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*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)])

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Rubi [A]  time = 0.903562, antiderivative size = 494, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, 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 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]))

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 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]

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*}

Mathematica [C]  time = 0.2876, size = 139, normalized size = 0.28 \[ \frac{8}{5} \text{RootSum}\left [32 \text{$\#$1}^5 a+\text{$\#$1}^{10} b+5 \text{$\#$1}^8 b+10 \text{$\#$1}^6 b+10 \text{$\#$1}^4 b+5 \text{$\#$1}^2 b+b\& ,\frac{\text{$\#$1}^3 x+2 \text{$\#$1}^3 \log \left (-\text{$\#$1} \sinh \left (\frac{x}{2}\right )+\text{$\#$1} \cosh \left (\frac{x}{2}\right )-\sinh \left (\frac{x}{2}\right )-\cosh \left (\frac{x}{2}\right )\right )}{16 \text{$\#$1}^3 a+\text{$\#$1}^8 b+4 \text{$\#$1}^6 b+6 \text{$\#$1}^4 b+4 \text{$\#$1}^2 b+b}\& \right ] \]

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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Maple [C]  time = 0.023, size = 156, normalized size = 0.3 \begin{align*}{\frac{1}{5}\sum _{{\it \_R}={\it RootOf} \left ( \left ( a-b \right ){{\it \_Z}}^{10}+ \left ( -5\,a-5\,b \right ){{\it \_Z}}^{8}+ \left ( 10\,a-10\,b \right ){{\it \_Z}}^{6}+ \left ( -10\,a-10\,b \right ){{\it \_Z}}^{4}+ \left ( 5\,a-5\,b \right ){{\it \_Z}}^{2}-a-b \right ) }{\frac{-{{\it \_R}}^{8}+4\,{{\it \_R}}^{6}-6\,{{\it \_R}}^{4}+4\,{{\it \_R}}^{2}-1}{{{\it \_R}}^{9}a-{{\it \_R}}^{9}b-4\,{{\it \_R}}^{7}a-4\,{{\it \_R}}^{7}b+6\,{{\it \_R}}^{5}a-6\,{{\it \_R}}^{5}b-4\,{{\it \_R}}^{3}a-4\,{{\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)^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., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{b \cosh \left (x\right )^{5} + a}\,{d x} \end{align*}

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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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: UnboundLocalError

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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)**5),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{b \cosh \left (x\right )^{5} + a}\,{d x} \end{align*}

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)