3.1.53 \(\int (f+g x)^3 \sqrt {d-c^2 d x^2} (a+b \cosh ^{-1}(c x)) \, dx\) [53]

Optimal. Leaf size=713 \[ \frac {b f^2 g x \sqrt {d-c^2 d x^2}}{c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b g^3 x \sqrt {d-c^2 d x^2}}{15 c^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c f^3 x^2 \sqrt {d-c^2 d x^2}}{4 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 b f g^2 x^2 \sqrt {d-c^2 d x^2}}{16 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c f^2 g x^3 \sqrt {d-c^2 d x^2}}{3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b g^3 x^3 \sqrt {d-c^2 d x^2}}{45 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {3 b c f g^2 x^4 \sqrt {d-c^2 d x^2}}{16 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c g^3 x^5 \sqrt {d-c^2 d x^2}}{25 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {1}{2} f^3 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac {3 f g^2 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 c^2}+\frac {3}{4} f g^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac {f^2 g (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{c^2}-\frac {2 g^3 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{15 c^4}-\frac {g^3 x^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{5 c^2}-\frac {f^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{4 b c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {3 f g^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{16 b c^3 \sqrt {-1+c x} \sqrt {1+c x}} \]

[Out]

1/2*f^3*x*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)-3/8*f*g^2*x*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/c^2+3/4*
f*g^2*x^3*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)-f^2*g*(-c*x+1)*(c*x+1)*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/
2)/c^2-2/15*g^3*(-c*x+1)*(c*x+1)*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/c^4-1/5*g^3*x^2*(-c*x+1)*(c*x+1)*(a+b
*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/c^2+b*f^2*g*x*(-c^2*d*x^2+d)^(1/2)/c/(c*x-1)^(1/2)/(c*x+1)^(1/2)+2/15*b*g^
3*x*(-c^2*d*x^2+d)^(1/2)/c^3/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/4*b*c*f^3*x^2*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c
*x+1)^(1/2)+3/16*b*f*g^2*x^2*(-c^2*d*x^2+d)^(1/2)/c/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/3*b*c*f^2*g*x^3*(-c^2*d*x^2+
d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/45*b*g^3*x^3*(-c^2*d*x^2+d)^(1/2)/c/(c*x-1)^(1/2)/(c*x+1)^(1/2)-3/16*b*
c*f*g^2*x^4*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/25*b*c*g^3*x^5*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/
2)/(c*x+1)^(1/2)-1/4*f^3*(a+b*arccosh(c*x))^2*(-c^2*d*x^2+d)^(1/2)/b/c/(c*x-1)^(1/2)/(c*x+1)^(1/2)-3/16*f*g^2*
(a+b*arccosh(c*x))^2*(-c^2*d*x^2+d)^(1/2)/b/c^3/(c*x-1)^(1/2)/(c*x+1)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.96, antiderivative size = 713, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 13, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.419, Rules used = {5972, 5975, 5896, 5893, 30, 5915, 41, 5927, 5939, 102, 12, 75, 5923} \begin {gather*} \frac {1}{2} f^3 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac {f^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{4 b c \sqrt {c x-1} \sqrt {c x+1}}-\frac {f^2 g (1-c x) (c x+1) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{c^2}-\frac {3 f g^2 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 c^2}+\frac {3}{4} f g^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac {g^3 x^2 (1-c x) (c x+1) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{5 c^2}-\frac {2 g^3 (1-c x) (c x+1) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{15 c^4}-\frac {3 f g^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{16 b c^3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c f^3 x^2 \sqrt {d-c^2 d x^2}}{4 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b f^2 g x \sqrt {d-c^2 d x^2}}{c \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c f^2 g x^3 \sqrt {d-c^2 d x^2}}{3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {3 b f g^2 x^2 \sqrt {d-c^2 d x^2}}{16 c \sqrt {c x-1} \sqrt {c x+1}}-\frac {3 b c f g^2 x^4 \sqrt {d-c^2 d x^2}}{16 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c g^3 x^5 \sqrt {d-c^2 d x^2}}{25 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b g^3 x^3 \sqrt {d-c^2 d x^2}}{45 c \sqrt {c x-1} \sqrt {c x+1}}+\frac {2 b g^3 x \sqrt {d-c^2 d x^2}}{15 c^3 \sqrt {c x-1} \sqrt {c x+1}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(b*f^2*g*x*Sqrt[d - c^2*d*x^2])/(c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (2*b*g^3*x*Sqrt[d - c^2*d*x^2])/(15*c^3*Sqr
t[-1 + c*x]*Sqrt[1 + c*x]) - (b*c*f^3*x^2*Sqrt[d - c^2*d*x^2])/(4*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (3*b*f*g^2*x
^2*Sqrt[d - c^2*d*x^2])/(16*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c*f^2*g*x^3*Sqrt[d - c^2*d*x^2])/(3*Sqrt[-1 +
c*x]*Sqrt[1 + c*x]) + (b*g^3*x^3*Sqrt[d - c^2*d*x^2])/(45*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (3*b*c*f*g^2*x^4*
Sqrt[d - c^2*d*x^2])/(16*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c*g^3*x^5*Sqrt[d - c^2*d*x^2])/(25*Sqrt[-1 + c*x]*
Sqrt[1 + c*x]) + (f^3*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/2 - (3*f*g^2*x*Sqrt[d - c^2*d*x^2]*(a + b*Ar
cCosh[c*x]))/(8*c^2) + (3*f*g^2*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/4 - (f^2*g*(1 - c*x)*(1 + c*x)*S
qrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/c^2 - (2*g^3*(1 - c*x)*(1 + c*x)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c
*x]))/(15*c^4) - (g^3*x^2*(1 - c*x)*(1 + c*x)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(5*c^2) - (f^3*Sqrt[d
- c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(4*b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (3*f*g^2*Sqrt[d - c^2*d*x^2]*(a +
b*ArcCosh[c*x])^2)/(16*b*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*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 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 41

Int[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[(a*c + b*d*x^2)^m, x] /; FreeQ[{a, b
, c, d, m}, x] && EqQ[b*c + a*d, 0] && (IntegerQ[m] || (GtQ[a, 0] && GtQ[c, 0]))

Rule 75

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

Rule 102

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

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 5896

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

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 5923

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*(x_)^(m_)*((d1_) + (e1_.)*(x_))^(p_)*((d2_) + (e2_.)*(x_))^(p_), x_Sym
bol] :> With[{u = IntHide[x^m*(d1 + e1*x)^p*(d2 + e2*x)^p, x]}, Dist[a + b*ArcCosh[c*x], u, x] - Dist[b*c*Simp
[Sqrt[d1 + e1*x]*(Sqrt[d2 + e2*x]/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]))], Int[SimplifyIntegrand[u/(Sqrt[d1 + e1*x]*S
qrt[d2 + e2*x]), x], x], x]] /; FreeQ[{a, b, c, d1, e1, d2, e2}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && In
tegerQ[p - 1/2] && NeQ[p, -2^(-1)] && (IGtQ[(m + 1)/2, 0] || ILtQ[(m + 2*p + 3)/2, 0])

Rule 5927

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

Rule 5939

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

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 5975

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

Rubi steps

\begin {align*} \int (f+g x)^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \, dx &=\frac {\sqrt {d-c^2 d x^2} \int \sqrt {-1+c x} \sqrt {1+c x} (f+g x)^3 \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {\sqrt {d-c^2 d x^2} \int \left (f^3 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )+3 f^2 g x \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )+3 f g^2 x^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )+g^3 x^3 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )\right ) \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {\left (f^3 \sqrt {d-c^2 d x^2}\right ) \int \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (3 f^2 g \sqrt {d-c^2 d x^2}\right ) \int x \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (3 f g^2 \sqrt {d-c^2 d x^2}\right ) \int x^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (g^3 \sqrt {d-c^2 d x^2}\right ) \int x^3 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {1}{2} f^3 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {3}{4} f g^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac {f^2 g (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{c^2}-\frac {2 g^3 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{15 c^4}-\frac {g^3 x^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{5 c^2}-\frac {\left (f^3 \sqrt {d-c^2 d x^2}\right ) \int \frac {a+b \cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (b c f^3 \sqrt {d-c^2 d x^2}\right ) \int x \, dx}{2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (b f^2 g \sqrt {d-c^2 d x^2}\right ) \int \left (-1+c^2 x^2\right ) \, dx}{c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (3 f g^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^2 \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{4 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (3 b c f g^2 \sqrt {d-c^2 d x^2}\right ) \int x^3 \, dx}{4 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (b c g^3 \sqrt {d-c^2 d x^2}\right ) \int \frac {-2-c^2 x^2+3 c^4 x^4}{15 c^4} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {b f^2 g x \sqrt {d-c^2 d x^2}}{c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c f^3 x^2 \sqrt {d-c^2 d x^2}}{4 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c f^2 g x^3 \sqrt {d-c^2 d x^2}}{3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {3 b c f g^2 x^4 \sqrt {d-c^2 d x^2}}{16 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {1}{2} f^3 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac {3 f g^2 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 c^2}+\frac {3}{4} f g^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac {f^2 g (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{c^2}-\frac {2 g^3 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{15 c^4}-\frac {g^3 x^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{5 c^2}-\frac {f^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{4 b c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (3 f g^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {a+b \cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{8 c^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (3 b f g^2 \sqrt {d-c^2 d x^2}\right ) \int x \, dx}{8 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (b g^3 \sqrt {d-c^2 d x^2}\right ) \int \left (-2-c^2 x^2+3 c^4 x^4\right ) \, dx}{15 c^3 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {b f^2 g x \sqrt {d-c^2 d x^2}}{c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b g^3 x \sqrt {d-c^2 d x^2}}{15 c^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c f^3 x^2 \sqrt {d-c^2 d x^2}}{4 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 b f g^2 x^2 \sqrt {d-c^2 d x^2}}{16 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c f^2 g x^3 \sqrt {d-c^2 d x^2}}{3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b g^3 x^3 \sqrt {d-c^2 d x^2}}{45 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {3 b c f g^2 x^4 \sqrt {d-c^2 d x^2}}{16 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c g^3 x^5 \sqrt {d-c^2 d x^2}}{25 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {1}{2} f^3 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac {3 f g^2 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 c^2}+\frac {3}{4} f g^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac {f^2 g (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{c^2}-\frac {2 g^3 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{15 c^4}-\frac {g^3 x^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{5 c^2}-\frac {f^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{4 b c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {3 f g^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{16 b c^3 \sqrt {-1+c x} \sqrt {1+c x}}\\ \end {align*}

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Mathematica [A]
time = 1.19, size = 491, normalized size = 0.69 \begin {gather*} \frac {240 a \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \sqrt {d-c^2 d x^2} \left (-16 g^3-c^2 g \left (120 f^2+45 f g x+8 g^2 x^2\right )+6 c^4 x \left (10 f^3+20 f^2 g x+15 f g^2 x^2+4 g^3 x^3\right )\right )-3600 a c \sqrt {d} f \left (4 c^2 f^2+3 g^2\right ) \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \text {ArcTan}\left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )+2400 b c^2 f^2 g \sqrt {d-c^2 d x^2} \left (9 c x+12 \left (\frac {-1+c x}{1+c x}\right )^{3/2} (1+c x)^3 \cosh ^{-1}(c x)-\cosh \left (3 \cosh ^{-1}(c x)\right )\right )-3600 b c^3 f^3 \sqrt {d-c^2 d x^2} \left (\cosh \left (2 \cosh ^{-1}(c x)\right )+2 \cosh ^{-1}(c x) \left (\cosh ^{-1}(c x)-\sinh \left (2 \cosh ^{-1}(c x)\right )\right )\right )-675 b c f g^2 \sqrt {d-c^2 d x^2} \left (8 \cosh ^{-1}(c x)^2+\cosh \left (4 \cosh ^{-1}(c x)\right )-4 \cosh ^{-1}(c x) \sinh \left (4 \cosh ^{-1}(c x)\right )\right )+8 b g^3 \sqrt {d-c^2 d x^2} \left (450 c x-450 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \cosh ^{-1}(c x)-25 \cosh \left (3 \cosh ^{-1}(c x)\right )-9 \cosh \left (5 \cosh ^{-1}(c x)\right )+75 \cosh ^{-1}(c x) \sinh \left (3 \cosh ^{-1}(c x)\right )+45 \cosh ^{-1}(c x) \sinh \left (5 \cosh ^{-1}(c x)\right )\right )}{28800 c^4 \sqrt {\frac {-1+c x}{1+c x}} (1+c x)} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(240*a*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*Sqrt[d - c^2*d*x^2]*(-16*g^3 - c^2*g*(120*f^2 + 45*f*g*x + 8*g^2*x
^2) + 6*c^4*x*(10*f^3 + 20*f^2*g*x + 15*f*g^2*x^2 + 4*g^3*x^3)) - 3600*a*c*Sqrt[d]*f*(4*c^2*f^2 + 3*g^2)*Sqrt[
(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] + 2400*b*c^2*f^2*g*
Sqrt[d - c^2*d*x^2]*(9*c*x + 12*((-1 + c*x)/(1 + c*x))^(3/2)*(1 + c*x)^3*ArcCosh[c*x] - Cosh[3*ArcCosh[c*x]])
- 3600*b*c^3*f^3*Sqrt[d - c^2*d*x^2]*(Cosh[2*ArcCosh[c*x]] + 2*ArcCosh[c*x]*(ArcCosh[c*x] - Sinh[2*ArcCosh[c*x
]])) - 675*b*c*f*g^2*Sqrt[d - c^2*d*x^2]*(8*ArcCosh[c*x]^2 + Cosh[4*ArcCosh[c*x]] - 4*ArcCosh[c*x]*Sinh[4*ArcC
osh[c*x]]) + 8*b*g^3*Sqrt[d - c^2*d*x^2]*(450*c*x - 450*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x] - 25
*Cosh[3*ArcCosh[c*x]] - 9*Cosh[5*ArcCosh[c*x]] + 75*ArcCosh[c*x]*Sinh[3*ArcCosh[c*x]] + 45*ArcCosh[c*x]*Sinh[5
*ArcCosh[c*x]]))/(28800*c^4*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1429\) vs. \(2(613)=1226\).
time = 7.53, size = 1430, normalized size = 2.01

method	result	size
default	\(\text {Expression too large to display}\)	\(1430\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((g*x+f)^3*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/5*a*g^3*x^2*(-c^2*d*x^2+d)^(3/2)/c^2/d-2/15*a*g^3/d/c^4*(-c^2*d*x^2+d)^(3/2)-3/4*a*f*g^2*x*(-c^2*d*x^2+d)^(
3/2)/c^2/d+3/8*a*f*g^2/c^2*x*(-c^2*d*x^2+d)^(1/2)+3/8*a*f*g^2/c^2*d/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2
*d*x^2+d)^(1/2))-a*f^2*g*(-c^2*d*x^2+d)^(3/2)/c^2/d+1/2*a*f^3*x*(-c^2*d*x^2+d)^(1/2)+1/2*a*f^3*d/(c^2*d)^(1/2)
*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))+b*(-1/16*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/c^3*
f*arccosh(c*x)^2*(4*c^2*f^2+3*g^2)+1/800*(-d*(c^2*x^2-1))^(1/2)*(16*x^6*c^6-28*c^4*x^4+16*(c*x+1)^(1/2)*(c*x-1
)^(1/2)*x^5*c^5+13*c^2*x^2-20*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^3*c^3+5*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c-1)*g^3*(-1
+5*arccosh(c*x))/(c*x+1)/c^4/(c*x-1)+3/256*(-d*(c^2*x^2-1))^(1/2)*(8*c^5*x^5-12*c^3*x^3+8*(c*x+1)^(1/2)*(c*x-1
)^(1/2)*x^4*c^4+4*c*x-8*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^2*c^2+(c*x-1)^(1/2)*(c*x+1)^(1/2))*f*g^2*(-1+4*arccosh(c
*x))/(c*x+1)/c^3/(c*x-1)+1/288*(-d*(c^2*x^2-1))^(1/2)*(4*c^4*x^4-5*c^2*x^2+4*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^3*c
^3-3*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c+1)*g*(36*arccosh(c*x)*c^2*f^2-12*c^2*f^2+3*arccosh(c*x)*g^2-g^2)/(c*x+1)/
c^4/(c*x-1)+1/16*(-d*(c^2*x^2-1))^(1/2)*(2*c^3*x^3-2*c*x+2*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^2*c^2-(c*x-1)^(1/2)*(
c*x+1)^(1/2))*f^3*(2*arccosh(c*x)-1)/(c*x+1)/c/(c*x-1)-1/16*(-d*(c^2*x^2-1))^(1/2)*((c*x+1)^(1/2)*(c*x-1)^(1/2
)*x*c+c^2*x^2-1)*g*(6*arccosh(c*x)*c^2*f^2-6*c^2*f^2+arccosh(c*x)*g^2-g^2)/(c*x+1)/c^4/(c*x-1)-1/16*(-d*(c^2*x
^2-1))^(1/2)*(-(c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c+c^2*x^2-1)*g*(6*arccosh(c*x)*c^2*f^2+6*c^2*f^2+arccosh(c*x)*g^2
+g^2)/(c*x+1)/c^4/(c*x-1)+1/16*(-d*(c^2*x^2-1))^(1/2)*(-2*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^2*c^2+2*c^3*x^3+(c*x-1
)^(1/2)*(c*x+1)^(1/2)-2*c*x)*f^3*(2*arccosh(c*x)+1)/(c*x+1)/c/(c*x-1)+1/288*(-d*(c^2*x^2-1))^(1/2)*(-4*(c*x+1)
^(1/2)*(c*x-1)^(1/2)*x^3*c^3+4*c^4*x^4+3*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c-5*c^2*x^2+1)*g*(36*arccosh(c*x)*c^2*f
^2+12*c^2*f^2+3*arccosh(c*x)*g^2+g^2)/(c*x+1)/c^4/(c*x-1)+3/256*(-d*(c^2*x^2-1))^(1/2)*(-8*(c*x+1)^(1/2)*(c*x-
1)^(1/2)*x^4*c^4+8*c^5*x^5+8*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^2*c^2-12*c^3*x^3-(c*x-1)^(1/2)*(c*x+1)^(1/2)+4*c*x)
*f*g^2*(1+4*arccosh(c*x))/(c*x+1)/c^3/(c*x-1)+1/800*(-d*(c^2*x^2-1))^(1/2)*(-16*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^
5*c^5+16*x^6*c^6+20*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^3*c^3-28*c^4*x^4-5*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c+13*c^2*x^
2-1)*g^3*(1+5*arccosh(c*x))/(c*x+1)/c^4/(c*x-1))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)^3*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2),x, algorithm="maxima")

[Out]

1/2*(sqrt(-c^2*d*x^2 + d)*x + sqrt(d)*arcsin(c*x)/c)*a*f^3 - 1/15*a*g^3*(3*(-c^2*d*x^2 + d)^(3/2)*x^2/(c^2*d)
+ 2*(-c^2*d*x^2 + d)^(3/2)/(c^4*d)) + 3/8*a*f*g^2*(sqrt(-c^2*d*x^2 + d)*x/c^2 - 2*(-c^2*d*x^2 + d)^(3/2)*x/(c^
2*d) + sqrt(d)*arcsin(c*x)/c^3) - (-c^2*d*x^2 + d)^(3/2)*a*f^2*g/(c^2*d) + integrate(sqrt(-c^2*d*x^2 + d)*b*g^
3*x^3*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1)) + 3*sqrt(-c^2*d*x^2 + d)*b*f*g^2*x^2*log(c*x + sqrt(c*x + 1)*sqrt
(c*x - 1)) + 3*sqrt(-c^2*d*x^2 + d)*b*f^2*g*x*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1)) + sqrt(-c^2*d*x^2 + d)*b*
f^3*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1)), x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {acosh}{\left (c x \right )}\right ) \left (f + g x\right )^{3}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)^3*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2),x, algorithm="giac")

[Out]

Exception raised: RuntimeError >> An error occurred running a Giac command:INPUT:sage2OUTPUT:sym2poly/r2sym(co
nst gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (f+g\,x\right )}^3\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,\sqrt {d-c^2\,d\,x^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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