3.1.92 \(\int \frac {x^4}{\sqrt {\sinh ^{-1}(a x)}} \, dx\) [92]

Optimal. Leaf size=163 \[ \frac {\sqrt {\pi } \text {Erf}\left (\sqrt {\sinh ^{-1}(a x)}\right )}{16 a^5}-\frac {\sqrt {3 \pi } \text {Erf}\left (\sqrt {3} \sqrt {\sinh ^{-1}(a x)}\right )}{32 a^5}+\frac {\sqrt {\frac {\pi }{5}} \text {Erf}\left (\sqrt {5} \sqrt {\sinh ^{-1}(a x)}\right )}{32 a^5}+\frac {\sqrt {\pi } \text {Erfi}\left (\sqrt {\sinh ^{-1}(a x)}\right )}{16 a^5}-\frac {\sqrt {3 \pi } \text {Erfi}\left (\sqrt {3} \sqrt {\sinh ^{-1}(a x)}\right )}{32 a^5}+\frac {\sqrt {\frac {\pi }{5}} \text {Erfi}\left (\sqrt {5} \sqrt {\sinh ^{-1}(a x)}\right )}{32 a^5} \]

[Out]

1/160*erf(5^(1/2)*arcsinh(a*x)^(1/2))*5^(1/2)*Pi^(1/2)/a^5+1/160*erfi(5^(1/2)*arcsinh(a*x)^(1/2))*5^(1/2)*Pi^(
1/2)/a^5+1/16*erf(arcsinh(a*x)^(1/2))*Pi^(1/2)/a^5+1/16*erfi(arcsinh(a*x)^(1/2))*Pi^(1/2)/a^5-1/32*erf(3^(1/2)
*arcsinh(a*x)^(1/2))*3^(1/2)*Pi^(1/2)/a^5-1/32*erfi(3^(1/2)*arcsinh(a*x)^(1/2))*3^(1/2)*Pi^(1/2)/a^5

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Rubi [A]
time = 0.14, antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 6, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5780, 5556, 3388, 2211, 2235, 2236} \begin {gather*} \frac {\sqrt {\pi } \text {Erf}\left (\sqrt {\sinh ^{-1}(a x)}\right )}{16 a^5}-\frac {\sqrt {3 \pi } \text {Erf}\left (\sqrt {3} \sqrt {\sinh ^{-1}(a x)}\right )}{32 a^5}+\frac {\sqrt {\frac {\pi }{5}} \text {Erf}\left (\sqrt {5} \sqrt {\sinh ^{-1}(a x)}\right )}{32 a^5}+\frac {\sqrt {\pi } \text {Erfi}\left (\sqrt {\sinh ^{-1}(a x)}\right )}{16 a^5}-\frac {\sqrt {3 \pi } \text {Erfi}\left (\sqrt {3} \sqrt {\sinh ^{-1}(a x)}\right )}{32 a^5}+\frac {\sqrt {\frac {\pi }{5}} \text {Erfi}\left (\sqrt {5} \sqrt {\sinh ^{-1}(a x)}\right )}{32 a^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[Pi]*Erf[Sqrt[ArcSinh[a*x]]])/(16*a^5) - (Sqrt[3*Pi]*Erf[Sqrt[3]*Sqrt[ArcSinh[a*x]]])/(32*a^5) + (Sqrt[Pi
/5]*Erf[Sqrt[5]*Sqrt[ArcSinh[a*x]]])/(32*a^5) + (Sqrt[Pi]*Erfi[Sqrt[ArcSinh[a*x]]])/(16*a^5) - (Sqrt[3*Pi]*Erf
i[Sqrt[3]*Sqrt[ArcSinh[a*x]]])/(32*a^5) + (Sqrt[Pi/5]*Erfi[Sqrt[5]*Sqrt[ArcSinh[a*x]]])/(32*a^5)

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 3388

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

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 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]

Rubi steps

\begin {align*} \int \frac {x^4}{\sqrt {\sinh ^{-1}(a x)}} \, dx &=\frac {\text {Subst}\left (\int \frac {\cosh (x) \sinh ^4(x)}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{a^5}\\ &=\frac {\text {Subst}\left (\int \left (\frac {\cosh (x)}{8 \sqrt {x}}-\frac {3 \cosh (3 x)}{16 \sqrt {x}}+\frac {\cosh (5 x)}{16 \sqrt {x}}\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{a^5}\\ &=\frac {\text {Subst}\left (\int \frac {\cosh (5 x)}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{16 a^5}+\frac {\text {Subst}\left (\int \frac {\cosh (x)}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{8 a^5}-\frac {3 \text {Subst}\left (\int \frac {\cosh (3 x)}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{16 a^5}\\ &=\frac {\text {Subst}\left (\int \frac {e^{-5 x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{32 a^5}+\frac {\text {Subst}\left (\int \frac {e^{5 x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{32 a^5}+\frac {\text {Subst}\left (\int \frac {e^{-x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{16 a^5}+\frac {\text {Subst}\left (\int \frac {e^x}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{16 a^5}-\frac {3 \text {Subst}\left (\int \frac {e^{-3 x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{32 a^5}-\frac {3 \text {Subst}\left (\int \frac {e^{3 x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{32 a^5}\\ &=\frac {\text {Subst}\left (\int e^{-5 x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{16 a^5}+\frac {\text {Subst}\left (\int e^{5 x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{16 a^5}+\frac {\text {Subst}\left (\int e^{-x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{8 a^5}+\frac {\text {Subst}\left (\int e^{x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{8 a^5}-\frac {3 \text {Subst}\left (\int e^{-3 x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{16 a^5}-\frac {3 \text {Subst}\left (\int e^{3 x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{16 a^5}\\ &=\frac {\sqrt {\pi } \text {erf}\left (\sqrt {\sinh ^{-1}(a x)}\right )}{16 a^5}-\frac {\sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {\sinh ^{-1}(a x)}\right )}{32 a^5}+\frac {\sqrt {\frac {\pi }{5}} \text {erf}\left (\sqrt {5} \sqrt {\sinh ^{-1}(a x)}\right )}{32 a^5}+\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {\sinh ^{-1}(a x)}\right )}{16 a^5}-\frac {\sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {\sinh ^{-1}(a x)}\right )}{32 a^5}+\frac {\sqrt {\frac {\pi }{5}} \text {erfi}\left (\sqrt {5} \sqrt {\sinh ^{-1}(a x)}\right )}{32 a^5}\\ \end {align*}

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Mathematica [A]
time = 0.07, size = 151, normalized size = 0.93 \begin {gather*} \frac {\frac {\sqrt {5} \sqrt {-\sinh ^{-1}(a x)} \Gamma \left (\frac {1}{2},-5 \sinh ^{-1}(a x)\right )}{\sqrt {\sinh ^{-1}(a x)}}+\frac {5 \sqrt {3} \sqrt {\sinh ^{-1}(a x)} \Gamma \left (\frac {1}{2},-3 \sinh ^{-1}(a x)\right )}{\sqrt {-\sinh ^{-1}(a x)}}+\frac {10 \sqrt {-\sinh ^{-1}(a x)} \Gamma \left (\frac {1}{2},-\sinh ^{-1}(a x)\right )}{\sqrt {\sinh ^{-1}(a x)}}-10 \Gamma \left (\frac {1}{2},\sinh ^{-1}(a x)\right )+5 \sqrt {3} \Gamma \left (\frac {1}{2},3 \sinh ^{-1}(a x)\right )-\sqrt {5} \Gamma \left (\frac {1}{2},5 \sinh ^{-1}(a x)\right )}{160 a^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((Sqrt[5]*Sqrt[-ArcSinh[a*x]]*Gamma[1/2, -5*ArcSinh[a*x]])/Sqrt[ArcSinh[a*x]] + (5*Sqrt[3]*Sqrt[ArcSinh[a*x]]*
Gamma[1/2, -3*ArcSinh[a*x]])/Sqrt[-ArcSinh[a*x]] + (10*Sqrt[-ArcSinh[a*x]]*Gamma[1/2, -ArcSinh[a*x]])/Sqrt[Arc
Sinh[a*x]] - 10*Gamma[1/2, ArcSinh[a*x]] + 5*Sqrt[3]*Gamma[1/2, 3*ArcSinh[a*x]] - Sqrt[5]*Gamma[1/2, 5*ArcSinh
[a*x]])/(160*a^5)

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Maple [F]
time = 7.84, size = 0, normalized size = 0.00 \[\int \frac {x^{4}}{\sqrt {\arcsinh \left (a x \right )}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/arcsinh(a*x)^(1/2),x)

[Out]

int(x^4/arcsinh(a*x)^(1/2),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^4/arcsinh(a*x)^(1/2),x, algorithm="maxima")

[Out]

integrate(x^4/sqrt(arcsinh(a*x)), 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^4/arcsinh(a*x)^(1/2),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 \frac {x^{4}}{\sqrt {\operatorname {asinh}{\left (a x \right )}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4/asinh(a*x)**(1/2),x)

[Out]

Integral(x**4/sqrt(asinh(a*x)), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/arcsinh(a*x)^(1/2),x, algorithm="giac")

[Out]

integrate(x^4/sqrt(arcsinh(a*x)), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^4}{\sqrt {\mathrm {asinh}\left (a\,x\right )}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/asinh(a*x)^(1/2),x)

[Out]

int(x^4/asinh(a*x)^(1/2), x)

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