3.5.91 \(\int \sqrt {a^2+x^2} \sinh ^{-1}(\frac {x}{a})^{3/2} \, dx\) [491]

Optimal. Leaf size=259 \[ -\frac {3 a \sqrt {a^2+x^2} \sqrt {\sinh ^{-1}\left (\frac {x}{a}\right )}}{16 \sqrt {1+\frac {x^2}{a^2}}}-\frac {3 x^2 \sqrt {a^2+x^2} \sqrt {\sinh ^{-1}\left (\frac {x}{a}\right )}}{8 a \sqrt {1+\frac {x^2}{a^2}}}+\frac {1}{2} x \sqrt {a^2+x^2} \sinh ^{-1}\left (\frac {x}{a}\right )^{3/2}+\frac {a \sqrt {a^2+x^2} \sinh ^{-1}\left (\frac {x}{a}\right )^{5/2}}{5 \sqrt {1+\frac {x^2}{a^2}}}+\frac {3 a \sqrt {\frac {\pi }{2}} \sqrt {a^2+x^2} \text {Erf}\left (\sqrt {2} \sqrt {\sinh ^{-1}\left (\frac {x}{a}\right )}\right )}{64 \sqrt {1+\frac {x^2}{a^2}}}+\frac {3 a \sqrt {\frac {\pi }{2}} \sqrt {a^2+x^2} \text {Erfi}\left (\sqrt {2} \sqrt {\sinh ^{-1}\left (\frac {x}{a}\right )}\right )}{64 \sqrt {1+\frac {x^2}{a^2}}} \]

[Out]

1/2*x*arcsinh(x/a)^(3/2)*(a^2+x^2)^(1/2)+1/5*a*arcsinh(x/a)^(5/2)*(a^2+x^2)^(1/2)/(1+x^2/a^2)^(1/2)+3/128*a*er
f(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)+3/128*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)-3/16*a*(a^2+x^2)^(1/2)*arcsinh(x/a)^(1/2)/(1+x^
2/a^2)^(1/2)-3/8*x^2*(a^2+x^2)^(1/2)*arcsinh(x/a)^(1/2)/a/(1+x^2/a^2)^(1/2)

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Rubi [A]
time = 0.21, antiderivative size = 259, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {5785, 5783, 5777, 5819, 3393, 3388, 2211, 2235, 2236} \begin {gather*} \frac {3 \sqrt {\frac {\pi }{2}} a \sqrt {a^2+x^2} \text {Erf}\left (\sqrt {2} \sqrt {\sinh ^{-1}\left (\frac {x}{a}\right )}\right )}{64 \sqrt {\frac {x^2}{a^2}+1}}+\frac {3 \sqrt {\frac {\pi }{2}} a \sqrt {a^2+x^2} \text {Erfi}\left (\sqrt {2} \sqrt {\sinh ^{-1}\left (\frac {x}{a}\right )}\right )}{64 \sqrt {\frac {x^2}{a^2}+1}}+\frac {a \sqrt {a^2+x^2} \sinh ^{-1}\left (\frac {x}{a}\right )^{5/2}}{5 \sqrt {\frac {x^2}{a^2}+1}}+\frac {1}{2} x \sqrt {a^2+x^2} \sinh ^{-1}\left (\frac {x}{a}\right )^{3/2}-\frac {3 x^2 \sqrt {a^2+x^2} \sqrt {\sinh ^{-1}\left (\frac {x}{a}\right )}}{8 a \sqrt {\frac {x^2}{a^2}+1}}-\frac {3 a \sqrt {a^2+x^2} \sqrt {\sinh ^{-1}\left (\frac {x}{a}\right )}}{16 \sqrt {\frac {x^2}{a^2}+1}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*a*Sqrt[a^2 + x^2]*Sqrt[ArcSinh[x/a]])/(16*Sqrt[1 + x^2/a^2]) - (3*x^2*Sqrt[a^2 + x^2]*Sqrt[ArcSinh[x/a]])/
(8*a*Sqrt[1 + x^2/a^2]) + (x*Sqrt[a^2 + x^2]*ArcSinh[x/a]^(3/2))/2 + (a*Sqrt[a^2 + x^2]*ArcSinh[x/a]^(5/2))/(5
*Sqrt[1 + x^2/a^2]) + (3*a*Sqrt[Pi/2]*Sqrt[a^2 + x^2]*Erf[Sqrt[2]*Sqrt[ArcSinh[x/a]]])/(64*Sqrt[1 + x^2/a^2])
+ (3*a*Sqrt[Pi/2]*Sqrt[a^2 + x^2]*Erfi[Sqrt[2]*Sqrt[ArcSinh[x/a]]])/(64*Sqrt[1 + x^2/a^2])

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 3393

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin
[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1])
)

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 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]

Rule 5819

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

Rubi steps

\begin {align*} \int \sqrt {a^2+x^2} \sinh ^{-1}\left (\frac {x}{a}\right )^{3/2} \, dx &=\frac {1}{2} x \sqrt {a^2+x^2} \sinh ^{-1}\left (\frac {x}{a}\right )^{3/2}+\frac {\sqrt {a^2+x^2} \int \frac {\sinh ^{-1}\left (\frac {x}{a}\right )^{3/2}}{\sqrt {1+\frac {x^2}{a^2}}} \, dx}{2 \sqrt {1+\frac {x^2}{a^2}}}-\frac {\left (3 \sqrt {a^2+x^2}\right ) \int x \sqrt {\sinh ^{-1}\left (\frac {x}{a}\right )} \, dx}{4 a \sqrt {1+\frac {x^2}{a^2}}}\\ &=-\frac {3 x^2 \sqrt {a^2+x^2} \sqrt {\sinh ^{-1}\left (\frac {x}{a}\right )}}{8 a \sqrt {1+\frac {x^2}{a^2}}}+\frac {1}{2} x \sqrt {a^2+x^2} \sinh ^{-1}\left (\frac {x}{a}\right )^{3/2}+\frac {a \sqrt {a^2+x^2} \sinh ^{-1}\left (\frac {x}{a}\right )^{5/2}}{5 \sqrt {1+\frac {x^2}{a^2}}}+\frac {\left (3 \sqrt {a^2+x^2}\right ) \int \frac {x^2}{\sqrt {1+\frac {x^2}{a^2}} \sqrt {\sinh ^{-1}\left (\frac {x}{a}\right )}} \, dx}{16 a^2 \sqrt {1+\frac {x^2}{a^2}}}\\ &=-\frac {3 x^2 \sqrt {a^2+x^2} \sqrt {\sinh ^{-1}\left (\frac {x}{a}\right )}}{8 a \sqrt {1+\frac {x^2}{a^2}}}+\frac {1}{2} x \sqrt {a^2+x^2} \sinh ^{-1}\left (\frac {x}{a}\right )^{3/2}+\frac {a \sqrt {a^2+x^2} \sinh ^{-1}\left (\frac {x}{a}\right )^{5/2}}{5 \sqrt {1+\frac {x^2}{a^2}}}+\frac {\left (3 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 )}{16 \sqrt {1+\frac {x^2}{a^2}}}\\ &=-\frac {3 x^2 \sqrt {a^2+x^2} \sqrt {\sinh ^{-1}\left (\frac {x}{a}\right )}}{8 a \sqrt {1+\frac {x^2}{a^2}}}+\frac {1}{2} x \sqrt {a^2+x^2} \sinh ^{-1}\left (\frac {x}{a}\right )^{3/2}+\frac {a \sqrt {a^2+x^2} \sinh ^{-1}\left (\frac {x}{a}\right )^{5/2}}{5 \sqrt {1+\frac {x^2}{a^2}}}-\frac {\left (3 a \sqrt {a^2+x^2}\right ) \text {Subst}\left (\int \left (\frac {1}{2 \sqrt {x}}-\frac {\cosh (2 x)}{2 \sqrt {x}}\right ) \, dx,x,\sinh ^{-1}\left (\frac {x}{a}\right )\right )}{16 \sqrt {1+\frac {x^2}{a^2}}}\\ &=-\frac {3 a \sqrt {a^2+x^2} \sqrt {\sinh ^{-1}\left (\frac {x}{a}\right )}}{16 \sqrt {1+\frac {x^2}{a^2}}}-\frac {3 x^2 \sqrt {a^2+x^2} \sqrt {\sinh ^{-1}\left (\frac {x}{a}\right )}}{8 a \sqrt {1+\frac {x^2}{a^2}}}+\frac {1}{2} x \sqrt {a^2+x^2} \sinh ^{-1}\left (\frac {x}{a}\right )^{3/2}+\frac {a \sqrt {a^2+x^2} \sinh ^{-1}\left (\frac {x}{a}\right )^{5/2}}{5 \sqrt {1+\frac {x^2}{a^2}}}+\frac {\left (3 a \sqrt {a^2+x^2}\right ) \text {Subst}\left (\int \frac {\cosh (2 x)}{\sqrt {x}} \, dx,x,\sinh ^{-1}\left (\frac {x}{a}\right )\right )}{32 \sqrt {1+\frac {x^2}{a^2}}}\\ &=-\frac {3 a \sqrt {a^2+x^2} \sqrt {\sinh ^{-1}\left (\frac {x}{a}\right )}}{16 \sqrt {1+\frac {x^2}{a^2}}}-\frac {3 x^2 \sqrt {a^2+x^2} \sqrt {\sinh ^{-1}\left (\frac {x}{a}\right )}}{8 a \sqrt {1+\frac {x^2}{a^2}}}+\frac {1}{2} x \sqrt {a^2+x^2} \sinh ^{-1}\left (\frac {x}{a}\right )^{3/2}+\frac {a \sqrt {a^2+x^2} \sinh ^{-1}\left (\frac {x}{a}\right )^{5/2}}{5 \sqrt {1+\frac {x^2}{a^2}}}+\frac {\left (3 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 )}{64 \sqrt {1+\frac {x^2}{a^2}}}+\frac {\left (3 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 )}{64 \sqrt {1+\frac {x^2}{a^2}}}\\ &=-\frac {3 a \sqrt {a^2+x^2} \sqrt {\sinh ^{-1}\left (\frac {x}{a}\right )}}{16 \sqrt {1+\frac {x^2}{a^2}}}-\frac {3 x^2 \sqrt {a^2+x^2} \sqrt {\sinh ^{-1}\left (\frac {x}{a}\right )}}{8 a \sqrt {1+\frac {x^2}{a^2}}}+\frac {1}{2} x \sqrt {a^2+x^2} \sinh ^{-1}\left (\frac {x}{a}\right )^{3/2}+\frac {a \sqrt {a^2+x^2} \sinh ^{-1}\left (\frac {x}{a}\right )^{5/2}}{5 \sqrt {1+\frac {x^2}{a^2}}}+\frac {\left (3 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 )}{32 \sqrt {1+\frac {x^2}{a^2}}}+\frac {\left (3 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 )}{32 \sqrt {1+\frac {x^2}{a^2}}}\\ &=-\frac {3 a \sqrt {a^2+x^2} \sqrt {\sinh ^{-1}\left (\frac {x}{a}\right )}}{16 \sqrt {1+\frac {x^2}{a^2}}}-\frac {3 x^2 \sqrt {a^2+x^2} \sqrt {\sinh ^{-1}\left (\frac {x}{a}\right )}}{8 a \sqrt {1+\frac {x^2}{a^2}}}+\frac {1}{2} x \sqrt {a^2+x^2} \sinh ^{-1}\left (\frac {x}{a}\right )^{3/2}+\frac {a \sqrt {a^2+x^2} \sinh ^{-1}\left (\frac {x}{a}\right )^{5/2}}{5 \sqrt {1+\frac {x^2}{a^2}}}+\frac {3 a \sqrt {\frac {\pi }{2}} \sqrt {a^2+x^2} \text {erf}\left (\sqrt {2} \sqrt {\sinh ^{-1}\left (\frac {x}{a}\right )}\right )}{64 \sqrt {1+\frac {x^2}{a^2}}}+\frac {3 a \sqrt {\frac {\pi }{2}} \sqrt {a^2+x^2} \text {erfi}\left (\sqrt {2} \sqrt {\sinh ^{-1}\left (\frac {x}{a}\right )}\right )}{64 \sqrt {1+\frac {x^2}{a^2}}}\\ \end {align*}

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Mathematica [A]
time = 0.17, size = 133, normalized size = 0.51 \begin {gather*} \frac {a \sqrt {a^2+x^2} \left (15 \sqrt {2 \pi } \text {Erf}\left (\sqrt {2} \sqrt {\sinh ^{-1}\left (\frac {x}{a}\right )}\right )+15 \sqrt {2 \pi } \text {Erfi}\left (\sqrt {2} \sqrt {\sinh ^{-1}\left (\frac {x}{a}\right )}\right )+8 \sqrt {\sinh ^{-1}\left (\frac {x}{a}\right )} \left (16 \sinh ^{-1}\left (\frac {x}{a}\right )^2-15 \cosh \left (2 \sinh ^{-1}\left (\frac {x}{a}\right )\right )+20 \sinh ^{-1}\left (\frac {x}{a}\right ) \sinh \left (2 \sinh ^{-1}\left (\frac {x}{a}\right )\right )\right )\right )}{640 \sqrt {1+\frac {x^2}{a^2}}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a*Sqrt[a^2 + x^2]*(15*Sqrt[2*Pi]*Erf[Sqrt[2]*Sqrt[ArcSinh[x/a]]] + 15*Sqrt[2*Pi]*Erfi[Sqrt[2]*Sqrt[ArcSinh[x/
a]]] + 8*Sqrt[ArcSinh[x/a]]*(16*ArcSinh[x/a]^2 - 15*Cosh[2*ArcSinh[x/a]] + 20*ArcSinh[x/a]*Sinh[2*ArcSinh[x/a]
])))/(640*Sqrt[1 + x^2/a^2])

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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \arcsinh \left (\frac {x}{a}\right )^{\frac {3}{2}} \sqrt {a^{2}+x^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arcsinh(x/a)^(3/2)*(a^2+x^2)^(1/2),x)

[Out]

int(arcsinh(x/a)^(3/2)*(a^2+x^2)^(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(arcsinh(x/a)^(3/2)*(a^2+x^2)^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(a^2 + x^2)*arcsinh(x/a)^(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(arcsinh(x/a)^(3/2)*(a^2+x^2)^(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}} \operatorname {asinh}^{\frac {3}{2}}{\left (\frac {x}{a} \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(asinh(x/a)**(3/2)*(a**2+x**2)**(1/2),x)

[Out]

Integral(sqrt(a**2 + x**2)*asinh(x/a)**(3/2), 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(arcsinh(x/a)^(3/2)*(a^2+x^2)^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(a^2 + x^2)*arcsinh(x/a)^(3/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(asinh(x/a)^(3/2)*(a^2 + x^2)^(1/2),x)

[Out]

int(asinh(x/a)^(3/2)*(a^2 + x^2)^(1/2), x)

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