3.127 \(\int \frac{(a+b \cosh ^{-1}(c+d x))^4}{(c e+d e x)^2} \, dx\)

Optimal. Leaf size=264 \[ \frac{24 i b^3 \text{PolyLog}\left (3,-i e^{\cosh ^{-1}(c+d x)}\right ) \left (a+b \cosh ^{-1}(c+d x)\right )}{d e^2}-\frac{24 i b^3 \text{PolyLog}\left (3,i e^{\cosh ^{-1}(c+d x)}\right ) \left (a+b \cosh ^{-1}(c+d x)\right )}{d e^2}-\frac{12 i b^2 \text{PolyLog}\left (2,-i e^{\cosh ^{-1}(c+d x)}\right ) \left (a+b \cosh ^{-1}(c+d x)\right )^2}{d e^2}+\frac{12 i b^2 \text{PolyLog}\left (2,i e^{\cosh ^{-1}(c+d x)}\right ) \left (a+b \cosh ^{-1}(c+d x)\right )^2}{d e^2}-\frac{24 i b^4 \text{PolyLog}\left (4,-i e^{\cosh ^{-1}(c+d x)}\right )}{d e^2}+\frac{24 i b^4 \text{PolyLog}\left (4,i e^{\cosh ^{-1}(c+d x)}\right )}{d e^2}-\frac{\left (a+b \cosh ^{-1}(c+d x)\right )^4}{d e^2 (c+d x)}+\frac{8 b \tan ^{-1}\left (e^{\cosh ^{-1}(c+d x)}\right ) \left (a+b \cosh ^{-1}(c+d x)\right )^3}{d e^2} \]

[Out]

-((a + b*ArcCosh[c + d*x])^4/(d*e^2*(c + d*x))) + (8*b*(a + b*ArcCosh[c + d*x])^3*ArcTan[E^ArcCosh[c + d*x]])/
(d*e^2) - ((12*I)*b^2*(a + b*ArcCosh[c + d*x])^2*PolyLog[2, (-I)*E^ArcCosh[c + d*x]])/(d*e^2) + ((12*I)*b^2*(a
+ b*ArcCosh[c + d*x])^2*PolyLog[2, I*E^ArcCosh[c + d*x]])/(d*e^2) + ((24*I)*b^3*(a + b*ArcCosh[c + d*x])*Poly
Log[3, (-I)*E^ArcCosh[c + d*x]])/(d*e^2) - ((24*I)*b^3*(a + b*ArcCosh[c + d*x])*PolyLog[3, I*E^ArcCosh[c + d*x
]])/(d*e^2) - ((24*I)*b^4*PolyLog[4, (-I)*E^ArcCosh[c + d*x]])/(d*e^2) + ((24*I)*b^4*PolyLog[4, I*E^ArcCosh[c
+ d*x]])/(d*e^2)

________________________________________________________________________________________

Rubi [A]  time = 0.412514, antiderivative size = 264, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 9, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.391, Rules used = {5866, 12, 5662, 5761, 4180, 2531, 6609, 2282, 6589} \[ \frac{24 i b^3 \text{PolyLog}\left (3,-i e^{\cosh ^{-1}(c+d x)}\right ) \left (a+b \cosh ^{-1}(c+d x)\right )}{d e^2}-\frac{24 i b^3 \text{PolyLog}\left (3,i e^{\cosh ^{-1}(c+d x)}\right ) \left (a+b \cosh ^{-1}(c+d x)\right )}{d e^2}-\frac{12 i b^2 \text{PolyLog}\left (2,-i e^{\cosh ^{-1}(c+d x)}\right ) \left (a+b \cosh ^{-1}(c+d x)\right )^2}{d e^2}+\frac{12 i b^2 \text{PolyLog}\left (2,i e^{\cosh ^{-1}(c+d x)}\right ) \left (a+b \cosh ^{-1}(c+d x)\right )^2}{d e^2}-\frac{24 i b^4 \text{PolyLog}\left (4,-i e^{\cosh ^{-1}(c+d x)}\right )}{d e^2}+\frac{24 i b^4 \text{PolyLog}\left (4,i e^{\cosh ^{-1}(c+d x)}\right )}{d e^2}-\frac{\left (a+b \cosh ^{-1}(c+d x)\right )^4}{d e^2 (c+d x)}+\frac{8 b \tan ^{-1}\left (e^{\cosh ^{-1}(c+d x)}\right ) \left (a+b \cosh ^{-1}(c+d x)\right )^3}{d e^2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcCosh[c + d*x])^4/(c*e + d*e*x)^2,x]

[Out]

-((a + b*ArcCosh[c + d*x])^4/(d*e^2*(c + d*x))) + (8*b*(a + b*ArcCosh[c + d*x])^3*ArcTan[E^ArcCosh[c + d*x]])/
(d*e^2) - ((12*I)*b^2*(a + b*ArcCosh[c + d*x])^2*PolyLog[2, (-I)*E^ArcCosh[c + d*x]])/(d*e^2) + ((12*I)*b^2*(a
+ b*ArcCosh[c + d*x])^2*PolyLog[2, I*E^ArcCosh[c + d*x]])/(d*e^2) + ((24*I)*b^3*(a + b*ArcCosh[c + d*x])*Poly
Log[3, (-I)*E^ArcCosh[c + d*x]])/(d*e^2) - ((24*I)*b^3*(a + b*ArcCosh[c + d*x])*PolyLog[3, I*E^ArcCosh[c + d*x
]])/(d*e^2) - ((24*I)*b^4*PolyLog[4, (-I)*E^ArcCosh[c + d*x]])/(d*e^2) + ((24*I)*b^4*PolyLog[4, I*E^ArcCosh[c
+ d*x]])/(d*e^2)

Rule 5866

Int[((a_.) + ArcCosh[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Dist[1/d, Subst[
Int[((d*e - c*f)/d + (f*x)/d)^m*(a + b*ArcCosh[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x
]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 5662

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))/(Sqr
t[-1 + c*x]*Sqrt[1 + c*x]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -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 2531

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> -Simp[((
f + g*x)^m*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)])/(b*c*n*Log[F]), x] + Dist[(g*m)/(b*c*n*Log[F]), Int[(f + g*x)
^(m - 1)*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 6609

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

Rule 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 6589

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rubi steps

\begin{align*} \int \frac{\left (a+b \cosh ^{-1}(c+d x)\right )^4}{(c e+d e x)^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (a+b \cosh ^{-1}(x)\right )^4}{e^2 x^2} \, dx,x,c+d x\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\left (a+b \cosh ^{-1}(x)\right )^4}{x^2} \, dx,x,c+d x\right )}{d e^2}\\ &=-\frac{\left (a+b \cosh ^{-1}(c+d x)\right )^4}{d e^2 (c+d x)}+\frac{(4 b) \operatorname{Subst}\left (\int \frac{\left (a+b \cosh ^{-1}(x)\right )^3}{\sqrt{-1+x} x \sqrt{1+x}} \, dx,x,c+d x\right )}{d e^2}\\ &=-\frac{\left (a+b \cosh ^{-1}(c+d x)\right )^4}{d e^2 (c+d x)}+\frac{(4 b) \operatorname{Subst}\left (\int (a+b x)^3 \text{sech}(x) \, dx,x,\cosh ^{-1}(c+d x)\right )}{d e^2}\\ &=-\frac{\left (a+b \cosh ^{-1}(c+d x)\right )^4}{d e^2 (c+d x)}+\frac{8 b \left (a+b \cosh ^{-1}(c+d x)\right )^3 \tan ^{-1}\left (e^{\cosh ^{-1}(c+d x)}\right )}{d e^2}-\frac{\left (12 i b^2\right ) \operatorname{Subst}\left (\int (a+b x)^2 \log \left (1-i e^x\right ) \, dx,x,\cosh ^{-1}(c+d x)\right )}{d e^2}+\frac{\left (12 i b^2\right ) \operatorname{Subst}\left (\int (a+b x)^2 \log \left (1+i e^x\right ) \, dx,x,\cosh ^{-1}(c+d x)\right )}{d e^2}\\ &=-\frac{\left (a+b \cosh ^{-1}(c+d x)\right )^4}{d e^2 (c+d x)}+\frac{8 b \left (a+b \cosh ^{-1}(c+d x)\right )^3 \tan ^{-1}\left (e^{\cosh ^{-1}(c+d x)}\right )}{d e^2}-\frac{12 i b^2 \left (a+b \cosh ^{-1}(c+d x)\right )^2 \text{Li}_2\left (-i e^{\cosh ^{-1}(c+d x)}\right )}{d e^2}+\frac{12 i b^2 \left (a+b \cosh ^{-1}(c+d x)\right )^2 \text{Li}_2\left (i e^{\cosh ^{-1}(c+d x)}\right )}{d e^2}+\frac{\left (24 i b^3\right ) \operatorname{Subst}\left (\int (a+b x) \text{Li}_2\left (-i e^x\right ) \, dx,x,\cosh ^{-1}(c+d x)\right )}{d e^2}-\frac{\left (24 i b^3\right ) \operatorname{Subst}\left (\int (a+b x) \text{Li}_2\left (i e^x\right ) \, dx,x,\cosh ^{-1}(c+d x)\right )}{d e^2}\\ &=-\frac{\left (a+b \cosh ^{-1}(c+d x)\right )^4}{d e^2 (c+d x)}+\frac{8 b \left (a+b \cosh ^{-1}(c+d x)\right )^3 \tan ^{-1}\left (e^{\cosh ^{-1}(c+d x)}\right )}{d e^2}-\frac{12 i b^2 \left (a+b \cosh ^{-1}(c+d x)\right )^2 \text{Li}_2\left (-i e^{\cosh ^{-1}(c+d x)}\right )}{d e^2}+\frac{12 i b^2 \left (a+b \cosh ^{-1}(c+d x)\right )^2 \text{Li}_2\left (i e^{\cosh ^{-1}(c+d x)}\right )}{d e^2}+\frac{24 i b^3 \left (a+b \cosh ^{-1}(c+d x)\right ) \text{Li}_3\left (-i e^{\cosh ^{-1}(c+d x)}\right )}{d e^2}-\frac{24 i b^3 \left (a+b \cosh ^{-1}(c+d x)\right ) \text{Li}_3\left (i e^{\cosh ^{-1}(c+d x)}\right )}{d e^2}-\frac{\left (24 i b^4\right ) \operatorname{Subst}\left (\int \text{Li}_3\left (-i e^x\right ) \, dx,x,\cosh ^{-1}(c+d x)\right )}{d e^2}+\frac{\left (24 i b^4\right ) \operatorname{Subst}\left (\int \text{Li}_3\left (i e^x\right ) \, dx,x,\cosh ^{-1}(c+d x)\right )}{d e^2}\\ &=-\frac{\left (a+b \cosh ^{-1}(c+d x)\right )^4}{d e^2 (c+d x)}+\frac{8 b \left (a+b \cosh ^{-1}(c+d x)\right )^3 \tan ^{-1}\left (e^{\cosh ^{-1}(c+d x)}\right )}{d e^2}-\frac{12 i b^2 \left (a+b \cosh ^{-1}(c+d x)\right )^2 \text{Li}_2\left (-i e^{\cosh ^{-1}(c+d x)}\right )}{d e^2}+\frac{12 i b^2 \left (a+b \cosh ^{-1}(c+d x)\right )^2 \text{Li}_2\left (i e^{\cosh ^{-1}(c+d x)}\right )}{d e^2}+\frac{24 i b^3 \left (a+b \cosh ^{-1}(c+d x)\right ) \text{Li}_3\left (-i e^{\cosh ^{-1}(c+d x)}\right )}{d e^2}-\frac{24 i b^3 \left (a+b \cosh ^{-1}(c+d x)\right ) \text{Li}_3\left (i e^{\cosh ^{-1}(c+d x)}\right )}{d e^2}-\frac{\left (24 i b^4\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(-i x)}{x} \, dx,x,e^{\cosh ^{-1}(c+d x)}\right )}{d e^2}+\frac{\left (24 i b^4\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(i x)}{x} \, dx,x,e^{\cosh ^{-1}(c+d x)}\right )}{d e^2}\\ &=-\frac{\left (a+b \cosh ^{-1}(c+d x)\right )^4}{d e^2 (c+d x)}+\frac{8 b \left (a+b \cosh ^{-1}(c+d x)\right )^3 \tan ^{-1}\left (e^{\cosh ^{-1}(c+d x)}\right )}{d e^2}-\frac{12 i b^2 \left (a+b \cosh ^{-1}(c+d x)\right )^2 \text{Li}_2\left (-i e^{\cosh ^{-1}(c+d x)}\right )}{d e^2}+\frac{12 i b^2 \left (a+b \cosh ^{-1}(c+d x)\right )^2 \text{Li}_2\left (i e^{\cosh ^{-1}(c+d x)}\right )}{d e^2}+\frac{24 i b^3 \left (a+b \cosh ^{-1}(c+d x)\right ) \text{Li}_3\left (-i e^{\cosh ^{-1}(c+d x)}\right )}{d e^2}-\frac{24 i b^3 \left (a+b \cosh ^{-1}(c+d x)\right ) \text{Li}_3\left (i e^{\cosh ^{-1}(c+d x)}\right )}{d e^2}-\frac{24 i b^4 \text{Li}_4\left (-i e^{\cosh ^{-1}(c+d x)}\right )}{d e^2}+\frac{24 i b^4 \text{Li}_4\left (i e^{\cosh ^{-1}(c+d x)}\right )}{d e^2}\\ \end{align*}

Mathematica [B]  time = 2.34485, size = 872, normalized size = 3.3 \[ \frac{-\frac{a^4}{c+d x}+4 b \left (2 \tan ^{-1}\left (\tanh \left (\frac{1}{2} \cosh ^{-1}(c+d x)\right )\right )-\frac{\cosh ^{-1}(c+d x)}{c+d x}\right ) a^3-6 i b^2 \left (\cosh ^{-1}(c+d x) \left (-\frac{i \cosh ^{-1}(c+d x)}{c+d x}+2 \log \left (1-i e^{-\cosh ^{-1}(c+d x)}\right )-2 \log \left (1+i e^{-\cosh ^{-1}(c+d x)}\right )\right )+2 \text{PolyLog}\left (2,-i e^{-\cosh ^{-1}(c+d x)}\right )-2 \text{PolyLog}\left (2,i e^{-\cosh ^{-1}(c+d x)}\right )\right ) a^2+4 b^3 \left (3 i \left (-\left (\log \left (1-i e^{-\cosh ^{-1}(c+d x)}\right )-\log \left (1+i e^{-\cosh ^{-1}(c+d x)}\right )\right ) \cosh ^{-1}(c+d x)^2-2 \left (\text{PolyLog}\left (2,-i e^{-\cosh ^{-1}(c+d x)}\right )-\text{PolyLog}\left (2,i e^{-\cosh ^{-1}(c+d x)}\right )\right ) \cosh ^{-1}(c+d x)-2 \text{PolyLog}\left (3,-i e^{-\cosh ^{-1}(c+d x)}\right )+2 \text{PolyLog}\left (3,i e^{-\cosh ^{-1}(c+d x)}\right )\right )-\frac{\cosh ^{-1}(c+d x)^3}{c+d x}\right ) a+b^4 \left (-\frac{\cosh ^{-1}(c+d x)^4}{c+d x}+i \cosh ^{-1}(c+d x)^4+4 i \log \left (1+i e^{-\cosh ^{-1}(c+d x)}\right ) \cosh ^{-1}(c+d x)^3-4 i \log \left (1+i e^{\cosh ^{-1}(c+d x)}\right ) \cosh ^{-1}(c+d x)^3-2 \pi \cosh ^{-1}(c+d x)^3-6 \pi \log \left (1+i e^{-\cosh ^{-1}(c+d x)}\right ) \cosh ^{-1}(c+d x)^2+6 \pi \log \left (1-i e^{\cosh ^{-1}(c+d x)}\right ) \cosh ^{-1}(c+d x)^2-12 i \text{PolyLog}\left (2,-i e^{\cosh ^{-1}(c+d x)}\right ) \cosh ^{-1}(c+d x)^2-\frac{3}{2} i \pi ^2 \cosh ^{-1}(c+d x)^2-3 i \pi ^2 \log \left (1+i e^{-\cosh ^{-1}(c+d x)}\right ) \cosh ^{-1}(c+d x)+3 i \pi ^2 \log \left (1-i e^{\cosh ^{-1}(c+d x)}\right ) \cosh ^{-1}(c+d x)+12 \pi \text{PolyLog}\left (2,i e^{\cosh ^{-1}(c+d x)}\right ) \cosh ^{-1}(c+d x)-24 i \text{PolyLog}\left (3,-i e^{-\cosh ^{-1}(c+d x)}\right ) \cosh ^{-1}(c+d x)+24 i \text{PolyLog}\left (3,-i e^{\cosh ^{-1}(c+d x)}\right ) \cosh ^{-1}(c+d x)+\frac{1}{2} \pi ^3 \cosh ^{-1}(c+d x)+\frac{1}{2} \pi ^3 \log \left (1+i e^{-\cosh ^{-1}(c+d x)}\right )-\frac{1}{2} \pi ^3 \log \left (1+i e^{\cosh ^{-1}(c+d x)}\right )+\frac{1}{2} \pi ^3 \log \left (\tan \left (\frac{1}{4} \left (2 i \cosh ^{-1}(c+d x)+\pi \right )\right )\right )+3 i \left (\pi -2 i \cosh ^{-1}(c+d x)\right )^2 \text{PolyLog}\left (2,-i e^{-\cosh ^{-1}(c+d x)}\right )+3 i \pi ^2 \text{PolyLog}\left (2,i e^{\cosh ^{-1}(c+d x)}\right )+12 \pi \text{PolyLog}\left (3,-i e^{-\cosh ^{-1}(c+d x)}\right )-12 \pi \text{PolyLog}\left (3,i e^{\cosh ^{-1}(c+d x)}\right )-24 i \text{PolyLog}\left (4,-i e^{-\cosh ^{-1}(c+d x)}\right )-24 i \text{PolyLog}\left (4,-i e^{\cosh ^{-1}(c+d x)}\right )-\frac{7 i \pi ^4}{16}\right )}{d e^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-(a^4/(c + d*x)) + 4*a^3*b*(-(ArcCosh[c + d*x]/(c + d*x)) + 2*ArcTan[Tanh[ArcCosh[c + d*x]/2]]) - (6*I)*a^2*b
^2*(ArcCosh[c + d*x]*(((-I)*ArcCosh[c + d*x])/(c + d*x) + 2*Log[1 - I/E^ArcCosh[c + d*x]] - 2*Log[1 + I/E^ArcC
osh[c + d*x]]) + 2*PolyLog[2, (-I)/E^ArcCosh[c + d*x]] - 2*PolyLog[2, I/E^ArcCosh[c + d*x]]) + 4*a*b^3*(-(ArcC
osh[c + d*x]^3/(c + d*x)) + (3*I)*(-(ArcCosh[c + d*x]^2*(Log[1 - I/E^ArcCosh[c + d*x]] - Log[1 + I/E^ArcCosh[c
+ d*x]])) - 2*ArcCosh[c + d*x]*(PolyLog[2, (-I)/E^ArcCosh[c + d*x]] - PolyLog[2, I/E^ArcCosh[c + d*x]]) - 2*P
olyLog[3, (-I)/E^ArcCosh[c + d*x]] + 2*PolyLog[3, I/E^ArcCosh[c + d*x]])) + b^4*(((-7*I)/16)*Pi^4 + (Pi^3*ArcC
osh[c + d*x])/2 - ((3*I)/2)*Pi^2*ArcCosh[c + d*x]^2 - 2*Pi*ArcCosh[c + d*x]^3 + I*ArcCosh[c + d*x]^4 - ArcCosh
[c + d*x]^4/(c + d*x) + (Pi^3*Log[1 + I/E^ArcCosh[c + d*x]])/2 - (3*I)*Pi^2*ArcCosh[c + d*x]*Log[1 + I/E^ArcCo
sh[c + d*x]] - 6*Pi*ArcCosh[c + d*x]^2*Log[1 + I/E^ArcCosh[c + d*x]] + (4*I)*ArcCosh[c + d*x]^3*Log[1 + I/E^Ar
cCosh[c + d*x]] + (3*I)*Pi^2*ArcCosh[c + d*x]*Log[1 - I*E^ArcCosh[c + d*x]] + 6*Pi*ArcCosh[c + d*x]^2*Log[1 -
I*E^ArcCosh[c + d*x]] - (Pi^3*Log[1 + I*E^ArcCosh[c + d*x]])/2 - (4*I)*ArcCosh[c + d*x]^3*Log[1 + I*E^ArcCosh[
c + d*x]] + (Pi^3*Log[Tan[(Pi + (2*I)*ArcCosh[c + d*x])/4]])/2 + (3*I)*(Pi - (2*I)*ArcCosh[c + d*x])^2*PolyLog
[2, (-I)/E^ArcCosh[c + d*x]] - (12*I)*ArcCosh[c + d*x]^2*PolyLog[2, (-I)*E^ArcCosh[c + d*x]] + (3*I)*Pi^2*Poly
Log[2, I*E^ArcCosh[c + d*x]] + 12*Pi*ArcCosh[c + d*x]*PolyLog[2, I*E^ArcCosh[c + d*x]] + 12*Pi*PolyLog[3, (-I)
/E^ArcCosh[c + d*x]] - (24*I)*ArcCosh[c + d*x]*PolyLog[3, (-I)/E^ArcCosh[c + d*x]] + (24*I)*ArcCosh[c + d*x]*P
olyLog[3, (-I)*E^ArcCosh[c + d*x]] - 12*Pi*PolyLog[3, I*E^ArcCosh[c + d*x]] - (24*I)*PolyLog[4, (-I)/E^ArcCosh
[c + d*x]] - (24*I)*PolyLog[4, (-I)*E^ArcCosh[c + d*x]]))/(d*e^2)

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Maple [F]  time = 0.086, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b{\rm arccosh} \left (dx+c\right ) \right ) ^{4}}{ \left ( dex+ce \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arccosh(d*x+c))^4/(d*e*x+c*e)^2,x)

[Out]

int((a+b*arccosh(d*x+c))^4/(d*e*x+c*e)^2,x)

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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((a+b*arccosh(d*x+c))^4/(d*e*x+c*e)^2,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{b^{4} \operatorname{arcosh}\left (d x + c\right )^{4} + 4 \, a b^{3} \operatorname{arcosh}\left (d x + c\right )^{3} + 6 \, a^{2} b^{2} \operatorname{arcosh}\left (d x + c\right )^{2} + 4 \, a^{3} b \operatorname{arcosh}\left (d x + c\right ) + a^{4}}{d^{2} e^{2} x^{2} + 2 \, c d e^{2} x + c^{2} e^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccosh(d*x+c))^4/(d*e*x+c*e)^2,x, algorithm="fricas")

[Out]

integral((b^4*arccosh(d*x + c)^4 + 4*a*b^3*arccosh(d*x + c)^3 + 6*a^2*b^2*arccosh(d*x + c)^2 + 4*a^3*b*arccosh
(d*x + c) + a^4)/(d^2*e^2*x^2 + 2*c*d*e^2*x + c^2*e^2), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{a^{4}}{c^{2} + 2 c d x + d^{2} x^{2}}\, dx + \int \frac{b^{4} \operatorname{acosh}^{4}{\left (c + d x \right )}}{c^{2} + 2 c d x + d^{2} x^{2}}\, dx + \int \frac{4 a b^{3} \operatorname{acosh}^{3}{\left (c + d x \right )}}{c^{2} + 2 c d x + d^{2} x^{2}}\, dx + \int \frac{6 a^{2} b^{2} \operatorname{acosh}^{2}{\left (c + d x \right )}}{c^{2} + 2 c d x + d^{2} x^{2}}\, dx + \int \frac{4 a^{3} b \operatorname{acosh}{\left (c + d x \right )}}{c^{2} + 2 c d x + d^{2} x^{2}}\, dx}{e^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*acosh(d*x+c))**4/(d*e*x+c*e)**2,x)

[Out]

(Integral(a**4/(c**2 + 2*c*d*x + d**2*x**2), x) + Integral(b**4*acosh(c + d*x)**4/(c**2 + 2*c*d*x + d**2*x**2)
, x) + Integral(4*a*b**3*acosh(c + d*x)**3/(c**2 + 2*c*d*x + d**2*x**2), x) + Integral(6*a**2*b**2*acosh(c + d
*x)**2/(c**2 + 2*c*d*x + d**2*x**2), x) + Integral(4*a**3*b*acosh(c + d*x)/(c**2 + 2*c*d*x + d**2*x**2), x))/e
**2

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccosh(d*x+c))^4/(d*e*x+c*e)^2,x, algorithm="giac")

[Out]

Exception raised: TypeError