt{3}}\right )+2 \log \left (\sinh \left (\frac{x}{2}\right )\right )-2 \log \left (\cosh \left (\frac{x}{2}\right )\right )+\log (1-2 \cosh (x))-\log (2 \cosh (x)+1)\right )$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(2*Sqrt[3]*ArcTanh[(2 - I*Tanh[x/2])/Sqrt[3]] + 2*Sqrt[3]*ArcTanh[(2 + I*Tanh[x/2])/Sqrt[3]] - 2*Log[Cosh[x/2]
] + Log[1 - 2*Cosh[x]] - Log[1 + 2*Cosh[x]] + 2*Log[Sinh[x/2]])/12

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Maple [B]  time = 0.046, size = 77, normalized size = 2.1 \begin{align*}{\frac{\ln \left ({{\rm e}^{x}}-1 \right ) }{6}}-{\frac{\ln \left ({{\rm e}^{x}}+1 \right ) }{6}}-{\frac{\ln \left ({{\rm e}^{2\,x}}+{{\rm e}^{x}}+1 \right ) }{12}}+{\frac{\sqrt{3}\ln \left ({{\rm e}^{2\,x}}+\sqrt{3}{{\rm e}^{x}}+1 \right ) }{12}}-{\frac{\sqrt{3}\ln \left ({{\rm e}^{2\,x}}-\sqrt{3}{{\rm e}^{x}}+1 \right ) }{12}}+{\frac{\ln \left ({{\rm e}^{2\,x}}-{{\rm e}^{x}}+1 \right ) }{12}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{e^{\left (3 \, x\right )} - e^{x}}{2 \,{\left (e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )} + 1\right )}}\,{d x} - \frac{1}{12} \, \log \left (e^{\left (2 \, x\right )} + e^{x} + 1\right ) + \frac{1}{12} \, \log \left (e^{\left (2 \, x\right )} - e^{x} + 1\right ) - \frac{1}{6} \, \log \left (e^{x} + 1\right ) + \frac{1}{6} \, \log \left (e^{x} - 1\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-integrate(1/2*(e^(3*x) - e^x)/(e^(4*x) - e^(2*x) + 1), x) - 1/12*log(e^(2*x) + e^x + 1) + 1/12*log(e^(2*x) -
e^x + 1) - 1/6*log(e^x + 1) + 1/6*log(e^x - 1)

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Fricas [B]  time = 2.17203, size = 358, normalized size = 9.94 \begin{align*} \frac{1}{12} \, \sqrt{3} \log \left (\frac{2 \, \cosh \left (x\right )^{2} + 2 \, \sinh \left (x\right )^{2} + 4 \, \sqrt{3} \cosh \left (x\right ) + 5}{2 \, \cosh \left (x\right )^{2} + 2 \, \sinh \left (x\right )^{2} - 1}\right ) - \frac{1}{12} \, \log \left (\frac{2 \, \cosh \left (x\right ) + 1}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) + \frac{1}{12} \, \log \left (\frac{2 \, \cosh \left (x\right ) - 1}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) - \frac{1}{6} \, \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) + \frac{1}{6} \, \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) - 1\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*sqrt(3)*log((2*cosh(x)^2 + 2*sinh(x)^2 + 4*sqrt(3)*cosh(x) + 5)/(2*cosh(x)^2 + 2*sinh(x)^2 - 1)) - 1/12*l
og((2*cosh(x) + 1)/(cosh(x) - sinh(x))) + 1/12*log((2*cosh(x) - 1)/(cosh(x) - sinh(x))) - 1/6*log(cosh(x) + si
nh(x) + 1) + 1/6*log(cosh(x) + sinh(x) - 1)

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(cosh(x)*csch(6*x), x)

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Giac [B]  time = 1.12417, size = 107, normalized size = 2.97 \begin{align*} -\frac{1}{12} \, \sqrt{3} \log \left (-\frac{\sqrt{3} - e^{\left (-x\right )} - e^{x}}{\sqrt{3} + e^{\left (-x\right )} + e^{x}}\right ) - \frac{1}{12} \, \log \left (e^{\left (-x\right )} + e^{x} + 2\right ) - \frac{1}{12} \, \log \left (e^{\left (-x\right )} + e^{x} + 1\right ) + \frac{1}{12} \, \log \left (e^{\left (-x\right )} + e^{x} - 1\right ) + \frac{1}{12} \, \log \left (e^{\left (-x\right )} + e^{x} - 2\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)*csch(6*x),x, algorithm="giac")

[Out]

-1/12*sqrt(3)*log(-(sqrt(3) - e^(-x) - e^x)/(sqrt(3) + e^(-x) + e^x)) - 1/12*log(e^(-x) + e^x + 2) - 1/12*log(
e^(-x) + e^x + 1) + 1/12*log(e^(-x) + e^x - 1) + 1/12*log(e^(-x) + e^x - 2)