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

Optimal. Leaf size=256 \[ -\frac{\sqrt{\pi } c \sqrt{a^2 c x^2+c} \text{Erf}\left (2 \sqrt{\sinh ^{-1}(a x)}\right )}{4 a \sqrt{a^2 x^2+1}}-\frac{\sqrt{\frac{\pi }{2}} c \sqrt{a^2 c x^2+c} \text{Erf}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )}{a \sqrt{a^2 x^2+1}}+\frac{\sqrt{\pi } c \sqrt{a^2 c x^2+c} \text{Erfi}\left (2 \sqrt{\sinh ^{-1}(a x)}\right )}{4 a \sqrt{a^2 x^2+1}}+\frac{\sqrt{\frac{\pi }{2}} c \sqrt{a^2 c x^2+c} \text{Erfi}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )}{a \sqrt{a^2 x^2+1}}-\frac{2 \sqrt{a^2 x^2+1} \left (a^2 c x^2+c\right )^{3/2}}{a \sqrt{\sinh ^{-1}(a x)}} \]

[Out]

(-2*Sqrt[1 + a^2*x^2]*(c + a^2*c*x^2)^(3/2))/(a*Sqrt[ArcSinh[a*x]]) - (c*Sqrt[Pi]*Sqrt[c + a^2*c*x^2]*Erf[2*Sq
rt[ArcSinh[a*x]]])/(4*a*Sqrt[1 + a^2*x^2]) - (c*Sqrt[Pi/2]*Sqrt[c + a^2*c*x^2]*Erf[Sqrt[2]*Sqrt[ArcSinh[a*x]]]
)/(a*Sqrt[1 + a^2*x^2]) + (c*Sqrt[Pi]*Sqrt[c + a^2*c*x^2]*Erfi[2*Sqrt[ArcSinh[a*x]]])/(4*a*Sqrt[1 + a^2*x^2])
+ (c*Sqrt[Pi/2]*Sqrt[c + a^2*c*x^2]*Erfi[Sqrt[2]*Sqrt[ArcSinh[a*x]]])/(a*Sqrt[1 + a^2*x^2])

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Rubi [A]  time = 0.208331, antiderivative size = 256, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 7, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.304, Rules used = {5696, 5779, 5448, 3308, 2180, 2204, 2205} \[ -\frac{\sqrt{\pi } c \sqrt{a^2 c x^2+c} \text{Erf}\left (2 \sqrt{\sinh ^{-1}(a x)}\right )}{4 a \sqrt{a^2 x^2+1}}-\frac{\sqrt{\frac{\pi }{2}} c \sqrt{a^2 c x^2+c} \text{Erf}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )}{a \sqrt{a^2 x^2+1}}+\frac{\sqrt{\pi } c \sqrt{a^2 c x^2+c} \text{Erfi}\left (2 \sqrt{\sinh ^{-1}(a x)}\right )}{4 a \sqrt{a^2 x^2+1}}+\frac{\sqrt{\frac{\pi }{2}} c \sqrt{a^2 c x^2+c} \text{Erfi}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )}{a \sqrt{a^2 x^2+1}}-\frac{2 \sqrt{a^2 x^2+1} \left (a^2 c x^2+c\right )^{3/2}}{a \sqrt{\sinh ^{-1}(a x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[1 + a^2*x^2]*(c + a^2*c*x^2)^(3/2))/(a*Sqrt[ArcSinh[a*x]]) - (c*Sqrt[Pi]*Sqrt[c + a^2*c*x^2]*Erf[2*Sq
rt[ArcSinh[a*x]]])/(4*a*Sqrt[1 + a^2*x^2]) - (c*Sqrt[Pi/2]*Sqrt[c + a^2*c*x^2]*Erf[Sqrt[2]*Sqrt[ArcSinh[a*x]]]
)/(a*Sqrt[1 + a^2*x^2]) + (c*Sqrt[Pi]*Sqrt[c + a^2*c*x^2]*Erfi[2*Sqrt[ArcSinh[a*x]]])/(4*a*Sqrt[1 + a^2*x^2])
+ (c*Sqrt[Pi/2]*Sqrt[c + a^2*c*x^2]*Erfi[Sqrt[2]*Sqrt[ArcSinh[a*x]]])/(a*Sqrt[1 + a^2*x^2])

Rule 5696

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(Sqrt[1 + c^2*x^2]
*(d + e*x^2)^p*(a + b*ArcSinh[c*x])^(n + 1))/(b*c*(n + 1)), x] - Dist[(c*(2*p + 1)*d^IntPart[p]*(d + e*x^2)^Fr
acPart[p])/(b*(n + 1)*(1 + c^2*x^2)^FracPart[p]), Int[x*(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] && LtQ[n, -1]

Rule 5779

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

Rule 5448

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 3308

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 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 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]

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]

Rubi steps

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

Mathematica [A]  time = 0.366563, size = 225, normalized size = 0.88 \[ -\frac{c \sqrt{a^2 c x^2+c} e^{-4 \sinh ^{-1}(a x)} \left (-2 e^{4 \sinh ^{-1}(a x)} \sqrt{-\sinh ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-4 \sinh ^{-1}(a x)\right )-2 e^{4 \sinh ^{-1}(a x)} \sqrt{\sinh ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},4 \sinh ^{-1}(a x)\right )+16 a^2 x^2 e^{4 \sinh ^{-1}(a x)}+4 \sqrt{2 \pi } e^{4 \sinh ^{-1}(a x)} \sqrt{\sinh ^{-1}(a x)} \text{Erf}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )-4 \sqrt{2 \pi } e^{4 \sinh ^{-1}(a x)} \sqrt{\sinh ^{-1}(a x)} \text{Erfi}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )+14 e^{4 \sinh ^{-1}(a x)}+e^{8 \sinh ^{-1}(a x)}+1\right )}{8 a \sqrt{a^2 x^2+1} \sqrt{\sinh ^{-1}(a x)}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(c*Sqrt[c + a^2*c*x^2]*(1 + 14*E^(4*ArcSinh[a*x]) + E^(8*ArcSinh[a*x]) + 16*a^2*E^(4*ArcSinh[a*x])*x^2 + 4*E^
(4*ArcSinh[a*x])*Sqrt[2*Pi]*Sqrt[ArcSinh[a*x]]*Erf[Sqrt[2]*Sqrt[ArcSinh[a*x]]] - 4*E^(4*ArcSinh[a*x])*Sqrt[2*P
i]*Sqrt[ArcSinh[a*x]]*Erfi[Sqrt[2]*Sqrt[ArcSinh[a*x]]] - 2*E^(4*ArcSinh[a*x])*Sqrt[-ArcSinh[a*x]]*Gamma[1/2, -
4*ArcSinh[a*x]] - 2*E^(4*ArcSinh[a*x])*Sqrt[ArcSinh[a*x]]*Gamma[1/2, 4*ArcSinh[a*x]]))/(8*a*E^(4*ArcSinh[a*x])
*Sqrt[1 + a^2*x^2]*Sqrt[ArcSinh[a*x]])

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Maple [F]  time = 0.164, size = 0, normalized size = 0. \begin{align*} \int{ \left ({a}^{2}c{x}^{2}+c \right ) ^{{\frac{3}{2}}} \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((a^2*c*x^2 + c)^(3/2)/arcsinh(a*x)^(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^2*c*x^2+c)^(3/2)/arcsinh(a*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 \frac{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac{3}{2}}}{\operatorname{asinh}^{\frac{3}{2}}{\left (a x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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