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

Optimal. Leaf size=336 \[ -\frac {d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^3}{8 b c \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {3}{8} d x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {b d \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 c \sqrt {c x-1} \sqrt {c x+1}}-\frac {3 b c d x^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 \sqrt {c x-1} \sqrt {c x+1}}+\frac {15}{64} b^2 d x \sqrt {d-c^2 d x^2}+\frac {1}{32} b^2 d x (1-c x) (c x+1) \sqrt {d-c^2 d x^2}+\frac {9 b^2 d \sqrt {d-c^2 d x^2} \cosh ^{-1}(c x)}{64 c \sqrt {c x-1} \sqrt {c x+1}} \]

[Out]

1/4*x*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))^2+15/64*b^2*d*x*(-c^2*d*x^2+d)^(1/2)+1/32*b^2*d*x*(-c*x+1)*(c*x+
1)*(-c^2*d*x^2+d)^(1/2)+3/8*d*x*(a+b*arccosh(c*x))^2*(-c^2*d*x^2+d)^(1/2)+9/64*b^2*d*arccosh(c*x)*(-c^2*d*x^2+
d)^(1/2)/c/(c*x-1)^(1/2)/(c*x+1)^(1/2)-3/8*b*c*d*x^2*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*
x+1)^(1/2)+1/8*b*d*(-c^2*x^2+1)^2*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/c/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/8*d*
(a+b*arccosh(c*x))^3*(-c^2*d*x^2+d)^(1/2)/b/c/(c*x-1)^(1/2)/(c*x+1)^(1/2)

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Rubi [A]  time = 0.61, antiderivative size = 348, normalized size of antiderivative = 1.04, number of steps used = 11, number of rules used = 9, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {5713, 5685, 5683, 5676, 5662, 90, 52, 5716, 38} \[ -\frac {d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^3}{8 b c \sqrt {c x-1} \sqrt {c x+1}}+\frac {3}{8} d x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {1}{4} d x (1-c x) (c x+1) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {b d \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 c \sqrt {c x-1} \sqrt {c x+1}}-\frac {3 b c d x^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 \sqrt {c x-1} \sqrt {c x+1}}+\frac {15}{64} b^2 d x \sqrt {d-c^2 d x^2}+\frac {1}{32} b^2 d x (1-c x) (c x+1) \sqrt {d-c^2 d x^2}+\frac {9 b^2 d \sqrt {d-c^2 d x^2} \cosh ^{-1}(c x)}{64 c \sqrt {c x-1} \sqrt {c x+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(15*b^2*d*x*Sqrt[d - c^2*d*x^2])/64 + (b^2*d*x*(1 - c*x)*(1 + c*x)*Sqrt[d - c^2*d*x^2])/32 + (9*b^2*d*Sqrt[d -
c^2*d*x^2]*ArcCosh[c*x])/(64*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (3*b*c*d*x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos
h[c*x]))/(8*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*d*(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(8*
c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (3*d*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/8 + (d*x*(1 - c*x)*(1 + c
*x)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/4 - (d*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^3)/(8*b*c*Sqrt
[-1 + c*x]*Sqrt[1 + c*x])

Rule 38

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[(x*(a + b*x)^m*(c + d*x)^m)/(2*m + 1)
, x] + Dist[(2*a*c*m)/(2*m + 1), Int[(a + b*x)^(m - 1)*(c + d*x)^(m - 1), x], x] /; FreeQ[{a, b, c, d}, x] &&
EqQ[b*c + a*d, 0] && IGtQ[m + 1/2, 0]

Rule 52

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ArcCosh[(b*x)/a]/b, x] /; FreeQ[{a,
b, c, d}, x] && EqQ[a + c, 0] && EqQ[b - d, 0] && GtQ[a, 0]

Rule 90

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a + b*
x)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 3)), x] + Dist[1/(d*f*(n + p + 3)), Int[(c + d*x)^n*(e +
f*x)^p*Simp[a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(n + p + 4) - b*(d*e*(
n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 3, 0]

Rule 5662

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))/(Sqr
t[-1 + c*x]*Sqrt[1 + c*x]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rule 5676

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

Rule 5683

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

Rule 5685

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

Rule 5713

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Dist[((-d)^IntPart[p]*(
d + e*x^2)^FracPart[p])/((1 + c*x)^FracPart[p]*(-1 + c*x)^FracPart[p]), Int[(1 + c*x)^p*(-1 + c*x)^p*(a + b*Ar
cCosh[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[c^2*d + e, 0] &&  !IntegerQ[p]

Rule 5716

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*(-d)^p)/(2*c*(p + 1)), 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] && IntegerQ[p]

Rubi steps

\begin {align*} \int \left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )^2 \, dx &=-\frac {\left (d \sqrt {d-c^2 d x^2}\right ) \int (-1+c x)^{3/2} (1+c x)^{3/2} \left (a+b \cosh ^{-1}(c x)\right )^2 \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {1}{4} d x (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {\left (3 d \sqrt {d-c^2 d x^2}\right ) \int \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2 \, dx}{4 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b c d \sqrt {d-c^2 d x^2}\right ) \int x \left (-1+c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{2 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {b d \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3}{8} d x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {1}{4} d x (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {\left (3 d \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{8 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (b^2 d \sqrt {d-c^2 d x^2}\right ) \int (-1+c x)^{3/2} (1+c x)^{3/2} \, dx}{8 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (3 b c d \sqrt {d-c^2 d x^2}\right ) \int x \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{4 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {1}{32} b^2 d x (1-c x) (1+c x) \sqrt {d-c^2 d x^2}-\frac {3 b c d x^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3}{8} d x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {1}{4} d x (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^3}{8 b c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (3 b^2 d \sqrt {d-c^2 d x^2}\right ) \int \sqrt {-1+c x} \sqrt {1+c x} \, dx}{32 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (3 b^2 c^2 d \sqrt {d-c^2 d x^2}\right ) \int \frac {x^2}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{8 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {15}{64} b^2 d x \sqrt {d-c^2 d x^2}+\frac {1}{32} b^2 d x (1-c x) (1+c x) \sqrt {d-c^2 d x^2}-\frac {3 b c d x^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3}{8} d x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {1}{4} d x (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^3}{8 b c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (3 b^2 d \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{64 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (3 b^2 d \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{16 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {15}{64} b^2 d x \sqrt {d-c^2 d x^2}+\frac {1}{32} b^2 d x (1-c x) (1+c x) \sqrt {d-c^2 d x^2}+\frac {9 b^2 d \sqrt {d-c^2 d x^2} \cosh ^{-1}(c x)}{64 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {3 b c d x^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3}{8} d x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {1}{4} d x (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^3}{8 b c \sqrt {-1+c x} \sqrt {1+c x}}\\ \end {align*}

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Mathematica [A]  time = 2.95, size = 374, normalized size = 1.11 \[ \frac {-288 a^2 d^{3/2} \sqrt {\frac {c x-1}{c x+1}} (c x+1) \tan ^{-1}\left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (c^2 x^2-1\right )}\right )-96 a^2 c d x \sqrt {\frac {c x-1}{c x+1}} (c x+1) \left (2 c^2 x^2-5\right ) \sqrt {d-c^2 d x^2}-192 a b d \sqrt {d-c^2 d x^2} \left (\cosh \left (2 \cosh ^{-1}(c x)\right )+2 \cosh ^{-1}(c x) \left (\cosh ^{-1}(c x)-\sinh \left (2 \cosh ^{-1}(c x)\right )\right )\right )+12 a b d \sqrt {d-c^2 d x^2} \left (8 \cosh ^{-1}(c x)^2+\cosh \left (4 \cosh ^{-1}(c x)\right )-4 \cosh ^{-1}(c x) \sinh \left (4 \cosh ^{-1}(c x)\right )\right )-32 b^2 d \sqrt {d-c^2 d x^2} \left (4 \cosh ^{-1}(c x)^3+6 \cosh \left (2 \cosh ^{-1}(c x)\right ) \cosh ^{-1}(c x)-3 \left (2 \cosh ^{-1}(c x)^2+1\right ) \sinh \left (2 \cosh ^{-1}(c x)\right )\right )+b^2 d \sqrt {d-c^2 d x^2} \left (32 \cosh ^{-1}(c x)^3+12 \cosh \left (4 \cosh ^{-1}(c x)\right ) \cosh ^{-1}(c x)-3 \left (8 \cosh ^{-1}(c x)^2+1\right ) \sinh \left (4 \cosh ^{-1}(c x)\right )\right )}{768 c \sqrt {\frac {c x-1}{c x+1}} (c x+1)} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-96*a^2*c*d*x*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*(-5 + 2*c^2*x^2)*Sqrt[d - c^2*d*x^2] - 288*a^2*d^(3/2)*Sqr
t[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] - 192*a*b*d*Sqrt[
d - c^2*d*x^2]*(Cosh[2*ArcCosh[c*x]] + 2*ArcCosh[c*x]*(ArcCosh[c*x] - Sinh[2*ArcCosh[c*x]])) - 32*b^2*d*Sqrt[d
- c^2*d*x^2]*(4*ArcCosh[c*x]^3 + 6*ArcCosh[c*x]*Cosh[2*ArcCosh[c*x]] - 3*(1 + 2*ArcCosh[c*x]^2)*Sinh[2*ArcCos
h[c*x]]) + 12*a*b*d*Sqrt[d - c^2*d*x^2]*(8*ArcCosh[c*x]^2 + Cosh[4*ArcCosh[c*x]] - 4*ArcCosh[c*x]*Sinh[4*ArcCo
sh[c*x]]) + b^2*d*Sqrt[d - c^2*d*x^2]*(32*ArcCosh[c*x]^3 + 12*ArcCosh[c*x]*Cosh[4*ArcCosh[c*x]] - 3*(1 + 8*Arc
Cosh[c*x]^2)*Sinh[4*ArcCosh[c*x]]))/(768*c*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x))

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fricas [F]  time = 0.62, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (a^{2} c^{2} d x^{2} - a^{2} d + {\left (b^{2} c^{2} d x^{2} - b^{2} d\right )} \operatorname {arcosh}\left (c x\right )^{2} + 2 \, {\left (a b c^{2} d x^{2} - a b d\right )} \operatorname {arcosh}\left (c x\right )\right )} \sqrt {-c^{2} d x^{2} + d}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-(a^2*c^2*d*x^2 - a^2*d + (b^2*c^2*d*x^2 - b^2*d)*arccosh(c*x)^2 + 2*(a*b*c^2*d*x^2 - a*b*d)*arccosh(
c*x))*sqrt(-c^2*d*x^2 + d), x)

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:sym2
poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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maple [B]  time = 0.38, size = 775, normalized size = 2.31 \[ \frac {x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} a^{2}}{4}+\frac {3 a^{2} d x \sqrt {-c^{2} d \,x^{2}+d}}{8}+\frac {3 a^{2} d^{2} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{8 \sqrt {c^{2} d}}+\frac {b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \,c^{3} \mathrm {arccosh}\left (c x \right ) x^{4}}{8 \sqrt {c x +1}\, \sqrt {c x -1}}-\frac {5 b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d c \,\mathrm {arccosh}\left (c x \right ) x^{2}}{8 \sqrt {c x +1}\, \sqrt {c x -1}}-\frac {b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right )^{3} d}{8 \sqrt {c x -1}\, \sqrt {c x +1}\, c}+\frac {17 b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \,\mathrm {arccosh}\left (c x \right )}{64 \sqrt {c x +1}\, \sqrt {c x -1}\, c}-\frac {17 b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d x}{64 \left (c x +1\right ) \left (c x -1\right )}-\frac {b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \,c^{4} \mathrm {arccosh}\left (c x \right )^{2} x^{5}}{4 \left (c x +1\right ) \left (c x -1\right )}+\frac {7 b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \,c^{2} \mathrm {arccosh}\left (c x \right )^{2} x^{3}}{8 \left (c x +1\right ) \left (c x -1\right )}-\frac {5 b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \mathrm {arccosh}\left (c x \right )^{2} x}{8 \left (c x +1\right ) \left (c x -1\right )}-\frac {b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \,c^{4} x^{5}}{32 \left (c x +1\right ) \left (c x -1\right )}+\frac {19 b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \,c^{2} x^{3}}{64 \left (c x +1\right ) \left (c x -1\right )}-\frac {3 a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right )^{2} d}{8 \sqrt {c x -1}\, \sqrt {c x +1}\, c}+\frac {a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \,c^{3} x^{4}}{8 \sqrt {c x +1}\, \sqrt {c x -1}}-\frac {5 a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d c \,x^{2}}{8 \sqrt {c x +1}\, \sqrt {c x -1}}-\frac {a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \,c^{4} \mathrm {arccosh}\left (c x \right ) x^{5}}{2 \left (c x +1\right ) \left (c x -1\right )}+\frac {7 a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \,c^{2} \mathrm {arccosh}\left (c x \right ) x^{3}}{4 \left (c x +1\right ) \left (c x -1\right )}-\frac {5 a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \,\mathrm {arccosh}\left (c x \right ) x}{4 \left (c x +1\right ) \left (c x -1\right )}+\frac {17 a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d}{64 \sqrt {c x +1}\, \sqrt {c x -1}\, c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*x*(-c^2*d*x^2+d)^(3/2)*a^2+3/8*a^2*d*x*(-c^2*d*x^2+d)^(1/2)+3/8*a^2*d^2/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)
*x/(-c^2*d*x^2+d)^(1/2))+1/8*b^2*(-d*(c^2*x^2-1))^(1/2)*d/(c*x+1)^(1/2)/(c*x-1)^(1/2)*c^3*arccosh(c*x)*x^4-5/8
*b^2*(-d*(c^2*x^2-1))^(1/2)*d/(c*x+1)^(1/2)/(c*x-1)^(1/2)*c*arccosh(c*x)*x^2-1/8*b^2*(-d*(c^2*x^2-1))^(1/2)/(c
*x-1)^(1/2)/(c*x+1)^(1/2)/c*arccosh(c*x)^3*d+17/64*b^2*(-d*(c^2*x^2-1))^(1/2)*d/(c*x+1)^(1/2)/(c*x-1)^(1/2)/c*
arccosh(c*x)-17/64*b^2*(-d*(c^2*x^2-1))^(1/2)*d/(c*x+1)/(c*x-1)*x-1/4*b^2*(-d*(c^2*x^2-1))^(1/2)*d/(c*x+1)/(c*
x-1)*c^4*arccosh(c*x)^2*x^5+7/8*b^2*(-d*(c^2*x^2-1))^(1/2)*d/(c*x+1)/(c*x-1)*c^2*arccosh(c*x)^2*x^3-5/8*b^2*(-
d*(c^2*x^2-1))^(1/2)*d/(c*x+1)/(c*x-1)*arccosh(c*x)^2*x-1/32*b^2*(-d*(c^2*x^2-1))^(1/2)*d/(c*x+1)/(c*x-1)*c^4*
x^5+19/64*b^2*(-d*(c^2*x^2-1))^(1/2)*d/(c*x+1)/(c*x-1)*c^2*x^3-3/8*a*b*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c
*x+1)^(1/2)/c*arccosh(c*x)^2*d+1/8*a*b*(-d*(c^2*x^2-1))^(1/2)*d/(c*x+1)^(1/2)/(c*x-1)^(1/2)*c^3*x^4-5/8*a*b*(-
d*(c^2*x^2-1))^(1/2)*d/(c*x+1)^(1/2)/(c*x-1)^(1/2)*c*x^2-1/2*a*b*(-d*(c^2*x^2-1))^(1/2)*d/(c*x+1)/(c*x-1)*c^4*
arccosh(c*x)*x^5+7/4*a*b*(-d*(c^2*x^2-1))^(1/2)*d/(c*x+1)/(c*x-1)*c^2*arccosh(c*x)*x^3-5/4*a*b*(-d*(c^2*x^2-1)
)^(1/2)*d/(c*x+1)/(c*x-1)*arccosh(c*x)*x+17/64*a*b*(-d*(c^2*x^2-1))^(1/2)*d/(c*x+1)^(1/2)/(c*x-1)^(1/2)/c

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{8} \, {\left (2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x + 3 \, \sqrt {-c^{2} d x^{2} + d} d x + \frac {3 \, d^{\frac {3}{2}} \arcsin \left (c x\right )}{c}\right )} a^{2} + \int {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} b^{2} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )^{2} + 2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} a b \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(2*(-c^2*d*x^2 + d)^(3/2)*x + 3*sqrt(-c^2*d*x^2 + d)*d*x + 3*d^(3/2)*arcsin(c*x)/c)*a^2 + integrate((-c^2*
d*x^2 + d)^(3/2)*b^2*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))^2 + 2*(-c^2*d*x^2 + d)^(3/2)*a*b*log(c*x + sqrt(c*
x + 1)*sqrt(c*x - 1)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^{3/2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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