3.1.66 \(\int \frac {1}{a+b \cosh ^8(x)} \, dx\) [66]

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]

-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 = 0.36, antiderivative size = 245, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3290, 3260, 212} \begin {gather*} -\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}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*Cosh[x]^8)^(-1),x]

[Out]

-1/4*ArcTanh[((-a)^(1/8)*Tanh[x])/Sqrt[(-a)^(1/4) - I*b^(1/4)]]/((-a)^(7/8)*Sqrt[(-a)^(1/4) - I*b^(1/4)]) - Ar
cTanh[((-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 212

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

Rule 3260

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 3290

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 {\text {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 {\text {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 {\text {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 {\text {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*}

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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 0.18, size = 158, normalized size = 0.64 \begin {gather*} 16 \text {RootSum}\left [b+8 b \text {$\#$1}+28 b \text {$\#$1}^2+56 b \text {$\#$1}^3+256 a \text {$\#$1}^4+70 b \text {$\#$1}^4+56 b \text {$\#$1}^5+28 b \text {$\#$1}^6+8 b \text {$\#$1}^7+b \text {$\#$1}^8\&,\frac {x \text {$\#$1}^3+\log (-\cosh (x)-\sinh (x)+\cosh (x) \text {$\#$1}-\sinh (x) \text {$\#$1}) \text {$\#$1}^3}{b+7 b \text {$\#$1}+21 b \text {$\#$1}^2+128 a \text {$\#$1}^3+35 b \text {$\#$1}^3+35 b \text {$\#$1}^4+21 b \text {$\#$1}^5+7 b \text {$\#$1}^6+b \text {$\#$1}^7}\&\right ] \end {gather*}

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] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 1.11, size = 233, normalized size = 0.95

method	result	size
risch	\(\munderset {\textit {\_R} =\RootOf \left (1+\left (16777216 a^{8}[phone] a^{7} b \right ) \textit {\_Z}^{8}-1048576 a^{6} \textit {\_Z}^{6}+24576 a^{4} \textit {\_Z}^{4}-256 \textit {\_Z}^{2} a^{2}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 x}+\left (-\frac {4194304 a^{8}}{b}-4194304 a^{7}\right ) \textit {\_R}^{7}+\left (\frac {524288 a^{7}}{b}+524288 a^{6}\right ) \textit {\_R}^{6}+\left (\frac {196608 a^{6}}{b}-65536 a^{5}\right ) \textit {\_R}^{5}+\left (-\frac {24576 a^{5}}{b}+8192 a^{4}\right ) \textit {\_R}^{4}+\left (-\frac {3072 a^{4}}{b}-1024 a^{3}\right ) \textit {\_R}^{3}+\left (\frac {384 a^{3}}{b}+128 a^{2}\right ) \textit {\_R}^{2}+\left (\frac {16 a^{2}}{b}-16 a \right ) \textit {\_R} -\frac {2 a}{b}+1\right )\)	\(184\)
default	\(\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\left (a +b \right ) \textit {\_Z}^{16}+\left (-8 a +8 b \right ) \textit {\_Z}^{14}+\left (28 a +28 b \right ) \textit {\_Z}^{12}+\left (-56 a +56 b \right ) \textit {\_Z}^{10}+\left (70 a +70 b \right ) \textit {\_Z}^{8}+\left (-56 a +56 b \right ) \textit {\_Z}^{6}+\left (28 a +28 b \right ) \textit {\_Z}^{4}+\left (-8 a +8 b \right ) \textit {\_Z}^{2}+a +b \right )}{\sum }\frac {\left (-\textit {\_R}^{14}+7 \textit {\_R}^{12}-21 \textit {\_R}^{10}+35 \textit {\_R}^{8}-35 \textit {\_R}^{6}+21 \textit {\_R}^{4}-7 \textit {\_R}^{2}+1\right ) \ln \left (\tanh \left (\frac {x}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{15} a +\textit {\_R}^{15} b -7 \textit {\_R}^{13} a +7 \textit {\_R}^{13} b +21 \textit {\_R}^{11} a +21 \textit {\_R}^{11} b -35 \textit {\_R}^{9} a +35 \textit {\_R}^{9} b +35 \textit {\_R}^{7} a +35 \textit {\_R}^{7} b -21 \textit {\_R}^{5} a +21 \textit {\_R}^{5} b +7 \textit {\_R}^{3} a +7 \textit {\_R}^{3} b -\textit {\_R} a +\textit {\_R} b}\right )}{8}\)	\(233\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b*cosh(x)^8),x,method=_RETURNVERBOSE)

[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 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 661324 vs. \(2 (165) = 330\).
time = 3.30, size = 661324, normalized size = 2699.28 \begin {gather*} \text {too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/192*sqrt(1/2)*sqrt((-I*sqrt(3) + 1)*(((a^3*sqrt(-(2*a*b*sqrt(-b/a) - a*b + b^2)/((a^6 + 2*a^5*b + a^4*b^2)*
sqrt(-b/a))) + a^2*b*sqrt(-(2*a*b*sqrt(-b/a) - a*b + b^2)/((a^6 + 2*a^5*b + a^4*b^2)*sqrt(-b/a))) + 3*a)*sqrt(
-b/a) - b)^2*a/((a^3 + a^2*b)^2*b) - 3*(2*a^2*b*sqrt(-(2*a*b*sqrt(-b/a) - a*b + b^2)/((a^6 + 2*a^5*b + a^4*b^2
)*sqrt(-b/a))) - (2*a^3*sqrt(-(2*a*b*sqrt(-b/a) - a*b + b^2)/((a^6 + 2*a^5*b + a^4*b^2)*sqrt(-b/a))) + 3*a)*sq
rt(-b/a) + b)/((a^5 + a^4*b)*sqrt(-b/a)))/(-1/1572864*(2*a^2*b*sqrt(-(2*a*b*sqrt(-b/a) - a*b + b^2)/((a^6 + 2*
a^5*b + a^4*b^2)*sqrt(-b/a))) - (2*a^3*sqrt(-(2*a*b*sqrt(-b/a) - a*b + b^2)/((a^6 + 2*a^5*b + a^4*b^2)*sqrt(-b
/a))) + 3*a)*sqrt(-b/a) + b)*((a^3*sqrt(-(2*a*b*sqrt(-b/a) - a*b + b^2)/((a^6 + 2*a^5*b + a^4*b^2)*sqrt(-b/a))
) + a^2*b*sqrt(-(2*a*b*sqrt(-b/a) - a*b + b^2)/((a^6 + 2*a^5*b + a^4*b^2)*sqrt(-b/a))) + 3*a)*sqrt(-b/a) - b)*
a/((a^5 + a^4*b)*(a^3 + a^2*b)*b) - 1/524288*(2*a^2*b*sqrt(-(2*a*b*sqrt(-b/a) - a*b + b^2)/((a^6 + 2*a^5*b + a
^4*b^2)*sq ...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{a + b \cosh ^{8}{\left (x \right )}}\, dx \end {gather*}

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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Giac [A]
time = 0.54, size = 1, normalized size = 0.00 \begin {gather*} 0 \end {gather*}

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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Mupad [F(-1)]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \text {Hanged} \end {gather*}

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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