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

Optimal. Leaf size=213 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt [8]{a} \tanh (x)}{\sqrt {\sqrt [4]{a}-\sqrt [4]{b}}}\right )}{4 a^{7/8} \sqrt {\sqrt [4]{a}-\sqrt [4]{b}}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [8]{a} \tanh (x)}{\sqrt {\sqrt [4]{a}-i \sqrt [4]{b}}}\right )}{4 a^{7/8} \sqrt {\sqrt [4]{a}-i \sqrt [4]{b}}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [8]{a} \tanh (x)}{\sqrt {\sqrt [4]{a}+i \sqrt [4]{b}}}\right )}{4 a^{7/8} \sqrt {\sqrt [4]{a}+i \sqrt [4]{b}}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [8]{a} \tanh (x)}{\sqrt {\sqrt [4]{a}+\sqrt [4]{b}}}\right )}{4 a^{7/8} \sqrt {\sqrt [4]{a}+\sqrt [4]{b}}} \]

[Out]

1/4*arctanh(a^(1/8)*tanh(x)/(a^(1/4)-b^(1/4))^(1/2))/a^(7/8)/(a^(1/4)-b^(1/4))^(1/2)+1/4*arctanh(a^(1/8)*tanh(
x)/(a^(1/4)-I*b^(1/4))^(1/2))/a^(7/8)/(a^(1/4)-I*b^(1/4))^(1/2)+1/4*arctanh(a^(1/8)*tanh(x)/(a^(1/4)+I*b^(1/4)
)^(1/2))/a^(7/8)/(a^(1/4)+I*b^(1/4))^(1/2)+1/4*arctanh(a^(1/8)*tanh(x)/(a^(1/4)+b^(1/4))^(1/2))/a^(7/8)/(a^(1/
4)+b^(1/4))^(1/2)

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Rubi [A]  time = 0.23, antiderivative size = 213, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 3, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {3211, 3181, 206} \[ \frac {\tanh ^{-1}\left (\frac {\sqrt [8]{a} \tanh (x)}{\sqrt {\sqrt [4]{a}-\sqrt [4]{b}}}\right )}{4 a^{7/8} \sqrt {\sqrt [4]{a}-\sqrt [4]{b}}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [8]{a} \tanh (x)}{\sqrt {\sqrt [4]{a}-i \sqrt [4]{b}}}\right )}{4 a^{7/8} \sqrt {\sqrt [4]{a}-i \sqrt [4]{b}}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [8]{a} \tanh (x)}{\sqrt {\sqrt [4]{a}+i \sqrt [4]{b}}}\right )}{4 a^{7/8} \sqrt {\sqrt [4]{a}+i \sqrt [4]{b}}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [8]{a} \tanh (x)}{\sqrt {\sqrt [4]{a}+\sqrt [4]{b}}}\right )}{4 a^{7/8} \sqrt {\sqrt [4]{a}+\sqrt [4]{b}}} \]

Antiderivative was successfully verified.

[In]

Int[(a - b*Cosh[x]^8)^(-1),x]

[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)])

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])

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 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]

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}-\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}}}\\ \end {align*}

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Mathematica [C]  time = 0.22, size = 158, normalized size = 0.74 \[ -16 \text {RootSum}\left [\text {$\#$1}^8 b+8 \text {$\#$1}^7 b+28 \text {$\#$1}^6 b+56 \text {$\#$1}^5 b-256 \text {$\#$1}^4 a+70 \text {$\#$1}^4 b+56 \text {$\#$1}^3 b+28 \text {$\#$1}^2 b+8 \text {$\#$1} b+b\& ,\frac {\text {$\#$1}^3 x+\text {$\#$1}^3 \log (-\text {$\#$1} \sinh (x)+\text {$\#$1} \cosh (x)-\sinh (x)-\cosh (x))}{\text {$\#$1}^7 b+7 \text {$\#$1}^6 b+21 \text {$\#$1}^5 b+35 \text {$\#$1}^4 b-128 \text {$\#$1}^3 a+35 \text {$\#$1}^3 b+21 \text {$\#$1}^2 b+7 \text {$\#$1} b+b}\& \right ] \]

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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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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giac [A]  time = 0.83, size = 1, normalized size = 0.00 \[ 0 \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

0

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maple [C]  time = 0.10, size = 239, normalized size = 1.12 \[ \frac {\left (\munderset {\textit {\_R} =\RootOf \left (\left (a -b \right ) \textit {\_Z}^{16}+\left (-8 a -8 b \right ) \textit {\_Z}^{14}+\left (28 a -28 b \right ) \textit {\_Z}^{12}+\left (-56 a -56 b \right ) \textit {\_Z}^{10}+\left (70 a -70 b \right ) \textit {\_Z}^{8}+\left (-56 a -56 b \right ) \textit {\_Z}^{6}+\left (28 a -28 b \right ) \textit {\_Z}^{4}+\left (-8 a -8 b \right ) \textit {\_Z}^{2}+a -b \right )}{\sum }\frac {\left (-\textit {\_R}^{14}+7 \textit {\_R}^{12}-21 \textit {\_R}^{10}+35 \textit {\_R}^{8}-35 \textit {\_R}^{6}+21 \textit {\_R}^{4}-7 \textit {\_R}^{2}+1\right ) \ln \left (\tanh \left (\frac {x}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{15} a -\textit {\_R}^{15} b -7 \textit {\_R}^{13} a -7 \textit {\_R}^{13} b +21 \textit {\_R}^{11} a -21 \textit {\_R}^{11} b -35 \textit {\_R}^{9} a -35 \textit {\_R}^{9} b +35 \textit {\_R}^{7} a -35 \textit {\_R}^{7} b -21 \textit {\_R}^{5} a -21 \textit {\_R}^{5} b +7 \textit {\_R}^{3} a -7 \textit {\_R}^{3} b -\textit {\_R} a -\textit {\_R} b}\right )}{8} \]

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 = 0.00, size = 0, normalized size = 0.00 \[ -\int \frac {1}{b \cosh \relax (x)^{8} - a}\,{d x} \]

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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mupad [F(-1)]  time = 0.00, size = -1, normalized size = -0.00 \[ \text {Hanged} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a - b*cosh(x)^8),x)

[Out]

\text{Hanged}

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{a - b \cosh ^{8}{\relax (x )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a-b*cosh(x)**8),x)

[Out]

Integral(1/(a - b*cosh(x)**8), x)

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