3.233 \(\int \cosh (x) \tanh (6 x) \, dx\)

Optimal. Leaf size=87 \[ \cosh (x)-\frac{\tanh ^{-1}\left (\sqrt{2} \cosh (x)\right )}{3 \sqrt{2}}-\frac{1}{6} \sqrt{2-\sqrt{3}} \tanh ^{-1}\left (\frac{2 \cosh (x)}{\sqrt{2-\sqrt{3}}}\right )-\frac{1}{6} \sqrt{2+\sqrt{3}} \tanh ^{-1}\left (\frac{2 \cosh (x)}{\sqrt{2+\sqrt{3}}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.251012, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 5, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.714, Rules used = {12, 6742, 2073, 207, 1166} \[ \cosh (x)-\frac{\tanh ^{-1}\left (\sqrt{2} \cosh (x)\right )}{3 \sqrt{2}}-\frac{1}{6} \sqrt{2-\sqrt{3}} \tanh ^{-1}\left (\frac{2 \cosh (x)}{\sqrt{2-\sqrt{3}}}\right )-\frac{1}{6} \sqrt{2+\sqrt{3}} \tanh ^{-1}\left (\frac{2 \cosh (x)}{\sqrt{2+\sqrt{3}}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-ArcTanh[Sqrt[2]*Cosh[x]]/(3*Sqrt[2]) - (Sqrt[2 - Sqrt[3]]*ArcTanh[(2*Cosh[x])/Sqrt[2 - Sqrt[3]]])/6 - (Sqrt[2
+ Sqrt[3]]*ArcTanh[(2*Cosh[x])/Sqrt[2 + Sqrt[3]]])/6 + Cosh[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 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

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]

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])

Rule 1166

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

Rubi steps

\begin{align*} \int \cosh (x) \tanh (6 x) \, dx &=\operatorname{Subst}\left (\int \frac{2 x^2 \left (-3+16 x^2-16 x^4\right )}{1-18 x^2+48 x^4-32 x^6} \, dx,x,\cosh (x)\right )\\ &=2 \operatorname{Subst}\left (\int \frac{x^2 \left (-3+16 x^2-16 x^4\right )}{1-18 x^2+48 x^4-32 x^6} \, dx,x,\cosh (x)\right )\\ &=2 \operatorname{Subst}\left (\int \left (\frac{1}{2}-\frac{1-12 x^2+16 x^4}{2 \left (1-18 x^2+48 x^4-32 x^6\right )}\right ) \, dx,x,\cosh (x)\right )\\ &=\cosh (x)-\operatorname{Subst}\left (\int \frac{1-12 x^2+16 x^4}{1-18 x^2+48 x^4-32 x^6} \, dx,x,\cosh (x)\right )\\ &=\cosh (x)-\operatorname{Subst}\left (\int \left (-\frac{1}{3 \left (-1+2 x^2\right )}-\frac{2 \left (-1+8 x^2\right )}{3 \left (1-16 x^2+16 x^4\right )}\right ) \, dx,x,\cosh (x)\right )\\ &=\cosh (x)+\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{-1+2 x^2} \, dx,x,\cosh (x)\right )+\frac{2}{3} \operatorname{Subst}\left (\int \frac{-1+8 x^2}{1-16 x^2+16 x^4} \, dx,x,\cosh (x)\right )\\ &=-\frac{\tanh ^{-1}\left (\sqrt{2} \cosh (x)\right )}{3 \sqrt{2}}+\cosh (x)+\frac{1}{3} \left (4 \left (2-\sqrt{3}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-8+4 \sqrt{3}+16 x^2} \, dx,x,\cosh (x)\right )+\frac{1}{3} \left (4 \left (2+\sqrt{3}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-8-4 \sqrt{3}+16 x^2} \, dx,x,\cosh (x)\right )\\ &=-\frac{\tanh ^{-1}\left (\sqrt{2} \cosh (x)\right )}{3 \sqrt{2}}-\frac{1}{6} \sqrt{2-\sqrt{3}} \tanh ^{-1}\left (\frac{2 \cosh (x)}{\sqrt{2-\sqrt{3}}}\right )-\frac{1}{6} \sqrt{2+\sqrt{3}} \tanh ^{-1}\left (\frac{2 \cosh (x)}{\sqrt{2+\sqrt{3}}}\right )+\cosh (x)\\ \end{align*}

Mathematica [C]  time = 0.28289, size = 395, normalized size = 4.54 \[ \frac{\sqrt{2} \text{RootSum}\left [\text{$\#$1}^8-\text{$\#$1}^4+1\& ,\frac{2 \text{$\#$1}^6 x+\text{$\#$1}^4 x-\text{$\#$1}^2 x+4 \text{$\#$1}^6 \log \left (-\text{$\#$1} \sinh \left (\frac{x}{2}\right )+\text{$\#$1} \cosh \left (\frac{x}{2}\right )-\sinh \left (\frac{x}{2}\right )-\cosh \left (\frac{x}{2}\right )\right )+2 \text{$\#$1}^4 \log \left (-\text{$\#$1} \sinh \left (\frac{x}{2}\right )+\text{$\#$1} \cosh \left (\frac{x}{2}\right )-\sinh \left (\frac{x}{2}\right )-\cosh \left (\frac{x}{2}\right )\right )-2 \text{$\#$1}^2 \log \left (-\text{$\#$1} \sinh \left (\frac{x}{2}\right )+\text{$\#$1} \cosh \left (\frac{x}{2}\right )-\sinh \left (\frac{x}{2}\right )-\cosh \left (\frac{x}{2}\right )\right )-4 \log \left (-\text{$\#$1} \sinh \left (\frac{x}{2}\right )+\text{$\#$1} \cosh \left (\frac{x}{2}\right )-\sinh \left (\frac{x}{2}\right )-\cosh \left (\frac{x}{2}\right )\right )-2 x}{2 \text{$\#$1}^7-\text{$\#$1}^3}\& \right ]+24 \sqrt{2} \cosh (x)-8 \tanh ^{-1}\left (\sqrt{2}-i \tanh \left (\frac{x}{2}\right )\right )+2 \log \left (\sqrt{2}-2 \cosh (x)\right )-2 \log \left (2 \cosh (x)+\sqrt{2}\right )-4 i \tan ^{-1}\left (\frac{\sinh \left (\frac{x}{2}\right )+\cosh \left (\frac{x}{2}\right )}{\left (1+\sqrt{2}\right ) \cosh \left (\frac{x}{2}\right )-\left (\sqrt{2}-1\right ) \sinh \left (\frac{x}{2}\right )}\right )+4 i \tan ^{-1}\left (\frac{\sinh \left (\frac{x}{2}\right )+\cosh \left (\frac{x}{2}\right )}{\left (\sqrt{2}-1\right ) \cosh \left (\frac{x}{2}\right )-\left (1+\sqrt{2}\right ) \sinh \left (\frac{x}{2}\right )}\right )}{24 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-4*I)*ArcTan[(Cosh[x/2] + Sinh[x/2])/((1 + Sqrt[2])*Cosh[x/2] - (-1 + Sqrt[2])*Sinh[x/2])] + (4*I)*ArcTan[(C
osh[x/2] + Sinh[x/2])/((-1 + Sqrt[2])*Cosh[x/2] - (1 + Sqrt[2])*Sinh[x/2])] - 8*ArcTanh[Sqrt[2] - I*Tanh[x/2]]
+ 24*Sqrt[2]*Cosh[x] + 2*Log[Sqrt[2] - 2*Cosh[x]] - 2*Log[Sqrt[2] + 2*Cosh[x]] + Sqrt[2]*RootSum[1 - #1^4 + #
1^8 & , (-2*x - 4*Log[-Cosh[x/2] - Sinh[x/2] + Cosh[x/2]*#1 - Sinh[x/2]*#1] - x*#1^2 - 2*Log[-Cosh[x/2] - Sinh
[x/2] + Cosh[x/2]*#1 - Sinh[x/2]*#1]*#1^2 + x*#1^4 + 2*Log[-Cosh[x/2] - Sinh[x/2] + Cosh[x/2]*#1 - Sinh[x/2]*#
1]*#1^4 + 2*x*#1^6 + 4*Log[-Cosh[x/2] - Sinh[x/2] + Cosh[x/2]*#1 - Sinh[x/2]*#1]*#1^6)/(-#1^3 + 2*#1^7) & ])/(
24*Sqrt[2])

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Maple [A]  time = 0.069, size = 102, normalized size = 1.2 \begin{align*} \cosh \left ( x \right ) -{\frac{2\,\sqrt{3} \left ( 3+2\,\sqrt{3} \right ) }{18\,\sqrt{6}+18\,\sqrt{2}}{\it Artanh} \left ( 8\,{\frac{\cosh \left ( x \right ) }{2\,\sqrt{6}+2\,\sqrt{2}}} \right ) }-{\frac{ \left ( -6+4\,\sqrt{3} \right ) \sqrt{3}}{18\,\sqrt{6}-18\,\sqrt{2}}{\it Artanh} \left ( 8\,{\frac{\cosh \left ( x \right ) }{2\,\sqrt{6}-2\,\sqrt{2}}} \right ) }-{\frac{{\it Artanh} \left ( \cosh \left ( x \right ) \sqrt{2} \right ) \sqrt{2}}{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

cosh(x)-2/9*3^(1/2)*(3+2*3^(1/2))/(2*6^(1/2)+2*2^(1/2))*arctanh(8*cosh(x)/(2*6^(1/2)+2*2^(1/2)))-2/9*(-3+2*3^(
1/2))*3^(1/2)/(2*6^(1/2)-2*2^(1/2))*arctanh(8*cosh(x)/(2*6^(1/2)-2*2^(1/2)))-1/6*arctanh(cosh(x)*2^(1/2))*2^(1
/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{2} \,{\left (e^{\left (2 \, x\right )} + 1\right )} e^{\left (-x\right )} - \frac{1}{12} \, \sqrt{2} \log \left (\sqrt{2} e^{x} + e^{\left (2 \, x\right )} + 1\right ) + \frac{1}{12} \, \sqrt{2} \log \left (-\sqrt{2} e^{x} + e^{\left (2 \, x\right )} + 1\right ) + \frac{1}{2} \, \int \frac{2 \,{\left (2 \, e^{\left (7 \, x\right )} + e^{\left (5 \, x\right )} - e^{\left (3 \, x\right )} - 2 \, e^{x}\right )}}{3 \,{\left (e^{\left (8 \, x\right )} - e^{\left (4 \, x\right )} + 1\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(e^(2*x) + 1)*e^(-x) - 1/12*sqrt(2)*log(sqrt(2)*e^x + e^(2*x) + 1) + 1/12*sqrt(2)*log(-sqrt(2)*e^x + e^(2*
x) + 1) + 1/2*integrate(2/3*(2*e^(7*x) + e^(5*x) - e^(3*x) - 2*e^x)/(e^(8*x) - e^(4*x) + 1), x)

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Fricas [B]  time = 2.19526, size = 976, normalized size = 11.22 \begin{align*} -\frac{\sqrt{\sqrt{3} + 2}{\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} \log \left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} + \sqrt{\sqrt{3} + 2}{\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} + 1\right ) - \sqrt{\sqrt{3} + 2}{\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} \log \left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} - \sqrt{\sqrt{3} + 2}{\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} + 1\right ) + \sqrt{-\sqrt{3} + 2}{\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} \log \left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} + \sqrt{-\sqrt{3} + 2}{\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} + 1\right ) - \sqrt{-\sqrt{3} + 2}{\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} \log \left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} - \sqrt{-\sqrt{3} + 2}{\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} + 1\right ) - 6 \, \cosh \left (x\right )^{2} -{\left (\sqrt{2} \cosh \left (x\right ) + \sqrt{2} \sinh \left (x\right )\right )} \log \left (\frac{\cosh \left (x\right )^{2} + \sinh \left (x\right )^{2} - 2 \, \sqrt{2} \cosh \left (x\right ) + 2}{\cosh \left (x\right )^{2} + \sinh \left (x\right )^{2}}\right ) - 12 \, \cosh \left (x\right ) \sinh \left (x\right ) - 6 \, \sinh \left (x\right )^{2} - 6}{12 \,{\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*(sqrt(sqrt(3) + 2)*(cosh(x) + sinh(x))*log(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 + sqrt(sqrt(3) + 2)
*(cosh(x) + sinh(x)) + 1) - sqrt(sqrt(3) + 2)*(cosh(x) + sinh(x))*log(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^
2 - sqrt(sqrt(3) + 2)*(cosh(x) + sinh(x)) + 1) + sqrt(-sqrt(3) + 2)*(cosh(x) + sinh(x))*log(cosh(x)^2 + 2*cosh
(x)*sinh(x) + sinh(x)^2 + sqrt(-sqrt(3) + 2)*(cosh(x) + sinh(x)) + 1) - sqrt(-sqrt(3) + 2)*(cosh(x) + sinh(x))
*log(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 - sqrt(-sqrt(3) + 2)*(cosh(x) + sinh(x)) + 1) - 6*cosh(x)^2 - (
sqrt(2)*cosh(x) + sqrt(2)*sinh(x))*log((cosh(x)^2 + sinh(x)^2 - 2*sqrt(2)*cosh(x) + 2)/(cosh(x)^2 + sinh(x)^2)
) - 12*cosh(x)*sinh(x) - 6*sinh(x)^2 - 6)/(cosh(x) + sinh(x))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(cosh(x)*tanh(6*x), x)

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Giac [B]  time = 1.22039, size = 212, normalized size = 2.44 \begin{align*} -\frac{1}{24} \,{\left (\sqrt{6} + \sqrt{2}\right )} \log \left (\frac{1}{2} \, \sqrt{6} + \frac{1}{2} \, \sqrt{2} + e^{\left (-x\right )} + e^{x}\right ) - \frac{1}{24} \,{\left (\sqrt{6} - \sqrt{2}\right )} \log \left (\frac{1}{2} \, \sqrt{6} - \frac{1}{2} \, \sqrt{2} + e^{\left (-x\right )} + e^{x}\right ) + \frac{1}{24} \,{\left (\sqrt{6} - \sqrt{2}\right )} \log \left (-\frac{1}{2} \, \sqrt{6} + \frac{1}{2} \, \sqrt{2} + e^{\left (-x\right )} + e^{x}\right ) + \frac{1}{24} \,{\left (\sqrt{6} + \sqrt{2}\right )} \log \left (-\frac{1}{2} \, \sqrt{6} - \frac{1}{2} \, \sqrt{2} + e^{\left (-x\right )} + e^{x}\right ) + \frac{1}{12} \, \sqrt{2} \log \left (-\frac{\sqrt{2} - e^{\left (-x\right )} - e^{x}}{\sqrt{2} + e^{\left (-x\right )} + e^{x}}\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(cosh(x)*tanh(6*x),x, algorithm="giac")

[Out]

-1/24*(sqrt(6) + sqrt(2))*log(1/2*sqrt(6) + 1/2*sqrt(2) + e^(-x) + e^x) - 1/24*(sqrt(6) - sqrt(2))*log(1/2*sqr
t(6) - 1/2*sqrt(2) + e^(-x) + e^x) + 1/24*(sqrt(6) - sqrt(2))*log(-1/2*sqrt(6) + 1/2*sqrt(2) + e^(-x) + e^x) +
1/24*(sqrt(6) + sqrt(2))*log(-1/2*sqrt(6) - 1/2*sqrt(2) + e^(-x) + e^x) + 1/12*sqrt(2)*log(-(sqrt(2) - e^(-x)
- e^x)/(sqrt(2) + e^(-x) + e^x)) + 1/2*e^(-x) + 1/2*e^x