3.219 \(\int \frac {(a+b \cosh ^{-1}(c x))^2}{(d-c^2 d x^2)^{5/2}} \, dx\)

Optimal. Leaf size=331 \[ \frac {b \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{3 c d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {2 x \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c d^2 \sqrt {d-c^2 d x^2}}-\frac {4 b \sqrt {c x-1} \sqrt {c x+1} \log \left (1-e^{2 \cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}+\frac {x \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {2 b^2 \sqrt {c x-1} \sqrt {c x+1} \text {Li}_2\left (e^{2 \cosh ^{-1}(c x)}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}-\frac {b^2 x}{3 d^2 \sqrt {d-c^2 d x^2}} \]

[Out]

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

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Rubi [A]  time = 0.56, antiderivative size = 346, normalized size of antiderivative = 1.05, number of steps used = 10, number of rules used = 10, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {5713, 5691, 5688, 5715, 3716, 2190, 2279, 2391, 5716, 39} \[ -\frac {2 b^2 \sqrt {c x-1} \sqrt {c x+1} \text {PolyLog}\left (2,e^{2 \cosh ^{-1}(c x)}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{3 c d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {2 x \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c d^2 \sqrt {d-c^2 d x^2}}+\frac {x \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d^2 (1-c x) (c x+1) \sqrt {d-c^2 d x^2}}-\frac {4 b \sqrt {c x-1} \sqrt {c x+1} \log \left (1-e^{2 \cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}-\frac {b^2 x}{3 d^2 \sqrt {d-c^2 d x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 39

Int[1/(((a_) + (b_.)*(x_))^(3/2)*((c_) + (d_.)*(x_))^(3/2)), x_Symbol] :> Simp[x/(a*c*Sqrt[a + b*x]*Sqrt[c + d
*x]), x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c + a*d, 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 3716

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (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)))/(E^(2*I*k*Pi)*(1 + E^(2*
(-(I*e) + f*fz*x))/E^(2*I*k*Pi))), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[4*k] && IGtQ[m, 0]

Rule 5688

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(((d1_) + (e1_.)*(x_))^(3/2)*((d2_) + (e2_.)*(x_))^(3/2)), x_Sym
bol] :> Simp[(x*(a + b*ArcCosh[c*x])^n)/(d1*d2*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x]), x] + Dist[(b*c*n*Sqrt[1 + c*x
]*Sqrt[-1 + c*x])/(d1*d2*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x]), Int[(x*(a + b*ArcCosh[c*x])^(n - 1))/(1 - c^2*x^2),
x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2}, x] && EqQ[e1, c*d1] && EqQ[e2, -(c*d2)] && GtQ[n, 0]

Rule 5691

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d1_) + (e1_.)*(x_))^(p_)*((d2_) + (e2_.)*(x_))^(p_), x_Symbol]
:> -Simp[(x*(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*(a + b*ArcCosh[c*x])^n)/(2*d1*d2*(p + 1)), x] + (Dist[(2*
p + 3)/(2*d1*d2*(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[1 + c*x]*Sqrt[-1 + c*x])/(2*(p + 1)*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*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] && LtQ[p, -1] && NeQ[p, -3/2] && 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 5715

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[1/e, Subst[Int[(
a + b*x)^n*Coth[x], x], x, ArcCosh[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0]

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 \frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx &=\frac {\left (\sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{(-1+c x)^{5/2} (1+c x)^{5/2}} \, dx}{d^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {x \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}-\frac {\left (2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (2 b c \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x \left (a+b \cosh ^{-1}(c x)\right )}{\left (-1+c^2 x^2\right )^2} \, dx}{3 d^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {b \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 c d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {2 x \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {x \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}+\frac {\left (b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {1}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (4 b c \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x \left (a+b \cosh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx}{3 d^2 \sqrt {d-c^2 d x^2}}\\ &=-\frac {b^2 x}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 c d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {2 x \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {x \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}-\frac {\left (4 b \sqrt {-1+c x} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int (a+b x) \coth (x) \, dx,x,\cosh ^{-1}(c x)\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}\\ &=-\frac {b^2 x}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 c d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {2 x \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {x \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}+\frac {2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (8 b \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 c d^2 \sqrt {d-c^2 d x^2}}\\ &=-\frac {b^2 x}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 c d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {2 x \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {x \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}+\frac {2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c d^2 \sqrt {d-c^2 d x^2}}-\frac {4 b \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 c d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (4 b^2 \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 c d^2 \sqrt {d-c^2 d x^2}}\\ &=-\frac {b^2 x}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 c d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {2 x \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {x \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}+\frac {2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c d^2 \sqrt {d-c^2 d x^2}}-\frac {4 b \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 c d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (2 b^2 \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 c d^2 \sqrt {d-c^2 d x^2}}\\ &=-\frac {b^2 x}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 c d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {2 x \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {x \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}+\frac {2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c d^2 \sqrt {d-c^2 d x^2}}-\frac {4 b \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 c d^2 \sqrt {d-c^2 d x^2}}-\frac {2 b^2 \sqrt {-1+c x} \sqrt {1+c x} \text {Li}_2\left (e^{2 \cosh ^{-1}(c x)}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}\\ \end {align*}

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Mathematica [A]  time = 1.52, size = 289, normalized size = 0.87 \[ \frac {\frac {a^2 c x \left (2 c^2 x^2-3\right )}{c^2 x^2-1}+a b \left (\frac {\sqrt {\frac {c x-1}{c x+1}} \left (\left (4-4 c^2 x^2\right ) \log \left (\sqrt {\frac {c x-1}{c x+1}} (c x+1)\right )-1\right )}{c x-1}+2 c x \left (\frac {1}{1-c^2 x^2}+2\right ) \cosh ^{-1}(c x)\right )+b^2 \left (-\frac {\cosh ^{-1}(c x) \left (\sqrt {\frac {c x-1}{c x+1}} (c x+1)+c x \cosh ^{-1}(c x)\right )}{c^2 x^2-1}+2 \sqrt {\frac {c x-1}{c x+1}} (c x+1) \text {Li}_2\left (e^{-2 \cosh ^{-1}(c x)}\right )+c x \left (2 \cosh ^{-1}(c x)^2-1\right )-2 \sqrt {\frac {c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x) \left (\cosh ^{-1}(c x)+2 \log \left (1-e^{-2 \cosh ^{-1}(c x)}\right )\right )\right )}{3 c d^2 \sqrt {d-c^2 d x^2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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fricas [F]  time = 0.73, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-c^{2} d x^{2} + d} {\left (b^{2} \operatorname {arcosh}\left (c x\right )^{2} + 2 \, a b \operatorname {arcosh}\left (c x\right ) + a^{2}\right )}}{c^{6} d^{3} x^{6} - 3 \, c^{4} d^{3} x^{4} + 3 \, c^{2} d^{3} x^{2} - d^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccosh(c*x))^2/(-c^2*d*x^2+d)^(5/2),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^6*d^3*x^6 - 3*c^4*d^3*x^4 +
3*c^2*d^3*x^2 - d^3), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*arccosh(c*x) + a)^2/(-c^2*d*x^2 + d)^(5/2), x)

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maple [B]  time = 0.44, size = 3050, normalized size = 9.21 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/3*a*b*(-d*(c^2*x^2-1))^(1/2)/(3*c^6*x^6-10*c^4*x^4+11*c^2*x^2-4)*c^6/d^3*x^7+14/3*a*b*(-d*(c^2*x^2-1))^(1/2
)/(3*c^6*x^6-10*c^4*x^4+11*c^2*x^2-4)*c^4/d^3*x^5-16/3*a*b*(-d*(c^2*x^2-1))^(1/2)/(3*c^6*x^6-10*c^4*x^4+11*c^2
*x^2-4)*c^2/d^3*x^3-8*a*b*(-d*(c^2*x^2-1))^(1/2)/(3*c^6*x^6-10*c^4*x^4+11*c^2*x^2-4)/d^3*arccosh(c*x)*x+2/3*b^
2*(-d*(c^2*x^2-1))^(1/2)/(3*c^6*x^6-10*c^4*x^4+11*c^2*x^2-4)/d^3*(c*x+1)*(c*x-1)*x-4/3*b^2*(-d*(c^2*x^2-1))^(1
/2)/(3*c^6*x^6-10*c^4*x^4+11*c^2*x^2-4)*c^6/d^3*arccosh(c*x)*x^7-2*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^6*x^6-10*c^
4*x^4+11*c^2*x^2-4)*c^4/d^3*arccosh(c*x)^2*x^5+14/3*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^6*x^6-10*c^4*x^4+11*c^2*x^
2-4)*c^4/d^3*arccosh(c*x)*x^5+17/3*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^6*x^6-10*c^4*x^4+11*c^2*x^2-4)*c^2/d^3*arcc
osh(c*x)^2*x^3-16/3*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^6*x^6-10*c^4*x^4+11*c^2*x^2-4)*c^2/d^3*arccosh(c*x)*x^3-4/
3*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^6*x^6-10*c^4*x^4+11*c^2*x^2-4)/c/d^3*(c*x-1)^(1/2)*(c*x+1)^(1/2)+2*b^2*(-d*(
c^2*x^2-1))^(1/2)/(3*c^6*x^6-10*c^4*x^4+11*c^2*x^2-4)/d^3*x-28/3*a*b*(-d*(c^2*x^2-1))^(1/2)/(3*c^6*x^6-10*c^4*
x^4+11*c^2*x^2-4)*c/d^3*arccosh(c*x)*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^2+4*a*b*(-d*(c^2*x^2-1))^(1/2)/(3*c^6*x^6-1
0*c^4*x^4+11*c^2*x^2-4)*c^3/d^3*arccosh(c*x)*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^4+34/3*a*b*(-d*(c^2*x^2-1))^(1/2)/(
3*c^6*x^6-10*c^4*x^4+11*c^2*x^2-4)*c^2/d^3*arccosh(c*x)*x^3+2*a*b*(-d*(c^2*x^2-1))^(1/2)/(3*c^6*x^6-10*c^4*x^4
+11*c^2*x^2-4)/d^3*(c*x+1)*(c*x-1)*x-4/3*a*b*(-d*(c^2*x^2-1))^(1/2)/(3*c^6*x^6-10*c^4*x^4+11*c^2*x^2-4)/c/d^3*
(c*x-1)^(1/2)*(c*x+1)^(1/2)+2*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^6*x^6-10*c^4*x^4+11*c^2*x^2-4)/d^3*arccosh(c*x)*
(c*x+1)*(c*x-1)*x+7/3*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^6*x^6-10*c^4*x^4+11*c^2*x^2-4)*c/d^3*(c*x+1)^(1/2)*(c*x-
1)^(1/2)*x^2+8/3*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^6*x^6-10*c^4*x^4+11*c^2*x^2-4)/c/d^3*arccosh(c*x)^2*(c*x+1)^(
1/2)*(c*x-1)^(1/2)-4/3*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^6*x^6-10*c^4*x^4+11*c^2*x^2-4)*c^2/d^3*(c*x+1)*(c*x-1)*
x^3-b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^6*x^6-10*c^4*x^4+11*c^2*x^2-4)*c^3/d^3*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^4+4/3
*b^2*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d^3/c/(c^2*x^2-1)*polylog(2,c*x+(c*x-1)^(1/2)*(c*x+1)^
(1/2))-4/3*b^2*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d^3/c/(c^2*x^2-1)*arccosh(c*x)^2-4/3*b^2*(-d
*(c^2*x^2-1))^(1/2)/(3*c^6*x^6-10*c^4*x^4+11*c^2*x^2-4)/c/d^3*arccosh(c*x)*(c*x+1)^(1/2)*(c*x-1)^(1/2)+2/3*b^2
*(-d*(c^2*x^2-1))^(1/2)/(3*c^6*x^6-10*c^4*x^4+11*c^2*x^2-4)*c^4/d^3*(c*x+1)*(c*x-1)*x^5+4/3*b^2*(-d*(c^2*x^2-1
))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d^3/c/(c^2*x^2-1)*polylog(2,-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))-4*a*b*(-d*(
c^2*x^2-1))^(1/2)/(3*c^6*x^6-10*c^4*x^4+11*c^2*x^2-4)*c^4/d^3*arccosh(c*x)*x^5+1/3*a^2/d*x/(-c^2*d*x^2+d)^(3/2
)+2/3*a^2/d^2*x/(-c^2*d*x^2+d)^(1/2)-4*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^6*x^6-10*c^4*x^4+11*c^2*x^2-4)/d^3*arcc
osh(c*x)^2*x+2*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^6*x^6-10*c^4*x^4+11*c^2*x^2-4)/d^3*arccosh(c*x)*x-2/3*b^2*(-d*(
c^2*x^2-1))^(1/2)/(3*c^6*x^6-10*c^4*x^4+11*c^2*x^2-4)*c^6/d^3*x^7+3*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^6*x^6-10*c
^4*x^4+11*c^2*x^2-4)*c^4/d^3*x^5-13/3*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^6*x^6-10*c^4*x^4+11*c^2*x^2-4)*c^2/d^3*x
^3+2*a*b*(-d*(c^2*x^2-1))^(1/2)/(3*c^6*x^6-10*c^4*x^4+11*c^2*x^2-4)/d^3*x+b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^6*x^
6-10*c^4*x^4+11*c^2*x^2-4)*c/d^3*arccosh(c*x)*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^2+4/3*b^2*(-d*(c^2*x^2-1))^(1/2)*(
c*x-1)^(1/2)*(c*x+1)^(1/2)/d^3/c/(c^2*x^2-1)*arccosh(c*x)*ln(1-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))+4/3*b^2*(-d*(c
^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d^3/c/(c^2*x^2-1)*arccosh(c*x)*ln(1+c*x+(c*x-1)^(1/2)*(c*x+1)^(1/
2))+4/3*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^6*x^6-10*c^4*x^4+11*c^2*x^2-4)*c^4/d^3*arccosh(c*x)*(c*x+1)*(c*x-1)*x^
5+2*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^6*x^6-10*c^4*x^4+11*c^2*x^2-4)*c^3/d^3*arccosh(c*x)^2*(c*x+1)^(1/2)*(c*x-1
)^(1/2)*x^4-10/3*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^6*x^6-10*c^4*x^4+11*c^2*x^2-4)*c^2/d^3*arccosh(c*x)*(c*x+1)*(
c*x-1)*x^3-14/3*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^6*x^6-10*c^4*x^4+11*c^2*x^2-4)*c/d^3*arccosh(c*x)^2*(c*x+1)^(1
/2)*(c*x-1)^(1/2)*x^2-8/3*a*b*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d^3/c/(c^2*x^2-1)*arccosh(c*x
)+4/3*a*b*(-d*(c^2*x^2-1))^(1/2)/(3*c^6*x^6-10*c^4*x^4+11*c^2*x^2-4)*c^4/d^3*(c*x+1)*(c*x-1)*x^5-10/3*a*b*(-d*
(c^2*x^2-1))^(1/2)/(3*c^6*x^6-10*c^4*x^4+11*c^2*x^2-4)*c^2/d^3*(c*x+1)*(c*x-1)*x^3+a*b*(-d*(c^2*x^2-1))^(1/2)/
(3*c^6*x^6-10*c^4*x^4+11*c^2*x^2-4)*c/d^3*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^2+16/3*a*b*(-d*(c^2*x^2-1))^(1/2)/(3*c
^6*x^6-10*c^4*x^4+11*c^2*x^2-4)/c/d^3*arccosh(c*x)*(c*x+1)^(1/2)*(c*x-1)^(1/2)+4/3*a*b*(-d*(c^2*x^2-1))^(1/2)*
(c*x-1)^(1/2)*(c*x+1)^(1/2)/d^3/c/(c^2*x^2-1)*ln((c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2-1)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{3} \, a b c {\left (\frac {\sqrt {-d}}{c^{4} d^{3} x^{2} - c^{2} d^{3}} + \frac {2 \, \sqrt {-d} \log \left (c x + 1\right )}{c^{2} d^{3}} + \frac {2 \, \sqrt {-d} \log \left (c x - 1\right )}{c^{2} d^{3}}\right )} + \frac {2}{3} \, a b {\left (\frac {2 \, x}{\sqrt {-c^{2} d x^{2} + d} d^{2}} + \frac {x}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} d}\right )} \operatorname {arcosh}\left (c x\right ) + \frac {1}{3} \, a^{2} {\left (\frac {2 \, x}{\sqrt {-c^{2} d x^{2} + d} d^{2}} + \frac {x}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} d}\right )} + b^{2} \int \frac {\log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*a*b*c*(sqrt(-d)/(c^4*d^3*x^2 - c^2*d^3) + 2*sqrt(-d)*log(c*x + 1)/(c^2*d^3) + 2*sqrt(-d)*log(c*x - 1)/(c^2
*d^3)) + 2/3*a*b*(2*x/(sqrt(-c^2*d*x^2 + d)*d^2) + x/((-c^2*d*x^2 + d)^(3/2)*d))*arccosh(c*x) + 1/3*a^2*(2*x/(
sqrt(-c^2*d*x^2 + d)*d^2) + x/((-c^2*d*x^2 + d)^(3/2)*d)) + b^2*integrate(log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1
))^2/(-c^2*d*x^2 + d)^(5/2), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2}{{\left (d-c^2\,d\,x^2\right )}^{5/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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