### 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]

-(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]

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Rubi [A]  time = 0.0867444, antiderivative size = 69, normalized size of antiderivative = 1., 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 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])

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 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) \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*}

Mathematica [C]  time = 0.0229881, 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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Maple [A]  time = 0.054, size = 66, normalized size = 1. \begin{align*} \cosh \left ( x \right ) -{\frac{ \left ( \sqrt{2}-1 \right ) \sqrt{2}}{4\,\sqrt{2-\sqrt{2}}}{\it Artanh} \left ( 2\,{\frac{\cosh \left ( x \right ) }{\sqrt{2-\sqrt{2}}}} \right ) }-{\frac{ \left ( 1+\sqrt{2} \right ) \sqrt{2}}{4\,\sqrt{2+\sqrt{2}}}{\it Artanh} \left ( 2\,{\frac{\cosh \left ( x \right ) }{\sqrt{2+\sqrt{2}}}} \right ) } \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

cosh(x)-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))-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))

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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}{2} \, \int \frac{2 \,{\left (e^{\left (7 \, x\right )} - e^{x}\right )}}{e^{\left (8 \, x\right )} + 1}\,{d x} \end{align*}

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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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \cosh{\left (x \right )} \tanh{\left (4 x \right )}\, dx \end{align*}

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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Giac [B]  time = 1.23464, size = 161, normalized size = 2.33 \begin{align*} -\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} \end{align*}

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