3.231 \(\int \cosh (x) \tanh (4 x) \, dx\)

Optimal. Leaf size=69 \[ \cosh (x)-\frac {1}{4} \sqrt {2-\sqrt {2}} \tanh ^{-1}\left (\frac {2 \cosh (x)}{\sqrt {2-\sqrt {2}}}\right )-\frac {1}{4} \sqrt {2+\sqrt {2}} \tanh ^{-1}\left (\frac {2 \cosh (x)}{\sqrt {2+\sqrt {2}}}\right ) \]

[Out]

cosh(x)-1/4*arctanh(2*cosh(x)/(2-2^(1/2))^(1/2))*(2-2^(1/2))^(1/2)-1/4*arctanh(2*cosh(x)/(2+2^(1/2))^(1/2))*(2
+2^(1/2))^(1/2)

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Rubi [A]  time = 0.09, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {12, 1279, 1166, 207} \[ \cosh (x)-\frac {1}{4} \sqrt {2-\sqrt {2}} \tanh ^{-1}\left (\frac {2 \cosh (x)}{\sqrt {2-\sqrt {2}}}\right )-\frac {1}{4} \sqrt {2+\sqrt {2}} \tanh ^{-1}\left (\frac {2 \cosh (x)}{\sqrt {2+\sqrt {2}}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Cosh[x]*Tanh[4*x],x]

[Out]

-(Sqrt[2 - Sqrt[2]]*ArcTanh[(2*Cosh[x])/Sqrt[2 - Sqrt[2]]])/4 - (Sqrt[2 + Sqrt[2]]*ArcTanh[(2*Cosh[x])/Sqrt[2
+ Sqrt[2]]])/4 + 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 1279

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[(e*f
*(f*x)^(m - 1)*(a + b*x^2 + c*x^4)^(p + 1))/(c*(m + 4*p + 3)), x] - Dist[f^2/(c*(m + 4*p + 3)), Int[(f*x)^(m -
2)*(a + b*x^2 + c*x^4)^p*Simp[a*e*(m - 1) + (b*e*(m + 2*p + 1) - c*d*(m + 4*p + 3))*x^2, x], x], x] /; FreeQ[
{a, b, c, d, e, f, p}, x] && NeQ[b^2 - 4*a*c, 0] && GtQ[m, 1] && NeQ[m + 4*p + 3, 0] && IntegerQ[2*p] && (Inte
gerQ[p] || IntegerQ[m])

Rubi steps

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

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Mathematica [C]  time = 0.02, size = 113, normalized size = 1.64 \[ \frac {1}{16} \text {RootSum}\left [\text {$\#$1}^8+1\& ,\frac {\text {$\#$1}^6 x+2 \text {$\#$1}^6 \log \left (-\text {$\#$1} \sinh \left (\frac {x}{2}\right )+\text {$\#$1} \cosh \left (\frac {x}{2}\right )-\sinh \left (\frac {x}{2}\right )-\cosh \left (\frac {x}{2}\right )\right )-2 \log \left (-\text {$\#$1} \sinh \left (\frac {x}{2}\right )+\text {$\#$1} \cosh \left (\frac {x}{2}\right )-\sinh \left (\frac {x}{2}\right )-\cosh \left (\frac {x}{2}\right )\right )-x}{\text {$\#$1}^7}\& \right ]+\cosh (x) \]

Antiderivative was successfully verified.

[In]

Integrate[Cosh[x]*Tanh[4*x],x]

[Out]

Cosh[x] + RootSum[1 + #1^8 & , (-x - 2*Log[-Cosh[x/2] - Sinh[x/2] + Cosh[x/2]*#1 - Sinh[x/2]*#1] + x*#1^6 + 2*
Log[-Cosh[x/2] - Sinh[x/2] + Cosh[x/2]*#1 - Sinh[x/2]*#1]*#1^6)/#1^7 & ]/16

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fricas [B]  time = 0.52, size = 213, normalized size = 3.09 \[ -\frac {\sqrt {\sqrt {2} + 2} {\left (\cosh \relax (x) + \sinh \relax (x)\right )} \log \left (\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2} + \sqrt {\sqrt {2} + 2} {\left (\cosh \relax (x) + \sinh \relax (x)\right )} + 1\right ) - \sqrt {\sqrt {2} + 2} {\left (\cosh \relax (x) + \sinh \relax (x)\right )} \log \left (\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2} - \sqrt {\sqrt {2} + 2} {\left (\cosh \relax (x) + \sinh \relax (x)\right )} + 1\right ) + \sqrt {-\sqrt {2} + 2} {\left (\cosh \relax (x) + \sinh \relax (x)\right )} \log \left (\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2} + \sqrt {-\sqrt {2} + 2} {\left (\cosh \relax (x) + \sinh \relax (x)\right )} + 1\right ) - \sqrt {-\sqrt {2} + 2} {\left (\cosh \relax (x) + \sinh \relax (x)\right )} \log \left (\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2} - \sqrt {-\sqrt {2} + 2} {\left (\cosh \relax (x) + \sinh \relax (x)\right )} + 1\right ) - 4 \, \cosh \relax (x)^{2} - 8 \, \cosh \relax (x) \sinh \relax (x) - 4 \, \sinh \relax (x)^{2} - 4}{8 \, {\left (\cosh \relax (x) + \sinh \relax (x)\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)*tanh(4*x),x, algorithm="fricas")

[Out]

-1/8*(sqrt(sqrt(2) + 2)*(cosh(x) + sinh(x))*log(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 + sqrt(sqrt(2) + 2)*
(cosh(x) + sinh(x)) + 1) - sqrt(sqrt(2) + 2)*(cosh(x) + sinh(x))*log(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2
- sqrt(sqrt(2) + 2)*(cosh(x) + sinh(x)) + 1) + sqrt(-sqrt(2) + 2)*(cosh(x) + sinh(x))*log(cosh(x)^2 + 2*cosh(
x)*sinh(x) + sinh(x)^2 + sqrt(-sqrt(2) + 2)*(cosh(x) + sinh(x)) + 1) - sqrt(-sqrt(2) + 2)*(cosh(x) + sinh(x))*
log(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 - sqrt(-sqrt(2) + 2)*(cosh(x) + sinh(x)) + 1) - 4*cosh(x)^2 - 8*
cosh(x)*sinh(x) - 4*sinh(x)^2 - 4)/(cosh(x) + sinh(x))

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giac [B]  time = 0.21, size = 119, normalized size = 1.72 \[ -\frac {1}{8} \, \sqrt {\sqrt {2} + 2} \log \left (\sqrt {\sqrt {2} + 2} + e^{\left (-x\right )} + e^{x}\right ) + \frac {1}{8} \, \sqrt {\sqrt {2} + 2} \log \left (-\sqrt {\sqrt {2} + 2} + e^{\left (-x\right )} + e^{x}\right ) - \frac {1}{8} \, \sqrt {-\sqrt {2} + 2} \log \left (\sqrt {-\sqrt {2} + 2} + e^{\left (-x\right )} + e^{x}\right ) + \frac {1}{8} \, \sqrt {-\sqrt {2} + 2} \log \left (-\sqrt {-\sqrt {2} + 2} + e^{\left (-x\right )} + e^{x}\right ) + \frac {1}{2} \, e^{\left (-x\right )} + \frac {1}{2} \, e^{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)*tanh(4*x),x, algorithm="giac")

[Out]

-1/8*sqrt(sqrt(2) + 2)*log(sqrt(sqrt(2) + 2) + e^(-x) + e^x) + 1/8*sqrt(sqrt(2) + 2)*log(-sqrt(sqrt(2) + 2) +
e^(-x) + e^x) - 1/8*sqrt(-sqrt(2) + 2)*log(sqrt(-sqrt(2) + 2) + e^(-x) + e^x) + 1/8*sqrt(-sqrt(2) + 2)*log(-sq
rt(-sqrt(2) + 2) + e^(-x) + e^x) + 1/2*e^(-x) + 1/2*e^x

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maple [A]  time = 0.16, size = 66, normalized size = 0.96 \[ \cosh \relax (x )-\frac {\left (1+\sqrt {2}\right ) \sqrt {2}\, \arctanh \left (\frac {2 \cosh \relax (x )}{\sqrt {2+\sqrt {2}}}\right )}{4 \sqrt {2+\sqrt {2}}}-\frac {\left (\sqrt {2}-1\right ) \sqrt {2}\, \arctanh \left (\frac {2 \cosh \relax (x )}{\sqrt {2-\sqrt {2}}}\right )}{4 \sqrt {2-\sqrt {2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(x)*tanh(4*x),x)

[Out]

cosh(x)-1/4*(1+2^(1/2))*2^(1/2)/(2+2^(1/2))^(1/2)*arctanh(2*cosh(x)/(2+2^(1/2))^(1/2))-1/4*(2^(1/2)-1)*2^(1/2)
/(2-2^(1/2))^(1/2)*arctanh(2*cosh(x)/(2-2^(1/2))^(1/2))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{2} \, {\left (e^{\left (2 \, x\right )} + 1\right )} e^{\left (-x\right )} + \frac {1}{2} \, \int \frac {2 \, {\left (e^{\left (7 \, x\right )} - e^{x}\right )}}{e^{\left (8 \, x\right )} + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)*tanh(4*x),x, algorithm="maxima")

[Out]

1/2*(e^(2*x) + 1)*e^(-x) + 1/2*integrate(2*(e^(7*x) - e^x)/(e^(8*x) + 1), x)

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mupad [B]  time = 0.08, size = 133, normalized size = 1.93 \[ \frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}+\ln \left ({\mathrm {e}}^{2\,x}-8\,{\mathrm {e}}^x\,\sqrt {\frac {1}{32}-\frac {\sqrt {2}}{64}}+1\right )\,\sqrt {\frac {1}{32}-\frac {\sqrt {2}}{64}}-\ln \left ({\mathrm {e}}^{2\,x}+8\,{\mathrm {e}}^x\,\sqrt {\frac {1}{32}-\frac {\sqrt {2}}{64}}+1\right )\,\sqrt {\frac {1}{32}-\frac {\sqrt {2}}{64}}+\ln \left ({\mathrm {e}}^{2\,x}-8\,{\mathrm {e}}^x\,\sqrt {\frac {\sqrt {2}}{64}+\frac {1}{32}}+1\right )\,\sqrt {\frac {\sqrt {2}}{64}+\frac {1}{32}}-\ln \left ({\mathrm {e}}^{2\,x}+8\,{\mathrm {e}}^x\,\sqrt {\frac {\sqrt {2}}{64}+\frac {1}{32}}+1\right )\,\sqrt {\frac {\sqrt {2}}{64}+\frac {1}{32}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tanh(4*x)*cosh(x),x)

[Out]

exp(-x)/2 + exp(x)/2 + log(exp(2*x) - 8*exp(x)*(1/32 - 2^(1/2)/64)^(1/2) + 1)*(1/32 - 2^(1/2)/64)^(1/2) - log(
exp(2*x) + 8*exp(x)*(1/32 - 2^(1/2)/64)^(1/2) + 1)*(1/32 - 2^(1/2)/64)^(1/2) + log(exp(2*x) - 8*exp(x)*(2^(1/2
)/64 + 1/32)^(1/2) + 1)*(2^(1/2)/64 + 1/32)^(1/2) - log(exp(2*x) + 8*exp(x)*(2^(1/2)/64 + 1/32)^(1/2) + 1)*(2^
(1/2)/64 + 1/32)^(1/2)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \cosh {\relax (x )} \tanh {\left (4 x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)*tanh(4*x),x)

[Out]

Integral(cosh(x)*tanh(4*x), x)

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