3.1.60 \(\int (f+g x) (d-c^2 d x^2)^{3/2} (a+b \cosh ^{-1}(c x)) \, dx\) [60]

Optimal. Leaf size=398 \[ \frac {b d g x \sqrt {d-c^2 d x^2}}{5 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {5 b c d f x^2 \sqrt {d-c^2 d x^2}}{16 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b c d g x^3 \sqrt {d-c^2 d x^2}}{15 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 d f x^4 \sqrt {d-c^2 d x^2}}{16 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 d g x^5 \sqrt {d-c^2 d x^2}}{25 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3}{8} d f x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{4} d f x (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac {d g (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{5 c^2}-\frac {3 d f \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{16 b c \sqrt {-1+c x} \sqrt {1+c x}} \]

[Out]

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

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Rubi [A]
time = 0.48, antiderivative size = 398, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 11, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.379, Rules used = {5972, 5975, 5898, 5896, 5893, 30, 74, 14, 5915, 41, 200} \begin {gather*} \frac {3}{8} d f x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{4} d f x (1-c x) (c x+1) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac {3 d f \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{16 b c \sqrt {c x-1} \sqrt {c x+1}}-\frac {d g (1-c x)^2 (c x+1)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{5 c^2}-\frac {5 b c d f x^2 \sqrt {d-c^2 d x^2}}{16 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b d g x \sqrt {d-c^2 d x^2}}{5 c \sqrt {c x-1} \sqrt {c x+1}}-\frac {2 b c d g x^3 \sqrt {d-c^2 d x^2}}{15 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b c^3 d f x^4 \sqrt {d-c^2 d x^2}}{16 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b c^3 d g x^5 \sqrt {d-c^2 d x^2}}{25 \sqrt {c x-1} \sqrt {c x+1}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

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 74

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

Rule 200

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n)^p, x], x] /; FreeQ[{a, b}, x]
&& IGtQ[n, 0] && IGtQ[p, 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 5898

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d1_) + (e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(x_))^(p_.), x_Symbo
l] :> Simp[x*(d1 + e1*x)^p*(d2 + e2*x)^p*((a + b*ArcCosh[c*x])^n/(2*p + 1)), x] + (Dist[2*d1*d2*(p/(2*p + 1)),
Int[(d1 + e1*x)^(p - 1)*(d2 + e2*x)^(p - 1)*(a + b*ArcCosh[c*x])^n, x], x] - Dist[b*c*(n/(2*p + 1))*Simp[(d1
+ e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*x)^p/(-1 + c*x)^p], Int[x*(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}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] &&
GtQ[n, 0] && GtQ[p, 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 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) \left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right ) \, dx &=-\frac {\left (d \sqrt {d-c^2 d x^2}\right ) \int (-1+c x)^{3/2} (1+c x)^{3/2} (f+g x) \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {\left (d \sqrt {d-c^2 d x^2}\right ) \int \left (f (-1+c x)^{3/2} (1+c x)^{3/2} \left (a+b \cosh ^{-1}(c x)\right )+g x (-1+c x)^{3/2} (1+c x)^{3/2} \left (a+b \cosh ^{-1}(c x)\right )\right ) \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {\left (d f \sqrt {d-c^2 d x^2}\right ) \int (-1+c x)^{3/2} (1+c x)^{3/2} \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (d g \sqrt {d-c^2 d x^2}\right ) \int x (-1+c x)^{3/2} (1+c x)^{3/2} \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {1}{4} d f x (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac {d g (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{5 c^2}+\frac {\left (3 d f \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}{4 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b c d f \sqrt {d-c^2 d x^2}\right ) \int x \left (-1+c^2 x^2\right ) \, dx}{4 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b d g \sqrt {d-c^2 d x^2}\right ) \int \left (-1+c^2 x^2\right )^2 \, dx}{5 c \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {3}{8} d f x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{4} d f x (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac {d g (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{5 c^2}-\frac {\left (3 d f \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 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b c d f \sqrt {d-c^2 d x^2}\right ) \int \left (-x+c^2 x^3\right ) \, dx}{4 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (3 b c d f \sqrt {d-c^2 d x^2}\right ) \int x \, dx}{8 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b d g \sqrt {d-c^2 d x^2}\right ) \int \left (1-2 c^2 x^2+c^4 x^4\right ) \, dx}{5 c \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {b d g x \sqrt {d-c^2 d x^2}}{5 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {5 b c d f x^2 \sqrt {d-c^2 d x^2}}{16 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b c d g x^3 \sqrt {d-c^2 d x^2}}{15 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 d f x^4 \sqrt {d-c^2 d x^2}}{16 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 d g x^5 \sqrt {d-c^2 d x^2}}{25 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3}{8} d f x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{4} d f x (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac {d g (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{5 c^2}-\frac {3 d f \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{16 b c \sqrt {-1+c x} \sqrt {1+c x}}\\ \end {align*}

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Mathematica [A]
time = 1.03, size = 432, normalized size = 1.09 \begin {gather*} \frac {-720 a d \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \sqrt {d-c^2 d x^2} \left (8 g \left (-1+c^2 x^2\right )^2+5 c^2 f x \left (-5+2 c^2 x^2\right )\right )-10800 a c d^{3/2} f \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 )+800 b d 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 d f \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 )+225 b c d f \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 d g \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^2 \sqrt {\frac {-1+c x}{1+c x}} (1+c x)} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-720*a*d*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*Sqrt[d - c^2*d*x^2]*(8*g*(-1 + c^2*x^2)^2 + 5*c^2*f*x*(-5 + 2*c
^2*x^2)) - 10800*a*c*d^(3/2)*f*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))] + 800*b*d*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*d*f*Sqrt[d - c^2*d*x^2]*(Cosh[2*ArcCosh[c*x]] + 2*ArcCosh[c*x]*(ArcCos
h[c*x] - Sinh[2*ArcCosh[c*x]])) + 225*b*c*d*f*Sqrt[d - c^2*d*x^2]*(8*ArcCosh[c*x]^2 + Cosh[4*ArcCosh[c*x]] - 4
*ArcCosh[c*x]*Sinh[4*ArcCosh[c*x]]) - 8*b*d*g*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^2*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. \(1175\) vs. \(2(338)=676\).
time = 6.85, size = 1176, normalized size = 2.95

method	result	size
default	\(-\frac {a g \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{5 c^{2} d}+\frac {a f x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{4}+\frac {3 a f d x \sqrt {-c^{2} d \,x^{2}+d}}{8}+\frac {3 a f \,d^{2} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{8 \sqrt {c^{2} d}}+b \left (-\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, f \mathrm {arccosh}\left (c x \right )^{2} d}{16 \sqrt {c x -1}\, \sqrt {c x +1}\, c}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (16 x^{6} c^{6}-28 c^{4} x^{4}+16 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{5} c^{5}+13 c^{2} x^{2}-20 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{3} c^{3}+5 \sqrt {c x +1}\, \sqrt {c x -1}\, x c -1\right ) g \left (-1+5 \,\mathrm {arccosh}\left (c x \right )\right ) d}{800 \left (c x +1\right ) c^{2} \left (c x -1\right )}-\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}\, x^{4} c^{4}+4 c x -8 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{2} c^{2}+\sqrt {c x -1}\, \sqrt {c x +1}\right ) f \left (-1+4 \,\mathrm {arccosh}\left (c x \right )\right ) d}{256 \left (c x +1\right ) \left (c x -1\right ) c}+\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}\, x^{3} c^{3}-3 \sqrt {c x +1}\, \sqrt {c x -1}\, x c +1\right ) g \left (-1+3 \,\mathrm {arccosh}\left (c x \right )\right ) d}{96 \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}\, x^{2} c^{2}-\sqrt {c x -1}\, \sqrt {c x +1}\right ) f \left (2 \,\mathrm {arccosh}\left (c x \right )-1\right ) d}{16 \left (c x +1\right ) \left (c x -1\right ) c}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (\sqrt {c x +1}\, \sqrt {c x -1}\, x c +c^{2} x^{2}-1\right ) g \left (-1+\mathrm {arccosh}\left (c x \right )\right ) d}{16 \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}\, x c +c^{2} x^{2}-1\right ) g \left (1+\mathrm {arccosh}\left (c x \right )\right ) d}{16 \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}\, x^{2} c^{2}+2 c^{3} x^{3}+\sqrt {c x -1}\, \sqrt {c x +1}-2 c x \right ) f \left (2 \,\mathrm {arccosh}\left (c x \right )+1\right ) d}{16 \left (c x +1\right ) \left (c x -1\right ) c}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-4 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{3} c^{3}+4 c^{4} x^{4}+3 \sqrt {c x +1}\, \sqrt {c x -1}\, x c -5 c^{2} x^{2}+1\right ) g \left (1+3 \,\mathrm {arccosh}\left (c x \right )\right ) d}{96 \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}\, x^{4} c^{4}+8 c^{5} x^{5}+8 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{2} c^{2}-12 c^{3} x^{3}-\sqrt {c x -1}\, \sqrt {c x +1}+4 c x \right ) f \left (1+4 \,\mathrm {arccosh}\left (c x \right )\right ) d}{256 \left (c x +1\right ) \left (c x -1\right ) c}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-16 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{5} c^{5}+16 x^{6} c^{6}+20 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{3} c^{3}-28 c^{4} x^{4}-5 \sqrt {c x +1}\, \sqrt {c x -1}\, x c +13 c^{2} x^{2}-1\right ) g \left (1+5 \,\mathrm {arccosh}\left (c x \right )\right ) d}{800 \left (c x +1\right ) c^{2} \left (c x -1\right )}\right )\)	\(1176\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/5*a*g*(-c^2*d*x^2+d)^(5/2)/c^2/d+1/4*a*f*x*(-c^2*d*x^2+d)^(3/2)+3/8*a*f*d*x*(-c^2*d*x^2+d)^(1/2)+3/8*a*f*d^
2/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))+b*(-3/16*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*
x+1)^(1/2)/c*f*arccosh(c*x)^2*d-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*(-1+5*ar
ccosh(c*x))*d/(c*x+1)/c^2/(c*x-1)-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)*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*(-1+4*arccosh(c*x))*d/
(c*x+1)/(c*x-1)/c+1/96*(-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*(-1+3*arccosh(c*x))*d/(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)*x^2*c^2-(c*x-1)^(1/2)*(c*x+1)^(1/2))*f*(2*arccosh(c*x)-1)*d/(c*x+1)/(
c*x-1)/c-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*(-1+arccosh(c*x))*d/(c*x+1)
/c^2/(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*(1+arccosh(c*x))*d/(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)*x^2*c^2+2*c^3*x^3+(c*x-1)^(1/2)*(
c*x+1)^(1/2)-2*c*x)*f*(2*arccosh(c*x)+1)*d/(c*x+1)/(c*x-1)/c+1/96*(-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*(1+3*arccosh(c*x))*d/(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)*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*(1+4*arccosh(c*x))*d/(c*x+1)/(c*x-1)/c-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*(1+5*arccosh(c*x))*d/(c*x+1)/c^2/(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)*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x)),x, algorithm="maxima")

[Out]

1/8*(2*(-c^2*d*x^2 + d)^(3/2)*x + 3*sqrt(-c^2*d*x^2 + d)*d*x + 3*d^(3/2)*arcsin(c*x)/c)*a*f - 1/5*(-c^2*d*x^2
+ d)^(5/2)*a*g/(c^2*d) + integrate((-c^2*d*x^2 + d)^(3/2)*b*g*x*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1)) + (-c^2
*d*x^2 + d)^(3/2)*b*f*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)*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x)),x, algorithm="fricas")

[Out]

integral(-(a*c^2*d*g*x^3 + a*c^2*d*f*x^2 - a*d*g*x - a*d*f + (b*c^2*d*g*x^3 + b*c^2*d*f*x^2 - b*d*g*x - b*d*f)
*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 \left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {acosh}{\left (c x \right )}\right ) \left (f + g x\right )\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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