\(\int (a+b \text {arccosh}(c x))^{3/2} \, dx\) [147]

Optimal result

Integrand size = 12, antiderivative size = 140 \[ \int (a+b \text {arccosh}(c x))^{3/2} \, dx=-\frac {3 b \sqrt {-1+c x} \sqrt {1+c x} \sqrt {a+b \text {arccosh}(c x)}}{2 c}+x (a+b \text {arccosh}(c x))^{3/2}-\frac {3 b^{3/2} e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{8 c}+\frac {3 b^{3/2} e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{8 c} \]

[Out]

x*(a+b*arccosh(c*x))^(3/2)-3/8*b^(3/2)*exp(a/b)*erf((a+b*arccosh(c*x))^(1/2)/b^(1/2))*Pi^(1/2)/c+3/8*b^(3/2)*e
rfi((a+b*arccosh(c*x))^(1/2)/b^(1/2))*Pi^(1/2)/c/exp(a/b)-3/2*b*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(a+b*arccosh(c*x))
^(1/2)/c

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {5879, 5915, 5881, 3389, 2211, 2236, 2235} \[ \int (a+b \text {arccosh}(c x))^{3/2} \, dx=-\frac {3 \sqrt {\pi } b^{3/2} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{8 c}+\frac {3 \sqrt {\pi } b^{3/2} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{8 c}-\frac {3 b \sqrt {c x-1} \sqrt {c x+1} \sqrt {a+b \text {arccosh}(c x)}}{2 c}+x (a+b \text {arccosh}(c x))^{3/2} \]

[In]

Int[(a + b*ArcCosh[c*x])^(3/2),x]

[Out]

(-3*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Sqrt[a + b*ArcCosh[c*x]])/(2*c) + x*(a + b*ArcCosh[c*x])^(3/2) - (3*b^(3/2)
*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(8*c) + (3*b^(3/2)*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c
*x]]/Sqrt[b]])/(8*c*E^(a/b))

Rule 2211

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] &&  !TrueQ[$UseGamma]

Rule 2235

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 2236

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 3389

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 5879

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 5881

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

Rule 5915

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d1_) + (e1_.)*(x_))^(p_)*((d2_) + (e2_.)*(x_))^(p_), x_Sy
mbol] :> 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/(2*c*(p + 1)))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*x)^p/(-1 + c*x)^p], Int[(1 + c*x)^(p + 1/2)*(-
1 + c*x)^(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] && EqQ[e2, (-c)*d2] && GtQ[n, 0] && NeQ[p, -1]

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

Mathematica [A] (warning: unable to verify)

Time = 0.41 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.92 \[ \int (a+b \text {arccosh}(c x))^{3/2} \, dx=\frac {a e^{-\frac {a}{b}} \sqrt {a+b \text {arccosh}(c x)} \left (\frac {e^{\frac {2 a}{b}} \Gamma \left (\frac {3}{2},\frac {a}{b}+\text {arccosh}(c x)\right )}{\sqrt {\frac {a}{b}+\text {arccosh}(c x)}}+\frac {\Gamma \left (\frac {3}{2},-\frac {a+b \text {arccosh}(c x)}{b}\right )}{\sqrt {-\frac {a+b \text {arccosh}(c x)}{b}}}\right )}{2 c}+\frac {b \left (-12 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \sqrt {a+b \text {arccosh}(c x)}+8 c x \text {arccosh}(c x) \sqrt {a+b \text {arccosh}(c x)}+\frac {(2 a+3 b) \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {a}{b}\right )-\sinh \left (\frac {a}{b}\right )\right )}{\sqrt {b}}+\frac {(2 a-3 b) \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {a}{b}\right )+\sinh \left (\frac {a}{b}\right )\right )}{\sqrt {b}}\right )}{8 c} \]

[In]

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

[Out]

(a*Sqrt[a + b*ArcCosh[c*x]]*((E^((2*a)/b)*Gamma[3/2, a/b + ArcCosh[c*x]])/Sqrt[a/b + ArcCosh[c*x]] + Gamma[3/2
, -((a + b*ArcCosh[c*x])/b)]/Sqrt[-((a + b*ArcCosh[c*x])/b)]))/(2*c*E^(a/b)) + (b*(-12*Sqrt[(-1 + c*x)/(1 + c*
x)]*(1 + c*x)*Sqrt[a + b*ArcCosh[c*x]] + 8*c*x*ArcCosh[c*x]*Sqrt[a + b*ArcCosh[c*x]] + ((2*a + 3*b)*Sqrt[Pi]*E
rfi[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]]*(Cosh[a/b] - Sinh[a/b]))/Sqrt[b] + ((2*a - 3*b)*Sqrt[Pi]*Erf[Sqrt[a + b*
ArcCosh[c*x]]/Sqrt[b]]*(Cosh[a/b] + Sinh[a/b]))/Sqrt[b]))/(8*c)

Maple [F]

\[\int \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{\frac {3}{2}}d x\]

[In]

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

[Out]

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

Fricas [F(-2)]

Exception generated. \[ \int (a+b \text {arccosh}(c x))^{3/2} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((a+b*arccosh(c*x))^(3/2),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

Sympy [F]

\[ \int (a+b \text {arccosh}(c x))^{3/2} \, dx=\int \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{\frac {3}{2}}\, dx \]

[In]

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

[Out]

Integral((a + b*acosh(c*x))**(3/2), x)

Maxima [F]

\[ \int (a+b \text {arccosh}(c x))^{3/2} \, dx=\int { {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{\frac {3}{2}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int (a+b \text {arccosh}(c x))^{3/2} \, dx=\int { {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{\frac {3}{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int (a+b \text {arccosh}(c x))^{3/2} \, dx=\int {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^{3/2} \,d x \]

[In]

int((a + b*acosh(c*x))^(3/2),x)

[Out]

int((a + b*acosh(c*x))^(3/2), x)