\(\int (c+d x^2)^3 \text {arccosh}(a x) \, dx\) [123]

Optimal result

Integrand size = 14, antiderivative size = 221 \[ \int \left (c+d x^2\right )^3 \text {arccosh}(a x) \, dx=-\frac {\left (35 a^6 c^3+35 a^4 c^2 d+21 a^2 c d^2+5 d^3\right ) \sqrt {-1+a x} \sqrt {1+a x}}{35 a^7}-\frac {d \left (35 a^4 c^2+42 a^2 c d+15 d^2\right ) (-1+a x)^{3/2} (1+a x)^{3/2}}{105 a^7}-\frac {3 d^2 \left (7 a^2 c+5 d\right ) (-1+a x)^{5/2} (1+a x)^{5/2}}{175 a^7}-\frac {d^3 (-1+a x)^{7/2} (1+a x)^{7/2}}{49 a^7}+c^3 x \text {arccosh}(a x)+c^2 d x^3 \text {arccosh}(a x)+\frac {3}{5} c d^2 x^5 \text {arccosh}(a x)+\frac {1}{7} d^3 x^7 \text {arccosh}(a x) \] Output:

-1/35*(35*a^6*c^3+35*a^4*c^2*d+21*a^2*c*d^2+5*d^3)*(a*x-1)^(1/2)*(a*x+1)^( 
1/2)/a^7-1/105*d*(35*a^4*c^2+42*a^2*c*d+15*d^2)*(a*x-1)^(3/2)*(a*x+1)^(3/2 
)/a^7-3/175*d^2*(7*a^2*c+5*d)*(a*x-1)^(5/2)*(a*x+1)^(5/2)/a^7-1/49*d^3*(a* 
x-1)^(7/2)*(a*x+1)^(7/2)/a^7+c^3*x*arccosh(a*x)+c^2*d*x^3*arccosh(a*x)+3/5 
*c*d^2*x^5*arccosh(a*x)+1/7*d^3*x^7*arccosh(a*x)
 

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.70 \[ \int \left (c+d x^2\right )^3 \text {arccosh}(a x) \, dx=-\frac {\sqrt {-1+a x} \sqrt {1+a x} \left (240 d^3+24 a^2 d^2 \left (49 c+5 d x^2\right )+2 a^4 d \left (1225 c^2+294 c d x^2+45 d^2 x^4\right )+a^6 \left (3675 c^3+1225 c^2 d x^2+441 c d^2 x^4+75 d^3 x^6\right )\right )}{3675 a^7}+\frac {1}{35} x \left (35 c^3+35 c^2 d x^2+21 c d^2 x^4+5 d^3 x^6\right ) \text {arccosh}(a x) \] Input:

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

Output:

-1/3675*(Sqrt[-1 + a*x]*Sqrt[1 + a*x]*(240*d^3 + 24*a^2*d^2*(49*c + 5*d*x^ 
2) + 2*a^4*d*(1225*c^2 + 294*c*d*x^2 + 45*d^2*x^4) + a^6*(3675*c^3 + 1225* 
c^2*d*x^2 + 441*c*d^2*x^4 + 75*d^3*x^6)))/a^7 + (x*(35*c^3 + 35*c^2*d*x^2 
+ 21*c*d^2*x^4 + 5*d^3*x^6)*ArcCosh[a*x])/35
 

Rubi [A] (verified)

Time = 1.12 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.07, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {6323, 27, 2113, 2331, 2389, 2009}

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.

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

\(\Big \downarrow \) 6323

\(\displaystyle -a \int \frac {x \left (5 d^3 x^6+21 c d^2 x^4+35 c^2 d x^2+35 c^3\right )}{35 \sqrt {a x-1} \sqrt {a x+1}}dx+c^3 x \text {arccosh}(a x)+c^2 d x^3 \text {arccosh}(a x)+\frac {3}{5} c d^2 x^5 \text {arccosh}(a x)+\frac {1}{7} d^3 x^7 \text {arccosh}(a x)\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {1}{35} a \int \frac {x \left (5 d^3 x^6+21 c d^2 x^4+35 c^2 d x^2+35 c^3\right )}{\sqrt {a x-1} \sqrt {a x+1}}dx+c^3 x \text {arccosh}(a x)+c^2 d x^3 \text {arccosh}(a x)+\frac {3}{5} c d^2 x^5 \text {arccosh}(a x)+\frac {1}{7} d^3 x^7 \text {arccosh}(a x)\)

\(\Big \downarrow \) 2113

\(\displaystyle -\frac {a \sqrt {a^2 x^2-1} \int \frac {x \left (5 d^3 x^6+21 c d^2 x^4+35 c^2 d x^2+35 c^3\right )}{\sqrt {a^2 x^2-1}}dx}{35 \sqrt {a x-1} \sqrt {a x+1}}+c^3 x \text {arccosh}(a x)+c^2 d x^3 \text {arccosh}(a x)+\frac {3}{5} c d^2 x^5 \text {arccosh}(a x)+\frac {1}{7} d^3 x^7 \text {arccosh}(a x)\)

\(\Big \downarrow \) 2331

\(\displaystyle -\frac {a \sqrt {a^2 x^2-1} \int \frac {5 d^3 x^6+21 c d^2 x^4+35 c^2 d x^2+35 c^3}{\sqrt {a^2 x^2-1}}dx^2}{70 \sqrt {a x-1} \sqrt {a x+1}}+c^3 x \text {arccosh}(a x)+c^2 d x^3 \text {arccosh}(a x)+\frac {3}{5} c d^2 x^5 \text {arccosh}(a x)+\frac {1}{7} d^3 x^7 \text {arccosh}(a x)\)

\(\Big \downarrow \) 2389

\(\displaystyle -\frac {a \sqrt {a^2 x^2-1} \int \left (\frac {5 \left (a^2 x^2-1\right )^{5/2} d^3}{a^6}+\frac {3 \left (7 c a^2+5 d\right ) \left (a^2 x^2-1\right )^{3/2} d^2}{a^6}+\frac {\left (35 c^2 a^4+42 c d a^2+15 d^2\right ) \sqrt {a^2 x^2-1} d}{a^6}+\frac {35 c^3 a^6+35 c^2 d a^4+21 c d^2 a^2+5 d^3}{a^6 \sqrt {a^2 x^2-1}}\right )dx^2}{70 \sqrt {a x-1} \sqrt {a x+1}}+c^3 x \text {arccosh}(a x)+c^2 d x^3 \text {arccosh}(a x)+\frac {3}{5} c d^2 x^5 \text {arccosh}(a x)+\frac {1}{7} d^3 x^7 \text {arccosh}(a x)\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {a \sqrt {a^2 x^2-1} \left (\frac {6 d^2 \left (a^2 x^2-1\right )^{5/2} \left (7 a^2 c+5 d\right )}{5 a^8}+\frac {10 d^3 \left (a^2 x^2-1\right )^{7/2}}{7 a^8}+\frac {2 d \left (a^2 x^2-1\right )^{3/2} \left (35 a^4 c^2+42 a^2 c d+15 d^2\right )}{3 a^8}+\frac {2 \sqrt {a^2 x^2-1} \left (35 a^6 c^3+35 a^4 c^2 d+21 a^2 c d^2+5 d^3\right )}{a^8}\right )}{70 \sqrt {a x-1} \sqrt {a x+1}}+c^3 x \text {arccosh}(a x)+c^2 d x^3 \text {arccosh}(a x)+\frac {3}{5} c d^2 x^5 \text {arccosh}(a x)+\frac {1}{7} d^3 x^7 \text {arccosh}(a x)\)

Input:

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

Output:

-1/70*(a*Sqrt[-1 + a^2*x^2]*((2*(35*a^6*c^3 + 35*a^4*c^2*d + 21*a^2*c*d^2 
+ 5*d^3)*Sqrt[-1 + a^2*x^2])/a^8 + (2*d*(35*a^4*c^2 + 42*a^2*c*d + 15*d^2) 
*(-1 + a^2*x^2)^(3/2))/(3*a^8) + (6*d^2*(7*a^2*c + 5*d)*(-1 + a^2*x^2)^(5/ 
2))/(5*a^8) + (10*d^3*(-1 + a^2*x^2)^(7/2))/(7*a^8)))/(Sqrt[-1 + a*x]*Sqrt 
[1 + a*x]) + c^3*x*ArcCosh[a*x] + c^2*d*x^3*ArcCosh[a*x] + (3*c*d^2*x^5*Ar 
cCosh[a*x])/5 + (d^3*x^7*ArcCosh[a*x])/7
 

Defintions of rubi rules used

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 2009

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

rule 2113

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

rule 2331

Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[1/2   S 
ubst[Int[x^((m - 1)/2)*SubstFor[x^2, Pq, x]*(a + b*x)^p, x], x, x^2], x] /; 
 FreeQ[{a, b, p}, x] && PolyQ[Pq, x^2] && IntegerQ[(m - 1)/2]
 

rule 2389

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand 
[Pq*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p 
, 0] || EqQ[n, 1])
 

rule 6323

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] && NeQ[c^2*d + e, 0] && (IGtQ[p, 0] 
|| ILtQ[p + 1/2, 0])
 

Maple [A] (verified)

Time = 0.14 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.76

method	result	size
parts	\(\frac {d^{3} x^{7} \operatorname {arccosh}\left (a x \right )}{7}+\frac {3 c \,d^{2} x^{5} \operatorname {arccosh}\left (a x \right )}{5}+c^{2} d \,x^{3} \operatorname {arccosh}\left (a x \right )+c^{3} x \,\operatorname {arccosh}\left (a x \right )-\frac {\sqrt {a x -1}\, \sqrt {a x +1}\, \left (75 a^{6} d^{3} x^{6}+441 a^{6} c \,d^{2} x^{4}+1225 a^{6} c^{2} d \,x^{2}+90 a^{4} d^{3} x^{4}+3675 a^{6} c^{3}+588 a^{4} c \,d^{2} x^{2}+2450 a^{4} c^{2} d +120 a^{2} d^{3} x^{2}+1176 a^{2} c \,d^{2}+240 d^{3}\right )}{3675 a^{7}}\)	\(168\)
derivativedivides	\(\frac {\operatorname {arccosh}\left (a x \right ) c^{3} a x +a \,\operatorname {arccosh}\left (a x \right ) c^{2} d \,x^{3}+\frac {3 a \,\operatorname {arccosh}\left (a x \right ) c \,d^{2} x^{5}}{5}+\frac {a \,\operatorname {arccosh}\left (a x \right ) d^{3} x^{7}}{7}-\frac {\sqrt {a x -1}\, \sqrt {a x +1}\, \left (75 a^{6} d^{3} x^{6}+441 a^{6} c \,d^{2} x^{4}+1225 a^{6} c^{2} d \,x^{2}+90 a^{4} d^{3} x^{4}+3675 a^{6} c^{3}+588 a^{4} c \,d^{2} x^{2}+2450 a^{4} c^{2} d +120 a^{2} d^{3} x^{2}+1176 a^{2} c \,d^{2}+240 d^{3}\right )}{3675 a^{6}}}{a}\)	\(176\)
default	\(\frac {\operatorname {arccosh}\left (a x \right ) c^{3} a x +a \,\operatorname {arccosh}\left (a x \right ) c^{2} d \,x^{3}+\frac {3 a \,\operatorname {arccosh}\left (a x \right ) c \,d^{2} x^{5}}{5}+\frac {a \,\operatorname {arccosh}\left (a x \right ) d^{3} x^{7}}{7}-\frac {\sqrt {a x -1}\, \sqrt {a x +1}\, \left (75 a^{6} d^{3} x^{6}+441 a^{6} c \,d^{2} x^{4}+1225 a^{6} c^{2} d \,x^{2}+90 a^{4} d^{3} x^{4}+3675 a^{6} c^{3}+588 a^{4} c \,d^{2} x^{2}+2450 a^{4} c^{2} d +120 a^{2} d^{3} x^{2}+1176 a^{2} c \,d^{2}+240 d^{3}\right )}{3675 a^{6}}}{a}\)	\(176\)
orering	\(\frac {x \left (325 a^{8} d^{4} x^{8}+1792 a^{8} c \,d^{3} x^{6}+4410 a^{8} c^{2} d^{2} x^{4}+30 a^{6} d^{4} x^{6}+9800 a^{8} c^{3} d \,x^{2}+294 a^{6} c \,d^{3} x^{4}+1225 a^{8} c^{4}+2450 a^{6} c^{2} d^{2} x^{2}+60 a^{4} d^{4} x^{4}-7350 a^{6} c^{3} d +1176 a^{4} c \,d^{3} x^{2}-4900 a^{4} c^{2} d^{2}+240 a^{2} d^{4} x^{2}-2352 a^{2} c \,d^{3}-480 d^{4}\right ) \operatorname {arccosh}\left (a x \right )}{1225 \left (d \,x^{2}+c \right ) a^{8}}-\frac {\left (75 a^{6} d^{3} x^{6}+441 a^{6} c \,d^{2} x^{4}+1225 a^{6} c^{2} d \,x^{2}+90 a^{4} d^{3} x^{4}+3675 a^{6} c^{3}+588 a^{4} c \,d^{2} x^{2}+2450 a^{4} c^{2} d +120 a^{2} d^{3} x^{2}+1176 a^{2} c \,d^{2}+240 d^{3}\right ) \left (a x -1\right ) \left (a x +1\right ) \left (6 \left (d \,x^{2}+c \right )^{2} \operatorname {arccosh}\left (a x \right ) d x +\frac {\left (d \,x^{2}+c \right )^{3} a}{\sqrt {a x -1}\, \sqrt {a x +1}}\right )}{3675 a^{8} \left (d \,x^{2}+c \right )^{3}}\)	\(352\)

Input:

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

Output:

1/7*d^3*x^7*arccosh(a*x)+3/5*c*d^2*x^5*arccosh(a*x)+c^2*d*x^3*arccosh(a*x) 
+c^3*x*arccosh(a*x)-1/3675/a^7*(a*x-1)^(1/2)*(a*x+1)^(1/2)*(75*a^6*d^3*x^6 
+441*a^6*c*d^2*x^4+1225*a^6*c^2*d*x^2+90*a^4*d^3*x^4+3675*a^6*c^3+588*a^4* 
c*d^2*x^2+2450*a^4*c^2*d+120*a^2*d^3*x^2+1176*a^2*c*d^2+240*d^3)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.81 \[ \int \left (c+d x^2\right )^3 \text {arccosh}(a x) \, dx=\frac {105 \, {\left (5 \, a^{7} d^{3} x^{7} + 21 \, a^{7} c d^{2} x^{5} + 35 \, a^{7} c^{2} d x^{3} + 35 \, a^{7} c^{3} x\right )} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right ) - {\left (75 \, a^{6} d^{3} x^{6} + 3675 \, a^{6} c^{3} + 2450 \, a^{4} c^{2} d + 1176 \, a^{2} c d^{2} + 9 \, {\left (49 \, a^{6} c d^{2} + 10 \, a^{4} d^{3}\right )} x^{4} + 240 \, d^{3} + {\left (1225 \, a^{6} c^{2} d + 588 \, a^{4} c d^{2} + 120 \, a^{2} d^{3}\right )} x^{2}\right )} \sqrt {a^{2} x^{2} - 1}}{3675 \, a^{7}} \] Input:

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

Output:

1/3675*(105*(5*a^7*d^3*x^7 + 21*a^7*c*d^2*x^5 + 35*a^7*c^2*d*x^3 + 35*a^7* 
c^3*x)*log(a*x + sqrt(a^2*x^2 - 1)) - (75*a^6*d^3*x^6 + 3675*a^6*c^3 + 245 
0*a^4*c^2*d + 1176*a^2*c*d^2 + 9*(49*a^6*c*d^2 + 10*a^4*d^3)*x^4 + 240*d^3 
 + (1225*a^6*c^2*d + 588*a^4*c*d^2 + 120*a^2*d^3)*x^2)*sqrt(a^2*x^2 - 1))/ 
a^7
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.16 \[ \int \left (c+d x^2\right )^3 \text {arccosh}(a x) \, dx=-\frac {1}{3675} \, {\left (\frac {75 \, \sqrt {a^{2} x^{2} - 1} d^{3} x^{6}}{a^{2}} + \frac {441 \, \sqrt {a^{2} x^{2} - 1} c d^{2} x^{4}}{a^{2}} + \frac {1225 \, \sqrt {a^{2} x^{2} - 1} c^{2} d x^{2}}{a^{2}} + \frac {90 \, \sqrt {a^{2} x^{2} - 1} d^{3} x^{4}}{a^{4}} + \frac {3675 \, \sqrt {a^{2} x^{2} - 1} c^{3}}{a^{2}} + \frac {588 \, \sqrt {a^{2} x^{2} - 1} c d^{2} x^{2}}{a^{4}} + \frac {2450 \, \sqrt {a^{2} x^{2} - 1} c^{2} d}{a^{4}} + \frac {120 \, \sqrt {a^{2} x^{2} - 1} d^{3} x^{2}}{a^{6}} + \frac {1176 \, \sqrt {a^{2} x^{2} - 1} c d^{2}}{a^{6}} + \frac {240 \, \sqrt {a^{2} x^{2} - 1} d^{3}}{a^{8}}\right )} a + \frac {1}{35} \, {\left (5 \, d^{3} x^{7} + 21 \, c d^{2} x^{5} + 35 \, c^{2} d x^{3} + 35 \, c^{3} x\right )} \operatorname {arcosh}\left (a x\right ) \] Input:

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

Output:

-1/3675*(75*sqrt(a^2*x^2 - 1)*d^3*x^6/a^2 + 441*sqrt(a^2*x^2 - 1)*c*d^2*x^ 
4/a^2 + 1225*sqrt(a^2*x^2 - 1)*c^2*d*x^2/a^2 + 90*sqrt(a^2*x^2 - 1)*d^3*x^ 
4/a^4 + 3675*sqrt(a^2*x^2 - 1)*c^3/a^2 + 588*sqrt(a^2*x^2 - 1)*c*d^2*x^2/a 
^4 + 2450*sqrt(a^2*x^2 - 1)*c^2*d/a^4 + 120*sqrt(a^2*x^2 - 1)*d^3*x^2/a^6 
+ 1176*sqrt(a^2*x^2 - 1)*c*d^2/a^6 + 240*sqrt(a^2*x^2 - 1)*d^3/a^8)*a + 1/ 
35*(5*d^3*x^7 + 21*c*d^2*x^5 + 35*c^2*d*x^3 + 35*c^3*x)*arccosh(a*x)
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 214, normalized size of antiderivative = 0.97 \[ \int \left (c+d x^2\right )^3 \text {arccosh}(a x) \, dx=\frac {1}{35} \, {\left (5 \, d^{3} x^{7} + 21 \, c d^{2} x^{5} + 35 \, c^{2} d x^{3} + 35 \, c^{3} x\right )} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right ) - \frac {{\left (35 \, a^{6} c^{3} + 35 \, a^{4} c^{2} d + 21 \, a^{2} c d^{2} + 5 \, d^{3}\right )} \sqrt {a^{2} x^{2} - 1}}{35 \, a^{7}} - \frac {1225 \, {\left (a^{2} x^{2} - 1\right )}^{\frac {3}{2}} a^{4} c^{2} d + 441 \, {\left (a^{2} x^{2} - 1\right )}^{\frac {5}{2}} a^{2} c d^{2} + 1470 \, {\left (a^{2} x^{2} - 1\right )}^{\frac {3}{2}} a^{2} c d^{2} + 75 \, {\left (a^{2} x^{2} - 1\right )}^{\frac {7}{2}} d^{3} + 315 \, {\left (a^{2} x^{2} - 1\right )}^{\frac {5}{2}} d^{3} + 525 \, {\left (a^{2} x^{2} - 1\right )}^{\frac {3}{2}} d^{3}}{3675 \, a^{7}} \] Input:

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

Output:

1/35*(5*d^3*x^7 + 21*c*d^2*x^5 + 35*c^2*d*x^3 + 35*c^3*x)*log(a*x + sqrt(a 
^2*x^2 - 1)) - 1/35*(35*a^6*c^3 + 35*a^4*c^2*d + 21*a^2*c*d^2 + 5*d^3)*sqr 
t(a^2*x^2 - 1)/a^7 - 1/3675*(1225*(a^2*x^2 - 1)^(3/2)*a^4*c^2*d + 441*(a^2 
*x^2 - 1)^(5/2)*a^2*c*d^2 + 1470*(a^2*x^2 - 1)^(3/2)*a^2*c*d^2 + 75*(a^2*x 
^2 - 1)^(7/2)*d^3 + 315*(a^2*x^2 - 1)^(5/2)*d^3 + 525*(a^2*x^2 - 1)^(3/2)* 
d^3)/a^7
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.21 \[ \int \left (c+d x^2\right )^3 \text {arccosh}(a x) \, dx=\frac {3675 \mathit {acosh} \left (a x \right ) a^{7} c^{3} x +3675 \mathit {acosh} \left (a x \right ) a^{7} c^{2} d \,x^{3}+2205 \mathit {acosh} \left (a x \right ) a^{7} c \,d^{2} x^{5}+525 \mathit {acosh} \left (a x \right ) a^{7} d^{3} x^{7}-1225 \sqrt {a^{2} x^{2}-1}\, a^{6} c^{2} d \,x^{2}-441 \sqrt {a^{2} x^{2}-1}\, a^{6} c \,d^{2} x^{4}-75 \sqrt {a^{2} x^{2}-1}\, a^{6} d^{3} x^{6}-2450 \sqrt {a^{2} x^{2}-1}\, a^{4} c^{2} d -588 \sqrt {a^{2} x^{2}-1}\, a^{4} c \,d^{2} x^{2}-90 \sqrt {a^{2} x^{2}-1}\, a^{4} d^{3} x^{4}-1176 \sqrt {a^{2} x^{2}-1}\, a^{2} c \,d^{2}-120 \sqrt {a^{2} x^{2}-1}\, a^{2} d^{3} x^{2}-240 \sqrt {a^{2} x^{2}-1}\, d^{3}-3675 \sqrt {a x +1}\, \sqrt {a x -1}\, a^{6} c^{3}}{3675 a^{7}} \] Input:

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

Output:

(3675*acosh(a*x)*a**7*c**3*x + 3675*acosh(a*x)*a**7*c**2*d*x**3 + 2205*aco 
sh(a*x)*a**7*c*d**2*x**5 + 525*acosh(a*x)*a**7*d**3*x**7 - 1225*sqrt(a**2* 
x**2 - 1)*a**6*c**2*d*x**2 - 441*sqrt(a**2*x**2 - 1)*a**6*c*d**2*x**4 - 75 
*sqrt(a**2*x**2 - 1)*a**6*d**3*x**6 - 2450*sqrt(a**2*x**2 - 1)*a**4*c**2*d 
 - 588*sqrt(a**2*x**2 - 1)*a**4*c*d**2*x**2 - 90*sqrt(a**2*x**2 - 1)*a**4* 
d**3*x**4 - 1176*sqrt(a**2*x**2 - 1)*a**2*c*d**2 - 120*sqrt(a**2*x**2 - 1) 
*a**2*d**3*x**2 - 240*sqrt(a**2*x**2 - 1)*d**3 - 3675*sqrt(a*x + 1)*sqrt(a 
*x - 1)*a**6*c**3)/(3675*a**7)