3.64 \(\int \frac {1}{a+b \cosh ^5(x)} \, dx\)

Optimal. Leaf size=494 \[ \frac {2 \tanh ^{-1}\left (\frac {\sqrt {\sqrt [5]{a}-\sqrt [5]{b}} \tanh \left (\frac {x}{2}\right )}{\sqrt {\sqrt [5]{a}+\sqrt [5]{b}}}\right )}{5 a^{4/5} \sqrt {\sqrt [5]{a}-\sqrt [5]{b}} \sqrt {\sqrt [5]{a}+\sqrt [5]{b}}}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt {\sqrt [5]{a}+\sqrt [5]{-1} \sqrt [5]{b}} \tanh \left (\frac {x}{2}\right )}{\sqrt {\sqrt [5]{a}-\sqrt [5]{-1} \sqrt [5]{b}}}\right )}{5 a^{4/5} \sqrt {\sqrt [5]{a}-\sqrt [5]{-1} \sqrt [5]{b}} \sqrt {\sqrt [5]{a}+\sqrt [5]{-1} \sqrt [5]{b}}}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt {\sqrt [5]{a}-(-1)^{2/5} \sqrt [5]{b}} \tanh \left (\frac {x}{2}\right )}{\sqrt {\sqrt [5]{a}+(-1)^{2/5} \sqrt [5]{b}}}\right )}{5 a^{4/5} \sqrt {\sqrt [5]{a}-(-1)^{2/5} \sqrt [5]{b}} \sqrt {\sqrt [5]{a}+(-1)^{2/5} \sqrt [5]{b}}}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt {\sqrt [5]{a}+(-1)^{3/5} \sqrt [5]{b}} \tanh \left (\frac {x}{2}\right )}{\sqrt {\sqrt [5]{a}-(-1)^{3/5} \sqrt [5]{b}}}\right )}{5 a^{4/5} \sqrt {\sqrt [5]{a}-(-1)^{3/5} \sqrt [5]{b}} \sqrt {\sqrt [5]{a}+(-1)^{3/5} \sqrt [5]{b}}}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt {\sqrt [5]{a}-(-1)^{4/5} \sqrt [5]{b}} \tanh \left (\frac {x}{2}\right )}{\sqrt {\sqrt [5]{a}+(-1)^{4/5} \sqrt [5]{b}}}\right )}{5 a^{4/5} \sqrt {\sqrt [5]{a}-(-1)^{4/5} \sqrt [5]{b}} \sqrt {\sqrt [5]{a}+(-1)^{4/5} \sqrt [5]{b}}} \]

[Out]

2/5*arctanh((a^(1/5)-b^(1/5))^(1/2)*tanh(1/2*x)/(a^(1/5)+b^(1/5))^(1/2))/a^(4/5)/(a^(1/5)-b^(1/5))^(1/2)/(a^(1
/5)+b^(1/5))^(1/2)+2/5*arctanh((a^(1/5)+(-1)^(1/5)*b^(1/5))^(1/2)*tanh(1/2*x)/(a^(1/5)-(-1)^(1/5)*b^(1/5))^(1/
2))/a^(4/5)/(a^(1/5)-(-1)^(1/5)*b^(1/5))^(1/2)/(a^(1/5)+(-1)^(1/5)*b^(1/5))^(1/2)+2/5*arctanh((a^(1/5)-(-1)^(2
/5)*b^(1/5))^(1/2)*tanh(1/2*x)/(a^(1/5)+(-1)^(2/5)*b^(1/5))^(1/2))/a^(4/5)/(a^(1/5)-(-1)^(2/5)*b^(1/5))^(1/2)/
(a^(1/5)+(-1)^(2/5)*b^(1/5))^(1/2)+2/5*arctanh((a^(1/5)+(-1)^(3/5)*b^(1/5))^(1/2)*tanh(1/2*x)/(a^(1/5)-(-1)^(3
/5)*b^(1/5))^(1/2))/a^(4/5)/(a^(1/5)-(-1)^(3/5)*b^(1/5))^(1/2)/(a^(1/5)+(-1)^(3/5)*b^(1/5))^(1/2)+2/5*arctanh(
(a^(1/5)-(-1)^(4/5)*b^(1/5))^(1/2)*tanh(1/2*x)/(a^(1/5)+(-1)^(4/5)*b^(1/5))^(1/2))/a^(4/5)/(a^(1/5)-(-1)^(4/5)
*b^(1/5))^(1/2)/(a^(1/5)+(-1)^(4/5)*b^(1/5))^(1/2)

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Rubi [A]  time = 0.90, antiderivative size = 494, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3213, 2659, 208} \[ \frac {2 \tanh ^{-1}\left (\frac {\sqrt {\sqrt [5]{a}-\sqrt [5]{b}} \tanh \left (\frac {x}{2}\right )}{\sqrt {\sqrt [5]{a}+\sqrt [5]{b}}}\right )}{5 a^{4/5} \sqrt {\sqrt [5]{a}-\sqrt [5]{b}} \sqrt {\sqrt [5]{a}+\sqrt [5]{b}}}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt {\sqrt [5]{a}+\sqrt [5]{-1} \sqrt [5]{b}} \tanh \left (\frac {x}{2}\right )}{\sqrt {\sqrt [5]{a}-\sqrt [5]{-1} \sqrt [5]{b}}}\right )}{5 a^{4/5} \sqrt {\sqrt [5]{a}-\sqrt [5]{-1} \sqrt [5]{b}} \sqrt {\sqrt [5]{a}+\sqrt [5]{-1} \sqrt [5]{b}}}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt {\sqrt [5]{a}-(-1)^{2/5} \sqrt [5]{b}} \tanh \left (\frac {x}{2}\right )}{\sqrt {\sqrt [5]{a}+(-1)^{2/5} \sqrt [5]{b}}}\right )}{5 a^{4/5} \sqrt {\sqrt [5]{a}-(-1)^{2/5} \sqrt [5]{b}} \sqrt {\sqrt [5]{a}+(-1)^{2/5} \sqrt [5]{b}}}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt {\sqrt [5]{a}+(-1)^{3/5} \sqrt [5]{b}} \tanh \left (\frac {x}{2}\right )}{\sqrt {\sqrt [5]{a}-(-1)^{3/5} \sqrt [5]{b}}}\right )}{5 a^{4/5} \sqrt {\sqrt [5]{a}-(-1)^{3/5} \sqrt [5]{b}} \sqrt {\sqrt [5]{a}+(-1)^{3/5} \sqrt [5]{b}}}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt {\sqrt [5]{a}-(-1)^{4/5} \sqrt [5]{b}} \tanh \left (\frac {x}{2}\right )}{\sqrt {\sqrt [5]{a}+(-1)^{4/5} \sqrt [5]{b}}}\right )}{5 a^{4/5} \sqrt {\sqrt [5]{a}-(-1)^{4/5} \sqrt [5]{b}} \sqrt {\sqrt [5]{a}+(-1)^{4/5} \sqrt [5]{b}}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Cosh[x]^5)^(-1),x]

[Out]

(2*ArcTanh[(Sqrt[a^(1/5) - b^(1/5)]*Tanh[x/2])/Sqrt[a^(1/5) + b^(1/5)]])/(5*a^(4/5)*Sqrt[a^(1/5) - b^(1/5)]*Sq
rt[a^(1/5) + b^(1/5)]) + (2*ArcTanh[(Sqrt[a^(1/5) + (-1)^(1/5)*b^(1/5)]*Tanh[x/2])/Sqrt[a^(1/5) - (-1)^(1/5)*b
^(1/5)]])/(5*a^(4/5)*Sqrt[a^(1/5) - (-1)^(1/5)*b^(1/5)]*Sqrt[a^(1/5) + (-1)^(1/5)*b^(1/5)]) + (2*ArcTanh[(Sqrt
[a^(1/5) - (-1)^(2/5)*b^(1/5)]*Tanh[x/2])/Sqrt[a^(1/5) + (-1)^(2/5)*b^(1/5)]])/(5*a^(4/5)*Sqrt[a^(1/5) - (-1)^
(2/5)*b^(1/5)]*Sqrt[a^(1/5) + (-1)^(2/5)*b^(1/5)]) + (2*ArcTanh[(Sqrt[a^(1/5) + (-1)^(3/5)*b^(1/5)]*Tanh[x/2])
/Sqrt[a^(1/5) - (-1)^(3/5)*b^(1/5)]])/(5*a^(4/5)*Sqrt[a^(1/5) - (-1)^(3/5)*b^(1/5)]*Sqrt[a^(1/5) + (-1)^(3/5)*
b^(1/5)]) + (2*ArcTanh[(Sqrt[a^(1/5) - (-1)^(4/5)*b^(1/5)]*Tanh[x/2])/Sqrt[a^(1/5) + (-1)^(4/5)*b^(1/5)]])/(5*
a^(4/5)*Sqrt[a^(1/5) - (-1)^(4/5)*b^(1/5)]*Sqrt[a^(1/5) + (-1)^(4/5)*b^(1/5)])

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]

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 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]))

Rubi steps

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

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Mathematica [C]  time = 0.29, size = 139, normalized size = 0.28 \[ \frac {8}{5} \text {RootSum}\left [\text {$\#$1}^{10} b+5 \text {$\#$1}^8 b+10 \text {$\#$1}^6 b+32 \text {$\#$1}^5 a+10 \text {$\#$1}^4 b+5 \text {$\#$1}^2 b+b\& ,\frac {\text {$\#$1}^3 x+2 \text {$\#$1}^3 \log \left (-\text {$\#$1} \sinh \left (\frac {x}{2}\right )+\text {$\#$1} \cosh \left (\frac {x}{2}\right )-\sinh \left (\frac {x}{2}\right )-\cosh \left (\frac {x}{2}\right )\right )}{\text {$\#$1}^8 b+4 \text {$\#$1}^6 b+6 \text {$\#$1}^4 b+16 \text {$\#$1}^3 a+4 \text {$\#$1}^2 b+b}\& \right ] \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Cosh[x]^5)^(-1),x]

[Out]

(8*RootSum[b + 5*b*#1^2 + 10*b*#1^4 + 32*a*#1^5 + 10*b*#1^6 + 5*b*#1^8 + b*#1^10 & , (x*#1^3 + 2*Log[-Cosh[x/2
] - Sinh[x/2] + Cosh[x/2]*#1 - Sinh[x/2]*#1]*#1^3)/(b + 4*b*#1^2 + 16*a*#1^3 + 6*b*#1^4 + 4*b*#1^6 + b*#1^8) &
])/5

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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*cosh(x)^5),x, algorithm="fricas")

[Out]

Exception raised: RuntimeError >> no explicit roots found

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{b \cosh \relax (x)^{5} + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*cosh(x)^5),x, algorithm="giac")

[Out]

integrate(1/(b*cosh(x)^5 + a), x)

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maple [C]  time = 0.11, size = 156, normalized size = 0.32 \[ \frac {\left (\munderset {\textit {\_R} =\RootOf \left (\left (a -b \right ) \textit {\_Z}^{10}+\left (-5 a -5 b \right ) \textit {\_Z}^{8}+\left (10 a -10 b \right ) \textit {\_Z}^{6}+\left (-10 a -10 b \right ) \textit {\_Z}^{4}+\left (5 a -5 b \right ) \textit {\_Z}^{2}-a -b \right )}{\sum }\frac {\left (-\textit {\_R}^{8}+4 \textit {\_R}^{6}-6 \textit {\_R}^{4}+4 \textit {\_R}^{2}-1\right ) \ln \left (\tanh \left (\frac {x}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{9} a -\textit {\_R}^{9} b -4 \textit {\_R}^{7} a -4 \textit {\_R}^{7} b +6 \textit {\_R}^{5} a -6 \textit {\_R}^{5} b -4 \textit {\_R}^{3} a -4 \textit {\_R}^{3} b +\textit {\_R} a -\textit {\_R} b}\right )}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b*cosh(x)^5),x)

[Out]

1/5*sum((-_R^8+4*_R^6-6*_R^4+4*_R^2-1)/(_R^9*a-_R^9*b-4*_R^7*a-4*_R^7*b+6*_R^5*a-6*_R^5*b-4*_R^3*a-4*_R^3*b+_R
*a-_R*b)*ln(tanh(1/2*x)-_R),_R=RootOf((a-b)*_Z^10+(-5*a-5*b)*_Z^8+(10*a-10*b)*_Z^6+(-10*a-10*b)*_Z^4+(5*a-5*b)
*_Z^2-a-b))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{b \cosh \relax (x)^{5} + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*cosh(x)^5),x, algorithm="maxima")

[Out]

integrate(1/(b*cosh(x)^5 + 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)^5),x)

[Out]

\text{Hanged}

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{a + b \cosh ^{5}{\relax (x )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*cosh(x)**5),x)

[Out]

Integral(1/(a + b*cosh(x)**5), x)

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