\(\int (d-c^2 d x^2) (a+b \text {arccosh}(c x))^3 \, dx\) [15]

Optimal result

Integrand size = 22, antiderivative size = 213 \[ \int \left (d-c^2 d x^2\right ) (a+b \text {arccosh}(c x))^3 \, dx=-\frac {122 b^3 d \sqrt {-1+c x} \sqrt {1+c x}}{27 c}+\frac {2}{27} b^3 c d x^2 \sqrt {-1+c x} \sqrt {1+c x}+\frac {14}{3} b^2 d x (a+b \text {arccosh}(c x))-\frac {2}{9} b^2 c^2 d x^3 (a+b \text {arccosh}(c x))-\frac {2 b d \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^2}{c}+\frac {b d (-1+c x)^{3/2} (1+c x)^{3/2} (a+b \text {arccosh}(c x))^2}{3 c}+\frac {2}{3} d x (a+b \text {arccosh}(c x))^3+\frac {1}{3} d x \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))^3 \] Output:

-122/27*b^3*d*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c+2/27*b^3*c*d*x^2*(c*x-1)^(1/2) 
*(c*x+1)^(1/2)+14/3*b^2*d*x*(a+b*arccosh(c*x))-2/9*b^2*c^2*d*x^3*(a+b*arcc 
osh(c*x))-2*b*d*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(a+b*arccosh(c*x))^2/c+1/3*b*d 
*(c*x-1)^(3/2)*(c*x+1)^(3/2)*(a+b*arccosh(c*x))^2/c+2/3*d*x*(a+b*arccosh(c 
*x))^3+1/3*d*x*(-c^2*x^2+1)*(a+b*arccosh(c*x))^3
 

Mathematica [A] (verified)

Time = 0.35 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.20 \[ \int \left (d-c^2 d x^2\right ) (a+b \text {arccosh}(c x))^3 \, dx=\frac {d \left (2 b^3 \sqrt {-1+c x} \sqrt {1+c x} \left (-61+c^2 x^2\right )-6 a b^2 c x \left (-21+c^2 x^2\right )+9 a^2 b \sqrt {-1+c x} \sqrt {1+c x} \left (-7+c^2 x^2\right )-9 a^3 c x \left (-3+c^2 x^2\right )+3 b \left (-2 b^2 c x \left (-21+c^2 x^2\right )+6 a b \sqrt {-1+c x} \sqrt {1+c x} \left (-7+c^2 x^2\right )-9 a^2 c x \left (-3+c^2 x^2\right )\right ) \text {arccosh}(c x)+9 b^2 \left (b \sqrt {-1+c x} \sqrt {1+c x} \left (-7+c^2 x^2\right )+a \left (9 c x-3 c^3 x^3\right )\right ) \text {arccosh}(c x)^2-9 b^3 c x \left (-3+c^2 x^2\right ) \text {arccosh}(c x)^3\right )}{27 c} \] Input:

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

Output:

(d*(2*b^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-61 + c^2*x^2) - 6*a*b^2*c*x*(-21 
+ c^2*x^2) + 9*a^2*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-7 + c^2*x^2) - 9*a^3*c 
*x*(-3 + c^2*x^2) + 3*b*(-2*b^2*c*x*(-21 + c^2*x^2) + 6*a*b*Sqrt[-1 + c*x] 
*Sqrt[1 + c*x]*(-7 + c^2*x^2) - 9*a^2*c*x*(-3 + c^2*x^2))*ArcCosh[c*x] + 9 
*b^2*(b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-7 + c^2*x^2) + a*(9*c*x - 3*c^3*x^3 
))*ArcCosh[c*x]^2 - 9*b^3*c*x*(-3 + c^2*x^2)*ArcCosh[c*x]^3))/(27*c)
 

Rubi [A] (verified)

Time = 2.12 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.21, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {6312, 6294, 6330, 25, 2009, 6304, 6309, 27, 960, 83}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \left (d-c^2 d x^2\right ) (a+b \text {arccosh}(c x))^3 \, dx\)

\(\Big \downarrow \) 6312

\(\displaystyle b c d \int x \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^2dx+\frac {2}{3} d \int (a+b \text {arccosh}(c x))^3dx+\frac {1}{3} d x \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))^3\)

\(\Big \downarrow \) 6294

\(\displaystyle \frac {2}{3} d \left (x (a+b \text {arccosh}(c x))^3-3 b c \int \frac {x (a+b \text {arccosh}(c x))^2}{\sqrt {c x-1} \sqrt {c x+1}}dx\right )+b c d \int x \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^2dx+\frac {1}{3} d x \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))^3\)

\(\Big \downarrow \) 6330

\(\displaystyle \frac {2}{3} d \left (x (a+b \text {arccosh}(c x))^3-3 b c \left (\frac {\sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^2}{c^2}-\frac {2 b \int (a+b \text {arccosh}(c x))dx}{c}\right )\right )+b c d \left (\frac {(c x-1)^{3/2} (c x+1)^{3/2} (a+b \text {arccosh}(c x))^2}{3 c^2}-\frac {2 b \int -((1-c x) (c x+1) (a+b \text {arccosh}(c x)))dx}{3 c}\right )+\frac {1}{3} d x \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))^3\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {2}{3} d \left (x (a+b \text {arccosh}(c x))^3-3 b c \left (\frac {\sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^2}{c^2}-\frac {2 b \int (a+b \text {arccosh}(c x))dx}{c}\right )\right )+b c d \left (\frac {2 b \int (1-c x) (c x+1) (a+b \text {arccosh}(c x))dx}{3 c}+\frac {(c x-1)^{3/2} (c x+1)^{3/2} (a+b \text {arccosh}(c x))^2}{3 c^2}\right )+\frac {1}{3} d x \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))^3\)

\(\Big \downarrow \) 2009

\(\displaystyle b c d \left (\frac {2 b \int (1-c x) (c x+1) (a+b \text {arccosh}(c x))dx}{3 c}+\frac {(c x-1)^{3/2} (c x+1)^{3/2} (a+b \text {arccosh}(c x))^2}{3 c^2}\right )+\frac {1}{3} d x \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))^3+\frac {2}{3} d \left (x (a+b \text {arccosh}(c x))^3-3 b c \left (\frac {\sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^2}{c^2}-\frac {2 b \left (a x+b x \text {arccosh}(c x)-\frac {b \sqrt {c x-1} \sqrt {c x+1}}{c}\right )}{c}\right )\right )\)

\(\Big \downarrow \) 6304

\(\displaystyle b c d \left (\frac {2 b \int \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))dx}{3 c}+\frac {(c x-1)^{3/2} (c x+1)^{3/2} (a+b \text {arccosh}(c x))^2}{3 c^2}\right )+\frac {1}{3} d x \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))^3+\frac {2}{3} d \left (x (a+b \text {arccosh}(c x))^3-3 b c \left (\frac {\sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^2}{c^2}-\frac {2 b \left (a x+b x \text {arccosh}(c x)-\frac {b \sqrt {c x-1} \sqrt {c x+1}}{c}\right )}{c}\right )\right )\)

\(\Big \downarrow \) 6309

\(\displaystyle b c d \left (\frac {2 b \left (-b c \int \frac {x \left (3-c^2 x^2\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}dx-\frac {1}{3} c^2 x^3 (a+b \text {arccosh}(c x))+x (a+b \text {arccosh}(c x))\right )}{3 c}+\frac {(c x-1)^{3/2} (c x+1)^{3/2} (a+b \text {arccosh}(c x))^2}{3 c^2}\right )+\frac {1}{3} d x \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))^3+\frac {2}{3} d \left (x (a+b \text {arccosh}(c x))^3-3 b c \left (\frac {\sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^2}{c^2}-\frac {2 b \left (a x+b x \text {arccosh}(c x)-\frac {b \sqrt {c x-1} \sqrt {c x+1}}{c}\right )}{c}\right )\right )\)

\(\Big \downarrow \) 27

\(\displaystyle b c d \left (\frac {2 b \left (-\frac {1}{3} b c \int \frac {x \left (3-c^2 x^2\right )}{\sqrt {c x-1} \sqrt {c x+1}}dx-\frac {1}{3} c^2 x^3 (a+b \text {arccosh}(c x))+x (a+b \text {arccosh}(c x))\right )}{3 c}+\frac {(c x-1)^{3/2} (c x+1)^{3/2} (a+b \text {arccosh}(c x))^2}{3 c^2}\right )+\frac {1}{3} d x \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))^3+\frac {2}{3} d \left (x (a+b \text {arccosh}(c x))^3-3 b c \left (\frac {\sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^2}{c^2}-\frac {2 b \left (a x+b x \text {arccosh}(c x)-\frac {b \sqrt {c x-1} \sqrt {c x+1}}{c}\right )}{c}\right )\right )\)

\(\Big \downarrow \) 960

\(\displaystyle b c d \left (\frac {2 b \left (-\frac {1}{3} b c \left (\frac {7}{3} \int \frac {x}{\sqrt {c x-1} \sqrt {c x+1}}dx-\frac {1}{3} x^2 \sqrt {c x-1} \sqrt {c x+1}\right )-\frac {1}{3} c^2 x^3 (a+b \text {arccosh}(c x))+x (a+b \text {arccosh}(c x))\right )}{3 c}+\frac {(c x-1)^{3/2} (c x+1)^{3/2} (a+b \text {arccosh}(c x))^2}{3 c^2}\right )+\frac {1}{3} d x \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))^3+\frac {2}{3} d \left (x (a+b \text {arccosh}(c x))^3-3 b c \left (\frac {\sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^2}{c^2}-\frac {2 b \left (a x+b x \text {arccosh}(c x)-\frac {b \sqrt {c x-1} \sqrt {c x+1}}{c}\right )}{c}\right )\right )\)

\(\Big \downarrow \) 83

\(\displaystyle \frac {1}{3} d x \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))^3+b c d \left (\frac {2 b \left (-\frac {1}{3} c^2 x^3 (a+b \text {arccosh}(c x))+x (a+b \text {arccosh}(c x))-\frac {1}{3} b c \left (\frac {7 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}-\frac {1}{3} x^2 \sqrt {c x-1} \sqrt {c x+1}\right )\right )}{3 c}+\frac {(c x-1)^{3/2} (c x+1)^{3/2} (a+b \text {arccosh}(c x))^2}{3 c^2}\right )+\frac {2}{3} d \left (x (a+b \text {arccosh}(c x))^3-3 b c \left (\frac {\sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^2}{c^2}-\frac {2 b \left (a x+b x \text {arccosh}(c x)-\frac {b \sqrt {c x-1} \sqrt {c x+1}}{c}\right )}{c}\right )\right )\)

Input:

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

Output:

(d*x*(1 - c^2*x^2)*(a + b*ArcCosh[c*x])^3)/3 + b*c*d*(((-1 + c*x)^(3/2)*(1 
 + c*x)^(3/2)*(a + b*ArcCosh[c*x])^2)/(3*c^2) + (2*b*(-1/3*(b*c*((7*Sqrt[- 
1 + c*x]*Sqrt[1 + c*x])/(3*c^2) - (x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/3)) + 
 x*(a + b*ArcCosh[c*x]) - (c^2*x^3*(a + b*ArcCosh[c*x]))/3))/(3*c)) + (2*d 
*(x*(a + b*ArcCosh[c*x])^3 - 3*b*c*((Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*A 
rcCosh[c*x])^2)/c^2 - (2*b*(a*x - (b*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/c + b*x 
*ArcCosh[c*x]))/c)))/3
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 83

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), 
 x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0] && EqQ[a*d*f 
*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)), 0]
 

rule 960

Int[((e_.)*(x_))^(m_.)*((a1_) + (b1_.)*(x_)^(non2_.))^(p_.)*((a2_) + (b2_.) 
*(x_)^(non2_.))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[d*(e*x)^( 
m + 1)*(a1 + b1*x^(n/2))^(p + 1)*((a2 + b2*x^(n/2))^(p + 1)/(b1*b2*e*(m + n 
*(p + 1) + 1))), x] - Simp[(a1*a2*d*(m + 1) - b1*b2*c*(m + n*(p + 1) + 1))/ 
(b1*b2*(m + n*(p + 1) + 1))   Int[(e*x)^m*(a1 + b1*x^(n/2))^p*(a2 + b2*x^(n 
/2))^p, x], x] /; FreeQ[{a1, b1, a2, b2, c, d, e, m, n, p}, x] && EqQ[non2, 
 n/2] && EqQ[a2*b1 + a1*b2, 0] && NeQ[m + n*(p + 1) + 1, 0]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 6294

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*A 
rcCosh[c*x])^n, x] - Simp[b*c*n   Int[x*((a + b*ArcCosh[c*x])^(n - 1)/(Sqrt 
[1 + c*x]*Sqrt[-1 + c*x])), x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]
 

rule 6304

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d1_) + (e1_.)*(x_))^(p_.)*( 
(d2_) + (e2_.)*(x_))^(p_.), x_Symbol] :> Int[(d1*d2 + e1*e2*x^2)^p*(a + b*A 
rcCosh[c*x])^n, x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[d2*e1 + 
 d1*e2, 0] && IntegerQ[p]
 

rule 6309

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(p_.), x_Symb 
ol] :> With[{u = IntHide[(d + e*x^2)^p, x]}, Simp[(a + b*ArcCosh[c*x])   u, 
 x] - Simp[b*c   Int[SimplifyIntegrand[u/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x] 
, x], x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]
 

rule 6312

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), 
x_Symbol] :> Simp[x*(d + e*x^2)^p*((a + b*ArcCosh[c*x])^n/(2*p + 1)), x] + 
(Simp[2*d*(p/(2*p + 1))   Int[(d + e*x^2)^(p - 1)*(a + b*ArcCosh[c*x])^n, x 
], x] - Simp[b*c*(n/(2*p + 1))*Simp[(d + e*x^2)^p/((1 + c*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, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 
0] && GtQ[p, 0]
 

rule 6330

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] - Simp[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] && E 
qQ[e2, (-c)*d2] && GtQ[n, 0] && NeQ[p, -1]
 

Maple [A] (verified)

Time = 0.20 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.47

method	result	size
derivativedivides	\(\frac {-d \,a^{3} \left (\frac {1}{3} c^{3} x^{3}-c x \right )-d \,b^{3} \left (-\frac {2 \operatorname {arccosh}\left (c x \right )^{3} c x}{3}+\frac {\operatorname {arccosh}\left (c x \right )^{3} c x \left (c x -1\right ) \left (c x +1\right )}{3}+2 \operatorname {arccosh}\left (c x \right )^{2} \sqrt {c x -1}\, \sqrt {c x +1}-\frac {40 c x \,\operatorname {arccosh}\left (c x \right )}{9}+\frac {40 \sqrt {c x -1}\, \sqrt {c x +1}}{9}-\frac {\operatorname {arccosh}\left (c x \right )^{2} \left (c x -1\right )^{\frac {3}{2}} \left (c x +1\right )^{\frac {3}{2}}}{3}+\frac {2 \,\operatorname {arccosh}\left (c x \right ) c x \left (c x -1\right ) \left (c x +1\right )}{9}-\frac {2 \left (c x -1\right )^{\frac {3}{2}} \left (c x +1\right )^{\frac {3}{2}}}{27}\right )-3 d a \,b^{2} \left (-\frac {2 \operatorname {arccosh}\left (c x \right )^{2} c x}{3}+\frac {\operatorname {arccosh}\left (c x \right )^{2} c x \left (c x -1\right ) \left (c x +1\right )}{3}+\frac {4 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}}{3}-\frac {40 c x}{27}-\frac {2 \,\operatorname {arccosh}\left (c x \right ) \left (c x -1\right )^{\frac {3}{2}} \left (c x +1\right )^{\frac {3}{2}}}{9}+\frac {2 c x \left (c x -1\right ) \left (c x +1\right )}{27}\right )-3 d \,a^{2} b \left (\frac {c^{3} x^{3} \operatorname {arccosh}\left (c x \right )}{3}-c x \,\operatorname {arccosh}\left (c x \right )-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (c^{2} x^{2}-7\right )}{9}\right )}{c}\)	\(313\)
default	\(\frac {-d \,a^{3} \left (\frac {1}{3} c^{3} x^{3}-c x \right )-d \,b^{3} \left (-\frac {2 \operatorname {arccosh}\left (c x \right )^{3} c x}{3}+\frac {\operatorname {arccosh}\left (c x \right )^{3} c x \left (c x -1\right ) \left (c x +1\right )}{3}+2 \operatorname {arccosh}\left (c x \right )^{2} \sqrt {c x -1}\, \sqrt {c x +1}-\frac {40 c x \,\operatorname {arccosh}\left (c x \right )}{9}+\frac {40 \sqrt {c x -1}\, \sqrt {c x +1}}{9}-\frac {\operatorname {arccosh}\left (c x \right )^{2} \left (c x -1\right )^{\frac {3}{2}} \left (c x +1\right )^{\frac {3}{2}}}{3}+\frac {2 \,\operatorname {arccosh}\left (c x \right ) c x \left (c x -1\right ) \left (c x +1\right )}{9}-\frac {2 \left (c x -1\right )^{\frac {3}{2}} \left (c x +1\right )^{\frac {3}{2}}}{27}\right )-3 d a \,b^{2} \left (-\frac {2 \operatorname {arccosh}\left (c x \right )^{2} c x}{3}+\frac {\operatorname {arccosh}\left (c x \right )^{2} c x \left (c x -1\right ) \left (c x +1\right )}{3}+\frac {4 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}}{3}-\frac {40 c x}{27}-\frac {2 \,\operatorname {arccosh}\left (c x \right ) \left (c x -1\right )^{\frac {3}{2}} \left (c x +1\right )^{\frac {3}{2}}}{9}+\frac {2 c x \left (c x -1\right ) \left (c x +1\right )}{27}\right )-3 d \,a^{2} b \left (\frac {c^{3} x^{3} \operatorname {arccosh}\left (c x \right )}{3}-c x \,\operatorname {arccosh}\left (c x \right )-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (c^{2} x^{2}-7\right )}{9}\right )}{c}\)	\(313\)
parts	\(-d \,a^{3} \left (\frac {1}{3} c^{2} x^{3}-x \right )-\frac {d \,b^{3} \left (-\frac {2 \operatorname {arccosh}\left (c x \right )^{3} c x}{3}+\frac {\operatorname {arccosh}\left (c x \right )^{3} c x \left (c x -1\right ) \left (c x +1\right )}{3}+2 \operatorname {arccosh}\left (c x \right )^{2} \sqrt {c x -1}\, \sqrt {c x +1}-\frac {40 c x \,\operatorname {arccosh}\left (c x \right )}{9}+\frac {40 \sqrt {c x -1}\, \sqrt {c x +1}}{9}-\frac {\operatorname {arccosh}\left (c x \right )^{2} \left (c x -1\right )^{\frac {3}{2}} \left (c x +1\right )^{\frac {3}{2}}}{3}+\frac {2 \,\operatorname {arccosh}\left (c x \right ) c x \left (c x -1\right ) \left (c x +1\right )}{9}-\frac {2 \left (c x -1\right )^{\frac {3}{2}} \left (c x +1\right )^{\frac {3}{2}}}{27}\right )}{c}-\frac {3 d a \,b^{2} \left (-\frac {2 \operatorname {arccosh}\left (c x \right )^{2} c x}{3}+\frac {\operatorname {arccosh}\left (c x \right )^{2} c x \left (c x -1\right ) \left (c x +1\right )}{3}+\frac {4 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}}{3}-\frac {40 c x}{27}-\frac {2 \,\operatorname {arccosh}\left (c x \right ) \left (c x -1\right )^{\frac {3}{2}} \left (c x +1\right )^{\frac {3}{2}}}{9}+\frac {2 c x \left (c x -1\right ) \left (c x +1\right )}{27}\right )}{c}-\frac {3 d \,a^{2} b \left (\frac {c^{3} x^{3} \operatorname {arccosh}\left (c x \right )}{3}-c x \,\operatorname {arccosh}\left (c x \right )-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (c^{2} x^{2}-7\right )}{9}\right )}{c}\)	\(317\)
orering	\(\frac {5 x \left (13 c^{4} x^{4}-194 c^{2} x^{2}-179\right ) \left (-c^{2} d \,x^{2}+d \right ) \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{3}}{81 \left (c^{2} x^{2}-1\right )^{2}}-\frac {\left (25 c^{4} x^{4}-683 c^{2} x^{2}-242\right ) \left (-2 c^{2} d x \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{3}+\frac {3 \left (-c^{2} d \,x^{2}+d \right ) \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{2} b c}{\sqrt {c x -1}\, \sqrt {c x +1}}\right )}{81 c^{2} \left (c^{2} x^{2}-1\right )}+\frac {2 x \left (c^{2} x^{2}-41\right ) \left (-2 c^{2} d \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{3}-\frac {12 c^{3} d x \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{2} b}{\sqrt {c x -1}\, \sqrt {c x +1}}+\frac {6 \left (-c^{2} d \,x^{2}+d \right ) \left (a +b \,\operatorname {arccosh}\left (c x \right )\right ) b^{2} c^{2}}{\left (c x -1\right ) \left (c x +1\right )}-\frac {3 \left (-c^{2} d \,x^{2}+d \right ) \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{2} b \,c^{2}}{2 \left (c x -1\right )^{\frac {3}{2}} \sqrt {c x +1}}-\frac {3 \left (-c^{2} d \,x^{2}+d \right ) \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{2} b \,c^{2}}{2 \sqrt {c x -1}\, \left (c x +1\right )^{\frac {3}{2}}}\right )}{27 c^{2}}-\frac {\left (c^{2} x^{2}-61\right ) \left (c x -1\right ) \left (c x +1\right ) \left (-\frac {18 c^{3} d \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{2} b}{\sqrt {c x -1}\, \sqrt {c x +1}}-\frac {36 c^{4} d x \left (a +b \,\operatorname {arccosh}\left (c x \right )\right ) b^{2}}{\left (c x -1\right ) \left (c x +1\right )}+\frac {9 c^{4} d x \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{2} b}{\left (c x -1\right )^{\frac {3}{2}} \sqrt {c x +1}}+\frac {9 c^{4} d x \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{2} b}{\sqrt {c x -1}\, \left (c x +1\right )^{\frac {3}{2}}}+\frac {6 \left (-c^{2} d \,x^{2}+d \right ) b^{3} c^{3}}{\left (c x -1\right )^{\frac {3}{2}} \left (c x +1\right )^{\frac {3}{2}}}-\frac {9 \left (-c^{2} d \,x^{2}+d \right ) \left (a +b \,\operatorname {arccosh}\left (c x \right )\right ) b^{2} c^{3}}{\left (c x -1\right )^{2} \left (c x +1\right )}-\frac {9 \left (-c^{2} d \,x^{2}+d \right ) \left (a +b \,\operatorname {arccosh}\left (c x \right )\right ) b^{2} c^{3}}{\left (c x -1\right ) \left (c x +1\right )^{2}}+\frac {9 \left (-c^{2} d \,x^{2}+d \right ) \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{2} b \,c^{3}}{4 \left (c x -1\right )^{\frac {5}{2}} \sqrt {c x +1}}+\frac {3 \left (-c^{2} d \,x^{2}+d \right ) \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{2} b \,c^{3}}{2 \left (c x -1\right )^{\frac {3}{2}} \left (c x +1\right )^{\frac {3}{2}}}+\frac {9 \left (-c^{2} d \,x^{2}+d \right ) \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{2} b \,c^{3}}{4 \sqrt {c x -1}\, \left (c x +1\right )^{\frac {5}{2}}}\right )}{81 c^{4}}\)	\(723\)

Input:

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

Output:

1/c*(-d*a^3*(1/3*c^3*x^3-c*x)-d*b^3*(-2/3*arccosh(c*x)^3*c*x+1/3*arccosh(c 
*x)^3*c*x*(c*x-1)*(c*x+1)+2*arccosh(c*x)^2*(c*x-1)^(1/2)*(c*x+1)^(1/2)-40/ 
9*c*x*arccosh(c*x)+40/9*(c*x-1)^(1/2)*(c*x+1)^(1/2)-1/3*arccosh(c*x)^2*(c* 
x-1)^(3/2)*(c*x+1)^(3/2)+2/9*arccosh(c*x)*c*x*(c*x-1)*(c*x+1)-2/27*(c*x-1) 
^(3/2)*(c*x+1)^(3/2))-3*d*a*b^2*(-2/3*arccosh(c*x)^2*c*x+1/3*arccosh(c*x)^ 
2*c*x*(c*x-1)*(c*x+1)+4/3*arccosh(c*x)*(c*x-1)^(1/2)*(c*x+1)^(1/2)-40/27*c 
*x-2/9*arccosh(c*x)*(c*x-1)^(3/2)*(c*x+1)^(3/2)+2/27*c*x*(c*x-1)*(c*x+1))- 
3*d*a^2*b*(1/3*c^3*x^3*arccosh(c*x)-c*x*arccosh(c*x)-1/9*(c*x-1)^(1/2)*(c* 
x+1)^(1/2)*(c^2*x^2-7)))
 

Fricas [A] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 296, normalized size of antiderivative = 1.39 \[ \int \left (d-c^2 d x^2\right ) (a+b \text {arccosh}(c x))^3 \, dx=-\frac {3 \, {\left (3 \, a^{3} + 2 \, a b^{2}\right )} c^{3} d x^{3} - 9 \, {\left (3 \, a^{3} + 14 \, a b^{2}\right )} c d x + 9 \, {\left (b^{3} c^{3} d x^{3} - 3 \, b^{3} c d x\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right )^{3} + 9 \, {\left (3 \, a b^{2} c^{3} d x^{3} - 9 \, a b^{2} c d x - {\left (b^{3} c^{2} d x^{2} - 7 \, b^{3} d\right )} \sqrt {c^{2} x^{2} - 1}\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right )^{2} + 3 \, {\left ({\left (9 \, a^{2} b + 2 \, b^{3}\right )} c^{3} d x^{3} - 3 \, {\left (9 \, a^{2} b + 14 \, b^{3}\right )} c d x - 6 \, {\left (a b^{2} c^{2} d x^{2} - 7 \, a b^{2} d\right )} \sqrt {c^{2} x^{2} - 1}\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left ({\left (9 \, a^{2} b + 2 \, b^{3}\right )} c^{2} d x^{2} - {\left (63 \, a^{2} b + 122 \, b^{3}\right )} d\right )} \sqrt {c^{2} x^{2} - 1}}{27 \, c} \] Input:

integrate((-c^2*d*x^2+d)*(a+b*arccosh(c*x))^3,x, algorithm="fricas")
 

Output:

-1/27*(3*(3*a^3 + 2*a*b^2)*c^3*d*x^3 - 9*(3*a^3 + 14*a*b^2)*c*d*x + 9*(b^3 
*c^3*d*x^3 - 3*b^3*c*d*x)*log(c*x + sqrt(c^2*x^2 - 1))^3 + 9*(3*a*b^2*c^3* 
d*x^3 - 9*a*b^2*c*d*x - (b^3*c^2*d*x^2 - 7*b^3*d)*sqrt(c^2*x^2 - 1))*log(c 
*x + sqrt(c^2*x^2 - 1))^2 + 3*((9*a^2*b + 2*b^3)*c^3*d*x^3 - 3*(9*a^2*b + 
14*b^3)*c*d*x - 6*(a*b^2*c^2*d*x^2 - 7*a*b^2*d)*sqrt(c^2*x^2 - 1))*log(c*x 
 + sqrt(c^2*x^2 - 1)) - ((9*a^2*b + 2*b^3)*c^2*d*x^2 - (63*a^2*b + 122*b^3 
)*d)*sqrt(c^2*x^2 - 1))/c
 

Sympy [F]

\[ \int \left (d-c^2 d x^2\right ) (a+b \text {arccosh}(c x))^3 \, dx=- d \left (\int \left (- a^{3}\right )\, dx + \int \left (- b^{3} \operatorname {acosh}^{3}{\left (c x \right )}\right )\, dx + \int \left (- 3 a b^{2} \operatorname {acosh}^{2}{\left (c x \right )}\right )\, dx + \int \left (- 3 a^{2} b \operatorname {acosh}{\left (c x \right )}\right )\, dx + \int a^{3} c^{2} x^{2}\, dx + \int b^{3} c^{2} x^{2} \operatorname {acosh}^{3}{\left (c x \right )}\, dx + \int 3 a b^{2} c^{2} x^{2} \operatorname {acosh}^{2}{\left (c x \right )}\, dx + \int 3 a^{2} b c^{2} x^{2} \operatorname {acosh}{\left (c x \right )}\, dx\right ) \] Input:

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

Output:

-d*(Integral(-a**3, x) + Integral(-b**3*acosh(c*x)**3, x) + Integral(-3*a* 
b**2*acosh(c*x)**2, x) + Integral(-3*a**2*b*acosh(c*x), x) + Integral(a**3 
*c**2*x**2, x) + Integral(b**3*c**2*x**2*acosh(c*x)**3, x) + Integral(3*a* 
b**2*c**2*x**2*acosh(c*x)**2, x) + Integral(3*a**2*b*c**2*x**2*acosh(c*x), 
 x))
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 437 vs. \(2 (182) = 364\).

Time = 0.05 (sec) , antiderivative size = 437, normalized size of antiderivative = 2.05 \[ \int \left (d-c^2 d x^2\right ) (a+b \text {arccosh}(c x))^3 \, dx=-\frac {1}{3} \, b^{3} c^{2} d x^{3} \operatorname {arcosh}\left (c x\right )^{3} - a b^{2} c^{2} d x^{3} \operatorname {arcosh}\left (c x\right )^{2} - \frac {1}{3} \, a^{3} c^{2} d x^{3} + b^{3} d x \operatorname {arcosh}\left (c x\right )^{3} - \frac {1}{3} \, {\left (3 \, x^{3} \operatorname {arcosh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} - 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {c^{2} x^{2} - 1}}{c^{4}}\right )}\right )} a^{2} b c^{2} d + \frac {2}{9} \, {\left (3 \, c {\left (\frac {\sqrt {c^{2} x^{2} - 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {c^{2} x^{2} - 1}}{c^{4}}\right )} \operatorname {arcosh}\left (c x\right ) - \frac {c^{2} x^{3} + 6 \, x}{c^{2}}\right )} a b^{2} c^{2} d + \frac {1}{27} \, {\left (9 \, c {\left (\frac {\sqrt {c^{2} x^{2} - 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {c^{2} x^{2} - 1}}{c^{4}}\right )} \operatorname {arcosh}\left (c x\right )^{2} + 2 \, c {\left (\frac {\sqrt {c^{2} x^{2} - 1} x^{2} + \frac {20 \, \sqrt {c^{2} x^{2} - 1}}{c^{2}}}{c^{2}} - \frac {3 \, {\left (c^{2} x^{3} + 6 \, x\right )} \operatorname {arcosh}\left (c x\right )}{c^{3}}\right )}\right )} b^{3} c^{2} d + 3 \, a b^{2} d x \operatorname {arcosh}\left (c x\right )^{2} - 3 \, {\left (\frac {\sqrt {c^{2} x^{2} - 1} \operatorname {arcosh}\left (c x\right )^{2}}{c} - \frac {2 \, {\left (c x \operatorname {arcosh}\left (c x\right ) - \sqrt {c^{2} x^{2} - 1}\right )}}{c}\right )} b^{3} d + 6 \, a b^{2} d {\left (x - \frac {\sqrt {c^{2} x^{2} - 1} \operatorname {arcosh}\left (c x\right )}{c}\right )} + a^{3} d x + \frac {3 \, {\left (c x \operatorname {arcosh}\left (c x\right ) - \sqrt {c^{2} x^{2} - 1}\right )} a^{2} b d}{c} \] Input:

integrate((-c^2*d*x^2+d)*(a+b*arccosh(c*x))^3,x, algorithm="maxima")
 

Output:

-1/3*b^3*c^2*d*x^3*arccosh(c*x)^3 - a*b^2*c^2*d*x^3*arccosh(c*x)^2 - 1/3*a 
^3*c^2*d*x^3 + b^3*d*x*arccosh(c*x)^3 - 1/3*(3*x^3*arccosh(c*x) - c*(sqrt( 
c^2*x^2 - 1)*x^2/c^2 + 2*sqrt(c^2*x^2 - 1)/c^4))*a^2*b*c^2*d + 2/9*(3*c*(s 
qrt(c^2*x^2 - 1)*x^2/c^2 + 2*sqrt(c^2*x^2 - 1)/c^4)*arccosh(c*x) - (c^2*x^ 
3 + 6*x)/c^2)*a*b^2*c^2*d + 1/27*(9*c*(sqrt(c^2*x^2 - 1)*x^2/c^2 + 2*sqrt( 
c^2*x^2 - 1)/c^4)*arccosh(c*x)^2 + 2*c*((sqrt(c^2*x^2 - 1)*x^2 + 20*sqrt(c 
^2*x^2 - 1)/c^2)/c^2 - 3*(c^2*x^3 + 6*x)*arccosh(c*x)/c^3))*b^3*c^2*d + 3* 
a*b^2*d*x*arccosh(c*x)^2 - 3*(sqrt(c^2*x^2 - 1)*arccosh(c*x)^2/c - 2*(c*x* 
arccosh(c*x) - sqrt(c^2*x^2 - 1))/c)*b^3*d + 6*a*b^2*d*(x - sqrt(c^2*x^2 - 
 1)*arccosh(c*x)/c) + a^3*d*x + 3*(c*x*arccosh(c*x) - sqrt(c^2*x^2 - 1))*a 
^2*b*d/c
 

Giac [F(-2)]

Exception generated. \[ \int \left (d-c^2 d x^2\right ) (a+b \text {arccosh}(c x))^3 \, dx=\text {Exception raised: TypeError} \] Input:

integrate((-c^2*d*x^2+d)*(a+b*arccosh(c*x))^3,x, algorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

Timed out. \[ \int \left (d-c^2 d x^2\right ) (a+b \text {arccosh}(c x))^3 \, dx=\int {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^3\,\left (d-c^2\,d\,x^2\right ) \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \left (d-c^2 d x^2\right ) (a+b \text {arccosh}(c x))^3 \, dx=\frac {d \left (-3 \mathit {acosh} \left (c x \right ) a^{2} b \,c^{3} x^{3}+9 \mathit {acosh} \left (c x \right ) a^{2} b c x +\sqrt {c^{2} x^{2}-1}\, a^{2} b \,c^{2} x^{2}+2 \sqrt {c^{2} x^{2}-1}\, a^{2} b -9 \sqrt {c x +1}\, \sqrt {c x -1}\, a^{2} b +3 \left (\int \mathit {acosh} \left (c x \right )^{3}d x \right ) b^{3} c +9 \left (\int \mathit {acosh} \left (c x \right )^{2}d x \right ) a \,b^{2} c -3 \left (\int \mathit {acosh} \left (c x \right )^{3} x^{2}d x \right ) b^{3} c^{3}-9 \left (\int \mathit {acosh} \left (c x \right )^{2} x^{2}d x \right ) a \,b^{2} c^{3}-a^{3} c^{3} x^{3}+3 a^{3} c x \right )}{3 c} \] Input:

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

Output:

(d*( - 3*acosh(c*x)*a**2*b*c**3*x**3 + 9*acosh(c*x)*a**2*b*c*x + sqrt(c**2 
*x**2 - 1)*a**2*b*c**2*x**2 + 2*sqrt(c**2*x**2 - 1)*a**2*b - 9*sqrt(c*x + 
1)*sqrt(c*x - 1)*a**2*b + 3*int(acosh(c*x)**3,x)*b**3*c + 9*int(acosh(c*x) 
**2,x)*a*b**2*c - 3*int(acosh(c*x)**3*x**2,x)*b**3*c**3 - 9*int(acosh(c*x) 
**2*x**2,x)*a*b**2*c**3 - a**3*c**3*x**3 + 3*a**3*c*x))/(3*c)