3.3.68 \(\int \frac {1}{a+b \sinh ^6(x)} \, dx\) [268]

Optimal. Leaf size=175 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {\sqrt [3]{a}-\sqrt [3]{b}} \tanh (x)}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt {\sqrt [3]{a}-\sqrt [3]{b}}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b}} \tanh (x)}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt {\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b}}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b}} \tanh (x)}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt {\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b}}} \]

[Out]

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

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Rubi [A]
time = 0.20, antiderivative size = 175, 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 = {3290, 3260, 212} \begin {gather*} \frac {\tanh ^{-1}\left (\frac {\sqrt {\sqrt [3]{a}-\sqrt [3]{b}} \tanh (x)}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt {\sqrt [3]{a}-\sqrt [3]{b}}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b}} \tanh (x)}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt {\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b}}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b}} \tanh (x)}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt {\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*Sinh[x]^6)^(-1),x]

[Out]

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

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 \sinh ^6(x)} \, dx &=\frac {\int \frac {1}{1+\frac {\sqrt [3]{b} \sinh ^2(x)}{\sqrt [3]{a}}} \, dx}{3 a}+\frac {\int \frac {1}{1-\frac {\sqrt [3]{-1} \sqrt [3]{b} \sinh ^2(x)}{\sqrt [3]{a}}} \, dx}{3 a}+\frac {\int \frac {1}{1+\frac {(-1)^{2/3} \sqrt [3]{b} \sinh ^2(x)}{\sqrt [3]{a}}} \, dx}{3 a}\\ &=\frac {\text {Subst}\left (\int \frac {1}{1-\left (1-\frac {\sqrt [3]{b}}{\sqrt [3]{a}}\right ) x^2} \, dx,x,\tanh (x)\right )}{3 a}+\frac {\text {Subst}\left (\int \frac {1}{1-\left (1+\frac {\sqrt [3]{-1} \sqrt [3]{b}}{\sqrt [3]{a}}\right ) x^2} \, dx,x,\tanh (x)\right )}{3 a}+\frac {\text {Subst}\left (\int \frac {1}{1-\left (1-\frac {(-1)^{2/3} \sqrt [3]{b}}{\sqrt [3]{a}}\right ) x^2} \, dx,x,\tanh (x)\right )}{3 a}\\ &=\frac {\tanh ^{-1}\left (\frac {\sqrt {\sqrt [3]{a}-\sqrt [3]{b}} \tanh (x)}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt {\sqrt [3]{a}-\sqrt [3]{b}}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b}} \tanh (x)}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt {\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b}}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b}} \tanh (x)}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt {\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{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 = 0.13, size = 134, normalized size = 0.77 \begin {gather*} \frac {16}{3} \text {RootSum}\left [b-6 b \text {$\#$1}+15 b \text {$\#$1}^2+64 a \text {$\#$1}^3-20 b \text {$\#$1}^3+15 b \text {$\#$1}^4-6 b \text {$\#$1}^5+b \text {$\#$1}^6\&,\frac {x \text {$\#$1}^2+\log (-\cosh (x)-\sinh (x)+\cosh (x) \text {$\#$1}-\sinh (x) \text {$\#$1}) \text {$\#$1}^2}{-b+5 b \text {$\#$1}+32 a \text {$\#$1}^2-10 b \text {$\#$1}^2+10 b \text {$\#$1}^3-5 b \text {$\#$1}^4+b \text {$\#$1}^5}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sinh[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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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.86, size = 128, normalized size = 0.73

method	result	size
default	\(\frac {\left (\munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{12}-6 a \,\textit {\_Z}^{10}+15 a \,\textit {\_Z}^{8}+\left (-20 a +64 b \right ) \textit {\_Z}^{6}+15 a \,\textit {\_Z}^{4}-6 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (-\textit {\_R}^{10}+5 \textit {\_R}^{8}-10 \textit {\_R}^{6}+10 \textit {\_R}^{4}-5 \textit {\_R}^{2}+1\right ) \ln \left (\tanh \left (\frac {x}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{11} a -5 \textit {\_R}^{9} a +10 \textit {\_R}^{7} a -10 \textit {\_R}^{5} a +32 \textit {\_R}^{5} b +5 \textit {\_R}^{3} a -\textit {\_R} a}\right )}{6}\)	\(128\)
risch	\(\munderset {\textit {\_R} =\RootOf \left (-1+\left (46656 a^{6}-46656 a^{5} b \right ) \textit {\_Z}^{6}-3888 a^{4} \textit {\_Z}^{4}+108 \textit {\_Z}^{2} a^{2}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 x}+\left (-\frac {15552 a^{6}}{b}+15552 a^{5}\right ) \textit {\_R}^{5}+\left (\frac {2592 a^{5}}{b}-2592 a^{4}\right ) \textit {\_R}^{4}+\left (\frac {864 a^{4}}{b}+432 a^{3}\right ) \textit {\_R}^{3}+\left (-\frac {144 a^{3}}{b}-72 a^{2}\right ) \textit {\_R}^{2}+\left (-\frac {12 a^{2}}{b}+12 a \right ) \textit {\_R} +\frac {2 a}{b}-1\right )\)	\(140\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b*sinh(x)^6),x,method=_RETURNVERBOSE)

[Out]

1/6*sum((-_R^10+5*_R^8-10*_R^6+10*_R^4-5*_R^2+1)/(_R^11*a-5*_R^9*a+10*_R^7*a-10*_R^5*a+32*_R^5*b+5*_R^3*a-_R*a
)*ln(tanh(1/2*x)-_R),_R=RootOf(a*_Z^12-6*a*_Z^10+15*a*_Z^8+(-20*a+64*b)*_Z^6+15*a*_Z^4-6*a*_Z^2+a))

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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*sinh(x)^6),x, algorithm="maxima")

[Out]

integrate(1/(b*sinh(x)^6 + a), x)

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Fricas [C] Result contains complex when optimal does not.
time = 1.42, size = 16401, normalized size = 93.72 \begin {gather*} \text {too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*sqrt(1/2)*sqrt(2*(1/2)^(2/3)*(-I*sqrt(3) + 1)*(1/(a^4 - a^3*b) - 1/(a^2 - a*b)^2)/(1/(a^6 - a^5*b) - 3/((
a^4 - a^3*b)*(a^2 - a*b)) + 2/(a^2 - a*b)^3 + b/((a - b)^2*a^5))^(1/3) - (1/2)^(1/3)*(I*sqrt(3) + 1)*(1/(a^6 -
a^5*b) - 3/((a^4 - a^3*b)*(a^2 - a*b)) + 2/(a^2 - a*b)^3 + b/((a - b)^2*a^5))^(1/3) + 2/(a^2 - a*b))*log(1/2*
(a^5 - a^4*b)*(2*(1/2)^(2/3)*(-I*sqrt(3) + 1)*(1/(a^4 - a^3*b) - 1/(a^2 - a*b)^2)/(1/(a^6 - a^5*b) - 3/((a^4 -
a^3*b)*(a^2 - a*b)) + 2/(a^2 - a*b)^3 + b/((a - b)^2*a^5))^(1/3) - (1/2)^(1/3)*(I*sqrt(3) + 1)*(1/(a^6 - a^5*
b) - 3/((a^4 - a^3*b)*(a^2 - a*b)) + 2/(a^2 - a*b)^3 + b/((a - b)^2*a^5))^(1/3) + 2/(a^2 - a*b))^2 + b*cosh(x)
^2 + 2*b*cosh(x)*sinh(x) + b*sinh(x)^2 + 1/2*sqrt(1/2)*((a^6 - a^5*b)*(2*(1/2)^(2/3)*(-I*sqrt(3) + 1)*(1/(a^4
- a^3*b) - 1/(a^2 - a*b)^2)/(1/(a^6 - a^5*b) - 3/((a^4 - a^3*b)*(a^2 - a*b)) + 2/(a^2 - a*b)^3 + b/((a - b)^2*
a^5))^(1/3) - (1/2)^(1/3)*(I*sqrt(3) + 1)*(1/(a^6 - a^5*b) - 3/((a^4 - a^3*b)*(a^2 - a*b)) + 2/(a^2 - a*b)^3 +
b/((a - b ...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{a + b \sinh ^{6}{\left (x \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(a + b*sinh(x)**6), x)

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Giac [A]
time = 0.46, size = 1, normalized size = 0.01 \begin {gather*} 0 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

0

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Mupad [B]
time = 58.56, size = 857, normalized size = 4.90 \begin {gather*} \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\,b-a^2\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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + b*sinh(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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