3.2.94 \(\int \frac {x^5 (a+b \cosh ^{-1}(c x))^2}{\sqrt {d-c^2 d x^2}} \, dx\) [194]

Optimal. Leaf size=421 \[ -\frac {16 a b x \sqrt {-1+c x} \sqrt {1+c x}}{15 c^5 \sqrt {d-c^2 d x^2}}-\frac {4144 b^2 (1-c x) (1+c x)}{3375 c^6 \sqrt {d-c^2 d x^2}}-\frac {272 b^2 x^2 (1-c x) (1+c x)}{3375 c^4 \sqrt {d-c^2 d x^2}}-\frac {2 b^2 x^4 (1-c x) (1+c x)}{125 c^2 \sqrt {d-c^2 d x^2}}-\frac {16 b^2 x \sqrt {-1+c x} \sqrt {1+c x} \cosh ^{-1}(c x)}{15 c^5 \sqrt {d-c^2 d x^2}}-\frac {8 b x^3 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{45 c^3 \sqrt {d-c^2 d x^2}}-\frac {2 b x^5 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{25 c \sqrt {d-c^2 d x^2}}-\frac {8 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{15 c^6 d}-\frac {4 x^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{15 c^4 d}-\frac {x^4 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{5 c^2 d} \]

[Out]

-4144/3375*b^2*(-c*x+1)*(c*x+1)/c^6/(-c^2*d*x^2+d)^(1/2)-272/3375*b^2*x^2*(-c*x+1)*(c*x+1)/c^4/(-c^2*d*x^2+d)^
(1/2)-2/125*b^2*x^4*(-c*x+1)*(c*x+1)/c^2/(-c^2*d*x^2+d)^(1/2)-16/15*a*b*x*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c^5/(-c^
2*d*x^2+d)^(1/2)-16/15*b^2*x*arccosh(c*x)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c^5/(-c^2*d*x^2+d)^(1/2)-8/45*b*x^3*(a+b
*arccosh(c*x))*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c^3/(-c^2*d*x^2+d)^(1/2)-2/25*b*x^5*(a+b*arccosh(c*x))*(c*x-1)^(1/2
)*(c*x+1)^(1/2)/c/(-c^2*d*x^2+d)^(1/2)-8/15*(a+b*arccosh(c*x))^2*(-c^2*d*x^2+d)^(1/2)/c^6/d-4/15*x^2*(a+b*arcc
osh(c*x))^2*(-c^2*d*x^2+d)^(1/2)/c^4/d-1/5*x^4*(a+b*arccosh(c*x))^2*(-c^2*d*x^2+d)^(1/2)/c^2/d

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Rubi [A]
time = 0.39, antiderivative size = 421, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 7, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {5938, 5914, 5879, 75, 5883, 102, 12} \begin {gather*} -\frac {2 b x^5 \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{25 c \sqrt {d-c^2 d x^2}}-\frac {x^4 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{5 c^2 d}-\frac {8 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{15 c^6 d}-\frac {16 a b x \sqrt {c x-1} \sqrt {c x+1}}{15 c^5 \sqrt {d-c^2 d x^2}}-\frac {4 x^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{15 c^4 d}-\frac {8 b x^3 \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{45 c^3 \sqrt {d-c^2 d x^2}}-\frac {2 b^2 x^4 (1-c x) (c x+1)}{125 c^2 \sqrt {d-c^2 d x^2}}-\frac {4144 b^2 (1-c x) (c x+1)}{3375 c^6 \sqrt {d-c^2 d x^2}}-\frac {16 b^2 x \sqrt {c x-1} \sqrt {c x+1} \cosh ^{-1}(c x)}{15 c^5 \sqrt {d-c^2 d x^2}}-\frac {272 b^2 x^2 (1-c x) (c x+1)}{3375 c^4 \sqrt {d-c^2 d x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^5*(a + b*ArcCosh[c*x])^2)/Sqrt[d - c^2*d*x^2],x]

[Out]

(-16*a*b*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(15*c^5*Sqrt[d - c^2*d*x^2]) - (4144*b^2*(1 - c*x)*(1 + c*x))/(3375*c
^6*Sqrt[d - c^2*d*x^2]) - (272*b^2*x^2*(1 - c*x)*(1 + c*x))/(3375*c^4*Sqrt[d - c^2*d*x^2]) - (2*b^2*x^4*(1 - c
*x)*(1 + c*x))/(125*c^2*Sqrt[d - c^2*d*x^2]) - (16*b^2*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*ArcCosh[c*x])/(15*c^5*Sq
rt[d - c^2*d*x^2]) - (8*b*x^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(45*c^3*Sqrt[d - c^2*d*x^2])
- (2*b*x^5*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(25*c*Sqrt[d - c^2*d*x^2]) - (8*Sqrt[d - c^2*d*x
^2]*(a + b*ArcCosh[c*x])^2)/(15*c^6*d) - (4*x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(15*c^4*d) - (x^4*
Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(5*c^2*d)

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 5914

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

Rule 5938

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

Rubi steps

\begin {align*} \int \frac {x^5 \left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt {d-c^2 d x^2}} \, dx &=\frac {\left (\sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x^5 \left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{\sqrt {d-c^2 d x^2}}\\ &=-\frac {x^4 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{5 c^2 \sqrt {d-c^2 d x^2}}+\frac {\left (4 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x^3 \left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{5 c^2 \sqrt {d-c^2 d x^2}}-\frac {\left (2 b \sqrt {-1+c x} \sqrt {1+c x}\right ) \int x^4 \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{5 c \sqrt {d-c^2 d x^2}}\\ &=-\frac {2 b x^5 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{25 c \sqrt {d-c^2 d x^2}}-\frac {4 x^2 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{15 c^4 \sqrt {d-c^2 d x^2}}-\frac {x^4 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{5 c^2 \sqrt {d-c^2 d x^2}}+\frac {\left (2 b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x^5}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{25 \sqrt {d-c^2 d x^2}}+\frac {\left (8 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x \left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{15 c^4 \sqrt {d-c^2 d x^2}}-\frac {\left (8 b \sqrt {-1+c x} \sqrt {1+c x}\right ) \int x^2 \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{15 c^3 \sqrt {d-c^2 d x^2}}\\ &=-\frac {2 b^2 x^4 (1-c x) (1+c x)}{125 c^2 \sqrt {d-c^2 d x^2}}-\frac {8 b x^3 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{45 c^3 \sqrt {d-c^2 d x^2}}-\frac {2 b x^5 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{25 c \sqrt {d-c^2 d x^2}}-\frac {8 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{15 c^6 \sqrt {d-c^2 d x^2}}-\frac {4 x^2 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{15 c^4 \sqrt {d-c^2 d x^2}}-\frac {x^4 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{5 c^2 \sqrt {d-c^2 d x^2}}-\frac {\left (16 b \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{15 c^5 \sqrt {d-c^2 d x^2}}+\frac {\left (2 b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {4 x^3}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{125 c^2 \sqrt {d-c^2 d x^2}}+\frac {\left (8 b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x^3}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{45 c^2 \sqrt {d-c^2 d x^2}}\\ &=-\frac {16 a b x \sqrt {-1+c x} \sqrt {1+c x}}{15 c^5 \sqrt {d-c^2 d x^2}}-\frac {8 b^2 x^2 (1-c x) (1+c x)}{135 c^4 \sqrt {d-c^2 d x^2}}-\frac {2 b^2 x^4 (1-c x) (1+c x)}{125 c^2 \sqrt {d-c^2 d x^2}}-\frac {8 b x^3 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{45 c^3 \sqrt {d-c^2 d x^2}}-\frac {2 b x^5 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{25 c \sqrt {d-c^2 d x^2}}-\frac {8 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{15 c^6 \sqrt {d-c^2 d x^2}}-\frac {4 x^2 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{15 c^4 \sqrt {d-c^2 d x^2}}-\frac {x^4 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{5 c^2 \sqrt {d-c^2 d x^2}}-\frac {\left (16 b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \cosh ^{-1}(c x) \, dx}{15 c^5 \sqrt {d-c^2 d x^2}}+\frac {\left (8 b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {2 x}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{135 c^4 \sqrt {d-c^2 d x^2}}+\frac {\left (8 b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x^3}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{125 c^2 \sqrt {d-c^2 d x^2}}\\ &=-\frac {16 a b x \sqrt {-1+c x} \sqrt {1+c x}}{15 c^5 \sqrt {d-c^2 d x^2}}-\frac {272 b^2 x^2 (1-c x) (1+c x)}{3375 c^4 \sqrt {d-c^2 d x^2}}-\frac {2 b^2 x^4 (1-c x) (1+c x)}{125 c^2 \sqrt {d-c^2 d x^2}}-\frac {16 b^2 x \sqrt {-1+c x} \sqrt {1+c x} \cosh ^{-1}(c x)}{15 c^5 \sqrt {d-c^2 d x^2}}-\frac {8 b x^3 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{45 c^3 \sqrt {d-c^2 d x^2}}-\frac {2 b x^5 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{25 c \sqrt {d-c^2 d x^2}}-\frac {8 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{15 c^6 \sqrt {d-c^2 d x^2}}-\frac {4 x^2 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{15 c^4 \sqrt {d-c^2 d x^2}}-\frac {x^4 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{5 c^2 \sqrt {d-c^2 d x^2}}+\frac {\left (8 b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {2 x}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{375 c^4 \sqrt {d-c^2 d x^2}}+\frac {\left (16 b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{135 c^4 \sqrt {d-c^2 d x^2}}+\frac {\left (16 b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{15 c^4 \sqrt {d-c^2 d x^2}}\\ &=-\frac {16 a b x \sqrt {-1+c x} \sqrt {1+c x}}{15 c^5 \sqrt {d-c^2 d x^2}}-\frac {32 b^2 (1-c x) (1+c x)}{27 c^6 \sqrt {d-c^2 d x^2}}-\frac {272 b^2 x^2 (1-c x) (1+c x)}{3375 c^4 \sqrt {d-c^2 d x^2}}-\frac {2 b^2 x^4 (1-c x) (1+c x)}{125 c^2 \sqrt {d-c^2 d x^2}}-\frac {16 b^2 x \sqrt {-1+c x} \sqrt {1+c x} \cosh ^{-1}(c x)}{15 c^5 \sqrt {d-c^2 d x^2}}-\frac {8 b x^3 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{45 c^3 \sqrt {d-c^2 d x^2}}-\frac {2 b x^5 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{25 c \sqrt {d-c^2 d x^2}}-\frac {8 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{15 c^6 \sqrt {d-c^2 d x^2}}-\frac {4 x^2 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{15 c^4 \sqrt {d-c^2 d x^2}}-\frac {x^4 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{5 c^2 \sqrt {d-c^2 d x^2}}+\frac {\left (16 b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{375 c^4 \sqrt {d-c^2 d x^2}}\\ &=-\frac {16 a b x \sqrt {-1+c x} \sqrt {1+c x}}{15 c^5 \sqrt {d-c^2 d x^2}}-\frac {4144 b^2 (1-c x) (1+c x)}{3375 c^6 \sqrt {d-c^2 d x^2}}-\frac {272 b^2 x^2 (1-c x) (1+c x)}{3375 c^4 \sqrt {d-c^2 d x^2}}-\frac {2 b^2 x^4 (1-c x) (1+c x)}{125 c^2 \sqrt {d-c^2 d x^2}}-\frac {16 b^2 x \sqrt {-1+c x} \sqrt {1+c x} \cosh ^{-1}(c x)}{15 c^5 \sqrt {d-c^2 d x^2}}-\frac {8 b x^3 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{45 c^3 \sqrt {d-c^2 d x^2}}-\frac {2 b x^5 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{25 c \sqrt {d-c^2 d x^2}}-\frac {8 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{15 c^6 \sqrt {d-c^2 d x^2}}-\frac {4 x^2 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{15 c^4 \sqrt {d-c^2 d x^2}}-\frac {x^4 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{5 c^2 \sqrt {d-c^2 d x^2}}\\ \end {align*}

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Mathematica [A]
time = 0.34, size = 255, normalized size = 0.61 \begin {gather*} \frac {\sqrt {d-c^2 d x^2} \left (30 a b c x \sqrt {-1+c x} \sqrt {1+c x} \left (120+20 c^2 x^2+9 c^4 x^4\right )-225 a^2 \left (-8+4 c^2 x^2+c^4 x^4+3 c^6 x^6\right )-2 b^2 \left (-2072+1936 c^2 x^2+109 c^4 x^4+27 c^6 x^6\right )+30 b \left (b c x \sqrt {-1+c x} \sqrt {1+c x} \left (120+20 c^2 x^2+9 c^4 x^4\right )-15 a \left (-8+4 c^2 x^2+c^4 x^4+3 c^6 x^6\right )\right ) \cosh ^{-1}(c x)-225 b^2 \left (-8+4 c^2 x^2+c^4 x^4+3 c^6 x^6\right ) \cosh ^{-1}(c x)^2\right )}{3375 c^6 d (-1+c x) (1+c x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^5*(a + b*ArcCosh[c*x])^2)/Sqrt[d - c^2*d*x^2],x]

[Out]

(Sqrt[d - c^2*d*x^2]*(30*a*b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(120 + 20*c^2*x^2 + 9*c^4*x^4) - 225*a^2*(-8 + 4
*c^2*x^2 + c^4*x^4 + 3*c^6*x^6) - 2*b^2*(-2072 + 1936*c^2*x^2 + 109*c^4*x^4 + 27*c^6*x^6) + 30*b*(b*c*x*Sqrt[-
1 + c*x]*Sqrt[1 + c*x]*(120 + 20*c^2*x^2 + 9*c^4*x^4) - 15*a*(-8 + 4*c^2*x^2 + c^4*x^4 + 3*c^6*x^6))*ArcCosh[c
*x] - 225*b^2*(-8 + 4*c^2*x^2 + c^4*x^4 + 3*c^6*x^6)*ArcCosh[c*x]^2))/(3375*c^6*d*(-1 + c*x)*(1 + c*x))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1313\) vs. \(2(365)=730\).
time = 4.26, size = 1314, normalized size = 3.12

method	result	size
default	\(\text {Expression too large to display}\)	\(1314\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(a+b*arccosh(c*x))^2/(-c^2*d*x^2+d)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.50, size = 407, normalized size = 0.97 \begin {gather*} -\frac {1}{15} \, {\left (\frac {3 \, \sqrt {-c^{2} d x^{2} + d} x^{4}}{c^{2} d} + \frac {4 \, \sqrt {-c^{2} d x^{2} + d} x^{2}}{c^{4} d} + \frac {8 \, \sqrt {-c^{2} d x^{2} + d}}{c^{6} d}\right )} b^{2} \operatorname {arcosh}\left (c x\right )^{2} - \frac {2}{15} \, {\left (\frac {3 \, \sqrt {-c^{2} d x^{2} + d} x^{4}}{c^{2} d} + \frac {4 \, \sqrt {-c^{2} d x^{2} + d} x^{2}}{c^{4} d} + \frac {8 \, \sqrt {-c^{2} d x^{2} + d}}{c^{6} d}\right )} a b \operatorname {arcosh}\left (c x\right ) - \frac {1}{15} \, {\left (\frac {3 \, \sqrt {-c^{2} d x^{2} + d} x^{4}}{c^{2} d} + \frac {4 \, \sqrt {-c^{2} d x^{2} + d} x^{2}}{c^{4} d} + \frac {8 \, \sqrt {-c^{2} d x^{2} + d}}{c^{6} d}\right )} a^{2} - \frac {2}{3375} \, b^{2} {\left (\frac {27 \, \sqrt {c^{2} x^{2} - 1} c^{2} \sqrt {-d} x^{4} + 136 \, \sqrt {c^{2} x^{2} - 1} \sqrt {-d} x^{2} + \frac {2072 \, \sqrt {c^{2} x^{2} - 1} \sqrt {-d}}{c^{2}}}{c^{4} d} - \frac {15 \, {\left (9 \, c^{4} \sqrt {-d} x^{5} + 20 \, c^{2} \sqrt {-d} x^{3} + 120 \, \sqrt {-d} x\right )} \operatorname {arcosh}\left (c x\right )}{c^{5} d}\right )} + \frac {2 \, {\left (9 \, c^{4} \sqrt {-d} x^{5} + 20 \, c^{2} \sqrt {-d} x^{3} + 120 \, \sqrt {-d} x\right )} a b}{225 \, c^{5} d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(a+b*arccosh(c*x))^2/(-c^2*d*x^2+d)^(1/2),x, algorithm="maxima")

[Out]

-1/15*(3*sqrt(-c^2*d*x^2 + d)*x^4/(c^2*d) + 4*sqrt(-c^2*d*x^2 + d)*x^2/(c^4*d) + 8*sqrt(-c^2*d*x^2 + d)/(c^6*d
))*b^2*arccosh(c*x)^2 - 2/15*(3*sqrt(-c^2*d*x^2 + d)*x^4/(c^2*d) + 4*sqrt(-c^2*d*x^2 + d)*x^2/(c^4*d) + 8*sqrt
(-c^2*d*x^2 + d)/(c^6*d))*a*b*arccosh(c*x) - 1/15*(3*sqrt(-c^2*d*x^2 + d)*x^4/(c^2*d) + 4*sqrt(-c^2*d*x^2 + d)
*x^2/(c^4*d) + 8*sqrt(-c^2*d*x^2 + d)/(c^6*d))*a^2 - 2/3375*b^2*((27*sqrt(c^2*x^2 - 1)*c^2*sqrt(-d)*x^4 + 136*
sqrt(c^2*x^2 - 1)*sqrt(-d)*x^2 + 2072*sqrt(c^2*x^2 - 1)*sqrt(-d)/c^2)/(c^4*d) - 15*(9*c^4*sqrt(-d)*x^5 + 20*c^
2*sqrt(-d)*x^3 + 120*sqrt(-d)*x)*arccosh(c*x)/(c^5*d)) + 2/225*(9*c^4*sqrt(-d)*x^5 + 20*c^2*sqrt(-d)*x^3 + 120
*sqrt(-d)*x)*a*b/(c^5*d)

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Fricas [A]
time = 0.46, size = 348, normalized size = 0.83 \begin {gather*} -\frac {225 \, {\left (3 \, b^{2} c^{6} x^{6} + b^{2} c^{4} x^{4} + 4 \, b^{2} c^{2} x^{2} - 8 \, 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} + 20 \, a b c^{3} x^{3} + 120 \, 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} + 20 \, b^{2} c^{3} x^{3} + 120 \, 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} + a b c^{4} x^{4} + 4 \, a b c^{2} x^{2} - 8 \, 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} + {\left (225 \, a^{2} + 218 \, b^{2}\right )} c^{4} x^{4} + 4 \, {\left (225 \, a^{2} + 968 \, b^{2}\right )} c^{2} x^{2} - 1800 \, a^{2} - 4144 \, b^{2}\right )} \sqrt {-c^{2} d x^{2} + d}}{3375 \, {\left (c^{8} d x^{2} - c^{6} d\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(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 + b^2*c^4*x^4 + 4*b^2*c^2*x^2 - 8*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 + 20*a*b*c^3*x^3 + 120*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 + 20*b^2*c^3*x^3 + 120*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 + a*b*c^
4*x^4 + 4*a*b*c^2*x^2 - 8*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 + (225*a^2 + 218*b^2)*c^4*x^4 + 4*(225*a^2 + 968*b^2)*c^2*x^2 - 1800*a^2 - 4144*b^2)*sqrt(-c^2*d*x^2 + d))/
(c^8*d*x^2 - c^6*d)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{5} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}}{\sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*(a+b*acosh(c*x))**2/(-c**2*d*x**2+d)**(1/2),x)

[Out]

Integral(x**5*(a + b*acosh(c*x))**2/sqrt(-d*(c*x - 1)*(c*x + 1)), x)

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^5\,{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2}{\sqrt {d-c^2\,d\,x^2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^5*(a + b*acosh(c*x))^2)/(d - c^2*d*x^2)^(1/2),x)

[Out]

int((x^5*(a + b*acosh(c*x))^2)/(d - c^2*d*x^2)^(1/2), x)

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