\(\int x^3 (d-c^2 d x^2)^2 (a+b \text {arccosh}(c x)) \, dx\) [11]

Optimal result

Integrand size = 25, antiderivative size = 200 \[ \int x^3 \left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=-\frac {73 b d^2 x \sqrt {-1+c x} \sqrt {1+c x}}{3072 c^3}-\frac {73 b d^2 x^3 \sqrt {-1+c x} \sqrt {1+c x}}{4608 c}+\frac {43 b c d^2 x^5 \sqrt {-1+c x} \sqrt {1+c x}}{1152}-\frac {1}{64} b c^3 d^2 x^7 \sqrt {-1+c x} \sqrt {1+c x}-\frac {73 b d^2 \text {arccosh}(c x)}{3072 c^4}+\frac {1}{4} d^2 x^4 (a+b \text {arccosh}(c x))-\frac {1}{3} c^2 d^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{8} c^4 d^2 x^8 (a+b \text {arccosh}(c x)) \]

[Out]

-73/3072*b*d^2*arccosh(c*x)/c^4+1/4*d^2*x^4*(a+b*arccosh(c*x))-1/3*c^2*d^2*x^6*(a+b*arccosh(c*x))+1/8*c^4*d^2*
x^8*(a+b*arccosh(c*x))-73/3072*b*d^2*x*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c^3-73/4608*b*d^2*x^3*(c*x-1)^(1/2)*(c*x+1)
^(1/2)/c+43/1152*b*c*d^2*x^5*(c*x-1)^(1/2)*(c*x+1)^(1/2)-1/64*b*c^3*d^2*x^7*(c*x-1)^(1/2)*(c*x+1)^(1/2)

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.42, number of steps used = 9, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {272, 45, 5921, 12, 534, 1281, 470, 327, 223, 212} \[ \int x^3 \left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=\frac {1}{8} c^4 d^2 x^8 (a+b \text {arccosh}(c x))-\frac {1}{3} c^2 d^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} d^2 x^4 (a+b \text {arccosh}(c x))-\frac {73 b d^2 \sqrt {c^2 x^2-1} \text {arctanh}\left (\frac {c x}{\sqrt {c^2 x^2-1}}\right )}{3072 c^4 \sqrt {c x-1} \sqrt {c x+1}}-\frac {43 b c d^2 x^5 \left (1-c^2 x^2\right )}{1152 \sqrt {c x-1} \sqrt {c x+1}}+\frac {73 b d^2 x^3 \left (1-c^2 x^2\right )}{4608 c \sqrt {c x-1} \sqrt {c x+1}}+\frac {73 b d^2 x \left (1-c^2 x^2\right )}{3072 c^3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b c^3 d^2 x^7 \left (1-c^2 x^2\right )}{64 \sqrt {c x-1} \sqrt {c x+1}} \]

[In]

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

[Out]

(73*b*d^2*x*(1 - c^2*x^2))/(3072*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (73*b*d^2*x^3*(1 - c^2*x^2))/(4608*c*Sqrt
[-1 + c*x]*Sqrt[1 + c*x]) - (43*b*c*d^2*x^5*(1 - c^2*x^2))/(1152*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*c^3*d^2*x^
7*(1 - c^2*x^2))/(64*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (d^2*x^4*(a + b*ArcCosh[c*x]))/4 - (c^2*d^2*x^6*(a + b*Ar
cCosh[c*x]))/3 + (c^4*d^2*x^8*(a + b*ArcCosh[c*x]))/8 - (73*b*d^2*Sqrt[-1 + c^2*x^2]*ArcTanh[(c*x)/Sqrt[-1 + c
^2*x^2]])/(3072*c^4*Sqrt[-1 + c*x]*Sqrt[1 + c*x])

Rule 12

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

Rule 45

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 212

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] && (GtQ[a, 0] || LtQ[b, 0])

Rule 223

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 272

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 327

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

Rule 470

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

Rule 534

Int[(u_.)*((c_) + (d_.)*(x_)^(n_.) + (e_.)*(x_)^(n2_.))^(q_.)*((a1_) + (b1_.)*(x_)^(non2_.))^(p_.)*((a2_) + (b
2_.)*(x_)^(non2_.))^(p_.), x_Symbol] :> Dist[(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 + e*x^(2*n))^q, x], x] /; FreeQ[{a1, b1,
a2, b2, c, d, e, n, p, q}, x] && EqQ[non2, n/2] && EqQ[n2, 2*n] && EqQ[a2*b1 + a1*b2, 0]

Rule 1281

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Si
mp[c^p*(f*x)^(m + 4*p - 1)*((d + e*x^2)^(q + 1)/(e*f^(4*p - 1)*(m + 4*p + 2*q + 1))), x] + Dist[1/(e*(m + 4*p
+ 2*q + 1)), Int[(f*x)^m*(d + e*x^2)^q*ExpandToSum[e*(m + 4*p + 2*q + 1)*((a + b*x^2 + c*x^4)^p - c^p*x^(4*p))
- d*c^p*(m + 4*p - 1)*x^(4*p - 2), x], x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && NeQ[b^2 - 4*a*c, 0] &&
IGtQ[p, 0] &&  !IntegerQ[q] && NeQ[m + 4*p + 2*q + 1, 0]

Rule 5921

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] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} d^2 x^4 (a+b \text {arccosh}(c x))-\frac {1}{3} c^2 d^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{8} c^4 d^2 x^8 (a+b \text {arccosh}(c x))-(b c) \int \frac {d^2 x^4 \left (6-8 c^2 x^2+3 c^4 x^4\right )}{24 \sqrt {-1+c x} \sqrt {1+c x}} \, dx \\ & = \frac {1}{4} d^2 x^4 (a+b \text {arccosh}(c x))-\frac {1}{3} c^2 d^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{8} c^4 d^2 x^8 (a+b \text {arccosh}(c x))-\frac {1}{24} \left (b c d^2\right ) \int \frac {x^4 \left (6-8 c^2 x^2+3 c^4 x^4\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx \\ & = \frac {1}{4} d^2 x^4 (a+b \text {arccosh}(c x))-\frac {1}{3} c^2 d^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{8} c^4 d^2 x^8 (a+b \text {arccosh}(c x))-\frac {\left (b c d^2 \sqrt {-1+c^2 x^2}\right ) \int \frac {x^4 \left (6-8 c^2 x^2+3 c^4 x^4\right )}{\sqrt {-1+c^2 x^2}} \, dx}{24 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {b c^3 d^2 x^7 \left (1-c^2 x^2\right )}{64 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {1}{4} d^2 x^4 (a+b \text {arccosh}(c x))-\frac {1}{3} c^2 d^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{8} c^4 d^2 x^8 (a+b \text {arccosh}(c x))-\frac {\left (b d^2 \sqrt {-1+c^2 x^2}\right ) \int \frac {x^4 \left (48 c^2-43 c^4 x^2\right )}{\sqrt {-1+c^2 x^2}} \, dx}{192 c \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {43 b c d^2 x^5 \left (1-c^2 x^2\right )}{1152 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 d^2 x^7 \left (1-c^2 x^2\right )}{64 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {1}{4} d^2 x^4 (a+b \text {arccosh}(c x))-\frac {1}{3} c^2 d^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{8} c^4 d^2 x^8 (a+b \text {arccosh}(c x))-\frac {\left (73 b c d^2 \sqrt {-1+c^2 x^2}\right ) \int \frac {x^4}{\sqrt {-1+c^2 x^2}} \, dx}{1152 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {73 b d^2 x^3 \left (1-c^2 x^2\right )}{4608 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {43 b c d^2 x^5 \left (1-c^2 x^2\right )}{1152 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 d^2 x^7 \left (1-c^2 x^2\right )}{64 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {1}{4} d^2 x^4 (a+b \text {arccosh}(c x))-\frac {1}{3} c^2 d^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{8} c^4 d^2 x^8 (a+b \text {arccosh}(c x))-\frac {\left (73 b d^2 \sqrt {-1+c^2 x^2}\right ) \int \frac {x^2}{\sqrt {-1+c^2 x^2}} \, dx}{1536 c \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {73 b d^2 x \left (1-c^2 x^2\right )}{3072 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {73 b d^2 x^3 \left (1-c^2 x^2\right )}{4608 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {43 b c d^2 x^5 \left (1-c^2 x^2\right )}{1152 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 d^2 x^7 \left (1-c^2 x^2\right )}{64 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {1}{4} d^2 x^4 (a+b \text {arccosh}(c x))-\frac {1}{3} c^2 d^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{8} c^4 d^2 x^8 (a+b \text {arccosh}(c x))-\frac {\left (73 b d^2 \sqrt {-1+c^2 x^2}\right ) \int \frac {1}{\sqrt {-1+c^2 x^2}} \, dx}{3072 c^3 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {73 b d^2 x \left (1-c^2 x^2\right )}{3072 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {73 b d^2 x^3 \left (1-c^2 x^2\right )}{4608 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {43 b c d^2 x^5 \left (1-c^2 x^2\right )}{1152 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 d^2 x^7 \left (1-c^2 x^2\right )}{64 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {1}{4} d^2 x^4 (a+b \text {arccosh}(c x))-\frac {1}{3} c^2 d^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{8} c^4 d^2 x^8 (a+b \text {arccosh}(c x))-\frac {\left (73 b d^2 \sqrt {-1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{1-c^2 x^2} \, dx,x,\frac {x}{\sqrt {-1+c^2 x^2}}\right )}{3072 c^3 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {73 b d^2 x \left (1-c^2 x^2\right )}{3072 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {73 b d^2 x^3 \left (1-c^2 x^2\right )}{4608 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {43 b c d^2 x^5 \left (1-c^2 x^2\right )}{1152 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 d^2 x^7 \left (1-c^2 x^2\right )}{64 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {1}{4} d^2 x^4 (a+b \text {arccosh}(c x))-\frac {1}{3} c^2 d^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{8} c^4 d^2 x^8 (a+b \text {arccosh}(c x))-\frac {73 b d^2 \sqrt {-1+c^2 x^2} \text {arctanh}\left (\frac {c x}{\sqrt {-1+c^2 x^2}}\right )}{3072 c^4 \sqrt {-1+c x} \sqrt {1+c x}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.97 \[ \int x^3 \left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=\frac {d^2 \left (2304 a c^4 x^4-3072 a c^6 x^6+1152 a c^8 x^8-219 b c x \sqrt {-1+c x} \sqrt {1+c x}-146 b c^3 x^3 \sqrt {-1+c x} \sqrt {1+c x}+344 b c^5 x^5 \sqrt {-1+c x} \sqrt {1+c x}-144 b c^7 x^7 \sqrt {-1+c x} \sqrt {1+c x}+384 b c^4 x^4 \left (6-8 c^2 x^2+3 c^4 x^4\right ) \text {arccosh}(c x)-438 b \text {arctanh}\left (\sqrt {\frac {-1+c x}{1+c x}}\right )\right )}{9216 c^4} \]

[In]

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

[Out]

(d^2*(2304*a*c^4*x^4 - 3072*a*c^6*x^6 + 1152*a*c^8*x^8 - 219*b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x] - 146*b*c^3*x^
3*Sqrt[-1 + c*x]*Sqrt[1 + c*x] + 344*b*c^5*x^5*Sqrt[-1 + c*x]*Sqrt[1 + c*x] - 144*b*c^7*x^7*Sqrt[-1 + c*x]*Sqr
t[1 + c*x] + 384*b*c^4*x^4*(6 - 8*c^2*x^2 + 3*c^4*x^4)*ArcCosh[c*x] - 438*b*ArcTanh[Sqrt[(-1 + c*x)/(1 + c*x)]
]))/(9216*c^4)

Maple [A] (verified)

Time = 0.51 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.96

method	result	size
parts	\(d^{2} a \left (\frac {1}{8} c^{4} x^{8}-\frac {1}{3} c^{2} x^{6}+\frac {1}{4} x^{4}\right )+\frac {d^{2} b \left (\frac {\operatorname {arccosh}\left (c x \right ) c^{8} x^{8}}{8}-\frac {\operatorname {arccosh}\left (c x \right ) c^{6} x^{6}}{3}+\frac {c^{4} x^{4} \operatorname {arccosh}\left (c x \right )}{4}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (144 c^{7} x^{7} \sqrt {c^{2} x^{2}-1}-344 c^{5} x^{5} \sqrt {c^{2} x^{2}-1}+146 \sqrt {c^{2} x^{2}-1}\, c^{3} x^{3}+219 c x \sqrt {c^{2} x^{2}-1}+219 \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )\right )}{9216 \sqrt {c^{2} x^{2}-1}}\right )}{c^{4}}\)	\(192\)
derivativedivides	\(\frac {d^{2} a \left (\frac {1}{8} c^{8} x^{8}-\frac {1}{3} c^{6} x^{6}+\frac {1}{4} c^{4} x^{4}\right )+d^{2} b \left (\frac {\operatorname {arccosh}\left (c x \right ) c^{8} x^{8}}{8}-\frac {\operatorname {arccosh}\left (c x \right ) c^{6} x^{6}}{3}+\frac {c^{4} x^{4} \operatorname {arccosh}\left (c x \right )}{4}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (144 c^{7} x^{7} \sqrt {c^{2} x^{2}-1}-344 c^{5} x^{5} \sqrt {c^{2} x^{2}-1}+146 \sqrt {c^{2} x^{2}-1}\, c^{3} x^{3}+219 c x \sqrt {c^{2} x^{2}-1}+219 \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )\right )}{9216 \sqrt {c^{2} x^{2}-1}}\right )}{c^{4}}\)	\(196\)
default	\(\frac {d^{2} a \left (\frac {1}{8} c^{8} x^{8}-\frac {1}{3} c^{6} x^{6}+\frac {1}{4} c^{4} x^{4}\right )+d^{2} b \left (\frac {\operatorname {arccosh}\left (c x \right ) c^{8} x^{8}}{8}-\frac {\operatorname {arccosh}\left (c x \right ) c^{6} x^{6}}{3}+\frac {c^{4} x^{4} \operatorname {arccosh}\left (c x \right )}{4}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (144 c^{7} x^{7} \sqrt {c^{2} x^{2}-1}-344 c^{5} x^{5} \sqrt {c^{2} x^{2}-1}+146 \sqrt {c^{2} x^{2}-1}\, c^{3} x^{3}+219 c x \sqrt {c^{2} x^{2}-1}+219 \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )\right )}{9216 \sqrt {c^{2} x^{2}-1}}\right )}{c^{4}}\)	\(196\)

[In]

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

[Out]

d^2*a*(1/8*c^4*x^8-1/3*c^2*x^6+1/4*x^4)+d^2*b/c^4*(1/8*arccosh(c*x)*c^8*x^8-1/3*arccosh(c*x)*c^6*x^6+1/4*c^4*x
^4*arccosh(c*x)-1/9216*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(144*c^7*x^7*(c^2*x^2-1)^(1/2)-344*c^5*x^5*(c^2*x^2-1)^(1/2
)+146*(c^2*x^2-1)^(1/2)*c^3*x^3+219*c*x*(c^2*x^2-1)^(1/2)+219*ln(c*x+(c^2*x^2-1)^(1/2)))/(c^2*x^2-1)^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.80 \[ \int x^3 \left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=\frac {1152 \, a c^{8} d^{2} x^{8} - 3072 \, a c^{6} d^{2} x^{6} + 2304 \, a c^{4} d^{2} x^{4} + 3 \, {\left (384 \, b c^{8} d^{2} x^{8} - 1024 \, b c^{6} d^{2} x^{6} + 768 \, b c^{4} d^{2} x^{4} - 73 \, b d^{2}\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (144 \, b c^{7} d^{2} x^{7} - 344 \, b c^{5} d^{2} x^{5} + 146 \, b c^{3} d^{2} x^{3} + 219 \, b c d^{2} x\right )} \sqrt {c^{2} x^{2} - 1}}{9216 \, c^{4}} \]

[In]

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

[Out]

1/9216*(1152*a*c^8*d^2*x^8 - 3072*a*c^6*d^2*x^6 + 2304*a*c^4*d^2*x^4 + 3*(384*b*c^8*d^2*x^8 - 1024*b*c^6*d^2*x
^6 + 768*b*c^4*d^2*x^4 - 73*b*d^2)*log(c*x + sqrt(c^2*x^2 - 1)) - (144*b*c^7*d^2*x^7 - 344*b*c^5*d^2*x^5 + 146
*b*c^3*d^2*x^3 + 219*b*c*d^2*x)*sqrt(c^2*x^2 - 1))/c^4

Sympy [F]

\[ \int x^3 \left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=d^{2} \left (\int a x^{3}\, dx + \int \left (- 2 a c^{2} x^{5}\right )\, dx + \int a c^{4} x^{7}\, dx + \int b x^{3} \operatorname {acosh}{\left (c x \right )}\, dx + \int \left (- 2 b c^{2} x^{5} \operatorname {acosh}{\left (c x \right )}\right )\, dx + \int b c^{4} x^{7} \operatorname {acosh}{\left (c x \right )}\, dx\right ) \]

[In]

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

[Out]

d**2*(Integral(a*x**3, x) + Integral(-2*a*c**2*x**5, x) + Integral(a*c**4*x**7, x) + Integral(b*x**3*acosh(c*x
), x) + Integral(-2*b*c**2*x**5*acosh(c*x), x) + Integral(b*c**4*x**7*acosh(c*x), x))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 346 vs. \(2 (168) = 336\).

Time = 0.25 (sec) , antiderivative size = 346, normalized size of antiderivative = 1.73 \[ \int x^3 \left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=\frac {1}{8} \, a c^{4} d^{2} x^{8} - \frac {1}{3} \, a c^{2} d^{2} x^{6} + \frac {1}{3072} \, {\left (384 \, x^{8} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {48 \, \sqrt {c^{2} x^{2} - 1} x^{7}}{c^{2}} + \frac {56 \, \sqrt {c^{2} x^{2} - 1} x^{5}}{c^{4}} + \frac {70 \, \sqrt {c^{2} x^{2} - 1} x^{3}}{c^{6}} + \frac {105 \, \sqrt {c^{2} x^{2} - 1} x}{c^{8}} + \frac {105 \, \log \left (2 \, c^{2} x + 2 \, \sqrt {c^{2} x^{2} - 1} c\right )}{c^{9}}\right )} c\right )} b c^{4} d^{2} + \frac {1}{4} \, a d^{2} x^{4} - \frac {1}{144} \, {\left (48 \, x^{6} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {8 \, \sqrt {c^{2} x^{2} - 1} x^{5}}{c^{2}} + \frac {10 \, \sqrt {c^{2} x^{2} - 1} x^{3}}{c^{4}} + \frac {15 \, \sqrt {c^{2} x^{2} - 1} x}{c^{6}} + \frac {15 \, \log \left (2 \, c^{2} x + 2 \, \sqrt {c^{2} x^{2} - 1} c\right )}{c^{7}}\right )} c\right )} b c^{2} d^{2} + \frac {1}{32} \, {\left (8 \, x^{4} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {2 \, \sqrt {c^{2} x^{2} - 1} x^{3}}{c^{2}} + \frac {3 \, \sqrt {c^{2} x^{2} - 1} x}{c^{4}} + \frac {3 \, \log \left (2 \, c^{2} x + 2 \, \sqrt {c^{2} x^{2} - 1} c\right )}{c^{5}}\right )} c\right )} b d^{2} \]

[In]

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

[Out]

1/8*a*c^4*d^2*x^8 - 1/3*a*c^2*d^2*x^6 + 1/3072*(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)*c)/c^9)*c)*b*c^4*d^2 + 1/4*a*d^2*x^4 - 1/144*(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)*c)/c^7)*c)*
b*c^2*d^2 + 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)*c)/c^5)*c)*b*d^2

Giac [F(-2)]

Exception generated. \[ \int x^3 \left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

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

[In]

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

[Out]

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