3.141 \(\int (a+b \sinh ^{-1}(c x))^{3/2} \, dx\)

Optimal. Leaf size=135 \[ \frac{3 \sqrt{\pi } b^{3/2} e^{a/b} \text{Erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{8 c}+\frac{3 \sqrt{\pi } b^{3/2} e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{8 c}-\frac{3 b \sqrt{c^2 x^2+1} \sqrt{a+b \sinh ^{-1}(c x)}}{2 c}+x \left (a+b \sinh ^{-1}(c x)\right )^{3/2} \]

[Out]

(-3*b*Sqrt[1 + c^2*x^2]*Sqrt[a + b*ArcSinh[c*x]])/(2*c) + x*(a + b*ArcSinh[c*x])^(3/2) + (3*b^(3/2)*E^(a/b)*Sq
rt[Pi]*Erf[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(8*c) + (3*b^(3/2)*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b
]])/(8*c*E^(a/b))

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Rubi [A]  time = 0.254633, antiderivative size = 135, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.583, Rules used = {5653, 5717, 5657, 3307, 2180, 2205, 2204} \[ \frac{3 \sqrt{\pi } b^{3/2} e^{a/b} \text{Erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{8 c}+\frac{3 \sqrt{\pi } b^{3/2} e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{8 c}-\frac{3 b \sqrt{c^2 x^2+1} \sqrt{a+b \sinh ^{-1}(c x)}}{2 c}+x \left (a+b \sinh ^{-1}(c x)\right )^{3/2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcSinh[c*x])^(3/2),x]

[Out]

(-3*b*Sqrt[1 + c^2*x^2]*Sqrt[a + b*ArcSinh[c*x]])/(2*c) + x*(a + b*ArcSinh[c*x])^(3/2) + (3*b^(3/2)*E^(a/b)*Sq
rt[Pi]*Erf[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(8*c) + (3*b^(3/2)*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b
]])/(8*c*E^(a/b))

Rule 5653

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

Rule 5717

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

Rule 5657

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

Rule 3307

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 2180

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] &&  !$UseGamma === True

Rule 2205

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 2204

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]

Rubi steps

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

Mathematica [A]  time = 1.03352, size = 251, normalized size = 1.86 \[ \frac{a e^{-\frac{a}{b}} \sqrt{a+b \sinh ^{-1}(c x)} \left (\frac{\text{Gamma}\left (\frac{3}{2},-\frac{a+b \sinh ^{-1}(c x)}{b}\right )}{\sqrt{-\frac{a+b \sinh ^{-1}(c x)}{b}}}-\frac{e^{\frac{2 a}{b}} \text{Gamma}\left (\frac{3}{2},\frac{a}{b}+\sinh ^{-1}(c x)\right )}{\sqrt{\frac{a}{b}+\sinh ^{-1}(c x)}}\right )}{2 c}+\frac{\sqrt{b} \left (4 \sqrt{b} \left (2 c x \sinh ^{-1}(c x)-3 \sqrt{c^2 x^2+1}\right ) \sqrt{a+b \sinh ^{-1}(c x)}+\sqrt{\pi } (3 b-2 a) \left (\sinh \left (\frac{a}{b}\right )+\cosh \left (\frac{a}{b}\right )\right ) \text{Erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )+\sqrt{\pi } (2 a+3 b) \left (\cosh \left (\frac{a}{b}\right )-\sinh \left (\frac{a}{b}\right )\right ) \text{Erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )\right )}{8 c} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(a*Sqrt[a + b*ArcSinh[c*x]]*(-((E^((2*a)/b)*Gamma[3/2, a/b + ArcSinh[c*x]])/Sqrt[a/b + ArcSinh[c*x]]) + Gamma[
3/2, -((a + b*ArcSinh[c*x])/b)]/Sqrt[-((a + b*ArcSinh[c*x])/b)]))/(2*c*E^(a/b)) + (Sqrt[b]*(4*Sqrt[b]*Sqrt[a +
b*ArcSinh[c*x]]*(-3*Sqrt[1 + c^2*x^2] + 2*c*x*ArcSinh[c*x]) + (2*a + 3*b)*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c*
x]]/Sqrt[b]]*(Cosh[a/b] - Sinh[a/b]) + (-2*a + 3*b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]]*(Cosh[a/b]
+ Sinh[a/b])))/(8*c)

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Maple [F]  time = 0.037, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b{\it Arcsinh} \left ( cx \right ) \right ) ^{{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \operatorname{arsinh}\left (c x\right ) + a\right )}^{\frac{3}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: UnboundLocalError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a + b*asinh(c*x))**(3/2), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \operatorname{arsinh}\left (c x\right ) + a\right )}^{\frac{3}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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