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

Optimal. Leaf size=379 \[ \frac{i b d^2 \sqrt{d-c^2 d x^2} \text{PolyLog}\left (2,-i e^{\cosh ^{-1}(c x)}\right )}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{i b d^2 \sqrt{d-c^2 d x^2} \text{PolyLog}\left (2,i e^{\cosh ^{-1}(c x)}\right )}{\sqrt{c x-1} \sqrt{c x+1}}+d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac{2 d^2 \sqrt{d-c^2 d x^2} \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt{c x-1} \sqrt{c x+1}}+\frac{1}{5} \left (d-c^2 d x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{3} d \left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )-\frac{b c^5 d^2 x^5 \sqrt{d-c^2 d x^2}}{25 \sqrt{c x-1} \sqrt{c x+1}}+\frac{11 b c^3 d^2 x^3 \sqrt{d-c^2 d x^2}}{45 \sqrt{c x-1} \sqrt{c x+1}}-\frac{23 b c d^2 x \sqrt{d-c^2 d x^2}}{15 \sqrt{c x-1} \sqrt{c x+1}} \]

[Out]

(-23*b*c*d^2*x*Sqrt[d - c^2*d*x^2])/(15*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (11*b*c^3*d^2*x^3*Sqrt[d - c^2*d*x^2])
/(45*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c^5*d^2*x^5*Sqrt[d - c^2*d*x^2])/(25*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + d
^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]) + (d*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]))/3 + ((d - c^2*d*x
^2)^(5/2)*(a + b*ArcCosh[c*x]))/5 - (2*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])*ArcTan[E^ArcCosh[c*x]])/(S
qrt[-1 + c*x]*Sqrt[1 + c*x]) + (I*b*d^2*Sqrt[d - c^2*d*x^2]*PolyLog[2, (-I)*E^ArcCosh[c*x]])/(Sqrt[-1 + c*x]*S
qrt[1 + c*x]) - (I*b*d^2*Sqrt[d - c^2*d*x^2]*PolyLog[2, I*E^ArcCosh[c*x]])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])

________________________________________________________________________________________

Rubi [A]  time = 1.0589, antiderivative size = 410, normalized size of antiderivative = 1.08, number of steps used = 14, number of rules used = 9, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {5798, 5745, 5743, 5761, 4180, 2279, 2391, 8, 194} \[ \frac{i b d^2 \sqrt{d-c^2 d x^2} \text{PolyLog}\left (2,-i e^{\cosh ^{-1}(c x)}\right )}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{i b d^2 \sqrt{d-c^2 d x^2} \text{PolyLog}\left (2,i e^{\cosh ^{-1}(c x)}\right )}{\sqrt{c x-1} \sqrt{c x+1}}+d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{5} d^2 (1-c x)^2 (c x+1)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{3} d^2 (1-c x) (c x+1) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac{2 d^2 \sqrt{d-c^2 d x^2} \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{b c^5 d^2 x^5 \sqrt{d-c^2 d x^2}}{25 \sqrt{c x-1} \sqrt{c x+1}}+\frac{11 b c^3 d^2 x^3 \sqrt{d-c^2 d x^2}}{45 \sqrt{c x-1} \sqrt{c x+1}}-\frac{23 b c d^2 x \sqrt{d-c^2 d x^2}}{15 \sqrt{c x-1} \sqrt{c x+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-23*b*c*d^2*x*Sqrt[d - c^2*d*x^2])/(15*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (11*b*c^3*d^2*x^3*Sqrt[d - c^2*d*x^2])
/(45*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c^5*d^2*x^5*Sqrt[d - c^2*d*x^2])/(25*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + d
^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]) + (d^2*(1 - c*x)*(1 + c*x)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]
))/3 + (d^2*(1 - c*x)^2*(1 + c*x)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/5 - (2*d^2*Sqrt[d - c^2*d*x^2]*(
a + b*ArcCosh[c*x])*ArcTan[E^ArcCosh[c*x]])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (I*b*d^2*Sqrt[d - c^2*d*x^2]*Poly
Log[2, (-I)*E^ArcCosh[c*x]])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (I*b*d^2*Sqrt[d - c^2*d*x^2]*PolyLog[2, I*E^ArcC
osh[c*x]])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])

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 5745

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*(d2 + e2*x)^p*(a + b*ArcCosh[c*x])^n)/(f*(m + 2*p + 1)
), x] + (Dist[(2*d1*d2*p)/(m + 2*p + 1), Int[(f*x)^m*(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])/(f*(m + 2*p + 1)*Sqrt[1 + c*
x]*Sqrt[-1 + c*x]), 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}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && GtQ[n, 0] && GtQ[p, 0] &&  !L
tQ[m, -1] && IntegerQ[p - 1/2] && (RationalQ[m] || EqQ[n, 1])

Rule 5743

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*
(x_)], x_Symbol] :> Simp[((f*x)^(m + 1)*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x]*(a + b*ArcCosh[c*x])^n)/(f*(m + 2)), x
] + (-Dist[(Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x])/((m + 2)*Sqrt[1 + c*x]*Sqrt[-1 + c*x]), Int[((f*x)^m*(a + b*ArcCo
sh[c*x])^n)/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x], x] - Dist[(b*c*n*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x])/(f*(m + 2)*S
qrt[1 + c*x]*Sqrt[-1 + c*x]), Int[(f*x)^(m + 1)*(a + b*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d1, e
1, d2, e2, f, m}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && GtQ[n, 0] &&  !LtQ[m, -1] && (RationalQ[m] |
| EqQ[n, 1])

Rule 5761

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

Rule 4180

Int[csc[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(-2*(c
+ d*x)^m*ArcTanh[E^(-(I*e) + f*fz*x)/E^(I*k*Pi)])/(f*fz*I), x] + (-Dist[(d*m)/(f*fz*I), Int[(c + d*x)^(m - 1)*
Log[1 - E^(-(I*e) + f*fz*x)/E^(I*k*Pi)], x], x] + Dist[(d*m)/(f*fz*I), Int[(c + d*x)^(m - 1)*Log[1 + E^(-(I*e)
+ f*fz*x)/E^(I*k*Pi)], x], x]) /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[2*k] && 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 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 194

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n)^p, x], x] /; FreeQ[{a, b}, x]
&& IGtQ[n, 0] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int \frac{\left (d-c^2 d x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right )}{x} \, dx &=\frac{\left (d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{(-1+c x)^{5/2} (1+c x)^{5/2} \left (a+b \cosh ^{-1}(c x)\right )}{x} \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{1}{5} d^2 (1-c x)^2 (1+c x)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac{\left (d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{(-1+c x)^{3/2} (1+c x)^{3/2} \left (a+b \cosh ^{-1}(c x)\right )}{x} \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (b c d^2 \sqrt{d-c^2 d x^2}\right ) \int \left (-1+c^2 x^2\right )^2 \, dx}{5 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{1}{3} d^2 (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{5} d^2 (1-c x)^2 (1+c x)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac{\left (d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{\sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{x} \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (b c d^2 \sqrt{d-c^2 d x^2}\right ) \int \left (1-2 c^2 x^2+c^4 x^4\right ) \, dx}{5 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (b c d^2 \sqrt{d-c^2 d x^2}\right ) \int \left (-1+c^2 x^2\right ) \, dx}{3 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{8 b c d^2 x \sqrt{d-c^2 d x^2}}{15 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{11 b c^3 d^2 x^3 \sqrt{d-c^2 d x^2}}{45 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c^5 d^2 x^5 \sqrt{d-c^2 d x^2}}{25 \sqrt{-1+c x} \sqrt{1+c x}}+d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{3} d^2 (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{5} d^2 (1-c x)^2 (1+c x)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac{\left (d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{a+b \cosh ^{-1}(c x)}{x \sqrt{-1+c x} \sqrt{1+c x}} \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (b c d^2 \sqrt{d-c^2 d x^2}\right ) \int 1 \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{23 b c d^2 x \sqrt{d-c^2 d x^2}}{15 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{11 b c^3 d^2 x^3 \sqrt{d-c^2 d x^2}}{45 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c^5 d^2 x^5 \sqrt{d-c^2 d x^2}}{25 \sqrt{-1+c x} \sqrt{1+c x}}+d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{3} d^2 (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{5} d^2 (1-c x)^2 (1+c x)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac{\left (d^2 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int (a+b x) \text{sech}(x) \, dx,x,\cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{23 b c d^2 x \sqrt{d-c^2 d x^2}}{15 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{11 b c^3 d^2 x^3 \sqrt{d-c^2 d x^2}}{45 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c^5 d^2 x^5 \sqrt{d-c^2 d x^2}}{25 \sqrt{-1+c x} \sqrt{1+c x}}+d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{3} d^2 (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{5} d^2 (1-c x)^2 (1+c x)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac{2 d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (i b d^2 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \log \left (1-i e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (i b d^2 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \log \left (1+i e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{23 b c d^2 x \sqrt{d-c^2 d x^2}}{15 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{11 b c^3 d^2 x^3 \sqrt{d-c^2 d x^2}}{45 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c^5 d^2 x^5 \sqrt{d-c^2 d x^2}}{25 \sqrt{-1+c x} \sqrt{1+c x}}+d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{3} d^2 (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{5} d^2 (1-c x)^2 (1+c x)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac{2 d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (i b d^2 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{\log (1-i x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (i b d^2 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{\log (1+i x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{23 b c d^2 x \sqrt{d-c^2 d x^2}}{15 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{11 b c^3 d^2 x^3 \sqrt{d-c^2 d x^2}}{45 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c^5 d^2 x^5 \sqrt{d-c^2 d x^2}}{25 \sqrt{-1+c x} \sqrt{1+c x}}+d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{3} d^2 (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{5} d^2 (1-c x)^2 (1+c x)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac{2 d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{i b d^2 \sqrt{d-c^2 d x^2} \text{Li}_2\left (-i e^{\cosh ^{-1}(c x)}\right )}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{i b d^2 \sqrt{d-c^2 d x^2} \text{Li}_2\left (i e^{\cosh ^{-1}(c x)}\right )}{\sqrt{-1+c x} \sqrt{1+c x}}\\ \end{align*}

Mathematica [A]  time = 3.57599, size = 471, normalized size = 1.24 \[ \frac{b d^2 \sqrt{d-c^2 d x^2} \left (i \text{PolyLog}\left (2,-i e^{-\cosh ^{-1}(c x)}\right )-i \text{PolyLog}\left (2,i e^{-\cosh ^{-1}(c x)}\right )-c x+c x \sqrt{\frac{c x-1}{c x+1}} \cosh ^{-1}(c x)+\sqrt{\frac{c x-1}{c x+1}} \cosh ^{-1}(c x)+i \cosh ^{-1}(c x) \log \left (1-i e^{-\cosh ^{-1}(c x)}\right )-i \cosh ^{-1}(c x) \log \left (1+i e^{-\cosh ^{-1}(c x)}\right )\right )}{\sqrt{\frac{c x-1}{c x+1}} (c x+1)}+\frac{1}{15} a d^2 \left (3 c^4 x^4-11 c^2 x^2+23\right ) \sqrt{d-c^2 d x^2}-a d^{5/2} \log \left (\sqrt{d} \sqrt{d-c^2 d x^2}+d\right )+a d^{5/2} \log (x)-\frac{b d^2 \sqrt{d-c^2 d x^2} \left (9 c x+12 \left (\frac{c x-1}{c x+1}\right )^{3/2} (c x+1)^3 \cosh ^{-1}(c x)-\cosh \left (3 \cosh ^{-1}(c x)\right )\right )}{18 \sqrt{\frac{c x-1}{c x+1}} (c x+1)}-\frac{b d^2 \sqrt{d-c^2 d x^2} \left (25 \cosh \left (3 \cosh ^{-1}(c x)\right )+9 \left (\cosh \left (5 \cosh ^{-1}(c x)\right )-50 c x\right )+15 \cosh ^{-1}(c x) \left (30 \sqrt{\frac{c x-1}{c x+1}} (c x+1)-5 \sinh \left (3 \cosh ^{-1}(c x)\right )-3 \sinh \left (5 \cosh ^{-1}(c x)\right )\right )\right )}{3600 \sqrt{\frac{c x-1}{c x+1}} (c x+1)} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(a*d^2*Sqrt[d - c^2*d*x^2]*(23 - 11*c^2*x^2 + 3*c^4*x^4))/15 - (b*d^2*Sqrt[d - c^2*d*x^2]*(9*c*x + 12*((-1 + c
*x)/(1 + c*x))^(3/2)*(1 + c*x)^3*ArcCosh[c*x] - Cosh[3*ArcCosh[c*x]]))/(18*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x
)) + a*d^(5/2)*Log[x] - a*d^(5/2)*Log[d + Sqrt[d]*Sqrt[d - c^2*d*x^2]] + (b*d^2*Sqrt[d - c^2*d*x^2]*(-(c*x) +
Sqrt[(-1 + c*x)/(1 + c*x)]*ArcCosh[c*x] + c*x*Sqrt[(-1 + c*x)/(1 + c*x)]*ArcCosh[c*x] + I*ArcCosh[c*x]*Log[1 -
I/E^ArcCosh[c*x]] - I*ArcCosh[c*x]*Log[1 + I/E^ArcCosh[c*x]] + I*PolyLog[2, (-I)/E^ArcCosh[c*x]] - I*PolyLog[
2, I/E^ArcCosh[c*x]]))/(Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)) - (b*d^2*Sqrt[d - c^2*d*x^2]*(25*Cosh[3*ArcCosh[
c*x]] + 9*(-50*c*x + Cosh[5*ArcCosh[c*x]]) + 15*ArcCosh[c*x]*(30*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x) - 5*Sinh
[3*ArcCosh[c*x]] - 3*Sinh[5*ArcCosh[c*x]])))/(3600*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x))

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Maple [A]  time = 0.258, size = 620, normalized size = 1.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*(-c^2*d*x^2+d)^(5/2)*a+1/3*a*d*(-c^2*d*x^2+d)^(3/2)-a*d^(5/2)*ln((2*d+2*d^(1/2)*(-c^2*d*x^2+d)^(1/2))/x)+a
*(-c^2*d*x^2+d)^(1/2)*d^2+I*b*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)*arccosh(c*x)*ln(1+I*(c*x+(c*x
-1)^(1/2)*(c*x+1)^(1/2)))*d^2-1/25*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(c*x+1)^(1/2)/(c*x-1)^(1/2)*x^5*c^5+11/45*b*(-
d*(c^2*x^2-1))^(1/2)*d^2/(c*x+1)^(1/2)/(c*x-1)^(1/2)*x^3*c^3-23/15*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(c*x+1)^(1/2)/
(c*x-1)^(1/2)*x*c+I*b*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)*dilog(1+I*(c*x+(c*x-1)^(1/2)*(c*x+1)^
(1/2)))*d^2+1/5*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(c*x+1)/(c*x-1)*arccosh(c*x)*x^6*c^6-14/15*b*(-d*(c^2*x^2-1))^(1/
2)*d^2/(c*x+1)/(c*x-1)*arccosh(c*x)*x^4*c^4+34/15*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(c*x+1)/(c*x-1)*arccosh(c*x)*x^
2*c^2-23/15*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(c*x+1)/(c*x-1)*arccosh(c*x)-I*b*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)
/(c*x+1)^(1/2)*dilog(1-I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))*d^2-I*b*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+
1)^(1/2)*arccosh(c*x)*ln(1-I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))*d^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((-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))/x,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{{\left (a c^{4} d^{2} x^{4} - 2 \, a c^{2} d^{2} x^{2} + a d^{2} +{\left (b c^{4} d^{2} x^{4} - 2 \, b c^{2} d^{2} x^{2} + b d^{2}\right )} \operatorname{arcosh}\left (c x\right )\right )} \sqrt{-c^{2} d x^{2} + d}}{x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (-c^{2} d x^{2} + d\right )}^{\frac{5}{2}}{\left (b \operatorname{arcosh}\left (c x\right ) + a\right )}}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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