3.1.18 \(\int \frac {(d-c^2 d x^2)^2 (a+b \text {arccosh}(c x))}{x^4} \, dx\) [18]

3.1.18.1 Optimal result

Integrand size = 25, antiderivative size = 142 \[ \int \frac {\left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x))}{x^4} \, dx=-b c^3 d^2 \sqrt {-1+c x} \sqrt {1+c x}+\frac {b c d^2 \sqrt {-1+c x} \sqrt {1+c x}}{6 x^2}-\frac {d^2 (a+b \text {arccosh}(c x))}{3 x^3}+\frac {2 c^2 d^2 (a+b \text {arccosh}(c x))}{x}+c^4 d^2 x (a+b \text {arccosh}(c x))-\frac {11}{6} b c^3 d^2 \arctan \left (\sqrt {-1+c x} \sqrt {1+c x}\right ) \]

output

-1/3*d^2*(a+b*arccosh(c*x))/x^3+2*c^2*d^2*(a+b*arccosh(c*x))/x+c^4*d^2*x*( 
a+b*arccosh(c*x))-11/6*b*c^3*d^2*arctan((c*x-1)^(1/2)*(c*x+1)^(1/2))-b*c^3 
*d^2*(c*x-1)^(1/2)*(c*x+1)^(1/2)+1/6*b*c*d^2*(c*x-1)^(1/2)*(c*x+1)^(1/2)/x 
^2
 

3.1.18.2 Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.95 \[ \int \frac {\left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x))}{x^4} \, dx=\frac {d^2 \left (-2 a+12 a c^2 x^2+6 a c^4 x^4+b c x \sqrt {-1+c x} \sqrt {1+c x}-6 b c^3 x^3 \sqrt {-1+c x} \sqrt {1+c x}+2 b \left (-1+6 c^2 x^2+3 c^4 x^4\right ) \text {arccosh}(c x)+11 b c^3 x^3 \arctan \left (\frac {1}{\sqrt {-1+c x} \sqrt {1+c x}}\right )\right )}{6 x^3} \]

input

Integrate[((d - c^2*d*x^2)^2*(a + b*ArcCosh[c*x]))/x^4,x]
 

output

(d^2*(-2*a + 12*a*c^2*x^2 + 6*a*c^4*x^4 + b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c* 
x] - 6*b*c^3*x^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x] + 2*b*(-1 + 6*c^2*x^2 + 3*c^ 
4*x^4)*ArcCosh[c*x] + 11*b*c^3*x^3*ArcTan[1/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) 
]))/(6*x^3)
 

3.1.18.3 Rubi [A] (warning: unable to verify)

Time = 0.52 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.15, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {6336, 27, 1905, 1578, 1192, 25, 1471, 25, 27, 299, 216}

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.

\(\displaystyle \int \frac {\left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x))}{x^4} \, dx\)

\(\Big \downarrow \) 6336

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1905

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

\(\Big \downarrow \) 1578

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

\(\Big \downarrow \) 1192

\(\displaystyle \frac {b d^2 \sqrt {c^2 x^2-1} \int -\frac {3 c^4 x^8+12 c^4 x^4+8 c^4}{\left (x^4+1\right )^2}d\sqrt {c^2 x^2-1}}{3 c \sqrt {c x-1} \sqrt {c x+1}}+c^4 d^2 x (a+b \text {arccosh}(c x))+\frac {2 c^2 d^2 (a+b \text {arccosh}(c x))}{x}-\frac {d^2 (a+b \text {arccosh}(c x))}{3 x^3}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {b d^2 \sqrt {c^2 x^2-1} \int \frac {3 c^4 x^8+12 c^4 x^4+8 c^4}{\left (x^4+1\right )^2}d\sqrt {c^2 x^2-1}}{3 c \sqrt {c x-1} \sqrt {c x+1}}+c^4 d^2 x (a+b \text {arccosh}(c x))+\frac {2 c^2 d^2 (a+b \text {arccosh}(c x))}{x}-\frac {d^2 (a+b \text {arccosh}(c x))}{3 x^3}\)

\(\Big \downarrow \) 1471

\(\displaystyle \frac {b d^2 \sqrt {c^2 x^2-1} \left (\frac {1}{2} \int -\frac {c^4 \left (6 x^4+17\right )}{x^4+1}d\sqrt {c^2 x^2-1}+\frac {c^4 \sqrt {c^2 x^2-1}}{2 \left (x^4+1\right )}\right )}{3 c \sqrt {c x-1} \sqrt {c x+1}}+c^4 d^2 x (a+b \text {arccosh}(c x))+\frac {2 c^2 d^2 (a+b \text {arccosh}(c x))}{x}-\frac {d^2 (a+b \text {arccosh}(c x))}{3 x^3}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {b d^2 \sqrt {c^2 x^2-1} \left (\frac {c^4 \sqrt {c^2 x^2-1}}{2 \left (x^4+1\right )}-\frac {1}{2} \int \frac {c^4 \left (6 x^4+17\right )}{x^4+1}d\sqrt {c^2 x^2-1}\right )}{3 c \sqrt {c x-1} \sqrt {c x+1}}+c^4 d^2 x (a+b \text {arccosh}(c x))+\frac {2 c^2 d^2 (a+b \text {arccosh}(c x))}{x}-\frac {d^2 (a+b \text {arccosh}(c x))}{3 x^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {b d^2 \sqrt {c^2 x^2-1} \left (\frac {c^4 \sqrt {c^2 x^2-1}}{2 \left (x^4+1\right )}-\frac {1}{2} c^4 \int \frac {6 x^4+17}{x^4+1}d\sqrt {c^2 x^2-1}\right )}{3 c \sqrt {c x-1} \sqrt {c x+1}}+c^4 d^2 x (a+b \text {arccosh}(c x))+\frac {2 c^2 d^2 (a+b \text {arccosh}(c x))}{x}-\frac {d^2 (a+b \text {arccosh}(c x))}{3 x^3}\)

\(\Big \downarrow \) 299

\(\displaystyle \frac {b d^2 \sqrt {c^2 x^2-1} \left (\frac {c^4 \sqrt {c^2 x^2-1}}{2 \left (x^4+1\right )}-\frac {1}{2} c^4 \left (11 \int \frac {1}{x^4+1}d\sqrt {c^2 x^2-1}+6 \sqrt {c^2 x^2-1}\right )\right )}{3 c \sqrt {c x-1} \sqrt {c x+1}}+c^4 d^2 x (a+b \text {arccosh}(c x))+\frac {2 c^2 d^2 (a+b \text {arccosh}(c x))}{x}-\frac {d^2 (a+b \text {arccosh}(c x))}{3 x^3}\)

\(\Big \downarrow \) 216

\(\displaystyle c^4 d^2 x (a+b \text {arccosh}(c x))+\frac {2 c^2 d^2 (a+b \text {arccosh}(c x))}{x}-\frac {d^2 (a+b \text {arccosh}(c x))}{3 x^3}+\frac {b d^2 \sqrt {c^2 x^2-1} \left (\frac {c^4 \sqrt {c^2 x^2-1}}{2 \left (x^4+1\right )}-\frac {1}{2} c^4 \left (11 \arctan \left (\sqrt {c^2 x^2-1}\right )+6 \sqrt {c^2 x^2-1}\right )\right )}{3 c \sqrt {c x-1} \sqrt {c x+1}}\)

input

Int[((d - c^2*d*x^2)^2*(a + b*ArcCosh[c*x]))/x^4,x]
 

output

-1/3*(d^2*(a + b*ArcCosh[c*x]))/x^3 + (2*c^2*d^2*(a + b*ArcCosh[c*x]))/x + 
 c^4*d^2*x*(a + b*ArcCosh[c*x]) + (b*d^2*Sqrt[-1 + c^2*x^2]*((c^4*Sqrt[-1 
+ c^2*x^2])/(2*(1 + x^4)) - (c^4*(6*Sqrt[-1 + c^2*x^2] + 11*ArcTan[Sqrt[-1 
 + c^2*x^2]]))/2))/(3*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])
 

3.1.18.3.1 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 216

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A 
rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a 
, 0] || GtQ[b, 0])
 

rule 299

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[d*x 
*((a + b*x^2)^(p + 1)/(b*(2*p + 3))), x] - Simp[(a*d - b*c*(2*p + 3))/(b*(2 
*p + 3))   Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - 
 a*d, 0] && NeQ[2*p + 3, 0]
 

rule 1192

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) 
 + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[2/e^(n + 2*p + 1)   Subst[Int[x^( 
2*m + 1)*(e*f - d*g + g*x^2)^n*(c*d^2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + 
 c*x^4)^p, x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && 
IGtQ[p, 0] && ILtQ[n, 0] && IntegerQ[m + 1/2]
 

rule 1471

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), 
x_Symbol] :> With[{Qx = PolynomialQuotient[(a + b*x^2 + c*x^4)^p, d + e*x^2 
, x], R = Coeff[PolynomialRemainder[(a + b*x^2 + c*x^4)^p, d + e*x^2, x], x 
, 0]}, Simp[(-R)*x*((d + e*x^2)^(q + 1)/(2*d*(q + 1))), x] + Simp[1/(2*d*(q 
 + 1))   Int[(d + e*x^2)^(q + 1)*ExpandToSum[2*d*(q + 1)*Qx + R*(2*q + 3), 
x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^ 
2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && LtQ[q, -1]
 

rule 1578

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_ 
)^4)^(p_.), x_Symbol] :> Simp[1/2   Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a 
+ b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && Int 
egerQ[(m - 1)/2]
 

rule 1905

Int[((f_.)*(x_))^(m_.)*((d1_) + (e1_.)*(x_)^(non2_.))^(q_.)*((d2_) + (e2_.) 
*(x_)^(non2_.))^(q_.)*((a_.) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_))^(p_.), x 
_Symbol] :> Simp[(d1 + e1*x^(n/2))^FracPart[q]*((d2 + e2*x^(n/2))^FracPart[ 
q]/(d1*d2 + e1*e2*x^n)^FracPart[q])   Int[(f*x)^m*(d1*d2 + e1*e2*x^n)^q*(a 
+ b*x^n + c*x^(2*n))^p, x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, f, n, p, 
q}, x] && EqQ[n2, 2*n] && EqQ[non2, n/2] && EqQ[d2*e1 + d1*e2, 0]
 

rule 6336

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]}, 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, f, m}, x] && E 
qQ[c^2*d + e, 0] && IGtQ[p, 0]
 

3.1.18.4 Maple [A] (verified)

Time = 0.48 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.06

method	result	size
parts	\(d^{2} a \left (c^{4} x +\frac {2 c^{2}}{x}-\frac {1}{3 x^{3}}\right )+d^{2} b \,c^{3} \left (c x \,\operatorname {arccosh}\left (c x \right )-\frac {\operatorname {arccosh}\left (c x \right )}{3 c^{3} x^{3}}+\frac {2 \,\operatorname {arccosh}\left (c x \right )}{c x}+\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (11 \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) c^{2} x^{2}-6 c^{2} x^{2} \sqrt {c^{2} x^{2}-1}+\sqrt {c^{2} x^{2}-1}\right )}{6 c^{2} x^{2} \sqrt {c^{2} x^{2}-1}}\right )\)	\(150\)
derivativedivides	\(c^{3} \left (d^{2} a \left (c x -\frac {1}{3 c^{3} x^{3}}+\frac {2}{c x}\right )+d^{2} b \left (c x \,\operatorname {arccosh}\left (c x \right )-\frac {\operatorname {arccosh}\left (c x \right )}{3 c^{3} x^{3}}+\frac {2 \,\operatorname {arccosh}\left (c x \right )}{c x}+\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (11 \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) c^{2} x^{2}-6 c^{2} x^{2} \sqrt {c^{2} x^{2}-1}+\sqrt {c^{2} x^{2}-1}\right )}{6 c^{2} x^{2} \sqrt {c^{2} x^{2}-1}}\right )\right )\)	\(152\)
default	\(c^{3} \left (d^{2} a \left (c x -\frac {1}{3 c^{3} x^{3}}+\frac {2}{c x}\right )+d^{2} b \left (c x \,\operatorname {arccosh}\left (c x \right )-\frac {\operatorname {arccosh}\left (c x \right )}{3 c^{3} x^{3}}+\frac {2 \,\operatorname {arccosh}\left (c x \right )}{c x}+\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (11 \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) c^{2} x^{2}-6 c^{2} x^{2} \sqrt {c^{2} x^{2}-1}+\sqrt {c^{2} x^{2}-1}\right )}{6 c^{2} x^{2} \sqrt {c^{2} x^{2}-1}}\right )\right )\)	\(152\)

input

int((-c^2*d*x^2+d)^2*(a+b*arccosh(c*x))/x^4,x,method=_RETURNVERBOSE)
 

output

d^2*a*(c^4*x+2*c^2/x-1/3/x^3)+d^2*b*c^3*(c*x*arccosh(c*x)-1/3/c^3/x^3*arcc 
osh(c*x)+2*arccosh(c*x)/c/x+1/6*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(11*arctan(1/( 
c^2*x^2-1)^(1/2))*c^2*x^2-6*c^2*x^2*(c^2*x^2-1)^(1/2)+(c^2*x^2-1)^(1/2))/c 
^2/x^2/(c^2*x^2-1)^(1/2))
 

3.1.18.5 Fricas [A] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.50 \[ \int \frac {\left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x))}{x^4} \, dx=\frac {6 \, a c^{4} d^{2} x^{4} - 22 \, b c^{3} d^{2} x^{3} \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) + 12 \, a c^{2} d^{2} x^{2} - 2 \, {\left (3 \, b c^{4} + 6 \, b c^{2} - b\right )} d^{2} x^{3} \log \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) - 2 \, a d^{2} + 2 \, {\left (3 \, b c^{4} d^{2} x^{4} + 6 \, b c^{2} d^{2} x^{2} - {\left (3 \, b c^{4} + 6 \, b c^{2} - b\right )} d^{2} x^{3} - b d^{2}\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (6 \, b c^{3} d^{2} x^{3} - b c d^{2} x\right )} \sqrt {c^{2} x^{2} - 1}}{6 \, x^{3}} \]

input

integrate((-c^2*d*x^2+d)^2*(a+b*arccosh(c*x))/x^4,x, algorithm="fricas")
 

output

1/6*(6*a*c^4*d^2*x^4 - 22*b*c^3*d^2*x^3*arctan(-c*x + sqrt(c^2*x^2 - 1)) + 
 12*a*c^2*d^2*x^2 - 2*(3*b*c^4 + 6*b*c^2 - b)*d^2*x^3*log(-c*x + sqrt(c^2* 
x^2 - 1)) - 2*a*d^2 + 2*(3*b*c^4*d^2*x^4 + 6*b*c^2*d^2*x^2 - (3*b*c^4 + 6* 
b*c^2 - b)*d^2*x^3 - b*d^2)*log(c*x + sqrt(c^2*x^2 - 1)) - (6*b*c^3*d^2*x^ 
3 - b*c*d^2*x)*sqrt(c^2*x^2 - 1))/x^3
 

3.1.18.6 Sympy [F]

\[ \int \frac {\left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x))}{x^4} \, dx=d^{2} \left (\int a c^{4}\, dx + \int \frac {a}{x^{4}}\, dx + \int \left (- \frac {2 a c^{2}}{x^{2}}\right )\, dx + \int b c^{4} \operatorname {acosh}{\left (c x \right )}\, dx + \int \frac {b \operatorname {acosh}{\left (c x \right )}}{x^{4}}\, dx + \int \left (- \frac {2 b c^{2} \operatorname {acosh}{\left (c x \right )}}{x^{2}}\right )\, dx\right ) \]

input

integrate((-c**2*d*x**2+d)**2*(a+b*acosh(c*x))/x**4,x)
 

output

d**2*(Integral(a*c**4, x) + Integral(a/x**4, x) + Integral(-2*a*c**2/x**2, 
 x) + Integral(b*c**4*acosh(c*x), x) + Integral(b*acosh(c*x)/x**4, x) + In 
tegral(-2*b*c**2*acosh(c*x)/x**2, x))
 

3.1.18.7 Maxima [A] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.96 \[ \int \frac {\left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x))}{x^4} \, dx=a c^{4} d^{2} x + {\left (c x \operatorname {arcosh}\left (c x\right ) - \sqrt {c^{2} x^{2} - 1}\right )} b c^{3} d^{2} + 2 \, {\left (c \arcsin \left (\frac {1}{c {\left | x \right |}}\right ) + \frac {\operatorname {arcosh}\left (c x\right )}{x}\right )} b c^{2} d^{2} - \frac {1}{6} \, {\left ({\left (c^{2} \arcsin \left (\frac {1}{c {\left | x \right |}}\right ) - \frac {\sqrt {c^{2} x^{2} - 1}}{x^{2}}\right )} c + \frac {2 \, \operatorname {arcosh}\left (c x\right )}{x^{3}}\right )} b d^{2} + \frac {2 \, a c^{2} d^{2}}{x} - \frac {a d^{2}}{3 \, x^{3}} \]

input

integrate((-c^2*d*x^2+d)^2*(a+b*arccosh(c*x))/x^4,x, algorithm="maxima")
 

output

a*c^4*d^2*x + (c*x*arccosh(c*x) - sqrt(c^2*x^2 - 1))*b*c^3*d^2 + 2*(c*arcs 
in(1/(c*abs(x))) + arccosh(c*x)/x)*b*c^2*d^2 - 1/6*((c^2*arcsin(1/(c*abs(x 
))) - sqrt(c^2*x^2 - 1)/x^2)*c + 2*arccosh(c*x)/x^3)*b*d^2 + 2*a*c^2*d^2/x 
 - 1/3*a*d^2/x^3
 

3.1.18.8 Giac [F(-2)]

Exception generated. \[ \int \frac {\left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x))}{x^4} \, dx=\text {Exception raised: TypeError} \]

input

integrate((-c^2*d*x^2+d)^2*(a+b*arccosh(c*x))/x^4,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
 

3.1.18.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x))}{x^4} \, dx=\int \frac {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^2}{x^4} \,d x \]

input

int(((a + b*acosh(c*x))*(d - c^2*d*x^2)^2)/x^4,x)
 

output

int(((a + b*acosh(c*x))*(d - c^2*d*x^2)^2)/x^4, x)