3.232 \(\int \cosh (x) \tanh (5 x) \, dx\)

Optimal. Leaf size=82 \[ \cosh (x)-\frac {1}{5} \sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \tanh ^{-1}\left (2 \sqrt {\frac {2}{5+\sqrt {5}}} \cosh (x)\right )-\frac {1}{5} \sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \tanh ^{-1}\left (\sqrt {\frac {2}{5} \left (5+\sqrt {5}\right )} \cosh (x)\right ) \]

[Out]

cosh(x)-1/10*arctanh(1/5*cosh(x)*(50+10*5^(1/2))^(1/2))*(10-2*5^(1/2))^(1/2)-1/10*arctanh(2*cosh(x)*2^(1/2)/(5
+5^(1/2))^(1/2))*(10+2*5^(1/2))^(1/2)

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Rubi [A]  time = 0.12, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {1676, 1166, 207} \[ \cosh (x)-\frac {1}{5} \sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \tanh ^{-1}\left (2 \sqrt {\frac {2}{5+\sqrt {5}}} \cosh (x)\right )-\frac {1}{5} \sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \tanh ^{-1}\left (\sqrt {\frac {2}{5} \left (5+\sqrt {5}\right )} \cosh (x)\right ) \]

Antiderivative was successfully verified.

[In]

Int[Cosh[x]*Tanh[5*x],x]

[Out]

-(Sqrt[(5 + Sqrt[5])/2]*ArcTanh[2*Sqrt[2/(5 + Sqrt[5])]*Cosh[x]])/5 - (Sqrt[(5 - Sqrt[5])/2]*ArcTanh[Sqrt[(2*(
5 + Sqrt[5]))/5]*Cosh[x]])/5 + Cosh[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 1676

Int[(Pq_)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> Int[ExpandIntegrand[Pq/(a + b*x^2 + c*x^4), x], x
] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x^2] && Expon[Pq, x^2] > 1

Rubi steps

\begin {align*} \int \cosh (x) \tanh (5 x) \, dx &=\operatorname {Subst}\left (\int \frac {1-12 x^2+16 x^4}{5-20 x^2+16 x^4} \, dx,x,\cosh (x)\right )\\ &=\operatorname {Subst}\left (\int \left (1-\frac {4 \left (1-2 x^2\right )}{5-20 x^2+16 x^4}\right ) \, dx,x,\cosh (x)\right )\\ &=\cosh (x)-4 \operatorname {Subst}\left (\int \frac {1-2 x^2}{5-20 x^2+16 x^4} \, dx,x,\cosh (x)\right )\\ &=\cosh (x)+\frac {1}{5} \left (4 \left (5-\sqrt {5}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-10+2 \sqrt {5}+16 x^2} \, dx,x,\cosh (x)\right )+\frac {1}{5} \left (4 \left (5+\sqrt {5}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-10-2 \sqrt {5}+16 x^2} \, dx,x,\cosh (x)\right )\\ &=-\frac {1}{5} \sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \tanh ^{-1}\left (2 \sqrt {\frac {2}{5+\sqrt {5}}} \cosh (x)\right )-\frac {1}{5} \sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \tanh ^{-1}\left (\sqrt {\frac {2}{5} \left (5+\sqrt {5}\right )} \cosh (x)\right )+\cosh (x)\\ \end {align*}

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Mathematica [C]  time = 0.03, size = 249, normalized size = 3.04 \[ \frac {1}{4} \text {RootSum}\left [\text {$\#$1}^8-\text {$\#$1}^6+\text {$\#$1}^4-\text {$\#$1}^2+1\& ,\frac {\text {$\#$1}^6 x+2 \text {$\#$1}^6 \log \left (-\text {$\#$1} \sinh \left (\frac {x}{2}\right )+\text {$\#$1} \cosh \left (\frac {x}{2}\right )-\sinh \left (\frac {x}{2}\right )-\cosh \left (\frac {x}{2}\right )\right )-\text {$\#$1}^4 x-2 \text {$\#$1}^4 \log \left (-\text {$\#$1} \sinh \left (\frac {x}{2}\right )+\text {$\#$1} \cosh \left (\frac {x}{2}\right )-\sinh \left (\frac {x}{2}\right )-\cosh \left (\frac {x}{2}\right )\right )+\text {$\#$1}^2 x+2 \text {$\#$1}^2 \log \left (-\text {$\#$1} \sinh \left (\frac {x}{2}\right )+\text {$\#$1} \cosh \left (\frac {x}{2}\right )-\sinh \left (\frac {x}{2}\right )-\cosh \left (\frac {x}{2}\right )\right )-2 \log \left (-\text {$\#$1} \sinh \left (\frac {x}{2}\right )+\text {$\#$1} \cosh \left (\frac {x}{2}\right )-\sinh \left (\frac {x}{2}\right )-\cosh \left (\frac {x}{2}\right )\right )-x}{4 \text {$\#$1}^7-3 \text {$\#$1}^5+2 \text {$\#$1}^3-\text {$\#$1}}\& \right ]+\cosh (x) \]

Antiderivative was successfully verified.

[In]

Integrate[Cosh[x]*Tanh[5*x],x]

[Out]

Cosh[x] + RootSum[1 - #1^2 + #1^4 - #1^6 + #1^8 & , (-x - 2*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 + x*#1^6 + 2*Log[-Cosh[x/2] - Sinh[x/2] + Cosh[x/2]*#1 - Sin
h[x/2]*#1]*#1^6)/(-#1 + 2*#1^3 - 3*#1^5 + 4*#1^7) & ]/4

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fricas [B]  time = 0.42, size = 293, normalized size = 3.57 \[ -\frac {{\left (\sqrt {2} \cosh \relax (x) + \sqrt {2} \sinh \relax (x)\right )} \sqrt {\sqrt {5} + 5} \log \left (2 \, \cosh \relax (x)^{2} + 4 \, \cosh \relax (x) \sinh \relax (x) + 2 \, \sinh \relax (x)^{2} + {\left (\sqrt {2} \cosh \relax (x) + \sqrt {2} \sinh \relax (x)\right )} \sqrt {\sqrt {5} + 5} + 2\right ) - {\left (\sqrt {2} \cosh \relax (x) + \sqrt {2} \sinh \relax (x)\right )} \sqrt {\sqrt {5} + 5} \log \left (2 \, \cosh \relax (x)^{2} + 4 \, \cosh \relax (x) \sinh \relax (x) + 2 \, \sinh \relax (x)^{2} - {\left (\sqrt {2} \cosh \relax (x) + \sqrt {2} \sinh \relax (x)\right )} \sqrt {\sqrt {5} + 5} + 2\right ) + {\left (\sqrt {2} \cosh \relax (x) + \sqrt {2} \sinh \relax (x)\right )} \sqrt {-\sqrt {5} + 5} \log \left (2 \, \cosh \relax (x)^{2} + 4 \, \cosh \relax (x) \sinh \relax (x) + 2 \, \sinh \relax (x)^{2} + {\left (\sqrt {2} \cosh \relax (x) + \sqrt {2} \sinh \relax (x)\right )} \sqrt {-\sqrt {5} + 5} + 2\right ) - {\left (\sqrt {2} \cosh \relax (x) + \sqrt {2} \sinh \relax (x)\right )} \sqrt {-\sqrt {5} + 5} \log \left (2 \, \cosh \relax (x)^{2} + 4 \, \cosh \relax (x) \sinh \relax (x) + 2 \, \sinh \relax (x)^{2} - {\left (\sqrt {2} \cosh \relax (x) + \sqrt {2} \sinh \relax (x)\right )} \sqrt {-\sqrt {5} + 5} + 2\right ) - 10 \, \cosh \relax (x)^{2} - 20 \, \cosh \relax (x) \sinh \relax (x) - 10 \, \sinh \relax (x)^{2} - 10}{20 \, {\left (\cosh \relax (x) + \sinh \relax (x)\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)*tanh(5*x),x, algorithm="fricas")

[Out]

-1/20*((sqrt(2)*cosh(x) + sqrt(2)*sinh(x))*sqrt(sqrt(5) + 5)*log(2*cosh(x)^2 + 4*cosh(x)*sinh(x) + 2*sinh(x)^2
+ (sqrt(2)*cosh(x) + sqrt(2)*sinh(x))*sqrt(sqrt(5) + 5) + 2) - (sqrt(2)*cosh(x) + sqrt(2)*sinh(x))*sqrt(sqrt(
5) + 5)*log(2*cosh(x)^2 + 4*cosh(x)*sinh(x) + 2*sinh(x)^2 - (sqrt(2)*cosh(x) + sqrt(2)*sinh(x))*sqrt(sqrt(5) +
5) + 2) + (sqrt(2)*cosh(x) + sqrt(2)*sinh(x))*sqrt(-sqrt(5) + 5)*log(2*cosh(x)^2 + 4*cosh(x)*sinh(x) + 2*sinh
(x)^2 + (sqrt(2)*cosh(x) + sqrt(2)*sinh(x))*sqrt(-sqrt(5) + 5) + 2) - (sqrt(2)*cosh(x) + sqrt(2)*sinh(x))*sqrt
(-sqrt(5) + 5)*log(2*cosh(x)^2 + 4*cosh(x)*sinh(x) + 2*sinh(x)^2 - (sqrt(2)*cosh(x) + sqrt(2)*sinh(x))*sqrt(-s
qrt(5) + 5) + 2) - 10*cosh(x)^2 - 20*cosh(x)*sinh(x) - 10*sinh(x)^2 - 10)/(cosh(x) + sinh(x))

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giac [B]  time = 0.18, size = 127, normalized size = 1.55 \[ -\frac {1}{20} \, \sqrt {2 \, \sqrt {5} + 10} \log \left (\sqrt {\frac {1}{2} \, \sqrt {5} + \frac {5}{2}} + e^{\left (-x\right )} + e^{x}\right ) + \frac {1}{20} \, \sqrt {2 \, \sqrt {5} + 10} \log \left (-\sqrt {\frac {1}{2} \, \sqrt {5} + \frac {5}{2}} + e^{\left (-x\right )} + e^{x}\right ) - \frac {1}{20} \, \sqrt {-2 \, \sqrt {5} + 10} \log \left (\sqrt {-\frac {1}{2} \, \sqrt {5} + \frac {5}{2}} + e^{\left (-x\right )} + e^{x}\right ) + \frac {1}{20} \, \sqrt {-2 \, \sqrt {5} + 10} \log \left (-\sqrt {-\frac {1}{2} \, \sqrt {5} + \frac {5}{2}} + e^{\left (-x\right )} + e^{x}\right ) + \frac {1}{2} \, e^{\left (-x\right )} + \frac {1}{2} \, e^{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)*tanh(5*x),x, algorithm="giac")

[Out]

-1/20*sqrt(2*sqrt(5) + 10)*log(sqrt(1/2*sqrt(5) + 5/2) + e^(-x) + e^x) + 1/20*sqrt(2*sqrt(5) + 10)*log(-sqrt(1
/2*sqrt(5) + 5/2) + e^(-x) + e^x) - 1/20*sqrt(-2*sqrt(5) + 10)*log(sqrt(-1/2*sqrt(5) + 5/2) + e^(-x) + e^x) +
1/20*sqrt(-2*sqrt(5) + 10)*log(-sqrt(-1/2*sqrt(5) + 5/2) + e^(-x) + e^x) + 1/2*e^(-x) + 1/2*e^x

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maple [A]  time = 0.17, size = 70, normalized size = 0.85 \[ \cosh \relax (x )-\frac {\sqrt {5}\, \left (\sqrt {5}-1\right ) \arctanh \left (\frac {4 \cosh \relax (x )}{\sqrt {10-2 \sqrt {5}}}\right )}{5 \sqrt {10-2 \sqrt {5}}}-\frac {\left (\sqrt {5}+1\right ) \sqrt {5}\, \arctanh \left (\frac {4 \cosh \relax (x )}{\sqrt {10+2 \sqrt {5}}}\right )}{5 \sqrt {10+2 \sqrt {5}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(x)*tanh(5*x),x)

[Out]

cosh(x)-1/5*5^(1/2)*(5^(1/2)-1)/(10-2*5^(1/2))^(1/2)*arctanh(4*cosh(x)/(10-2*5^(1/2))^(1/2))-1/5*(5^(1/2)+1)*5
^(1/2)/(10+2*5^(1/2))^(1/2)*arctanh(4*cosh(x)/(10+2*5^(1/2))^(1/2))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{2} \, {\left (e^{\left (2 \, x\right )} + 1\right )} e^{\left (-x\right )} + \frac {1}{2} \, \int \frac {2 \, {\left (e^{\left (7 \, x\right )} - e^{\left (5 \, x\right )} + e^{\left (3 \, x\right )} - e^{x}\right )}}{e^{\left (8 \, x\right )} - e^{\left (6 \, x\right )} + e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )} + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)*tanh(5*x),x, algorithm="maxima")

[Out]

1/2*(e^(2*x) + 1)*e^(-x) + 1/2*integrate(2*(e^(7*x) - e^(5*x) + e^(3*x) - e^x)/(e^(8*x) - e^(6*x) + e^(4*x) -
e^(2*x) + 1), x)

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mupad [B]  time = 0.09, size = 141, normalized size = 1.72 \[ \frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}+\ln \left (4\,{\mathrm {e}}^{2\,x}-40\,{\mathrm {e}}^x\,\sqrt {\frac {1}{40}-\frac {\sqrt {5}}{200}}+4\right )\,\sqrt {\frac {1}{40}-\frac {\sqrt {5}}{200}}-\ln \left (4\,{\mathrm {e}}^{2\,x}+40\,{\mathrm {e}}^x\,\sqrt {\frac {1}{40}-\frac {\sqrt {5}}{200}}+4\right )\,\sqrt {\frac {1}{40}-\frac {\sqrt {5}}{200}}+\ln \left (4\,{\mathrm {e}}^{2\,x}-40\,{\mathrm {e}}^x\,\sqrt {\frac {\sqrt {5}}{200}+\frac {1}{40}}+4\right )\,\sqrt {\frac {\sqrt {5}}{200}+\frac {1}{40}}-\ln \left (4\,{\mathrm {e}}^{2\,x}+40\,{\mathrm {e}}^x\,\sqrt {\frac {\sqrt {5}}{200}+\frac {1}{40}}+4\right )\,\sqrt {\frac {\sqrt {5}}{200}+\frac {1}{40}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tanh(5*x)*cosh(x),x)

[Out]

exp(-x)/2 + exp(x)/2 + log(4*exp(2*x) - 40*exp(x)*(1/40 - 5^(1/2)/200)^(1/2) + 4)*(1/40 - 5^(1/2)/200)^(1/2) -
log(4*exp(2*x) + 40*exp(x)*(1/40 - 5^(1/2)/200)^(1/2) + 4)*(1/40 - 5^(1/2)/200)^(1/2) + log(4*exp(2*x) - 40*e
xp(x)*(5^(1/2)/200 + 1/40)^(1/2) + 4)*(5^(1/2)/200 + 1/40)^(1/2) - log(4*exp(2*x) + 40*exp(x)*(5^(1/2)/200 + 1
/40)^(1/2) + 4)*(5^(1/2)/200 + 1/40)^(1/2)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \cosh {\relax (x )} \tanh {\left (5 x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)*tanh(5*x),x)

[Out]

Integral(cosh(x)*tanh(5*x), x)

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