3.639 \(\int \frac{d+e x^2}{\sqrt{a+b \sinh ^{-1}(c x)}} \, dx\)

Optimal. Leaf size=287 \[ -\frac{\sqrt{\pi } e e^{a/b} \text{Erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{8 \sqrt{b} c^3}+\frac{\sqrt{\frac{\pi }{3}} e e^{\frac{3 a}{b}} \text{Erf}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{8 \sqrt{b} c^3}-\frac{\sqrt{\pi } e e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{8 \sqrt{b} c^3}+\frac{\sqrt{\frac{\pi }{3}} e e^{-\frac{3 a}{b}} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{8 \sqrt{b} c^3}+\frac{\sqrt{\pi } d e^{a/b} \text{Erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{2 \sqrt{b} c}+\frac{\sqrt{\pi } d e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{2 \sqrt{b} c} \]

[Out]

(d*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(2*Sqrt[b]*c) - (e*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*
ArcSinh[c*x]]/Sqrt[b]])/(8*Sqrt[b]*c^3) + (e*E^((3*a)/b)*Sqrt[Pi/3]*Erf[(Sqrt[3]*Sqrt[a + b*ArcSinh[c*x]])/Sqr
t[b]])/(8*Sqrt[b]*c^3) + (d*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(2*Sqrt[b]*c*E^(a/b)) - (e*Sqrt[P
i]*Erfi[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(8*Sqrt[b]*c^3*E^(a/b)) + (e*Sqrt[Pi/3]*Erfi[(Sqrt[3]*Sqrt[a + b*Ar
cSinh[c*x]])/Sqrt[b]])/(8*Sqrt[b]*c^3*E^((3*a)/b))

________________________________________________________________________________________

Rubi [A]  time = 0.571743, antiderivative size = 287, normalized size of antiderivative = 1., number of steps used = 21, number of rules used = 8, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {5706, 5657, 3307, 2180, 2205, 2204, 5669, 5448} \[ -\frac{\sqrt{\pi } e e^{a/b} \text{Erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{8 \sqrt{b} c^3}+\frac{\sqrt{\frac{\pi }{3}} e e^{\frac{3 a}{b}} \text{Erf}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{8 \sqrt{b} c^3}-\frac{\sqrt{\pi } e e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{8 \sqrt{b} c^3}+\frac{\sqrt{\frac{\pi }{3}} e e^{-\frac{3 a}{b}} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{8 \sqrt{b} c^3}+\frac{\sqrt{\pi } d e^{a/b} \text{Erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{2 \sqrt{b} c}+\frac{\sqrt{\pi } d e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{2 \sqrt{b} c} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x^2)/Sqrt[a + b*ArcSinh[c*x]],x]

[Out]

(d*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(2*Sqrt[b]*c) - (e*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*
ArcSinh[c*x]]/Sqrt[b]])/(8*Sqrt[b]*c^3) + (e*E^((3*a)/b)*Sqrt[Pi/3]*Erf[(Sqrt[3]*Sqrt[a + b*ArcSinh[c*x]])/Sqr
t[b]])/(8*Sqrt[b]*c^3) + (d*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(2*Sqrt[b]*c*E^(a/b)) - (e*Sqrt[P
i]*Erfi[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(8*Sqrt[b]*c^3*E^(a/b)) + (e*Sqrt[Pi/3]*Erfi[(Sqrt[3]*Sqrt[a + b*Ar
cSinh[c*x]])/Sqrt[b]])/(8*Sqrt[b]*c^3*E^((3*a)/b))

Rule 5706

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a
+ b*ArcSinh[c*x])^n, (d + e*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, n}, x] && NeQ[e, c^2*d] && IntegerQ[p] &&
(p > 0 || IGtQ[n, 0])

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]

Rule 5669

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

Rubi steps

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

Mathematica [A]  time = 0.66194, size = 218, normalized size = 0.76 \[ \frac{e^{-\frac{3 a}{b}} \left (-3 e^{\frac{4 a}{b}} \left (4 c^2 d-e\right ) \sqrt{\frac{a}{b}+\sinh ^{-1}(c x)} \text{Gamma}\left (\frac{1}{2},\frac{a}{b}+\sinh ^{-1}(c x)\right )+3 e^{\frac{2 a}{b}} \left (4 c^2 d-e\right ) \sqrt{-\frac{a+b \sinh ^{-1}(c x)}{b}} \text{Gamma}\left (\frac{1}{2},-\frac{a+b \sinh ^{-1}(c x)}{b}\right )+\sqrt{3} e \sqrt{-\frac{a+b \sinh ^{-1}(c x)}{b}} \text{Gamma}\left (\frac{1}{2},-\frac{3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )-\sqrt{3} e e^{\frac{6 a}{b}} \sqrt{\frac{a}{b}+\sinh ^{-1}(c x)} \text{Gamma}\left (\frac{1}{2},\frac{3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )\right )}{24 c^3 \sqrt{a+b \sinh ^{-1}(c x)}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-3*(4*c^2*d - e)*E^((4*a)/b)*Sqrt[a/b + ArcSinh[c*x]]*Gamma[1/2, a/b + ArcSinh[c*x]] + Sqrt[3]*e*Sqrt[-((a +
b*ArcSinh[c*x])/b)]*Gamma[1/2, (-3*(a + b*ArcSinh[c*x]))/b] + 3*(4*c^2*d - e)*E^((2*a)/b)*Sqrt[-((a + b*ArcSin
h[c*x])/b)]*Gamma[1/2, -((a + b*ArcSinh[c*x])/b)] - Sqrt[3]*e*E^((6*a)/b)*Sqrt[a/b + ArcSinh[c*x]]*Gamma[1/2,
(3*(a + b*ArcSinh[c*x]))/b])/(24*c^3*E^((3*a)/b)*Sqrt[a + b*ArcSinh[c*x]])

________________________________________________________________________________________

Maple [F]  time = 0.192, size = 0, normalized size = 0. \begin{align*} \int{(e{x}^{2}+d){\frac{1}{\sqrt{a+b{\it Arcsinh} \left ( cx \right ) }}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x^2+d)/(a+b*arcsinh(c*x))^(1/2),x)

[Out]

int((e*x^2+d)/(a+b*arcsinh(c*x))^(1/2),x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)/(a+b*arcsinh(c*x))^(1/2),x, algorithm="maxima")

[Out]

integrate((e*x^2 + d)/sqrt(b*arcsinh(c*x) + a), x)

________________________________________________________________________________________

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((e*x^2+d)/(a+b*arcsinh(c*x))^(1/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x**2+d)/(a+b*asinh(c*x))**(1/2),x)

[Out]

Integral((d + e*x**2)/sqrt(a + b*asinh(c*x)), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)/(a+b*arcsinh(c*x))^(1/2),x, algorithm="giac")

[Out]

integrate((e*x^2 + d)/sqrt(b*arcsinh(c*x) + a), x)