3.68 \(\int \frac{x^3}{\sinh ^{-1}(a x)^4} \, dx\)
Optimal. Leaf size=141 \[ -\frac{\text{Chi}\left (2 \sinh ^{-1}(a x)\right )}{3 a^4}+\frac{4 \text{Chi}\left (4 \sinh ^{-1}(a x)\right )}{3 a^4}-\frac{8 x^3 \sqrt{a^2 x^2+1}}{3 a \sinh ^{-1}(a x)}-\frac{x^3 \sqrt{a^2 x^2+1}}{3 a \sinh ^{-1}(a x)^3}-\frac{x^2}{2 a^2 \sinh ^{-1}(a x)^2}-\frac{x \sqrt{a^2 x^2+1}}{a^3 \sinh ^{-1}(a x)}-\frac{2 x^4}{3 \sinh ^{-1}(a x)^2} \]
[Out]
-(x^3*Sqrt[1 + a^2*x^2])/(3*a*ArcSinh[a*x]^3) - x^2/(2*a^2*ArcSinh[a*x]^2) - (2*x^4)/(3*ArcSinh[a*x]^2) - (x*S
qrt[1 + a^2*x^2])/(a^3*ArcSinh[a*x]) - (8*x^3*Sqrt[1 + a^2*x^2])/(3*a*ArcSinh[a*x]) - CoshIntegral[2*ArcSinh[a
*x]]/(3*a^4) + (4*CoshIntegral[4*ArcSinh[a*x]])/(3*a^4)
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Rubi [A] time = 0.283297, antiderivative size = 141, normalized size of antiderivative = 1.,
number of steps used = 9, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used =
{5667, 5774, 5665, 3301} \[ -\frac{\text{Chi}\left (2 \sinh ^{-1}(a x)\right )}{3 a^4}+\frac{4 \text{Chi}\left (4 \sinh ^{-1}(a x)\right )}{3 a^4}-\frac{8 x^3 \sqrt{a^2 x^2+1}}{3 a \sinh ^{-1}(a x)}-\frac{x^3 \sqrt{a^2 x^2+1}}{3 a \sinh ^{-1}(a x)^3}-\frac{x^2}{2 a^2 \sinh ^{-1}(a x)^2}-\frac{x \sqrt{a^2 x^2+1}}{a^3 \sinh ^{-1}(a x)}-\frac{2 x^4}{3 \sinh ^{-1}(a x)^2} \]
Antiderivative was successfully verified.
[In]
Int[x^3/ArcSinh[a*x]^4,x]
[Out]
-(x^3*Sqrt[1 + a^2*x^2])/(3*a*ArcSinh[a*x]^3) - x^2/(2*a^2*ArcSinh[a*x]^2) - (2*x^4)/(3*ArcSinh[a*x]^2) - (x*S
qrt[1 + a^2*x^2])/(a^3*ArcSinh[a*x]) - (8*x^3*Sqrt[1 + a^2*x^2])/(3*a*ArcSinh[a*x]) - CoshIntegral[2*ArcSinh[a
*x]]/(3*a^4) + (4*CoshIntegral[4*ArcSinh[a*x]])/(3*a^4)
Rule 5667
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[(x^m*Sqrt[1 + c^2*x^2]*(a + b*ArcSi
nh[c*x])^(n + 1))/(b*c*(n + 1)), x] + (-Dist[(c*(m + 1))/(b*(n + 1)), Int[(x^(m + 1)*(a + b*ArcSinh[c*x])^(n +
1))/Sqrt[1 + c^2*x^2], x], x] - Dist[m/(b*c*(n + 1)), Int[(x^(m - 1)*(a + b*ArcSinh[c*x])^(n + 1))/Sqrt[1 + c
^2*x^2], x], x]) /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && LtQ[n, -2]
Rule 5774
Int[(((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*((f_.)*(x_))^(m_.))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp
[((f*x)^m*(a + b*ArcSinh[c*x])^(n + 1))/(b*c*Sqrt[d]*(n + 1)), x] - Dist[(f*m)/(b*c*Sqrt[d]*(n + 1)), Int[(f*x
)^(m - 1)*(a + b*ArcSinh[c*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && LtQ[n, -
1] && GtQ[d, 0]
Rule 5665
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[(x^m*Sqrt[1 + c^2*x^2]*(a + b*ArcSi
nh[c*x])^(n + 1))/(b*c*(n + 1)), x] - Dist[1/(b*c^(m + 1)*(n + 1)), Subst[Int[ExpandTrigReduce[(a + b*x)^(n +
1), Sinh[x]^(m - 1)*(m + (m + 1)*Sinh[x]^2), x], x], x, ArcSinh[c*x]], x] /; FreeQ[{a, b, c}, x] && IGtQ[m, 0]
&& GeQ[n, -2] && LtQ[n, -1]
Rule 3301
Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CoshIntegral[(c*f*fz)/d
+ f*fz*x]/d, x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*(e - Pi/2) - c*f*fz*I, 0]
Rubi steps
\begin{align*} \int \frac{x^3}{\sinh ^{-1}(a x)^4} \, dx &=-\frac{x^3 \sqrt{1+a^2 x^2}}{3 a \sinh ^{-1}(a x)^3}+\frac{\int \frac{x^2}{\sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^3} \, dx}{a}+\frac{1}{3} (4 a) \int \frac{x^4}{\sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^3} \, dx\\ &=-\frac{x^3 \sqrt{1+a^2 x^2}}{3 a \sinh ^{-1}(a x)^3}-\frac{x^2}{2 a^2 \sinh ^{-1}(a x)^2}-\frac{2 x^4}{3 \sinh ^{-1}(a x)^2}+\frac{8}{3} \int \frac{x^3}{\sinh ^{-1}(a x)^2} \, dx+\frac{\int \frac{x}{\sinh ^{-1}(a x)^2} \, dx}{a^2}\\ &=-\frac{x^3 \sqrt{1+a^2 x^2}}{3 a \sinh ^{-1}(a x)^3}-\frac{x^2}{2 a^2 \sinh ^{-1}(a x)^2}-\frac{2 x^4}{3 \sinh ^{-1}(a x)^2}-\frac{x \sqrt{1+a^2 x^2}}{a^3 \sinh ^{-1}(a x)}-\frac{8 x^3 \sqrt{1+a^2 x^2}}{3 a \sinh ^{-1}(a x)}+\frac{\operatorname{Subst}\left (\int \frac{\cosh (2 x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{a^4}+\frac{8 \operatorname{Subst}\left (\int \left (-\frac{\cosh (2 x)}{2 x}+\frac{\cosh (4 x)}{2 x}\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{3 a^4}\\ &=-\frac{x^3 \sqrt{1+a^2 x^2}}{3 a \sinh ^{-1}(a x)^3}-\frac{x^2}{2 a^2 \sinh ^{-1}(a x)^2}-\frac{2 x^4}{3 \sinh ^{-1}(a x)^2}-\frac{x \sqrt{1+a^2 x^2}}{a^3 \sinh ^{-1}(a x)}-\frac{8 x^3 \sqrt{1+a^2 x^2}}{3 a \sinh ^{-1}(a x)}+\frac{\text{Chi}\left (2 \sinh ^{-1}(a x)\right )}{a^4}-\frac{4 \operatorname{Subst}\left (\int \frac{\cosh (2 x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{3 a^4}+\frac{4 \operatorname{Subst}\left (\int \frac{\cosh (4 x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{3 a^4}\\ &=-\frac{x^3 \sqrt{1+a^2 x^2}}{3 a \sinh ^{-1}(a x)^3}-\frac{x^2}{2 a^2 \sinh ^{-1}(a x)^2}-\frac{2 x^4}{3 \sinh ^{-1}(a x)^2}-\frac{x \sqrt{1+a^2 x^2}}{a^3 \sinh ^{-1}(a x)}-\frac{8 x^3 \sqrt{1+a^2 x^2}}{3 a \sinh ^{-1}(a x)}-\frac{\text{Chi}\left (2 \sinh ^{-1}(a x)\right )}{3 a^4}+\frac{4 \text{Chi}\left (4 \sinh ^{-1}(a x)\right )}{3 a^4}\\ \end{align*}
Mathematica [A] time = 0.372958, size = 105, normalized size = 0.74 \[ -\frac{\frac{a x \left (2 a^2 x^2 \sqrt{a^2 x^2+1}+a x \left (4 a^2 x^2+3\right ) \sinh ^{-1}(a x)+2 \sqrt{a^2 x^2+1} \left (8 a^2 x^2+3\right ) \sinh ^{-1}(a x)^2\right )}{\sinh ^{-1}(a x)^3}+2 \text{Chi}\left (2 \sinh ^{-1}(a x)\right )-8 \text{Chi}\left (4 \sinh ^{-1}(a x)\right )}{6 a^4} \]
Antiderivative was successfully verified.
[In]
Integrate[x^3/ArcSinh[a*x]^4,x]
[Out]
-((a*x*(2*a^2*x^2*Sqrt[1 + a^2*x^2] + a*x*(3 + 4*a^2*x^2)*ArcSinh[a*x] + 2*Sqrt[1 + a^2*x^2]*(3 + 8*a^2*x^2)*A
rcSinh[a*x]^2))/ArcSinh[a*x]^3 + 2*CoshIntegral[2*ArcSinh[a*x]] - 8*CoshIntegral[4*ArcSinh[a*x]])/(6*a^4)
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Maple [A] time = 0.031, size = 114, normalized size = 0.8 \begin{align*}{\frac{1}{{a}^{4}} \left ({\frac{\sinh \left ( 2\,{\it Arcsinh} \left ( ax \right ) \right ) }{12\, \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{3}}}+{\frac{\cosh \left ( 2\,{\it Arcsinh} \left ( ax \right ) \right ) }{12\, \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{2}}}+{\frac{\sinh \left ( 2\,{\it Arcsinh} \left ( ax \right ) \right ) }{6\,{\it Arcsinh} \left ( ax \right ) }}-{\frac{{\it Chi} \left ( 2\,{\it Arcsinh} \left ( ax \right ) \right ) }{3}}-{\frac{\sinh \left ( 4\,{\it Arcsinh} \left ( ax \right ) \right ) }{24\, \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{3}}}-{\frac{\cosh \left ( 4\,{\it Arcsinh} \left ( ax \right ) \right ) }{12\, \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{2}}}-{\frac{\sinh \left ( 4\,{\it Arcsinh} \left ( ax \right ) \right ) }{3\,{\it Arcsinh} \left ( ax \right ) }}+{\frac{4\,{\it Chi} \left ( 4\,{\it Arcsinh} \left ( ax \right ) \right ) }{3}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^3/arcsinh(a*x)^4,x)
[Out]
1/a^4*(1/12/arcsinh(a*x)^3*sinh(2*arcsinh(a*x))+1/12/arcsinh(a*x)^2*cosh(2*arcsinh(a*x))+1/6/arcsinh(a*x)*sinh
(2*arcsinh(a*x))-1/3*Chi(2*arcsinh(a*x))-1/24/arcsinh(a*x)^3*sinh(4*arcsinh(a*x))-1/12/arcsinh(a*x)^2*cosh(4*a
rcsinh(a*x))-1/3/arcsinh(a*x)*sinh(4*arcsinh(a*x))+4/3*Chi(4*arcsinh(a*x)))
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^3/arcsinh(a*x)^4,x, algorithm="maxima")
[Out]
-1/6*(2*a^13*x^14 + 10*a^11*x^12 + 20*a^9*x^10 + 20*a^7*x^8 + 10*a^5*x^6 + 2*a^3*x^4 + 2*(a^8*x^9 + a^6*x^7)*(
a^2*x^2 + 1)^(5/2) + 2*(5*a^9*x^10 + 9*a^7*x^8 + 4*a^5*x^6)*(a^2*x^2 + 1)^2 + (16*a^13*x^14 + 80*a^11*x^12 + 1
60*a^9*x^10 + 160*a^7*x^8 + 80*a^5*x^6 + 16*a^3*x^4 + 4*(4*a^8*x^9 + 7*a^6*x^7 + 3*a^4*x^5)*(a^2*x^2 + 1)^(5/2
) + (80*a^9*x^10 + 192*a^7*x^8 + 154*a^5*x^6 + 45*a^3*x^4 + 3*a*x^2)*(a^2*x^2 + 1)^2 + (160*a^10*x^11 + 488*a^
8*x^9 + 550*a^6*x^7 + 279*a^4*x^5 + 63*a^2*x^3 + 6*x)*(a^2*x^2 + 1)^(3/2) + (160*a^11*x^12 + 592*a^9*x^10 + 84
6*a^7*x^8 + 583*a^5*x^6 + 196*a^3*x^4 + 27*a*x^2)*(a^2*x^2 + 1) + (80*a^12*x^13 + 348*a^10*x^11 + 598*a^8*x^9
+ 509*a^6*x^7 + 216*a^4*x^5 + 37*a^2*x^3)*sqrt(a^2*x^2 + 1))*log(a*x + sqrt(a^2*x^2 + 1))^2 + 4*(5*a^10*x^11 +
13*a^8*x^9 + 11*a^6*x^7 + 3*a^4*x^5)*(a^2*x^2 + 1)^(3/2) + 4*(5*a^11*x^12 + 17*a^9*x^10 + 21*a^7*x^8 + 11*a^5
*x^6 + 2*a^3*x^4)*(a^2*x^2 + 1) + (4*a^13*x^14 + 20*a^11*x^12 + 40*a^9*x^10 + 40*a^7*x^8 + 20*a^5*x^6 + 4*a^3*
x^4 + 2*(2*a^8*x^9 + 3*a^6*x^7 + a^4*x^5)*(a^2*x^2 + 1)^(5/2) + (20*a^9*x^10 + 44*a^7*x^8 + 31*a^5*x^6 + 7*a^3
*x^4)*(a^2*x^2 + 1)^2 + (40*a^10*x^11 + 116*a^8*x^9 + 121*a^6*x^7 + 53*a^4*x^5 + 8*a^2*x^3)*(a^2*x^2 + 1)^(3/2
) + (40*a^11*x^12 + 144*a^9*x^10 + 197*a^7*x^8 + 125*a^5*x^6 + 35*a^3*x^4 + 3*a*x^2)*(a^2*x^2 + 1) + (20*a^12*
x^13 + 86*a^10*x^11 + 145*a^8*x^9 + 119*a^6*x^7 + 47*a^4*x^5 + 7*a^2*x^3)*sqrt(a^2*x^2 + 1))*log(a*x + sqrt(a^
2*x^2 + 1)) + 2*(5*a^12*x^13 + 21*a^10*x^11 + 34*a^8*x^9 + 26*a^6*x^7 + 9*a^4*x^5 + a^2*x^3)*sqrt(a^2*x^2 + 1)
)/((a^13*x^10 + 5*a^11*x^8 + (a^2*x^2 + 1)^(5/2)*a^8*x^5 + 10*a^9*x^6 + 10*a^7*x^4 + 5*a^5*x^2 + 5*(a^9*x^6 +
a^7*x^4)*(a^2*x^2 + 1)^2 + a^3 + 10*(a^10*x^7 + 2*a^8*x^5 + a^6*x^3)*(a^2*x^2 + 1)^(3/2) + 10*(a^11*x^8 + 3*a^
9*x^6 + 3*a^7*x^4 + a^5*x^2)*(a^2*x^2 + 1) + 5*(a^12*x^9 + 4*a^10*x^7 + 6*a^8*x^5 + 4*a^6*x^3 + a^4*x)*sqrt(a^
2*x^2 + 1))*log(a*x + sqrt(a^2*x^2 + 1))^3) + integrate(1/6*(64*a^15*x^15 + 384*a^13*x^13 + 960*a^11*x^11 + 12
80*a^9*x^9 + 960*a^7*x^7 + 384*a^5*x^5 + 64*a^3*x^3 + 8*(8*a^9*x^9 + 7*a^7*x^7)*(a^2*x^2 + 1)^3 + (384*a^10*x^
10 + 664*a^8*x^8 + 308*a^6*x^6 + 12*a^4*x^4 - 9*a^2*x^2)*(a^2*x^2 + 1)^(5/2) + 2*(480*a^11*x^11 + 1240*a^9*x^9
+ 1096*a^7*x^7 + 360*a^5*x^5 + 15*a^3*x^3 - 9*a*x)*(a^2*x^2 + 1)^2 + 2*(640*a^12*x^12 + 2200*a^10*x^10 + 2844
*a^8*x^8 + 1684*a^6*x^6 + 433*a^4*x^4 + 36*a^2*x^2 + 3)*(a^2*x^2 + 1)^(3/2) + 2*(480*a^13*x^13 + 2060*a^11*x^1
1 + 3496*a^9*x^9 + 2952*a^7*x^7 + 1283*a^5*x^5 + 274*a^3*x^3 + 27*a*x)*(a^2*x^2 + 1) + (384*a^14*x^14 + 1976*a
^12*x^12 + 4148*a^10*x^10 + 4524*a^8*x^8 + 2699*a^6*x^6 + 842*a^4*x^4 + 111*a^2*x^2)*sqrt(a^2*x^2 + 1))/((a^15
*x^12 + 6*a^13*x^10 + 15*a^11*x^8 + (a^2*x^2 + 1)^3*a^9*x^6 + 20*a^9*x^6 + 15*a^7*x^4 + 6*a^5*x^2 + 6*(a^10*x^
7 + a^8*x^5)*(a^2*x^2 + 1)^(5/2) + 15*(a^11*x^8 + 2*a^9*x^6 + a^7*x^4)*(a^2*x^2 + 1)^2 + a^3 + 20*(a^12*x^9 +
3*a^10*x^7 + 3*a^8*x^5 + a^6*x^3)*(a^2*x^2 + 1)^(3/2) + 15*(a^13*x^10 + 4*a^11*x^8 + 6*a^9*x^6 + 4*a^7*x^4 + a
^5*x^2)*(a^2*x^2 + 1) + 6*(a^14*x^11 + 5*a^12*x^9 + 10*a^10*x^7 + 10*a^8*x^5 + 5*a^6*x^3 + a^4*x)*sqrt(a^2*x^2
+ 1))*log(a*x + sqrt(a^2*x^2 + 1))), x)
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x^{3}}{\operatorname{arsinh}\left (a x\right )^{4}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^3/arcsinh(a*x)^4,x, algorithm="fricas")
[Out]
integral(x^3/arcsinh(a*x)^4, x)
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3}}{\operatorname{asinh}^{4}{\left (a x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x**3/asinh(a*x)**4,x)
[Out]
Integral(x**3/asinh(a*x)**4, x)
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3}}{\operatorname{arsinh}\left (a x\right )^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^3/arcsinh(a*x)^4,x, algorithm="giac")
[Out]
integrate(x^3/arcsinh(a*x)^4, x)