3.3.3 \(\int \frac {(a+b \cosh ^{-1}(c x))^2}{x^4 \sqrt {d-c^2 d x^2}} \, dx\) [203]

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

[Out]

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

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Rubi [A]
time = 0.33, antiderivative size = 328, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.310, Rules used = {5932, 5917, 5882, 3799, 2221, 2317, 2438, 5883, 97} \begin {gather*} -\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 {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 {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}}+\frac {2 b^2 c^3 \sqrt {c x-1} \sqrt {c x+1} \text {Li}_2\left (-e^{-2 \cosh ^{-1}(c x)}\right )}{3 \sqrt {d-c^2 d x^2}} \end {gather*}

Antiderivative was successfully verified.

[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]) - (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])

Rule 97

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]

Rule 2221

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/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], 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 2317

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 2438

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 3799

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 5882

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> Dist[1/b, Subst[Int[x^n*Tanh[-a/b + x/b], x],
x, a + b*ArcCosh[c*x]], x] /; FreeQ[{a, b, c}, x] && IGtQ[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 5917

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

Rule 5932

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[
(f*x)^(m + 1)*(d + e*x^2)^(p + 1)*((a + b*ArcCosh[c*x])^n/(d*f*(m + 1))), x] + (Dist[c^2*((m + 2*p + 3)/(f^2*(
m + 1))), Int[(f*x)^(m + 2)*(d + e*x^2)^p*(a + b*ArcCosh[c*x])^n, x], x] + Dist[b*c*(n/(f*(m + 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*ArcCos
h[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && ILtQ[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 ) \text {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 ) \text {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 ) \text {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 ) \text {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*}

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Mathematica [A]
time = 0.89, size = 370, normalized size = 1.13 \begin {gather*} \frac {-a^2-a^2 c^2 x^2+b^2 c^2 x^2+2 a^2 c^4 x^4-b^2 c^4 x^4+a b c x \sqrt {-1+c x} \sqrt {1+c x}-b^2 (1+c x) \left (1-c x+2 c^2 x^2+2 c^3 x^3 \left (-1+\sqrt {\frac {-1+c x}{1+c x}}\right )\right ) \cosh ^{-1}(c x)^2+b (1+c x) \cosh ^{-1}(c x) \left (b c x \sqrt {\frac {-1+c x}{1+c x}}+2 a \left (-1+c x-2 c^2 x^2+2 c^3 x^3\right )-4 b c^3 x^3 \sqrt {\frac {-1+c x}{1+c x}} \log \left (1+e^{-2 \cosh ^{-1}(c x)}\right )\right )-4 a b c^3 x^3 \sqrt {-1+c x} \sqrt {1+c x} \log \left (-1+\sqrt {1+c x}\right )-4 a b c^3 x^3 \sqrt {-1+c x} \sqrt {1+c x} \log \left (1+\sqrt {1+c x}\right )+2 b^2 c^3 x^3 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \text {PolyLog}\left (2,-e^{-2 \cosh ^{-1}(c x)}\right )}{3 x^3 \sqrt {d-c^2 d x^2}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(2196\) vs. \(2(310)=620\).
time = 3.76, size = 2197, normalized size = 6.70

method	result	size
default	\(\text {Expression too large to display}\)	\(2197\)

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,method=_RETURNVERBOSE)

[Out]

-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-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)*(c*x-1)^(1/2)*arccosh(c*x)^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)*(c
*x-1)^(1/2)*arccosh(c*x)*(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+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(1+(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2)*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-
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)*(c*x-1)^(1/2)*arccosh(c*x)*(c*x+1)^(1/2)*c^3-1/3*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*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)*arccosh(c*x)*(c*x+1)*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*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*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)*arccosh(c*
x)*(c*x+1)^(1/2)*c^5+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(
1+(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^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^5*arccos
h(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+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*(c*x-1)^(1/2)*arccosh(c*x)^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*(c*x-1)*arccosh(c*x)*(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*(c*x-1)^(1/2)*
arccosh(c*x)*(c*x+1)^(1/2)*c-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*arcco
sh(c*x)*c^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)*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*a
rccosh(c*x)^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)+a^2*(-1/3/d/x^3*(-c^2*d*x^2+d)^(1/2)-2/3*c^2/d/x*
(-c^2*d*x^2+d)^(1/2))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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]

1/3*(4*c^2*sqrt(-d)*log(x)/d - sqrt(-d)/(d*x^2))*a*b*c - 2/3*a*b*(2*sqrt(-c^2*d*x^2 + d)*c^2/(d*x) + sqrt(-c^2
*d*x^2 + d)/(d*x^3))*arccosh(c*x) - 1/3*a^2*(2*sqrt(-c^2*d*x^2 + d)*c^2/(d*x) + sqrt(-c^2*d*x^2 + d)/(d*x^3))
+ b^2*integrate(log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))^2/(sqrt(-c^2*d*x^2 + d)*x^4), x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2}{x^4\,\sqrt {d-c^2\,d\,x^2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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