\(\int \frac {(f+g x)^3 (a+b \text {arccosh}(c x))}{(d-c^2 d x^2)^{3/2}} \, dx\) [71]

Optimal result

Integrand size = 31, antiderivative size = 549 \[ \int \frac {(f+g x)^3 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {b g^3 x \sqrt {-1+c x} \sqrt {1+c x}}{c^3 d \sqrt {d-c^2 d x^2}}-\frac {(c f-g)^3 (1-c x) (a+b \text {arccosh}(c x))}{2 c^4 d \sqrt {d-c^2 d x^2}}+\frac {(c f+g)^3 (1+c x) (a+b \text {arccosh}(c x))}{2 c^4 d \sqrt {d-c^2 d x^2}}+\frac {g^3 (1-c x) (1+c x) (a+b \text {arccosh}(c x))}{c^4 d \sqrt {d-c^2 d x^2}}-\frac {3 f g^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^2}{2 b c^3 d \sqrt {d-c^2 d x^2}}+\frac {b (c f+g)^3 \sqrt {(1-c x) (1+c x)} \sqrt {1-c^2 x^2} \log \left (\sqrt {-\frac {1-c x}{1+c x}}\right )}{c^4 d \sqrt {-\frac {1-c x}{1+c x}} (1+c x) \sqrt {d-c^2 d x^2}}-\frac {b (c f-g)^3 \sqrt {(1-c x) (1+c x)} \sqrt {1-c^2 x^2} \log \left (\frac {2}{1+c x}\right )}{2 c^4 d \sqrt {-\frac {1-c x}{1+c x}} (1+c x) \sqrt {d-c^2 d x^2}}-\frac {b (c f+g)^3 \sqrt {(1-c x) (1+c x)} \sqrt {1-c^2 x^2} \log \left (\frac {2}{1+c x}\right )}{2 c^4 d \sqrt {-\frac {1-c x}{1+c x}} (1+c x) \sqrt {d-c^2 d x^2}} \]

[Out]

-1/2*(c*f-g)^3*(-c*x+1)*(a+b*arccosh(c*x))/c^4/d/(-c^2*d*x^2+d)^(1/2)+1/2*(c*f+g)^3*(c*x+1)*(a+b*arccosh(c*x))
/c^4/d/(-c^2*d*x^2+d)^(1/2)+g^3*(-c*x+1)*(c*x+1)*(a+b*arccosh(c*x))/c^4/d/(-c^2*d*x^2+d)^(1/2)+b*g^3*x*(c*x-1)
^(1/2)*(c*x+1)^(1/2)/c^3/d/(-c^2*d*x^2+d)^(1/2)-3/2*f*g^2*(a+b*arccosh(c*x))^2*(c*x-1)^(1/2)*(c*x+1)^(1/2)/b/c
^3/d/(-c^2*d*x^2+d)^(1/2)-1/2*b*(c*f-g)^3*ln(2/(c*x+1))*((-c*x+1)*(c*x+1))^(1/2)*(-c^2*x^2+1)^(1/2)/c^4/d/(c*x
+1)/((c*x-1)/(c*x+1))^(1/2)/(-c^2*d*x^2+d)^(1/2)-1/2*b*(c*f+g)^3*ln(2/(c*x+1))*((-c*x+1)*(c*x+1))^(1/2)*(-c^2*
x^2+1)^(1/2)/c^4/d/(c*x+1)/((c*x-1)/(c*x+1))^(1/2)/(-c^2*d*x^2+d)^(1/2)+1/2*b*(c*f+g)^3*ln((c*x-1)/(c*x+1))*((
-c*x+1)*(c*x+1))^(1/2)*(-c^2*x^2+1)^(1/2)/c^4/d/(c*x+1)/((c*x-1)/(c*x+1))^(1/2)/(-c^2*d*x^2+d)^(1/2)

Rubi [A] (verified)

Time = 1.00 (sec) , antiderivative size = 549, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.484, Rules used = {5972, 5981, 37, 5987, 12, 1986, 15, 266, 272, 36, 31, 29, 5893, 5915, 8} \[ \int \frac {(f+g x)^3 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=-\frac {(1-c x) (c f-g)^3 (a+b \text {arccosh}(c x))}{2 c^4 d \sqrt {d-c^2 d x^2}}+\frac {(c x+1) (c f+g)^3 (a+b \text {arccosh}(c x))}{2 c^4 d \sqrt {d-c^2 d x^2}}+\frac {g^3 (1-c x) (c x+1) (a+b \text {arccosh}(c x))}{c^4 d \sqrt {d-c^2 d x^2}}-\frac {3 f g^2 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^2}{2 b c^3 d \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {(1-c x) (c x+1)} \sqrt {1-c^2 x^2} (c f-g)^3 \log \left (\frac {2}{c x+1}\right )}{2 c^4 d \sqrt {-\frac {1-c x}{c x+1}} (c x+1) \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {(1-c x) (c x+1)} \sqrt {1-c^2 x^2} (c f+g)^3 \log \left (\sqrt {-\frac {1-c x}{c x+1}}\right )}{c^4 d \sqrt {-\frac {1-c x}{c x+1}} (c x+1) \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {(1-c x) (c x+1)} \sqrt {1-c^2 x^2} (c f+g)^3 \log \left (\frac {2}{c x+1}\right )}{2 c^4 d \sqrt {-\frac {1-c x}{c x+1}} (c x+1) \sqrt {d-c^2 d x^2}}+\frac {b g^3 x \sqrt {c x-1} \sqrt {c x+1}}{c^3 d \sqrt {d-c^2 d x^2}} \]

[In]

Int[((f + g*x)^3*(a + b*ArcCosh[c*x]))/(d - c^2*d*x^2)^(3/2),x]

[Out]

(b*g^3*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(c^3*d*Sqrt[d - c^2*d*x^2]) - ((c*f - g)^3*(1 - c*x)*(a + b*ArcCosh[c*x
]))/(2*c^4*d*Sqrt[d - c^2*d*x^2]) + ((c*f + g)^3*(1 + c*x)*(a + b*ArcCosh[c*x]))/(2*c^4*d*Sqrt[d - c^2*d*x^2])
+ (g^3*(1 - c*x)*(1 + c*x)*(a + b*ArcCosh[c*x]))/(c^4*d*Sqrt[d - c^2*d*x^2]) - (3*f*g^2*Sqrt[-1 + c*x]*Sqrt[1
+ c*x]*(a + b*ArcCosh[c*x])^2)/(2*b*c^3*d*Sqrt[d - c^2*d*x^2]) + (b*(c*f + g)^3*Sqrt[(1 - c*x)*(1 + c*x)]*Sqr
t[1 - c^2*x^2]*Log[Sqrt[-((1 - c*x)/(1 + c*x))]])/(c^4*d*Sqrt[-((1 - c*x)/(1 + c*x))]*(1 + c*x)*Sqrt[d - c^2*d
*x^2]) - (b*(c*f - g)^3*Sqrt[(1 - c*x)*(1 + c*x)]*Sqrt[1 - c^2*x^2]*Log[2/(1 + c*x)])/(2*c^4*d*Sqrt[-((1 - c*x
)/(1 + c*x))]*(1 + c*x)*Sqrt[d - c^2*d*x^2]) - (b*(c*f + g)^3*Sqrt[(1 - c*x)*(1 + c*x)]*Sqrt[1 - c^2*x^2]*Log[
2/(1 + c*x)])/(2*c^4*d*Sqrt[-((1 - c*x)/(1 + c*x))]*(1 + c*x)*Sqrt[d - c^2*d*x^2])

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 15

Int[(u_.)*((a_.)*(x_)^(n_))^(m_), x_Symbol] :> Dist[a^IntPart[m]*((a*x^n)^FracPart[m]/x^(n*FracPart[m])), Int[
u*x^(m*n), x], x] /; FreeQ[{a, m, n}, x] &&  !IntegerQ[m]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n +
1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rule 266

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 1986

Int[(u_.)*((e_.)*((a_.) + (b_.)*(x_)^(n_.))^(q_.)*((c_) + (d_.)*(x_)^(n_))^(r_.))^(p_), x_Symbol] :> Dist[Simp
[(e*(a + b*x^n)^q*(c + d*x^n)^r)^p/((a + b*x^n)^(p*q)*(c + d*x^n)^(p*r))], Int[u*(a + b*x^n)^(p*q)*(c + d*x^n)
^(p*r), x], x] /; FreeQ[{a, b, c, d, e, n, p, q, r}, x]

Rule 5893

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

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]

Rule 5972

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol]
:> Dist[(-d)^IntPart[p]*((d + e*x^2)^FracPart[p]/((-1 + c*x)^FracPart[p]*(1 + c*x)^FracPart[p])), Int[(f + g*x
)^m*(-1 + c*x)^p*(1 + c*x)^p*(a + b*ArcCosh[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && EqQ[c^2*d
+ e, 0] && IntegerQ[p - 1/2] && IntegerQ[m]

Rule 5981

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

Rule 5987

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*(u_), x_Symbol] :> With[{v = IntHide[u, x]}, Dist[a + b*ArcCosh[c*x],
v, x] - Dist[b*c*(Sqrt[1 - c^2*x^2]/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])), Int[SimplifyIntegrand[v/Sqrt[1 - c^2*x^2]
, x], x], x] /; InverseFunctionFreeQ[v, x]] /; FreeQ[{a, b, c}, x]

Rubi steps \begin{align*} \text {integral}& = -\frac {\left (\sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {(f+g x)^3 (a+b \text {arccosh}(c x))}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{d \sqrt {d-c^2 d x^2}} \\ & = -\frac {\left (\sqrt {-1+c x} \sqrt {1+c x}\right ) \int \left (-\frac {(c f-g)^3 (a+b \text {arccosh}(c x))}{2 c^3 \sqrt {-1+c x} (1+c x)^{3/2}}+\frac {(c f+g)^3 (a+b \text {arccosh}(c x))}{2 c^3 (-1+c x)^{3/2} \sqrt {1+c x}}+\frac {3 f g^2 (a+b \text {arccosh}(c x))}{c^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {g^3 x (a+b \text {arccosh}(c x))}{c^2 \sqrt {-1+c x} \sqrt {1+c x}}\right ) \, dx}{d \sqrt {d-c^2 d x^2}} \\ & = \frac {\left ((c f-g)^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {a+b \text {arccosh}(c x)}{\sqrt {-1+c x} (1+c x)^{3/2}} \, dx}{2 c^3 d \sqrt {d-c^2 d x^2}}-\frac {\left (3 f g^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {a+b \text {arccosh}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {\left (g^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x (a+b \text {arccosh}(c x))}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {\left ((c f+g)^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {a+b \text {arccosh}(c x)}{(-1+c x)^{3/2} \sqrt {1+c x}} \, dx}{2 c^3 d \sqrt {d-c^2 d x^2}} \\ & = -\frac {(c f-g)^3 (1-c x) (a+b \text {arccosh}(c x))}{2 c^4 d \sqrt {d-c^2 d x^2}}+\frac {(c f+g)^3 (1+c x) (a+b \text {arccosh}(c x))}{2 c^4 d \sqrt {d-c^2 d x^2}}+\frac {g^3 (1-c x) (1+c x) (a+b \text {arccosh}(c x))}{c^4 d \sqrt {d-c^2 d x^2}}-\frac {3 f g^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^2}{2 b c^3 d \sqrt {d-c^2 d x^2}}+\frac {\left (b g^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int 1 \, dx}{c^3 d \sqrt {d-c^2 d x^2}}-\frac {\left (b (c f-g)^3 \sqrt {1-c^2 x^2}\right ) \int \frac {\sqrt {\frac {-1+c x}{1+c x}}}{c \sqrt {1-c^2 x^2}} \, dx}{2 c^2 d \sqrt {d-c^2 d x^2}}-\frac {\left (b (c f+g)^3 \sqrt {1-c^2 x^2}\right ) \int \frac {1}{c \sqrt {\frac {-1+c x}{1+c x}} \sqrt {1-c^2 x^2}} \, dx}{2 c^2 d \sqrt {d-c^2 d x^2}} \\ & = \frac {b g^3 x \sqrt {-1+c x} \sqrt {1+c x}}{c^3 d \sqrt {d-c^2 d x^2}}-\frac {(c f-g)^3 (1-c x) (a+b \text {arccosh}(c x))}{2 c^4 d \sqrt {d-c^2 d x^2}}+\frac {(c f+g)^3 (1+c x) (a+b \text {arccosh}(c x))}{2 c^4 d \sqrt {d-c^2 d x^2}}+\frac {g^3 (1-c x) (1+c x) (a+b \text {arccosh}(c x))}{c^4 d \sqrt {d-c^2 d x^2}}-\frac {3 f g^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^2}{2 b c^3 d \sqrt {d-c^2 d x^2}}-\frac {\left (b (c f-g)^3 \sqrt {1-c^2 x^2}\right ) \int \frac {\sqrt {\frac {-1+c x}{1+c x}}}{\sqrt {1-c^2 x^2}} \, dx}{2 c^3 d \sqrt {d-c^2 d x^2}}-\frac {\left (b (c f+g)^3 \sqrt {1-c^2 x^2}\right ) \int \frac {1}{\sqrt {\frac {-1+c x}{1+c x}} \sqrt {1-c^2 x^2}} \, dx}{2 c^3 d \sqrt {d-c^2 d x^2}} \\ & = \frac {b g^3 x \sqrt {-1+c x} \sqrt {1+c x}}{c^3 d \sqrt {d-c^2 d x^2}}-\frac {(c f-g)^3 (1-c x) (a+b \text {arccosh}(c x))}{2 c^4 d \sqrt {d-c^2 d x^2}}+\frac {(c f+g)^3 (1+c x) (a+b \text {arccosh}(c x))}{2 c^4 d \sqrt {d-c^2 d x^2}}+\frac {g^3 (1-c x) (1+c x) (a+b \text {arccosh}(c x))}{c^4 d \sqrt {d-c^2 d x^2}}-\frac {3 f g^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^2}{2 b c^3 d \sqrt {d-c^2 d x^2}}+\frac {\left (b (c f-g)^3 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \sqrt {-\frac {x^2}{\left (-1+x^2\right )^2}} \, dx,x,\sqrt {\frac {-1+c x}{1+c x}}\right )}{c^4 d \sqrt {d-c^2 d x^2}}+\frac {\left (b (c f+g)^3 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\sqrt {-\frac {x^2}{\left (-1+x^2\right )^2}}}{x^2} \, dx,x,\sqrt {\frac {-1+c x}{1+c x}}\right )}{c^4 d \sqrt {d-c^2 d x^2}} \\ & = \frac {b g^3 x \sqrt {-1+c x} \sqrt {1+c x}}{c^3 d \sqrt {d-c^2 d x^2}}-\frac {(c f-g)^3 (1-c x) (a+b \text {arccosh}(c x))}{2 c^4 d \sqrt {d-c^2 d x^2}}+\frac {(c f+g)^3 (1+c x) (a+b \text {arccosh}(c x))}{2 c^4 d \sqrt {d-c^2 d x^2}}+\frac {g^3 (1-c x) (1+c x) (a+b \text {arccosh}(c x))}{c^4 d \sqrt {d-c^2 d x^2}}-\frac {3 f g^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^2}{2 b c^3 d \sqrt {d-c^2 d x^2}}-\frac {\left (b (c f-g)^3 \sqrt {-((-1+c x) (1+c x))} \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\sqrt {x^2}}{-1+x^2} \, dx,x,\sqrt {\frac {-1+c x}{1+c x}}\right )}{c^4 d \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \sqrt {d-c^2 d x^2}}-\frac {\left (b (c f+g)^3 \sqrt {-((-1+c x) (1+c x))} \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {x^2} \left (-1+x^2\right )} \, dx,x,\sqrt {\frac {-1+c x}{1+c x}}\right )}{c^4 d \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \sqrt {d-c^2 d x^2}} \\ & = \frac {b g^3 x \sqrt {-1+c x} \sqrt {1+c x}}{c^3 d \sqrt {d-c^2 d x^2}}-\frac {(c f-g)^3 (1-c x) (a+b \text {arccosh}(c x))}{2 c^4 d \sqrt {d-c^2 d x^2}}+\frac {(c f+g)^3 (1+c x) (a+b \text {arccosh}(c x))}{2 c^4 d \sqrt {d-c^2 d x^2}}+\frac {g^3 (1-c x) (1+c x) (a+b \text {arccosh}(c x))}{c^4 d \sqrt {d-c^2 d x^2}}-\frac {3 f g^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^2}{2 b c^3 d \sqrt {d-c^2 d x^2}}-\frac {\left (b (c f-g)^3 \sqrt {-((-1+c x) (1+c x))} \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {x}{-1+x^2} \, dx,x,\sqrt {\frac {-1+c x}{1+c x}}\right )}{c^4 d \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \sqrt {d-c^2 d x^2}}-\frac {\left (b (c f+g)^3 \sqrt {-((-1+c x) (1+c x))} \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{x \left (-1+x^2\right )} \, dx,x,\sqrt {\frac {-1+c x}{1+c x}}\right )}{c^4 d \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \sqrt {d-c^2 d x^2}} \\ & = \frac {b g^3 x \sqrt {-1+c x} \sqrt {1+c x}}{c^3 d \sqrt {d-c^2 d x^2}}-\frac {(c f-g)^3 (1-c x) (a+b \text {arccosh}(c x))}{2 c^4 d \sqrt {d-c^2 d x^2}}+\frac {(c f+g)^3 (1+c x) (a+b \text {arccosh}(c x))}{2 c^4 d \sqrt {d-c^2 d x^2}}+\frac {g^3 (1-c x) (1+c x) (a+b \text {arccosh}(c x))}{c^4 d \sqrt {d-c^2 d x^2}}-\frac {3 f g^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^2}{2 b c^3 d \sqrt {d-c^2 d x^2}}+\frac {b (c f-g)^3 \sqrt {(1-c x) (1+c x)} \sqrt {1-c^2 x^2} \log (1+c x)}{2 c^4 d \sqrt {-\frac {1-c x}{1+c x}} (1+c x) \sqrt {d-c^2 d x^2}}-\frac {\left (b (c f+g)^3 \sqrt {-((-1+c x) (1+c x))} \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{(-1+x) x} \, dx,x,\frac {-1+c x}{1+c x}\right )}{2 c^4 d \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \sqrt {d-c^2 d x^2}} \\ & = \frac {b g^3 x \sqrt {-1+c x} \sqrt {1+c x}}{c^3 d \sqrt {d-c^2 d x^2}}-\frac {(c f-g)^3 (1-c x) (a+b \text {arccosh}(c x))}{2 c^4 d \sqrt {d-c^2 d x^2}}+\frac {(c f+g)^3 (1+c x) (a+b \text {arccosh}(c x))}{2 c^4 d \sqrt {d-c^2 d x^2}}+\frac {g^3 (1-c x) (1+c x) (a+b \text {arccosh}(c x))}{c^4 d \sqrt {d-c^2 d x^2}}-\frac {3 f g^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^2}{2 b c^3 d \sqrt {d-c^2 d x^2}}+\frac {b (c f-g)^3 \sqrt {(1-c x) (1+c x)} \sqrt {1-c^2 x^2} \log (1+c x)}{2 c^4 d \sqrt {-\frac {1-c x}{1+c x}} (1+c x) \sqrt {d-c^2 d x^2}}-\frac {\left (b (c f+g)^3 \sqrt {-((-1+c x) (1+c x))} \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{-1+x} \, dx,x,\frac {-1+c x}{1+c x}\right )}{2 c^4 d \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \sqrt {d-c^2 d x^2}}+\frac {\left (b (c f+g)^3 \sqrt {-((-1+c x) (1+c x))} \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{x} \, dx,x,\frac {-1+c x}{1+c x}\right )}{2 c^4 d \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \sqrt {d-c^2 d x^2}} \\ & = \frac {b g^3 x \sqrt {-1+c x} \sqrt {1+c x}}{c^3 d \sqrt {d-c^2 d x^2}}-\frac {(c f-g)^3 (1-c x) (a+b \text {arccosh}(c x))}{2 c^4 d \sqrt {d-c^2 d x^2}}+\frac {(c f+g)^3 (1+c x) (a+b \text {arccosh}(c x))}{2 c^4 d \sqrt {d-c^2 d x^2}}+\frac {g^3 (1-c x) (1+c x) (a+b \text {arccosh}(c x))}{c^4 d \sqrt {d-c^2 d x^2}}-\frac {3 f g^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^2}{2 b c^3 d \sqrt {d-c^2 d x^2}}+\frac {b (c f+g)^3 \sqrt {(1-c x) (1+c x)} \sqrt {1-c^2 x^2} \log \left (-\frac {1-c x}{1+c x}\right )}{2 c^4 d \sqrt {-\frac {1-c x}{1+c x}} (1+c x) \sqrt {d-c^2 d x^2}}+\frac {b (c f-g)^3 \sqrt {(1-c x) (1+c x)} \sqrt {1-c^2 x^2} \log (1+c x)}{2 c^4 d \sqrt {-\frac {1-c x}{1+c x}} (1+c x) \sqrt {d-c^2 d x^2}}+\frac {b (c f+g)^3 \sqrt {(1-c x) (1+c x)} \sqrt {1-c^2 x^2} \log (1+c x)}{2 c^4 d \sqrt {-\frac {1-c x}{1+c x}} (1+c x) \sqrt {d-c^2 d x^2}} \\ \end{align*}

Mathematica [A] (warning: unable to verify)

Time = 2.11 (sec) , antiderivative size = 501, normalized size of antiderivative = 0.91 \[ \int \frac {(f+g x)^3 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\sqrt {-d \left (-1+c^2 x^2\right )} \left (\frac {a g^3}{c^4 d^2}-\frac {a \left (3 c^2 f^2 g+g^3+c^4 f^3 x+3 c^2 f g^2 x\right )}{c^4 d^2 \left (-1+c^2 x^2\right )}\right )+\frac {3 a f g^2 \arctan \left (\frac {c x \sqrt {-d \left (-1+c^2 x^2\right )}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )}{c^3 d^{3/2}}-\frac {b f^3 \left (-c x \text {arccosh}(c x)+\sqrt {\frac {-1+c x}{1+c x}} (1+c x) \log \left (\sqrt {\frac {-1+c x}{1+c x}} (1+c x)\right )\right )}{c d \sqrt {-d (-1+c x) (1+c x)}}-\frac {3 b f g^2 \left (-2 c x \text {arccosh}(c x)+\sqrt {\frac {-1+c x}{1+c x}} (1+c x) \left (\text {arccosh}(c x)^2+2 \log \left (\sqrt {\frac {-1+c x}{1+c x}} (1+c x)\right )\right )\right )}{2 c^3 d \sqrt {-d (-1+c x) (1+c x)}}+\frac {3 b f^2 g \left (\text {arccosh}(c x)+\sqrt {\frac {-1+c x}{1+c x}} (1+c x) \left (\log \left (\cosh \left (\frac {1}{2} \text {arccosh}(c x)\right )\right )-\log \left (\sinh \left (\frac {1}{2} \text {arccosh}(c x)\right )\right )\right )\right )}{c^2 d \sqrt {-d (-1+c x) (1+c x)}}-\frac {b g^3 \left (-3 \text {arccosh}(c x)+\text {arccosh}(c x) \cosh (2 \text {arccosh}(c x))-2 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \left (\log \left (\cosh \left (\frac {1}{2} \text {arccosh}(c x)\right )\right )-\log \left (\sinh \left (\frac {1}{2} \text {arccosh}(c x)\right )\right )\right )-\sinh (2 \text {arccosh}(c x))\right )}{2 c^4 d \sqrt {-d (-1+c x) (1+c x)}} \]

[In]

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

[Out]

Sqrt[-(d*(-1 + c^2*x^2))]*((a*g^3)/(c^4*d^2) - (a*(3*c^2*f^2*g + g^3 + c^4*f^3*x + 3*c^2*f*g^2*x))/(c^4*d^2*(-
1 + c^2*x^2))) + (3*a*f*g^2*ArcTan[(c*x*Sqrt[-(d*(-1 + c^2*x^2))])/(Sqrt[d]*(-1 + c^2*x^2))])/(c^3*d^(3/2)) -
(b*f^3*(-(c*x*ArcCosh[c*x]) + Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*Log[Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)]))
/(c*d*Sqrt[-(d*(-1 + c*x)*(1 + c*x))]) - (3*b*f*g^2*(-2*c*x*ArcCosh[c*x] + Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x
)*(ArcCosh[c*x]^2 + 2*Log[Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)])))/(2*c^3*d*Sqrt[-(d*(-1 + c*x)*(1 + c*x))]) +
(3*b*f^2*g*(ArcCosh[c*x] + Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*(Log[Cosh[ArcCosh[c*x]/2]] - Log[Sinh[ArcCosh
[c*x]/2]])))/(c^2*d*Sqrt[-(d*(-1 + c*x)*(1 + c*x))]) - (b*g^3*(-3*ArcCosh[c*x] + ArcCosh[c*x]*Cosh[2*ArcCosh[c
*x]] - 2*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*(Log[Cosh[ArcCosh[c*x]/2]] - Log[Sinh[ArcCosh[c*x]/2]]) - Sinh[2
*ArcCosh[c*x]]))/(2*c^4*d*Sqrt[-(d*(-1 + c*x)*(1 + c*x))])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1231\) vs. \(2(486)=972\).

Time = 1.94 (sec) , antiderivative size = 1232, normalized size of antiderivative = 2.24

method	result	size
default	\(\text {Expression too large to display}\)	\(1232\)
parts	\(\text {Expression too large to display}\)	\(1232\)

[In]

int((g*x+f)^3*(a+b*arccosh(c*x))/(-c^2*d*x^2+d)^(3/2),x,method=_RETURNVERBOSE)

[Out]

a*(f^3/d*x/(-c^2*d*x^2+d)^(1/2)+g^3*(-x^2/c^2/d/(-c^2*d*x^2+d)^(1/2)+2/d/c^4/(-c^2*d*x^2+d)^(1/2))+3*f*g^2*(x/
c^2/d/(-c^2*d*x^2+d)^(1/2)-1/c^2/d/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2)))+3*f^2*g/c^2/d/(
-c^2*d*x^2+d)^(1/2))-b*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d^2/c/(c^2*x^2-1)*f^3*arccosh(c*x)-3
*b*(-d*(c^2*x^2-1))^(1/2)*arccosh(c*x)/d^2/(c^2*x^2-1)*x^2*f^2*g+b*(-d*(c^2*x^2-1))^(1/2)*arccosh(c*x)/d^2/c^4
/(c^2*x^2-1)*(c*x-1)*(c*x+1)*g^3-3*b*(-d*(c^2*x^2-1))^(1/2)*arccosh(c*x)/d^2/c^2/(c^2*x^2-1)*x*f*g^2+3/2*b*(-d
*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d^2/c^3/(c^2*x^2-1)*arccosh(c*x)^2*f*g^2-b*(-d*(c^2*x^2-1))^(1
/2)*arccosh(c*x)/d^2/(c^2*x^2-1)*x*f^3-b*(-d*(c^2*x^2-1))^(1/2)*g^3/d^2/c^3/(c^2*x^2-1)*(c*x+1)^(1/2)*(c*x-1)^
(1/2)*x+b*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d^2/c/(c^2*x^2-1)*ln((c*x-1)^(1/2)*(c*x+1)^(1/2)+
c*x-1)*f^3+b*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d^2/c^4/(c^2*x^2-1)*ln((c*x-1)^(1/2)*(c*x+1)^(
1/2)+c*x-1)*g^3+b*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)*ln(1+c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))/d^2
/c/(c^2*x^2-1)*f^3-b*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)*ln(1+c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))/
d^2/c^4/(c^2*x^2-1)*g^3-3*b*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d^2/c^3/(c^2*x^2-1)*f*arccosh(c
*x)*g^2+3*b*(-d*(c^2*x^2-1))^(1/2)*arccosh(c*x)/d^2/c^2/(c^2*x^2-1)*(c*x-1)*(c*x+1)*f^2*g+3*b*(-d*(c^2*x^2-1))
^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d^2/c^2/(c^2*x^2-1)*ln((c*x-1)^(1/2)*(c*x+1)^(1/2)+c*x-1)*f^2*g+3*b*(-d*(c^
2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d^2/c^3/(c^2*x^2-1)*ln((c*x-1)^(1/2)*(c*x+1)^(1/2)+c*x-1)*f*g^2-3*
b*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)*ln(1+c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))/d^2/c^2/(c^2*x^2-1)
*f^2*g+3*b*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)*ln(1+c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))/d^2/c^3/(c
^2*x^2-1)*f*g^2-b*(-d*(c^2*x^2-1))^(1/2)*g^3/d^2/c^4/(c^2*x^2-1)*arccosh(c*x)

Fricas [F]

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

[In]

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

[Out]

integral((a*g^3*x^3 + 3*a*f*g^2*x^2 + 3*a*f^2*g*x + a*f^3 + (b*g^3*x^3 + 3*b*f*g^2*x^2 + 3*b*f^2*g*x + b*f^3)*
arccosh(c*x))*sqrt(-c^2*d*x^2 + d)/(c^4*d^2*x^4 - 2*c^2*d^2*x^2 + d^2), x)

Sympy [F]

\[ \int \frac {(f+g x)^3 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {\left (a + b \operatorname {acosh}{\left (c x \right )}\right ) \left (f + g x\right )^{3}}{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}}}\, dx \]

[In]

integrate((g*x+f)**3*(a+b*acosh(c*x))/(-c**2*d*x**2+d)**(3/2),x)

[Out]

Integral((a + b*acosh(c*x))*(f + g*x)**3/(-d*(c*x - 1)*(c*x + 1))**(3/2), x)

Maxima [F]

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

[In]

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

[Out]

-1/2*b*c*f^3*sqrt(-1/(c^4*d))*log(x^2 - 1/c^2)/d - a*g^3*(x^2/(sqrt(-c^2*d*x^2 + d)*c^2*d) - 2/(sqrt(-c^2*d*x^
2 + d)*c^4*d)) + 3*a*f*g^2*(x/(sqrt(-c^2*d*x^2 + d)*c^2*d) - arcsin(c*x)/(c^3*d^(3/2))) + b*f^3*x*arccosh(c*x)
/(sqrt(-c^2*d*x^2 + d)*d) + a*f^3*x/(sqrt(-c^2*d*x^2 + d)*d) + 3*a*f^2*g/(sqrt(-c^2*d*x^2 + d)*c^2*d) + integr
ate(b*g^3*x^3*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))/(-c^2*d*x^2 + d)^(3/2) + 3*b*f*g^2*x^2*log(c*x + sqrt(c*x
+ 1)*sqrt(c*x - 1))/(-c^2*d*x^2 + d)^(3/2) + 3*b*f^2*g*x*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))/(-c^2*d*x^2 +
d)^(3/2), x)

Giac [F]

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

[In]

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

[Out]

integrate((g*x + f)^3*(b*arccosh(c*x) + a)/(-c^2*d*x^2 + d)^(3/2), x)

Mupad [F(-1)]

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

[In]

int(((f + g*x)^3*(a + b*acosh(c*x)))/(d - c^2*d*x^2)^(3/2),x)

[Out]

int(((f + g*x)^3*(a + b*acosh(c*x)))/(d - c^2*d*x^2)^(3/2), x)