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

Optimal. Leaf size=435 \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt [5]{b}-\sqrt [5]{a} \tanh \left (\frac{x}{2}\right )}{\sqrt{a^{2/5}+b^{2/5}}}\right )}{5 a^{4/5} \sqrt{a^{2/5}+b^{2/5}}}+\frac{2 (-1)^{9/10} \tanh ^{-1}\left (\frac{(-1)^{9/10} \left (\sqrt [5]{a} \tanh \left (\frac{x}{2}\right )+\sqrt [5]{-1} \sqrt [5]{b}\right )}{\sqrt{\sqrt [5]{-1} b^{2/5}-(-1)^{4/5} a^{2/5}}}\right )}{5 a^{4/5} \sqrt{\sqrt [5]{-1} b^{2/5}-(-1)^{4/5} a^{2/5}}}+\frac{2 \sqrt [5]{-1} \tanh ^{-1}\left (\frac{\sqrt [5]{-1} \sqrt [5]{a} \tanh \left (\frac{x}{2}\right )+\sqrt [5]{b}}{\sqrt{(-1)^{2/5} a^{2/5}+b^{2/5}}}\right )}{5 a^{4/5} \sqrt{(-1)^{2/5} a^{2/5}+b^{2/5}}}+\frac{2 (-1)^{9/10} \tanh ^{-1}\left (\frac{(-1)^{3/10} \left ((-1)^{3/5} \sqrt [5]{a} \tanh \left (\frac{x}{2}\right )+\sqrt [5]{b}\right )}{\sqrt{(-1)^{3/5} b^{2/5}-(-1)^{4/5} a^{2/5}}}\right )}{5 a^{4/5} \sqrt{(-1)^{3/5} b^{2/5}-(-1)^{4/5} a^{2/5}}}-\frac{2 (-1)^{9/10} \tanh ^{-1}\left (\frac{-(-1)^{9/10} \sqrt [5]{a} \tanh \left (\frac{x}{2}\right )+i \sqrt [5]{b}}{\sqrt{-(-1)^{4/5} a^{2/5}-b^{2/5}}}\right )}{5 a^{4/5} \sqrt{-(-1)^{4/5} a^{2/5}-b^{2/5}}} \]

[Out]

(-2*ArcTanh[(b^(1/5) - a^(1/5)*Tanh[x/2])/Sqrt[a^(2/5) + b^(2/5)]])/(5*a^(4/5)*Sqrt[a^(2/5) + b^(2/5)]) + (2*(
-1)^(9/10)*ArcTanh[((-1)^(9/10)*((-1)^(1/5)*b^(1/5) + a^(1/5)*Tanh[x/2]))/Sqrt[-((-1)^(4/5)*a^(2/5)) + (-1)^(1
/5)*b^(2/5)]])/(5*a^(4/5)*Sqrt[-((-1)^(4/5)*a^(2/5)) + (-1)^(1/5)*b^(2/5)]) + (2*(-1)^(1/5)*ArcTanh[(b^(1/5) +
(-1)^(1/5)*a^(1/5)*Tanh[x/2])/Sqrt[(-1)^(2/5)*a^(2/5) + b^(2/5)]])/(5*a^(4/5)*Sqrt[(-1)^(2/5)*a^(2/5) + b^(2/
5)]) + (2*(-1)^(9/10)*ArcTanh[((-1)^(3/10)*(b^(1/5) + (-1)^(3/5)*a^(1/5)*Tanh[x/2]))/Sqrt[-((-1)^(4/5)*a^(2/5)
) + (-1)^(3/5)*b^(2/5)]])/(5*a^(4/5)*Sqrt[-((-1)^(4/5)*a^(2/5)) + (-1)^(3/5)*b^(2/5)]) - (2*(-1)^(9/10)*ArcTan
h[(I*b^(1/5) - (-1)^(9/10)*a^(1/5)*Tanh[x/2])/Sqrt[-((-1)^(4/5)*a^(2/5)) - b^(2/5)]])/(5*a^(4/5)*Sqrt[-((-1)^(
4/5)*a^(2/5)) - b^(2/5)])

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Rubi [A]  time = 0.974696, antiderivative size = 435, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 5, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {3213, 2660, 618, 206, 204} \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt [5]{b}-\sqrt [5]{a} \tanh \left (\frac{x}{2}\right )}{\sqrt{a^{2/5}+b^{2/5}}}\right )}{5 a^{4/5} \sqrt{a^{2/5}+b^{2/5}}}+\frac{2 (-1)^{9/10} \tanh ^{-1}\left (\frac{(-1)^{9/10} \left (\sqrt [5]{a} \tanh \left (\frac{x}{2}\right )+\sqrt [5]{-1} \sqrt [5]{b}\right )}{\sqrt{\sqrt [5]{-1} b^{2/5}-(-1)^{4/5} a^{2/5}}}\right )}{5 a^{4/5} \sqrt{\sqrt [5]{-1} b^{2/5}-(-1)^{4/5} a^{2/5}}}+\frac{2 \sqrt [5]{-1} \tanh ^{-1}\left (\frac{\sqrt [5]{-1} \sqrt [5]{a} \tanh \left (\frac{x}{2}\right )+\sqrt [5]{b}}{\sqrt{(-1)^{2/5} a^{2/5}+b^{2/5}}}\right )}{5 a^{4/5} \sqrt{(-1)^{2/5} a^{2/5}+b^{2/5}}}+\frac{2 (-1)^{9/10} \tanh ^{-1}\left (\frac{(-1)^{3/10} \left ((-1)^{3/5} \sqrt [5]{a} \tanh \left (\frac{x}{2}\right )+\sqrt [5]{b}\right )}{\sqrt{(-1)^{3/5} b^{2/5}-(-1)^{4/5} a^{2/5}}}\right )}{5 a^{4/5} \sqrt{(-1)^{3/5} b^{2/5}-(-1)^{4/5} a^{2/5}}}-\frac{2 (-1)^{9/10} \tanh ^{-1}\left (\frac{-(-1)^{9/10} \sqrt [5]{a} \tanh \left (\frac{x}{2}\right )+i \sqrt [5]{b}}{\sqrt{-(-1)^{4/5} a^{2/5}-b^{2/5}}}\right )}{5 a^{4/5} \sqrt{-(-1)^{4/5} a^{2/5}-b^{2/5}}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sinh[x]^5)^(-1),x]

[Out]

(-2*ArcTanh[(b^(1/5) - a^(1/5)*Tanh[x/2])/Sqrt[a^(2/5) + b^(2/5)]])/(5*a^(4/5)*Sqrt[a^(2/5) + b^(2/5)]) + (2*(
-1)^(9/10)*ArcTanh[((-1)^(9/10)*((-1)^(1/5)*b^(1/5) + a^(1/5)*Tanh[x/2]))/Sqrt[-((-1)^(4/5)*a^(2/5)) + (-1)^(1
/5)*b^(2/5)]])/(5*a^(4/5)*Sqrt[-((-1)^(4/5)*a^(2/5)) + (-1)^(1/5)*b^(2/5)]) + (2*(-1)^(1/5)*ArcTanh[(b^(1/5) +
(-1)^(1/5)*a^(1/5)*Tanh[x/2])/Sqrt[(-1)^(2/5)*a^(2/5) + b^(2/5)]])/(5*a^(4/5)*Sqrt[(-1)^(2/5)*a^(2/5) + b^(2/
5)]) + (2*(-1)^(9/10)*ArcTanh[((-1)^(3/10)*(b^(1/5) + (-1)^(3/5)*a^(1/5)*Tanh[x/2]))/Sqrt[-((-1)^(4/5)*a^(2/5)
) + (-1)^(3/5)*b^(2/5)]])/(5*a^(4/5)*Sqrt[-((-1)^(4/5)*a^(2/5)) + (-1)^(3/5)*b^(2/5)]) - (2*(-1)^(9/10)*ArcTan
h[(I*b^(1/5) - (-1)^(9/10)*a^(1/5)*Tanh[x/2])/Sqrt[-((-1)^(4/5)*a^(2/5)) - b^(2/5)]])/(5*a^(4/5)*Sqrt[-((-1)^(
4/5)*a^(2/5)) - b^(2/5)])

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 2660

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x]}, Dis
t[(2*e)/d, Subst[Int[1/(a + 2*b*e*x + a*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 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

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 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{1}{a+b \sinh ^5(x)} \, dx &=\int \left (-\frac{(-1)^{9/10}}{5 a^{4/5} \left (-(-1)^{9/10} \sqrt [5]{a}-i \sqrt [5]{b} \sinh (x)\right )}-\frac{(-1)^{9/10}}{5 a^{4/5} \left (-(-1)^{9/10} \sqrt [5]{a}-\sqrt [10]{-1} \sqrt [5]{b} \sinh (x)\right )}-\frac{(-1)^{9/10}}{5 a^{4/5} \left (-(-1)^{9/10} \sqrt [5]{a}+(-1)^{3/10} \sqrt [5]{b} \sinh (x)\right )}-\frac{(-1)^{9/10}}{5 a^{4/5} \left (-(-1)^{9/10} \sqrt [5]{a}+(-1)^{7/10} \sqrt [5]{b} \sinh (x)\right )}-\frac{(-1)^{9/10}}{5 a^{4/5} \left (-(-1)^{9/10} \sqrt [5]{a}-(-1)^{9/10} \sqrt [5]{b} \sinh (x)\right )}\right ) \, dx\\ &=-\frac{(-1)^{9/10} \int \frac{1}{-(-1)^{9/10} \sqrt [5]{a}-i \sqrt [5]{b} \sinh (x)} \, dx}{5 a^{4/5}}-\frac{(-1)^{9/10} \int \frac{1}{-(-1)^{9/10} \sqrt [5]{a}-\sqrt [10]{-1} \sqrt [5]{b} \sinh (x)} \, dx}{5 a^{4/5}}-\frac{(-1)^{9/10} \int \frac{1}{-(-1)^{9/10} \sqrt [5]{a}+(-1)^{3/10} \sqrt [5]{b} \sinh (x)} \, dx}{5 a^{4/5}}-\frac{(-1)^{9/10} \int \frac{1}{-(-1)^{9/10} \sqrt [5]{a}+(-1)^{7/10} \sqrt [5]{b} \sinh (x)} \, dx}{5 a^{4/5}}-\frac{(-1)^{9/10} \int \frac{1}{-(-1)^{9/10} \sqrt [5]{a}-(-1)^{9/10} \sqrt [5]{b} \sinh (x)} \, dx}{5 a^{4/5}}\\ &=-\frac{\left (2 (-1)^{9/10}\right ) \operatorname{Subst}\left (\int \frac{1}{-(-1)^{9/10} \sqrt [5]{a}-2 i \sqrt [5]{b} x+(-1)^{9/10} \sqrt [5]{a} x^2} \, dx,x,\tanh \left (\frac{x}{2}\right )\right )}{5 a^{4/5}}-\frac{\left (2 (-1)^{9/10}\right ) \operatorname{Subst}\left (\int \frac{1}{-(-1)^{9/10} \sqrt [5]{a}-2 \sqrt [10]{-1} \sqrt [5]{b} x+(-1)^{9/10} \sqrt [5]{a} x^2} \, dx,x,\tanh \left (\frac{x}{2}\right )\right )}{5 a^{4/5}}-\frac{\left (2 (-1)^{9/10}\right ) \operatorname{Subst}\left (\int \frac{1}{-(-1)^{9/10} \sqrt [5]{a}+2 (-1)^{3/10} \sqrt [5]{b} x+(-1)^{9/10} \sqrt [5]{a} x^2} \, dx,x,\tanh \left (\frac{x}{2}\right )\right )}{5 a^{4/5}}-\frac{\left (2 (-1)^{9/10}\right ) \operatorname{Subst}\left (\int \frac{1}{-(-1)^{9/10} \sqrt [5]{a}+2 (-1)^{7/10} \sqrt [5]{b} x+(-1)^{9/10} \sqrt [5]{a} x^2} \, dx,x,\tanh \left (\frac{x}{2}\right )\right )}{5 a^{4/5}}-\frac{\left (2 (-1)^{9/10}\right ) \operatorname{Subst}\left (\int \frac{1}{-(-1)^{9/10} \sqrt [5]{a}-2 (-1)^{9/10} \sqrt [5]{b} x+(-1)^{9/10} \sqrt [5]{a} x^2} \, dx,x,\tanh \left (\frac{x}{2}\right )\right )}{5 a^{4/5}}\\ &=\frac{\left (4 (-1)^{9/10}\right ) \operatorname{Subst}\left (\int \frac{1}{-4 (-1)^{4/5} \left (a^{2/5}+b^{2/5}\right )-x^2} \, dx,x,-2 (-1)^{9/10} \sqrt [5]{b}+2 (-1)^{9/10} \sqrt [5]{a} \tanh \left (\frac{x}{2}\right )\right )}{5 a^{4/5}}+\frac{\left (4 (-1)^{9/10}\right ) \operatorname{Subst}\left (\int \frac{1}{-4 (-1)^{2/5} \left ((-1)^{2/5} a^{2/5}+b^{2/5}\right )-x^2} \, dx,x,2 (-1)^{7/10} \sqrt [5]{b}+2 (-1)^{9/10} \sqrt [5]{a} \tanh \left (\frac{x}{2}\right )\right )}{5 a^{4/5}}+\frac{\left (4 (-1)^{9/10}\right ) \operatorname{Subst}\left (\int \frac{1}{-4 \left ((-1)^{4/5} a^{2/5}+b^{2/5}\right )-x^2} \, dx,x,-2 i \sqrt [5]{b}+2 (-1)^{9/10} \sqrt [5]{a} \tanh \left (\frac{x}{2}\right )\right )}{5 a^{4/5}}+\frac{\left (4 (-1)^{9/10}\right ) \operatorname{Subst}\left (\int \frac{1}{-4 \left ((-1)^{4/5} a^{2/5}-\sqrt [5]{-1} b^{2/5}\right )-x^2} \, dx,x,-2 \sqrt [10]{-1} \sqrt [5]{b}+2 (-1)^{9/10} \sqrt [5]{a} \tanh \left (\frac{x}{2}\right )\right )}{5 a^{4/5}}+\frac{\left (4 (-1)^{9/10}\right ) \operatorname{Subst}\left (\int \frac{1}{-4 \left ((-1)^{4/5} a^{2/5}-(-1)^{3/5} b^{2/5}\right )-x^2} \, dx,x,2 (-1)^{3/10} \sqrt [5]{b}+2 (-1)^{9/10} \sqrt [5]{a} \tanh \left (\frac{x}{2}\right )\right )}{5 a^{4/5}}\\ &=-\frac{2 (-1)^{7/10} \tan ^{-1}\left (\frac{i \sqrt [5]{b}+(-1)^{7/10} \sqrt [5]{a} \tanh \left (\frac{x}{2}\right )}{\sqrt{(-1)^{2/5} a^{2/5}+b^{2/5}}}\right )}{5 a^{4/5} \sqrt{(-1)^{2/5} a^{2/5}+b^{2/5}}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt [5]{b}-\sqrt [5]{a} \tanh \left (\frac{x}{2}\right )}{\sqrt{a^{2/5}+b^{2/5}}}\right )}{5 a^{4/5} \sqrt{a^{2/5}+b^{2/5}}}-\frac{2 (-1)^{9/10} \tanh ^{-1}\left (\frac{i \sqrt [5]{b}-(-1)^{9/10} \sqrt [5]{a} \tanh \left (\frac{x}{2}\right )}{\sqrt{-(-1)^{4/5} a^{2/5}-b^{2/5}}}\right )}{5 a^{4/5} \sqrt{-(-1)^{4/5} a^{2/5}-b^{2/5}}}-\frac{2 (-1)^{9/10} \tanh ^{-1}\left (\frac{\sqrt [10]{-1} \sqrt [5]{b}-(-1)^{9/10} \sqrt [5]{a} \tanh \left (\frac{x}{2}\right )}{\sqrt{-(-1)^{4/5} a^{2/5}+\sqrt [5]{-1} b^{2/5}}}\right )}{5 a^{4/5} \sqrt{-(-1)^{4/5} a^{2/5}+\sqrt [5]{-1} b^{2/5}}}+\frac{2 (-1)^{9/10} \tanh ^{-1}\left (\frac{(-1)^{3/10} \sqrt [5]{b}+(-1)^{9/10} \sqrt [5]{a} \tanh \left (\frac{x}{2}\right )}{\sqrt{-(-1)^{4/5} a^{2/5}+(-1)^{3/5} b^{2/5}}}\right )}{5 a^{4/5} \sqrt{-(-1)^{4/5} a^{2/5}+(-1)^{3/5} b^{2/5}}}\\ \end{align*}

Mathematica [C]  time = 0.325722, size = 141, normalized size = 0.32 \[ \frac{8}{5} \text{RootSum}\left [32 \text{$\#$1}^5 a+\text{$\#$1}^{10} b-5 \text{$\#$1}^8 b+10 \text{$\#$1}^6 b-10 \text{$\#$1}^4 b+5 \text{$\#$1}^2 b-b\& ,\frac{\text{$\#$1}^3 x+2 \text{$\#$1}^3 \log \left (-\text{$\#$1} \sinh \left (\frac{x}{2}\right )+\text{$\#$1} \cosh \left (\frac{x}{2}\right )-\sinh \left (\frac{x}{2}\right )-\cosh \left (\frac{x}{2}\right )\right )}{16 \text{$\#$1}^3 a+\text{$\#$1}^8 b-4 \text{$\#$1}^6 b+6 \text{$\#$1}^4 b-4 \text{$\#$1}^2 b+b}\& \right ] \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sinh[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 = 0.032, size = 113, normalized size = 0.3 \begin{align*}{\frac{1}{5}\sum _{{\it \_R}={\it RootOf} \left ( a{{\it \_Z}}^{10}-5\,a{{\it \_Z}}^{8}+10\,a{{\it \_Z}}^{6}-32\,b{{\it \_Z}}^{5}-10\,a{{\it \_Z}}^{4}+5\,a{{\it \_Z}}^{2}-a \right ) }{\frac{-{{\it \_R}}^{8}+4\,{{\it \_R}}^{6}-6\,{{\it \_R}}^{4}+4\,{{\it \_R}}^{2}-1}{{{\it \_R}}^{9}a-4\,{{\it \_R}}^{7}a+6\,{{\it \_R}}^{5}a-16\,{{\it \_R}}^{4}b-4\,{{\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)^5),x)

[Out]

1/5*sum((-_R^8+4*_R^6-6*_R^4+4*_R^2-1)/(_R^9*a-4*_R^7*a+6*_R^5*a-16*_R^4*b-4*_R^3*a+_R*a)*ln(tanh(1/2*x)-_R),_
R=RootOf(_Z^10*a-5*_Z^8*a+10*_Z^6*a-32*_Z^5*b-10*_Z^4*a+5*_Z^2*a-a))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(b*sinh(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*sinh(x)^5),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(a + b*sinh(x)**5), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(b*sinh(x)^5 + a), x)