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

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]

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)])

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Rubi [A]  time = 0.279326, antiderivative size = 175, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {3211, 3181, 206} \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{\sqrt [3]{a}-\sqrt [3]{b}} \tanh (x)}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt{\sqrt [3]{a}-\sqrt [3]{b}}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b}} \tanh (x)}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt{\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b}}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b}} \tanh (x)}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt{\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b}}} \]

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 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]

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 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])

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{\operatorname{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{\operatorname{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{\operatorname{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*}

Mathematica [C]  time = 0.16913, size = 134, normalized size = 0.77 \[ \frac{16}{3} \text{RootSum}\left [64 \text{$\#$1}^3 a+\text{$\#$1}^6 b-6 \text{$\#$1}^5 b+15 \text{$\#$1}^4 b-20 \text{$\#$1}^3 b+15 \text{$\#$1}^2 b-6 \text{$\#$1} b+b\& ,\frac{\text{$\#$1}^2 x+\text{$\#$1}^2 \log (-\text{$\#$1} \sinh (x)+\text{$\#$1} \cosh (x)-\sinh (x)-\cosh (x))}{32 \text{$\#$1}^2 a+\text{$\#$1}^5 b-5 \text{$\#$1}^4 b+10 \text{$\#$1}^3 b-10 \text{$\#$1}^2 b+5 \text{$\#$1} b-b}\& \right ] \]

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]  time = 0.03, size = 128, normalized size = 0.7 \begin{align*}{\frac{1}{6}\sum _{{\it \_R}={\it RootOf} \left ( a{{\it \_Z}}^{12}-6\,a{{\it \_Z}}^{10}+15\,a{{\it \_Z}}^{8}+ \left ( -20\,a+64\,b \right ){{\it \_Z}}^{6}+15\,a{{\it \_Z}}^{4}-6\,a{{\it \_Z}}^{2}+a \right ) }{\frac{-{{\it \_R}}^{10}+5\,{{\it \_R}}^{8}-10\,{{\it \_R}}^{6}+10\,{{\it \_R}}^{4}-5\,{{\it \_R}}^{2}+1}{{{\it \_R}}^{11}a-5\,{{\it \_R}}^{9}a+10\,{{\it \_R}}^{7}a-10\,{{\it \_R}}^{5}a+32\,{{\it \_R}}^{5}b+5\,{{\it \_R}}^{3}a-{\it \_R}\,a}\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*sinh(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-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., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{b \sinh \left (x\right )^{6} + a}\,{d x} \end{align*}

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 [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*sinh(x)^6),x, algorithm="fricas")

[Out]

Timed out

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

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 = 1.37914, size = 1, normalized size = 0.01 \begin{align*} 0 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

0