3.149 \(\int x (a+b \cosh ^{-1}(c x))^{5/2} \, dx\)

Optimal. Leaf size=228 \[ -\frac{15 \sqrt{\frac{\pi }{2}} b^{5/2} e^{\frac{2 a}{b}} \text{Erf}\left (\frac{\sqrt{2} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{256 c^2}-\frac{15 \sqrt{\frac{\pi }{2}} b^{5/2} e^{-\frac{2 a}{b}} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{256 c^2}-\frac{15 b^2 \sqrt{a+b \cosh ^{-1}(c x)}}{64 c^2}+\frac{15}{32} b^2 x^2 \sqrt{a+b \cosh ^{-1}(c x)}-\frac{\left (a+b \cosh ^{-1}(c x)\right )^{5/2}}{4 c^2}+\frac{1}{2} x^2 \left (a+b \cosh ^{-1}(c x)\right )^{5/2}-\frac{5 b x \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}{8 c} \]

[Out]

(-15*b^2*Sqrt[a + b*ArcCosh[c*x]])/(64*c^2) + (15*b^2*x^2*Sqrt[a + b*ArcCosh[c*x]])/32 - (5*b*x*Sqrt[-1 + c*x]
*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^(3/2))/(8*c) - (a + b*ArcCosh[c*x])^(5/2)/(4*c^2) + (x^2*(a + b*ArcCosh[c*
x])^(5/2))/2 - (15*b^(5/2)*E^((2*a)/b)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(256*c^2) -
(15*b^(5/2)*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(256*c^2*E^((2*a)/b))

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Rubi [A]  time = 1.3278, antiderivative size = 228, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.643, Rules used = {5664, 5759, 5676, 5781, 3312, 3307, 2180, 2204, 2205} \[ -\frac{15 \sqrt{\frac{\pi }{2}} b^{5/2} e^{\frac{2 a}{b}} \text{Erf}\left (\frac{\sqrt{2} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{256 c^2}-\frac{15 \sqrt{\frac{\pi }{2}} b^{5/2} e^{-\frac{2 a}{b}} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{256 c^2}-\frac{15 b^2 \sqrt{a+b \cosh ^{-1}(c x)}}{64 c^2}+\frac{15}{32} b^2 x^2 \sqrt{a+b \cosh ^{-1}(c x)}-\frac{\left (a+b \cosh ^{-1}(c x)\right )^{5/2}}{4 c^2}+\frac{1}{2} x^2 \left (a+b \cosh ^{-1}(c x)\right )^{5/2}-\frac{5 b x \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}{8 c} \]

Antiderivative was successfully verified.

[In]

Int[x*(a + b*ArcCosh[c*x])^(5/2),x]

[Out]

(-15*b^2*Sqrt[a + b*ArcCosh[c*x]])/(64*c^2) + (15*b^2*x^2*Sqrt[a + b*ArcCosh[c*x]])/32 - (5*b*x*Sqrt[-1 + c*x]
*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^(3/2))/(8*c) - (a + b*ArcCosh[c*x])^(5/2)/(4*c^2) + (x^2*(a + b*ArcCosh[c*
x])^(5/2))/2 - (15*b^(5/2)*E^((2*a)/b)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(256*c^2) -
(15*b^(5/2)*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(256*c^2*E^((2*a)/b))

Rule 5664

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

Rule 5759

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_))/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_
.)*(x_)]), x_Symbol] :> Simp[(f*(f*x)^(m - 1)*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x]*(a + b*ArcCosh[c*x])^n)/(e1*e2*m
), x] + (Dist[(f^2*(m - 1))/(c^2*m), Int[((f*x)^(m - 2)*(a + b*ArcCosh[c*x])^n)/(Sqrt[d1 + e1*x]*Sqrt[d2 + e2*
x]), x], x] + Dist[(b*f*n*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x])/(c*d1*d2*m*Sqrt[1 + c*x]*Sqrt[-1 + c*x]), Int[(f*x)
^(m - 1)*(a + b*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d1, e1, d2, e2, f}, x] && EqQ[e1 - c*d1, 0]
&& EqQ[e2 + c*d2, 0] && GtQ[n, 0] && GtQ[m, 1] && IntegerQ[m]

Rule 5676

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)]), x_Symbol]
:> Simp[(a + b*ArcCosh[c*x])^(n + 1)/(b*c*Sqrt[-(d1*d2)]*(n + 1)), x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n},
x] && EqQ[e1, c*d1] && EqQ[e2, -(c*d2)] && GtQ[d1, 0] && LtQ[d2, 0] && NeQ[n, -1]

Rule 5781

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d1_) + (e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(x_))^(p_
.), x_Symbol] :> Dist[(-(d1*d2))^p/c^(m + 1), Subst[Int[(a + b*x)^n*Cosh[x]^m*Sinh[x]^(2*p + 1), x], x, ArcCos
h[c*x]], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && IntegerQ[p
+ 1/2] && GtQ[p, -1] && IGtQ[m, 0] && (GtQ[d1, 0] && LtQ[d2, 0])

Rule 3312

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 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 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 x \left (a+b \cosh ^{-1}(c x)\right )^{5/2} \, dx &=\frac{1}{2} x^2 \left (a+b \cosh ^{-1}(c x)\right )^{5/2}-\frac{1}{4} (5 b c) \int \frac{x^2 \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx\\ &=-\frac{5 b x \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}{8 c}+\frac{1}{2} x^2 \left (a+b \cosh ^{-1}(c x)\right )^{5/2}+\frac{1}{16} \left (15 b^2\right ) \int x \sqrt{a+b \cosh ^{-1}(c x)} \, dx-\frac{(5 b) \int \frac{\left (a+b \cosh ^{-1}(c x)\right )^{3/2}}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{8 c}\\ &=\frac{15}{32} b^2 x^2 \sqrt{a+b \cosh ^{-1}(c x)}-\frac{5 b x \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}{8 c}-\frac{\left (a+b \cosh ^{-1}(c x)\right )^{5/2}}{4 c^2}+\frac{1}{2} x^2 \left (a+b \cosh ^{-1}(c x)\right )^{5/2}-\frac{1}{64} \left (15 b^3 c\right ) \int \frac{x^2}{\sqrt{-1+c x} \sqrt{1+c x} \sqrt{a+b \cosh ^{-1}(c x)}} \, dx\\ &=\frac{15}{32} b^2 x^2 \sqrt{a+b \cosh ^{-1}(c x)}-\frac{5 b x \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}{8 c}-\frac{\left (a+b \cosh ^{-1}(c x)\right )^{5/2}}{4 c^2}+\frac{1}{2} x^2 \left (a+b \cosh ^{-1}(c x)\right )^{5/2}-\frac{\left (15 b^3\right ) \operatorname{Subst}\left (\int \frac{\cosh ^2(x)}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{64 c^2}\\ &=\frac{15}{32} b^2 x^2 \sqrt{a+b \cosh ^{-1}(c x)}-\frac{5 b x \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}{8 c}-\frac{\left (a+b \cosh ^{-1}(c x)\right )^{5/2}}{4 c^2}+\frac{1}{2} x^2 \left (a+b \cosh ^{-1}(c x)\right )^{5/2}-\frac{\left (15 b^3\right ) \operatorname{Subst}\left (\int \left (\frac{1}{2 \sqrt{a+b x}}+\frac{\cosh (2 x)}{2 \sqrt{a+b x}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{64 c^2}\\ &=-\frac{15 b^2 \sqrt{a+b \cosh ^{-1}(c x)}}{64 c^2}+\frac{15}{32} b^2 x^2 \sqrt{a+b \cosh ^{-1}(c x)}-\frac{5 b x \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}{8 c}-\frac{\left (a+b \cosh ^{-1}(c x)\right )^{5/2}}{4 c^2}+\frac{1}{2} x^2 \left (a+b \cosh ^{-1}(c x)\right )^{5/2}-\frac{\left (15 b^3\right ) \operatorname{Subst}\left (\int \frac{\cosh (2 x)}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{128 c^2}\\ &=-\frac{15 b^2 \sqrt{a+b \cosh ^{-1}(c x)}}{64 c^2}+\frac{15}{32} b^2 x^2 \sqrt{a+b \cosh ^{-1}(c x)}-\frac{5 b x \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}{8 c}-\frac{\left (a+b \cosh ^{-1}(c x)\right )^{5/2}}{4 c^2}+\frac{1}{2} x^2 \left (a+b \cosh ^{-1}(c x)\right )^{5/2}-\frac{\left (15 b^3\right ) \operatorname{Subst}\left (\int \frac{e^{-2 x}}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{256 c^2}-\frac{\left (15 b^3\right ) \operatorname{Subst}\left (\int \frac{e^{2 x}}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{256 c^2}\\ &=-\frac{15 b^2 \sqrt{a+b \cosh ^{-1}(c x)}}{64 c^2}+\frac{15}{32} b^2 x^2 \sqrt{a+b \cosh ^{-1}(c x)}-\frac{5 b x \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}{8 c}-\frac{\left (a+b \cosh ^{-1}(c x)\right )^{5/2}}{4 c^2}+\frac{1}{2} x^2 \left (a+b \cosh ^{-1}(c x)\right )^{5/2}-\frac{\left (15 b^2\right ) \operatorname{Subst}\left (\int e^{\frac{2 a}{b}-\frac{2 x^2}{b}} \, dx,x,\sqrt{a+b \cosh ^{-1}(c x)}\right )}{128 c^2}-\frac{\left (15 b^2\right ) \operatorname{Subst}\left (\int e^{-\frac{2 a}{b}+\frac{2 x^2}{b}} \, dx,x,\sqrt{a+b \cosh ^{-1}(c x)}\right )}{128 c^2}\\ &=-\frac{15 b^2 \sqrt{a+b \cosh ^{-1}(c x)}}{64 c^2}+\frac{15}{32} b^2 x^2 \sqrt{a+b \cosh ^{-1}(c x)}-\frac{5 b x \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}{8 c}-\frac{\left (a+b \cosh ^{-1}(c x)\right )^{5/2}}{4 c^2}+\frac{1}{2} x^2 \left (a+b \cosh ^{-1}(c x)\right )^{5/2}-\frac{15 b^{5/2} e^{\frac{2 a}{b}} \sqrt{\frac{\pi }{2}} \text{erf}\left (\frac{\sqrt{2} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{256 c^2}-\frac{15 b^{5/2} e^{-\frac{2 a}{b}} \sqrt{\frac{\pi }{2}} \text{erfi}\left (\frac{\sqrt{2} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{256 c^2}\\ \end{align*}

Mathematica [A]  time = 1.87998, size = 207, normalized size = 0.91 \[ \frac{8 \sqrt{a+b \cosh ^{-1}(c x)} \left (\left (16 a^2+15 b^2\right ) \cosh \left (2 \cosh ^{-1}(c x)\right )+4 b \cosh ^{-1}(c x) \left (8 a \cosh \left (2 \cosh ^{-1}(c x)\right )-5 b \sinh \left (2 \cosh ^{-1}(c x)\right )\right )-20 a b \sinh \left (2 \cosh ^{-1}(c x)\right )+16 b^2 \cosh \left (2 \cosh ^{-1}(c x)\right ) \cosh ^{-1}(c x)^2\right )-15 \sqrt{2 \pi } b^{5/2} \left (\sinh \left (\frac{2 a}{b}\right )+\cosh \left (\frac{2 a}{b}\right )\right ) \text{Erf}\left (\frac{\sqrt{2} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )-15 \sqrt{2 \pi } b^{5/2} \left (\cosh \left (\frac{2 a}{b}\right )-\sinh \left (\frac{2 a}{b}\right )\right ) \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{512 c^2} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x*(a + b*ArcCosh[c*x])^(5/2),x]

[Out]

(-15*b^(5/2)*Sqrt[2*Pi]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]]*(Cosh[(2*a)/b] - Sinh[(2*a)/b]) - 15*
b^(5/2)*Sqrt[2*Pi]*Erf[(Sqrt[2]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]]*(Cosh[(2*a)/b] + Sinh[(2*a)/b]) + 8*Sqrt[a
+ b*ArcCosh[c*x]]*((16*a^2 + 15*b^2)*Cosh[2*ArcCosh[c*x]] + 16*b^2*ArcCosh[c*x]^2*Cosh[2*ArcCosh[c*x]] - 20*a*
b*Sinh[2*ArcCosh[c*x]] + 4*b*ArcCosh[c*x]*(8*a*Cosh[2*ArcCosh[c*x]] - 5*b*Sinh[2*ArcCosh[c*x]])))/(512*c^2)

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Maple [F]  time = 0.124, size = 0, normalized size = 0. \begin{align*} \int x \left ( a+b{\rm arccosh} \left (cx\right ) \right ) ^{{\frac{5}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a+b*arccosh(c*x))^(5/2),x)

[Out]

int(x*(a+b*arccosh(c*x))^(5/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \operatorname{arcosh}\left (c x\right ) + a\right )}^{\frac{5}{2}} x\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*arccosh(c*x))^(5/2),x, algorithm="maxima")

[Out]

integrate((b*arccosh(c*x) + a)^(5/2)*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*arccosh(c*x))^(5/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*acosh(c*x))**(5/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*arccosh(c*x))^(5/2),x, algorithm="giac")

[Out]

sage0*x