\(\int (f+g x)^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \, dx\) [54]

Optimal result

Integrand size = 31, antiderivative size = 479 \[ \int (f+g x)^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \, dx=\frac {2 b f g x \sqrt {d-c^2 d x^2}}{3 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c f^2 x^2 \sqrt {d-c^2 d x^2}}{4 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b g^2 x^2 \sqrt {d-c^2 d x^2}}{16 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b c f g x^3 \sqrt {d-c^2 d x^2}}{9 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c g^2 x^4 \sqrt {d-c^2 d x^2}}{16 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {1}{2} f^2 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {g^2 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{8 c^2}+\frac {1}{4} g^2 x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {2 f g (1-c x) (1+c x) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{3 c^2}-\frac {f^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{4 b c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {g^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{16 b c^3 \sqrt {-1+c x} \sqrt {1+c x}} \]

[Out]

1/2*f^2*x*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)-1/8*g^2*x*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/c^2+1/4*g^
2*x^3*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)-2/3*f*g*(-c*x+1)*(c*x+1)*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)
/c^2+2/3*b*f*g*x*(-c^2*d*x^2+d)^(1/2)/c/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/4*b*c*f^2*x^2*(-c^2*d*x^2+d)^(1/2)/(c*x-
1)^(1/2)/(c*x+1)^(1/2)+1/16*b*g^2*x^2*(-c^2*d*x^2+d)^(1/2)/c/(c*x-1)^(1/2)/(c*x+1)^(1/2)-2/9*b*c*f*g*x^3*(-c^2
*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/16*b*c*g^2*x^4*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)-
1/4*f^2*(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)-1/16*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] (verified)

Time = 0.76 (sec) , antiderivative size = 479, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.290, Rules used = {5972, 5975, 5896, 5893, 30, 5915, 41, 5927, 5939} \[ \int (f+g x)^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \, dx=\frac {1}{2} f^2 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {f^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{4 b c \sqrt {c x-1} \sqrt {c x+1}}-\frac {2 f g (1-c x) (c x+1) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{3 c^2}-\frac {g^2 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{8 c^2}+\frac {1}{4} g^2 x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {g^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{16 b c^3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c f^2 x^2 \sqrt {d-c^2 d x^2}}{4 \sqrt {c x-1} \sqrt {c x+1}}+\frac {2 b f g x \sqrt {d-c^2 d x^2}}{3 c \sqrt {c x-1} \sqrt {c x+1}}-\frac {2 b c f g x^3 \sqrt {d-c^2 d x^2}}{9 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b g^2 x^2 \sqrt {d-c^2 d x^2}}{16 c \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c g^2 x^4 \sqrt {d-c^2 d x^2}}{16 \sqrt {c x-1} \sqrt {c x+1}} \]

[In]

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

[Out]

(2*b*f*g*x*Sqrt[d - c^2*d*x^2])/(3*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c*f^2*x^2*Sqrt[d - c^2*d*x^2])/(4*Sqrt
[-1 + c*x]*Sqrt[1 + c*x]) + (b*g^2*x^2*Sqrt[d - c^2*d*x^2])/(16*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (2*b*c*f*g*x
^3*Sqrt[d - c^2*d*x^2])/(9*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c*g^2*x^4*Sqrt[d - c^2*d*x^2])/(16*Sqrt[-1 + c*x
]*Sqrt[1 + c*x]) + (f^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/2 - (g^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcC
osh[c*x]))/(8*c^2) + (g^2*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/4 - (2*f*g*(1 - c*x)*(1 + c*x)*Sqrt[d
- c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(3*c^2) - (f^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(4*b*c*Sqrt[-1 +
c*x]*Sqrt[1 + c*x]) - (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 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 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 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*} \text {integral}& = \frac {\sqrt {d-c^2 d x^2} \int \sqrt {-1+c x} \sqrt {1+c x} (f+g x)^2 (a+b \text {arccosh}(c x)) \, dx}{\sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {\sqrt {d-c^2 d x^2} \int \left (f^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))+2 f g x \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))+g^2 x^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))\right ) \, dx}{\sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {\left (f^2 \sqrt {d-c^2 d x^2}\right ) \int \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (2 f g \sqrt {d-c^2 d x^2}\right ) \int x \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (g^2 \sqrt {d-c^2 d x^2}\right ) \int x^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \, dx}{\sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {1}{2} f^2 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))+\frac {1}{4} g^2 x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {2 f g (1-c x) (1+c x) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{3 c^2}-\frac {\left (f^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {a+b \text {arccosh}(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^2 \sqrt {d-c^2 d x^2}\right ) \int x \, dx}{2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (2 b f g \sqrt {d-c^2 d x^2}\right ) \int (-1+c x) (1+c x) \, dx}{3 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (g^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^2 (a+b \text {arccosh}(c x))}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{4 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (b c g^2 \sqrt {d-c^2 d x^2}\right ) \int x^3 \, dx}{4 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {b c f^2 x^2 \sqrt {d-c^2 d x^2}}{4 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c g^2 x^4 \sqrt {d-c^2 d x^2}}{16 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {1}{2} f^2 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {g^2 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{8 c^2}+\frac {1}{4} g^2 x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {2 f g (1-c x) (1+c x) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{3 c^2}-\frac {f^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{4 b c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (2 b f g \sqrt {d-c^2 d x^2}\right ) \int \left (-1+c^2 x^2\right ) \, dx}{3 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (g^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {a+b \text {arccosh}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{8 c^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b g^2 \sqrt {d-c^2 d x^2}\right ) \int x \, dx}{8 c \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {2 b f g x \sqrt {d-c^2 d x^2}}{3 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c f^2 x^2 \sqrt {d-c^2 d x^2}}{4 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b g^2 x^2 \sqrt {d-c^2 d x^2}}{16 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b c f g x^3 \sqrt {d-c^2 d x^2}}{9 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c g^2 x^4 \sqrt {d-c^2 d x^2}}{16 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {1}{2} f^2 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {g^2 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{8 c^2}+\frac {1}{4} g^2 x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {2 f g (1-c x) (1+c x) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{3 c^2}-\frac {f^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{4 b c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {g^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{16 b c^3 \sqrt {-1+c x} \sqrt {1+c x}} \\ \end{align*}

Mathematica [A] (warning: unable to verify)

Time = 0.91 (sec) , antiderivative size = 356, normalized size of antiderivative = 0.74 \[ \int (f+g x)^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \, dx=\frac {48 a c \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \sqrt {d-c^2 d x^2} \left (12 c^2 f^2 x+16 f g \left (-1+c^2 x^2\right )+3 g^2 x \left (-1+2 c^2 x^2\right )\right )-144 a \sqrt {d} \left (4 c^2 f^2+g^2\right ) \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )+64 b c f g \sqrt {d-c^2 d x^2} \left (9 c x+12 \left (\frac {-1+c x}{1+c x}\right )^{3/2} (1+c x)^3 \text {arccosh}(c x)-\cosh (3 \text {arccosh}(c x))\right )-144 b c^2 f^2 \sqrt {d-c^2 d x^2} (\cosh (2 \text {arccosh}(c x))+2 \text {arccosh}(c x) (\text {arccosh}(c x)-\sinh (2 \text {arccosh}(c x))))-9 b g^2 \sqrt {d-c^2 d x^2} \left (8 \text {arccosh}(c x)^2+\cosh (4 \text {arccosh}(c x))-4 \text {arccosh}(c x) \sinh (4 \text {arccosh}(c x))\right )}{1152 c^3 \sqrt {\frac {-1+c x}{1+c x}} (1+c x)} \]

[In]

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

[Out]

(48*a*c*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*Sqrt[d - c^2*d*x^2]*(12*c^2*f^2*x + 16*f*g*(-1 + c^2*x^2) + 3*g^2
*x*(-1 + 2*c^2*x^2)) - 144*a*Sqrt[d]*(4*c^2*f^2 + 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))] + 64*b*c*f*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]]) - 144*b*c^2*f^2*Sqrt[d - c^2*d*x^2]*(Cosh[2*ArcCosh[c*x]
] + 2*ArcCosh[c*x]*(ArcCosh[c*x] - Sinh[2*ArcCosh[c*x]])) - 9*b*g^2*Sqrt[d - c^2*d*x^2]*(8*ArcCosh[c*x]^2 + Co
sh[4*ArcCosh[c*x]] - 4*ArcCosh[c*x]*Sinh[4*ArcCosh[c*x]]))/(1152*c^3*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(992\) vs. \(2(407)=814\).

Time = 1.17 (sec) , antiderivative size = 993, normalized size of antiderivative = 2.07

method	result	size
default	\(a \left (f^{2} \left (\frac {x \sqrt {-c^{2} d \,x^{2}+d}}{2}+\frac {d \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{2 \sqrt {c^{2} d}}\right )+g^{2} \left (-\frac {x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{4 c^{2} d}+\frac {\frac {x \sqrt {-c^{2} d \,x^{2}+d}}{2}+\frac {d \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{2 \sqrt {c^{2} d}}}{4 c^{2}}\right )-\frac {2 f g \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{3 c^{2} d}\right )+b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right )^{2} \left (4 c^{2} f^{2}+g^{2}\right )}{16 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3}}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (8 c^{5} x^{5}-12 c^{3} x^{3}+8 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{4} x^{4}+4 c x -8 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}+\sqrt {c x -1}\, \sqrt {c x +1}\right ) g^{2} \left (-1+4 \,\operatorname {arccosh}\left (c x \right )\right )}{256 \left (c x +1\right ) c^{3} \left (c x -1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (4 c^{4} x^{4}-5 c^{2} x^{2}+4 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{3}-3 \sqrt {c x -1}\, \sqrt {c x +1}\, c x +1\right ) f g \left (-1+3 \,\operatorname {arccosh}\left (c x \right )\right )}{36 \left (c x +1\right ) c^{2} \left (c x -1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 c^{3} x^{3}-2 c x +2 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-\sqrt {c x -1}\, \sqrt {c x +1}\right ) f^{2} \left (-1+2 \,\operatorname {arccosh}\left (c x \right )\right )}{16 \left (c x +1\right ) c \left (c x -1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) f g \left (-1+\operatorname {arccosh}\left (c x \right )\right )}{4 \left (c x +1\right ) c^{2} \left (c x -1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) f g \left (1+\operatorname {arccosh}\left (c x \right )\right )}{4 \left (c x +1\right ) c^{2} \left (c x -1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-2 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}+2 c^{3} x^{3}+\sqrt {c x -1}\, \sqrt {c x +1}-2 c x \right ) f^{2} \left (1+2 \,\operatorname {arccosh}\left (c x \right )\right )}{16 \left (c x +1\right ) c \left (c x -1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-4 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{3}+4 c^{4} x^{4}+3 \sqrt {c x -1}\, \sqrt {c x +1}\, c x -5 c^{2} x^{2}+1\right ) f g \left (1+3 \,\operatorname {arccosh}\left (c x \right )\right )}{36 \left (c x +1\right ) c^{2} \left (c x -1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-8 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{4} x^{4}+8 c^{5} x^{5}+8 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-12 c^{3} x^{3}-\sqrt {c x -1}\, \sqrt {c x +1}+4 c x \right ) g^{2} \left (1+4 \,\operatorname {arccosh}\left (c x \right )\right )}{256 \left (c x +1\right ) c^{3} \left (c x -1\right )}\right )\)	\(993\)
parts	\(a \left (f^{2} \left (\frac {x \sqrt {-c^{2} d \,x^{2}+d}}{2}+\frac {d \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{2 \sqrt {c^{2} d}}\right )+g^{2} \left (-\frac {x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{4 c^{2} d}+\frac {\frac {x \sqrt {-c^{2} d \,x^{2}+d}}{2}+\frac {d \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{2 \sqrt {c^{2} d}}}{4 c^{2}}\right )-\frac {2 f g \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{3 c^{2} d}\right )+b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right )^{2} \left (4 c^{2} f^{2}+g^{2}\right )}{16 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3}}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (8 c^{5} x^{5}-12 c^{3} x^{3}+8 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{4} x^{4}+4 c x -8 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}+\sqrt {c x -1}\, \sqrt {c x +1}\right ) g^{2} \left (-1+4 \,\operatorname {arccosh}\left (c x \right )\right )}{256 \left (c x +1\right ) c^{3} \left (c x -1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (4 c^{4} x^{4}-5 c^{2} x^{2}+4 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{3}-3 \sqrt {c x -1}\, \sqrt {c x +1}\, c x +1\right ) f g \left (-1+3 \,\operatorname {arccosh}\left (c x \right )\right )}{36 \left (c x +1\right ) c^{2} \left (c x -1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 c^{3} x^{3}-2 c x +2 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-\sqrt {c x -1}\, \sqrt {c x +1}\right ) f^{2} \left (-1+2 \,\operatorname {arccosh}\left (c x \right )\right )}{16 \left (c x +1\right ) c \left (c x -1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) f g \left (-1+\operatorname {arccosh}\left (c x \right )\right )}{4 \left (c x +1\right ) c^{2} \left (c x -1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) f g \left (1+\operatorname {arccosh}\left (c x \right )\right )}{4 \left (c x +1\right ) c^{2} \left (c x -1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-2 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}+2 c^{3} x^{3}+\sqrt {c x -1}\, \sqrt {c x +1}-2 c x \right ) f^{2} \left (1+2 \,\operatorname {arccosh}\left (c x \right )\right )}{16 \left (c x +1\right ) c \left (c x -1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-4 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{3}+4 c^{4} x^{4}+3 \sqrt {c x -1}\, \sqrt {c x +1}\, c x -5 c^{2} x^{2}+1\right ) f g \left (1+3 \,\operatorname {arccosh}\left (c x \right )\right )}{36 \left (c x +1\right ) c^{2} \left (c x -1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-8 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{4} x^{4}+8 c^{5} x^{5}+8 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-12 c^{3} x^{3}-\sqrt {c x -1}\, \sqrt {c x +1}+4 c x \right ) g^{2} \left (1+4 \,\operatorname {arccosh}\left (c x \right )\right )}{256 \left (c x +1\right ) c^{3} \left (c x -1\right )}\right )\)	\(993\)

[In]

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

[Out]

a*(f^2*(1/2*x*(-c^2*d*x^2+d)^(1/2)+1/2*d/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2)))+g^2*(-1/4
*x*(-c^2*d*x^2+d)^(3/2)/c^2/d+1/4/c^2*(1/2*x*(-c^2*d*x^2+d)^(1/2)+1/2*d/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(
-c^2*d*x^2+d)^(1/2))))-2/3*f*g*(-c^2*d*x^2+d)^(3/2)/c^2/d)+b*(-1/16*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+
1)^(1/2)/c^3*arccosh(c*x)^2*(4*c^2*f^2+g^2)+1/256*(-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)*c^4*x^4+4*c*x-8*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^2*x^2+(c*x-1)^(1/2)*(c*x+1)^(1/2))*g^2*(-1+4*arcc
osh(c*x))/(c*x+1)/c^3/(c*x-1)+1/36*(-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)*c
^3*x^3-3*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x+1)*f*g*(-1+3*arccosh(c*x))/(c*x+1)/c^2/(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)*c^2*x^2-(c*x-1)^(1/2)*(c*x+1)^(1/2))*f^2*(-1+2*arccosh(c*
x))/(c*x+1)/c/(c*x-1)-1/4*(-d*(c^2*x^2-1))^(1/2)*((c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x+c^2*x^2-1)*f*g*(-1+arccosh(c
*x))/(c*x+1)/c^2/(c*x-1)-1/4*(-d*(c^2*x^2-1))^(1/2)*(-(c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x+c^2*x^2-1)*f*g*(1+arccos
h(c*x))/(c*x+1)/c^2/(c*x-1)+1/16*(-d*(c^2*x^2-1))^(1/2)*(-2*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^2*x^2+2*c^3*x^3+(c*x
-1)^(1/2)*(c*x+1)^(1/2)-2*c*x)*f^2*(1+2*arccosh(c*x))/(c*x+1)/c/(c*x-1)+1/36*(-d*(c^2*x^2-1))^(1/2)*(-4*(c*x-1
)^(1/2)*(c*x+1)^(1/2)*c^3*x^3+4*c^4*x^4+3*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x-5*c^2*x^2+1)*f*g*(1+3*arccosh(c*x))/
(c*x+1)/c^2/(c*x-1)+1/256*(-d*(c^2*x^2-1))^(1/2)*(-8*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^4*x^4+8*c^5*x^5+8*(c*x-1)^(
1/2)*(c*x+1)^(1/2)*c^2*x^2-12*c^3*x^3-(c*x-1)^(1/2)*(c*x+1)^(1/2)+4*c*x)*g^2*(1+4*arccosh(c*x))/(c*x+1)/c^3/(c
*x-1))

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

integrate((g*x+f)**2*(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)**2, x)

Maxima [F]

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

[In]

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

Giac [F(-2)]

Exception generated. \[ \int (f+g x)^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate((g*x+f)^2*(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

Mupad [F(-1)]

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

[In]

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

[Out]

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