\(\int x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2 \, dx\) [170]

Optimal result

Integrand size = 29, antiderivative size = 371 \[ \int x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2 \, dx=-\frac {856 b^2 \sqrt {d-c^2 d x^2}}{3375 c^4}+\frac {22 b^2 x^2 \sqrt {d-c^2 d x^2}}{3375 c^2}+\frac {2}{125} b^2 x^4 \sqrt {d-c^2 d x^2}+\frac {4 a b x \sqrt {d-c^2 d x^2}}{15 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {4 b^2 x \sqrt {d-c^2 d x^2} \text {arccosh}(c x)}{15 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{45 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b c x^5 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{25 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{15 c^4}-\frac {x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{15 c^2}+\frac {1}{5} x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2 \]

[Out]

-856/3375*b^2*(-c^2*d*x^2+d)^(1/2)/c^4+22/3375*b^2*x^2*(-c^2*d*x^2+d)^(1/2)/c^2+2/125*b^2*x^4*(-c^2*d*x^2+d)^(
1/2)-2/15*(a+b*arccosh(c*x))^2*(-c^2*d*x^2+d)^(1/2)/c^4-1/15*x^2*(a+b*arccosh(c*x))^2*(-c^2*d*x^2+d)^(1/2)/c^2
+1/5*x^4*(a+b*arccosh(c*x))^2*(-c^2*d*x^2+d)^(1/2)+4/15*a*b*x*(-c^2*d*x^2+d)^(1/2)/c^3/(c*x-1)^(1/2)/(c*x+1)^(
1/2)+4/15*b^2*x*arccosh(c*x)*(-c^2*d*x^2+d)^(1/2)/c^3/(c*x-1)^(1/2)/(c*x+1)^(1/2)+2/45*b*x^3*(a+b*arccosh(c*x)
)*(-c^2*d*x^2+d)^(1/2)/c/(c*x-1)^(1/2)/(c*x+1)^(1/2)-2/25*b*c*x^5*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/(c*x
-1)^(1/2)/(c*x+1)^(1/2)

Rubi [A] (verified)

Time = 0.53 (sec) , antiderivative size = 371, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {5926, 5939, 5915, 5879, 75, 5883, 102, 12} \[ \int x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2 \, dx=-\frac {x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{15 c^2}-\frac {2 b c x^5 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{25 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{5} x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2+\frac {2 b x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{45 c \sqrt {c x-1} \sqrt {c x+1}}-\frac {2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{15 c^4}+\frac {4 a b x \sqrt {d-c^2 d x^2}}{15 c^3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {4 b^2 x \text {arccosh}(c x) \sqrt {d-c^2 d x^2}}{15 c^3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {22 b^2 x^2 \sqrt {d-c^2 d x^2}}{3375 c^2}+\frac {2}{125} b^2 x^4 \sqrt {d-c^2 d x^2}-\frac {856 b^2 \sqrt {d-c^2 d x^2}}{3375 c^4} \]

[In]

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

[Out]

(-856*b^2*Sqrt[d - c^2*d*x^2])/(3375*c^4) + (22*b^2*x^2*Sqrt[d - c^2*d*x^2])/(3375*c^2) + (2*b^2*x^4*Sqrt[d -
c^2*d*x^2])/125 + (4*a*b*x*Sqrt[d - c^2*d*x^2])/(15*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (4*b^2*x*Sqrt[d - c^2*
d*x^2]*ArcCosh[c*x])/(15*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (2*b*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])
)/(45*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (2*b*c*x^5*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(25*Sqrt[-1 + c*x
]*Sqrt[1 + c*x]) - (2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(15*c^4) - (x^2*Sqrt[d - c^2*d*x^2]*(a + b*A
rcCosh[c*x])^2)/(15*c^2) + (x^4*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/5

Rule 12

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

Rule 75

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

Rule 102

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(a +
b*x)^(m - 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(m + n + p + 1))), x] + Dist[1/(d*f*(m + n + p + 1)), I
nt[(a + b*x)^(m - 2)*(c + d*x)^n*(e + f*x)^p*Simp[a^2*d*f*(m + n + p + 1) - b*(b*c*e*(m - 1) + a*(d*e*(n + 1)
+ c*f*(p + 1))) + b*(a*d*f*(2*m + n + p) - b*(d*e*(m + n) + c*f*(m + p)))*x, x], x], x] /; FreeQ[{a, b, c, d,
e, f, n, p}, x] && GtQ[m, 1] && NeQ[m + n + p + 1, 0] && IntegerQ[m]

Rule 5879

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*ArcCosh[c*x])^n, x] - Dist[b*c*n, In
t[x*((a + b*ArcCosh[c*x])^(n - 1)/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])), x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]

Rule 5883

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

Rule 5915

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d1_) + (e1_.)*(x_))^(p_)*((d2_) + (e2_.)*(x_))^(p_), x_Sy
mbol] :> Simp[(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*((a + b*ArcCosh[c*x])^n/(2*e1*e2*(p + 1))), x] - Dist[b*
(n/(2*c*(p + 1)))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*x)^p/(-1 + c*x)^p], Int[(1 + c*x)^(p + 1/2)*(-
1 + c*x)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, p}, x] && EqQ[e1, c
*d1] && EqQ[e2, (-c)*d2] && GtQ[n, 0] && NeQ[p, -1]

Rule 5926

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(
f*x)^(m + 1)*Sqrt[d + e*x^2]*((a + b*ArcCosh[c*x])^n/(f*(m + 2))), x] + (-Dist[(1/(m + 2))*Simp[Sqrt[d + e*x^2
]/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])], Int[(f*x)^m*((a + b*ArcCosh[c*x])^n/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])), x], x]
- Dist[b*c*(n/(f*(m + 2)))*Simp[Sqrt[d + e*x^2]/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])], Int[(f*x)^(m + 1)*(a + b*Arc
Cosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0] && (IGtQ[m,
-2] || EqQ[n, 1])

Rule 5939

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d1_) + (e1_.)*(x_))^(p_)*((d2_) + (e2_.)*(x_
))^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 1)*(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*((a + b*ArcCosh[c*x])^n/(e1
*e2*(m + 2*p + 1))), x] + (Dist[f^2*((m - 1)/(c^2*(m + 2*p + 1))), Int[(f*x)^(m - 2)*(d1 + e1*x)^p*(d2 + e2*x)
^p*(a + b*ArcCosh[c*x])^n, x], x] - Dist[b*f*(n/(c*(m + 2*p + 1)))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 +
e2*x)^p/(-1 + c*x)^p], Int[(f*x)^(m - 1)*(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1)
, x], x]) /; FreeQ[{a, b, c, d1, e1, d2, e2, f, p}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && GtQ[n, 0] && IG
tQ[m, 1] && NeQ[m + 2*p + 1, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{5} x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {\sqrt {d-c^2 d x^2} \int \frac {x^3 (a+b \text {arccosh}(c x))^2}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{5 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (2 b c \sqrt {d-c^2 d x^2}\right ) \int x^4 (a+b \text {arccosh}(c x)) \, dx}{5 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {2 b c x^5 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{25 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{15 c^2}+\frac {1}{5} x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {\left (2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x (a+b \text {arccosh}(c x))^2}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{15 c^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (2 b \sqrt {d-c^2 d x^2}\right ) \int x^2 (a+b \text {arccosh}(c x)) \, dx}{15 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (2 b^2 c^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^5}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{25 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {2}{125} b^2 x^4 \sqrt {d-c^2 d x^2}+\frac {2 b x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{45 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b c x^5 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{25 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{15 c^4}-\frac {x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{15 c^2}+\frac {1}{5} x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2+\frac {\left (2 b^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {4 x^3}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{125 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (2 b^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^3}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{45 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (4 b \sqrt {d-c^2 d x^2}\right ) \int (a+b \text {arccosh}(c x)) \, dx}{15 c^3 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {2 b^2 x^2 \sqrt {d-c^2 d x^2}}{135 c^2}+\frac {2}{125} b^2 x^4 \sqrt {d-c^2 d x^2}+\frac {4 a b x \sqrt {d-c^2 d x^2}}{15 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{45 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b c x^5 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{25 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{15 c^4}-\frac {x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{15 c^2}+\frac {1}{5} x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2+\frac {\left (8 b^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^3}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{125 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (4 b^2 \sqrt {d-c^2 d x^2}\right ) \int \text {arccosh}(c x) \, dx}{15 c^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (2 b^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {2 x}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{135 c^2 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {22 b^2 x^2 \sqrt {d-c^2 d x^2}}{3375 c^2}+\frac {2}{125} b^2 x^4 \sqrt {d-c^2 d x^2}+\frac {4 a b x \sqrt {d-c^2 d x^2}}{15 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {4 b^2 x \sqrt {d-c^2 d x^2} \text {arccosh}(c x)}{15 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{45 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b c x^5 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{25 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{15 c^4}-\frac {x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{15 c^2}+\frac {1}{5} x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2+\frac {\left (8 b^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {2 x}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{375 c^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (4 b^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{135 c^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (4 b^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{15 c^2 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {8 b^2 \sqrt {d-c^2 d x^2}}{27 c^4}+\frac {22 b^2 x^2 \sqrt {d-c^2 d x^2}}{3375 c^2}+\frac {2}{125} b^2 x^4 \sqrt {d-c^2 d x^2}+\frac {4 a b x \sqrt {d-c^2 d x^2}}{15 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {4 b^2 x \sqrt {d-c^2 d x^2} \text {arccosh}(c x)}{15 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{45 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b c x^5 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{25 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{15 c^4}-\frac {x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{15 c^2}+\frac {1}{5} x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2+\frac {\left (16 b^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{375 c^2 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {856 b^2 \sqrt {d-c^2 d x^2}}{3375 c^4}+\frac {22 b^2 x^2 \sqrt {d-c^2 d x^2}}{3375 c^2}+\frac {2}{125} b^2 x^4 \sqrt {d-c^2 d x^2}+\frac {4 a b x \sqrt {d-c^2 d x^2}}{15 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {4 b^2 x \sqrt {d-c^2 d x^2} \text {arccosh}(c x)}{15 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{45 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b c x^5 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{25 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{15 c^4}-\frac {x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{15 c^2}+\frac {1}{5} x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2 \\ \end{align*}

Mathematica [A] (verified)

Time = 0.43 (sec) , antiderivative size = 237, normalized size of antiderivative = 0.64 \[ \int x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2 \, dx=\frac {\sqrt {d-c^2 d x^2} \left (225 a^2 \left (-1+c^2 x^2\right )^2 \left (2+3 c^2 x^2\right )-30 a b c x \sqrt {-1+c x} \sqrt {1+c x} \left (-30-5 c^2 x^2+9 c^4 x^4\right )+2 b^2 \left (428-439 c^2 x^2-16 c^4 x^4+27 c^6 x^6\right )+30 b \left (15 a \left (-1+c^2 x^2\right )^2 \left (2+3 c^2 x^2\right )+b c x \sqrt {-1+c x} \sqrt {1+c x} \left (30+5 c^2 x^2-9 c^4 x^4\right )\right ) \text {arccosh}(c x)+225 b^2 \left (-1+c^2 x^2\right )^2 \left (2+3 c^2 x^2\right ) \text {arccosh}(c x)^2\right )}{3375 c^4 \left (-1+c^2 x^2\right )} \]

[In]

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

[Out]

(Sqrt[d - c^2*d*x^2]*(225*a^2*(-1 + c^2*x^2)^2*(2 + 3*c^2*x^2) - 30*a*b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-30
- 5*c^2*x^2 + 9*c^4*x^4) + 2*b^2*(428 - 439*c^2*x^2 - 16*c^4*x^4 + 27*c^6*x^6) + 30*b*(15*a*(-1 + c^2*x^2)^2*(
2 + 3*c^2*x^2) + b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(30 + 5*c^2*x^2 - 9*c^4*x^4))*ArcCosh[c*x] + 225*b^2*(-1 +
c^2*x^2)^2*(2 + 3*c^2*x^2)*ArcCosh[c*x]^2))/(3375*c^4*(-1 + c^2*x^2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1283\) vs. \(2(315)=630\).

Time = 1.49 (sec) , antiderivative size = 1284, normalized size of antiderivative = 3.46

method	result	size
default	\(\text {Expression too large to display}\)	\(1284\)
parts	\(\text {Expression too large to display}\)	\(1284\)

[In]

int(x^3*(a+b*arccosh(c*x))^2*(-c^2*d*x^2+d)^(1/2),x,method=_RETURNVERBOSE)

[Out]

a^2*(-1/5*x^2*(-c^2*d*x^2+d)^(3/2)/c^2/d-2/15/d/c^4*(-c^2*d*x^2+d)^(3/2))+b^2*(1/4000*(-d*(c^2*x^2-1))^(1/2)*(
16*c^6*x^6-28*c^4*x^4+16*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^5*x^5+13*c^2*x^2-20*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^3*x^3
+5*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x-1)*(25*arccosh(c*x)^2-10*arccosh(c*x)+2)/(c*x+1)/c^4/(c*x-1)+1/864*(-d*(c^2
*x^2-1))^(1/2)*(4*c^4*x^4-5*c^2*x^2+4*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^3*x^3-3*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x+1)
*(9*arccosh(c*x)^2-6*arccosh(c*x)+2)/(c*x+1)/c^4/(c*x-1)-1/16*(-d*(c^2*x^2-1))^(1/2)*((c*x-1)^(1/2)*(c*x+1)^(1
/2)*c*x+c^2*x^2-1)*(arccosh(c*x)^2-2*arccosh(c*x)+2)/(c*x+1)/c^4/(c*x-1)-1/16*(-d*(c^2*x^2-1))^(1/2)*(-(c*x-1)
^(1/2)*(c*x+1)^(1/2)*c*x+c^2*x^2-1)*(arccosh(c*x)^2+2*arccosh(c*x)+2)/(c*x+1)/c^4/(c*x-1)+1/864*(-d*(c^2*x^2-1
))^(1/2)*(-4*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^3*x^3+4*c^4*x^4+3*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x-5*c^2*x^2+1)*(9*a
rccosh(c*x)^2+6*arccosh(c*x)+2)/(c*x+1)/c^4/(c*x-1)+1/4000*(-d*(c^2*x^2-1))^(1/2)*(-16*(c*x+1)^(1/2)*(c*x-1)^(
1/2)*c^5*x^5+16*c^6*x^6+20*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^3*x^3-28*c^4*x^4-5*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x+13
*c^2*x^2-1)*(25*arccosh(c*x)^2+10*arccosh(c*x)+2)/(c*x+1)/c^4/(c*x-1))+2*a*b*(1/800*(-d*(c^2*x^2-1))^(1/2)*(16
*c^6*x^6-28*c^4*x^4+16*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^5*x^5+13*c^2*x^2-20*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^3*x^3+5
*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x-1)*(-1+5*arccosh(c*x))/(c*x+1)/c^4/(c*x-1)+1/288*(-d*(c^2*x^2-1))^(1/2)*(4*c^
4*x^4-5*c^2*x^2+4*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^3*x^3-3*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x+1)*(-1+3*arccosh(c*x))
/(c*x+1)/c^4/(c*x-1)-1/16*(-d*(c^2*x^2-1))^(1/2)*((c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x+c^2*x^2-1)*(-1+arccosh(c*x))
/(c*x+1)/c^4/(c*x-1)-1/16*(-d*(c^2*x^2-1))^(1/2)*(-(c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x+c^2*x^2-1)*(1+arccosh(c*x))
/(c*x+1)/c^4/(c*x-1)+1/288*(-d*(c^2*x^2-1))^(1/2)*(-4*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^3*x^3+4*c^4*x^4+3*(c*x-1)^
(1/2)*(c*x+1)^(1/2)*c*x-5*c^2*x^2+1)*(1+3*arccosh(c*x))/(c*x+1)/c^4/(c*x-1)+1/800*(-d*(c^2*x^2-1))^(1/2)*(-16*
(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^5*x^5+16*c^6*x^6+20*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^3*x^3-28*c^4*x^4-5*(c*x-1)^(1/
2)*(c*x+1)^(1/2)*c*x+13*c^2*x^2-1)*(1+5*arccosh(c*x))/(c*x+1)/c^4/(c*x-1))

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 349, normalized size of antiderivative = 0.94 \[ \int x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2 \, dx=\frac {225 \, {\left (3 \, b^{2} c^{6} x^{6} - 4 \, b^{2} c^{4} x^{4} - b^{2} c^{2} x^{2} + 2 \, b^{2}\right )} \sqrt {-c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right )^{2} - 30 \, {\left (9 \, a b c^{5} x^{5} - 5 \, a b c^{3} x^{3} - 30 \, a b c x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} - 30 \, {\left ({\left (9 \, b^{2} c^{5} x^{5} - 5 \, b^{2} c^{3} x^{3} - 30 \, b^{2} c x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} - 15 \, {\left (3 \, a b c^{6} x^{6} - 4 \, a b c^{4} x^{4} - a b c^{2} x^{2} + 2 \, a b\right )} \sqrt {-c^{2} d x^{2} + d}\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) + {\left (27 \, {\left (25 \, a^{2} + 2 \, b^{2}\right )} c^{6} x^{6} - 4 \, {\left (225 \, a^{2} + 8 \, b^{2}\right )} c^{4} x^{4} - {\left (225 \, a^{2} + 878 \, b^{2}\right )} c^{2} x^{2} + 450 \, a^{2} + 856 \, b^{2}\right )} \sqrt {-c^{2} d x^{2} + d}}{3375 \, {\left (c^{6} x^{2} - c^{4}\right )}} \]

[In]

integrate(x^3*(a+b*arccosh(c*x))^2*(-c^2*d*x^2+d)^(1/2),x, algorithm="fricas")

[Out]

1/3375*(225*(3*b^2*c^6*x^6 - 4*b^2*c^4*x^4 - b^2*c^2*x^2 + 2*b^2)*sqrt(-c^2*d*x^2 + d)*log(c*x + sqrt(c^2*x^2
- 1))^2 - 30*(9*a*b*c^5*x^5 - 5*a*b*c^3*x^3 - 30*a*b*c*x)*sqrt(-c^2*d*x^2 + d)*sqrt(c^2*x^2 - 1) - 30*((9*b^2*
c^5*x^5 - 5*b^2*c^3*x^3 - 30*b^2*c*x)*sqrt(-c^2*d*x^2 + d)*sqrt(c^2*x^2 - 1) - 15*(3*a*b*c^6*x^6 - 4*a*b*c^4*x
^4 - a*b*c^2*x^2 + 2*a*b)*sqrt(-c^2*d*x^2 + d))*log(c*x + sqrt(c^2*x^2 - 1)) + (27*(25*a^2 + 2*b^2)*c^6*x^6 -
4*(225*a^2 + 8*b^2)*c^4*x^4 - (225*a^2 + 878*b^2)*c^2*x^2 + 450*a^2 + 856*b^2)*sqrt(-c^2*d*x^2 + d))/(c^6*x^2
- c^4)

Sympy [F]

\[ \int x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2 \, dx=\int x^{3} \sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}\, dx \]

[In]

integrate(x**3*(a+b*acosh(c*x))**2*(-c**2*d*x**2+d)**(1/2),x)

[Out]

Integral(x**3*sqrt(-d*(c*x - 1)*(c*x + 1))*(a + b*acosh(c*x))**2, x)

Maxima [A] (verification not implemented)

none

Time = 0.45 (sec) , antiderivative size = 326, normalized size of antiderivative = 0.88 \[ \int x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2 \, dx=-\frac {1}{15} \, b^{2} {\left (\frac {3 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x^{2}}{c^{2} d} + \frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}}{c^{4} d}\right )} \operatorname {arcosh}\left (c x\right )^{2} - \frac {2}{15} \, a b {\left (\frac {3 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x^{2}}{c^{2} d} + \frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}}{c^{4} d}\right )} \operatorname {arcosh}\left (c x\right ) - \frac {1}{15} \, a^{2} {\left (\frac {3 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x^{2}}{c^{2} d} + \frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}}{c^{4} d}\right )} + \frac {2}{3375} \, b^{2} {\left (\frac {27 \, \sqrt {c^{2} x^{2} - 1} c^{2} \sqrt {-d} x^{4} + 11 \, \sqrt {c^{2} x^{2} - 1} \sqrt {-d} x^{2} - \frac {428 \, \sqrt {c^{2} x^{2} - 1} \sqrt {-d}}{c^{2}}}{c^{2}} - \frac {15 \, {\left (9 \, c^{4} \sqrt {-d} x^{5} - 5 \, c^{2} \sqrt {-d} x^{3} - 30 \, \sqrt {-d} x\right )} \operatorname {arcosh}\left (c x\right )}{c^{3}}\right )} - \frac {2 \, {\left (9 \, c^{4} \sqrt {-d} x^{5} - 5 \, c^{2} \sqrt {-d} x^{3} - 30 \, \sqrt {-d} x\right )} a b}{225 \, c^{3}} \]

[In]

integrate(x^3*(a+b*arccosh(c*x))^2*(-c^2*d*x^2+d)^(1/2),x, algorithm="maxima")

[Out]

-1/15*b^2*(3*(-c^2*d*x^2 + d)^(3/2)*x^2/(c^2*d) + 2*(-c^2*d*x^2 + d)^(3/2)/(c^4*d))*arccosh(c*x)^2 - 2/15*a*b*
(3*(-c^2*d*x^2 + d)^(3/2)*x^2/(c^2*d) + 2*(-c^2*d*x^2 + d)^(3/2)/(c^4*d))*arccosh(c*x) - 1/15*a^2*(3*(-c^2*d*x
^2 + d)^(3/2)*x^2/(c^2*d) + 2*(-c^2*d*x^2 + d)^(3/2)/(c^4*d)) + 2/3375*b^2*((27*sqrt(c^2*x^2 - 1)*c^2*sqrt(-d)
*x^4 + 11*sqrt(c^2*x^2 - 1)*sqrt(-d)*x^2 - 428*sqrt(c^2*x^2 - 1)*sqrt(-d)/c^2)/c^2 - 15*(9*c^4*sqrt(-d)*x^5 -
5*c^2*sqrt(-d)*x^3 - 30*sqrt(-d)*x)*arccosh(c*x)/c^3) - 2/225*(9*c^4*sqrt(-d)*x^5 - 5*c^2*sqrt(-d)*x^3 - 30*sq
rt(-d)*x)*a*b/c^3

Giac [F(-2)]

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

[In]

integrate(x^3*(a+b*arccosh(c*x))^2*(-c^2*d*x^2+d)^(1/2),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. \[ \int x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2 \, dx=\int x^3\,{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2\,\sqrt {d-c^2\,d\,x^2} \,d x \]

[In]

int(x^3*(a + b*acosh(c*x))^2*(d - c^2*d*x^2)^(1/2),x)

[Out]

int(x^3*(a + b*acosh(c*x))^2*(d - c^2*d*x^2)^(1/2), x)