3.480 \(\int x^3 (d+e x^2)^3 (a+b \cosh ^{-1}(c x)) \, dx\)

Optimal. Leaf size=494 \[ \frac{\left (d+e x^2\right )^5 \left (a+b \cosh ^{-1}(c x)\right )}{10 e^2}-\frac{d \left (d+e x^2\right )^4 \left (a+b \cosh ^{-1}(c x)\right )}{8 e^2}+\frac{b x \left (1-c^2 x^2\right ) \left (26 c^4 d^2+201 c^2 d e+126 e^2\right ) \left (d+e x^2\right )^2}{9600 c^5 e \sqrt{c x-1} \sqrt{c x+1}}-\frac{b x \left (1-c^2 x^2\right ) \left (-1096 c^4 d^2 e+136 c^6 d^3-1617 c^2 d e^2-630 e^3\right ) \left (d+e x^2\right )}{38400 c^7 e \sqrt{c x-1} \sqrt{c x+1}}-\frac{b x \left (1-c^2 x^2\right ) \left (-7758 c^4 d^2 e^2-2536 c^6 d^3 e+1232 c^8 d^4-6615 c^2 d e^3-1890 e^4\right )}{76800 c^9 e \sqrt{c x-1} \sqrt{c x+1}}+\frac{b \sqrt{c^2 x^2-1} \left (-480 c^6 d^3 e^2-800 c^4 d^2 e^3+128 c^{10} d^5-525 c^2 d e^4-126 e^5\right ) \tanh ^{-1}\left (\frac{c x}{\sqrt{c^2 x^2-1}}\right )}{5120 c^{10} e^2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b x \left (1-c^2 x^2\right ) \left (d+e x^2\right )^4}{100 c e \sqrt{c x-1} \sqrt{c x+1}}+\frac{b x \left (1-c^2 x^2\right ) \left (11 c^2 d+18 e\right ) \left (d+e x^2\right )^3}{1600 c^3 e \sqrt{c x-1} \sqrt{c x+1}} \]

[Out]

-(b*(1232*c^8*d^4 - 2536*c^6*d^3*e - 7758*c^4*d^2*e^2 - 6615*c^2*d*e^3 - 1890*e^4)*x*(1 - c^2*x^2))/(76800*c^9
*e*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*(136*c^6*d^3 - 1096*c^4*d^2*e - 1617*c^2*d*e^2 - 630*e^3)*x*(1 - c^2*x^2
)*(d + e*x^2))/(38400*c^7*e*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*(26*c^4*d^2 + 201*c^2*d*e + 126*e^2)*x*(1 - c^2
*x^2)*(d + e*x^2)^2)/(9600*c^5*e*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*(11*c^2*d + 18*e)*x*(1 - c^2*x^2)*(d + e*x
^2)^3)/(1600*c^3*e*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*x*(1 - c^2*x^2)*(d + e*x^2)^4)/(100*c*e*Sqrt[-1 + c*x]*S
qrt[1 + c*x]) - (d*(d + e*x^2)^4*(a + b*ArcCosh[c*x]))/(8*e^2) + ((d + e*x^2)^5*(a + b*ArcCosh[c*x]))/(10*e^2)
+ (b*(128*c^10*d^5 - 480*c^6*d^3*e^2 - 800*c^4*d^2*e^3 - 525*c^2*d*e^4 - 126*e^5)*Sqrt[-1 + c^2*x^2]*ArcTanh[
(c*x)/Sqrt[-1 + c^2*x^2]])/(5120*c^10*e^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])

________________________________________________________________________________________

Rubi [A]  time = 0.647944, antiderivative size = 494, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {266, 43, 5790, 12, 566, 528, 388, 217, 206} \[ \frac{\left (d+e x^2\right )^5 \left (a+b \cosh ^{-1}(c x)\right )}{10 e^2}-\frac{d \left (d+e x^2\right )^4 \left (a+b \cosh ^{-1}(c x)\right )}{8 e^2}+\frac{b x \left (1-c^2 x^2\right ) \left (26 c^4 d^2+201 c^2 d e+126 e^2\right ) \left (d+e x^2\right )^2}{9600 c^5 e \sqrt{c x-1} \sqrt{c x+1}}-\frac{b x \left (1-c^2 x^2\right ) \left (-1096 c^4 d^2 e+136 c^6 d^3-1617 c^2 d e^2-630 e^3\right ) \left (d+e x^2\right )}{38400 c^7 e \sqrt{c x-1} \sqrt{c x+1}}-\frac{b x \left (1-c^2 x^2\right ) \left (-7758 c^4 d^2 e^2-2536 c^6 d^3 e+1232 c^8 d^4-6615 c^2 d e^3-1890 e^4\right )}{76800 c^9 e \sqrt{c x-1} \sqrt{c x+1}}+\frac{b \sqrt{c^2 x^2-1} \left (-480 c^6 d^3 e^2-800 c^4 d^2 e^3+128 c^{10} d^5-525 c^2 d e^4-126 e^5\right ) \tanh ^{-1}\left (\frac{c x}{\sqrt{c^2 x^2-1}}\right )}{5120 c^{10} e^2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b x \left (1-c^2 x^2\right ) \left (d+e x^2\right )^4}{100 c e \sqrt{c x-1} \sqrt{c x+1}}+\frac{b x \left (1-c^2 x^2\right ) \left (11 c^2 d+18 e\right ) \left (d+e x^2\right )^3}{1600 c^3 e \sqrt{c x-1} \sqrt{c x+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b*(1232*c^8*d^4 - 2536*c^6*d^3*e - 7758*c^4*d^2*e^2 - 6615*c^2*d*e^3 - 1890*e^4)*x*(1 - c^2*x^2))/(76800*c^9
*e*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*(136*c^6*d^3 - 1096*c^4*d^2*e - 1617*c^2*d*e^2 - 630*e^3)*x*(1 - c^2*x^2
)*(d + e*x^2))/(38400*c^7*e*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*(26*c^4*d^2 + 201*c^2*d*e + 126*e^2)*x*(1 - c^2
*x^2)*(d + e*x^2)^2)/(9600*c^5*e*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*(11*c^2*d + 18*e)*x*(1 - c^2*x^2)*(d + e*x
^2)^3)/(1600*c^3*e*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*x*(1 - c^2*x^2)*(d + e*x^2)^4)/(100*c*e*Sqrt[-1 + c*x]*S
qrt[1 + c*x]) - (d*(d + e*x^2)^4*(a + b*ArcCosh[c*x]))/(8*e^2) + ((d + e*x^2)^5*(a + b*ArcCosh[c*x]))/(10*e^2)
+ (b*(128*c^10*d^5 - 480*c^6*d^3*e^2 - 800*c^4*d^2*e^3 - 525*c^2*d*e^4 - 126*e^5)*Sqrt[-1 + c^2*x^2]*ArcTanh[
(c*x)/Sqrt[-1 + c^2*x^2]])/(5120*c^10*e^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 5790

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]}, Dist[a + b*ArcCosh[c*x], u, x] - Dist[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] && LeQ[m + p, 0]))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 566

Int[((e1_) + (f1_.)*(x_)^(n2_.))^(r_.)*((e2_) + (f2_.)*(x_)^(n2_.))^(r_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_)
+ (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[((e1 + f1*x^(n/2))^FracPart[r]*(e2 + f2*x^(n/2))^FracPart[r])/(e1
*e2 + f1*f2*x^n)^FracPart[r], Int[(a + b*x^n)^p*(c + d*x^n)^q*(e1*e2 + f1*f2*x^n)^r, x], x] /; FreeQ[{a, b, c,
d, e1, f1, e2, f2, n, p, q, r}, x] && EqQ[n2, n/2] && EqQ[e2*f1 + e1*f2, 0]

Rule 528

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

Rule 388

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

Rule 217

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]

Rule 206

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

Rubi steps

\begin{align*} \int x^3 \left (d+e x^2\right )^3 \left (a+b \cosh ^{-1}(c x)\right ) \, dx &=-\frac{d \left (d+e x^2\right )^4 \left (a+b \cosh ^{-1}(c x)\right )}{8 e^2}+\frac{\left (d+e x^2\right )^5 \left (a+b \cosh ^{-1}(c x)\right )}{10 e^2}-(b c) \int \frac{\left (d+e x^2\right )^4 \left (-d+4 e x^2\right )}{40 e^2 \sqrt{-1+c x} \sqrt{1+c x}} \, dx\\ &=-\frac{d \left (d+e x^2\right )^4 \left (a+b \cosh ^{-1}(c x)\right )}{8 e^2}+\frac{\left (d+e x^2\right )^5 \left (a+b \cosh ^{-1}(c x)\right )}{10 e^2}-\frac{(b c) \int \frac{\left (d+e x^2\right )^4 \left (-d+4 e x^2\right )}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{40 e^2}\\ &=-\frac{d \left (d+e x^2\right )^4 \left (a+b \cosh ^{-1}(c x)\right )}{8 e^2}+\frac{\left (d+e x^2\right )^5 \left (a+b \cosh ^{-1}(c x)\right )}{10 e^2}-\frac{\left (b c \sqrt{-1+c^2 x^2}\right ) \int \frac{\left (d+e x^2\right )^4 \left (-d+4 e x^2\right )}{\sqrt{-1+c^2 x^2}} \, dx}{40 e^2 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{b x \left (1-c^2 x^2\right ) \left (d+e x^2\right )^4}{100 c e \sqrt{-1+c x} \sqrt{1+c x}}-\frac{d \left (d+e x^2\right )^4 \left (a+b \cosh ^{-1}(c x)\right )}{8 e^2}+\frac{\left (d+e x^2\right )^5 \left (a+b \cosh ^{-1}(c x)\right )}{10 e^2}-\frac{\left (b \sqrt{-1+c^2 x^2}\right ) \int \frac{\left (d+e x^2\right )^3 \left (-2 d \left (5 c^2 d-2 e\right )+2 e \left (11 c^2 d+18 e\right ) x^2\right )}{\sqrt{-1+c^2 x^2}} \, dx}{400 c e^2 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{b \left (11 c^2 d+18 e\right ) x \left (1-c^2 x^2\right ) \left (d+e x^2\right )^3}{1600 c^3 e \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b x \left (1-c^2 x^2\right ) \left (d+e x^2\right )^4}{100 c e \sqrt{-1+c x} \sqrt{1+c x}}-\frac{d \left (d+e x^2\right )^4 \left (a+b \cosh ^{-1}(c x)\right )}{8 e^2}+\frac{\left (d+e x^2\right )^5 \left (a+b \cosh ^{-1}(c x)\right )}{10 e^2}-\frac{\left (b \sqrt{-1+c^2 x^2}\right ) \int \frac{\left (d+e x^2\right )^2 \left (-2 d \left (40 c^4 d^2-27 c^2 d e-18 e^2\right )+2 e \left (26 c^4 d^2+201 c^2 d e+126 e^2\right ) x^2\right )}{\sqrt{-1+c^2 x^2}} \, dx}{3200 c^3 e^2 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{b \left (26 c^4 d^2+201 c^2 d e+126 e^2\right ) x \left (1-c^2 x^2\right ) \left (d+e x^2\right )^2}{9600 c^5 e \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b \left (11 c^2 d+18 e\right ) x \left (1-c^2 x^2\right ) \left (d+e x^2\right )^3}{1600 c^3 e \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b x \left (1-c^2 x^2\right ) \left (d+e x^2\right )^4}{100 c e \sqrt{-1+c x} \sqrt{1+c x}}-\frac{d \left (d+e x^2\right )^4 \left (a+b \cosh ^{-1}(c x)\right )}{8 e^2}+\frac{\left (d+e x^2\right )^5 \left (a+b \cosh ^{-1}(c x)\right )}{10 e^2}-\frac{\left (b \sqrt{-1+c^2 x^2}\right ) \int \frac{\left (d+e x^2\right ) \left (-2 d \left (240 c^6 d^3-188 c^4 d^2 e-309 c^2 d e^2-126 e^3\right )-2 e \left (136 c^6 d^3-1096 c^4 d^2 e-1617 c^2 d e^2-630 e^3\right ) x^2\right )}{\sqrt{-1+c^2 x^2}} \, dx}{19200 c^5 e^2 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{b \left (136 c^6 d^3-1096 c^4 d^2 e-1617 c^2 d e^2-630 e^3\right ) x \left (1-c^2 x^2\right ) \left (d+e x^2\right )}{38400 c^7 e \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b \left (26 c^4 d^2+201 c^2 d e+126 e^2\right ) x \left (1-c^2 x^2\right ) \left (d+e x^2\right )^2}{9600 c^5 e \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b \left (11 c^2 d+18 e\right ) x \left (1-c^2 x^2\right ) \left (d+e x^2\right )^3}{1600 c^3 e \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b x \left (1-c^2 x^2\right ) \left (d+e x^2\right )^4}{100 c e \sqrt{-1+c x} \sqrt{1+c x}}-\frac{d \left (d+e x^2\right )^4 \left (a+b \cosh ^{-1}(c x)\right )}{8 e^2}+\frac{\left (d+e x^2\right )^5 \left (a+b \cosh ^{-1}(c x)\right )}{10 e^2}-\frac{\left (b \sqrt{-1+c^2 x^2}\right ) \int \frac{-2 d \left (960 c^8 d^4-616 c^6 d^3 e-2332 c^4 d^2 e^2-2121 c^2 d e^3-630 e^4\right )-2 e \left (1232 c^8 d^4-2536 c^6 d^3 e-7758 c^4 d^2 e^2-6615 c^2 d e^3-1890 e^4\right ) x^2}{\sqrt{-1+c^2 x^2}} \, dx}{76800 c^7 e^2 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{b \left (1232 c^8 d^4-2536 c^6 d^3 e-7758 c^4 d^2 e^2-6615 c^2 d e^3-1890 e^4\right ) x \left (1-c^2 x^2\right )}{76800 c^9 e \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b \left (136 c^6 d^3-1096 c^4 d^2 e-1617 c^2 d e^2-630 e^3\right ) x \left (1-c^2 x^2\right ) \left (d+e x^2\right )}{38400 c^7 e \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b \left (26 c^4 d^2+201 c^2 d e+126 e^2\right ) x \left (1-c^2 x^2\right ) \left (d+e x^2\right )^2}{9600 c^5 e \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b \left (11 c^2 d+18 e\right ) x \left (1-c^2 x^2\right ) \left (d+e x^2\right )^3}{1600 c^3 e \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b x \left (1-c^2 x^2\right ) \left (d+e x^2\right )^4}{100 c e \sqrt{-1+c x} \sqrt{1+c x}}-\frac{d \left (d+e x^2\right )^4 \left (a+b \cosh ^{-1}(c x)\right )}{8 e^2}+\frac{\left (d+e x^2\right )^5 \left (a+b \cosh ^{-1}(c x)\right )}{10 e^2}+\frac{\left (b \left (2 e \left (1232 c^8 d^4-2536 c^6 d^3 e-7758 c^4 d^2 e^2-6615 c^2 d e^3-1890 e^4\right )+4 c^2 d \left (960 c^8 d^4-616 c^6 d^3 e-2332 c^4 d^2 e^2-2121 c^2 d e^3-630 e^4\right )\right ) \sqrt{-1+c^2 x^2}\right ) \int \frac{1}{\sqrt{-1+c^2 x^2}} \, dx}{153600 c^9 e^2 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{b \left (1232 c^8 d^4-2536 c^6 d^3 e-7758 c^4 d^2 e^2-6615 c^2 d e^3-1890 e^4\right ) x \left (1-c^2 x^2\right )}{76800 c^9 e \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b \left (136 c^6 d^3-1096 c^4 d^2 e-1617 c^2 d e^2-630 e^3\right ) x \left (1-c^2 x^2\right ) \left (d+e x^2\right )}{38400 c^7 e \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b \left (26 c^4 d^2+201 c^2 d e+126 e^2\right ) x \left (1-c^2 x^2\right ) \left (d+e x^2\right )^2}{9600 c^5 e \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b \left (11 c^2 d+18 e\right ) x \left (1-c^2 x^2\right ) \left (d+e x^2\right )^3}{1600 c^3 e \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b x \left (1-c^2 x^2\right ) \left (d+e x^2\right )^4}{100 c e \sqrt{-1+c x} \sqrt{1+c x}}-\frac{d \left (d+e x^2\right )^4 \left (a+b \cosh ^{-1}(c x)\right )}{8 e^2}+\frac{\left (d+e x^2\right )^5 \left (a+b \cosh ^{-1}(c x)\right )}{10 e^2}+\frac{\left (b \left (2 e \left (1232 c^8 d^4-2536 c^6 d^3 e-7758 c^4 d^2 e^2-6615 c^2 d e^3-1890 e^4\right )+4 c^2 d \left (960 c^8 d^4-616 c^6 d^3 e-2332 c^4 d^2 e^2-2121 c^2 d e^3-630 e^4\right )\right ) \sqrt{-1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{1-c^2 x^2} \, dx,x,\frac{x}{\sqrt{-1+c^2 x^2}}\right )}{153600 c^9 e^2 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{b \left (1232 c^8 d^4-2536 c^6 d^3 e-7758 c^4 d^2 e^2-6615 c^2 d e^3-1890 e^4\right ) x \left (1-c^2 x^2\right )}{76800 c^9 e \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b \left (136 c^6 d^3-1096 c^4 d^2 e-1617 c^2 d e^2-630 e^3\right ) x \left (1-c^2 x^2\right ) \left (d+e x^2\right )}{38400 c^7 e \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b \left (26 c^4 d^2+201 c^2 d e+126 e^2\right ) x \left (1-c^2 x^2\right ) \left (d+e x^2\right )^2}{9600 c^5 e \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b \left (11 c^2 d+18 e\right ) x \left (1-c^2 x^2\right ) \left (d+e x^2\right )^3}{1600 c^3 e \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b x \left (1-c^2 x^2\right ) \left (d+e x^2\right )^4}{100 c e \sqrt{-1+c x} \sqrt{1+c x}}-\frac{d \left (d+e x^2\right )^4 \left (a+b \cosh ^{-1}(c x)\right )}{8 e^2}+\frac{\left (d+e x^2\right )^5 \left (a+b \cosh ^{-1}(c x)\right )}{10 e^2}+\frac{b \left (128 c^{10} d^5-480 c^6 d^3 e^2-800 c^4 d^2 e^3-525 c^2 d e^4-126 e^5\right ) \sqrt{-1+c^2 x^2} \tanh ^{-1}\left (\frac{c x}{\sqrt{-1+c^2 x^2}}\right )}{5120 c^{10} e^2 \sqrt{-1+c x} \sqrt{1+c x}}\\ \end{align*}

Mathematica [A]  time = 0.52861, size = 294, normalized size = 0.6 \[ \frac{1920 a x^4 \left (20 d^2 e x^2+10 d^3+15 d e^2 x^4+4 e^3 x^6\right )-\frac{b x \sqrt{c x-1} \sqrt{c x+1} \left (16 c^8 \left (400 d^2 e x^4+300 d^3 x^2+225 d e^2 x^6+48 e^3 x^8\right )+8 c^6 \left (1000 d^2 e x^2+900 d^3+525 d e^2 x^4+108 e^3 x^6\right )+6 c^4 e \left (2000 d^2+875 d e x^2+168 e^2 x^4\right )+315 c^2 e^2 \left (25 d+4 e x^2\right )+1890 e^3\right )}{c^9}-\frac{30 b \left (800 c^4 d^2 e+480 c^6 d^3+525 c^2 d e^2+126 e^3\right ) \tanh ^{-1}\left (\sqrt{\frac{c x-1}{c x+1}}\right )}{c^{10}}+1920 b x^4 \cosh ^{-1}(c x) \left (20 d^2 e x^2+10 d^3+15 d e^2 x^4+4 e^3 x^6\right )}{76800} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(1920*a*x^4*(10*d^3 + 20*d^2*e*x^2 + 15*d*e^2*x^4 + 4*e^3*x^6) - (b*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(1890*e^3 +
315*c^2*e^2*(25*d + 4*e*x^2) + 6*c^4*e*(2000*d^2 + 875*d*e*x^2 + 168*e^2*x^4) + 8*c^6*(900*d^3 + 1000*d^2*e*x
^2 + 525*d*e^2*x^4 + 108*e^3*x^6) + 16*c^8*(300*d^3*x^2 + 400*d^2*e*x^4 + 225*d*e^2*x^6 + 48*e^3*x^8)))/c^9 +
1920*b*x^4*(10*d^3 + 20*d^2*e*x^2 + 15*d*e^2*x^4 + 4*e^3*x^6)*ArcCosh[c*x] - (30*b*(480*c^6*d^3 + 800*c^4*d^2*
e + 525*c^2*d*e^2 + 126*e^3)*ArcTanh[Sqrt[(-1 + c*x)/(1 + c*x)]])/c^10)/76800

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Maple [A]  time = 0.02, size = 659, normalized size = 1.3 \begin{align*} -{\frac{5\,b{x}^{3}{d}^{2}e}{48\,{c}^{3}}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{3\,b{d}^{3}}{32\,{c}^{4}}\sqrt{cx-1}\sqrt{cx+1}\ln \left ( cx+\sqrt{{c}^{2}{x}^{2}-1} \right ){\frac{1}{\sqrt{{c}^{2}{x}^{2}-1}}}}-{\frac{105\,bxd{e}^{2}}{1024\,{c}^{7}}\sqrt{cx-1}\sqrt{cx+1}}+{\frac{{d}^{3}b{\rm arccosh} \left (cx\right ){x}^{4}}{4}}-{\frac{b{d}^{3}{x}^{3}}{16\,c}\sqrt{cx-1}\sqrt{cx+1}}+{\frac{{d}^{3}a{x}^{4}}{4}}+{\frac{b{\rm arccosh} \left (cx\right ){e}^{3}{x}^{10}}{10}}+{\frac{3\,ad{e}^{2}{x}^{8}}{8}}+{\frac{a{d}^{2}e{x}^{6}}{2}}+{\frac{a{e}^{3}{x}^{10}}{10}}-{\frac{105\,bd{e}^{2}}{1024\,{c}^{8}}\sqrt{cx-1}\sqrt{cx+1}\ln \left ( cx+\sqrt{{c}^{2}{x}^{2}-1} \right ){\frac{1}{\sqrt{{c}^{2}{x}^{2}-1}}}}-{\frac{5\,b{d}^{2}e}{32\,{c}^{6}}\sqrt{cx-1}\sqrt{cx+1}\ln \left ( cx+\sqrt{{c}^{2}{x}^{2}-1} \right ){\frac{1}{\sqrt{{c}^{2}{x}^{2}-1}}}}-{\frac{21\,b{e}^{3}{x}^{3}}{1280\,{c}^{7}}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{63\,b{e}^{3}x}{2560\,{c}^{9}}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{b{e}^{3}{x}^{9}}{100\,c}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{9\,b{e}^{3}{x}^{7}}{800\,{c}^{3}}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{21\,b{e}^{3}{x}^{5}}{1600\,{c}^{5}}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{5\,bx{d}^{2}e}{32\,{c}^{5}}\sqrt{cx-1}\sqrt{cx+1}}+{\frac{3\,b{\rm arccosh} \left (cx\right )d{e}^{2}{x}^{8}}{8}}+{\frac{b{\rm arccosh} \left (cx\right ){d}^{2}e{x}^{6}}{2}}-{\frac{3\,b{x}^{7}d{e}^{2}}{64\,c}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{b{x}^{5}{d}^{2}e}{12\,c}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{7\,b{x}^{5}d{e}^{2}}{128\,{c}^{3}}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{63\,b{e}^{3}}{2560\,{c}^{10}}\sqrt{cx-1}\sqrt{cx+1}\ln \left ( cx+\sqrt{{c}^{2}{x}^{2}-1} \right ){\frac{1}{\sqrt{{c}^{2}{x}^{2}-1}}}}-{\frac{35\,b{x}^{3}d{e}^{2}}{512\,{c}^{5}}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{3\,b{d}^{3}x}{32\,{c}^{3}}\sqrt{cx-1}\sqrt{cx+1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-5/48/c^3*b*(c*x-1)^(1/2)*(c*x+1)^(1/2)*x^3*d^2*e-3/32/c^4*d^3*b*(c*x-1)^(1/2)*(c*x+1)^(1/2)/(c^2*x^2-1)^(1/2)
*ln(c*x+(c^2*x^2-1)^(1/2))-105/1024/c^7*b*(c*x-1)^(1/2)*(c*x+1)^(1/2)*x*d*e^2+1/4*d^3*b*arccosh(c*x)*x^4-1/16/
c*d^3*b*(c*x-1)^(1/2)*(c*x+1)^(1/2)*x^3+1/4*d^3*a*x^4+1/10*b*arccosh(c*x)*e^3*x^10+3/8*a*d*e^2*x^8+1/2*a*d^2*e
*x^6+1/10*a*e^3*x^10-105/1024/c^8*b*(c*x-1)^(1/2)*(c*x+1)^(1/2)/(c^2*x^2-1)^(1/2)*ln(c*x+(c^2*x^2-1)^(1/2))*d*
e^2-5/32/c^6*b*(c*x-1)^(1/2)*(c*x+1)^(1/2)/(c^2*x^2-1)^(1/2)*ln(c*x+(c^2*x^2-1)^(1/2))*d^2*e-21/1280/c^7*b*(c*
x-1)^(1/2)*(c*x+1)^(1/2)*e^3*x^3-63/2560/c^9*b*(c*x-1)^(1/2)*(c*x+1)^(1/2)*e^3*x-1/100/c*b*(c*x-1)^(1/2)*(c*x+
1)^(1/2)*e^3*x^9-9/800/c^3*b*(c*x-1)^(1/2)*(c*x+1)^(1/2)*e^3*x^7-21/1600/c^5*b*(c*x-1)^(1/2)*(c*x+1)^(1/2)*e^3
*x^5-5/32/c^5*b*(c*x-1)^(1/2)*(c*x+1)^(1/2)*x*d^2*e+3/8*b*arccosh(c*x)*d*e^2*x^8+1/2*b*arccosh(c*x)*d^2*e*x^6-
3/64/c*b*(c*x-1)^(1/2)*(c*x+1)^(1/2)*x^7*d*e^2-1/12/c*b*(c*x-1)^(1/2)*(c*x+1)^(1/2)*x^5*d^2*e-7/128/c^3*b*(c*x
-1)^(1/2)*(c*x+1)^(1/2)*x^5*d*e^2-63/2560/c^10*b*(c*x-1)^(1/2)*(c*x+1)^(1/2)/(c^2*x^2-1)^(1/2)*e^3*ln(c*x+(c^2
*x^2-1)^(1/2))-35/512/c^5*b*(c*x-1)^(1/2)*(c*x+1)^(1/2)*x^3*d*e^2-3/32*b*d^3*x*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c^3

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Maxima [A]  time = 1.19067, size = 706, normalized size = 1.43 \begin{align*} \frac{1}{10} \, a e^{3} x^{10} + \frac{3}{8} \, a d e^{2} x^{8} + \frac{1}{2} \, a d^{2} e x^{6} + \frac{1}{4} \, a d^{3} x^{4} + \frac{1}{32} \,{\left (8 \, x^{4} \operatorname{arcosh}\left (c x\right ) -{\left (\frac{2 \, \sqrt{c^{2} x^{2} - 1} x^{3}}{c^{2}} + \frac{3 \, \sqrt{c^{2} x^{2} - 1} x}{c^{4}} + \frac{3 \, \log \left (2 \, c^{2} x + 2 \, \sqrt{c^{2} x^{2} - 1} \sqrt{c^{2}}\right )}{\sqrt{c^{2}} c^{4}}\right )} c\right )} b d^{3} + \frac{1}{96} \,{\left (48 \, x^{6} \operatorname{arcosh}\left (c x\right ) -{\left (\frac{8 \, \sqrt{c^{2} x^{2} - 1} x^{5}}{c^{2}} + \frac{10 \, \sqrt{c^{2} x^{2} - 1} x^{3}}{c^{4}} + \frac{15 \, \sqrt{c^{2} x^{2} - 1} x}{c^{6}} + \frac{15 \, \log \left (2 \, c^{2} x + 2 \, \sqrt{c^{2} x^{2} - 1} \sqrt{c^{2}}\right )}{\sqrt{c^{2}} c^{6}}\right )} c\right )} b d^{2} e + \frac{1}{1024} \,{\left (384 \, x^{8} \operatorname{arcosh}\left (c x\right ) -{\left (\frac{48 \, \sqrt{c^{2} x^{2} - 1} x^{7}}{c^{2}} + \frac{56 \, \sqrt{c^{2} x^{2} - 1} x^{5}}{c^{4}} + \frac{70 \, \sqrt{c^{2} x^{2} - 1} x^{3}}{c^{6}} + \frac{105 \, \sqrt{c^{2} x^{2} - 1} x}{c^{8}} + \frac{105 \, \log \left (2 \, c^{2} x + 2 \, \sqrt{c^{2} x^{2} - 1} \sqrt{c^{2}}\right )}{\sqrt{c^{2}} c^{8}}\right )} c\right )} b d e^{2} + \frac{1}{12800} \,{\left (1280 \, x^{10} \operatorname{arcosh}\left (c x\right ) -{\left (\frac{128 \, \sqrt{c^{2} x^{2} - 1} x^{9}}{c^{2}} + \frac{144 \, \sqrt{c^{2} x^{2} - 1} x^{7}}{c^{4}} + \frac{168 \, \sqrt{c^{2} x^{2} - 1} x^{5}}{c^{6}} + \frac{210 \, \sqrt{c^{2} x^{2} - 1} x^{3}}{c^{8}} + \frac{315 \, \sqrt{c^{2} x^{2} - 1} x}{c^{10}} + \frac{315 \, \log \left (2 \, c^{2} x + 2 \, \sqrt{c^{2} x^{2} - 1} \sqrt{c^{2}}\right )}{\sqrt{c^{2}} c^{10}}\right )} c\right )} b e^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*a*e^3*x^10 + 3/8*a*d*e^2*x^8 + 1/2*a*d^2*e*x^6 + 1/4*a*d^3*x^4 + 1/32*(8*x^4*arccosh(c*x) - (2*sqrt(c^2*x
^2 - 1)*x^3/c^2 + 3*sqrt(c^2*x^2 - 1)*x/c^4 + 3*log(2*c^2*x + 2*sqrt(c^2*x^2 - 1)*sqrt(c^2))/(sqrt(c^2)*c^4))*
c)*b*d^3 + 1/96*(48*x^6*arccosh(c*x) - (8*sqrt(c^2*x^2 - 1)*x^5/c^2 + 10*sqrt(c^2*x^2 - 1)*x^3/c^4 + 15*sqrt(c
^2*x^2 - 1)*x/c^6 + 15*log(2*c^2*x + 2*sqrt(c^2*x^2 - 1)*sqrt(c^2))/(sqrt(c^2)*c^6))*c)*b*d^2*e + 1/1024*(384*
x^8*arccosh(c*x) - (48*sqrt(c^2*x^2 - 1)*x^7/c^2 + 56*sqrt(c^2*x^2 - 1)*x^5/c^4 + 70*sqrt(c^2*x^2 - 1)*x^3/c^6
+ 105*sqrt(c^2*x^2 - 1)*x/c^8 + 105*log(2*c^2*x + 2*sqrt(c^2*x^2 - 1)*sqrt(c^2))/(sqrt(c^2)*c^8))*c)*b*d*e^2
+ 1/12800*(1280*x^10*arccosh(c*x) - (128*sqrt(c^2*x^2 - 1)*x^9/c^2 + 144*sqrt(c^2*x^2 - 1)*x^7/c^4 + 168*sqrt(
c^2*x^2 - 1)*x^5/c^6 + 210*sqrt(c^2*x^2 - 1)*x^3/c^8 + 315*sqrt(c^2*x^2 - 1)*x/c^10 + 315*log(2*c^2*x + 2*sqrt
(c^2*x^2 - 1)*sqrt(c^2))/(sqrt(c^2)*c^10))*c)*b*e^3

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Fricas [A]  time = 2.53091, size = 798, normalized size = 1.62 \begin{align*} \frac{7680 \, a c^{10} e^{3} x^{10} + 28800 \, a c^{10} d e^{2} x^{8} + 38400 \, a c^{10} d^{2} e x^{6} + 19200 \, a c^{10} d^{3} x^{4} + 15 \,{\left (512 \, b c^{10} e^{3} x^{10} + 1920 \, b c^{10} d e^{2} x^{8} + 2560 \, b c^{10} d^{2} e x^{6} + 1280 \, b c^{10} d^{3} x^{4} - 480 \, b c^{6} d^{3} - 800 \, b c^{4} d^{2} e - 525 \, b c^{2} d e^{2} - 126 \, b e^{3}\right )} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) -{\left (768 \, b c^{9} e^{3} x^{9} + 144 \,{\left (25 \, b c^{9} d e^{2} + 6 \, b c^{7} e^{3}\right )} x^{7} + 8 \,{\left (800 \, b c^{9} d^{2} e + 525 \, b c^{7} d e^{2} + 126 \, b c^{5} e^{3}\right )} x^{5} + 10 \,{\left (480 \, b c^{9} d^{3} + 800 \, b c^{7} d^{2} e + 525 \, b c^{5} d e^{2} + 126 \, b c^{3} e^{3}\right )} x^{3} + 15 \,{\left (480 \, b c^{7} d^{3} + 800 \, b c^{5} d^{2} e + 525 \, b c^{3} d e^{2} + 126 \, b c e^{3}\right )} x\right )} \sqrt{c^{2} x^{2} - 1}}{76800 \, c^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/76800*(7680*a*c^10*e^3*x^10 + 28800*a*c^10*d*e^2*x^8 + 38400*a*c^10*d^2*e*x^6 + 19200*a*c^10*d^3*x^4 + 15*(5
12*b*c^10*e^3*x^10 + 1920*b*c^10*d*e^2*x^8 + 2560*b*c^10*d^2*e*x^6 + 1280*b*c^10*d^3*x^4 - 480*b*c^6*d^3 - 800
*b*c^4*d^2*e - 525*b*c^2*d*e^2 - 126*b*e^3)*log(c*x + sqrt(c^2*x^2 - 1)) - (768*b*c^9*e^3*x^9 + 144*(25*b*c^9*
d*e^2 + 6*b*c^7*e^3)*x^7 + 8*(800*b*c^9*d^2*e + 525*b*c^7*d*e^2 + 126*b*c^5*e^3)*x^5 + 10*(480*b*c^9*d^3 + 800
*b*c^7*d^2*e + 525*b*c^5*d*e^2 + 126*b*c^3*e^3)*x^3 + 15*(480*b*c^7*d^3 + 800*b*c^5*d^2*e + 525*b*c^3*d*e^2 +
126*b*c*e^3)*x)*sqrt(c^2*x^2 - 1))/c^10

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Sympy [A]  time = 50.6566, size = 604, normalized size = 1.22 \begin{align*} \begin{cases} \frac{a d^{3} x^{4}}{4} + \frac{a d^{2} e x^{6}}{2} + \frac{3 a d e^{2} x^{8}}{8} + \frac{a e^{3} x^{10}}{10} + \frac{b d^{3} x^{4} \operatorname{acosh}{\left (c x \right )}}{4} + \frac{b d^{2} e x^{6} \operatorname{acosh}{\left (c x \right )}}{2} + \frac{3 b d e^{2} x^{8} \operatorname{acosh}{\left (c x \right )}}{8} + \frac{b e^{3} x^{10} \operatorname{acosh}{\left (c x \right )}}{10} - \frac{b d^{3} x^{3} \sqrt{c^{2} x^{2} - 1}}{16 c} - \frac{b d^{2} e x^{5} \sqrt{c^{2} x^{2} - 1}}{12 c} - \frac{3 b d e^{2} x^{7} \sqrt{c^{2} x^{2} - 1}}{64 c} - \frac{b e^{3} x^{9} \sqrt{c^{2} x^{2} - 1}}{100 c} - \frac{3 b d^{3} x \sqrt{c^{2} x^{2} - 1}}{32 c^{3}} - \frac{5 b d^{2} e x^{3} \sqrt{c^{2} x^{2} - 1}}{48 c^{3}} - \frac{7 b d e^{2} x^{5} \sqrt{c^{2} x^{2} - 1}}{128 c^{3}} - \frac{9 b e^{3} x^{7} \sqrt{c^{2} x^{2} - 1}}{800 c^{3}} - \frac{3 b d^{3} \operatorname{acosh}{\left (c x \right )}}{32 c^{4}} - \frac{5 b d^{2} e x \sqrt{c^{2} x^{2} - 1}}{32 c^{5}} - \frac{35 b d e^{2} x^{3} \sqrt{c^{2} x^{2} - 1}}{512 c^{5}} - \frac{21 b e^{3} x^{5} \sqrt{c^{2} x^{2} - 1}}{1600 c^{5}} - \frac{5 b d^{2} e \operatorname{acosh}{\left (c x \right )}}{32 c^{6}} - \frac{105 b d e^{2} x \sqrt{c^{2} x^{2} - 1}}{1024 c^{7}} - \frac{21 b e^{3} x^{3} \sqrt{c^{2} x^{2} - 1}}{1280 c^{7}} - \frac{105 b d e^{2} \operatorname{acosh}{\left (c x \right )}}{1024 c^{8}} - \frac{63 b e^{3} x \sqrt{c^{2} x^{2} - 1}}{2560 c^{9}} - \frac{63 b e^{3} \operatorname{acosh}{\left (c x \right )}}{2560 c^{10}} & \text{for}\: c \neq 0 \\\left (a + \frac{i \pi b}{2}\right ) \left (\frac{d^{3} x^{4}}{4} + \frac{d^{2} e x^{6}}{2} + \frac{3 d e^{2} x^{8}}{8} + \frac{e^{3} x^{10}}{10}\right ) & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a*d**3*x**4/4 + a*d**2*e*x**6/2 + 3*a*d*e**2*x**8/8 + a*e**3*x**10/10 + b*d**3*x**4*acosh(c*x)/4 +
b*d**2*e*x**6*acosh(c*x)/2 + 3*b*d*e**2*x**8*acosh(c*x)/8 + b*e**3*x**10*acosh(c*x)/10 - b*d**3*x**3*sqrt(c**2
*x**2 - 1)/(16*c) - b*d**2*e*x**5*sqrt(c**2*x**2 - 1)/(12*c) - 3*b*d*e**2*x**7*sqrt(c**2*x**2 - 1)/(64*c) - b*
e**3*x**9*sqrt(c**2*x**2 - 1)/(100*c) - 3*b*d**3*x*sqrt(c**2*x**2 - 1)/(32*c**3) - 5*b*d**2*e*x**3*sqrt(c**2*x
**2 - 1)/(48*c**3) - 7*b*d*e**2*x**5*sqrt(c**2*x**2 - 1)/(128*c**3) - 9*b*e**3*x**7*sqrt(c**2*x**2 - 1)/(800*c
**3) - 3*b*d**3*acosh(c*x)/(32*c**4) - 5*b*d**2*e*x*sqrt(c**2*x**2 - 1)/(32*c**5) - 35*b*d*e**2*x**3*sqrt(c**2
*x**2 - 1)/(512*c**5) - 21*b*e**3*x**5*sqrt(c**2*x**2 - 1)/(1600*c**5) - 5*b*d**2*e*acosh(c*x)/(32*c**6) - 105
*b*d*e**2*x*sqrt(c**2*x**2 - 1)/(1024*c**7) - 21*b*e**3*x**3*sqrt(c**2*x**2 - 1)/(1280*c**7) - 105*b*d*e**2*ac
osh(c*x)/(1024*c**8) - 63*b*e**3*x*sqrt(c**2*x**2 - 1)/(2560*c**9) - 63*b*e**3*acosh(c*x)/(2560*c**10), Ne(c,
0)), ((a + I*pi*b/2)*(d**3*x**4/4 + d**2*e*x**6/2 + 3*d*e**2*x**8/8 + e**3*x**10/10), True))

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Giac [A]  time = 1.57286, size = 612, normalized size = 1.24 \begin{align*} \frac{1}{4} \, a d^{3} x^{4} + \frac{1}{32} \,{\left (8 \, x^{4} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) -{\left (\sqrt{c^{2} x^{2} - 1} x{\left (\frac{2 \, x^{2}}{c^{2}} + \frac{3}{c^{4}}\right )} - \frac{3 \, \log \left ({\left | -x{\left | c \right |} + \sqrt{c^{2} x^{2} - 1} \right |}\right )}{c^{4}{\left | c \right |}}\right )} c\right )} b d^{3} + \frac{1}{12800} \,{\left (1280 \, a x^{10} +{\left (1280 \, x^{10} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) -{\left (\sqrt{c^{2} x^{2} - 1}{\left (2 \,{\left (4 \,{\left (2 \, x^{2}{\left (\frac{8 \, x^{2}}{c^{2}} + \frac{9}{c^{4}}\right )} + \frac{21}{c^{6}}\right )} x^{2} + \frac{105}{c^{8}}\right )} x^{2} + \frac{315}{c^{10}}\right )} x - \frac{315 \, \log \left ({\left | -x{\left | c \right |} + \sqrt{c^{2} x^{2} - 1} \right |}\right )}{c^{10}{\left | c \right |}}\right )} c\right )} b\right )} e^{3} + \frac{1}{1024} \,{\left (384 \, a d x^{8} +{\left (384 \, x^{8} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) -{\left (\sqrt{c^{2} x^{2} - 1}{\left (2 \,{\left (4 \, x^{2}{\left (\frac{6 \, x^{2}}{c^{2}} + \frac{7}{c^{4}}\right )} + \frac{35}{c^{6}}\right )} x^{2} + \frac{105}{c^{8}}\right )} x - \frac{105 \, \log \left ({\left | -x{\left | c \right |} + \sqrt{c^{2} x^{2} - 1} \right |}\right )}{c^{8}{\left | c \right |}}\right )} c\right )} b d\right )} e^{2} + \frac{1}{96} \,{\left (48 \, a d^{2} x^{6} +{\left (48 \, x^{6} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) -{\left (\sqrt{c^{2} x^{2} - 1}{\left (2 \, x^{2}{\left (\frac{4 \, x^{2}}{c^{2}} + \frac{5}{c^{4}}\right )} + \frac{15}{c^{6}}\right )} x - \frac{15 \, \log \left ({\left | -x{\left | c \right |} + \sqrt{c^{2} x^{2} - 1} \right |}\right )}{c^{6}{\left | c \right |}}\right )} c\right )} b d^{2}\right )} e \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*a*d^3*x^4 + 1/32*(8*x^4*log(c*x + sqrt(c^2*x^2 - 1)) - (sqrt(c^2*x^2 - 1)*x*(2*x^2/c^2 + 3/c^4) - 3*log(ab
s(-x*abs(c) + sqrt(c^2*x^2 - 1)))/(c^4*abs(c)))*c)*b*d^3 + 1/12800*(1280*a*x^10 + (1280*x^10*log(c*x + sqrt(c^
2*x^2 - 1)) - (sqrt(c^2*x^2 - 1)*(2*(4*(2*x^2*(8*x^2/c^2 + 9/c^4) + 21/c^6)*x^2 + 105/c^8)*x^2 + 315/c^10)*x -
315*log(abs(-x*abs(c) + sqrt(c^2*x^2 - 1)))/(c^10*abs(c)))*c)*b)*e^3 + 1/1024*(384*a*d*x^8 + (384*x^8*log(c*x
+ sqrt(c^2*x^2 - 1)) - (sqrt(c^2*x^2 - 1)*(2*(4*x^2*(6*x^2/c^2 + 7/c^4) + 35/c^6)*x^2 + 105/c^8)*x - 105*log(
abs(-x*abs(c) + sqrt(c^2*x^2 - 1)))/(c^8*abs(c)))*c)*b*d)*e^2 + 1/96*(48*a*d^2*x^6 + (48*x^6*log(c*x + sqrt(c^
2*x^2 - 1)) - (sqrt(c^2*x^2 - 1)*(2*x^2*(4*x^2/c^2 + 5/c^4) + 15/c^6)*x - 15*log(abs(-x*abs(c) + sqrt(c^2*x^2
- 1)))/(c^6*abs(c)))*c)*b*d^2)*e