3.65 \(\int \frac{1}{a+b \cosh ^6(x)} \, dx\)

Optimal. Leaf size=171 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt [6]{a} \tanh (x)}{\sqrt{\sqrt [3]{a}+\sqrt [3]{b}}}\right )}{3 a^{5/6} \sqrt{\sqrt [3]{a}+\sqrt [3]{b}}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [6]{a} \tanh (x)}{\sqrt{\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b}}}\right )}{3 a^{5/6} \sqrt{\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b}}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [6]{a} \tanh (x)}{\sqrt{\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b}}}\right )}{3 a^{5/6} \sqrt{\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b}}} \]

[Out]

ArcTanh[(a^(1/6)*Tanh[x])/Sqrt[a^(1/3) + b^(1/3)]]/(3*a^(5/6)*Sqrt[a^(1/3) + b^(1/3)]) + ArcTanh[(a^(1/6)*Tanh
[x])/Sqrt[a^(1/3) - (-1)^(1/3)*b^(1/3)]]/(3*a^(5/6)*Sqrt[a^(1/3) - (-1)^(1/3)*b^(1/3)]) + ArcTanh[(a^(1/6)*Tan
h[x])/Sqrt[a^(1/3) + (-1)^(2/3)*b^(1/3)]]/(3*a^(5/6)*Sqrt[a^(1/3) + (-1)^(2/3)*b^(1/3)])

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Rubi [A]  time = 0.225911, antiderivative size = 171, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {3211, 3181, 206} \[ \frac{\tanh ^{-1}\left (\frac{\sqrt [6]{a} \tanh (x)}{\sqrt{\sqrt [3]{a}+\sqrt [3]{b}}}\right )}{3 a^{5/6} \sqrt{\sqrt [3]{a}+\sqrt [3]{b}}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [6]{a} \tanh (x)}{\sqrt{\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b}}}\right )}{3 a^{5/6} \sqrt{\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b}}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [6]{a} \tanh (x)}{\sqrt{\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b}}}\right )}{3 a^{5/6} \sqrt{\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b}}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Cosh[x]^6)^(-1),x]

[Out]

ArcTanh[(a^(1/6)*Tanh[x])/Sqrt[a^(1/3) + b^(1/3)]]/(3*a^(5/6)*Sqrt[a^(1/3) + b^(1/3)]) + ArcTanh[(a^(1/6)*Tanh
[x])/Sqrt[a^(1/3) - (-1)^(1/3)*b^(1/3)]]/(3*a^(5/6)*Sqrt[a^(1/3) - (-1)^(1/3)*b^(1/3)]) + ArcTanh[(a^(1/6)*Tan
h[x])/Sqrt[a^(1/3) + (-1)^(2/3)*b^(1/3)]]/(3*a^(5/6)*Sqrt[a^(1/3) + (-1)^(2/3)*b^(1/3)])

Rule 3211

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^(n_))^(-1), x_Symbol] :> Module[{k}, Dist[2/(a*n), Sum[Int[1/(1 - Si
n[e + f*x]^2/((-1)^((4*k)/n)*Rt[-(a/b), n/2])), x], {k, 1, n/2}], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[n/
2]

Rule 3181

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(-1), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist
[ff/f, Subst[Int[1/(a + (a + b)*ff^2*x^2), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x]

Rule 206

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

Rubi steps

\begin{align*} \int \frac{1}{a+b \cosh ^6(x)} \, dx &=\frac{\int \frac{1}{1+\frac{\sqrt [3]{b} \cosh ^2(x)}{\sqrt [3]{a}}} \, dx}{3 a}+\frac{\int \frac{1}{1-\frac{\sqrt [3]{-1} \sqrt [3]{b} \cosh ^2(x)}{\sqrt [3]{a}}} \, dx}{3 a}+\frac{\int \frac{1}{1+\frac{(-1)^{2/3} \sqrt [3]{b} \cosh ^2(x)}{\sqrt [3]{a}}} \, dx}{3 a}\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{1-\left (1+\frac{\sqrt [3]{b}}{\sqrt [3]{a}}\right ) x^2} \, dx,x,\coth (x)\right )}{3 a}+\frac{\operatorname{Subst}\left (\int \frac{1}{1-\left (1-\frac{\sqrt [3]{-1} \sqrt [3]{b}}{\sqrt [3]{a}}\right ) x^2} \, dx,x,\coth (x)\right )}{3 a}+\frac{\operatorname{Subst}\left (\int \frac{1}{1-\left (1+\frac{(-1)^{2/3} \sqrt [3]{b}}{\sqrt [3]{a}}\right ) x^2} \, dx,x,\coth (x)\right )}{3 a}\\ &=\frac{\tanh ^{-1}\left (\frac{\sqrt [6]{a} \tanh (x)}{\sqrt{\sqrt [3]{a}+\sqrt [3]{b}}}\right )}{3 a^{5/6} \sqrt{\sqrt [3]{a}+\sqrt [3]{b}}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [6]{a} \tanh (x)}{\sqrt{\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b}}}\right )}{3 a^{5/6} \sqrt{\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b}}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [6]{a} \tanh (x)}{\sqrt{\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b}}}\right )}{3 a^{5/6} \sqrt{\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b}}}\\ \end{align*}

Mathematica [C]  time = 0.214289, size = 132, normalized size = 0.77 \[ \frac{16}{3} \text{RootSum}\left [64 \text{$\#$1}^3 a+\text{$\#$1}^6 b+6 \text{$\#$1}^5 b+15 \text{$\#$1}^4 b+20 \text{$\#$1}^3 b+15 \text{$\#$1}^2 b+6 \text{$\#$1} b+b\& ,\frac{\text{$\#$1}^2 x+\text{$\#$1}^2 \log (-\text{$\#$1} \sinh (x)+\text{$\#$1} \cosh (x)-\sinh (x)-\cosh (x))}{32 \text{$\#$1}^2 a+\text{$\#$1}^5 b+5 \text{$\#$1}^4 b+10 \text{$\#$1}^3 b+10 \text{$\#$1}^2 b+5 \text{$\#$1} b+b}\& \right ] \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Cosh[x]^6)^(-1),x]

[Out]

(16*RootSum[b + 6*b*#1 + 15*b*#1^2 + 64*a*#1^3 + 20*b*#1^3 + 15*b*#1^4 + 6*b*#1^5 + b*#1^6 & , (x*#1^2 + Log[-
Cosh[x] - Sinh[x] + Cosh[x]*#1 - Sinh[x]*#1]*#1^2)/(b + 5*b*#1 + 32*a*#1^2 + 10*b*#1^2 + 10*b*#1^3 + 5*b*#1^4
+ b*#1^5) & ])/3

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Maple [C]  time = 0.026, size = 177, normalized size = 1. \begin{align*}{\frac{1}{6}\sum _{{\it \_R}={\it RootOf} \left ( \left ( a+b \right ){{\it \_Z}}^{12}+ \left ( -6\,a+6\,b \right ){{\it \_Z}}^{10}+ \left ( 15\,a+15\,b \right ){{\it \_Z}}^{8}+ \left ( -20\,a+20\,b \right ){{\it \_Z}}^{6}+ \left ( 15\,a+15\,b \right ){{\it \_Z}}^{4}+ \left ( -6\,a+6\,b \right ){{\it \_Z}}^{2}+a+b \right ) }{\frac{-{{\it \_R}}^{10}+5\,{{\it \_R}}^{8}-10\,{{\it \_R}}^{6}+10\,{{\it \_R}}^{4}-5\,{{\it \_R}}^{2}+1}{{{\it \_R}}^{11}a+{{\it \_R}}^{11}b-5\,{{\it \_R}}^{9}a+5\,{{\it \_R}}^{9}b+10\,{{\it \_R}}^{7}a+10\,{{\it \_R}}^{7}b-10\,{{\it \_R}}^{5}a+10\,{{\it \_R}}^{5}b+5\,{{\it \_R}}^{3}a+5\,{{\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)^6),x)

[Out]

1/6*sum((-_R^10+5*_R^8-10*_R^6+10*_R^4-5*_R^2+1)/(_R^11*a+_R^11*b-5*_R^9*a+5*_R^9*b+10*_R^7*a+10*_R^7*b-10*_R^
5*a+10*_R^5*b+5*_R^3*a+5*_R^3*b-_R*a+_R*b)*ln(tanh(1/2*x)-_R),_R=RootOf((a+b)*_Z^12+(-6*a+6*b)*_Z^10+(15*a+15*
b)*_Z^8+(-20*a+20*b)*_Z^6+(15*a+15*b)*_Z^4+(-6*a+6*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 )^{6} + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*cosh(x)^6),x, algorithm="maxima")

[Out]

integrate(1/(b*cosh(x)^6 + 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)^6),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)**6),x)

[Out]

Timed out

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Giac [A]  time = 1.47506, size = 1, normalized size = 0.01 \begin{align*} 0 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*cosh(x)^6),x, algorithm="giac")

[Out]

0