3.66 \(\int \frac{1}{a+b \cosh ^8(x)} \, dx\)

Optimal. Leaf size=245 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt [8]{-a} \tanh (x)}{\sqrt{\sqrt [4]{-a}-\sqrt [4]{b}}}\right )}{4 (-a)^{7/8} \sqrt{\sqrt [4]{-a}-\sqrt [4]{b}}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [8]{-a} \tanh (x)}{\sqrt{\sqrt [4]{-a}-i \sqrt [4]{b}}}\right )}{4 (-a)^{7/8} \sqrt{\sqrt [4]{-a}-i \sqrt [4]{b}}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [8]{-a} \tanh (x)}{\sqrt{\sqrt [4]{-a}+i \sqrt [4]{b}}}\right )}{4 (-a)^{7/8} \sqrt{\sqrt [4]{-a}+i \sqrt [4]{b}}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [8]{-a} \tanh (x)}{\sqrt{\sqrt [4]{-a}+\sqrt [4]{b}}}\right )}{4 (-a)^{7/8} \sqrt{\sqrt [4]{-a}+\sqrt [4]{b}}} \]

[Out]

-ArcTanh[((-a)^(1/8)*Tanh[x])/Sqrt[(-a)^(1/4) - b^(1/4)]]/(4*(-a)^(7/8)*Sqrt[(-a)^(1/4) - b^(1/4)]) - ArcTanh[
((-a)^(1/8)*Tanh[x])/Sqrt[(-a)^(1/4) - I*b^(1/4)]]/(4*(-a)^(7/8)*Sqrt[(-a)^(1/4) - I*b^(1/4)]) - ArcTanh[((-a)
^(1/8)*Tanh[x])/Sqrt[(-a)^(1/4) + I*b^(1/4)]]/(4*(-a)^(7/8)*Sqrt[(-a)^(1/4) + I*b^(1/4)]) - ArcTanh[((-a)^(1/8
)*Tanh[x])/Sqrt[(-a)^(1/4) + b^(1/4)]]/(4*(-a)^(7/8)*Sqrt[(-a)^(1/4) + b^(1/4)])

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Rubi [A]  time = 0.486965, antiderivative size = 245, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {3211, 3181, 206} \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt [8]{-a} \tanh (x)}{\sqrt{\sqrt [4]{-a}-i \sqrt [4]{b}}}\right )}{4 (-a)^{7/8} \sqrt{\sqrt [4]{-a}-i \sqrt [4]{b}}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [8]{-a} \tanh (x)}{\sqrt{\sqrt [4]{-a}+i \sqrt [4]{b}}}\right )}{4 (-a)^{7/8} \sqrt{\sqrt [4]{-a}+i \sqrt [4]{b}}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [8]{-a} \tanh (x)}{\sqrt{\sqrt [4]{-a}+\sqrt [4]{b}}}\right )}{4 (-a)^{7/8} \sqrt{\sqrt [4]{-a}+\sqrt [4]{b}}}-\frac{\tanh ^{-1}\left (\frac{(-a)^{5/8} \tanh (x)}{\sqrt{a \sqrt [4]{b}+(-a)^{5/4}}}\right )}{4 (-a)^{3/8} \sqrt{a \sqrt [4]{b}+(-a)^{5/4}}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Cosh[x]^8)^(-1),x]

[Out]

-ArcTanh[((-a)^(1/8)*Tanh[x])/Sqrt[(-a)^(1/4) - I*b^(1/4)]]/(4*(-a)^(7/8)*Sqrt[(-a)^(1/4) - I*b^(1/4)]) - ArcT
anh[((-a)^(1/8)*Tanh[x])/Sqrt[(-a)^(1/4) + I*b^(1/4)]]/(4*(-a)^(7/8)*Sqrt[(-a)^(1/4) + I*b^(1/4)]) - ArcTanh[(
(-a)^(1/8)*Tanh[x])/Sqrt[(-a)^(1/4) + b^(1/4)]]/(4*(-a)^(7/8)*Sqrt[(-a)^(1/4) + b^(1/4)]) - ArcTanh[((-a)^(5/8
)*Tanh[x])/Sqrt[(-a)^(5/4) + a*b^(1/4)]]/(4*(-a)^(3/8)*Sqrt[(-a)^(5/4) + a*b^(1/4)])

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 ^8(x)} \, dx &=\frac{\int \frac{1}{1-\frac{\sqrt [4]{b} \cosh ^2(x)}{\sqrt [4]{-a}}} \, dx}{4 a}+\frac{\int \frac{1}{1-\frac{i \sqrt [4]{b} \cosh ^2(x)}{\sqrt [4]{-a}}} \, dx}{4 a}+\frac{\int \frac{1}{1+\frac{i \sqrt [4]{b} \cosh ^2(x)}{\sqrt [4]{-a}}} \, dx}{4 a}+\frac{\int \frac{1}{1+\frac{\sqrt [4]{b} \cosh ^2(x)}{\sqrt [4]{-a}}} \, dx}{4 a}\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{1-\left (1-\frac{\sqrt [4]{b}}{\sqrt [4]{-a}}\right ) x^2} \, dx,x,\coth (x)\right )}{4 a}+\frac{\operatorname{Subst}\left (\int \frac{1}{1-\left (1-\frac{i \sqrt [4]{b}}{\sqrt [4]{-a}}\right ) x^2} \, dx,x,\coth (x)\right )}{4 a}+\frac{\operatorname{Subst}\left (\int \frac{1}{1-\left (1+\frac{i \sqrt [4]{b}}{\sqrt [4]{-a}}\right ) x^2} \, dx,x,\coth (x)\right )}{4 a}+\frac{\operatorname{Subst}\left (\int \frac{1}{1-\left (1+\frac{\sqrt [4]{b}}{\sqrt [4]{-a}}\right ) x^2} \, dx,x,\coth (x)\right )}{4 a}\\ &=-\frac{\tanh ^{-1}\left (\frac{\sqrt [8]{-a} \tanh (x)}{\sqrt{\sqrt [4]{-a}-i \sqrt [4]{b}}}\right )}{4 (-a)^{7/8} \sqrt{\sqrt [4]{-a}-i \sqrt [4]{b}}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [8]{-a} \tanh (x)}{\sqrt{\sqrt [4]{-a}+i \sqrt [4]{b}}}\right )}{4 (-a)^{7/8} \sqrt{\sqrt [4]{-a}+i \sqrt [4]{b}}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [8]{-a} \tanh (x)}{\sqrt{\sqrt [4]{-a}+\sqrt [4]{b}}}\right )}{4 (-a)^{7/8} \sqrt{\sqrt [4]{-a}+\sqrt [4]{b}}}-\frac{\tanh ^{-1}\left (\frac{(-a)^{5/8} \tanh (x)}{\sqrt{(-a)^{5/4}+a \sqrt [4]{b}}}\right )}{4 (-a)^{3/8} \sqrt{(-a)^{5/4}+a \sqrt [4]{b}}}\\ \end{align*}

Mathematica [C]  time = 0.260313, size = 158, normalized size = 0.64 \[ 16 \text{RootSum}\left [256 \text{$\#$1}^4 a+\text{$\#$1}^8 b+8 \text{$\#$1}^7 b+28 \text{$\#$1}^6 b+56 \text{$\#$1}^5 b+70 \text{$\#$1}^4 b+56 \text{$\#$1}^3 b+28 \text{$\#$1}^2 b+8 \text{$\#$1} b+b\& ,\frac{\text{$\#$1}^3 x+\text{$\#$1}^3 \log (-\text{$\#$1} \sinh (x)+\text{$\#$1} \cosh (x)-\sinh (x)-\cosh (x))}{128 \text{$\#$1}^3 a+\text{$\#$1}^7 b+7 \text{$\#$1}^6 b+21 \text{$\#$1}^5 b+35 \text{$\#$1}^4 b+35 \text{$\#$1}^3 b+21 \text{$\#$1}^2 b+7 \text{$\#$1} b+b}\& \right ] \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Cosh[x]^8)^(-1),x]

[Out]

16*RootSum[b + 8*b*#1 + 28*b*#1^2 + 56*b*#1^3 + 256*a*#1^4 + 70*b*#1^4 + 56*b*#1^5 + 28*b*#1^6 + 8*b*#1^7 + b*
#1^8 & , (x*#1^3 + Log[-Cosh[x] - Sinh[x] + Cosh[x]*#1 - Sinh[x]*#1]*#1^3)/(b + 7*b*#1 + 21*b*#1^2 + 128*a*#1^
3 + 35*b*#1^3 + 35*b*#1^4 + 21*b*#1^5 + 7*b*#1^6 + b*#1^7) & ]

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Maple [C]  time = 0.03, size = 233, normalized size = 1. \begin{align*}{\frac{1}{8}\sum _{{\it \_R}={\it RootOf} \left ( \left ( a+b \right ){{\it \_Z}}^{16}+ \left ( -8\,a+8\,b \right ){{\it \_Z}}^{14}+ \left ( 28\,a+28\,b \right ){{\it \_Z}}^{12}+ \left ( -56\,a+56\,b \right ){{\it \_Z}}^{10}+ \left ( 70\,a+70\,b \right ){{\it \_Z}}^{8}+ \left ( -56\,a+56\,b \right ){{\it \_Z}}^{6}+ \left ( 28\,a+28\,b \right ){{\it \_Z}}^{4}+ \left ( -8\,a+8\,b \right ){{\it \_Z}}^{2}+a+b \right ) }{\frac{-{{\it \_R}}^{14}+7\,{{\it \_R}}^{12}-21\,{{\it \_R}}^{10}+35\,{{\it \_R}}^{8}-35\,{{\it \_R}}^{6}+21\,{{\it \_R}}^{4}-7\,{{\it \_R}}^{2}+1}{{{\it \_R}}^{15}a+{{\it \_R}}^{15}b-7\,{{\it \_R}}^{13}a+7\,{{\it \_R}}^{13}b+21\,{{\it \_R}}^{11}a+21\,{{\it \_R}}^{11}b-35\,{{\it \_R}}^{9}a+35\,{{\it \_R}}^{9}b+35\,{{\it \_R}}^{7}a+35\,{{\it \_R}}^{7}b-21\,{{\it \_R}}^{5}a+21\,{{\it \_R}}^{5}b+7\,{{\it \_R}}^{3}a+7\,{{\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)^8),x)

[Out]

1/8*sum((-_R^14+7*_R^12-21*_R^10+35*_R^8-35*_R^6+21*_R^4-7*_R^2+1)/(_R^15*a+_R^15*b-7*_R^13*a+7*_R^13*b+21*_R^
11*a+21*_R^11*b-35*_R^9*a+35*_R^9*b+35*_R^7*a+35*_R^7*b-21*_R^5*a+21*_R^5*b+7*_R^3*a+7*_R^3*b-_R*a+_R*b)*ln(ta
nh(1/2*x)-_R),_R=RootOf((a+b)*_Z^16+(-8*a+8*b)*_Z^14+(28*a+28*b)*_Z^12+(-56*a+56*b)*_Z^10+(70*a+70*b)*_Z^8+(-5
6*a+56*b)*_Z^6+(28*a+28*b)*_Z^4+(-8*a+8*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 )^{8} + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*cosh(x)^8),x, algorithm="maxima")

[Out]

integrate(1/(b*cosh(x)^8 + 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)^8),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)**8),x)

[Out]

Timed out

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Giac [A]  time = 2.1572, size = 1, normalized size = 0. \begin{align*} 0 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*cosh(x)^8),x, algorithm="giac")

[Out]

0