3.5.86 \(\int \sqrt {a^2+x^2} \sqrt {\sinh ^{-1}(\frac {x}{a})} \, dx\) [486]

Optimal. Leaf size=176 \[ \frac {1}{2} x \sqrt {a^2+x^2} \sqrt {\sinh ^{-1}\left (\frac {x}{a}\right )}+\frac {a \sqrt {a^2+x^2} \sinh ^{-1}\left (\frac {x}{a}\right )^{3/2}}{3 \sqrt {1+\frac {x^2}{a^2}}}+\frac {a \sqrt {\frac {\pi }{2}} \sqrt {a^2+x^2} \text {Erf}\left (\sqrt {2} \sqrt {\sinh ^{-1}\left (\frac {x}{a}\right )}\right )}{16 \sqrt {1+\frac {x^2}{a^2}}}-\frac {a \sqrt {\frac {\pi }{2}} \sqrt {a^2+x^2} \text {Erfi}\left (\sqrt {2} \sqrt {\sinh ^{-1}\left (\frac {x}{a}\right )}\right )}{16 \sqrt {1+\frac {x^2}{a^2}}} \]

[Out]

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

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Rubi [A]
time = 0.10, antiderivative size = 176, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {5785, 5783, 5780, 5556, 12, 3389, 2211, 2235, 2236} \begin {gather*} \frac {\sqrt {\frac {\pi }{2}} a \sqrt {a^2+x^2} \text {Erf}\left (\sqrt {2} \sqrt {\sinh ^{-1}\left (\frac {x}{a}\right )}\right )}{16 \sqrt {\frac {x^2}{a^2}+1}}-\frac {\sqrt {\frac {\pi }{2}} a \sqrt {a^2+x^2} \text {Erfi}\left (\sqrt {2} \sqrt {\sinh ^{-1}\left (\frac {x}{a}\right )}\right )}{16 \sqrt {\frac {x^2}{a^2}+1}}+\frac {a \sqrt {a^2+x^2} \sinh ^{-1}\left (\frac {x}{a}\right )^{3/2}}{3 \sqrt {\frac {x^2}{a^2}+1}}+\frac {1}{2} x \sqrt {a^2+x^2} \sqrt {\sinh ^{-1}\left (\frac {x}{a}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[a^2 + x^2]*Sqrt[ArcSinh[x/a]],x]

[Out]

(x*Sqrt[a^2 + x^2]*Sqrt[ArcSinh[x/a]])/2 + (a*Sqrt[a^2 + x^2]*ArcSinh[x/a]^(3/2))/(3*Sqrt[1 + x^2/a^2]) + (a*S
qrt[Pi/2]*Sqrt[a^2 + x^2]*Erf[Sqrt[2]*Sqrt[ArcSinh[x/a]]])/(16*Sqrt[1 + x^2/a^2]) - (a*Sqrt[Pi/2]*Sqrt[a^2 + x
^2]*Erfi[Sqrt[2]*Sqrt[ArcSinh[x/a]]])/(16*Sqrt[1 + x^2/a^2])

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 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 5785

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

Rubi steps

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

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Mathematica [A]
time = 0.06, size = 110, normalized size = 0.62 \begin {gather*} \frac {a \sqrt {a^2+x^2} \left (16 \sinh ^{-1}\left (\frac {x}{a}\right )^2-3 \sqrt {2} \sqrt {-\sinh ^{-1}\left (\frac {x}{a}\right )} \Gamma \left (\frac {3}{2},-2 \sinh ^{-1}\left (\frac {x}{a}\right )\right )-3 \sqrt {2} \sqrt {\sinh ^{-1}\left (\frac {x}{a}\right )} \Gamma \left (\frac {3}{2},2 \sinh ^{-1}\left (\frac {x}{a}\right )\right )\right )}{48 \sqrt {1+\frac {x^2}{a^2}} \sqrt {\sinh ^{-1}\left (\frac {x}{a}\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a^2 + x^2]*Sqrt[ArcSinh[x/a]],x]

[Out]

(a*Sqrt[a^2 + x^2]*(16*ArcSinh[x/a]^2 - 3*Sqrt[2]*Sqrt[-ArcSinh[x/a]]*Gamma[3/2, -2*ArcSinh[x/a]] - 3*Sqrt[2]*
Sqrt[ArcSinh[x/a]]*Gamma[3/2, 2*ArcSinh[x/a]]))/(48*Sqrt[1 + x^2/a^2]*Sqrt[ArcSinh[x/a]])

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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \sqrt {a^{2}+x^{2}}\, \sqrt {\arcsinh \left (\frac {x}{a}\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(a**2 + x**2)*sqrt(asinh(x/a)), 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((a^2+x^2)^(1/2)*arcsinh(x/a)^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(a^2 + x^2)*sqrt(arcsinh(x/a)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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