3.26 \(\int \frac{(a+b \cosh ^{-1}(c x))^2}{(d+e x)^3} \, dx\)

Optimal. Leaf size=380 \[ \frac{b^2 c^3 d \text{PolyLog}\left (2,-\frac{e e^{\cosh ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right )}{e \left (c^2 d^2-e^2\right )^{3/2}}-\frac{b^2 c^3 d \text{PolyLog}\left (2,-\frac{e e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 d^2-e^2}+c d}\right )}{e \left (c^2 d^2-e^2\right )^{3/2}}-\frac{b c \sqrt{-\frac{1-c x}{c x+1}} (c x+1) \left (a+b \cosh ^{-1}(c x)\right )}{\left (c^2 d^2-e^2\right ) (d+e x)}+\frac{b c^3 d \left (a+b \cosh ^{-1}(c x)\right ) \log \left (\frac{e e^{\cosh ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}+1\right )}{e \left (c^2 d^2-e^2\right )^{3/2}}-\frac{b c^3 d \left (a+b \cosh ^{-1}(c x)\right ) \log \left (\frac{e e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 d^2-e^2}+c d}+1\right )}{e \left (c^2 d^2-e^2\right )^{3/2}}-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{2 e (d+e x)^2}+\frac{b^2 c^2 \log (d+e x)}{e \left (c^2 d^2-e^2\right )} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.746648, antiderivative size = 380, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.556, Rules used = {5802, 5832, 3324, 3320, 2264, 2190, 2279, 2391, 2668, 31} \[ \frac{b^2 c^3 d \text{PolyLog}\left (2,-\frac{e e^{\cosh ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right )}{e \left (c^2 d^2-e^2\right )^{3/2}}-\frac{b^2 c^3 d \text{PolyLog}\left (2,-\frac{e e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 d^2-e^2}+c d}\right )}{e \left (c^2 d^2-e^2\right )^{3/2}}-\frac{b c \sqrt{-\frac{1-c x}{c x+1}} (c x+1) \left (a+b \cosh ^{-1}(c x)\right )}{\left (c^2 d^2-e^2\right ) (d+e x)}+\frac{b c^3 d \left (a+b \cosh ^{-1}(c x)\right ) \log \left (\frac{e e^{\cosh ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}+1\right )}{e \left (c^2 d^2-e^2\right )^{3/2}}-\frac{b c^3 d \left (a+b \cosh ^{-1}(c x)\right ) \log \left (\frac{e e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 d^2-e^2}+c d}+1\right )}{e \left (c^2 d^2-e^2\right )^{3/2}}-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{2 e (d+e x)^2}+\frac{b^2 c^2 \log (d+e x)}{e \left (c^2 d^2-e^2\right )} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcCosh[c*x])^2/(d + e*x)^3,x]

[Out]

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

Rule 5802

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Simp[((d + e*x)^(m + 1)
*(a + b*ArcCosh[c*x])^n)/(e*(m + 1)), x] - Dist[(b*c*n)/(e*(m + 1)), Int[((d + e*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, e, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rule 5832

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(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*(c*f + g*Cosh[x])^m,
x], x, ArcCosh[c*x]], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, f, g, n}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2
, 0] && IntegerQ[m] && GtQ[d1, 0] && LtQ[d2, 0] && (GtQ[m, 0] || IGtQ[n, 0])

Rule 3324

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^2, x_Symbol] :> Simp[(b*(c + d*x)^m*Cos[
e + f*x])/(f*(a^2 - b^2)*(a + b*Sin[e + f*x])), x] + (Dist[a/(a^2 - b^2), Int[(c + d*x)^m/(a + b*Sin[e + f*x])
, x], x] - Dist[(b*d*m)/(f*(a^2 - b^2)), Int[((c + d*x)^(m - 1)*Cos[e + f*x])/(a + b*Sin[e + f*x]), x], x]) /;
FreeQ[{a, b, c, d, e, f}, x] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]

Rule 3320

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)]), x_Symbol]
:> Dist[2, Int[((c + d*x)^m*E^(-(I*e) + f*fz*x))/(E^(I*Pi*(k - 1/2))*(b + (2*a*E^(-(I*e) + f*fz*x))/E^(I*Pi*(k
- 1/2)) - (b*E^(2*(-(I*e) + f*fz*x)))/E^(2*I*k*Pi))), x], x] /; FreeQ[{a, b, c, d, e, f, fz}, x] && IntegerQ[
2*k] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]

Rule 2264

Int[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.)*(F_)^(v_)), x_Symbol] :> With[{q =
Rt[b^2 - 4*a*c, 2]}, Dist[(2*c)/q, Int[((f + g*x)^m*F^u)/(b - q + 2*c*F^u), x], x] - Dist[(2*c)/q, Int[((f +
g*x)^m*F^u)/(b + q + 2*c*F^u), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[v, 2*u] && LinearQ[u, x] && NeQ[
b^2 - 4*a*c, 0] && 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 2668

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(b^p*f), S
ubst[Int[(a + x)^m*(b^2 - x^2)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && Integer
Q[(p - 1)/2] && NeQ[a^2 - b^2, 0]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

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

Mathematica [C]  time = 8.40222, size = 1089, normalized size = 2.87 \[ -\frac{a^2}{2 e (d+e x)^2}+b c^2 \left (-\frac{\cosh ^{-1}(c x)}{e (c d+c e x)^2}-\frac{2 c d \tan ^{-1}\left (\frac{\sqrt{e-c d} \sqrt{\frac{c x-1}{c x+1}}}{\sqrt{c d+e}}\right )}{e (e-c d)^{3/2} (c d+e)^{3/2}}+\frac{\sqrt{c x-1} \sqrt{c x+1}}{(e-c d) (c d+e) (c d+c e x)}\right ) a+b^2 c^2 \left (-\frac{\cosh ^{-1}(c x)^2}{2 e (c d+c e x)^2}-\frac{\sqrt{\frac{c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x)}{(c d-e) (c d+e) (c d+c e x)}+\frac{\log \left (\frac{e x}{d}+1\right )}{c^2 d^2 e-e^3}+\frac{c d \left (2 \cosh ^{-1}(c x) \tan ^{-1}\left (\frac{(c d+e) \coth \left (\frac{1}{2} \cosh ^{-1}(c x)\right )}{\sqrt{e^2-c^2 d^2}}\right )-2 i \cos ^{-1}\left (-\frac{c d}{e}\right ) \tan ^{-1}\left (\frac{(e-c d) \tanh \left (\frac{1}{2} \cosh ^{-1}(c x)\right )}{\sqrt{e^2-c^2 d^2}}\right )+\left (\cos ^{-1}\left (-\frac{c d}{e}\right )+2 \left (\tan ^{-1}\left (\frac{(c d+e) \coth \left (\frac{1}{2} \cosh ^{-1}(c x)\right )}{\sqrt{e^2-c^2 d^2}}\right )+\tan ^{-1}\left (\frac{(e-c d) \tanh \left (\frac{1}{2} \cosh ^{-1}(c x)\right )}{\sqrt{e^2-c^2 d^2}}\right )\right )\right ) \log \left (\frac{\sqrt{e^2-c^2 d^2} e^{-\frac{1}{2} \cosh ^{-1}(c x)}}{\sqrt{2} \sqrt{e} \sqrt{c (d+e x)}}\right )+\left (\cos ^{-1}\left (-\frac{c d}{e}\right )-2 \left (\tan ^{-1}\left (\frac{(c d+e) \coth \left (\frac{1}{2} \cosh ^{-1}(c x)\right )}{\sqrt{e^2-c^2 d^2}}\right )+\tan ^{-1}\left (\frac{(e-c d) \tanh \left (\frac{1}{2} \cosh ^{-1}(c x)\right )}{\sqrt{e^2-c^2 d^2}}\right )\right )\right ) \log \left (\frac{\sqrt{e^2-c^2 d^2} e^{\frac{1}{2} \cosh ^{-1}(c x)}}{\sqrt{2} \sqrt{e} \sqrt{c (d+e x)}}\right )-\left (\cos ^{-1}\left (-\frac{c d}{e}\right )+2 \tan ^{-1}\left (\frac{(e-c d) \tanh \left (\frac{1}{2} \cosh ^{-1}(c x)\right )}{\sqrt{e^2-c^2 d^2}}\right )\right ) \log \left (\frac{(c d+e) \left (c d-e+i \sqrt{e^2-c^2 d^2}\right ) \left (\tanh \left (\frac{1}{2} \cosh ^{-1}(c x)\right )-1\right )}{e \left (c d+e+i \sqrt{e^2-c^2 d^2} \tanh \left (\frac{1}{2} \cosh ^{-1}(c x)\right )\right )}\right )-\left (\cos ^{-1}\left (-\frac{c d}{e}\right )-2 \tan ^{-1}\left (\frac{(e-c d) \tanh \left (\frac{1}{2} \cosh ^{-1}(c x)\right )}{\sqrt{e^2-c^2 d^2}}\right )\right ) \log \left (\frac{(c d+e) \left (-c d+e+i \sqrt{e^2-c^2 d^2}\right ) \left (\tanh \left (\frac{1}{2} \cosh ^{-1}(c x)\right )+1\right )}{e \left (c d+e+i \sqrt{e^2-c^2 d^2} \tanh \left (\frac{1}{2} \cosh ^{-1}(c x)\right )\right )}\right )+i \left (\text{PolyLog}\left (2,\frac{\left (c d-i \sqrt{e^2-c^2 d^2}\right ) \left (c d+e-i \sqrt{e^2-c^2 d^2} \tanh \left (\frac{1}{2} \cosh ^{-1}(c x)\right )\right )}{e \left (c d+e+i \sqrt{e^2-c^2 d^2} \tanh \left (\frac{1}{2} \cosh ^{-1}(c x)\right )\right )}\right )-\text{PolyLog}\left (2,\frac{\left (c d+i \sqrt{e^2-c^2 d^2}\right ) \left (c d+e-i \sqrt{e^2-c^2 d^2} \tanh \left (\frac{1}{2} \cosh ^{-1}(c x)\right )\right )}{e \left (c d+e+i \sqrt{e^2-c^2 d^2} \tanh \left (\frac{1}{2} \cosh ^{-1}(c x)\right )\right )}\right )\right )\right )}{e \left (e^2-c^2 d^2\right )^{3/2}}\right ) \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + b*ArcCosh[c*x])^2/(d + e*x)^3,x]

[Out]

-a^2/(2*e*(d + e*x)^2) + a*b*c^2*((Sqrt[-1 + c*x]*Sqrt[1 + c*x])/((-(c*d) + e)*(c*d + e)*(c*d + c*e*x)) - ArcC
osh[c*x]/(e*(c*d + c*e*x)^2) - (2*c*d*ArcTan[(Sqrt[-(c*d) + e]*Sqrt[(-1 + c*x)/(1 + c*x)])/Sqrt[c*d + e]])/(e*
(-(c*d) + e)^(3/2)*(c*d + e)^(3/2))) + b^2*c^2*(-((Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x])/((c*d -
e)*(c*d + e)*(c*d + c*e*x))) - ArcCosh[c*x]^2/(2*e*(c*d + c*e*x)^2) + Log[1 + (e*x)/d]/(c^2*d^2*e - e^3) + (c*
d*(2*ArcCosh[c*x]*ArcTan[((c*d + e)*Coth[ArcCosh[c*x]/2])/Sqrt[-(c^2*d^2) + e^2]] - (2*I)*ArcCos[-((c*d)/e)]*A
rcTan[((-(c*d) + e)*Tanh[ArcCosh[c*x]/2])/Sqrt[-(c^2*d^2) + e^2]] + (ArcCos[-((c*d)/e)] + 2*(ArcTan[((c*d + e)
*Coth[ArcCosh[c*x]/2])/Sqrt[-(c^2*d^2) + e^2]] + ArcTan[((-(c*d) + e)*Tanh[ArcCosh[c*x]/2])/Sqrt[-(c^2*d^2) +
e^2]]))*Log[Sqrt[-(c^2*d^2) + e^2]/(Sqrt[2]*Sqrt[e]*E^(ArcCosh[c*x]/2)*Sqrt[c*(d + e*x)])] + (ArcCos[-((c*d)/e
)] - 2*(ArcTan[((c*d + e)*Coth[ArcCosh[c*x]/2])/Sqrt[-(c^2*d^2) + e^2]] + ArcTan[((-(c*d) + e)*Tanh[ArcCosh[c*
x]/2])/Sqrt[-(c^2*d^2) + e^2]]))*Log[(Sqrt[-(c^2*d^2) + e^2]*E^(ArcCosh[c*x]/2))/(Sqrt[2]*Sqrt[e]*Sqrt[c*(d +
e*x)])] - (ArcCos[-((c*d)/e)] + 2*ArcTan[((-(c*d) + e)*Tanh[ArcCosh[c*x]/2])/Sqrt[-(c^2*d^2) + e^2]])*Log[((c*
d + e)*(c*d - e + I*Sqrt[-(c^2*d^2) + e^2])*(-1 + Tanh[ArcCosh[c*x]/2]))/(e*(c*d + e + I*Sqrt[-(c^2*d^2) + e^2
]*Tanh[ArcCosh[c*x]/2]))] - (ArcCos[-((c*d)/e)] - 2*ArcTan[((-(c*d) + e)*Tanh[ArcCosh[c*x]/2])/Sqrt[-(c^2*d^2)
+ e^2]])*Log[((c*d + e)*(-(c*d) + e + I*Sqrt[-(c^2*d^2) + e^2])*(1 + Tanh[ArcCosh[c*x]/2]))/(e*(c*d + e + I*S
qrt[-(c^2*d^2) + e^2]*Tanh[ArcCosh[c*x]/2]))] + I*(PolyLog[2, ((c*d - I*Sqrt[-(c^2*d^2) + e^2])*(c*d + e - I*S
qrt[-(c^2*d^2) + e^2]*Tanh[ArcCosh[c*x]/2]))/(e*(c*d + e + I*Sqrt[-(c^2*d^2) + e^2]*Tanh[ArcCosh[c*x]/2]))] -
PolyLog[2, ((c*d + I*Sqrt[-(c^2*d^2) + e^2])*(c*d + e - I*Sqrt[-(c^2*d^2) + e^2]*Tanh[ArcCosh[c*x]/2]))/(e*(c*
d + e + I*Sqrt[-(c^2*d^2) + e^2]*Tanh[ArcCosh[c*x]/2]))])))/(e*(-(c^2*d^2) + e^2)^(3/2)))

________________________________________________________________________________________

Maple [B]  time = 0.112, size = 1170, normalized size = 3.1 \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/(e*x+d)^3,x)

[Out]

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

________________________________________________________________________________________

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/(e*x+d)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b^{2} \operatorname{arcosh}\left (c x\right )^{2} + 2 \, a b \operatorname{arcosh}\left (c x\right ) + a^{2}}{e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccosh(c*x))^2/(e*x+d)^3,x, algorithm="fricas")

[Out]

integral((b^2*arccosh(c*x)^2 + 2*a*b*arccosh(c*x) + a^2)/(e^3*x^3 + 3*d*e^2*x^2 + 3*d^2*e*x + d^3), x)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b \operatorname{acosh}{\left (c x \right )}\right )^{2}}{\left (d + e x\right )^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*acosh(c*x))**2/(e*x+d)**3,x)

[Out]

Integral((a + b*acosh(c*x))**2/(d + e*x)**3, x)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \operatorname{arcosh}\left (c x\right ) + a\right )}^{2}}{{\left (e x + d\right )}^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccosh(c*x))^2/(e*x+d)^3,x, algorithm="giac")

[Out]

integrate((b*arccosh(c*x) + a)^2/(e*x + d)^3, x)