3.66 \(\int \frac {(f+g x)^3 (a+b \cosh ^{-1}(c x))}{\sqrt {d-c^2 d x^2}} \, dx\)

Optimal. Leaf size=478 \[ \frac {f^3 \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c \sqrt {d-c^2 d x^2}}-\frac {3 f^2 g (1-c x) (c x+1) \left (a+b \cosh ^{-1}(c x)\right )}{c^2 \sqrt {d-c^2 d x^2}}-\frac {3 f g^2 x (1-c x) (c x+1) \left (a+b \cosh ^{-1}(c x)\right )}{2 c^2 \sqrt {d-c^2 d x^2}}-\frac {g^3 x^2 (1-c x) (c x+1) \left (a+b \cosh ^{-1}(c x)\right )}{3 c^2 \sqrt {d-c^2 d x^2}}-\frac {2 g^3 (1-c x) (c x+1) \left (a+b \cosh ^{-1}(c x)\right )}{3 c^4 \sqrt {d-c^2 d x^2}}+\frac {3 f g^2 \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )^2}{4 b c^3 \sqrt {d-c^2 d x^2}}-\frac {3 b f^2 g x \sqrt {c x-1} \sqrt {c x+1}}{c \sqrt {d-c^2 d x^2}}-\frac {3 b f g^2 x^2 \sqrt {c x-1} \sqrt {c x+1}}{4 c \sqrt {d-c^2 d x^2}}-\frac {b g^3 x^3 \sqrt {c x-1} \sqrt {c x+1}}{9 c \sqrt {d-c^2 d x^2}}-\frac {2 b g^3 x \sqrt {c x-1} \sqrt {c x+1}}{3 c^3 \sqrt {d-c^2 d x^2}} \]

[Out]

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

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Rubi [A]  time = 1.29, antiderivative size = 478, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 7, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {5836, 5822, 5676, 5718, 8, 5759, 30} \[ -\frac {3 f^2 g (1-c x) (c x+1) \left (a+b \cosh ^{-1}(c x)\right )}{c^2 \sqrt {d-c^2 d x^2}}+\frac {f^3 \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c \sqrt {d-c^2 d x^2}}+\frac {3 f g^2 \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )^2}{4 b c^3 \sqrt {d-c^2 d x^2}}-\frac {3 f g^2 x (1-c x) (c x+1) \left (a+b \cosh ^{-1}(c x)\right )}{2 c^2 \sqrt {d-c^2 d x^2}}-\frac {2 g^3 (1-c x) (c x+1) \left (a+b \cosh ^{-1}(c x)\right )}{3 c^4 \sqrt {d-c^2 d x^2}}-\frac {g^3 x^2 (1-c x) (c x+1) \left (a+b \cosh ^{-1}(c x)\right )}{3 c^2 \sqrt {d-c^2 d x^2}}-\frac {3 b f^2 g x \sqrt {c x-1} \sqrt {c x+1}}{c \sqrt {d-c^2 d x^2}}-\frac {3 b f g^2 x^2 \sqrt {c x-1} \sqrt {c x+1}}{4 c \sqrt {d-c^2 d x^2}}-\frac {b g^3 x^3 \sqrt {c x-1} \sqrt {c x+1}}{9 c \sqrt {d-c^2 d x^2}}-\frac {2 b g^3 x \sqrt {c x-1} \sqrt {c x+1}}{3 c^3 \sqrt {d-c^2 d x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*b*f^2*g*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(c*Sqrt[d - c^2*d*x^2]) - (2*b*g^3*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])
/(3*c^3*Sqrt[d - c^2*d*x^2]) - (3*b*f*g^2*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(4*c*Sqrt[d - c^2*d*x^2]) - (b*g^3
*x^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(9*c*Sqrt[d - c^2*d*x^2]) - (3*f^2*g*(1 - c*x)*(1 + c*x)*(a + b*ArcCosh[c*x
]))/(c^2*Sqrt[d - c^2*d*x^2]) - (2*g^3*(1 - c*x)*(1 + c*x)*(a + b*ArcCosh[c*x]))/(3*c^4*Sqrt[d - c^2*d*x^2]) -
(3*f*g^2*x*(1 - c*x)*(1 + c*x)*(a + b*ArcCosh[c*x]))/(2*c^2*Sqrt[d - c^2*d*x^2]) - (g^3*x^2*(1 - c*x)*(1 + c*
x)*(a + b*ArcCosh[c*x]))/(3*c^2*Sqrt[d - c^2*d*x^2]) + (f^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^
2)/(2*b*c*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)/(4*b*c^3*Sqrt[d
- c^2*d*x^2])

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, 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 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 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 \frac {(f+g x)^3 \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt {d-c^2 d x^2}} \, dx &=\frac {\left (\sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {(f+g x)^3 \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{\sqrt {d-c^2 d x^2}}\\ &=\frac {\left (\sqrt {-1+c x} \sqrt {1+c x}\right ) \int \left (\frac {f^3 \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 f^2 g x \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 f g^2 x^2 \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {g^3 x^3 \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}}\right ) \, dx}{\sqrt {d-c^2 d x^2}}\\ &=\frac {\left (f^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {a+b \cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{\sqrt {d-c^2 d x^2}}+\frac {\left (3 f^2 g \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{\sqrt {d-c^2 d x^2}}+\frac {\left (3 f g^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x^2 \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{\sqrt {d-c^2 d x^2}}+\frac {\left (g^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x^3 \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{\sqrt {d-c^2 d x^2}}\\ &=-\frac {3 f^2 g (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{c^2 \sqrt {d-c^2 d x^2}}-\frac {3 f g^2 x (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{2 c^2 \sqrt {d-c^2 d x^2}}-\frac {g^3 x^2 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{3 c^2 \sqrt {d-c^2 d x^2}}+\frac {f^3 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c \sqrt {d-c^2 d x^2}}-\frac {\left (3 b f^2 g \sqrt {-1+c x} \sqrt {1+c x}\right ) \int 1 \, dx}{c \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 \cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{2 c^2 \sqrt {d-c^2 d x^2}}-\frac {\left (3 b f g^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int x \, dx}{2 c \sqrt {d-c^2 d x^2}}+\frac {\left (2 g^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{3 c^2 \sqrt {d-c^2 d x^2}}-\frac {\left (b g^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int x^2 \, dx}{3 c \sqrt {d-c^2 d x^2}}\\ &=-\frac {3 b f^2 g x \sqrt {-1+c x} \sqrt {1+c x}}{c \sqrt {d-c^2 d x^2}}-\frac {3 b f g^2 x^2 \sqrt {-1+c x} \sqrt {1+c x}}{4 c \sqrt {d-c^2 d x^2}}-\frac {b g^3 x^3 \sqrt {-1+c x} \sqrt {1+c x}}{9 c \sqrt {d-c^2 d x^2}}-\frac {3 f^2 g (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{c^2 \sqrt {d-c^2 d x^2}}-\frac {2 g^3 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{3 c^4 \sqrt {d-c^2 d x^2}}-\frac {3 f g^2 x (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{2 c^2 \sqrt {d-c^2 d x^2}}-\frac {g^3 x^2 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{3 c^2 \sqrt {d-c^2 d x^2}}+\frac {f^3 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c \sqrt {d-c^2 d x^2}}+\frac {3 f g^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{4 b c^3 \sqrt {d-c^2 d x^2}}-\frac {\left (2 b g^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int 1 \, dx}{3 c^3 \sqrt {d-c^2 d x^2}}\\ &=-\frac {3 b f^2 g x \sqrt {-1+c x} \sqrt {1+c x}}{c \sqrt {d-c^2 d x^2}}-\frac {2 b g^3 x \sqrt {-1+c x} \sqrt {1+c x}}{3 c^3 \sqrt {d-c^2 d x^2}}-\frac {3 b f g^2 x^2 \sqrt {-1+c x} \sqrt {1+c x}}{4 c \sqrt {d-c^2 d x^2}}-\frac {b g^3 x^3 \sqrt {-1+c x} \sqrt {1+c x}}{9 c \sqrt {d-c^2 d x^2}}-\frac {3 f^2 g (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{c^2 \sqrt {d-c^2 d x^2}}-\frac {2 g^3 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{3 c^4 \sqrt {d-c^2 d x^2}}-\frac {3 f g^2 x (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{2 c^2 \sqrt {d-c^2 d x^2}}-\frac {g^3 x^2 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{3 c^2 \sqrt {d-c^2 d x^2}}+\frac {f^3 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c \sqrt {d-c^2 d x^2}}+\frac {3 f g^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{4 b c^3 \sqrt {d-c^2 d x^2}}\\ \end {align*}

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Mathematica [A]  time = 2.09, size = 405, normalized size = 0.85 \[ \frac {-\frac {36 a c f \left (2 c^2 f^2+3 g^2\right ) \tan ^{-1}\left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (c^2 x^2-1\right )}\right )}{\sqrt {d}}-\frac {12 a \sqrt {d-c^2 d x^2} \left (c^2 g \left (18 f^2+9 f g x+2 g^2 x^2\right )+4 g^3\right )}{d}+\frac {216 b c^2 f^2 g \sqrt {d-c^2 d x^2} \left (c x-\sqrt {c x-1} \sqrt {c x+1} \cosh ^{-1}(c x)\right )}{d \sqrt {c x-1} \sqrt {c x+1}}+\frac {27 b c f g^2 \sqrt {\frac {c x-1}{c x+1}} (c x+1) \left (2 \cosh ^{-1}(c x) \left (\cosh ^{-1}(c x)+\sinh \left (2 \cosh ^{-1}(c x)\right )\right )-\cosh \left (2 \cosh ^{-1}(c x)\right )\right )}{\sqrt {d-c^2 d x^2}}+\frac {8 b g^3 \sqrt {d-c^2 d x^2} \left (c x \left (c^2 x^2+6\right )-3 \sqrt {c x-1} \sqrt {c x+1} \left (c^2 x^2+2\right ) \cosh ^{-1}(c x)\right )}{d \sqrt {c x-1} \sqrt {c x+1}}+\frac {36 b c^3 f^3 \sqrt {\frac {c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x)^2}{\sqrt {d-c^2 d x^2}}}{72 c^4} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((-12*a*Sqrt[d - c^2*d*x^2]*(4*g^3 + c^2*g*(18*f^2 + 9*f*g*x + 2*g^2*x^2)))/d + (36*b*c^3*f^3*Sqrt[(-1 + c*x)/
(1 + c*x)]*(1 + c*x)*ArcCosh[c*x]^2)/Sqrt[d - c^2*d*x^2] + (216*b*c^2*f^2*g*Sqrt[d - c^2*d*x^2]*(c*x - Sqrt[-1
+ c*x]*Sqrt[1 + c*x]*ArcCosh[c*x]))/(d*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (8*b*g^3*Sqrt[d - c^2*d*x^2]*(c*x*(6 +
c^2*x^2) - 3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(2 + c^2*x^2)*ArcCosh[c*x]))/(d*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (36
*a*c*f*(2*c^2*f^2 + 3*g^2)*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))])/Sqrt[d] + (27*b*c*f*g^2
*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*(-Cosh[2*ArcCosh[c*x]] + 2*ArcCosh[c*x]*(ArcCosh[c*x] + Sinh[2*ArcCosh[c
*x]])))/Sqrt[d - c^2*d*x^2])/(72*c^4)

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fricas [F]  time = 0.63, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (a g^{3} x^{3} + 3 \, a f g^{2} x^{2} + 3 \, a f^{2} g x + a f^{3} + {\left (b g^{3} x^{3} + 3 \, b f g^{2} x^{2} + 3 \, b f^{2} g x + b f^{3}\right )} \operatorname {arcosh}\left (c x\right )\right )} \sqrt {-c^{2} d x^{2} + d}}{c^{2} d x^{2} - d}, x\right ) \]

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)/(c^2*d*x^2 - d), x)

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

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]

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

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maple [B]  time = 1.04, size = 859, normalized size = 1.80 \[ -\frac {a \,g^{3} x^{2} \sqrt {-c^{2} d \,x^{2}+d}}{3 c^{2} d}-\frac {2 a \,g^{3} \sqrt {-c^{2} d \,x^{2}+d}}{3 d \,c^{4}}-\frac {3 a f \,g^{2} x \sqrt {-c^{2} d \,x^{2}+d}}{2 c^{2} d}+\frac {3 a f \,g^{2} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{2 c^{2} \sqrt {c^{2} d}}-\frac {3 a \,f^{2} g \sqrt {-c^{2} d \,x^{2}+d}}{c^{2} d}+\frac {a \,f^{3} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{\sqrt {c^{2} d}}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, f^{3} \mathrm {arccosh}\left (c x \right )^{2}}{2 d c \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, g^{3} \sqrt {c x +1}\, \sqrt {c x -1}\, x^{3}}{9 c d \left (c^{2} x^{2}-1\right )}+\frac {2 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, g^{3} \sqrt {c x +1}\, \sqrt {c x -1}\, x}{3 c^{3} d \left (c^{2} x^{2}-1\right )}-\frac {3 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, f \,g^{2} \mathrm {arccosh}\left (c x \right ) x^{3}}{2 d \left (c^{2} x^{2}-1\right )}+\frac {3 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, f \,g^{2} \mathrm {arccosh}\left (c x \right ) x}{2 d \,c^{2} \left (c^{2} x^{2}-1\right )}-\frac {3 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, f \,g^{2} \sqrt {c x -1}\, \sqrt {c x +1}}{8 d \,c^{3} \left (c^{2} x^{2}-1\right )}-\frac {3 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, g \,\mathrm {arccosh}\left (c x \right ) x^{2} f^{2}}{d \left (c^{2} x^{2}-1\right )}-\frac {3 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, f \mathrm {arccosh}\left (c x \right )^{2} g^{2}}{4 d \,c^{3} \left (c^{2} x^{2}-1\right )}+\frac {3 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, f \,g^{2} \sqrt {c x +1}\, \sqrt {c x -1}\, x^{2}}{4 d c \left (c^{2} x^{2}-1\right )}+\frac {3 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, g \sqrt {c x +1}\, \sqrt {c x -1}\, x \,f^{2}}{c d \left (c^{2} x^{2}-1\right )}+\frac {2 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, g^{3} \mathrm {arccosh}\left (c x \right )}{3 c^{4} d \left (c^{2} x^{2}-1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, g^{3} \mathrm {arccosh}\left (c x \right ) x^{4}}{3 d \left (c^{2} x^{2}-1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, g^{3} \mathrm {arccosh}\left (c x \right ) x^{2}}{3 c^{2} d \left (c^{2} x^{2}-1\right )}+\frac {3 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, g \,\mathrm {arccosh}\left (c x \right ) f^{2}}{c^{2} d \left (c^{2} x^{2}-1\right )} \]

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)

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {1}{3} \, a g^{3} {\left (\frac {\sqrt {-c^{2} d x^{2} + d} x^{2}}{c^{2} d} + \frac {2 \, \sqrt {-c^{2} d x^{2} + d}}{c^{4} d}\right )} - \frac {3}{2} \, a f g^{2} {\left (\frac {\sqrt {-c^{2} d x^{2} + d} x}{c^{2} d} - \frac {\arcsin \left (c x\right )}{c^{3} \sqrt {d}}\right )} + \frac {a f^{3} \arcsin \left (c x\right )}{c \sqrt {d}} - \frac {3 \, \sqrt {-c^{2} d x^{2} + d} a f^{2} g}{c^{2} d} + \int \frac {b g^{3} x^{3} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )}{\sqrt {-c^{2} d x^{2} + d}} + \frac {3 \, b f g^{2} x^{2} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )}{\sqrt {-c^{2} d x^{2} + d}} + \frac {3 \, b f^{2} g x \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )}{\sqrt {-c^{2} d x^{2} + d}} + \frac {b f^{3} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )}{\sqrt {-c^{2} d x^{2} + d}}\,{d x} \]

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]

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

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

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

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

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