3.58 \(\int \frac{1}{1+\cosh ^3(x)} \, dx\)

Optimal. Leaf size=91 \[ -\frac{2 \sqrt [4]{-\frac{1}{3}} \tanh ^{-1}\left ((-1)^{3/4} \sqrt [4]{3} \tanh \left (\frac{x}{2}\right )\right )}{3 \left (1+(-1)^{2/3}\right )}-\frac{2 \sqrt [4]{-\frac{1}{3}} \tan ^{-1}\left ((-1)^{3/4} \sqrt [4]{3} \tanh \left (\frac{x}{2}\right )\right )}{3 \left (1-\sqrt [3]{-1}\right )}+\frac{\sinh (x)}{3 (\cosh (x)+1)} \]

[Out]

(-2*(-1/3)^(1/4)*ArcTan[(-1)^(3/4)*3^(1/4)*Tanh[x/2]])/(3*(1 - (-1)^(1/3))) - (2*(-1/3)^(1/4)*ArcTanh[(-1)^(3/
4)*3^(1/4)*Tanh[x/2]])/(3*(1 + (-1)^(2/3))) + Sinh[x]/(3*(1 + Cosh[x]))

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Rubi [A]  time = 0.130056, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.625, Rules used = {3213, 2648, 2659, 205, 208} \[ -\frac{2 \sqrt [4]{-\frac{1}{3}} \tanh ^{-1}\left ((-1)^{3/4} \sqrt [4]{3} \tanh \left (\frac{x}{2}\right )\right )}{3 \left (1+(-1)^{2/3}\right )}-\frac{2 \sqrt [4]{-\frac{1}{3}} \tan ^{-1}\left ((-1)^{3/4} \sqrt [4]{3} \tanh \left (\frac{x}{2}\right )\right )}{3 \left (1-\sqrt [3]{-1}\right )}+\frac{\sinh (x)}{3 (\cosh (x)+1)} \]

Antiderivative was successfully verified.

[In]

Int[(1 + Cosh[x]^3)^(-1),x]

[Out]

(-2*(-1/3)^(1/4)*ArcTan[(-1)^(3/4)*3^(1/4)*Tanh[x/2]])/(3*(1 - (-1)^(1/3))) - (2*(-1/3)^(1/4)*ArcTanh[(-1)^(3/
4)*3^(1/4)*Tanh[x/2]])/(3*(1 + (-1)^(2/3))) + Sinh[x]/(3*(1 + Cosh[x]))

Rule 3213

Int[((a_) + (b_.)*((c_.)*sin[(e_.) + (f_.)*(x_)])^(n_))^(p_), x_Symbol] :> Int[ExpandTrig[(a + b*(c*sin[e + f*
x])^n)^p, x], x] /; FreeQ[{a, b, c, e, f, n}, x] && (IGtQ[p, 0] || (EqQ[p, -1] && IntegerQ[n]))

Rule 2648

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> -Simp[Cos[c + d*x]/(d*(b + a*Sin[c + d*x])), x]
/; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]

Rule 2659

Int[((a_) + (b_.)*sin[Pi/2 + (c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x
]}, Dist[(2*e)/d, Subst[Int[1/(a + b + (a - b)*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}
, x] && NeQ[a^2 - b^2, 0]

Rule 205

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

Rule 208

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

Rubi steps

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

Mathematica [C]  time = 0.846407, size = 133, normalized size = 1.46 \[ \frac{1}{18} \left (6 \tanh \left (\frac{x}{2}\right )-\sqrt{6+2 i \sqrt{3}} \left (\sqrt{3}+3 i\right ) \tan ^{-1}\left (\frac{\left (3+i \sqrt{3}\right ) \tanh \left (\frac{x}{2}\right )}{\sqrt{6-2 i \sqrt{3}}}\right )-\sqrt{6-2 i \sqrt{3}} \left (\sqrt{3}-3 i\right ) \tan ^{-1}\left (\frac{\left (3-i \sqrt{3}\right ) \tanh \left (\frac{x}{2}\right )}{\sqrt{6+2 i \sqrt{3}}}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + Cosh[x]^3)^(-1),x]

[Out]

(-(Sqrt[6 + (2*I)*Sqrt[3]]*(3*I + Sqrt[3])*ArcTan[((3 + I*Sqrt[3])*Tanh[x/2])/Sqrt[6 - (2*I)*Sqrt[3]]]) - Sqrt
[6 - (2*I)*Sqrt[3]]*(-3*I + Sqrt[3])*ArcTan[((3 - I*Sqrt[3])*Tanh[x/2])/Sqrt[6 + (2*I)*Sqrt[3]]] + 6*Tanh[x/2]
)/18

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Maple [B]  time = 0.016, size = 216, normalized size = 2.4 \begin{align*}{\frac{1}{3}\tanh \left ({\frac{x}{2}} \right ) }+{\frac{{3}^{{\frac{3}{4}}}\sqrt{2}}{18}\arctan \left ( \sqrt{2}\sqrt [4]{3}\tanh \left ({\frac{x}{2}} \right ) -1 \right ) }+{\frac{{3}^{{\frac{3}{4}}}\sqrt{2}}{36}\ln \left ({ \left ( \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}+{\frac{{3}^{{\frac{3}{4}}}\sqrt{2}}{3}\tanh \left ({\frac{x}{2}} \right ) }+{\frac{\sqrt{3}}{3}} \right ) \left ( \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}-{\frac{{3}^{{\frac{3}{4}}}\sqrt{2}}{3}\tanh \left ({\frac{x}{2}} \right ) }+{\frac{\sqrt{3}}{3}} \right ) ^{-1}} \right ) }+{\frac{{3}^{{\frac{3}{4}}}\sqrt{2}}{18}\arctan \left ( \sqrt{2}\sqrt [4]{3}\tanh \left ({\frac{x}{2}} \right ) +1 \right ) }-{\frac{\sqrt{2}\sqrt [4]{3}}{12}\ln \left ({ \left ( \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}-{\frac{{3}^{{\frac{3}{4}}}\sqrt{2}}{3}\tanh \left ({\frac{x}{2}} \right ) }+{\frac{\sqrt{3}}{3}} \right ) \left ( \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}+{\frac{{3}^{{\frac{3}{4}}}\sqrt{2}}{3}\tanh \left ({\frac{x}{2}} \right ) }+{\frac{\sqrt{3}}{3}} \right ) ^{-1}} \right ) }-{\frac{\sqrt{2}\sqrt [4]{3}}{6}\arctan \left ( \sqrt{2}\sqrt [4]{3}\tanh \left ({\frac{x}{2}} \right ) -1 \right ) }-{\frac{\sqrt{2}\sqrt [4]{3}}{6}\arctan \left ( \sqrt{2}\sqrt [4]{3}\tanh \left ({\frac{x}{2}} \right ) +1 \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1+cosh(x)^3),x)

[Out]

1/3*tanh(1/2*x)+1/18*3^(3/4)*2^(1/2)*arctan(2^(1/2)*3^(1/4)*tanh(1/2*x)-1)+1/36*3^(3/4)*2^(1/2)*ln((tanh(1/2*x
)^2+1/3*3^(3/4)*tanh(1/2*x)*2^(1/2)+1/3*3^(1/2))/(tanh(1/2*x)^2-1/3*3^(3/4)*tanh(1/2*x)*2^(1/2)+1/3*3^(1/2)))+
1/18*3^(3/4)*2^(1/2)*arctan(2^(1/2)*3^(1/4)*tanh(1/2*x)+1)-1/12*2^(1/2)*3^(1/4)*ln((tanh(1/2*x)^2-1/3*3^(3/4)*
tanh(1/2*x)*2^(1/2)+1/3*3^(1/2))/(tanh(1/2*x)^2+1/3*3^(3/4)*tanh(1/2*x)*2^(1/2)+1/3*3^(1/2)))-1/6*2^(1/2)*3^(1
/4)*arctan(2^(1/2)*3^(1/4)*tanh(1/2*x)-1)-1/6*2^(1/2)*3^(1/4)*arctan(2^(1/2)*3^(1/4)*tanh(1/2*x)+1)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{2}{3 \,{\left (e^{x} + 1\right )}} - \int \frac{2 \,{\left (e^{\left (3 \, x\right )} - 4 \, e^{\left (2 \, x\right )} + e^{x}\right )}}{3 \,{\left (e^{\left (4 \, x\right )} - 2 \, e^{\left (3 \, x\right )} + 6 \, e^{\left (2 \, x\right )} - 2 \, e^{x} + 1\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+cosh(x)^3),x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 2.49769, size = 1922, normalized size = 21.12 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+cosh(x)^3),x, algorithm="fricas")

[Out]

1/36*(4*(3^(3/4)*e^x + 3^(3/4))*sqrt(-4*sqrt(3) + 8)*arctan(1/12*(sqrt(3)*(sqrt(3) + 3) - 3*sqrt(3) + 9)*e^x -
1/48*(2*sqrt(3)*(sqrt(3) + 3) - (3^(3/4)*(3*sqrt(3) + 5) + 3*3^(1/4)*(sqrt(3) + 1))*sqrt(-4*sqrt(3) + 8) - 6*
sqrt(3) + 18)*sqrt(2*(3^(1/4)*(sqrt(3) + 2) - 3^(1/4)*e^x)*sqrt(-4*sqrt(3) + 8) + 4*sqrt(3) + 4*e^(2*x) - 4*e^
x + 4) + 1/12*sqrt(3)*(sqrt(3) - 3) - 1/24*((3^(3/4)*(3*sqrt(3) + 5) + 3*3^(1/4)*(sqrt(3) + 1))*e^x - 3^(3/4)*
(sqrt(3) + 1) - 3*3^(1/4)*(sqrt(3) - 1))*sqrt(-4*sqrt(3) + 8) + 1/4*sqrt(3) - 1/4) + 4*(3^(3/4)*e^x + 3^(3/4))
*sqrt(-4*sqrt(3) + 8)*arctan(-1/12*(sqrt(3)*(sqrt(3) + 3) - 3*sqrt(3) + 9)*e^x + 1/48*(2*sqrt(3)*(sqrt(3) + 3)
+ (3^(3/4)*(3*sqrt(3) + 5) + 3*3^(1/4)*(sqrt(3) + 1))*sqrt(-4*sqrt(3) + 8) - 6*sqrt(3) + 18)*sqrt(-2*(3^(1/4)
*(sqrt(3) + 2) - 3^(1/4)*e^x)*sqrt(-4*sqrt(3) + 8) + 4*sqrt(3) + 4*e^(2*x) - 4*e^x + 4) - 1/12*sqrt(3)*(sqrt(3
) - 3) - 1/24*((3^(3/4)*(3*sqrt(3) + 5) + 3*3^(1/4)*(sqrt(3) + 1))*e^x - 3^(3/4)*(sqrt(3) + 1) - 3*3^(1/4)*(sq
rt(3) - 1))*sqrt(-4*sqrt(3) + 8) - 1/4*sqrt(3) + 1/4) - (3^(1/4)*(2*sqrt(3) + 3)*e^x + 3^(1/4)*(2*sqrt(3) + 3)
)*sqrt(-4*sqrt(3) + 8)*log(2*(3^(1/4)*(sqrt(3) + 2) - 3^(1/4)*e^x)*sqrt(-4*sqrt(3) + 8) + 4*sqrt(3) + 4*e^(2*x
) - 4*e^x + 4) + (3^(1/4)*(2*sqrt(3) + 3)*e^x + 3^(1/4)*(2*sqrt(3) + 3))*sqrt(-4*sqrt(3) + 8)*log(-2*(3^(1/4)*
(sqrt(3) + 2) - 3^(1/4)*e^x)*sqrt(-4*sqrt(3) + 8) + 4*sqrt(3) + 4*e^(2*x) - 4*e^x + 4) - 24)/(e^x + 1)

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Sympy [B]  time = 4.35318, size = 330, normalized size = 3.63 \begin{align*} - \frac{2 \sqrt{2} \cdot 3^{\frac{3}{4}} \log{\left (36 \tanh ^{2}{\left (\frac{x}{2} \right )} - 12 \sqrt{2} \cdot 3^{\frac{3}{4}} \tanh{\left (\frac{x}{2} \right )} + 12 \sqrt{3} \right )}}{18 + 18 \sqrt{3}} - \frac{3 \sqrt{2} \sqrt [4]{3} \log{\left (36 \tanh ^{2}{\left (\frac{x}{2} \right )} - 12 \sqrt{2} \cdot 3^{\frac{3}{4}} \tanh{\left (\frac{x}{2} \right )} + 12 \sqrt{3} \right )}}{18 + 18 \sqrt{3}} + \frac{3 \sqrt{2} \sqrt [4]{3} \log{\left (36 \tanh ^{2}{\left (\frac{x}{2} \right )} + 12 \sqrt{2} \cdot 3^{\frac{3}{4}} \tanh{\left (\frac{x}{2} \right )} + 12 \sqrt{3} \right )}}{18 + 18 \sqrt{3}} + \frac{2 \sqrt{2} \cdot 3^{\frac{3}{4}} \log{\left (36 \tanh ^{2}{\left (\frac{x}{2} \right )} + 12 \sqrt{2} \cdot 3^{\frac{3}{4}} \tanh{\left (\frac{x}{2} \right )} + 12 \sqrt{3} \right )}}{18 + 18 \sqrt{3}} + \frac{6 \tanh{\left (\frac{x}{2} \right )}}{18 + 18 \sqrt{3}} + \frac{6 \sqrt{3} \tanh{\left (\frac{x}{2} \right )}}{18 + 18 \sqrt{3}} - \frac{2 \sqrt{2} \cdot 3^{\frac{3}{4}} \operatorname{atan}{\left (\sqrt{2} \sqrt [4]{3} \tanh{\left (\frac{x}{2} \right )} - 1 \right )}}{18 + 18 \sqrt{3}} - \frac{2 \sqrt{2} \cdot 3^{\frac{3}{4}} \operatorname{atan}{\left (\sqrt{2} \sqrt [4]{3} \tanh{\left (\frac{x}{2} \right )} + 1 \right )}}{18 + 18 \sqrt{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+cosh(x)**3),x)

[Out]

-2*sqrt(2)*3**(3/4)*log(36*tanh(x/2)**2 - 12*sqrt(2)*3**(3/4)*tanh(x/2) + 12*sqrt(3))/(18 + 18*sqrt(3)) - 3*sq
rt(2)*3**(1/4)*log(36*tanh(x/2)**2 - 12*sqrt(2)*3**(3/4)*tanh(x/2) + 12*sqrt(3))/(18 + 18*sqrt(3)) + 3*sqrt(2)
*3**(1/4)*log(36*tanh(x/2)**2 + 12*sqrt(2)*3**(3/4)*tanh(x/2) + 12*sqrt(3))/(18 + 18*sqrt(3)) + 2*sqrt(2)*3**(
3/4)*log(36*tanh(x/2)**2 + 12*sqrt(2)*3**(3/4)*tanh(x/2) + 12*sqrt(3))/(18 + 18*sqrt(3)) + 6*tanh(x/2)/(18 + 1
8*sqrt(3)) + 6*sqrt(3)*tanh(x/2)/(18 + 18*sqrt(3)) - 2*sqrt(2)*3**(3/4)*atan(sqrt(2)*3**(1/4)*tanh(x/2) - 1)/(
18 + 18*sqrt(3)) - 2*sqrt(2)*3**(3/4)*atan(sqrt(2)*3**(1/4)*tanh(x/2) + 1)/(18 + 18*sqrt(3))

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+cosh(x)^3),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError