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]

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

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Rubi [A]  time = 0.25, antiderivative size = 87, normalized size of antiderivative = 1.00, 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 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]

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 6742

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

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*}

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Mathematica [C]  time = 0.31, size = 395, normalized size = 4.54 \[ \frac {\sqrt {2} \text {RootSum}\left [\text {$\#$1}^8-\text {$\#$1}^4+1\& ,\frac {2 \text {$\#$1}^6 x+4 \text {$\#$1}^6 \log \left (-\text {$\#$1} \sinh \left (\frac {x}{2}\right )+\text {$\#$1} \cosh \left (\frac {x}{2}\right )-\sinh \left (\frac {x}{2}\right )-\cosh \left (\frac {x}{2}\right )\right )+\text {$\#$1}^4 x+2 \text {$\#$1}^4 \log \left (-\text {$\#$1} \sinh \left (\frac {x}{2}\right )+\text {$\#$1} \cosh \left (\frac {x}{2}\right )-\sinh \left (\frac {x}{2}\right )-\cosh \left (\frac {x}{2}\right )\right )-\text {$\#$1}^2 x-2 \text {$\#$1}^2 \log \left (-\text {$\#$1} \sinh \left (\frac {x}{2}\right )+\text {$\#$1} \cosh \left (\frac {x}{2}\right )-\sinh \left (\frac {x}{2}\right )-\cosh \left (\frac {x}{2}\right )\right )-4 \log \left (-\text {$\#$1} \sinh \left (\frac {x}{2}\right )+\text {$\#$1} \cosh \left (\frac {x}{2}\right )-\sinh \left (\frac {x}{2}\right )-\cosh \left (\frac {x}{2}\right )\right )-2 x}{2 \text {$\#$1}^7-\text {$\#$1}^3}\& \right ]+24 \sqrt {2} \cosh (x)-8 \tanh ^{-1}\left (\sqrt {2}-i \tanh \left (\frac {x}{2}\right )\right )+2 \log \left (\sqrt {2}-2 \cosh (x)\right )-2 \log \left (2 \cosh (x)+\sqrt {2}\right )-4 i \tan ^{-1}\left (\frac {\sinh \left (\frac {x}{2}\right )+\cosh \left (\frac {x}{2}\right )}{\left (1+\sqrt {2}\right ) \cosh \left (\frac {x}{2}\right )-\left (\sqrt {2}-1\right ) \sinh \left (\frac {x}{2}\right )}\right )+4 i \tan ^{-1}\left (\frac {\sinh \left (\frac {x}{2}\right )+\cosh \left (\frac {x}{2}\right )}{\left (\sqrt {2}-1\right ) \cosh \left (\frac {x}{2}\right )-\left (1+\sqrt {2}\right ) \sinh \left (\frac {x}{2}\right )}\right )}{24 \sqrt {2}} \]

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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fricas [B]  time = 0.48, size = 258, normalized size = 2.97 \[ -\frac {\sqrt {\sqrt {3} + 2} {\left (\cosh \relax (x) + \sinh \relax (x)\right )} \log \left (\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2} + \sqrt {\sqrt {3} + 2} {\left (\cosh \relax (x) + \sinh \relax (x)\right )} + 1\right ) - \sqrt {\sqrt {3} + 2} {\left (\cosh \relax (x) + \sinh \relax (x)\right )} \log \left (\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2} - \sqrt {\sqrt {3} + 2} {\left (\cosh \relax (x) + \sinh \relax (x)\right )} + 1\right ) + \sqrt {-\sqrt {3} + 2} {\left (\cosh \relax (x) + \sinh \relax (x)\right )} \log \left (\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2} + \sqrt {-\sqrt {3} + 2} {\left (\cosh \relax (x) + \sinh \relax (x)\right )} + 1\right ) - \sqrt {-\sqrt {3} + 2} {\left (\cosh \relax (x) + \sinh \relax (x)\right )} \log \left (\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2} - \sqrt {-\sqrt {3} + 2} {\left (\cosh \relax (x) + \sinh \relax (x)\right )} + 1\right ) - 6 \, \cosh \relax (x)^{2} - {\left (\sqrt {2} \cosh \relax (x) + \sqrt {2} \sinh \relax (x)\right )} \log \left (\frac {\cosh \relax (x)^{2} + \sinh \relax (x)^{2} - 2 \, \sqrt {2} \cosh \relax (x) + 2}{\cosh \relax (x)^{2} + \sinh \relax (x)^{2}}\right ) - 12 \, \cosh \relax (x) \sinh \relax (x) - 6 \, \sinh \relax (x)^{2} - 6}{12 \, {\left (\cosh \relax (x) + \sinh \relax (x)\right )}} \]

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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giac [B]  time = 0.16, size = 157, normalized size = 1.80 \[ -\frac {1}{24} \, {\left (\sqrt {6} + \sqrt {2}\right )} \log \left (\frac {1}{2} \, \sqrt {6} + \frac {1}{2} \, \sqrt {2} + e^{\left (-x\right )} + e^{x}\right ) - \frac {1}{24} \, {\left (\sqrt {6} - \sqrt {2}\right )} \log \left (\frac {1}{2} \, \sqrt {6} - \frac {1}{2} \, \sqrt {2} + e^{\left (-x\right )} + e^{x}\right ) + \frac {1}{24} \, {\left (\sqrt {6} - \sqrt {2}\right )} \log \left (-\frac {1}{2} \, \sqrt {6} + \frac {1}{2} \, \sqrt {2} + e^{\left (-x\right )} + e^{x}\right ) + \frac {1}{24} \, {\left (\sqrt {6} + \sqrt {2}\right )} \log \left (-\frac {1}{2} \, \sqrt {6} - \frac {1}{2} \, \sqrt {2} + e^{\left (-x\right )} + e^{x}\right ) + \frac {1}{12} \, \sqrt {2} \log \left (-\frac {\sqrt {2} - e^{\left (-x\right )} - e^{x}}{\sqrt {2} + e^{\left (-x\right )} + e^{x}}\right ) + \frac {1}{2} \, e^{\left (-x\right )} + \frac {1}{2} \, e^{x} \]

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

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maple [A]  time = 0.18, size = 102, normalized size = 1.17 \[ \cosh \relax (x )-\frac {2 \left (3+2 \sqrt {3}\right ) \sqrt {3}\, \arctanh \left (\frac {8 \cosh \relax (x )}{2 \sqrt {6}+2 \sqrt {2}}\right )}{9 \left (2 \sqrt {6}+2 \sqrt {2}\right )}-\frac {2 \left (-3+2 \sqrt {3}\right ) \sqrt {3}\, \arctanh \left (\frac {8 \cosh \relax (x )}{2 \sqrt {6}-2 \sqrt {2}}\right )}{9 \left (2 \sqrt {6}-2 \sqrt {2}\right )}-\frac {\arctanh \left (\cosh \relax (x ) \sqrt {2}\right ) \sqrt {2}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

cosh(x)-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)))-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 = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{2} \, {\left (e^{\left (2 \, x\right )} + 1\right )} e^{\left (-x\right )} - \frac {1}{12} \, \sqrt {2} \log \left (\sqrt {2} e^{x} + e^{\left (2 \, x\right )} + 1\right ) + \frac {1}{12} \, \sqrt {2} \log \left (-\sqrt {2} e^{x} + e^{\left (2 \, x\right )} + 1\right ) + \frac {1}{2} \, \int \frac {2 \, {\left (2 \, e^{\left (7 \, x\right )} + e^{\left (5 \, x\right )} - e^{\left (3 \, x\right )} - 2 \, e^{x}\right )}}{3 \, {\left (e^{\left (8 \, x\right )} - e^{\left (4 \, x\right )} + 1\right )}}\,{d x} \]

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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mupad [B]  time = 0.10, size = 170, normalized size = 1.95 \[ \frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}-\frac {\sqrt {2}\,\ln \left ({\mathrm {e}}^{2\,x}+\sqrt {2}\,{\mathrm {e}}^x+1\right )}{12}+\frac {\sqrt {2}\,\ln \left ({\mathrm {e}}^{2\,x}-\sqrt {2}\,{\mathrm {e}}^x+1\right )}{12}+\ln \left ({\mathrm {e}}^{2\,x}-12\,{\mathrm {e}}^x\,\sqrt {\frac {1}{72}-\frac {\sqrt {3}}{144}}+1\right )\,\sqrt {\frac {1}{72}-\frac {\sqrt {3}}{144}}-\ln \left ({\mathrm {e}}^{2\,x}+12\,{\mathrm {e}}^x\,\sqrt {\frac {1}{72}-\frac {\sqrt {3}}{144}}+1\right )\,\sqrt {\frac {1}{72}-\frac {\sqrt {3}}{144}}+\ln \left ({\mathrm {e}}^{2\,x}-12\,{\mathrm {e}}^x\,\sqrt {\frac {\sqrt {3}}{144}+\frac {1}{72}}+1\right )\,\sqrt {\frac {\sqrt {3}}{144}+\frac {1}{72}}-\ln \left ({\mathrm {e}}^{2\,x}+12\,{\mathrm {e}}^x\,\sqrt {\frac {\sqrt {3}}{144}+\frac {1}{72}}+1\right )\,\sqrt {\frac {\sqrt {3}}{144}+\frac {1}{72}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp(-x)/2 + exp(x)/2 - (2^(1/2)*log(exp(2*x) + 2^(1/2)*exp(x) + 1))/12 + (2^(1/2)*log(exp(2*x) - 2^(1/2)*exp(x
) + 1))/12 + log(exp(2*x) - 12*exp(x)*(1/72 - 3^(1/2)/144)^(1/2) + 1)*(1/72 - 3^(1/2)/144)^(1/2) - log(exp(2*x
) + 12*exp(x)*(1/72 - 3^(1/2)/144)^(1/2) + 1)*(1/72 - 3^(1/2)/144)^(1/2) + log(exp(2*x) - 12*exp(x)*(3^(1/2)/1
44 + 1/72)^(1/2) + 1)*(3^(1/2)/144 + 1/72)^(1/2) - log(exp(2*x) + 12*exp(x)*(3^(1/2)/144 + 1/72)^(1/2) + 1)*(3
^(1/2)/144 + 1/72)^(1/2)

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

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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