3.589 \(\int \frac{\cosh ^3(x)-\sinh ^3(x)}{\cosh ^3(x)+\sinh ^3(x)} \, dx\)

Optimal. Leaf size=33 \[ -\frac{1}{3 (\tanh (x)+1)}-\frac{4 \tan ^{-1}\left (\frac{1-2 \tanh (x)}{\sqrt{3}}\right )}{3 \sqrt{3}} \]

[Out]

(-4*ArcTan[(1 - 2*Tanh[x])/Sqrt[3]])/(3*Sqrt[3]) - 1/(3*(1 + Tanh[x]))

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Rubi [A]  time = 0.136227, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {2074, 618, 204} \[ -\frac{1}{3 (\tanh (x)+1)}-\frac{4 \tan ^{-1}\left (\frac{1-2 \tanh (x)}{\sqrt{3}}\right )}{3 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

Int[(Cosh[x]^3 - Sinh[x]^3)/(Cosh[x]^3 + Sinh[x]^3),x]

[Out]

(-4*ArcTan[(1 - 2*Tanh[x])/Sqrt[3]])/(3*Sqrt[3]) - 1/(3*(1 + Tanh[x]))

Rule 2074

Int[(P_)^(p_)*(Q_)^(q_.), x_Symbol] :> With[{PP = Factor[P]}, Int[ExpandIntegrand[PP^p*Q^q, x], x] /;  !SumQ[N
onfreeFactors[PP, x]]] /; FreeQ[q, x] && PolyQ[P, x] && PolyQ[Q, x] && IntegerQ[p] && NeQ[P, x]

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{\cosh ^3(x)-\sinh ^3(x)}{\cosh ^3(x)+\sinh ^3(x)} \, dx &=\operatorname{Subst}\left (\int \frac{1+x+x^2}{1+x+x^3+x^4} \, dx,x,\tanh (x)\right )\\ &=\operatorname{Subst}\left (\int \left (\frac{1}{3 (1+x)^2}+\frac{2}{3 \left (1-x+x^2\right )}\right ) \, dx,x,\tanh (x)\right )\\ &=-\frac{1}{3 (1+\tanh (x))}+\frac{2}{3} \operatorname{Subst}\left (\int \frac{1}{1-x+x^2} \, dx,x,\tanh (x)\right )\\ &=-\frac{1}{3 (1+\tanh (x))}-\frac{4}{3} \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,-1+2 \tanh (x)\right )\\ &=-\frac{4 \tan ^{-1}\left (\frac{1-2 \tanh (x)}{\sqrt{3}}\right )}{3 \sqrt{3}}-\frac{1}{3 (1+\tanh (x))}\\ \end{align*}

Mathematica [A]  time = 0.128968, size = 37, normalized size = 1.12 \[ \frac{1}{18} \left (3 \sinh (2 x)-3 \cosh (2 x)+8 \sqrt{3} \tan ^{-1}\left (\frac{2 \tanh (x)-1}{\sqrt{3}}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(Cosh[x]^3 - Sinh[x]^3)/(Cosh[x]^3 + Sinh[x]^3),x]

[Out]

(8*Sqrt[3]*ArcTan[(-1 + 2*Tanh[x])/Sqrt[3]] - 3*Cosh[2*x] + 3*Sinh[2*x])/18

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Maple [C]  time = 0.084, size = 78, normalized size = 2.4 \begin{align*} -{\frac{2}{3} \left ( 1+\tanh \left ({\frac{x}{2}} \right ) \right ) ^{-2}}+{\frac{2}{3} \left ( 1+\tanh \left ({\frac{x}{2}} \right ) \right ) ^{-1}}+{\frac{2\,i}{9}}\sqrt{3}\ln \left ( \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}+ \left ( -1-i\sqrt{3} \right ) \tanh \left ({\frac{x}{2}} \right ) +1 \right ) -{\frac{2\,i}{9}}\sqrt{3}\ln \left ( \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}+ \left ( -1+i\sqrt{3} \right ) \tanh \left ({\frac{x}{2}} \right ) +1 \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((cosh(x)^3-sinh(x)^3)/(cosh(x)^3+sinh(x)^3),x)

[Out]

-2/3/(1+tanh(1/2*x))^2+2/3/(1+tanh(1/2*x))+2/9*I*3^(1/2)*ln(tanh(1/2*x)^2+(-1-I*3^(1/2))*tanh(1/2*x)+1)-2/9*I*
3^(1/2)*ln(tanh(1/2*x)^2+(-1+I*3^(1/2))*tanh(1/2*x)+1)

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Maxima [B]  time = 1.46238, size = 95, normalized size = 2.88 \begin{align*} \frac{4}{9} \, \sqrt{3} \arctan \left (\frac{1}{6} \cdot 3^{\frac{3}{4}} \sqrt{2}{\left (2 \, \sqrt{3} e^{\left (-x\right )} + 3^{\frac{1}{4}} \sqrt{2}\right )}\right ) - \frac{4}{9} \, \sqrt{3} \arctan \left (\frac{1}{6} \cdot 3^{\frac{3}{4}} \sqrt{2}{\left (2 \, \sqrt{3} e^{\left (-x\right )} - 3^{\frac{1}{4}} \sqrt{2}\right )}\right ) - \frac{1}{6} \, e^{\left (-2 \, x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((cosh(x)^3-sinh(x)^3)/(cosh(x)^3+sinh(x)^3),x, algorithm="maxima")

[Out]

4/9*sqrt(3)*arctan(1/6*3^(3/4)*sqrt(2)*(2*sqrt(3)*e^(-x) + 3^(1/4)*sqrt(2))) - 4/9*sqrt(3)*arctan(1/6*3^(3/4)*
sqrt(2)*(2*sqrt(3)*e^(-x) - 3^(1/4)*sqrt(2))) - 1/6*e^(-2*x)

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Fricas [B]  time = 2.22686, size = 265, normalized size = 8.03 \begin{align*} -\frac{8 \,{\left (\sqrt{3} \cosh \left (x\right )^{2} + 2 \, \sqrt{3} \cosh \left (x\right ) \sinh \left (x\right ) + \sqrt{3} \sinh \left (x\right )^{2}\right )} \arctan \left (-\frac{\sqrt{3} \cosh \left (x\right ) + \sqrt{3} \sinh \left (x\right )}{3 \,{\left (\cosh \left (x\right ) - \sinh \left (x\right )\right )}}\right ) + 3}{18 \,{\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((cosh(x)^3-sinh(x)^3)/(cosh(x)^3+sinh(x)^3),x, algorithm="fricas")

[Out]

-1/18*(8*(sqrt(3)*cosh(x)^2 + 2*sqrt(3)*cosh(x)*sinh(x) + sqrt(3)*sinh(x)^2)*arctan(-1/3*(sqrt(3)*cosh(x) + sq
rt(3)*sinh(x))/(cosh(x) - sinh(x))) + 3)/(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2)

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Sympy [B]  time = 1.99149, size = 102, normalized size = 3.09 \begin{align*} \frac{4 \sqrt{3} \sinh{\left (x \right )} \operatorname{atan}{\left (\frac{2 \sqrt{3} \sinh{\left (x \right )}}{3 \cosh{\left (x \right )}} - \frac{\sqrt{3}}{3} \right )}}{9 \sinh{\left (x \right )} + 9 \cosh{\left (x \right )}} + \frac{4 \sqrt{3} \cosh{\left (x \right )} \operatorname{atan}{\left (\frac{2 \sqrt{3} \sinh{\left (x \right )}}{3 \cosh{\left (x \right )}} - \frac{\sqrt{3}}{3} \right )}}{9 \sinh{\left (x \right )} + 9 \cosh{\left (x \right )}} - \frac{3 \cosh{\left (x \right )}}{9 \sinh{\left (x \right )} + 9 \cosh{\left (x \right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((cosh(x)**3-sinh(x)**3)/(cosh(x)**3+sinh(x)**3),x)

[Out]

4*sqrt(3)*sinh(x)*atan(2*sqrt(3)*sinh(x)/(3*cosh(x)) - sqrt(3)/3)/(9*sinh(x) + 9*cosh(x)) + 4*sqrt(3)*cosh(x)*
atan(2*sqrt(3)*sinh(x)/(3*cosh(x)) - sqrt(3)/3)/(9*sinh(x) + 9*cosh(x)) - 3*cosh(x)/(9*sinh(x) + 9*cosh(x))

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Giac [A]  time = 1.08848, size = 30, normalized size = 0.91 \begin{align*} \frac{4}{9} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3} e^{\left (2 \, x\right )}\right ) - \frac{1}{6} \, e^{\left (-2 \, x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((cosh(x)^3-sinh(x)^3)/(cosh(x)^3+sinh(x)^3),x, algorithm="giac")

[Out]

4/9*sqrt(3)*arctan(1/3*sqrt(3)*e^(2*x)) - 1/6*e^(-2*x)