\(\int \frac {(d+e x^2)^3 (a+b \text {arccosh}(c x))}{x^2} \, dx\) [384]

Optimal result

Integrand size = 21, antiderivative size = 214 \[ \int \frac {\left (d+e x^2\right )^3 (a+b \text {arccosh}(c x))}{x^2} \, dx=-\frac {b e \left (225 c^4 d^2+50 c^2 d e+8 e^2\right ) \sqrt {-1+c x} \sqrt {1+c x}}{75 c^5}-\frac {b e^2 \left (25 c^2 d+4 e\right ) x^2 \sqrt {-1+c x} \sqrt {1+c x}}{75 c^3}-\frac {b e^3 x^4 \sqrt {-1+c x} \sqrt {1+c x}}{25 c}-\frac {d^3 (a+b \text {arccosh}(c x))}{x}+3 d^2 e x (a+b \text {arccosh}(c x))+d e^2 x^3 (a+b \text {arccosh}(c x))+\frac {1}{5} e^3 x^5 (a+b \text {arccosh}(c x))+b c d^3 \arctan \left (\sqrt {-1+c x} \sqrt {1+c x}\right ) \] Output:

-1/75*b*e*(225*c^4*d^2+50*c^2*d*e+8*e^2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c^5-1 
/75*b*e^2*(25*c^2*d+4*e)*x^2*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c^3-1/25*b*e^3*x^ 
4*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c-d^3*(a+b*arccosh(c*x))/x+3*d^2*e*x*(a+b*ar 
ccosh(c*x))+d*e^2*x^3*(a+b*arccosh(c*x))+1/5*e^3*x^5*(a+b*arccosh(c*x))+b* 
c*d^3*arctan((c*x-1)^(1/2)*(c*x+1)^(1/2))
 

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.85 \[ \int \frac {\left (d+e x^2\right )^3 (a+b \text {arccosh}(c x))}{x^2} \, dx=-\frac {a d^3}{x}+3 a d^2 e x+a d e^2 x^3+\frac {1}{5} a e^3 x^5-\frac {b e \sqrt {-1+c x} \sqrt {1+c x} \left (8 e^2+2 c^2 e \left (25 d+2 e x^2\right )+c^4 \left (225 d^2+25 d e x^2+3 e^2 x^4\right )\right )}{75 c^5}+\frac {b \left (-5 d^3+15 d^2 e x^2+5 d e^2 x^4+e^3 x^6\right ) \text {arccosh}(c x)}{5 x}-b c d^3 \arctan \left (\frac {1}{\sqrt {-1+c x} \sqrt {1+c x}}\right ) \] Input:

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

Output:

-((a*d^3)/x) + 3*a*d^2*e*x + a*d*e^2*x^3 + (a*e^3*x^5)/5 - (b*e*Sqrt[-1 + 
c*x]*Sqrt[1 + c*x]*(8*e^2 + 2*c^2*e*(25*d + 2*e*x^2) + c^4*(225*d^2 + 25*d 
*e*x^2 + 3*e^2*x^4)))/(75*c^5) + (b*(-5*d^3 + 15*d^2*e*x^2 + 5*d*e^2*x^4 + 
 e^3*x^6)*ArcCosh[c*x])/(5*x) - b*c*d^3*ArcTan[1/(Sqrt[-1 + c*x]*Sqrt[1 + 
c*x])]
 

Rubi [A] (verified)

Time = 0.81 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.01, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {6373, 27, 2113, 2331, 2123, 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.

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

\(\Big \downarrow \) 6373

\(\displaystyle -b c \int -\frac {-e^3 x^6-5 d e^2 x^4-15 d^2 e x^2+5 d^3}{5 x \sqrt {c x-1} \sqrt {c x+1}}dx-\frac {d^3 (a+b \text {arccosh}(c x))}{x}+3 d^2 e x (a+b \text {arccosh}(c x))+d e^2 x^3 (a+b \text {arccosh}(c x))+\frac {1}{5} e^3 x^5 (a+b \text {arccosh}(c x))\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{5} b c \int \frac {-e^3 x^6-5 d e^2 x^4-15 d^2 e x^2+5 d^3}{x \sqrt {c x-1} \sqrt {c x+1}}dx-\frac {d^3 (a+b \text {arccosh}(c x))}{x}+3 d^2 e x (a+b \text {arccosh}(c x))+d e^2 x^3 (a+b \text {arccosh}(c x))+\frac {1}{5} e^3 x^5 (a+b \text {arccosh}(c x))\)

\(\Big \downarrow \) 2113

\(\displaystyle \frac {b c \sqrt {c^2 x^2-1} \int \frac {-e^3 x^6-5 d e^2 x^4-15 d^2 e x^2+5 d^3}{x \sqrt {c^2 x^2-1}}dx}{5 \sqrt {c x-1} \sqrt {c x+1}}-\frac {d^3 (a+b \text {arccosh}(c x))}{x}+3 d^2 e x (a+b \text {arccosh}(c x))+d e^2 x^3 (a+b \text {arccosh}(c x))+\frac {1}{5} e^3 x^5 (a+b \text {arccosh}(c x))\)

\(\Big \downarrow \) 2331

\(\displaystyle \frac {b c \sqrt {c^2 x^2-1} \int \frac {-e^3 x^6-5 d e^2 x^4-15 d^2 e x^2+5 d^3}{x^2 \sqrt {c^2 x^2-1}}dx^2}{10 \sqrt {c x-1} \sqrt {c x+1}}-\frac {d^3 (a+b \text {arccosh}(c x))}{x}+3 d^2 e x (a+b \text {arccosh}(c x))+d e^2 x^3 (a+b \text {arccosh}(c x))+\frac {1}{5} e^3 x^5 (a+b \text {arccosh}(c x))\)

\(\Big \downarrow \) 2123

\(\displaystyle \frac {b c \sqrt {c^2 x^2-1} \int \left (\frac {5 d^3}{x^2 \sqrt {c^2 x^2-1}}-\frac {e^3 \left (c^2 x^2-1\right )^{3/2}}{c^4}-\frac {e^2 \left (5 d c^2+2 e\right ) \sqrt {c^2 x^2-1}}{c^4}-\frac {e \left (15 d^2 c^4+5 d e c^2+e^2\right )}{c^4 \sqrt {c^2 x^2-1}}\right )dx^2}{10 \sqrt {c x-1} \sqrt {c x+1}}-\frac {d^3 (a+b \text {arccosh}(c x))}{x}+3 d^2 e x (a+b \text {arccosh}(c x))+d e^2 x^3 (a+b \text {arccosh}(c x))+\frac {1}{5} e^3 x^5 (a+b \text {arccosh}(c x))\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {d^3 (a+b \text {arccosh}(c x))}{x}+3 d^2 e x (a+b \text {arccosh}(c x))+d e^2 x^3 (a+b \text {arccosh}(c x))+\frac {1}{5} e^3 x^5 (a+b \text {arccosh}(c x))+\frac {b c \sqrt {c^2 x^2-1} \left (10 d^3 \arctan \left (\sqrt {c^2 x^2-1}\right )-\frac {2 e^2 \left (c^2 x^2-1\right )^{3/2} \left (5 c^2 d+2 e\right )}{3 c^6}-\frac {2 e^3 \left (c^2 x^2-1\right )^{5/2}}{5 c^6}-\frac {2 e \sqrt {c^2 x^2-1} \left (15 c^4 d^2+5 c^2 d e+e^2\right )}{c^6}\right )}{10 \sqrt {c x-1} \sqrt {c x+1}}\)

Input:

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

Output:

-((d^3*(a + b*ArcCosh[c*x]))/x) + 3*d^2*e*x*(a + b*ArcCosh[c*x]) + d*e^2*x 
^3*(a + b*ArcCosh[c*x]) + (e^3*x^5*(a + b*ArcCosh[c*x]))/5 + (b*c*Sqrt[-1 
+ c^2*x^2]*((-2*e*(15*c^4*d^2 + 5*c^2*d*e + e^2)*Sqrt[-1 + c^2*x^2])/c^6 - 
 (2*e^2*(5*c^2*d + 2*e)*(-1 + c^2*x^2)^(3/2))/(3*c^6) - (2*e^3*(-1 + c^2*x 
^2)^(5/2))/(5*c^6) + 10*d^3*ArcTan[Sqrt[-1 + c^2*x^2]]))/(10*Sqrt[-1 + c*x 
]*Sqrt[1 + c*x])
 

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 2123

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] 
:> Int[ExpandIntegrand[Px*(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c 
, d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2])
 

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 6373

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x 
_)^2)^(p_.), x_Symbol] :> With[{u = IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Sim 
p[(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, f, m}, x] && 
NeQ[c^2*d + e, 0] && IntegerQ[p] && (GtQ[p, 0] || (IGtQ[(m - 1)/2, 0] && Le 
Q[m + p, 0]))
 

Maple [A] (verified)

Time = 0.20 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.28

method	result	size
parts	\(a \left (\frac {e^{3} x^{5}}{5}+d \,e^{2} x^{3}+3 d^{2} e x -\frac {d^{3}}{x}\right )+b c \left (\frac {\operatorname {arccosh}\left (c x \right ) e^{3} x^{5}}{5 c}+\frac {\operatorname {arccosh}\left (c x \right ) d \,e^{2} x^{3}}{c}+\frac {3 \,\operatorname {arccosh}\left (c x \right ) d^{2} e x}{c}-\frac {\operatorname {arccosh}\left (c x \right ) d^{3}}{c x}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (75 c^{6} d^{3} \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )+225 c^{4} d^{2} e \sqrt {c^{2} x^{2}-1}+25 c^{4} d \,e^{2} x^{2} \sqrt {c^{2} x^{2}-1}+3 e^{3} c^{4} x^{4} \sqrt {c^{2} x^{2}-1}+50 c^{2} d \,e^{2} \sqrt {c^{2} x^{2}-1}+4 e^{3} c^{2} x^{2} \sqrt {c^{2} x^{2}-1}+8 e^{3} \sqrt {c^{2} x^{2}-1}\right )}{75 c^{6} \sqrt {c^{2} x^{2}-1}}\right )\)	\(273\)
derivativedivides	\(c \left (\frac {a \left (3 c^{5} d^{2} e x +c^{5} d \,e^{2} x^{3}+\frac {e^{3} c^{5} x^{5}}{5}-\frac {c^{5} d^{3}}{x}\right )}{c^{6}}+\frac {b \left (3 \,\operatorname {arccosh}\left (c x \right ) c^{5} d^{2} e x +\operatorname {arccosh}\left (c x \right ) c^{5} d \,e^{2} x^{3}+\frac {\operatorname {arccosh}\left (c x \right ) e^{3} c^{5} x^{5}}{5}-\frac {\operatorname {arccosh}\left (c x \right ) c^{5} d^{3}}{x}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (75 c^{6} d^{3} \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )+225 c^{4} d^{2} e \sqrt {c^{2} x^{2}-1}+25 c^{4} d \,e^{2} x^{2} \sqrt {c^{2} x^{2}-1}+3 e^{3} c^{4} x^{4} \sqrt {c^{2} x^{2}-1}+50 c^{2} d \,e^{2} \sqrt {c^{2} x^{2}-1}+4 e^{3} c^{2} x^{2} \sqrt {c^{2} x^{2}-1}+8 e^{3} \sqrt {c^{2} x^{2}-1}\right )}{75 \sqrt {c^{2} x^{2}-1}}\right )}{c^{6}}\right )\)	\(289\)
default	\(c \left (\frac {a \left (3 c^{5} d^{2} e x +c^{5} d \,e^{2} x^{3}+\frac {e^{3} c^{5} x^{5}}{5}-\frac {c^{5} d^{3}}{x}\right )}{c^{6}}+\frac {b \left (3 \,\operatorname {arccosh}\left (c x \right ) c^{5} d^{2} e x +\operatorname {arccosh}\left (c x \right ) c^{5} d \,e^{2} x^{3}+\frac {\operatorname {arccosh}\left (c x \right ) e^{3} c^{5} x^{5}}{5}-\frac {\operatorname {arccosh}\left (c x \right ) c^{5} d^{3}}{x}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (75 c^{6} d^{3} \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )+225 c^{4} d^{2} e \sqrt {c^{2} x^{2}-1}+25 c^{4} d \,e^{2} x^{2} \sqrt {c^{2} x^{2}-1}+3 e^{3} c^{4} x^{4} \sqrt {c^{2} x^{2}-1}+50 c^{2} d \,e^{2} \sqrt {c^{2} x^{2}-1}+4 e^{3} c^{2} x^{2} \sqrt {c^{2} x^{2}-1}+8 e^{3} \sqrt {c^{2} x^{2}-1}\right )}{75 \sqrt {c^{2} x^{2}-1}}\right )}{c^{6}}\right )\)	\(289\)

Input:

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

Output:

a*(1/5*e^3*x^5+d*e^2*x^3+3*d^2*e*x-d^3/x)+b*c*(1/5/c*arccosh(c*x)*e^3*x^5+ 
1/c*arccosh(c*x)*d*e^2*x^3+3/c*arccosh(c*x)*d^2*e*x-arccosh(c*x)*d^3/c/x-1 
/75/c^6*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(75*c^6*d^3*arctan(1/(c^2*x^2-1)^(1/2) 
)+225*c^4*d^2*e*(c^2*x^2-1)^(1/2)+25*c^4*d*e^2*x^2*(c^2*x^2-1)^(1/2)+3*e^3 
*c^4*x^4*(c^2*x^2-1)^(1/2)+50*c^2*d*e^2*(c^2*x^2-1)^(1/2)+4*e^3*c^2*x^2*(c 
^2*x^2-1)^(1/2)+8*e^3*(c^2*x^2-1)^(1/2))/(c^2*x^2-1)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 327, normalized size of antiderivative = 1.53 \[ \int \frac {\left (d+e x^2\right )^3 (a+b \text {arccosh}(c x))}{x^2} \, dx=\frac {15 \, a c^{5} e^{3} x^{6} + 75 \, a c^{5} d e^{2} x^{4} + 150 \, b c^{6} d^{3} x \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) + 225 \, a c^{5} d^{2} e x^{2} - 75 \, a c^{5} d^{3} + 15 \, {\left (5 \, b c^{5} d^{3} - 15 \, b c^{5} d^{2} e - 5 \, b c^{5} d e^{2} - b c^{5} e^{3}\right )} x \log \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) + 15 \, {\left (b c^{5} e^{3} x^{6} + 5 \, b c^{5} d e^{2} x^{4} + 15 \, b c^{5} d^{2} e x^{2} - 5 \, b c^{5} d^{3} + {\left (5 \, b c^{5} d^{3} - 15 \, b c^{5} d^{2} e - 5 \, b c^{5} d e^{2} - b c^{5} e^{3}\right )} x\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (3 \, b c^{4} e^{3} x^{5} + {\left (25 \, b c^{4} d e^{2} + 4 \, b c^{2} e^{3}\right )} x^{3} + {\left (225 \, b c^{4} d^{2} e + 50 \, b c^{2} d e^{2} + 8 \, b e^{3}\right )} x\right )} \sqrt {c^{2} x^{2} - 1}}{75 \, c^{5} x} \] Input:

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

Output:

1/75*(15*a*c^5*e^3*x^6 + 75*a*c^5*d*e^2*x^4 + 150*b*c^6*d^3*x*arctan(-c*x 
+ sqrt(c^2*x^2 - 1)) + 225*a*c^5*d^2*e*x^2 - 75*a*c^5*d^3 + 15*(5*b*c^5*d^ 
3 - 15*b*c^5*d^2*e - 5*b*c^5*d*e^2 - b*c^5*e^3)*x*log(-c*x + sqrt(c^2*x^2 
- 1)) + 15*(b*c^5*e^3*x^6 + 5*b*c^5*d*e^2*x^4 + 15*b*c^5*d^2*e*x^2 - 5*b*c 
^5*d^3 + (5*b*c^5*d^3 - 15*b*c^5*d^2*e - 5*b*c^5*d*e^2 - b*c^5*e^3)*x)*log 
(c*x + sqrt(c^2*x^2 - 1)) - (3*b*c^4*e^3*x^5 + (25*b*c^4*d*e^2 + 4*b*c^2*e 
^3)*x^3 + (225*b*c^4*d^2*e + 50*b*c^2*d*e^2 + 8*b*e^3)*x)*sqrt(c^2*x^2 - 1 
))/(c^5*x)
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.04 \[ \int \frac {\left (d+e x^2\right )^3 (a+b \text {arccosh}(c x))}{x^2} \, dx=\frac {1}{5} \, a e^{3} x^{5} + a d e^{2} x^{3} - {\left (c \arcsin \left (\frac {1}{c {\left | x \right |}}\right ) + \frac {\operatorname {arcosh}\left (c x\right )}{x}\right )} b d^{3} + \frac {1}{3} \, {\left (3 \, x^{3} \operatorname {arcosh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} - 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {c^{2} x^{2} - 1}}{c^{4}}\right )}\right )} b d e^{2} + \frac {1}{75} \, {\left (15 \, x^{5} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {3 \, \sqrt {c^{2} x^{2} - 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {c^{2} x^{2} - 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} - 1}}{c^{6}}\right )} c\right )} b e^{3} + 3 \, a d^{2} e x + \frac {3 \, {\left (c x \operatorname {arcosh}\left (c x\right ) - \sqrt {c^{2} x^{2} - 1}\right )} b d^{2} e}{c} - \frac {a d^{3}}{x} \] Input:

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

Output:

1/5*a*e^3*x^5 + a*d*e^2*x^3 - (c*arcsin(1/(c*abs(x))) + arccosh(c*x)/x)*b* 
d^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))*b*d*e^2 + 1/75*(15*x^5*arccosh(c*x) - (3*sqrt(c^2*x^2 - 1)* 
x^4/c^2 + 4*sqrt(c^2*x^2 - 1)*x^2/c^4 + 8*sqrt(c^2*x^2 - 1)/c^6)*c)*b*e^3 
+ 3*a*d^2*e*x + 3*(c*x*arccosh(c*x) - sqrt(c^2*x^2 - 1))*b*d^2*e/c - a*d^3 
/x
 

Giac [F]

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

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

Output:

integrate((e*x^2 + d)^3*(b*arccosh(c*x) + a)/x^2, x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.27 \[ \int \frac {\left (d+e x^2\right )^3 (a+b \text {arccosh}(c x))}{x^2} \, dx=\frac {-75 \mathit {acosh} \left (c x \right ) b \,c^{5} d^{3}+225 \mathit {acosh} \left (c x \right ) b \,c^{5} d^{2} e \,x^{2}+75 \mathit {acosh} \left (c x \right ) b \,c^{5} d \,e^{2} x^{4}+15 \mathit {acosh} \left (c x \right ) b \,c^{5} e^{3} x^{6}-150 \mathit {atan} \left (\sqrt {c^{2} x^{2}-1}+c x \right ) b \,c^{6} d^{3} x -25 \sqrt {c^{2} x^{2}-1}\, b \,c^{4} d \,e^{2} x^{3}-3 \sqrt {c^{2} x^{2}-1}\, b \,c^{4} e^{3} x^{5}-50 \sqrt {c^{2} x^{2}-1}\, b \,c^{2} d \,e^{2} x -4 \sqrt {c^{2} x^{2}-1}\, b \,c^{2} e^{3} x^{3}-8 \sqrt {c^{2} x^{2}-1}\, b \,e^{3} x -225 \sqrt {c x +1}\, \sqrt {c x -1}\, b \,c^{4} d^{2} e x -75 a \,c^{5} d^{3}+225 a \,c^{5} d^{2} e \,x^{2}+75 a \,c^{5} d \,e^{2} x^{4}+15 a \,c^{5} e^{3} x^{6}}{75 c^{5} x} \] Input:

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

Output:

( - 75*acosh(c*x)*b*c**5*d**3 + 225*acosh(c*x)*b*c**5*d**2*e*x**2 + 75*aco 
sh(c*x)*b*c**5*d*e**2*x**4 + 15*acosh(c*x)*b*c**5*e**3*x**6 - 150*atan(sqr 
t(c**2*x**2 - 1) + c*x)*b*c**6*d**3*x - 25*sqrt(c**2*x**2 - 1)*b*c**4*d*e* 
*2*x**3 - 3*sqrt(c**2*x**2 - 1)*b*c**4*e**3*x**5 - 50*sqrt(c**2*x**2 - 1)* 
b*c**2*d*e**2*x - 4*sqrt(c**2*x**2 - 1)*b*c**2*e**3*x**3 - 8*sqrt(c**2*x** 
2 - 1)*b*e**3*x - 225*sqrt(c*x + 1)*sqrt(c*x - 1)*b*c**4*d**2*e*x - 75*a*c 
**5*d**3 + 225*a*c**5*d**2*e*x**2 + 75*a*c**5*d*e**2*x**4 + 15*a*c**5*e**3 
*x**6)/(75*c**5*x)