3.203 \(\int \frac{(a+b \cosh ^{-1}(c x))^2}{x^4 \sqrt{d-c^2 d x^2}} \, dx\)

Optimal. Leaf size=328 \[ \frac{2 b^2 c^3 \sqrt{c x-1} \sqrt{c x+1} \text{PolyLog}\left (2,-e^{-2 \cosh ^{-1}(c x)}\right )}{3 \sqrt{d-c^2 d x^2}}-\frac{2 c^3 \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 \sqrt{d-c^2 d x^2}}-\frac{2 c^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d x}+\frac{b c \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{3 x^2 \sqrt{d-c^2 d x^2}}-\frac{\sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d x^3}-\frac{4 b c^3 \sqrt{c x-1} \sqrt{c x+1} \log \left (e^{-2 \cosh ^{-1}(c x)}+1\right ) \left (a+b \cosh ^{-1}(c x)\right )}{3 \sqrt{d-c^2 d x^2}}+\frac{b^2 c^2 (1-c x) (c x+1)}{3 x \sqrt{d-c^2 d x^2}} \]

[Out]

(b^2*c^2*(1 - c*x)*(1 + c*x))/(3*x*Sqrt[d - c^2*d*x^2]) + (b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x
]))/(3*x^2*Sqrt[d - c^2*d*x^2]) - (2*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^2)/(3*Sqrt[d - c^2*
d*x^2]) - (Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(3*d*x^3) - (2*c^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c
*x])^2)/(3*d*x) - (4*b*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])*Log[1 + E^(-2*ArcCosh[c*x])])/(3*
Sqrt[d - c^2*d*x^2]) + (2*b^2*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[2, -E^(-2*ArcCosh[c*x])])/(3*Sqrt[d - c
^2*d*x^2])

________________________________________________________________________________________

Rubi [A]  time = 0.868904, antiderivative size = 344, normalized size of antiderivative = 1.05, number of steps used = 10, number of rules used = 10, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.345, Rules used = {5798, 5748, 5724, 5660, 3718, 2190, 2279, 2391, 5662, 95} \[ -\frac{2 b^2 c^3 \sqrt{c x-1} \sqrt{c x+1} \text{PolyLog}\left (2,-e^{2 \cosh ^{-1}(c x)}\right )}{3 \sqrt{d-c^2 d x^2}}+\frac{2 c^3 \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 \sqrt{d-c^2 d x^2}}-\frac{2 c^2 (1-c x) (c x+1) \left (a+b \cosh ^{-1}(c x)\right )^2}{3 x \sqrt{d-c^2 d x^2}}+\frac{b c \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{3 x^2 \sqrt{d-c^2 d x^2}}-\frac{(1-c x) (c x+1) \left (a+b \cosh ^{-1}(c x)\right )^2}{3 x^3 \sqrt{d-c^2 d x^2}}-\frac{4 b c^3 \sqrt{c x-1} \sqrt{c x+1} \log \left (e^{2 \cosh ^{-1}(c x)}+1\right ) \left (a+b \cosh ^{-1}(c x)\right )}{3 \sqrt{d-c^2 d x^2}}+\frac{b^2 c^2 (1-c x) (c x+1)}{3 x \sqrt{d-c^2 d x^2}} \]

Warning: Unable to verify antiderivative.

[In]

Int[(a + b*ArcCosh[c*x])^2/(x^4*Sqrt[d - c^2*d*x^2]),x]

[Out]

(b^2*c^2*(1 - c*x)*(1 + c*x))/(3*x*Sqrt[d - c^2*d*x^2]) + (b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x
]))/(3*x^2*Sqrt[d - c^2*d*x^2]) + (2*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^2)/(3*Sqrt[d - c^2*
d*x^2]) - ((1 - c*x)*(1 + c*x)*(a + b*ArcCosh[c*x])^2)/(3*x^3*Sqrt[d - c^2*d*x^2]) - (2*c^2*(1 - c*x)*(1 + c*x
)*(a + b*ArcCosh[c*x])^2)/(3*x*Sqrt[d - c^2*d*x^2]) - (4*b*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x
])*Log[1 + E^(2*ArcCosh[c*x])])/(3*Sqrt[d - c^2*d*x^2]) - (2*b^2*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[2, -
E^(2*ArcCosh[c*x])])/(3*Sqrt[d - c^2*d*x^2])

Rule 5798

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_.)*((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[(f*x)^m*(1 + c*
x)^p*(-1 + c*x)^p*(a + b*ArcCosh[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[c^2*d + e, 0]
&&  !IntegerQ[p]

Rule 5748

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d1_) + (e1_.)*(x_))^(p_)*((d2_) + (e2_.)*(x_
))^(p_), x_Symbol] :> Simp[((f*x)^(m + 1)*(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*(a + b*ArcCosh[c*x])^n)/(d1*
d2*f*(m + 1)), x] + (Dist[(c^2*(m + 2*p + 3))/(f^2*(m + 1)), Int[(f*x)^(m + 2)*(d1 + e1*x)^p*(d2 + e2*x)^p*(a
+ b*ArcCosh[c*x])^n, x], x] + Dist[(b*c*n*(-(d1*d2))^IntPart[p]*(d1 + e1*x)^FracPart[p]*(d2 + e2*x)^FracPart[p
])/(f*(m + 1)*(1 + c*x)^FracPart[p]*(-1 + c*x)^FracPart[p]), Int[(f*x)^(m + 1)*(-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, f, p}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 +
c*d2, 0] && GtQ[n, 0] && LtQ[m, -1] && IntegerQ[m] && IntegerQ[p + 1/2]

Rule 5724

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d1_) + (e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(x
_))^(p_.), x_Symbol] :> Simp[((f*x)^(m + 1)*(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*(a + b*ArcCosh[c*x])^n)/(d
1*d2*f*(m + 1)), x] + Dist[(b*c*n*(-(d1*d2))^IntPart[p]*(d1 + e1*x)^FracPart[p]*(d2 + e2*x)^FracPart[p])/(f*(m
+ 1)*(1 + c*x)^FracPart[p]*(-1 + c*x)^FracPart[p]), Int[(f*x)^(m + 1)*(-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, f, m, p}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2,
0] && GtQ[n, 0] && EqQ[m + 2*p + 3, 0] && NeQ[m, -1] && IntegerQ[p + 1/2]

Rule 5660

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> Subst[Int[(a + b*x)^n/Coth[x], x], x, ArcCosh
[c*x]] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0]

Rule 3718

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (Complex[0, fz_])*(f_.)*(x_)], x_Symbol] :> -Simp[(I*(c + d*x)^(m +
1))/(d*(m + 1)), x] + Dist[2*I, Int[((c + d*x)^m*E^(2*(-(I*e) + f*fz*x)))/(1 + E^(2*(-(I*e) + f*fz*x))), x],
x] /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]

Rule 2190

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), x]
- Dist[(d*m)/(b*f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*Log[1 + (b*(F^(g*(e + f*x)))^n)/a], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2279

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

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 95

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))/((m + 1)*(b*c - a*d)*(b*e - a*f)), x] /; FreeQ[{a, b, c, d,
e, f, m, n, p}, x] && EqQ[Simplify[m + n + p + 3], 0] && EqQ[a*d*f*(m + 1) + b*c*f*(n + 1) + b*d*e*(p + 1), 0
] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{x^4 \sqrt{d-c^2 d x^2}} \, dx &=\frac{\left (\sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{x^4 \sqrt{-1+c x} \sqrt{1+c x}} \, dx}{\sqrt{d-c^2 d x^2}}\\ &=-\frac{(1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{3 x^3 \sqrt{d-c^2 d x^2}}-\frac{\left (2 b c \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{a+b \cosh ^{-1}(c x)}{x^3} \, dx}{3 \sqrt{d-c^2 d x^2}}+\frac{\left (2 c^2 \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{x^2 \sqrt{-1+c x} \sqrt{1+c x}} \, dx}{3 \sqrt{d-c^2 d x^2}}\\ &=\frac{b c \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 x^2 \sqrt{d-c^2 d x^2}}-\frac{(1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{3 x^3 \sqrt{d-c^2 d x^2}}-\frac{2 c^2 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{3 x \sqrt{d-c^2 d x^2}}-\frac{\left (b^2 c^2 \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{1}{x^2 \sqrt{-1+c x} \sqrt{1+c x}} \, dx}{3 \sqrt{d-c^2 d x^2}}-\frac{\left (4 b c^3 \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{a+b \cosh ^{-1}(c x)}{x} \, dx}{3 \sqrt{d-c^2 d x^2}}\\ &=\frac{b^2 c^2 (1-c x) (1+c x)}{3 x \sqrt{d-c^2 d x^2}}+\frac{b c \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 x^2 \sqrt{d-c^2 d x^2}}-\frac{(1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{3 x^3 \sqrt{d-c^2 d x^2}}-\frac{2 c^2 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{3 x \sqrt{d-c^2 d x^2}}-\frac{\left (4 b c^3 \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int (a+b x) \tanh (x) \, dx,x,\cosh ^{-1}(c x)\right )}{3 \sqrt{d-c^2 d x^2}}\\ &=\frac{b^2 c^2 (1-c x) (1+c x)}{3 x \sqrt{d-c^2 d x^2}}+\frac{b c \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 x^2 \sqrt{d-c^2 d x^2}}+\frac{2 c^3 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 \sqrt{d-c^2 d x^2}}-\frac{(1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{3 x^3 \sqrt{d-c^2 d x^2}}-\frac{2 c^2 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{3 x \sqrt{d-c^2 d x^2}}-\frac{\left (8 b c^3 \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \frac{e^{2 x} (a+b x)}{1+e^{2 x}} \, dx,x,\cosh ^{-1}(c x)\right )}{3 \sqrt{d-c^2 d x^2}}\\ &=\frac{b^2 c^2 (1-c x) (1+c x)}{3 x \sqrt{d-c^2 d x^2}}+\frac{b c \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 x^2 \sqrt{d-c^2 d x^2}}+\frac{2 c^3 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 \sqrt{d-c^2 d x^2}}-\frac{(1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{3 x^3 \sqrt{d-c^2 d x^2}}-\frac{2 c^2 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{3 x \sqrt{d-c^2 d x^2}}-\frac{4 b c^3 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+e^{2 \cosh ^{-1}(c x)}\right )}{3 \sqrt{d-c^2 d x^2}}+\frac{\left (4 b^2 c^3 \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{3 \sqrt{d-c^2 d x^2}}\\ &=\frac{b^2 c^2 (1-c x) (1+c x)}{3 x \sqrt{d-c^2 d x^2}}+\frac{b c \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 x^2 \sqrt{d-c^2 d x^2}}+\frac{2 c^3 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 \sqrt{d-c^2 d x^2}}-\frac{(1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{3 x^3 \sqrt{d-c^2 d x^2}}-\frac{2 c^2 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{3 x \sqrt{d-c^2 d x^2}}-\frac{4 b c^3 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+e^{2 \cosh ^{-1}(c x)}\right )}{3 \sqrt{d-c^2 d x^2}}+\frac{\left (2 b^2 c^3 \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{2 \cosh ^{-1}(c x)}\right )}{3 \sqrt{d-c^2 d x^2}}\\ &=\frac{b^2 c^2 (1-c x) (1+c x)}{3 x \sqrt{d-c^2 d x^2}}+\frac{b c \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 x^2 \sqrt{d-c^2 d x^2}}+\frac{2 c^3 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 \sqrt{d-c^2 d x^2}}-\frac{(1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{3 x^3 \sqrt{d-c^2 d x^2}}-\frac{2 c^2 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{3 x \sqrt{d-c^2 d x^2}}-\frac{4 b c^3 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+e^{2 \cosh ^{-1}(c x)}\right )}{3 \sqrt{d-c^2 d x^2}}-\frac{2 b^2 c^3 \sqrt{-1+c x} \sqrt{1+c x} \text{Li}_2\left (-e^{2 \cosh ^{-1}(c x)}\right )}{3 \sqrt{d-c^2 d x^2}}\\ \end{align*}

Mathematica [A]  time = 1.57971, size = 346, normalized size = 1.05 \[ \frac{\frac{2 b^2 c^3 x^3 (c x-1)^{3/2} \text{PolyLog}\left (2,-e^{-2 \cosh ^{-1}(c x)}\right )}{\sqrt{\frac{c x-1}{c x+1}}}+(c x-1) \sqrt{c x+1} \left (a^2 \sqrt{c x-1} \sqrt{c x+1} \left (2 c^2 x^2+1\right )-4 a b c^3 x^3 \log (c x)+a b c x-b^2 c^2 x^2 \sqrt{c x-1} \sqrt{c x+1}\right )+b \sqrt{c x-1} (c x+1) \cosh ^{-1}(c x) \left (2 a \left (2 c^3 x^3-2 c^2 x^2+c x-1\right )-4 b c^3 x^3 \sqrt{\frac{c x-1}{c x+1}} \log \left (e^{-2 \cosh ^{-1}(c x)}+1\right )+b c x \sqrt{\frac{c x-1}{c x+1}}\right )-b^2 \sqrt{c x-1} (c x+1) \left (2 c^3 x^3 \left (\sqrt{\frac{c x-1}{c x+1}}-1\right )+2 c^2 x^2-c x+1\right ) \cosh ^{-1}(c x)^2}{3 x^3 \sqrt{c x-1} \sqrt{d-c^2 d x^2}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + b*ArcCosh[c*x])^2/(x^4*Sqrt[d - c^2*d*x^2]),x]

[Out]

(-(b^2*Sqrt[-1 + c*x]*(1 + c*x)*(1 - c*x + 2*c^2*x^2 + 2*c^3*x^3*(-1 + Sqrt[(-1 + c*x)/(1 + c*x)]))*ArcCosh[c*
x]^2) + b*Sqrt[-1 + c*x]*(1 + c*x)*ArcCosh[c*x]*(b*c*x*Sqrt[(-1 + c*x)/(1 + c*x)] + 2*a*(-1 + c*x - 2*c^2*x^2
+ 2*c^3*x^3) - 4*b*c^3*x^3*Sqrt[(-1 + c*x)/(1 + c*x)]*Log[1 + E^(-2*ArcCosh[c*x])]) + (-1 + c*x)*Sqrt[1 + c*x]
*(a*b*c*x - b^2*c^2*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x] + a^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(1 + 2*c^2*x^2) - 4*a*
b*c^3*x^3*Log[c*x]) + (2*b^2*c^3*x^3*(-1 + c*x)^(3/2)*PolyLog[2, -E^(-2*ArcCosh[c*x])])/Sqrt[(-1 + c*x)/(1 + c
*x)])/(3*x^3*Sqrt[-1 + c*x]*Sqrt[d - c^2*d*x^2])

________________________________________________________________________________________

Maple [B]  time = 0.338, size = 2198, normalized size = 6.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arccosh(c*x))^2/x^4/(-c^2*d*x^2+d)^(1/2),x)

[Out]

-2/3*a^2*c^2/d/x*(-c^2*d*x^2+d)^(1/2)+2/3*b^2*(-d*(c^2*x^2-1))^(1/2)/d/(3*c^4*x^4-2*c^2*x^2-1)*x^5*c^8-1/3*b^2
*(-d*(c^2*x^2-1))^(1/2)/d/(3*c^4*x^4-2*c^2*x^2-1)*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^3+4/3*b^2*(-d*(c^2*x^2-1))^(1/
2)/d/(3*c^4*x^4-2*c^2*x^2-1)/x*arccosh(c*x)^2*c^2+4/3*a*b*(-d*(c^2*x^2-1))^(1/2)/d/(3*c^4*x^4-2*c^2*x^2-1)*x^5
*c^8-2/3*a*b*(-d*(c^2*x^2-1))^(1/2)/d/(3*c^4*x^4-2*c^2*x^2-1)*x^3*c^6-2/3*a*b*(-d*(c^2*x^2-1))^(1/2)/d/(3*c^4*
x^4-2*c^2*x^2-1)*x*c^4+2/3*a*b*(-d*(c^2*x^2-1))^(1/2)/d/(3*c^4*x^4-2*c^2*x^2-1)/x^3*arccosh(c*x)+4/3*b^2*(-d*(
c^2*x^2-1))^(1/2)/d/(3*c^4*x^4-2*c^2*x^2-1)*x^5*arccosh(c*x)*c^8-2*b^2*(-d*(c^2*x^2-1))^(1/2)/d/(3*c^4*x^4-2*c
^2*x^2-1)*x^3*arccosh(c*x)^2*c^6-2/3*b^2*(-d*(c^2*x^2-1))^(1/2)/d/(3*c^4*x^4-2*c^2*x^2-1)*x^3*arccosh(c*x)*c^6
+1/3*b^2*(-d*(c^2*x^2-1))^(1/2)/d/(3*c^4*x^4-2*c^2*x^2-1)*x*arccosh(c*x)^2*c^4-2/3*b^2*(-d*(c^2*x^2-1))^(1/2)/
d/(3*c^4*x^4-2*c^2*x^2-1)*x*arccosh(c*x)*c^4-1/3*a^2/d/x^3*(-c^2*d*x^2+d)^(1/2)+4*a*b*(-d*(c^2*x^2-1))^(1/2)/d
/(3*c^4*x^4-2*c^2*x^2-1)*x^2*arccosh(c*x)*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^5-4/3*b^2*(-d*(c^2*x^2-1))^(1/2)/d/(3*
c^4*x^4-2*c^2*x^2-1)*x^3*arccosh(c*x)*(c*x+1)*(c*x-1)*c^6+2*b^2*(-d*(c^2*x^2-1))^(1/2)/d/(3*c^4*x^4-2*c^2*x^2-
1)*x^2*arccosh(c*x)^2*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^5-2/3*b^2*(-d*(c^2*x^2-1))^(1/2)/d/(3*c^4*x^4-2*c^2*x^2-1)
*x*arccosh(c*x)*(c*x+1)*(c*x-1)*c^4-1/3*b^2*(-d*(c^2*x^2-1))^(1/2)/d/(3*c^4*x^4-2*c^2*x^2-1)/x^2*arccosh(c*x)*
(c*x+1)^(1/2)*(c*x-1)^(1/2)*c+4/3*b^2*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d/(c^2*x^2-1)*arccosh
(c*x)*ln((c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2+1)*c^3-8/3*a*b*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)
/d/(c^2*x^2-1)*arccosh(c*x)*c^3-4/3*a*b*(-d*(c^2*x^2-1))^(1/2)/d/(3*c^4*x^4-2*c^2*x^2-1)*x^3*(c*x+1)*(c*x-1)*c
^6-1/3*a*b*(-d*(c^2*x^2-1))^(1/2)/d/(3*c^4*x^4-2*c^2*x^2-1)/x^2*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c+4/3*a*b*(-d*(c^2
*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d/(c^2*x^2-1)*ln((c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2+1)*c^3-2/3*a*b
*(-d*(c^2*x^2-1))^(1/2)/d/(3*c^4*x^4-2*c^2*x^2-1)*x*(c*x+1)*(c*x-1)*c^4+4/3*a*b*(-d*(c^2*x^2-1))^(1/2)/d/(3*c^
4*x^4-2*c^2*x^2-1)*arccosh(c*x)*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^3-b^2*(-d*(c^2*x^2-1))^(1/2)/d/(3*c^4*x^4-2*c^2*
x^2-1)*x^2*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^5+2/3*b^2*(-d*(c^2*x^2-1))^(1/2)/d/(3*c^4*x^4-2*c^2*x^2-1)*arccosh(c*
x)^2*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^3-b^2*(-d*(c^2*x^2-1))^(1/2)/d/(3*c^4*x^4-2*c^2*x^2-1)*arccosh(c*x)*(c*x+1)
^(1/2)*(c*x-1)^(1/2)*c^3-4/3*b^2*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d/(c^2*x^2-1)*arccosh(c*x)
^2*c^3+2/3*b^2*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d/(c^2*x^2-1)*polylog(2,-(c*x+(c*x-1)^(1/2)*
(c*x+1)^(1/2))^2)*c^3-2/3*b^2*(-d*(c^2*x^2-1))^(1/2)/d/(3*c^4*x^4-2*c^2*x^2-1)*x^3*(c*x+1)*(c*x-1)*c^6-4*a*b*(
-d*(c^2*x^2-1))^(1/2)/d/(3*c^4*x^4-2*c^2*x^2-1)*x^3*arccosh(c*x)*c^6+2/3*a*b*(-d*(c^2*x^2-1))^(1/2)/d/(3*c^4*x
^4-2*c^2*x^2-1)*x*arccosh(c*x)*c^4-a*b*(-d*(c^2*x^2-1))^(1/2)/d/(3*c^4*x^4-2*c^2*x^2-1)*(c*x+1)^(1/2)*(c*x-1)^
(1/2)*c^3+8/3*a*b*(-d*(c^2*x^2-1))^(1/2)/d/(3*c^4*x^4-2*c^2*x^2-1)/x*arccosh(c*x)*c^2+1/3*b^2*(-d*(c^2*x^2-1))
^(1/2)/d/(3*c^4*x^4-2*c^2*x^2-1)*x^3*c^6-2/3*b^2*(-d*(c^2*x^2-1))^(1/2)/d/(3*c^4*x^4-2*c^2*x^2-1)*x*c^4-1/3*b^
2*(-d*(c^2*x^2-1))^(1/2)/d/(3*c^4*x^4-2*c^2*x^2-1)/x*c^2+1/3*b^2*(-d*(c^2*x^2-1))^(1/2)/d/(3*c^4*x^4-2*c^2*x^2
-1)/x^3*arccosh(c*x)^2

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccosh(c*x))^2/x^4/(-c^2*d*x^2+d)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-c^{2} d x^{2} + d}{\left (b^{2} \operatorname{arcosh}\left (c x\right )^{2} + 2 \, a b \operatorname{arcosh}\left (c x\right ) + a^{2}\right )}}{c^{2} d x^{6} - d x^{4}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccosh(c*x))^2/x^4/(-c^2*d*x^2+d)^(1/2),x, algorithm="fricas")

[Out]

integral(-sqrt(-c^2*d*x^2 + d)*(b^2*arccosh(c*x)^2 + 2*a*b*arccosh(c*x) + a^2)/(c^2*d*x^6 - d*x^4), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b \operatorname{acosh}{\left (c x \right )}\right )^{2}}{x^{4} \sqrt{- d \left (c x - 1\right ) \left (c x + 1\right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*acosh(c*x))**2/x**4/(-c**2*d*x**2+d)**(1/2),x)

[Out]

Integral((a + b*acosh(c*x))**2/(x**4*sqrt(-d*(c*x - 1)*(c*x + 1))), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \operatorname{arcosh}\left (c x\right ) + a\right )}^{2}}{\sqrt{-c^{2} d x^{2} + d} x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccosh(c*x))^2/x^4/(-c^2*d*x^2+d)^(1/2),x, algorithm="giac")

[Out]

integrate((b*arccosh(c*x) + a)^2/(sqrt(-c^2*d*x^2 + d)*x^4), x)