3.67 \(\int \frac{x^4}{\sinh ^{-1}(a x)^4} \, dx\)

Optimal. Leaf size=155 \[ \frac{\text{Shi}\left (\sinh ^{-1}(a x)\right )}{48 a^5}-\frac{27 \text{Shi}\left (3 \sinh ^{-1}(a x)\right )}{32 a^5}+\frac{125 \text{Shi}\left (5 \sinh ^{-1}(a x)\right )}{96 a^5}-\frac{25 x^4 \sqrt{a^2 x^2+1}}{6 a \sinh ^{-1}(a x)}-\frac{x^4 \sqrt{a^2 x^2+1}}{3 a \sinh ^{-1}(a x)^3}-\frac{2 x^3}{3 a^2 \sinh ^{-1}(a x)^2}-\frac{2 x^2 \sqrt{a^2 x^2+1}}{a^3 \sinh ^{-1}(a x)}-\frac{5 x^5}{6 \sinh ^{-1}(a x)^2} \]

[Out]

-(x^4*Sqrt[1 + a^2*x^2])/(3*a*ArcSinh[a*x]^3) - (2*x^3)/(3*a^2*ArcSinh[a*x]^2) - (5*x^5)/(6*ArcSinh[a*x]^2) -
(2*x^2*Sqrt[1 + a^2*x^2])/(a^3*ArcSinh[a*x]) - (25*x^4*Sqrt[1 + a^2*x^2])/(6*a*ArcSinh[a*x]) + SinhIntegral[Ar
cSinh[a*x]]/(48*a^5) - (27*SinhIntegral[3*ArcSinh[a*x]])/(32*a^5) + (125*SinhIntegral[5*ArcSinh[a*x]])/(96*a^5
)

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Rubi [A]  time = 0.319825, antiderivative size = 155, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {5667, 5774, 5665, 3298} \[ \frac{\text{Shi}\left (\sinh ^{-1}(a x)\right )}{48 a^5}-\frac{27 \text{Shi}\left (3 \sinh ^{-1}(a x)\right )}{32 a^5}+\frac{125 \text{Shi}\left (5 \sinh ^{-1}(a x)\right )}{96 a^5}-\frac{25 x^4 \sqrt{a^2 x^2+1}}{6 a \sinh ^{-1}(a x)}-\frac{x^4 \sqrt{a^2 x^2+1}}{3 a \sinh ^{-1}(a x)^3}-\frac{2 x^3}{3 a^2 \sinh ^{-1}(a x)^2}-\frac{2 x^2 \sqrt{a^2 x^2+1}}{a^3 \sinh ^{-1}(a x)}-\frac{5 x^5}{6 \sinh ^{-1}(a x)^2} \]

Antiderivative was successfully verified.

[In]

Int[x^4/ArcSinh[a*x]^4,x]

[Out]

-(x^4*Sqrt[1 + a^2*x^2])/(3*a*ArcSinh[a*x]^3) - (2*x^3)/(3*a^2*ArcSinh[a*x]^2) - (5*x^5)/(6*ArcSinh[a*x]^2) -
(2*x^2*Sqrt[1 + a^2*x^2])/(a^3*ArcSinh[a*x]) - (25*x^4*Sqrt[1 + a^2*x^2])/(6*a*ArcSinh[a*x]) + SinhIntegral[Ar
cSinh[a*x]]/(48*a^5) - (27*SinhIntegral[3*ArcSinh[a*x]])/(32*a^5) + (125*SinhIntegral[5*ArcSinh[a*x]])/(96*a^5
)

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 3298

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(I*SinhIntegral[(c*f*fz)
/d + f*fz*x])/d, x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*e - c*f*fz*I, 0]

Rubi steps

\begin{align*} \int \frac{x^4}{\sinh ^{-1}(a x)^4} \, dx &=-\frac{x^4 \sqrt{1+a^2 x^2}}{3 a \sinh ^{-1}(a x)^3}+\frac{4 \int \frac{x^3}{\sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^3} \, dx}{3 a}+\frac{1}{3} (5 a) \int \frac{x^5}{\sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^3} \, dx\\ &=-\frac{x^4 \sqrt{1+a^2 x^2}}{3 a \sinh ^{-1}(a x)^3}-\frac{2 x^3}{3 a^2 \sinh ^{-1}(a x)^2}-\frac{5 x^5}{6 \sinh ^{-1}(a x)^2}+\frac{25}{6} \int \frac{x^4}{\sinh ^{-1}(a x)^2} \, dx+\frac{2 \int \frac{x^2}{\sinh ^{-1}(a x)^2} \, dx}{a^2}\\ &=-\frac{x^4 \sqrt{1+a^2 x^2}}{3 a \sinh ^{-1}(a x)^3}-\frac{2 x^3}{3 a^2 \sinh ^{-1}(a x)^2}-\frac{5 x^5}{6 \sinh ^{-1}(a x)^2}-\frac{2 x^2 \sqrt{1+a^2 x^2}}{a^3 \sinh ^{-1}(a x)}-\frac{25 x^4 \sqrt{1+a^2 x^2}}{6 a \sinh ^{-1}(a x)}+\frac{2 \operatorname{Subst}\left (\int \left (-\frac{\sinh (x)}{4 x}+\frac{3 \sinh (3 x)}{4 x}\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{a^5}+\frac{25 \operatorname{Subst}\left (\int \left (\frac{\sinh (x)}{8 x}-\frac{9 \sinh (3 x)}{16 x}+\frac{5 \sinh (5 x)}{16 x}\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{6 a^5}\\ &=-\frac{x^4 \sqrt{1+a^2 x^2}}{3 a \sinh ^{-1}(a x)^3}-\frac{2 x^3}{3 a^2 \sinh ^{-1}(a x)^2}-\frac{5 x^5}{6 \sinh ^{-1}(a x)^2}-\frac{2 x^2 \sqrt{1+a^2 x^2}}{a^3 \sinh ^{-1}(a x)}-\frac{25 x^4 \sqrt{1+a^2 x^2}}{6 a \sinh ^{-1}(a x)}-\frac{\operatorname{Subst}\left (\int \frac{\sinh (x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{2 a^5}+\frac{25 \operatorname{Subst}\left (\int \frac{\sinh (x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{48 a^5}+\frac{125 \operatorname{Subst}\left (\int \frac{\sinh (5 x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{96 a^5}+\frac{3 \operatorname{Subst}\left (\int \frac{\sinh (3 x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{2 a^5}-\frac{75 \operatorname{Subst}\left (\int \frac{\sinh (3 x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{32 a^5}\\ &=-\frac{x^4 \sqrt{1+a^2 x^2}}{3 a \sinh ^{-1}(a x)^3}-\frac{2 x^3}{3 a^2 \sinh ^{-1}(a x)^2}-\frac{5 x^5}{6 \sinh ^{-1}(a x)^2}-\frac{2 x^2 \sqrt{1+a^2 x^2}}{a^3 \sinh ^{-1}(a x)}-\frac{25 x^4 \sqrt{1+a^2 x^2}}{6 a \sinh ^{-1}(a x)}+\frac{\text{Shi}\left (\sinh ^{-1}(a x)\right )}{48 a^5}-\frac{27 \text{Shi}\left (3 \sinh ^{-1}(a x)\right )}{32 a^5}+\frac{125 \text{Shi}\left (5 \sinh ^{-1}(a x)\right )}{96 a^5}\\ \end{align*}

Mathematica [A]  time = 0.328406, size = 156, normalized size = 1.01 \[ -\frac{32 a^4 x^4 \sqrt{a^2 x^2+1}+80 a^5 x^5 \sinh ^{-1}(a x)+400 a^4 x^4 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^2+64 a^3 x^3 \sinh ^{-1}(a x)+192 a^2 x^2 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^2-2 \sinh ^{-1}(a x)^3 \text{Shi}\left (\sinh ^{-1}(a x)\right )+81 \sinh ^{-1}(a x)^3 \text{Shi}\left (3 \sinh ^{-1}(a x)\right )-125 \sinh ^{-1}(a x)^3 \text{Shi}\left (5 \sinh ^{-1}(a x)\right )}{96 a^5 \sinh ^{-1}(a x)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4/ArcSinh[a*x]^4,x]

[Out]

-(32*a^4*x^4*Sqrt[1 + a^2*x^2] + 64*a^3*x^3*ArcSinh[a*x] + 80*a^5*x^5*ArcSinh[a*x] + 192*a^2*x^2*Sqrt[1 + a^2*
x^2]*ArcSinh[a*x]^2 + 400*a^4*x^4*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^2 - 2*ArcSinh[a*x]^3*SinhIntegral[ArcSinh[a*x
]] + 81*ArcSinh[a*x]^3*SinhIntegral[3*ArcSinh[a*x]] - 125*ArcSinh[a*x]^3*SinhIntegral[5*ArcSinh[a*x]])/(96*a^5
*ArcSinh[a*x]^3)

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Maple [A]  time = 0.046, size = 169, normalized size = 1.1 \begin{align*}{\frac{1}{{a}^{5}} \left ( -{\frac{1}{24\, \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{3}}\sqrt{{a}^{2}{x}^{2}+1}}-{\frac{ax}{48\, \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{2}}}-{\frac{1}{48\,{\it Arcsinh} \left ( ax \right ) }\sqrt{{a}^{2}{x}^{2}+1}}+{\frac{{\it Shi} \left ({\it Arcsinh} \left ( ax \right ) \right ) }{48}}+{\frac{\cosh \left ( 3\,{\it Arcsinh} \left ( ax \right ) \right ) }{16\, \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{3}}}+{\frac{3\,\sinh \left ( 3\,{\it Arcsinh} \left ( ax \right ) \right ) }{32\, \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{2}}}+{\frac{9\,\cosh \left ( 3\,{\it Arcsinh} \left ( ax \right ) \right ) }{32\,{\it Arcsinh} \left ( ax \right ) }}-{\frac{27\,{\it Shi} \left ( 3\,{\it Arcsinh} \left ( ax \right ) \right ) }{32}}-{\frac{\cosh \left ( 5\,{\it Arcsinh} \left ( ax \right ) \right ) }{48\, \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{3}}}-{\frac{5\,\sinh \left ( 5\,{\it Arcsinh} \left ( ax \right ) \right ) }{96\, \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{2}}}-{\frac{25\,\cosh \left ( 5\,{\it Arcsinh} \left ( ax \right ) \right ) }{96\,{\it Arcsinh} \left ( ax \right ) }}+{\frac{125\,{\it Shi} \left ( 5\,{\it Arcsinh} \left ( ax \right ) \right ) }{96}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/arcsinh(a*x)^4,x)

[Out]

1/a^5*(-1/24/arcsinh(a*x)^3*(a^2*x^2+1)^(1/2)-1/48*a*x/arcsinh(a*x)^2-1/48/arcsinh(a*x)*(a^2*x^2+1)^(1/2)+1/48
*Shi(arcsinh(a*x))+1/16/arcsinh(a*x)^3*cosh(3*arcsinh(a*x))+3/32/arcsinh(a*x)^2*sinh(3*arcsinh(a*x))+9/32/arcs
inh(a*x)*cosh(3*arcsinh(a*x))-27/32*Shi(3*arcsinh(a*x))-1/48/arcsinh(a*x)^3*cosh(5*arcsinh(a*x))-5/96/arcsinh(
a*x)^2*sinh(5*arcsinh(a*x))-25/96/arcsinh(a*x)*cosh(5*arcsinh(a*x))+125/96*Shi(5*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^4/arcsinh(a*x)^4,x, algorithm="maxima")

[Out]

-1/6*(2*a^13*x^15 + 10*a^11*x^13 + 20*a^9*x^11 + 20*a^7*x^9 + 10*a^5*x^7 + 2*a^3*x^5 + 2*(a^8*x^10 + a^6*x^8)*
(a^2*x^2 + 1)^(5/2) + 2*(5*a^9*x^11 + 9*a^7*x^9 + 4*a^5*x^7)*(a^2*x^2 + 1)^2 + (25*a^13*x^15 + 125*a^11*x^13 +
250*a^9*x^11 + 250*a^7*x^9 + 125*a^5*x^7 + 25*a^3*x^5 + (25*a^8*x^10 + 49*a^6*x^8 + 27*a^4*x^6 + 3*a^2*x^4)*(
a^2*x^2 + 1)^(5/2) + (125*a^9*x^11 + 321*a^7*x^9 + 286*a^5*x^7 + 102*a^3*x^5 + 12*a*x^3)*(a^2*x^2 + 1)^2 + (25
0*a^10*x^12 + 794*a^8*x^10 + 946*a^6*x^8 + 519*a^4*x^6 + 129*a^2*x^4 + 12*x^2)*(a^2*x^2 + 1)^(3/2) + 2*(125*a^
11*x^13 + 473*a^9*x^11 + 696*a^7*x^9 + 497*a^5*x^7 + 173*a^3*x^5 + 24*a*x^3)*(a^2*x^2 + 1) + (125*a^12*x^14 +
549*a^10*x^12 + 955*a^8*x^10 + 824*a^6*x^8 + 354*a^4*x^6 + 61*a^2*x^4)*sqrt(a^2*x^2 + 1))*log(a*x + sqrt(a^2*x
^2 + 1))^2 + 4*(5*a^10*x^12 + 13*a^8*x^10 + 11*a^6*x^8 + 3*a^4*x^6)*(a^2*x^2 + 1)^(3/2) + 4*(5*a^11*x^13 + 17*
a^9*x^11 + 21*a^7*x^9 + 11*a^5*x^7 + 2*a^3*x^5)*(a^2*x^2 + 1) + (5*a^13*x^15 + 25*a^11*x^13 + 50*a^9*x^11 + 50
*a^7*x^9 + 25*a^5*x^7 + 5*a^3*x^5 + (5*a^8*x^10 + 8*a^6*x^8 + 3*a^4*x^6)*(a^2*x^2 + 1)^(5/2) + (25*a^9*x^11 +
57*a^7*x^9 + 42*a^5*x^7 + 10*a^3*x^5)*(a^2*x^2 + 1)^2 + (50*a^10*x^12 + 148*a^8*x^10 + 158*a^6*x^8 + 71*a^4*x^
6 + 11*a^2*x^4)*(a^2*x^2 + 1)^(3/2) + 2*(25*a^11*x^13 + 91*a^9*x^11 + 126*a^7*x^9 + 81*a^5*x^7 + 23*a^3*x^5 +
2*a*x^3)*(a^2*x^2 + 1) + (25*a^12*x^14 + 108*a^10*x^12 + 183*a^8*x^10 + 151*a^6*x^8 + 60*a^4*x^6 + 9*a^2*x^4)*
sqrt(a^2*x^2 + 1))*log(a*x + sqrt(a^2*x^2 + 1)) + 2*(5*a^12*x^14 + 21*a^10*x^12 + 34*a^8*x^10 + 26*a^6*x^8 + 9
*a^4*x^6 + a^2*x^4)*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 + 1
0*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*(125*a^15*x^1
6 + 750*a^13*x^14 + 1875*a^11*x^12 + 2500*a^9*x^10 + 1875*a^7*x^8 + 750*a^5*x^6 + 125*a^3*x^4 + (125*a^9*x^10
+ 147*a^7*x^8 + 27*a^5*x^6 - 3*a^3*x^4)*(a^2*x^2 + 1)^3 + (750*a^10*x^11 + 1485*a^8*x^9 + 901*a^6*x^7 + 147*a^
4*x^5 - 12*a^2*x^3)*(a^2*x^2 + 1)^(5/2) + (1875*a^11*x^12 + 5220*a^9*x^10 + 5209*a^7*x^8 + 2185*a^5*x^6 + 321*
a^3*x^4)*(a^2*x^2 + 1)^2 + (2500*a^12*x^13 + 8970*a^10*x^11 + 12366*a^8*x^9 + 8143*a^6*x^7 + 2583*a^4*x^5 + 36
0*a^2*x^3 + 24*x)*(a^2*x^2 + 1)^(3/2) + (1875*a^13*x^14 + 8235*a^11*x^12 + 14449*a^9*x^10 + 12834*a^7*x^8 + 60
30*a^5*x^6 + 1429*a^3*x^4 + 144*a*x^2)*(a^2*x^2 + 1) + (750*a^14*x^15 + 3897*a^12*x^13 + 8293*a^10*x^11 + 9226
*a^8*x^9 + 5655*a^6*x^7 + 1819*a^4*x^5 + 244*a^2*x^3)*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^{4}}{\operatorname{arsinh}\left (a x\right )^{4}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/arcsinh(a*x)^4,x, algorithm="fricas")

[Out]

integral(x^4/arcsinh(a*x)^4, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4}}{\operatorname{asinh}^{4}{\left (a x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4/asinh(a*x)**4,x)

[Out]

Integral(x**4/asinh(a*x)**4, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4}}{\operatorname{arsinh}\left (a x\right )^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/arcsinh(a*x)^4,x, algorithm="giac")

[Out]

integrate(x^4/arcsinh(a*x)^4, x)