3.56 \(\int \frac{1}{a+b \cosh ^3(x)} \, dx\)

Optimal. Leaf size=288 \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{\sqrt [3]{a}-\sqrt [3]{b}} \tanh \left (\frac{x}{2}\right )}{\sqrt{\sqrt [3]{a}+\sqrt [3]{b}}}\right )}{3 a^{2/3} \sqrt{\sqrt [3]{a}-\sqrt [3]{b}} \sqrt{\sqrt [3]{a}+\sqrt [3]{b}}}+\frac{2 \tanh ^{-1}\left (\frac{\sqrt{\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b}} \tanh \left (\frac{x}{2}\right )}{\sqrt{\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b}}}\right )}{3 a^{2/3} \sqrt{\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b}} \sqrt{\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b}}}+\frac{2 \tanh ^{-1}\left (\frac{\sqrt{\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b}} \tanh \left (\frac{x}{2}\right )}{\sqrt{\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b}}}\right )}{3 a^{2/3} \sqrt{\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b}} \sqrt{\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b}}} \]

[Out]

(2*ArcTanh[(Sqrt[a^(1/3) - b^(1/3)]*Tanh[x/2])/Sqrt[a^(1/3) + b^(1/3)]])/(3*a^(2/3)*Sqrt[a^(1/3) - b^(1/3)]*Sq
rt[a^(1/3) + b^(1/3)]) + (2*ArcTanh[(Sqrt[a^(1/3) + (-1)^(1/3)*b^(1/3)]*Tanh[x/2])/Sqrt[a^(1/3) - (-1)^(1/3)*b
^(1/3)]])/(3*a^(2/3)*Sqrt[a^(1/3) - (-1)^(1/3)*b^(1/3)]*Sqrt[a^(1/3) + (-1)^(1/3)*b^(1/3)]) + (2*ArcTanh[(Sqrt
[a^(1/3) - (-1)^(2/3)*b^(1/3)]*Tanh[x/2])/Sqrt[a^(1/3) + (-1)^(2/3)*b^(1/3)]])/(3*a^(2/3)*Sqrt[a^(1/3) - (-1)^
(2/3)*b^(1/3)]*Sqrt[a^(1/3) + (-1)^(2/3)*b^(1/3)])

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Rubi [A]  time = 0.478395, antiderivative size = 288, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {3213, 2659, 208} \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{\sqrt [3]{a}-\sqrt [3]{b}} \tanh \left (\frac{x}{2}\right )}{\sqrt{\sqrt [3]{a}+\sqrt [3]{b}}}\right )}{3 a^{2/3} \sqrt{\sqrt [3]{a}-\sqrt [3]{b}} \sqrt{\sqrt [3]{a}+\sqrt [3]{b}}}+\frac{2 \tanh ^{-1}\left (\frac{\sqrt{\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b}} \tanh \left (\frac{x}{2}\right )}{\sqrt{\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b}}}\right )}{3 a^{2/3} \sqrt{\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b}} \sqrt{\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b}}}+\frac{2 \tanh ^{-1}\left (\frac{\sqrt{\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b}} \tanh \left (\frac{x}{2}\right )}{\sqrt{\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b}}}\right )}{3 a^{2/3} \sqrt{\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b}} \sqrt{\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b}}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Cosh[x]^3)^(-1),x]

[Out]

(2*ArcTanh[(Sqrt[a^(1/3) - b^(1/3)]*Tanh[x/2])/Sqrt[a^(1/3) + b^(1/3)]])/(3*a^(2/3)*Sqrt[a^(1/3) - b^(1/3)]*Sq
rt[a^(1/3) + b^(1/3)]) + (2*ArcTanh[(Sqrt[a^(1/3) + (-1)^(1/3)*b^(1/3)]*Tanh[x/2])/Sqrt[a^(1/3) - (-1)^(1/3)*b
^(1/3)]])/(3*a^(2/3)*Sqrt[a^(1/3) - (-1)^(1/3)*b^(1/3)]*Sqrt[a^(1/3) + (-1)^(1/3)*b^(1/3)]) + (2*ArcTanh[(Sqrt
[a^(1/3) - (-1)^(2/3)*b^(1/3)]*Tanh[x/2])/Sqrt[a^(1/3) + (-1)^(2/3)*b^(1/3)]])/(3*a^(2/3)*Sqrt[a^(1/3) - (-1)^
(2/3)*b^(1/3)]*Sqrt[a^(1/3) + (-1)^(2/3)*b^(1/3)])

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

Mathematica [C]  time = 0.105394, size = 105, normalized size = 0.36 \[ \frac{2}{3} \text{RootSum}\left [8 \text{$\#$1}^3 a+\text{$\#$1}^6 b+3 \text{$\#$1}^4 b+3 \text{$\#$1}^2 b+b\& ,\frac{\text{$\#$1} x+2 \text{$\#$1} \log \left (-\text{$\#$1} \sinh \left (\frac{x}{2}\right )+\text{$\#$1} \cosh \left (\frac{x}{2}\right )-\sinh \left (\frac{x}{2}\right )-\cosh \left (\frac{x}{2}\right )\right )}{\text{$\#$1}^4 b+2 \text{$\#$1}^2 b+4 \text{$\#$1} a+b}\& \right ] \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Cosh[x]^3)^(-1),x]

[Out]

(2*RootSum[b + 3*b*#1^2 + 8*a*#1^3 + 3*b*#1^4 + b*#1^6 & , (x*#1 + 2*Log[-Cosh[x/2] - Sinh[x/2] + Cosh[x/2]*#1
- Sinh[x/2]*#1]*#1)/(b + 4*a*#1 + 2*b*#1^2 + b*#1^4) & ])/3

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Maple [C]  time = 0.292, size = 100, normalized size = 0.4 \begin{align*}{\frac{1}{3}\sum _{{\it \_R}={\it RootOf} \left ( \left ( a-b \right ){{\it \_Z}}^{6}+ \left ( -3\,a-3\,b \right ){{\it \_Z}}^{4}+ \left ( 3\,a-3\,b \right ){{\it \_Z}}^{2}-a-b \right ) }{\frac{-{{\it \_R}}^{4}+2\,{{\it \_R}}^{2}-1}{{{\it \_R}}^{5}a-{{\it \_R}}^{5}b-2\,{{\it \_R}}^{3}a-2\,{{\it \_R}}^{3}b+{\it \_R}\,a-{\it \_R}\,b}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -{\it \_R} \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b*cosh(x)^3),x)

[Out]

1/3*sum((-_R^4+2*_R^2-1)/(_R^5*a-_R^5*b-2*_R^3*a-2*_R^3*b+_R*a-_R*b)*ln(tanh(1/2*x)-_R),_R=RootOf((a-b)*_Z^6+(
-3*a-3*b)*_Z^4+(3*a-3*b)*_Z^2-a-b))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{b \cosh \left (x\right )^{3} + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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Sympy [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(1/(a+b*cosh(x)**3),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{b \cosh \left (x\right )^{3} + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(b*cosh(x)^3 + a), x)