3.99 \(\int x \sqrt{a+b \sinh ^{-1}(c+d x)} \, dx\)

Optimal. Leaf size=259 \[ -\frac{\sqrt{\pi } \sqrt{b} c e^{a/b} \text{Erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{4 d^2}-\frac{\sqrt{\frac{\pi }{2}} \sqrt{b} e^{\frac{2 a}{b}} \text{Erf}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{16 d^2}+\frac{\sqrt{\pi } \sqrt{b} c e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{4 d^2}-\frac{\sqrt{\frac{\pi }{2}} \sqrt{b} e^{-\frac{2 a}{b}} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{16 d^2}-\frac{c (c+d x) \sqrt{a+b \sinh ^{-1}(c+d x)}}{d^2}+\frac{\cosh \left (2 \sinh ^{-1}(c+d x)\right ) \sqrt{a+b \sinh ^{-1}(c+d x)}}{4 d^2} \]

[Out]

-((c*(c + d*x)*Sqrt[a + b*ArcSinh[c + d*x]])/d^2) + (Sqrt[a + b*ArcSinh[c + d*x]]*Cosh[2*ArcSinh[c + d*x]])/(4
*d^2) - (Sqrt[b]*c*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(4*d^2) - (Sqrt[b]*E^((2*a)/b)*
Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(16*d^2) + (Sqrt[b]*c*Sqrt[Pi]*Erfi[Sqrt[a + b
*ArcSinh[c + d*x]]/Sqrt[b]])/(4*d^2*E^(a/b)) - (Sqrt[b]*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])
/Sqrt[b]])/(16*d^2*E^((2*a)/b))

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Rubi [A]  time = 0.699855, antiderivative size = 259, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 10, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.625, Rules used = {5865, 5805, 6741, 6742, 5325, 5298, 2205, 2204, 5324, 5299} \[ -\frac{\sqrt{\pi } \sqrt{b} c e^{a/b} \text{Erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{4 d^2}-\frac{\sqrt{\frac{\pi }{2}} \sqrt{b} e^{\frac{2 a}{b}} \text{Erf}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{16 d^2}+\frac{\sqrt{\pi } \sqrt{b} c e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{4 d^2}-\frac{\sqrt{\frac{\pi }{2}} \sqrt{b} e^{-\frac{2 a}{b}} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{16 d^2}-\frac{c (c+d x) \sqrt{a+b \sinh ^{-1}(c+d x)}}{d^2}+\frac{\cosh \left (2 \sinh ^{-1}(c+d x)\right ) \sqrt{a+b \sinh ^{-1}(c+d x)}}{4 d^2} \]

Antiderivative was successfully verified.

[In]

Int[x*Sqrt[a + b*ArcSinh[c + d*x]],x]

[Out]

-((c*(c + d*x)*Sqrt[a + b*ArcSinh[c + d*x]])/d^2) + (Sqrt[a + b*ArcSinh[c + d*x]]*Cosh[2*ArcSinh[c + d*x]])/(4
*d^2) - (Sqrt[b]*c*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(4*d^2) - (Sqrt[b]*E^((2*a)/b)*
Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(16*d^2) + (Sqrt[b]*c*Sqrt[Pi]*Erfi[Sqrt[a + b
*ArcSinh[c + d*x]]/Sqrt[b]])/(4*d^2*E^(a/b)) - (Sqrt[b]*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])
/Sqrt[b]])/(16*d^2*E^((2*a)/b))

Rule 5865

Int[((a_.) + ArcSinh[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Dist[1/d, Subst[
Int[((d*e - c*f)/d + (f*x)/d)^m*(a + b*ArcSinh[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x
]

Rule 5805

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

Rule 6741

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =!= u]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rule 5325

Int[Cosh[(c_.) + (d_.)*(x_)^(n_)]*((e_.)*(x_))^(m_.), x_Symbol] :> Simp[(e^(n - 1)*(e*x)^(m - n + 1)*Sinh[c +
d*x^n])/(d*n), x] - Dist[(e^n*(m - n + 1))/(d*n), Int[(e*x)^(m - n)*Sinh[c + d*x^n], x], x] /; FreeQ[{c, d, e}
, x] && IGtQ[n, 0] && LtQ[0, n, m + 1]

Rule 5298

Int[Sinh[(c_.) + (d_.)*(x_)^(n_)], x_Symbol] :> Dist[1/2, Int[E^(c + d*x^n), x], x] - Dist[1/2, Int[E^(-c - d*
x^n), x], x] /; FreeQ[{c, d}, x] && IGtQ[n, 1]

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 5324

Int[((e_.)*(x_))^(m_.)*Sinh[(c_.) + (d_.)*(x_)^(n_)], x_Symbol] :> Simp[(e^(n - 1)*(e*x)^(m - n + 1)*Cosh[c +
d*x^n])/(d*n), x] - Dist[(e^n*(m - n + 1))/(d*n), Int[(e*x)^(m - n)*Cosh[c + d*x^n], x], x] /; FreeQ[{c, d, e}
, x] && IGtQ[n, 0] && LtQ[0, n, m + 1]

Rule 5299

Int[Cosh[(c_.) + (d_.)*(x_)^(n_)], x_Symbol] :> Dist[1/2, Int[E^(c + d*x^n), x], x] + Dist[1/2, Int[E^(-c - d*
x^n), x], x] /; FreeQ[{c, d}, x] && IGtQ[n, 1]

Rubi steps

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

Mathematica [A]  time = 1.76584, size = 251, normalized size = 0.97 \[ \frac{-16 c e^{-\frac{a}{b}} \sqrt{a+b \sinh ^{-1}(c+d x)} \left (\frac{\text{Gamma}\left (\frac{3}{2},-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right )}{\sqrt{-\frac{a+b \sinh ^{-1}(c+d x)}{b}}}-\frac{e^{\frac{2 a}{b}} \text{Gamma}\left (\frac{3}{2},\frac{a}{b}+\sinh ^{-1}(c+d x)\right )}{\sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)}}\right )-\sqrt{2 \pi } \sqrt{b} \left (\sinh \left (\frac{2 a}{b}\right )+\cosh \left (\frac{2 a}{b}\right )\right ) \text{Erf}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )+\sqrt{2 \pi } \sqrt{b} \left (\sinh \left (\frac{2 a}{b}\right )-\cosh \left (\frac{2 a}{b}\right )\right ) \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )+8 \cosh \left (2 \sinh ^{-1}(c+d x)\right ) \sqrt{a+b \sinh ^{-1}(c+d x)}}{32 d^2} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x*Sqrt[a + b*ArcSinh[c + d*x]],x]

[Out]

(8*Sqrt[a + b*ArcSinh[c + d*x]]*Cosh[2*ArcSinh[c + d*x]] - (16*c*Sqrt[a + b*ArcSinh[c + d*x]]*(-((E^((2*a)/b)*
Gamma[3/2, a/b + ArcSinh[c + d*x]])/Sqrt[a/b + ArcSinh[c + d*x]]) + Gamma[3/2, -((a + b*ArcSinh[c + d*x])/b)]/
Sqrt[-((a + b*ArcSinh[c + d*x])/b)]))/E^(a/b) + Sqrt[b]*Sqrt[2*Pi]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])
/Sqrt[b]]*(-Cosh[(2*a)/b] + Sinh[(2*a)/b]) - Sqrt[b]*Sqrt[2*Pi]*Erf[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqr
t[b]]*(Cosh[(2*a)/b] + Sinh[(2*a)/b]))/(32*d^2)

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Maple [F]  time = 0.161, size = 0, normalized size = 0. \begin{align*} \int x\sqrt{a+b{\it Arcsinh} \left ( dx+c \right ) }\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: UnboundLocalError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(b*arcsinh(d*x + c) + a)*x, x)