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

Optimal. Leaf size=298 \[ \frac {b x \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{3 c d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {b^2 \sqrt {c x-1} \sqrt {c x+1} \text {Li}_2\left (-e^{\cosh ^{-1}(c x)}\right )}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}-\frac {b^2 \sqrt {c x-1} \sqrt {c x+1} \text {Li}_2\left (e^{\cosh ^{-1}(c x)}\right )}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}-\frac {b^2}{3 c^2 d^2 \sqrt {d-c^2 d x^2}} \]

[Out]

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

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Rubi [A]  time = 0.47, antiderivative size = 313, normalized size of antiderivative = 1.05, number of steps used = 10, number of rules used = 8, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {5798, 5718, 5689, 74, 5694, 4182, 2279, 2391} \[ \frac {b^2 \sqrt {c x-1} \sqrt {c x+1} \text {PolyLog}\left (2,-e^{\cosh ^{-1}(c x)}\right )}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}-\frac {b^2 \sqrt {c x-1} \sqrt {c x+1} \text {PolyLog}\left (2,e^{\cosh ^{-1}(c x)}\right )}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}+\frac {b x \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{3 c d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^2 d^2 (1-c x) (c x+1) \sqrt {d-c^2 d x^2}}+\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}-\frac {b^2}{3 c^2 d^2 \sqrt {d-c^2 d x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 74

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 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 4182

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

Rule 5689

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

Rule 5694

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

Rule 5718

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

Rubi steps

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

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Mathematica [A]  time = 2.54, size = 332, normalized size = 1.11 \[ \frac {4 a^2+a b \left (8 \cosh ^{-1}(c x)+2 \sinh \left (2 \cosh ^{-1}(c x)\right )+\left (\sinh \left (3 \cosh ^{-1}(c x)\right )-3 \sqrt {\frac {c x-1}{c x+1}} (c x+1)\right ) \log \left (\tanh \left (\frac {1}{2} \cosh ^{-1}(c x)\right )\right )\right )+b^2 \left (4 \left (\frac {c x-1}{c x+1}\right )^{3/2} (c x+1)^3 \text {Li}_2\left (-e^{-\cosh ^{-1}(c x)}\right )-4 \left (\frac {c x-1}{c x+1}\right )^{3/2} (c x+1)^3 \text {Li}_2\left (e^{-\cosh ^{-1}(c x)}\right )+4 \cosh ^{-1}(c x)^2+2 \cosh \left (2 \cosh ^{-1}(c x)\right )-3 \sqrt {\frac {c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x) \log \left (1-e^{-\cosh ^{-1}(c x)}\right )+3 \sqrt {\frac {c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x) \log \left (e^{-\cosh ^{-1}(c x)}+1\right )+2 \cosh ^{-1}(c x) \sinh \left (2 \cosh ^{-1}(c x)\right )+\cosh ^{-1}(c x) \log \left (1-e^{-\cosh ^{-1}(c x)}\right ) \sinh \left (3 \cosh ^{-1}(c x)\right )-\cosh ^{-1}(c x) \log \left (e^{-\cosh ^{-1}(c x)}+1\right ) \sinh \left (3 \cosh ^{-1}(c x)\right )-2\right )}{12 c^2 d \left (d-c^2 d x^2\right )^{3/2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(4*a^2 + b^2*(-2 + 4*ArcCosh[c*x]^2 + 2*Cosh[2*ArcCosh[c*x]] - 3*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[
c*x]*Log[1 - E^(-ArcCosh[c*x])] + 3*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x]*Log[1 + E^(-ArcCosh[c*x]
)] + 4*((-1 + c*x)/(1 + c*x))^(3/2)*(1 + c*x)^3*PolyLog[2, -E^(-ArcCosh[c*x])] - 4*((-1 + c*x)/(1 + c*x))^(3/2
)*(1 + c*x)^3*PolyLog[2, E^(-ArcCosh[c*x])] + 2*ArcCosh[c*x]*Sinh[2*ArcCosh[c*x]] + ArcCosh[c*x]*Log[1 - E^(-A
rcCosh[c*x])]*Sinh[3*ArcCosh[c*x]] - ArcCosh[c*x]*Log[1 + E^(-ArcCosh[c*x])]*Sinh[3*ArcCosh[c*x]]) + a*b*(8*Ar
cCosh[c*x] + 2*Sinh[2*ArcCosh[c*x]] + Log[Tanh[ArcCosh[c*x]/2]]*(-3*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x) + Sin
h[3*ArcCosh[c*x]])))/(12*c^2*d*(d - c^2*d*x^2)^(3/2))

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fricas [F]  time = 0.67, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-c^{2} d x^{2} + d} {\left (b^{2} x \operatorname {arcosh}\left (c x\right )^{2} + 2 \, a b x \operatorname {arcosh}\left (c x\right ) + a^{2} x\right )}}{c^{6} d^{3} x^{6} - 3 \, c^{4} d^{3} x^{4} + 3 \, c^{2} d^{3} x^{2} - d^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(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*x*arccosh(c*x)^2 + 2*a*b*x*arccosh(c*x) + a^2*x)/(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 \[ \int \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2} x}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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maple [B]  time = 0.43, size = 720, normalized size = 2.42 \[ \frac {a^{2}}{3 c^{2} d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right ) \sqrt {c x +1}\, \sqrt {c x -1}\, x}{3 d^{3} \left (c^{2} x^{2}-1\right )^{2} c}+\frac {b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x^{2}}{3 d^{3} \left (c^{2} x^{2}-1\right )^{2}}+\frac {b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right )^{2}}{3 d^{3} \left (c^{2} x^{2}-1\right )^{2} c^{2}}-\frac {b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}}{3 d^{3} \left (c^{2} x^{2}-1\right )^{2} c^{2}}+\frac {b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \mathrm {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{3 d^{3} c^{2} \left (c^{2} x^{2}-1\right )}+\frac {b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \polylog \left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{3 d^{3} c^{2} \left (c^{2} x^{2}-1\right )}-\frac {b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \mathrm {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{3 d^{3} c^{2} \left (c^{2} x^{2}-1\right )}-\frac {b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \polylog \left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{3 d^{3} c^{2} \left (c^{2} x^{2}-1\right )}+\frac {a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x +1}\, \sqrt {c x -1}\, x}{3 d^{3} \left (c^{2} x^{2}-1\right )^{2} c}+\frac {2 a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right )}{3 d^{3} \left (c^{2} x^{2}-1\right )^{2} c^{2}}+\frac {a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}-1\right )}{3 d^{3} c^{2} \left (c^{2} x^{2}-1\right )}-\frac {a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{3 d^{3} c^{2} \left (c^{2} x^{2}-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((x*(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 \[ \int \frac {x \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}}{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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