Integrand size = 21, antiderivative size = 231 \[ \int \frac {x^3 (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^3} \, dx=-\frac {b c x \left (1-c^2 x^2\right )}{8 e \left (c^2 d+e\right ) \sqrt {-1+c x} \sqrt {1+c x} \left (d+e x^2\right )}+\frac {x^4 (a+b \text {arccosh}(c x))}{4 d \left (d+e x^2\right )^2}-\frac {b \sqrt {1-c^2 x^2} \arcsin (c x)}{4 d e^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c \left (2 c^2 d+3 e\right ) \sqrt {1-c^2 x^2} \arctan \left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{8 \sqrt {d} e^2 \left (c^2 d+e\right )^{3/2} \sqrt {-1+c x} \sqrt {1+c x}} \]
1/4*x^4*(a+b*arccosh(c*x))/d/(e*x^2+d)^2-1/8*b*c*x*(-c^2*x^2+1)/e/(c^2*d+e )/(e*x^2+d)/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/4*b*arcsin(c*x)*(-c^2*x^2+1)^(1/ 2)/d/e^2/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/8*b*c*(2*c^2*d+3*e)*arctan(x*(c^2*d +e)^(1/2)/d^(1/2)/(-c^2*x^2+1)^(1/2))*(-c^2*x^2+1)^(1/2)/e^2/(c^2*d+e)^(3/ 2)/d^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)
Time = 0.67 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.83 \[ \int \frac {x^3 (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^3} \, dx=\frac {\frac {\frac {b c e x \sqrt {-1+c x} \sqrt {1+c x} \left (d+e x^2\right )}{c^2 d+e}-2 a \left (d+2 e x^2\right )}{\left (d+e x^2\right )^2}-\frac {2 b \left (d+2 e x^2\right ) \text {arccosh}(c x)}{\left (d+e x^2\right )^2}-\frac {b c \left (2 c^2 d+3 e\right ) \sqrt {-1+c x} \sqrt {1+c x} \arctan \left (\frac {\sqrt {-c^2 d-e} x}{\sqrt {d} \sqrt {-1+c^2 x^2}}\right )}{\sqrt {d} \left (-c^2 d-e\right )^{3/2} \sqrt {-1+c^2 x^2}}}{8 e^2} \]
(((b*c*e*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(d + e*x^2))/(c^2*d + e) - 2*a*(d + 2*e*x^2))/(d + e*x^2)^2 - (2*b*(d + 2*e*x^2)*ArcCosh[c*x])/(d + e*x^2)^2 - (b*c*(2*c^2*d + 3*e)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*ArcTan[(Sqrt[-(c^2*d) - e]*x)/(Sqrt[d]*Sqrt[-1 + c^2*x^2])])/(Sqrt[d]*(-(c^2*d) - e)^(3/2)*Sqrt [-1 + c^2*x^2]))/(8*e^2)
Time = 0.59 (sec) , antiderivative size = 220, normalized size of antiderivative = 0.95, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {6373, 27, 2038, 372, 25, 398, 224, 219, 291, 221}
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.
| \(\displaystyle \int \frac {x^3 (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 6373 |
| \(\displaystyle \frac {x^4 (a+b \text {arccosh}(c x))}{4 d \left (d+e x^2\right )^2}-b c \int \frac {x^4}{4 d \sqrt {c x-1} \sqrt {c x+1} \left (e x^2+d\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x^4 (a+b \text {arccosh}(c x))}{4 d \left (d+e x^2\right )^2}-\frac {b c \int \frac {x^4}{\sqrt {c x-1} \sqrt {c x+1} \left (e x^2+d\right )^2}dx}{4 d}\) |
| \(\Big \downarrow \) 2038 |
| \(\displaystyle \frac {x^4 (a+b \text {arccosh}(c x))}{4 d \left (d+e x^2\right )^2}-\frac {b c \sqrt {c^2 x^2-1} \int \frac {x^4}{\sqrt {c^2 x^2-1} \left (e x^2+d\right )^2}dx}{4 d \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 372 |
| \(\displaystyle \frac {x^4 (a+b \text {arccosh}(c x))}{4 d \left (d+e x^2\right )^2}-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {\int -\frac {d-2 \left (d c^2+e\right ) x^2}{\sqrt {c^2 x^2-1} \left (e x^2+d\right )}dx}{2 e \left (c^2 d+e\right )}-\frac {d x \sqrt {c^2 x^2-1}}{2 e \left (c^2 d+e\right ) \left (d+e x^2\right )}\right )}{4 d \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {x^4 (a+b \text {arccosh}(c x))}{4 d \left (d+e x^2\right )^2}-\frac {b c \sqrt {c^2 x^2-1} \left (-\frac {\int \frac {d-2 \left (d c^2+e\right ) x^2}{\sqrt {c^2 x^2-1} \left (e x^2+d\right )}dx}{2 e \left (c^2 d+e\right )}-\frac {d x \sqrt {c^2 x^2-1}}{2 e \left (c^2 d+e\right ) \left (d+e x^2\right )}\right )}{4 d \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 398 |
| \(\displaystyle \frac {x^4 (a+b \text {arccosh}(c x))}{4 d \left (d+e x^2\right )^2}-\frac {b c \sqrt {c^2 x^2-1} \left (-\frac {\frac {d \left (2 c^2 d+3 e\right ) \int \frac {1}{\sqrt {c^2 x^2-1} \left (e x^2+d\right )}dx}{e}-\frac {2 \left (c^2 d+e\right ) \int \frac {1}{\sqrt {c^2 x^2-1}}dx}{e}}{2 e \left (c^2 d+e\right )}-\frac {d x \sqrt {c^2 x^2-1}}{2 e \left (c^2 d+e\right ) \left (d+e x^2\right )}\right )}{4 d \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {x^4 (a+b \text {arccosh}(c x))}{4 d \left (d+e x^2\right )^2}-\frac {b c \sqrt {c^2 x^2-1} \left (-\frac {\frac {d \left (2 c^2 d+3 e\right ) \int \frac {1}{\sqrt {c^2 x^2-1} \left (e x^2+d\right )}dx}{e}-\frac {2 \left (c^2 d+e\right ) \int \frac {1}{1-\frac {c^2 x^2}{c^2 x^2-1}}d\frac {x}{\sqrt {c^2 x^2-1}}}{e}}{2 e \left (c^2 d+e\right )}-\frac {d x \sqrt {c^2 x^2-1}}{2 e \left (c^2 d+e\right ) \left (d+e x^2\right )}\right )}{4 d \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {x^4 (a+b \text {arccosh}(c x))}{4 d \left (d+e x^2\right )^2}-\frac {b c \sqrt {c^2 x^2-1} \left (-\frac {\frac {d \left (2 c^2 d+3 e\right ) \int \frac {1}{\sqrt {c^2 x^2-1} \left (e x^2+d\right )}dx}{e}-\frac {2 \text {arctanh}\left (\frac {c x}{\sqrt {c^2 x^2-1}}\right ) \left (c^2 d+e\right )}{c e}}{2 e \left (c^2 d+e\right )}-\frac {d x \sqrt {c^2 x^2-1}}{2 e \left (c^2 d+e\right ) \left (d+e x^2\right )}\right )}{4 d \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 291 |
| \(\displaystyle \frac {x^4 (a+b \text {arccosh}(c x))}{4 d \left (d+e x^2\right )^2}-\frac {b c \sqrt {c^2 x^2-1} \left (-\frac {\frac {d \left (2 c^2 d+3 e\right ) \int \frac {1}{d-\frac {\left (d c^2+e\right ) x^2}{c^2 x^2-1}}d\frac {x}{\sqrt {c^2 x^2-1}}}{e}-\frac {2 \text {arctanh}\left (\frac {c x}{\sqrt {c^2 x^2-1}}\right ) \left (c^2 d+e\right )}{c e}}{2 e \left (c^2 d+e\right )}-\frac {d x \sqrt {c^2 x^2-1}}{2 e \left (c^2 d+e\right ) \left (d+e x^2\right )}\right )}{4 d \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {x^4 (a+b \text {arccosh}(c x))}{4 d \left (d+e x^2\right )^2}-\frac {b c \sqrt {c^2 x^2-1} \left (-\frac {\frac {\sqrt {d} \left (2 c^2 d+3 e\right ) \text {arctanh}\left (\frac {x \sqrt {c^2 d+e}}{\sqrt {d} \sqrt {c^2 x^2-1}}\right )}{e \sqrt {c^2 d+e}}-\frac {2 \text {arctanh}\left (\frac {c x}{\sqrt {c^2 x^2-1}}\right ) \left (c^2 d+e\right )}{c e}}{2 e \left (c^2 d+e\right )}-\frac {d x \sqrt {c^2 x^2-1}}{2 e \left (c^2 d+e\right ) \left (d+e x^2\right )}\right )}{4 d \sqrt {c x-1} \sqrt {c x+1}}\) |
(x^4*(a + b*ArcCosh[c*x]))/(4*d*(d + e*x^2)^2) - (b*c*Sqrt[-1 + c^2*x^2]*( -1/2*(d*x*Sqrt[-1 + c^2*x^2])/(e*(c^2*d + e)*(d + e*x^2)) - ((-2*(c^2*d + e)*ArcTanh[(c*x)/Sqrt[-1 + c^2*x^2]])/(c*e) + (Sqrt[d]*(2*c^2*d + 3*e)*Arc Tanh[(Sqrt[c^2*d + e]*x)/(Sqrt[d]*Sqrt[-1 + c^2*x^2])])/(e*Sqrt[c^2*d + e] ))/(2*e*(c^2*d + e))))/(4*d*Sqrt[-1 + c*x]*Sqrt[1 + c*x])
3.6.7.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 0]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Subst [Int[1/(c - (b*c - a*d)*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ ), x_Symbol] :> Simp[(-a)*e^3*(e*x)^(m - 3)*(a + b*x^2)^(p + 1)*((c + d*x^2 )^(q + 1)/(2*b*(b*c - a*d)*(p + 1))), x] + Simp[e^4/(2*b*(b*c - a*d)*(p + 1 )) Int[(e*x)^(m - 4)*(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[a*c*(m - 3) + (a*d*(m + 2*q - 1) + 2*b*c*(p + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && GtQ[m, 3] && IntBinomialQ[a , b, c, d, e, m, 2, p, q, x]
Int[((e_) + (f_.)*(x_)^2)/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]) , x_Symbol] :> Simp[f/b Int[1/Sqrt[c + d*x^2], x], x] + Simp[(b*e - a*f)/ b Int[1/((a + b*x^2)*Sqrt[c + d*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f} , x]
Int[(u_.)*((c_) + (d_.)*(x_)^(n_.))^(q_.)*((a1_) + (b1_.)*(x_)^(non2_.))^(p _)*((a2_) + (b2_.)*(x_)^(non2_.))^(p_), x_Symbol] :> Simp[(a1 + b1*x^(n/2)) ^FracPart[p]*((a2 + b2*x^(n/2))^FracPart[p]/(a1*a2 + b1*b2*x^n)^FracPart[p] ) Int[u*(a1*a2 + b1*b2*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a1, b1, a2, b2, c, d, n, p, q}, x] && EqQ[non2, n/2] && EqQ[a2*b1 + a1*b2, 0] && !(Eq Q[n, 2] && IGtQ[q, 0])
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]))
Leaf count of result is larger than twice the leaf count of optimal. \(1168\) vs. \(2(197)=394\).
Time = 0.55 (sec) , antiderivative size = 1169, normalized size of antiderivative = 5.06
| method | result | size |
| parts | \(\text {Expression too large to display}\) | \(1169\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(1199\) |
| default | \(\text {Expression too large to display}\) | \(1199\) |
a*(1/4*d/e^2/(e*x^2+d)^2-1/2/e^2/(e*x^2+d))+b/c^4*(1/4*c^8*arccosh(c*x)/e^ 2*d/(c^2*e*x^2+c^2*d)^2-1/2*c^6*arccosh(c*x)/e^2/(c^2*e*x^2+c^2*d)+1/16*c^ 6*e*(2*ln(-2*(-(-(c^2*d+e)/e)^(1/2)*(c^2*x^2-1)^(1/2)*e+(-c^2*d*e)^(1/2)*c *x+e)/(e*c*x+(-c^2*d*e)^(1/2)))*c^6*x^2*d^2*e+2*ln(-2*(-(-(c^2*d+e)/e)^(1/ 2)*(c^2*x^2-1)^(1/2)*e+(-c^2*d*e)^(1/2)*c*x+e)/(e*c*x+(-c^2*d*e)^(1/2)))*c ^6*d^3-2*ln(2*((-(c^2*d+e)/e)^(1/2)*(c^2*x^2-1)^(1/2)*e+(-c^2*d*e)^(1/2)*c *x-e)/(e*c*x-(-c^2*d*e)^(1/2)))*c^6*d^2*e*x^2-2*ln(2*((-(c^2*d+e)/e)^(1/2) *(c^2*x^2-1)^(1/2)*e+(-c^2*d*e)^(1/2)*c*x-e)/(e*c*x-(-c^2*d*e)^(1/2)))*c^6 *d^3+5*ln(-2*(-(-(c^2*d+e)/e)^(1/2)*(c^2*x^2-1)^(1/2)*e+(-c^2*d*e)^(1/2)*c *x+e)/(e*c*x+(-c^2*d*e)^(1/2)))*c^4*x^2*d*e^2+5*ln(-2*(-(-(c^2*d+e)/e)^(1/ 2)*(c^2*x^2-1)^(1/2)*e+(-c^2*d*e)^(1/2)*c*x+e)/(e*c*x+(-c^2*d*e)^(1/2)))*c ^4*d^2*e-5*ln(2*((-(c^2*d+e)/e)^(1/2)*(c^2*x^2-1)^(1/2)*e+(-c^2*d*e)^(1/2) *c*x-e)/(e*c*x-(-c^2*d*e)^(1/2)))*c^4*d*e^2*x^2-5*ln(2*((-(c^2*d+e)/e)^(1/ 2)*(c^2*x^2-1)^(1/2)*e+(-c^2*d*e)^(1/2)*c*x-e)/(e*c*x-(-c^2*d*e)^(1/2)))*c ^4*d^2*e+2*c^3*d*e*(-c^2*d*e)^(1/2)*(c^2*x^2-1)^(1/2)*(-(c^2*d+e)/e)^(1/2) *x+3*ln(-2*(-(-(c^2*d+e)/e)^(1/2)*(c^2*x^2-1)^(1/2)*e+(-c^2*d*e)^(1/2)*c*x +e)/(e*c*x+(-c^2*d*e)^(1/2)))*c^2*x^2*e^3+3*ln(-2*(-(-(c^2*d+e)/e)^(1/2)*( c^2*x^2-1)^(1/2)*e+(-c^2*d*e)^(1/2)*c*x+e)/(e*c*x+(-c^2*d*e)^(1/2)))*c^2*d *e^2-3*ln(2*((-(c^2*d+e)/e)^(1/2)*(c^2*x^2-1)^(1/2)*e+(-c^2*d*e)^(1/2)*c*x -e)/(e*c*x-(-c^2*d*e)^(1/2)))*e^3*c^2*x^2-3*ln(2*((-(c^2*d+e)/e)^(1/2)*...
Leaf count of result is larger than twice the leaf count of optimal. 577 vs. \(2 (196) = 392\).
Time = 0.36 (sec) , antiderivative size = 1217, normalized size of antiderivative = 5.27 \[ \int \frac {x^3 (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^3} \, dx=\text {Too large to display} \]
[-1/16*(2*(2*a - b)*c^4*d^4 + 2*(4*a - b)*c^2*d^3*e - 4*(b*c^4*d^2*e^2 + 2 *b*c^2*d*e^3 + b*e^4)*x^4*log(c*x + sqrt(c^2*x^2 - 1)) + 4*a*d^2*e^2 - 2*( b*c^4*d^2*e^2 + b*c^2*d*e^3)*x^4 + 4*((2*a - b)*c^4*d^3*e + (4*a - b)*c^2* d^2*e^2 + 2*a*d*e^3)*x^2 - (2*b*c^3*d^3 + 3*b*c*d^2*e + (2*b*c^3*d*e^2 + 3 *b*c*e^3)*x^4 + 2*(2*b*c^3*d^2*e + 3*b*c*d*e^2)*x^2)*sqrt(c^2*d^2 + d*e)*l og(-(2*c^2*d^2 - (4*c^4*d^2 + 4*c^2*d*e + e^2)*x^2 + d*e - 2*sqrt(c^2*d^2 + d*e)*((2*c^3*d + c*e)*x^2 - c*d) - 2*sqrt(c^2*x^2 - 1)*(sqrt(c^2*d^2 + d *e)*(2*c^2*d + e)*x + 2*(c^3*d^2 + c*d*e)*x))/(e*x^2 + d)) - 4*(b*c^4*d^4 + 2*b*c^2*d^3*e + b*d^2*e^2 + (b*c^4*d^2*e^2 + 2*b*c^2*d*e^3 + b*e^4)*x^4 + 2*(b*c^4*d^3*e + 2*b*c^2*d^2*e^2 + b*d*e^3)*x^2)*log(-c*x + sqrt(c^2*x^2 - 1)) - 2*sqrt(c^2*x^2 - 1)*((b*c^3*d^2*e^2 + b*c*d*e^3)*x^3 + (b*c^3*d^3 *e + b*c*d^2*e^2)*x))/(c^4*d^5*e^2 + 2*c^2*d^4*e^3 + d^3*e^4 + (c^4*d^3*e^ 4 + 2*c^2*d^2*e^5 + d*e^6)*x^4 + 2*(c^4*d^4*e^3 + 2*c^2*d^3*e^4 + d^2*e^5) *x^2), -1/8*((2*a - b)*c^4*d^4 + (4*a - b)*c^2*d^3*e - 2*(b*c^4*d^2*e^2 + 2*b*c^2*d*e^3 + b*e^4)*x^4*log(c*x + sqrt(c^2*x^2 - 1)) + 2*a*d^2*e^2 - (b *c^4*d^2*e^2 + b*c^2*d*e^3)*x^4 + 2*((2*a - b)*c^4*d^3*e + (4*a - b)*c^2*d ^2*e^2 + 2*a*d*e^3)*x^2 - (2*b*c^3*d^3 + 3*b*c*d^2*e + (2*b*c^3*d*e^2 + 3* b*c*e^3)*x^4 + 2*(2*b*c^3*d^2*e + 3*b*c*d*e^2)*x^2)*sqrt(-c^2*d^2 - d*e)*a rctan((sqrt(-c^2*d^2 - d*e)*sqrt(c^2*x^2 - 1)*e*x - sqrt(-c^2*d^2 - d*e)*( c*e*x^2 + c*d))/(c^2*d^2 + d*e)) - 2*(b*c^4*d^4 + 2*b*c^2*d^3*e + b*d^2...
Timed out. \[ \int \frac {x^3 (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^3} \, dx=\text {Timed out} \]
\[ \int \frac {x^3 (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^3} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x^{3}}{{\left (e x^{2} + d\right )}^{3}} \,d x } \]
-1/8*b*((c^4*d + 2*c^2*e)*log(e*x^2 + d)/(c^4*d^2*e^2 + 2*c^2*d*e^3 + e^4) + (c^4*d^3 + c^2*d^2*e + (c^4*d^2*e + c^2*d*e^2)*x^2 + 2*(c^4*d^3 + 2*c^2 *d^2*e + d*e^2 + 2*(c^4*d^2*e + 2*c^2*d*e^2 + e^3)*x^2)*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1)) - (c^4*d^3 + 2*c^2*d^2*e + (c^4*d*e^2 + 2*c^2*e^3)*x^ 4 + 2*(c^4*d^2*e + 2*c^2*d*e^2)*x^2)*log(c*x + 1) - (c^4*d^3 + 2*c^2*d^2*e + (c^4*d*e^2 + 2*c^2*e^3)*x^4 + 2*(c^4*d^2*e + 2*c^2*d*e^2)*x^2)*log(c*x - 1))/(c^4*d^4*e^2 + 2*c^2*d^3*e^3 + d^2*e^4 + (c^4*d^2*e^4 + 2*c^2*d*e^5 + e^6)*x^4 + 2*(c^4*d^3*e^3 + 2*c^2*d^2*e^4 + d*e^5)*x^2) + 8*integrate(1/ 4*(2*c*e*x^2 + c*d)/(c^3*e^4*x^7 - c*d^2*e^2*x + (2*c^3*d*e^3 - c*e^4)*x^5 + (c^3*d^2*e^2 - 2*c*d*e^3)*x^3 + (c^2*e^4*x^6 + (2*c^2*d*e^3 - e^4)*x^4 - d^2*e^2 + (c^2*d^2*e^2 - 2*d*e^3)*x^2)*e^(1/2*log(c*x + 1) + 1/2*log(c*x - 1))), x)) - 1/4*(2*e*x^2 + d)*a/(e^4*x^4 + 2*d*e^3*x^2 + d^2*e^2)
Exception generated. \[ \int \frac {x^3 (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^3} \, dx=\text {Exception raised: TypeError} \]
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
Timed out. \[ \int \frac {x^3 (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^3} \, dx=\int \frac {x^3\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}{{\left (e\,x^2+d\right )}^3} \,d x \]