3.65 \(\int \frac {1}{a+b \cosh ^6(x)} \, dx\)

Optimal. Leaf size=171 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt [6]{a} \tanh (x)}{\sqrt {\sqrt [3]{a}+\sqrt [3]{b}}}\right )}{3 a^{5/6} \sqrt {\sqrt [3]{a}+\sqrt [3]{b}}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [6]{a} \tanh (x)}{\sqrt {\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b}}}\right )}{3 a^{5/6} \sqrt {\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b}}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [6]{a} \tanh (x)}{\sqrt {\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b}}}\right )}{3 a^{5/6} \sqrt {\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b}}} \]

[Out]

1/3*arctanh(a^(1/6)*tanh(x)/(a^(1/3)+b^(1/3))^(1/2))/a^(5/6)/(a^(1/3)+b^(1/3))^(1/2)+1/3*arctanh(a^(1/6)*tanh(
x)/(a^(1/3)-(-1)^(1/3)*b^(1/3))^(1/2))/a^(5/6)/(a^(1/3)-(-1)^(1/3)*b^(1/3))^(1/2)+1/3*arctanh(a^(1/6)*tanh(x)/
(a^(1/3)+(-1)^(2/3)*b^(1/3))^(1/2))/a^(5/6)/(a^(1/3)+(-1)^(2/3)*b^(1/3))^(1/2)

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Rubi [A]  time = 0.23, antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3211, 3181, 206} \[ \frac {\tanh ^{-1}\left (\frac {\sqrt [6]{a} \tanh (x)}{\sqrt {\sqrt [3]{a}+\sqrt [3]{b}}}\right )}{3 a^{5/6} \sqrt {\sqrt [3]{a}+\sqrt [3]{b}}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [6]{a} \tanh (x)}{\sqrt {\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b}}}\right )}{3 a^{5/6} \sqrt {\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b}}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [6]{a} \tanh (x)}{\sqrt {\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b}}}\right )}{3 a^{5/6} \sqrt {\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b}}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Cosh[x]^6)^(-1),x]

[Out]

ArcTanh[(a^(1/6)*Tanh[x])/Sqrt[a^(1/3) + b^(1/3)]]/(3*a^(5/6)*Sqrt[a^(1/3) + b^(1/3)]) + ArcTanh[(a^(1/6)*Tanh
[x])/Sqrt[a^(1/3) - (-1)^(1/3)*b^(1/3)]]/(3*a^(5/6)*Sqrt[a^(1/3) - (-1)^(1/3)*b^(1/3)]) + ArcTanh[(a^(1/6)*Tan
h[x])/Sqrt[a^(1/3) + (-1)^(2/3)*b^(1/3)]]/(3*a^(5/6)*Sqrt[a^(1/3) + (-1)^(2/3)*b^(1/3)])

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 ^6(x)} \, dx &=\frac {\int \frac {1}{1+\frac {\sqrt [3]{b} \cosh ^2(x)}{\sqrt [3]{a}}} \, dx}{3 a}+\frac {\int \frac {1}{1-\frac {\sqrt [3]{-1} \sqrt [3]{b} \cosh ^2(x)}{\sqrt [3]{a}}} \, dx}{3 a}+\frac {\int \frac {1}{1+\frac {(-1)^{2/3} \sqrt [3]{b} \cosh ^2(x)}{\sqrt [3]{a}}} \, dx}{3 a}\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{1-\left (1+\frac {\sqrt [3]{b}}{\sqrt [3]{a}}\right ) x^2} \, dx,x,\coth (x)\right )}{3 a}+\frac {\operatorname {Subst}\left (\int \frac {1}{1-\left (1-\frac {\sqrt [3]{-1} \sqrt [3]{b}}{\sqrt [3]{a}}\right ) x^2} \, dx,x,\coth (x)\right )}{3 a}+\frac {\operatorname {Subst}\left (\int \frac {1}{1-\left (1+\frac {(-1)^{2/3} \sqrt [3]{b}}{\sqrt [3]{a}}\right ) x^2} \, dx,x,\coth (x)\right )}{3 a}\\ &=\frac {\tanh ^{-1}\left (\frac {\sqrt [6]{a} \tanh (x)}{\sqrt {\sqrt [3]{a}+\sqrt [3]{b}}}\right )}{3 a^{5/6} \sqrt {\sqrt [3]{a}+\sqrt [3]{b}}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [6]{a} \tanh (x)}{\sqrt {\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b}}}\right )}{3 a^{5/6} \sqrt {\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b}}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [6]{a} \tanh (x)}{\sqrt {\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b}}}\right )}{3 a^{5/6} \sqrt {\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b}}}\\ \end {align*}

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Mathematica [C]  time = 0.22, size = 132, normalized size = 0.77 \[ \frac {16}{3} \text {RootSum}\left [\text {$\#$1}^6 b+6 \text {$\#$1}^5 b+15 \text {$\#$1}^4 b+64 \text {$\#$1}^3 a+20 \text {$\#$1}^3 b+15 \text {$\#$1}^2 b+6 \text {$\#$1} b+b\& ,\frac {\text {$\#$1}^2 x+\text {$\#$1}^2 \log (-\text {$\#$1} \sinh (x)+\text {$\#$1} \cosh (x)-\sinh (x)-\cosh (x))}{\text {$\#$1}^5 b+5 \text {$\#$1}^4 b+10 \text {$\#$1}^3 b+32 \text {$\#$1}^2 a+10 \text {$\#$1}^2 b+5 \text {$\#$1} b+b}\& \right ] \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Cosh[x]^6)^(-1),x]

[Out]

(16*RootSum[b + 6*b*#1 + 15*b*#1^2 + 64*a*#1^3 + 20*b*#1^3 + 15*b*#1^4 + 6*b*#1^5 + b*#1^6 & , (x*#1^2 + Log[-
Cosh[x] - Sinh[x] + Cosh[x]*#1 - Sinh[x]*#1]*#1^2)/(b + 5*b*#1 + 32*a*#1^2 + 10*b*#1^2 + 10*b*#1^3 + 5*b*#1^4
+ b*#1^5) & ])/3

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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)^6),x, algorithm="fricas")

[Out]

Timed out

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giac [A]  time = 0.32, size = 1, normalized size = 0.01 \[ 0 \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*cosh(x)^6),x, algorithm="giac")

[Out]

0

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maple [C]  time = 0.10, size = 177, normalized size = 1.04 \[ \frac {\left (\munderset {\textit {\_R} =\RootOf \left (\left (a +b \right ) \textit {\_Z}^{12}+\left (-6 a +6 b \right ) \textit {\_Z}^{10}+\left (15 a +15 b \right ) \textit {\_Z}^{8}+\left (-20 a +20 b \right ) \textit {\_Z}^{6}+\left (15 a +15 b \right ) \textit {\_Z}^{4}+\left (-6 a +6 b \right ) \textit {\_Z}^{2}+a +b \right )}{\sum }\frac {\left (-\textit {\_R}^{10}+5 \textit {\_R}^{8}-10 \textit {\_R}^{6}+10 \textit {\_R}^{4}-5 \textit {\_R}^{2}+1\right ) \ln \left (\tanh \left (\frac {x}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{11} a +\textit {\_R}^{11} b -5 \textit {\_R}^{9} a +5 \textit {\_R}^{9} b +10 \textit {\_R}^{7} a +10 \textit {\_R}^{7} b -10 \textit {\_R}^{5} a +10 \textit {\_R}^{5} b +5 \textit {\_R}^{3} a +5 \textit {\_R}^{3} b -\textit {\_R} a +\textit {\_R} b}\right )}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b*cosh(x)^6),x)

[Out]

1/6*sum((-_R^10+5*_R^8-10*_R^6+10*_R^4-5*_R^2+1)/(_R^11*a+_R^11*b-5*_R^9*a+5*_R^9*b+10*_R^7*a+10*_R^7*b-10*_R^
5*a+10*_R^5*b+5*_R^3*a+5*_R^3*b-_R*a+_R*b)*ln(tanh(1/2*x)-_R),_R=RootOf((a+b)*_Z^12+(-6*a+6*b)*_Z^10+(15*a+15*
b)*_Z^8+(-20*a+20*b)*_Z^6+(15*a+15*b)*_Z^4+(-6*a+6*b)*_Z^2+a+b))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{b \cosh \relax (x)^{6} + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*cosh(x)^6),x, algorithm="maxima")

[Out]

integrate(1/(b*cosh(x)^6 + a), x)

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mupad [B]  time = 58.39, size = 844, normalized size = 4.94 \[ \sum _{k=1}^6\ln \left (\mathrm {root}\left (46656\,a^5\,b\,d^6+46656\,a^6\,d^6-3888\,a^4\,d^4+108\,a^2\,d^2-1,d,k\right )\,\left (\mathrm {root}\left (46656\,a^5\,b\,d^6+46656\,a^6\,d^6-3888\,a^4\,d^4+108\,a^2\,d^2-1,d,k\right )\,\left (\mathrm {root}\left (46656\,a^5\,b\,d^6+46656\,a^6\,d^6-3888\,a^4\,d^4+108\,a^2\,d^2-1,d,k\right )\,\left (\mathrm {root}\left (46656\,a^5\,b\,d^6+46656\,a^6\,d^6-3888\,a^4\,d^4+108\,a^2\,d^2-1,d,k\right )\,\left (\frac {1459166279268040704\,\left (327680\,a^7\,{\mathrm {e}}^{2\,x}+298496\,a^6\,b+65536\,a^7+158\,a^2\,b^5+91315\,a^3\,b^4+348176\,a^4\,b^3+489952\,a^5\,b^2+196\,a^2\,b^5\,{\mathrm {e}}^{2\,x}+274019\,a^3\,b^4\,{\mathrm {e}}^{2\,x}+1132876\,a^4\,b^3\,{\mathrm {e}}^{2\,x}+1770440\,a^5\,b^2\,{\mathrm {e}}^{2\,x}+1239040\,a^6\,b\,{\mathrm {e}}^{2\,x}\right )}{b^{10}\,{\left (a+b\right )}^3}+\frac {\mathrm {root}\left (46656\,a^5\,b\,d^6+46656\,a^6\,d^6-3888\,a^4\,d^4+108\,a^2\,d^2-1,d,k\right )\,\left (262144\,a^7\,{\mathrm {e}}^{2\,x}+203520\,a^6\,b+65536\,a^7+453\,a^3\,b^4+72022\,a^4\,b^3+209472\,a^5\,b^2+630\,a^3\,b^4\,{\mathrm {e}}^{2\,x}+254512\,a^4\,b^3\,{\mathrm {e}}^{2\,x}+767508\,a^5\,b^2\,{\mathrm {e}}^{2\,x}+775680\,a^6\,b\,{\mathrm {e}}^{2\,x}\right )\,17509995351216488448}{b^{10}\,{\left (a+b\right )}^2}\right )-\frac {486388759756013568\,\left (655360\,a^5\,{\mathrm {e}}^{2\,x}-9\,a\,b^4+370176\,a^4\,b+196608\,a^5-24408\,a^2\,b^3+149088\,a^3\,b^2-63676\,a^2\,b^3\,{\mathrm {e}}^{2\,x}+526248\,a^3\,b^2\,{\mathrm {e}}^{2\,x}-10\,a\,b^4\,{\mathrm {e}}^{2\,x}+1245184\,a^4\,b\,{\mathrm {e}}^{2\,x}\right )}{b^{10}\,{\left (a+b\right )}^2}\right )-\frac {40532396646334464\,\left (655360\,a^5\,{\mathrm {e}}^{2\,x}-b^5\,{\mathrm {e}}^{2\,x}-24677\,a\,b^4+773120\,a^4\,b+262144\,a^5-b^5+198071\,a^2\,b^3+733696\,a^3\,b^2+477713\,a^2\,b^3\,{\mathrm {e}}^{2\,x}+1770640\,a^3\,b^2\,{\mathrm {e}}^{2\,x}-53861\,a\,b^4\,{\mathrm {e}}^{2\,x}+1894400\,a^4\,b\,{\mathrm {e}}^{2\,x}\right )}{b^{10}\,{\left (a+b\right )}^3}\right )+\frac {13510798882111488\,\left (655360\,a^3\,{\mathrm {e}}^{2\,x}+11382\,b^3\,{\mathrm {e}}^{2\,x}+144416\,a\,b^2+269056\,a^2\,b+131072\,a^3+6459\,b^3+677524\,a\,b^2\,{\mathrm {e}}^{2\,x}+1321472\,a^2\,b\,{\mathrm {e}}^{2\,x}\right )}{b^{10}\,{\left (a+b\right )}^2}\right )+\frac {1125899906842624\,\left (851968\,a^4\,{\mathrm {e}}^{2\,x}+6006\,b^4\,{\mathrm {e}}^{2\,x}+211497\,a\,b^3+597504\,a^3\,b+196608\,a^4+3840\,b^4+608544\,a^2\,b^2+2562504\,a^2\,b^2\,{\mathrm {e}}^{2\,x}+864565\,a\,b^3\,{\mathrm {e}}^{2\,x}+2555904\,a^3\,b\,{\mathrm {e}}^{2\,x}\right )}{b^{10}\,{\left (a+b\right )}^2\,\left (a^2+b\,a\right )}\right )\,\mathrm {root}\left (46656\,a^5\,b\,d^6+46656\,a^6\,d^6-3888\,a^4\,d^4+108\,a^2\,d^2-1,d,k\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + b*cosh(x)^6),x)

[Out]

symsum(log(root(46656*a^5*b*d^6 + 46656*a^6*d^6 - 3888*a^4*d^4 + 108*a^2*d^2 - 1, d, k)*(root(46656*a^5*b*d^6
+ 46656*a^6*d^6 - 3888*a^4*d^4 + 108*a^2*d^2 - 1, d, k)*(root(46656*a^5*b*d^6 + 46656*a^6*d^6 - 3888*a^4*d^4 +
108*a^2*d^2 - 1, d, k)*(root(46656*a^5*b*d^6 + 46656*a^6*d^6 - 3888*a^4*d^4 + 108*a^2*d^2 - 1, d, k)*((145916
6279268040704*(327680*a^7*exp(2*x) + 298496*a^6*b + 65536*a^7 + 158*a^2*b^5 + 91315*a^3*b^4 + 348176*a^4*b^3 +
489952*a^5*b^2 + 196*a^2*b^5*exp(2*x) + 274019*a^3*b^4*exp(2*x) + 1132876*a^4*b^3*exp(2*x) + 1770440*a^5*b^2*
exp(2*x) + 1239040*a^6*b*exp(2*x)))/(b^10*(a + b)^3) + (17509995351216488448*root(46656*a^5*b*d^6 + 46656*a^6*
d^6 - 3888*a^4*d^4 + 108*a^2*d^2 - 1, d, k)*(262144*a^7*exp(2*x) + 203520*a^6*b + 65536*a^7 + 453*a^3*b^4 + 72
022*a^4*b^3 + 209472*a^5*b^2 + 630*a^3*b^4*exp(2*x) + 254512*a^4*b^3*exp(2*x) + 767508*a^5*b^2*exp(2*x) + 7756
80*a^6*b*exp(2*x)))/(b^10*(a + b)^2)) - (486388759756013568*(655360*a^5*exp(2*x) - 9*a*b^4 + 370176*a^4*b + 19
6608*a^5 - 24408*a^2*b^3 + 149088*a^3*b^2 - 63676*a^2*b^3*exp(2*x) + 526248*a^3*b^2*exp(2*x) - 10*a*b^4*exp(2*
x) + 1245184*a^4*b*exp(2*x)))/(b^10*(a + b)^2)) - (40532396646334464*(655360*a^5*exp(2*x) - b^5*exp(2*x) - 246
77*a*b^4 + 773120*a^4*b + 262144*a^5 - b^5 + 198071*a^2*b^3 + 733696*a^3*b^2 + 477713*a^2*b^3*exp(2*x) + 17706
40*a^3*b^2*exp(2*x) - 53861*a*b^4*exp(2*x) + 1894400*a^4*b*exp(2*x)))/(b^10*(a + b)^3)) + (13510798882111488*(
655360*a^3*exp(2*x) + 11382*b^3*exp(2*x) + 144416*a*b^2 + 269056*a^2*b + 131072*a^3 + 6459*b^3 + 677524*a*b^2*
exp(2*x) + 1321472*a^2*b*exp(2*x)))/(b^10*(a + b)^2)) + (1125899906842624*(851968*a^4*exp(2*x) + 6006*b^4*exp(
2*x) + 211497*a*b^3 + 597504*a^3*b + 196608*a^4 + 3840*b^4 + 608544*a^2*b^2 + 2562504*a^2*b^2*exp(2*x) + 86456
5*a*b^3*exp(2*x) + 2555904*a^3*b*exp(2*x)))/(b^10*(a + b)^2*(a*b + a^2)))*root(46656*a^5*b*d^6 + 46656*a^6*d^6
- 3888*a^4*d^4 + 108*a^2*d^2 - 1, d, k), k, 1, 6)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{a + b \cosh ^{6}{\relax (x )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*cosh(x)**6),x)

[Out]

Integral(1/(a + b*cosh(x)**6), x)

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