3.1.71 \(\int \frac {(f+g x)^3 (a+b \cosh ^{-1}(c x))}{(d-c^2 d x^2)^{3/2}} \, dx\) [71]

Optimal. Leaf size=549 \[ \frac {b g^3 x \sqrt {-1+c x} \sqrt {1+c x}}{c^3 d \sqrt {d-c^2 d x^2}}-\frac {(c f-g)^3 (1-c x) \left (a+b \cosh ^{-1}(c x)\right )}{2 c^4 d \sqrt {d-c^2 d x^2}}+\frac {(c f+g)^3 (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{2 c^4 d \sqrt {d-c^2 d x^2}}+\frac {g^3 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{c^4 d \sqrt {d-c^2 d x^2}}-\frac {3 f g^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c^3 d \sqrt {d-c^2 d x^2}}+\frac {b (c f+g)^3 \sqrt {(1-c x) (1+c x)} \sqrt {1-c^2 x^2} \log \left (\sqrt {-\frac {1-c x}{1+c x}}\right )}{c^4 d \sqrt {-\frac {1-c x}{1+c x}} (1+c x) \sqrt {d-c^2 d x^2}}-\frac {b (c f-g)^3 \sqrt {(1-c x) (1+c x)} \sqrt {1-c^2 x^2} \log \left (\frac {2}{1+c x}\right )}{2 c^4 d \sqrt {-\frac {1-c x}{1+c x}} (1+c x) \sqrt {d-c^2 d x^2}}-\frac {b (c f+g)^3 \sqrt {(1-c x) (1+c x)} \sqrt {1-c^2 x^2} \log \left (\frac {2}{1+c x}\right )}{2 c^4 d \sqrt {-\frac {1-c x}{1+c x}} (1+c x) \sqrt {d-c^2 d x^2}} \]

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

Antiderivative was successfully verified.

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

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Mathematica [A]
time = 1.12, size = 353, normalized size = 0.64 \begin {gather*} \frac {-3 b c \sqrt {d} f g^2 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \cosh ^{-1}(c x)^2+6 a c f g^2 \sqrt {d-c^2 d x^2} \text {ArcTan}\left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )+b \sqrt {d} \cosh ^{-1}(c x) \left (3 g^3+2 c^4 f^3 x+6 c^2 f g (f+g x)-g^3 \cosh \left (2 \cosh ^{-1}(c x)\right )\right )-2 \sqrt {d} \left (-a \left (2 g^3+c^4 f^3 x+c^2 g \left (3 f^2+3 f g x-g^2 x^2\right )\right )+b c f \left (c^2 f^2+3 g^2\right ) \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 )+b g \left (3 c^2 f^2+g^2\right ) \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \log \left (\tanh \left (\frac {1}{2} \cosh ^{-1}(c x)\right )\right )\right )+b \sqrt {d} g^3 \sinh \left (2 \cosh ^{-1}(c x)\right )}{2 c^4 d^{3/2} \sqrt {d-c^2 d x^2}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-3*b*c*Sqrt[d]*f*g^2*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x]^2 + 6*a*c*f*g^2*Sqrt[d - c^2*d*x^2]*Ar
cTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] + b*Sqrt[d]*ArcCosh[c*x]*(3*g^3 + 2*c^4*f^3*x + 6*c^2
*f*g*(f + g*x) - g^3*Cosh[2*ArcCosh[c*x]]) - 2*Sqrt[d]*(-(a*(2*g^3 + c^4*f^3*x + c^2*g*(3*f^2 + 3*f*g*x - g^2*
x^2))) + b*c*f*(c^2*f^2 + 3*g^2)*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*Log[Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)
] + b*g*(3*c^2*f^2 + g^2)*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*Log[Tanh[ArcCosh[c*x]/2]]) + b*Sqrt[d]*g^3*Sinh
[2*ArcCosh[c*x]])/(2*c^4*d^(3/2)*Sqrt[d - c^2*d*x^2])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1237\) vs. \(2(486)=972\).
time = 9.01, size = 1238, normalized size = 2.26

method	result	size
default	\(-\frac {a \,g^{3} x^{2}}{c^{2} d \sqrt {-c^{2} d \,x^{2}+d}}+\frac {2 a \,g^{3}}{d \,c^{4} \sqrt {-c^{2} d \,x^{2}+d}}+\frac {3 a f \,g^{2} x}{c^{2} d \sqrt {-c^{2} d \,x^{2}+d}}-\frac {3 a f \,g^{2} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{c^{2} d \sqrt {c^{2} d}}+\frac {3 a \,f^{2} g}{c^{2} d \sqrt {-c^{2} d \,x^{2}+d}}+\frac {a \,f^{3} x}{d \sqrt {-c^{2} d \,x^{2}+d}}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}-1\right ) g^{3}}{c^{4} d^{2} \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) f^{3}}{c \,d^{2} \left (c^{2} x^{2}-1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) g^{3}}{c^{4} d^{2} \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}\, \mathrm {arccosh}\left (c x \right )^{2} f \,g^{2}}{2 d^{2} c^{3} \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 ) g^{2}}{d^{2} c^{3} \left (c^{2} x^{2}-1\right )}+\frac {3 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right ) \left (c x +1\right ) \left (c x -1\right ) f^{2} g}{c^{2} d^{2} \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}\, \ln \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}-1\right ) f^{2} g}{c^{2} d^{2} \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}\, \ln \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}-1\right ) f \,g^{2}}{c^{3} d^{2} \left (c^{2} x^{2}-1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right ) x \,f^{3}}{d^{2} \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}{c^{3} d^{2} \left (c^{2} x^{2}-1\right )}-\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 )}{d^{2} c \left (c^{2} x^{2}-1\right )}-\frac {3 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right ) x^{2} f^{2} g}{d^{2} \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right ) \left (c x +1\right ) \left (c x -1\right ) g^{3}}{c^{4} d^{2} \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}\, \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) f^{2} g}{c^{2} d^{2} \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}\, \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) f \,g^{2}}{c^{3} d^{2} \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}-1\right ) f^{3}}{c \,d^{2} \left (c^{2} x^{2}-1\right )}-\frac {3 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right ) x f \,g^{2}}{c^{2} d^{2} \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 )}{c^{4} d^{2} \left (c^{2} x^{2}-1\right )}\)	\(1238\)

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)^(3/2),x,method=_RETURNVERBOSE)

[Out]

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

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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)^(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)

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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)^(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)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 \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)**(3/2),x)

[Out]

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

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Giac [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)^(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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \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 \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)^(3/2),x)

[Out]

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

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