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

Optimal. Leaf size=186 \[ -\frac{15 \sqrt{\pi } b^{5/2} e^{a/b} \text{Erf}\left (\frac{\sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{16 d}-\frac{15 \sqrt{\pi } b^{5/2} e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{16 d}+\frac{15 b^2 (c+d x) \sqrt{a+b \cosh ^{-1}(c+d x)}}{4 d}-\frac{5 b \sqrt{c+d x-1} \sqrt{c+d x+1} \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{2 d}+\frac{(c+d x) \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}{d} \]

[Out]

(15*b^2*(c + d*x)*Sqrt[a + b*ArcCosh[c + d*x]])/(4*d) - (5*b*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x]*(a + b*ArcCo
sh[c + d*x])^(3/2))/(2*d) + ((c + d*x)*(a + b*ArcCosh[c + d*x])^(5/2))/d - (15*b^(5/2)*E^(a/b)*Sqrt[Pi]*Erf[Sq
rt[a + b*ArcCosh[c + d*x]]/Sqrt[b]])/(16*d) - (15*b^(5/2)*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]])
/(16*d*E^(a/b))

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Rubi [A]  time = 0.609113, antiderivative size = 186, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.571, Rules used = {5864, 5654, 5718, 5781, 3307, 2180, 2204, 2205} \[ -\frac{15 \sqrt{\pi } b^{5/2} e^{a/b} \text{Erf}\left (\frac{\sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{16 d}-\frac{15 \sqrt{\pi } b^{5/2} e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{16 d}+\frac{15 b^2 (c+d x) \sqrt{a+b \cosh ^{-1}(c+d x)}}{4 d}-\frac{5 b \sqrt{c+d x-1} \sqrt{c+d x+1} \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{2 d}+\frac{(c+d x) \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}{d} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcCosh[c + d*x])^(5/2),x]

[Out]

(15*b^2*(c + d*x)*Sqrt[a + b*ArcCosh[c + d*x]])/(4*d) - (5*b*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x]*(a + b*ArcCo
sh[c + d*x])^(3/2))/(2*d) + ((c + d*x)*(a + b*ArcCosh[c + d*x])^(5/2))/d - (15*b^(5/2)*E^(a/b)*Sqrt[Pi]*Erf[Sq
rt[a + b*ArcCosh[c + d*x]]/Sqrt[b]])/(16*d) - (15*b^(5/2)*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]])
/(16*d*E^(a/b))

Rule 5864

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

Rule 5654

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

Rule 5718

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d1_) + (e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(x_))^(p_.), x_
Symbol] :> Simp[((d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*(a + b*ArcCosh[c*x])^n)/(2*e1*e2*(p + 1)), x] - Dist[
(b*n*(-(d1*d2))^IntPart[p]*(d1 + e1*x)^FracPart[p]*(d2 + e2*x)^FracPart[p])/(2*c*(p + 1)*(1 + c*x)^FracPart[p]
*(-1 + c*x)^FracPart[p]), Int[(-1 + c^2*x^2)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c,
d1, e1, d2, e2, p}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && GtQ[n, 0] && NeQ[p, -1] && IntegerQ[p + 1
/2]

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

Mathematica [B]  time = 3.30878, size = 494, normalized size = 2.66 \[ \frac{8 a^2 e^{-\frac{a}{b}} \sqrt{a+b \cosh ^{-1}(c+d x)} \left (\frac{e^{\frac{2 a}{b}} \text{Gamma}\left (\frac{3}{2},\frac{a}{b}+\cosh ^{-1}(c+d x)\right )}{\sqrt{\frac{a}{b}+\cosh ^{-1}(c+d x)}}+\frac{\text{Gamma}\left (\frac{3}{2},-\frac{a+b \cosh ^{-1}(c+d x)}{b}\right )}{\sqrt{-\frac{a+b \cosh ^{-1}(c+d x)}{b}}}\right )-\sqrt{\pi } \sqrt{b} \left (4 a^2-12 a b+15 b^2\right ) \left (\sinh \left (\frac{a}{b}\right )+\cosh \left (\frac{a}{b}\right )\right ) \text{Erf}\left (\frac{\sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )-\sqrt{\pi } \sqrt{b} \left (4 a^2+12 a b+15 b^2\right ) \left (\cosh \left (\frac{a}{b}\right )-\sinh \left (\frac{a}{b}\right )\right ) \text{Erfi}\left (\frac{\sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )+4 a b \left (\frac{\sqrt{\pi } (2 a-3 b) \left (\sinh \left (\frac{a}{b}\right )+\cosh \left (\frac{a}{b}\right )\right ) \text{Erf}\left (\frac{\sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{\sqrt{b}}+\frac{\sqrt{\pi } (2 a+3 b) \left (\cosh \left (\frac{a}{b}\right )-\sinh \left (\frac{a}{b}\right )\right ) \text{Erfi}\left (\frac{\sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{\sqrt{b}}-12 \sqrt{\frac{c+d x-1}{c+d x+1}} (c+d x+1) \sqrt{a+b \cosh ^{-1}(c+d x)}+8 (c+d x) \cosh ^{-1}(c+d x) \sqrt{a+b \cosh ^{-1}(c+d x)}\right )+4 b \sqrt{a+b \cosh ^{-1}(c+d x)} \left (2 \sqrt{\frac{c+d x-1}{c+d x+1}} (c+d x+1) \left (a-5 b \cosh ^{-1}(c+d x)\right )+b (c+d x) \left (4 \cosh ^{-1}(c+d x)^2+15\right )\right )}{16 d} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(4*b*Sqrt[a + b*ArcCosh[c + d*x]]*(2*Sqrt[(-1 + c + d*x)/(1 + c + d*x)]*(1 + c + d*x)*(a - 5*b*ArcCosh[c + d*x
]) + b*(c + d*x)*(15 + 4*ArcCosh[c + d*x]^2)) + (8*a^2*Sqrt[a + b*ArcCosh[c + d*x]]*((E^((2*a)/b)*Gamma[3/2, a
/b + ArcCosh[c + d*x]])/Sqrt[a/b + ArcCosh[c + d*x]] + Gamma[3/2, -((a + b*ArcCosh[c + d*x])/b)]/Sqrt[-((a + b
*ArcCosh[c + d*x])/b)]))/E^(a/b) - Sqrt[b]*(4*a^2 + 12*a*b + 15*b^2)*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c + d*x]
]/Sqrt[b]]*(Cosh[a/b] - Sinh[a/b]) - Sqrt[b]*(4*a^2 - 12*a*b + 15*b^2)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c + d*x
]]/Sqrt[b]]*(Cosh[a/b] + Sinh[a/b]) + 4*a*b*(-12*Sqrt[(-1 + c + d*x)/(1 + c + d*x)]*(1 + c + d*x)*Sqrt[a + b*A
rcCosh[c + d*x]] + 8*(c + d*x)*ArcCosh[c + d*x]*Sqrt[a + b*ArcCosh[c + d*x]] + ((2*a + 3*b)*Sqrt[Pi]*Erfi[Sqrt
[a + b*ArcCosh[c + d*x]]/Sqrt[b]]*(Cosh[a/b] - Sinh[a/b]))/Sqrt[b] + ((2*a - 3*b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcC
osh[c + d*x]]/Sqrt[b]]*(Cosh[a/b] + Sinh[a/b]))/Sqrt[b]))/(16*d)

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Maple [F]  time = 0.122, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b{\rm arccosh} \left (dx+c\right ) \right ) ^{{\frac{5}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

sage0*x