3.60 \(\int \frac {x^4}{\sinh ^{-1}(a x)^3} \, dx\)

Optimal. Leaf size=97 \[ \frac {\text {Chi}\left (\sinh ^{-1}(a x)\right )}{16 a^5}-\frac {27 \text {Chi}\left (3 \sinh ^{-1}(a x)\right )}{32 a^5}+\frac {25 \text {Chi}\left (5 \sinh ^{-1}(a x)\right )}{32 a^5}-\frac {2 x^3}{a^2 \sinh ^{-1}(a x)}-\frac {x^4 \sqrt {a^2 x^2+1}}{2 a \sinh ^{-1}(a x)^2}-\frac {5 x^5}{2 \sinh ^{-1}(a x)} \]

[Out]

-2*x^3/a^2/arcsinh(a*x)-5/2*x^5/arcsinh(a*x)+1/16*Chi(arcsinh(a*x))/a^5-27/32*Chi(3*arcsinh(a*x))/a^5+25/32*Ch
i(5*arcsinh(a*x))/a^5-1/2*x^4*(a^2*x^2+1)^(1/2)/a/arcsinh(a*x)^2

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Rubi [A]  time = 0.35, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5667, 5774, 5669, 5448, 3301} \[ \frac {\text {Chi}\left (\sinh ^{-1}(a x)\right )}{16 a^5}-\frac {27 \text {Chi}\left (3 \sinh ^{-1}(a x)\right )}{32 a^5}+\frac {25 \text {Chi}\left (5 \sinh ^{-1}(a x)\right )}{32 a^5}-\frac {x^4 \sqrt {a^2 x^2+1}}{2 a \sinh ^{-1}(a x)^2}-\frac {2 x^3}{a^2 \sinh ^{-1}(a x)}-\frac {5 x^5}{2 \sinh ^{-1}(a x)} \]

Antiderivative was successfully verified.

[In]

Int[x^4/ArcSinh[a*x]^3,x]

[Out]

-(x^4*Sqrt[1 + a^2*x^2])/(2*a*ArcSinh[a*x]^2) - (2*x^3)/(a^2*ArcSinh[a*x]) - (5*x^5)/(2*ArcSinh[a*x]) + CoshIn
tegral[ArcSinh[a*x]]/(16*a^5) - (27*CoshIntegral[3*ArcSinh[a*x]])/(32*a^5) + (25*CoshIntegral[5*ArcSinh[a*x]])
/(32*a^5)

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]

Rule 5448

Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int
[ExpandTrigReduce[(c + d*x)^m, Sinh[a + b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n,
0] && IGtQ[p, 0]

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 5669

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Dist[1/c^(m + 1), Subst[Int[(a + b*x)^n*
Sinh[x]^m*Cosh[x], x], x, ArcSinh[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[m, 0]

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]

Rubi steps

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

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Mathematica [A]  time = 0.16, size = 102, normalized size = 1.05 \[ -\frac {80 a^5 x^5 \sinh ^{-1}(a x)+64 a^3 x^3 \sinh ^{-1}(a x)+16 a^4 x^4 \sqrt {a^2 x^2+1}-2 \sinh ^{-1}(a x)^2 \text {Chi}\left (\sinh ^{-1}(a x)\right )+27 \sinh ^{-1}(a x)^2 \text {Chi}\left (3 \sinh ^{-1}(a x)\right )-25 \sinh ^{-1}(a x)^2 \text {Chi}\left (5 \sinh ^{-1}(a x)\right )}{32 a^5 \sinh ^{-1}(a x)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4/ArcSinh[a*x]^3,x]

[Out]

-1/32*(16*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] - 2*ArcSinh[a*x]^2*Cos
hIntegral[ArcSinh[a*x]] + 27*ArcSinh[a*x]^2*CoshIntegral[3*ArcSinh[a*x]] - 25*ArcSinh[a*x]^2*CoshIntegral[5*Ar
cSinh[a*x]])/(a^5*ArcSinh[a*x]^2)

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fricas [F]  time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{4}}{\operatorname {arsinh}\left (a x\right )^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/arcsinh(a*x)^3,x, algorithm="fricas")

[Out]

integral(x^4/arcsinh(a*x)^3, x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{4}}{\operatorname {arsinh}\left (a x\right )^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/arcsinh(a*x)^3,x, algorithm="giac")

[Out]

integrate(x^4/arcsinh(a*x)^3, x)

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maple [A]  time = 0.17, size = 120, normalized size = 1.24 \[ \frac {-\frac {\sqrt {a^{2} x^{2}+1}}{16 \arcsinh \left (a x \right )^{2}}-\frac {a x}{16 \arcsinh \left (a x \right )}+\frac {\Chi \left (\arcsinh \left (a x \right )\right )}{16}+\frac {3 \cosh \left (3 \arcsinh \left (a x \right )\right )}{32 \arcsinh \left (a x \right )^{2}}+\frac {9 \sinh \left (3 \arcsinh \left (a x \right )\right )}{32 \arcsinh \left (a x \right )}-\frac {27 \Chi \left (3 \arcsinh \left (a x \right )\right )}{32}-\frac {\cosh \left (5 \arcsinh \left (a x \right )\right )}{32 \arcsinh \left (a x \right )^{2}}-\frac {5 \sinh \left (5 \arcsinh \left (a x \right )\right )}{32 \arcsinh \left (a x \right )}+\frac {25 \Chi \left (5 \arcsinh \left (a x \right )\right )}{32}}{a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/arcsinh(a*x)^3,x)

[Out]

1/a^5*(-1/16/arcsinh(a*x)^2*(a^2*x^2+1)^(1/2)-1/16*a*x/arcsinh(a*x)+1/16*Chi(arcsinh(a*x))+3/32/arcsinh(a*x)^2
*cosh(3*arcsinh(a*x))+9/32/arcsinh(a*x)*sinh(3*arcsinh(a*x))-27/32*Chi(3*arcsinh(a*x))-1/32/arcsinh(a*x)^2*cos
h(5*arcsinh(a*x))-5/32/arcsinh(a*x)*sinh(5*arcsinh(a*x))+25/32*Chi(5*arcsinh(a*x)))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {a^{8} x^{11} + 3 \, a^{6} x^{9} + 3 \, a^{4} x^{7} + a^{2} x^{5} + {\left (a^{5} x^{8} + a^{3} x^{6}\right )} {\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}} + {\left (3 \, a^{6} x^{9} + 5 \, a^{4} x^{7} + 2 \, a^{2} x^{5}\right )} {\left (a^{2} x^{2} + 1\right )} + {\left (5 \, a^{8} x^{11} + 15 \, a^{6} x^{9} + 15 \, a^{4} x^{7} + 5 \, a^{2} x^{5} + {\left (5 \, a^{5} x^{8} + 8 \, a^{3} x^{6} + 3 \, a x^{4}\right )} {\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}} + {\left (15 \, a^{6} x^{9} + 31 \, a^{4} x^{7} + 20 \, a^{2} x^{5} + 4 \, x^{3}\right )} {\left (a^{2} x^{2} + 1\right )} + {\left (15 \, a^{7} x^{10} + 38 \, a^{5} x^{8} + 32 \, a^{3} x^{6} + 9 \, a x^{4}\right )} \sqrt {a^{2} x^{2} + 1}\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right ) + {\left (3 \, a^{7} x^{10} + 7 \, a^{5} x^{8} + 5 \, a^{3} x^{6} + a x^{4}\right )} \sqrt {a^{2} x^{2} + 1}}{2 \, {\left (a^{8} x^{6} + 3 \, a^{6} x^{4} + {\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}} a^{5} x^{3} + 3 \, a^{4} x^{2} + 3 \, {\left (a^{6} x^{4} + a^{4} x^{2}\right )} {\left (a^{2} x^{2} + 1\right )} + a^{2} + 3 \, {\left (a^{7} x^{5} + 2 \, a^{5} x^{3} + a^{3} x\right )} \sqrt {a^{2} x^{2} + 1}\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{2}} + \int \frac {25 \, a^{10} x^{12} + 100 \, a^{8} x^{10} + 150 \, a^{6} x^{8} + 100 \, a^{4} x^{6} + 25 \, a^{2} x^{4} + {\left (25 \, a^{6} x^{8} + 24 \, a^{4} x^{6} + 3 \, a^{2} x^{4}\right )} {\left (a^{2} x^{2} + 1\right )}^{2} + {\left (100 \, a^{7} x^{9} + 172 \, a^{5} x^{7} + 87 \, a^{3} x^{5} + 12 \, a x^{3}\right )} {\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}} + 3 \, {\left (50 \, a^{8} x^{10} + 124 \, a^{6} x^{8} + 105 \, a^{4} x^{6} + 35 \, a^{2} x^{4} + 4 \, x^{2}\right )} {\left (a^{2} x^{2} + 1\right )} + {\left (100 \, a^{9} x^{11} + 324 \, a^{7} x^{9} + 381 \, a^{5} x^{7} + 193 \, a^{3} x^{5} + 36 \, a x^{3}\right )} \sqrt {a^{2} x^{2} + 1}}{2 \, {\left (a^{10} x^{8} + 4 \, a^{8} x^{6} + {\left (a^{2} x^{2} + 1\right )}^{2} a^{6} x^{4} + 6 \, a^{6} x^{4} + 4 \, a^{4} x^{2} + 4 \, {\left (a^{7} x^{5} + a^{5} x^{3}\right )} {\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}} + 6 \, {\left (a^{8} x^{6} + 2 \, a^{6} x^{4} + a^{4} x^{2}\right )} {\left (a^{2} x^{2} + 1\right )} + a^{2} + 4 \, {\left (a^{9} x^{7} + 3 \, a^{7} x^{5} + 3 \, a^{5} x^{3} + a^{3} x\right )} \sqrt {a^{2} x^{2} + 1}\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/arcsinh(a*x)^3,x, algorithm="maxima")

[Out]

-1/2*(a^8*x^11 + 3*a^6*x^9 + 3*a^4*x^7 + a^2*x^5 + (a^5*x^8 + a^3*x^6)*(a^2*x^2 + 1)^(3/2) + (3*a^6*x^9 + 5*a^
4*x^7 + 2*a^2*x^5)*(a^2*x^2 + 1) + (5*a^8*x^11 + 15*a^6*x^9 + 15*a^4*x^7 + 5*a^2*x^5 + (5*a^5*x^8 + 8*a^3*x^6
+ 3*a*x^4)*(a^2*x^2 + 1)^(3/2) + (15*a^6*x^9 + 31*a^4*x^7 + 20*a^2*x^5 + 4*x^3)*(a^2*x^2 + 1) + (15*a^7*x^10 +
38*a^5*x^8 + 32*a^3*x^6 + 9*a*x^4)*sqrt(a^2*x^2 + 1))*log(a*x + sqrt(a^2*x^2 + 1)) + (3*a^7*x^10 + 7*a^5*x^8
+ 5*a^3*x^6 + a*x^4)*sqrt(a^2*x^2 + 1))/((a^8*x^6 + 3*a^6*x^4 + (a^2*x^2 + 1)^(3/2)*a^5*x^3 + 3*a^4*x^2 + 3*(a
^6*x^4 + a^4*x^2)*(a^2*x^2 + 1) + a^2 + 3*(a^7*x^5 + 2*a^5*x^3 + a^3*x)*sqrt(a^2*x^2 + 1))*log(a*x + sqrt(a^2*
x^2 + 1))^2) + integrate(1/2*(25*a^10*x^12 + 100*a^8*x^10 + 150*a^6*x^8 + 100*a^4*x^6 + 25*a^2*x^4 + (25*a^6*x
^8 + 24*a^4*x^6 + 3*a^2*x^4)*(a^2*x^2 + 1)^2 + (100*a^7*x^9 + 172*a^5*x^7 + 87*a^3*x^5 + 12*a*x^3)*(a^2*x^2 +
1)^(3/2) + 3*(50*a^8*x^10 + 124*a^6*x^8 + 105*a^4*x^6 + 35*a^2*x^4 + 4*x^2)*(a^2*x^2 + 1) + (100*a^9*x^11 + 32
4*a^7*x^9 + 381*a^5*x^7 + 193*a^3*x^5 + 36*a*x^3)*sqrt(a^2*x^2 + 1))/((a^10*x^8 + 4*a^8*x^6 + (a^2*x^2 + 1)^2*
a^6*x^4 + 6*a^6*x^4 + 4*a^4*x^2 + 4*(a^7*x^5 + a^5*x^3)*(a^2*x^2 + 1)^(3/2) + 6*(a^8*x^6 + 2*a^6*x^4 + a^4*x^2
)*(a^2*x^2 + 1) + a^2 + 4*(a^9*x^7 + 3*a^7*x^5 + 3*a^5*x^3 + a^3*x)*sqrt(a^2*x^2 + 1))*log(a*x + sqrt(a^2*x^2
+ 1))), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^4}{{\mathrm {asinh}\left (a\,x\right )}^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/asinh(a*x)^3,x)

[Out]

int(x^4/asinh(a*x)^3, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{4}}{\operatorname {asinh}^{3}{\left (a x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4/asinh(a*x)**3,x)

[Out]

Integral(x**4/asinh(a*x)**3, x)

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