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]

-2/5*(-1)^(9/10)*arctanh((I*b^(1/5)-(-1)^(9/10)*a^(1/5)*tanh(1/2*x))/(-(-1)^(4/5)*a^(2/5)-b^(2/5))^(1/2))/a^(4
/5)/(-(-1)^(4/5)*a^(2/5)-b^(2/5))^(1/2)-2/5*arctanh((b^(1/5)-a^(1/5)*tanh(1/2*x))/(a^(2/5)+b^(2/5))^(1/2))/a^(
4/5)/(a^(2/5)+b^(2/5))^(1/2)+2/5*(-1)^(1/5)*arctanh((b^(1/5)+(-1)^(1/5)*a^(1/5)*tanh(1/2*x))/((-1)^(2/5)*a^(2/
5)+b^(2/5))^(1/2))/a^(4/5)/((-1)^(2/5)*a^(2/5)+b^(2/5))^(1/2)+2/5*(-1)^(9/10)*arctanh((-1)^(9/10)*((-1)^(1/5)*
b^(1/5)+a^(1/5)*tanh(1/2*x))/(-(-1)^(4/5)*a^(2/5)+(-1)^(1/5)*b^(2/5))^(1/2))/a^(4/5)/(-(-1)^(4/5)*a^(2/5)+(-1)
^(1/5)*b^(2/5))^(1/2)+2/5*(-1)^(9/10)*arctanh((-1)^(3/10)*(b^(1/5)+(-1)^(3/5)*a^(1/5)*tanh(1/2*x))/(-(-1)^(4/5
)*a^(2/5)+(-1)^(3/5)*b^(2/5))^(1/2))/a^(4/5)/(-(-1)^(4/5)*a^(2/5)+(-1)^(3/5)*b^(2/5))^(1/2)

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Rubi [A]  time = 0.97, antiderivative size = 435, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, 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 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])

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 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 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 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]))

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*}

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Mathematica [C]  time = 0.33, size = 141, normalized size = 0.32 \[ \frac {8}{5} \text {RootSum}\left [\text {$\#$1}^{10} b-5 \text {$\#$1}^8 b+10 \text {$\#$1}^6 b+32 \text {$\#$1}^5 a-10 \text {$\#$1}^4 b+5 \text {$\#$1}^2 b-b\& ,\frac {\text {$\#$1}^3 x+2 \text {$\#$1}^3 \log \left (-\text {$\#$1} \sinh \left (\frac {x}{2}\right )+\text {$\#$1} \cosh \left (\frac {x}{2}\right )-\sinh \left (\frac {x}{2}\right )-\cosh \left (\frac {x}{2}\right )\right )}{\text {$\#$1}^8 b-4 \text {$\#$1}^6 b+6 \text {$\#$1}^4 b+16 \text {$\#$1}^3 a-4 \text {$\#$1}^2 b+b}\& \right ] \]

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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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError >> no explicit roots found

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{b \sinh \relax (x)^{5} + a}\,{d x} \]

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)

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maple [C]  time = 0.06, size = 113, normalized size = 0.26 \[ \frac {\left (\munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{10}-5 a \,\textit {\_Z}^{8}+10 a \,\textit {\_Z}^{6}-32 b \,\textit {\_Z}^{5}-10 a \,\textit {\_Z}^{4}+5 a \,\textit {\_Z}^{2}-a \right )}{\sum }\frac {\left (-\textit {\_R}^{8}+4 \textit {\_R}^{6}-6 \textit {\_R}^{4}+4 \textit {\_R}^{2}-1\right ) \ln \left (\tanh \left (\frac {x}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{9} a -4 \textit {\_R}^{7} a +6 \textit {\_R}^{5} a -16 \textit {\_R}^{4} b -4 \textit {\_R}^{3} a +\textit {\_R} a}\right )}{5} \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{b \sinh \relax (x)^{5} + a}\,{d x} \]

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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mupad [F(-1)]  time = 0.00, size = -1, normalized size = -0.00 \[ \text {Hanged} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + b*sinh(x)^5),x)

[Out]

\text{Hanged}

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{a + b \sinh ^{5}{\relax (x )}}\, dx \]

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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