3.54 \(\int (f+g x)^2 \sqrt {d-c^2 d x^2} (a+b \cosh ^{-1}(c x)) \, dx\)

Optimal. Leaf size=479 \[ \frac {1}{2} f^2 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac {f^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{4 b c \sqrt {c x-1} \sqrt {c x+1}}-\frac {2 f g (1-c x) (c x+1) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 c^2}-\frac {g^2 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 c^2}+\frac {1}{4} g^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac {g^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{16 b c^3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c f^2 x^2 \sqrt {d-c^2 d x^2}}{4 \sqrt {c x-1} \sqrt {c x+1}}+\frac {2 b f g x \sqrt {d-c^2 d x^2}}{3 c \sqrt {c x-1} \sqrt {c x+1}}-\frac {2 b c f g x^3 \sqrt {d-c^2 d x^2}}{9 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b g^2 x^2 \sqrt {d-c^2 d x^2}}{16 c \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c g^2 x^4 \sqrt {d-c^2 d x^2}}{16 \sqrt {c x-1} \sqrt {c x+1}} \]

[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]  time = 1.18, antiderivative size = 479, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 8, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.258, Rules used = {5836, 5822, 5683, 5676, 30, 5718, 5743, 5759} \[ \frac {1}{2} f^2 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac {f^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{4 b c \sqrt {c x-1} \sqrt {c x+1}}-\frac {2 f g (1-c x) (c x+1) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 c^2}+\frac {1}{4} g^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac {g^2 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 c^2}-\frac {g^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{16 b c^3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c f^2 x^2 \sqrt {d-c^2 d x^2}}{4 \sqrt {c x-1} \sqrt {c x+1}}-\frac {2 b c f g x^3 \sqrt {d-c^2 d x^2}}{9 \sqrt {c x-1} \sqrt {c x+1}}+\frac {2 b f g x \sqrt {d-c^2 d x^2}}{3 c \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c g^2 x^4 \sqrt {d-c^2 d x^2}}{16 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b g^2 x^2 \sqrt {d-c^2 d x^2}}{16 c \sqrt {c x-1} \sqrt {c x+1}} \]

Antiderivative was successfully verified.

[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 5676

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

Rule 5683

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[(Sqrt[d1 + e1*x]*Sqrt[d2 + e2
*x])/(2*Sqrt[1 + c*x]*Sqrt[-1 + c*x]), Int[(a + b*ArcCosh[c*x])^n/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x], x] - Dis
t[(b*c*n*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x])/(2*Sqrt[1 + c*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 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 5743

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[(Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x])/((m + 2)*Sqrt[1 + c*x]*Sqrt[-1 + c*x]), Int[((f*x)^m*(a + b*ArcCo
sh[c*x])^n)/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x], x] - Dist[(b*c*n*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x])/(f*(m + 2)*S
qrt[1 + c*x]*Sqrt[-1 + c*x]), Int[(f*x)^(m + 1)*(a + b*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d1, e
1, d2, e2, f, m}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && GtQ[n, 0] &&  !LtQ[m, -1] && (RationalQ[m] |
| EqQ[n, 1])

Rule 5759

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

Rule 5822

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]))

Rule 5836

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[m] && IntegerQ[p - 1/2]

Rubi steps

\begin {align*} \int (f+g x)^2 \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)^2 \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^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )+2 f g x \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )+g^2 x^2 \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^2 \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 (2 f 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 (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 {1}{2} f^2 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{4} g^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac {2 f g (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 c^2}-\frac {\left (f^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}{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 \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 {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 (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 {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 {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} \left (a+b \cosh ^{-1}(c x)\right )-\frac {g^2 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 c^2}+\frac {1}{4} g^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac {2 f g (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 c^2}-\frac {f^2 \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 (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 (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} \left (a+b \cosh ^{-1}(c x)\right )-\frac {g^2 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 c^2}+\frac {1}{4} g^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac {2 f g (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 c^2}-\frac {f^2 \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 {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*}

________________________________________________________________________________________

Mathematica [A]  time = 1.21, size = 356, normalized size = 0.74 \[ \frac {48 a c \sqrt {\frac {c x-1}{c x+1}} (c x+1) \sqrt {d-c^2 d x^2} \left (12 c^2 f^2 x+16 f g \left (c^2 x^2-1\right )+3 g^2 x \left (2 c^2 x^2-1\right )\right )-144 a \sqrt {d} \sqrt {\frac {c x-1}{c x+1}} (c x+1) \left (4 c^2 f^2+g^2\right ) \tan ^{-1}\left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (c^2 x^2-1\right )}\right )-144 b c^2 f^2 \sqrt {d-c^2 d x^2} \left (\cosh \left (2 \cosh ^{-1}(c x)\right )+2 \cosh ^{-1}(c x) \left (\cosh ^{-1}(c x)-\sinh \left (2 \cosh ^{-1}(c x)\right )\right )\right )+64 b c f g \sqrt {d-c^2 d x^2} \left (9 c x+12 \left (\frac {c x-1}{c x+1}\right )^{3/2} (c x+1)^3 \cosh ^{-1}(c x)-\cosh \left (3 \cosh ^{-1}(c x)\right )\right )-9 b g^2 \sqrt {d-c^2 d x^2} \left (8 \cosh ^{-1}(c x)^2+\cosh \left (4 \cosh ^{-1}(c x)\right )-4 \cosh ^{-1}(c x) \sinh \left (4 \cosh ^{-1}(c x)\right )\right )}{1152 c^3 \sqrt {\frac {c x-1}{c x+1}} (c x+1)} \]

Warning: Unable to verify antiderivative.

[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))

________________________________________________________________________________________

fricas [F]  time = 0.68, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {-c^{2} d x^{2} + d} {\left (a g^{2} x^{2} + 2 \, a f g x + a f^{2} + {\left (b g^{2} x^{2} + 2 \, b f g x + b f^{2}\right )} \operatorname {arcosh}\left (c x\right )\right )}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

________________________________________________________________________________________

maple [B]  time = 0.94, size = 855, normalized size = 1.78 \[ -\frac {a \,g^{2} x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{4 c^{2} d}+\frac {a \,g^{2} x \sqrt {-c^{2} d \,x^{2}+d}}{8 c^{2}}+\frac {a \,g^{2} d \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{8 c^{2} \sqrt {c^{2} d}}-\frac {2 a f g \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{3 c^{2} d}+\frac {a \,f^{2} x \sqrt {-c^{2} d \,x^{2}+d}}{2}+\frac {a \,f^{2} d \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{2 \sqrt {c^{2} d}}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right )^{2} f^{2}}{4 \sqrt {c x -1}\, \sqrt {c x +1}\, c}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right )^{2} g^{2}}{16 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3}}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, g^{2} c^{2} \mathrm {arccosh}\left (c x \right ) x^{5}}{4 \left (c x +1\right ) \left (c x -1\right )}-\frac {3 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, g^{2} \mathrm {arccosh}\left (c x \right ) x^{3}}{8 \left (c x +1\right ) \left (c x -1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, g^{2} \mathrm {arccosh}\left (c x \right ) x}{8 \left (c x +1\right ) c^{2} \left (c x -1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, g^{2}}{128 \sqrt {c x +1}\, c^{3} \sqrt {c x -1}}+\frac {2 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, f g \,\mathrm {arccosh}\left (c x \right )}{3 \left (c x +1\right ) c^{2} \left (c x -1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, f^{2} c^{2} \mathrm {arccosh}\left (c x \right ) x^{3}}{2 \left (c x +1\right ) \left (c x -1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, f^{2} \mathrm {arccosh}\left (c x \right ) x}{2 \left (c x +1\right ) \left (c x -1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, f^{2}}{8 \sqrt {c x +1}\, c \sqrt {c x -1}}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, f^{2} c \,x^{2}}{4 \sqrt {c x +1}\, \sqrt {c x -1}}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, g^{2} c \,x^{4}}{16 \sqrt {c x +1}\, \sqrt {c x -1}}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, g^{2} x^{2}}{16 \sqrt {c x +1}\, c \sqrt {c x -1}}+\frac {2 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, f g \,c^{2} \mathrm {arccosh}\left (c x \right ) x^{4}}{3 \left (c x +1\right ) \left (c x -1\right )}-\frac {4 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, f g \,\mathrm {arccosh}\left (c x \right ) x^{2}}{3 \left (c x +1\right ) \left (c x -1\right )}-\frac {2 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, f g c \,x^{3}}{9 \sqrt {c x +1}\, \sqrt {c x -1}}+\frac {2 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, f g x}{3 \sqrt {c x +1}\, c \sqrt {c x -1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*a*g^2*x*(-c^2*d*x^2+d)^(3/2)/c^2/d+1/8*a*g^2/c^2*x*(-c^2*d*x^2+d)^(1/2)+1/8*a*g^2/c^2*d/(c^2*d)^(1/2)*arc
tan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))-2/3*a*f*g/c^2/d*(-c^2*d*x^2+d)^(3/2)+1/2*a*f^2*x*(-c^2*d*x^2+d)^(1/2
)+1/2*a*f^2*d/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))-1/4*b*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^
(1/2)/(c*x+1)^(1/2)/c*arccosh(c*x)^2*f^2-1/16*b*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/c^3*arccosh
(c*x)^2*g^2+1/4*b*(-d*(c^2*x^2-1))^(1/2)*g^2/(c*x+1)*c^2/(c*x-1)*arccosh(c*x)*x^5-3/8*b*(-d*(c^2*x^2-1))^(1/2)
*g^2/(c*x+1)/(c*x-1)*arccosh(c*x)*x^3+1/8*b*(-d*(c^2*x^2-1))^(1/2)*g^2/(c*x+1)/c^2/(c*x-1)*arccosh(c*x)*x-1/12
8*b*(-d*(c^2*x^2-1))^(1/2)*g^2/(c*x+1)^(1/2)/c^3/(c*x-1)^(1/2)+2/3*b*(-d*(c^2*x^2-1))^(1/2)*f*g/(c*x+1)/c^2/(c
*x-1)*arccosh(c*x)+1/2*b*(-d*(c^2*x^2-1))^(1/2)*f^2/(c*x+1)*c^2/(c*x-1)*arccosh(c*x)*x^3-1/2*b*(-d*(c^2*x^2-1)
)^(1/2)*f^2/(c*x+1)/(c*x-1)*arccosh(c*x)*x+1/8*b*(-d*(c^2*x^2-1))^(1/2)*f^2/(c*x+1)^(1/2)/c/(c*x-1)^(1/2)-1/4*
b*(-d*(c^2*x^2-1))^(1/2)*f^2/(c*x+1)^(1/2)*c/(c*x-1)^(1/2)*x^2-1/16*b*(-d*(c^2*x^2-1))^(1/2)*g^2/(c*x+1)^(1/2)
*c/(c*x-1)^(1/2)*x^4+1/16*b*(-d*(c^2*x^2-1))^(1/2)*g^2/(c*x+1)^(1/2)/c/(c*x-1)^(1/2)*x^2+2/3*b*(-d*(c^2*x^2-1)
)^(1/2)*f*g/(c*x+1)*c^2/(c*x-1)*arccosh(c*x)*x^4-4/3*b*(-d*(c^2*x^2-1))^(1/2)*f*g/(c*x+1)/(c*x-1)*arccosh(c*x)
*x^2-2/9*b*(-d*(c^2*x^2-1))^(1/2)*f*g/(c*x+1)^(1/2)*c/(c*x-1)^(1/2)*x^3+2/3*b*(-d*(c^2*x^2-1))^(1/2)*f*g/(c*x+
1)^(1/2)/c/(c*x-1)^(1/2)*x

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{2} \, {\left (\sqrt {-c^{2} d x^{2} + d} x + \frac {\sqrt {d} \arcsin \left (c x\right )}{c}\right )} a f^{2} + \frac {1}{8} \, a g^{2} {\left (\frac {\sqrt {-c^{2} d x^{2} + d} x}{c^{2}} - \frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x}{c^{2} d} + \frac {\sqrt {d} \arcsin \left (c x\right )}{c^{3}}\right )} - \frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} a f g}{3 \, c^{2} d} + \int \sqrt {-c^{2} d x^{2} + d} b g^{2} x^{2} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right ) + 2 \, \sqrt {-c^{2} d x^{2} + d} b f g x \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right ) + \sqrt {-c^{2} d x^{2} + d} b f^{2} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {acosh}{\left (c x \right )}\right ) \left (f + g x\right )^{2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________