3.1.81 \(\int x^3 \sinh ^{-1}(a x)^{3/2} \, dx\) [81]

Optimal. Leaf size=199 \[ \frac {9 x \sqrt {1+a^2 x^2} \sqrt {\sinh ^{-1}(a x)}}{64 a^3}-\frac {3 x^3 \sqrt {1+a^2 x^2} \sqrt {\sinh ^{-1}(a x)}}{32 a}-\frac {3 \sinh ^{-1}(a x)^{3/2}}{32 a^4}+\frac {1}{4} x^4 \sinh ^{-1}(a x)^{3/2}-\frac {3 \sqrt {\pi } \text {Erf}\left (2 \sqrt {\sinh ^{-1}(a x)}\right )}{2048 a^4}+\frac {3 \sqrt {\frac {\pi }{2}} \text {Erf}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{128 a^4}+\frac {3 \sqrt {\pi } \text {Erfi}\left (2 \sqrt {\sinh ^{-1}(a x)}\right )}{2048 a^4}-\frac {3 \sqrt {\frac {\pi }{2}} \text {Erfi}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{128 a^4} \]

[Out]

-3/32*arcsinh(a*x)^(3/2)/a^4+1/4*x^4*arcsinh(a*x)^(3/2)+3/256*erf(2^(1/2)*arcsinh(a*x)^(1/2))*2^(1/2)*Pi^(1/2)
/a^4-3/256*erfi(2^(1/2)*arcsinh(a*x)^(1/2))*2^(1/2)*Pi^(1/2)/a^4-3/2048*erf(2*arcsinh(a*x)^(1/2))*Pi^(1/2)/a^4
+3/2048*erfi(2*arcsinh(a*x)^(1/2))*Pi^(1/2)/a^4+9/64*x*(a^2*x^2+1)^(1/2)*arcsinh(a*x)^(1/2)/a^3-3/32*x^3*(a^2*
x^2+1)^(1/2)*arcsinh(a*x)^(1/2)/a

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Rubi [A]
time = 0.34, antiderivative size = 199, normalized size of antiderivative = 1.00, number of steps used = 25, number of rules used = 10, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {5777, 5812, 5783, 5780, 5556, 12, 3389, 2211, 2235, 2236} \begin {gather*} -\frac {3 \sqrt {\pi } \text {Erf}\left (2 \sqrt {\sinh ^{-1}(a x)}\right )}{2048 a^4}+\frac {3 \sqrt {\frac {\pi }{2}} \text {Erf}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{128 a^4}+\frac {3 \sqrt {\pi } \text {Erfi}\left (2 \sqrt {\sinh ^{-1}(a x)}\right )}{2048 a^4}-\frac {3 \sqrt {\frac {\pi }{2}} \text {Erfi}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{128 a^4}-\frac {3 \sinh ^{-1}(a x)^{3/2}}{32 a^4}-\frac {3 x^3 \sqrt {a^2 x^2+1} \sqrt {\sinh ^{-1}(a x)}}{32 a}+\frac {9 x \sqrt {a^2 x^2+1} \sqrt {\sinh ^{-1}(a x)}}{64 a^3}+\frac {1}{4} x^4 \sinh ^{-1}(a x)^{3/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3*ArcSinh[a*x]^(3/2),x]

[Out]

(9*x*Sqrt[1 + a^2*x^2]*Sqrt[ArcSinh[a*x]])/(64*a^3) - (3*x^3*Sqrt[1 + a^2*x^2]*Sqrt[ArcSinh[a*x]])/(32*a) - (3
*ArcSinh[a*x]^(3/2))/(32*a^4) + (x^4*ArcSinh[a*x]^(3/2))/4 - (3*Sqrt[Pi]*Erf[2*Sqrt[ArcSinh[a*x]]])/(2048*a^4)
+ (3*Sqrt[Pi/2]*Erf[Sqrt[2]*Sqrt[ArcSinh[a*x]]])/(128*a^4) + (3*Sqrt[Pi]*Erfi[2*Sqrt[ArcSinh[a*x]]])/(2048*a^
4) - (3*Sqrt[Pi/2]*Erfi[Sqrt[2]*Sqrt[ArcSinh[a*x]]])/(128*a^4)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2211

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[F^(g*(e - c*(
f/d)) + f*g*(x^2/d)), x], x, Sqrt[c + d*x]], x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !TrueQ[$UseGamma]

Rule 2235

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt[Pi]*(Erfi[(c + d*x)*Rt[b*Log[F], 2
]]/(2*d*Rt[b*Log[F], 2])), x] /; FreeQ[{F, a, b, c, d}, x] && PosQ[b]

Rule 2236

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt[Pi]*(Erf[(c + d*x)*Rt[(-b)*Log[F],
2]]/(2*d*Rt[(-b)*Log[F], 2])), x] /; FreeQ[{F, a, b, c, d}, x] && NegQ[b]

Rule 3389

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Dist[I/2, Int[(c + d*x)^m/E^(I*(e + f*x))
, x], x] - Dist[I/2, Int[(c + d*x)^m*E^(I*(e + f*x)), x], x] /; FreeQ[{c, d, e, f, m}, x]

Rule 5556

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 5777

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*((a + b*ArcSinh[c*x])^n/(
m + 1)), x] - Dist[b*c*(n/(m + 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] && GtQ[n, 0]

Rule 5780

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

Rule 5783

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(1/(b*c*(n + 1)))*S
imp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]]*(a + b*ArcSinh[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ
[e, c^2*d] && NeQ[n, -1]

Rule 5812

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[
f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*((a + b*ArcSinh[c*x])^n/(e*(m + 2*p + 1))), x] + (-Dist[f^2*((m - 1)/(c^2*
(m + 2*p + 1))), Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcSinh[c*x])^n, x], x] - Dist[b*f*(n/(c*(m + 2*p + 1)
))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p], Int[(f*x)^(m - 1)*(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1)
, x], x]) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && IGtQ[m, 1] && NeQ[m + 2*p + 1, 0
]

Rubi steps

\begin {align*} \int x^3 \sinh ^{-1}(a x)^{3/2} \, dx &=\frac {1}{4} x^4 \sinh ^{-1}(a x)^{3/2}-\frac {1}{8} (3 a) \int \frac {x^4 \sqrt {\sinh ^{-1}(a x)}}{\sqrt {1+a^2 x^2}} \, dx\\ &=-\frac {3 x^3 \sqrt {1+a^2 x^2} \sqrt {\sinh ^{-1}(a x)}}{32 a}+\frac {1}{4} x^4 \sinh ^{-1}(a x)^{3/2}+\frac {3}{64} \int \frac {x^3}{\sqrt {\sinh ^{-1}(a x)}} \, dx+\frac {9 \int \frac {x^2 \sqrt {\sinh ^{-1}(a x)}}{\sqrt {1+a^2 x^2}} \, dx}{32 a}\\ &=\frac {9 x \sqrt {1+a^2 x^2} \sqrt {\sinh ^{-1}(a x)}}{64 a^3}-\frac {3 x^3 \sqrt {1+a^2 x^2} \sqrt {\sinh ^{-1}(a x)}}{32 a}+\frac {1}{4} x^4 \sinh ^{-1}(a x)^{3/2}+\frac {3 \text {Subst}\left (\int \frac {\cosh (x) \sinh ^3(x)}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{64 a^4}-\frac {9 \int \frac {\sqrt {\sinh ^{-1}(a x)}}{\sqrt {1+a^2 x^2}} \, dx}{64 a^3}-\frac {9 \int \frac {x}{\sqrt {\sinh ^{-1}(a x)}} \, dx}{128 a^2}\\ &=\frac {9 x \sqrt {1+a^2 x^2} \sqrt {\sinh ^{-1}(a x)}}{64 a^3}-\frac {3 x^3 \sqrt {1+a^2 x^2} \sqrt {\sinh ^{-1}(a x)}}{32 a}-\frac {3 \sinh ^{-1}(a x)^{3/2}}{32 a^4}+\frac {1}{4} x^4 \sinh ^{-1}(a x)^{3/2}+\frac {3 \text {Subst}\left (\int \left (-\frac {\sinh (2 x)}{4 \sqrt {x}}+\frac {\sinh (4 x)}{8 \sqrt {x}}\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{64 a^4}-\frac {9 \text {Subst}\left (\int \frac {\cosh (x) \sinh (x)}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{128 a^4}\\ &=\frac {9 x \sqrt {1+a^2 x^2} \sqrt {\sinh ^{-1}(a x)}}{64 a^3}-\frac {3 x^3 \sqrt {1+a^2 x^2} \sqrt {\sinh ^{-1}(a x)}}{32 a}-\frac {3 \sinh ^{-1}(a x)^{3/2}}{32 a^4}+\frac {1}{4} x^4 \sinh ^{-1}(a x)^{3/2}+\frac {3 \text {Subst}\left (\int \frac {\sinh (4 x)}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{512 a^4}-\frac {3 \text {Subst}\left (\int \frac {\sinh (2 x)}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{256 a^4}-\frac {9 \text {Subst}\left (\int \frac {\sinh (2 x)}{2 \sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{128 a^4}\\ &=\frac {9 x \sqrt {1+a^2 x^2} \sqrt {\sinh ^{-1}(a x)}}{64 a^3}-\frac {3 x^3 \sqrt {1+a^2 x^2} \sqrt {\sinh ^{-1}(a x)}}{32 a}-\frac {3 \sinh ^{-1}(a x)^{3/2}}{32 a^4}+\frac {1}{4} x^4 \sinh ^{-1}(a x)^{3/2}-\frac {3 \text {Subst}\left (\int \frac {e^{-4 x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{1024 a^4}+\frac {3 \text {Subst}\left (\int \frac {e^{4 x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{1024 a^4}+\frac {3 \text {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{512 a^4}-\frac {3 \text {Subst}\left (\int \frac {e^{2 x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{512 a^4}-\frac {9 \text {Subst}\left (\int \frac {\sinh (2 x)}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{256 a^4}\\ &=\frac {9 x \sqrt {1+a^2 x^2} \sqrt {\sinh ^{-1}(a x)}}{64 a^3}-\frac {3 x^3 \sqrt {1+a^2 x^2} \sqrt {\sinh ^{-1}(a x)}}{32 a}-\frac {3 \sinh ^{-1}(a x)^{3/2}}{32 a^4}+\frac {1}{4} x^4 \sinh ^{-1}(a x)^{3/2}-\frac {3 \text {Subst}\left (\int e^{-4 x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{512 a^4}+\frac {3 \text {Subst}\left (\int e^{4 x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{512 a^4}+\frac {3 \text {Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{256 a^4}-\frac {3 \text {Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{256 a^4}+\frac {9 \text {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{512 a^4}-\frac {9 \text {Subst}\left (\int \frac {e^{2 x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{512 a^4}\\ &=\frac {9 x \sqrt {1+a^2 x^2} \sqrt {\sinh ^{-1}(a x)}}{64 a^3}-\frac {3 x^3 \sqrt {1+a^2 x^2} \sqrt {\sinh ^{-1}(a x)}}{32 a}-\frac {3 \sinh ^{-1}(a x)^{3/2}}{32 a^4}+\frac {1}{4} x^4 \sinh ^{-1}(a x)^{3/2}-\frac {3 \sqrt {\pi } \text {erf}\left (2 \sqrt {\sinh ^{-1}(a x)}\right )}{2048 a^4}+\frac {3 \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{512 a^4}+\frac {3 \sqrt {\pi } \text {erfi}\left (2 \sqrt {\sinh ^{-1}(a x)}\right )}{2048 a^4}-\frac {3 \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{512 a^4}+\frac {9 \text {Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{256 a^4}-\frac {9 \text {Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{256 a^4}\\ &=\frac {9 x \sqrt {1+a^2 x^2} \sqrt {\sinh ^{-1}(a x)}}{64 a^3}-\frac {3 x^3 \sqrt {1+a^2 x^2} \sqrt {\sinh ^{-1}(a x)}}{32 a}-\frac {3 \sinh ^{-1}(a x)^{3/2}}{32 a^4}+\frac {1}{4} x^4 \sinh ^{-1}(a x)^{3/2}-\frac {3 \sqrt {\pi } \text {erf}\left (2 \sqrt {\sinh ^{-1}(a x)}\right )}{2048 a^4}+\frac {3 \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{128 a^4}+\frac {3 \sqrt {\pi } \text {erfi}\left (2 \sqrt {\sinh ^{-1}(a x)}\right )}{2048 a^4}-\frac {3 \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{128 a^4}\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 102, normalized size = 0.51 \begin {gather*} \frac {-\sqrt {\sinh ^{-1}(a x)} \Gamma \left (\frac {5}{2},-4 \sinh ^{-1}(a x)\right )+8 \sqrt {2} \sqrt {\sinh ^{-1}(a x)} \Gamma \left (\frac {5}{2},-2 \sinh ^{-1}(a x)\right )+\sqrt {-\sinh ^{-1}(a x)} \left (-8 \sqrt {2} \Gamma \left (\frac {5}{2},2 \sinh ^{-1}(a x)\right )+\Gamma \left (\frac {5}{2},4 \sinh ^{-1}(a x)\right )\right )}{512 a^4 \sqrt {-\sinh ^{-1}(a x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^3*ArcSinh[a*x]^(3/2),x]

[Out]

(-(Sqrt[ArcSinh[a*x]]*Gamma[5/2, -4*ArcSinh[a*x]]) + 8*Sqrt[2]*Sqrt[ArcSinh[a*x]]*Gamma[5/2, -2*ArcSinh[a*x]]
+ Sqrt[-ArcSinh[a*x]]*(-8*Sqrt[2]*Gamma[5/2, 2*ArcSinh[a*x]] + Gamma[5/2, 4*ArcSinh[a*x]]))/(512*a^4*Sqrt[-Arc
Sinh[a*x]])

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Maple [F]
time = 5.14, size = 0, normalized size = 0.00 \[\int x^{3} \arcsinh \left (a x \right )^{\frac {3}{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*arcsinh(a*x)^(3/2),x)

[Out]

int(x^3*arcsinh(a*x)^(3/2),x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*arcsinh(a*x)^(3/2),x, algorithm="maxima")

[Out]

integrate(x^3*arcsinh(a*x)^(3/2), x)

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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*arcsinh(a*x)^(3/2),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{3} \operatorname {asinh}^{\frac {3}{2}}{\left (a x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*asinh(a*x)**(3/2),x)

[Out]

Integral(x**3*asinh(a*x)**(3/2), x)

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*arcsinh(a*x)^(3/2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x^3\,{\mathrm {asinh}\left (a\,x\right )}^{3/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*asinh(a*x)^(3/2),x)

[Out]

int(x^3*asinh(a*x)^(3/2), x)

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