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

Optimal. Leaf size=432 \[ \frac {4 i b^3 \text {Li}_3\left (-i e^{\cosh ^{-1}(c+d x)}\right ) \left (a+b \cosh ^{-1}(c+d x)\right )}{d e^4}-\frac {4 i b^3 \text {Li}_3\left (i e^{\cosh ^{-1}(c+d x)}\right ) \left (a+b \cosh ^{-1}(c+d x)\right )}{d e^4}-\frac {8 b^3 \tan ^{-1}\left (e^{\cosh ^{-1}(c+d x)}\right ) \left (a+b \cosh ^{-1}(c+d x)\right )}{d e^4}-\frac {2 i b^2 \text {Li}_2\left (-i e^{\cosh ^{-1}(c+d x)}\right ) \left (a+b \cosh ^{-1}(c+d x)\right )^2}{d e^4}+\frac {2 i b^2 \text {Li}_2\left (i e^{\cosh ^{-1}(c+d x)}\right ) \left (a+b \cosh ^{-1}(c+d x)\right )^2}{d e^4}+\frac {2 b^2 \left (a+b \cosh ^{-1}(c+d x)\right )^2}{d e^4 (c+d x)}+\frac {2 b \sqrt {c+d x-1} \sqrt {c+d x+1} \left (a+b \cosh ^{-1}(c+d x)\right )^3}{3 d e^4 (c+d x)^2}-\frac {\left (a+b \cosh ^{-1}(c+d x)\right )^4}{3 d e^4 (c+d x)^3}+\frac {4 b \tan ^{-1}\left (e^{\cosh ^{-1}(c+d x)}\right ) \left (a+b \cosh ^{-1}(c+d x)\right )^3}{3 d e^4}+\frac {4 i b^4 \text {Li}_2\left (-i e^{\cosh ^{-1}(c+d x)}\right )}{d e^4}-\frac {4 i b^4 \text {Li}_2\left (i e^{\cosh ^{-1}(c+d x)}\right )}{d e^4}-\frac {4 i b^4 \text {Li}_4\left (-i e^{\cosh ^{-1}(c+d x)}\right )}{d e^4}+\frac {4 i b^4 \text {Li}_4\left (i e^{\cosh ^{-1}(c+d x)}\right )}{d e^4} \]

[Out]

2*b^2*(a+b*arccosh(d*x+c))^2/d/e^4/(d*x+c)-1/3*(a+b*arccosh(d*x+c))^4/d/e^4/(d*x+c)^3-8*b^3*(a+b*arccosh(d*x+c
))*arctan(d*x+c+(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2))/d/e^4+4/3*b*(a+b*arccosh(d*x+c))^3*arctan(d*x+c+(d*x+c-1)^(1/
2)*(d*x+c+1)^(1/2))/d/e^4+4*I*b^4*polylog(2,-I*(d*x+c+(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)))/d/e^4-2*I*b^2*(a+b*arc
cosh(d*x+c))^2*polylog(2,-I*(d*x+c+(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)))/d/e^4-4*I*b^4*polylog(2,I*(d*x+c+(d*x+c-1
)^(1/2)*(d*x+c+1)^(1/2)))/d/e^4+2*I*b^2*(a+b*arccosh(d*x+c))^2*polylog(2,I*(d*x+c+(d*x+c-1)^(1/2)*(d*x+c+1)^(1
/2)))/d/e^4+4*I*b^3*(a+b*arccosh(d*x+c))*polylog(3,-I*(d*x+c+(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)))/d/e^4-4*I*b^3*(
a+b*arccosh(d*x+c))*polylog(3,I*(d*x+c+(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)))/d/e^4-4*I*b^4*polylog(4,-I*(d*x+c+(d*
x+c-1)^(1/2)*(d*x+c+1)^(1/2)))/d/e^4+4*I*b^4*polylog(4,I*(d*x+c+(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)))/d/e^4+2/3*b*
(a+b*arccosh(d*x+c))^3*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)/d/e^4/(d*x+c)^2

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Rubi [A]  time = 0.84, antiderivative size = 432, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 12, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {5866, 12, 5662, 5748, 5761, 4180, 2531, 6609, 2282, 6589, 2279, 2391} \[ \frac {4 i b^3 \text {PolyLog}\left (3,-i e^{\cosh ^{-1}(c+d x)}\right ) \left (a+b \cosh ^{-1}(c+d x)\right )}{d e^4}-\frac {4 i b^3 \text {PolyLog}\left (3,i e^{\cosh ^{-1}(c+d x)}\right ) \left (a+b \cosh ^{-1}(c+d x)\right )}{d e^4}-\frac {2 i b^2 \text {PolyLog}\left (2,-i e^{\cosh ^{-1}(c+d x)}\right ) \left (a+b \cosh ^{-1}(c+d x)\right )^2}{d e^4}+\frac {2 i b^2 \text {PolyLog}\left (2,i e^{\cosh ^{-1}(c+d x)}\right ) \left (a+b \cosh ^{-1}(c+d x)\right )^2}{d e^4}+\frac {4 i b^4 \text {PolyLog}\left (2,-i e^{\cosh ^{-1}(c+d x)}\right )}{d e^4}-\frac {4 i b^4 \text {PolyLog}\left (2,i e^{\cosh ^{-1}(c+d x)}\right )}{d e^4}-\frac {4 i b^4 \text {PolyLog}\left (4,-i e^{\cosh ^{-1}(c+d x)}\right )}{d e^4}+\frac {4 i b^4 \text {PolyLog}\left (4,i e^{\cosh ^{-1}(c+d x)}\right )}{d e^4}+\frac {2 b^2 \left (a+b \cosh ^{-1}(c+d x)\right )^2}{d e^4 (c+d x)}-\frac {8 b^3 \tan ^{-1}\left (e^{\cosh ^{-1}(c+d x)}\right ) \left (a+b \cosh ^{-1}(c+d x)\right )}{d e^4}+\frac {2 b \sqrt {c+d x-1} \sqrt {c+d x+1} \left (a+b \cosh ^{-1}(c+d x)\right )^3}{3 d e^4 (c+d x)^2}-\frac {\left (a+b \cosh ^{-1}(c+d x)\right )^4}{3 d e^4 (c+d x)^3}+\frac {4 b \tan ^{-1}\left (e^{\cosh ^{-1}(c+d x)}\right ) \left (a+b \cosh ^{-1}(c+d x)\right )^3}{3 d e^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*b^2*(a + b*ArcCosh[c + d*x])^2)/(d*e^4*(c + d*x)) + (2*b*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x]*(a + b*ArcCos
h[c + d*x])^3)/(3*d*e^4*(c + d*x)^2) - (a + b*ArcCosh[c + d*x])^4/(3*d*e^4*(c + d*x)^3) - (8*b^3*(a + b*ArcCos
h[c + d*x])*ArcTan[E^ArcCosh[c + d*x]])/(d*e^4) + (4*b*(a + b*ArcCosh[c + d*x])^3*ArcTan[E^ArcCosh[c + d*x]])/
(3*d*e^4) + ((4*I)*b^4*PolyLog[2, (-I)*E^ArcCosh[c + d*x]])/(d*e^4) - ((2*I)*b^2*(a + b*ArcCosh[c + d*x])^2*Po
lyLog[2, (-I)*E^ArcCosh[c + d*x]])/(d*e^4) - ((4*I)*b^4*PolyLog[2, I*E^ArcCosh[c + d*x]])/(d*e^4) + ((2*I)*b^2
*(a + b*ArcCosh[c + d*x])^2*PolyLog[2, I*E^ArcCosh[c + d*x]])/(d*e^4) + ((4*I)*b^3*(a + b*ArcCosh[c + d*x])*Po
lyLog[3, (-I)*E^ArcCosh[c + d*x]])/(d*e^4) - ((4*I)*b^3*(a + b*ArcCosh[c + d*x])*PolyLog[3, I*E^ArcCosh[c + d*
x]])/(d*e^4) - ((4*I)*b^4*PolyLog[4, (-I)*E^ArcCosh[c + d*x]])/(d*e^4) + ((4*I)*b^4*PolyLog[4, I*E^ArcCosh[c +
d*x]])/(d*e^4)

Rule 12

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

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 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 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 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 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 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 5748

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 + 1)*(d2 + e2*x)^(p + 1)*(a + b*ArcCosh[c*x])^n)/(d1*
d2*f*(m + 1)), x] + (Dist[(c^2*(m + 2*p + 3))/(f^2*(m + 1)), Int[(f*x)^(m + 2)*(d1 + e1*x)^p*(d2 + e2*x)^p*(a
+ b*ArcCosh[c*x])^n, x], x] + Dist[(b*c*n*(-(d1*d2))^IntPart[p]*(d1 + e1*x)^FracPart[p]*(d2 + e2*x)^FracPart[p
])/(f*(m + 1)*(1 + c*x)^FracPart[p]*(-1 + c*x)^FracPart[p]), 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, p}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 +
c*d2, 0] && GtQ[n, 0] && LtQ[m, -1] && IntegerQ[m] && IntegerQ[p + 1/2]

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 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 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]

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]

Rubi steps

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

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Mathematica [B]  time = 9.36, size = 1374, normalized size = 3.18 \[ \text {result too large to display} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-1/3*a^4/(d*e^4*(c + d*x)^3) + (4*a^3*b*Sqrt[-1 + c + d*x]*((Sqrt[(-1 + c + d*x)/(1 + c + d*x)]*(1 + c + d*x))
/(6*(c + d*x)^2) - ArcCosh[c + d*x]/(3*(c + d*x)^3) + ArcTan[Tanh[ArcCosh[c + d*x]/2]]/3))/(d*e^4*Sqrt[(-1 + c
+ d*x)/(1 + c + d*x)]*Sqrt[1 + c + d*x]) + (2*a^2*b^2*Sqrt[-1 + c + d*x]*((c + d*x)^(-1) + (Sqrt[(-1 + c + d*
x)/(1 + c + d*x)]*(1 + c + d*x)*ArcCosh[c + d*x])/(c + d*x)^2 - ArcCosh[c + d*x]^2/(c + d*x)^3 - I*ArcCosh[c +
d*x]*Log[1 - I/E^ArcCosh[c + d*x]] + I*ArcCosh[c + d*x]*Log[1 + I/E^ArcCosh[c + d*x]] - I*PolyLog[2, (-I)/E^A
rcCosh[c + d*x]] + I*PolyLog[2, I/E^ArcCosh[c + d*x]]))/(d*e^4*Sqrt[(-1 + c + d*x)/(1 + c + d*x)]*Sqrt[1 + c +
d*x]) + (4*a*b^3*Sqrt[-1 + c + d*x]*(ArcCosh[c + d*x]/(c + d*x) + (Sqrt[(-1 + c + d*x)/(1 + c + d*x)]*(1 + c
+ d*x)*ArcCosh[c + d*x]^2)/(2*(c + d*x)^2) - ArcCosh[c + d*x]^3/(3*(c + d*x)^3) - (I/2)*((-4*I)*ArcTan[Tanh[Ar
cCosh[c + d*x]/2]] + ArcCosh[c + d*x]^2*Log[1 - I/E^ArcCosh[c + d*x]] - ArcCosh[c + d*x]^2*Log[1 + I/E^ArcCosh
[c + d*x]] + 2*ArcCosh[c + d*x]*PolyLog[2, (-I)/E^ArcCosh[c + d*x]] - 2*ArcCosh[c + d*x]*PolyLog[2, I/E^ArcCos
h[c + d*x]] + 2*PolyLog[3, (-I)/E^ArcCosh[c + d*x]] - 2*PolyLog[3, I/E^ArcCosh[c + d*x]])))/(d*e^4*Sqrt[(-1 +
c + d*x)/(1 + c + d*x)]*Sqrt[1 + c + d*x]) + (b^4*Sqrt[-1 + c + d*x]*((I/2)*(8 + Pi^2 - (4*I)*Pi*ArcCosh[c + d
*x] - 4*ArcCosh[c + d*x]^2)*PolyLog[2, (-I)/E^ArcCosh[c + d*x]] - (I/96)*(7*Pi^4 + (8*I)*Pi^3*ArcCosh[c + d*x]
+ 24*Pi^2*ArcCosh[c + d*x]^2 + ((192*I)*ArcCosh[c + d*x]^2)/(c + d*x) - (32*I)*Pi*ArcCosh[c + d*x]^3 + ((64*I
)*Sqrt[(-1 + c + d*x)/(1 + c + d*x)]*(1 + c + d*x)*ArcCosh[c + d*x]^3)/(c + d*x)^2 - 16*ArcCosh[c + d*x]^4 - (
(32*I)*ArcCosh[c + d*x]^4)/(c + d*x)^3 - 384*ArcCosh[c + d*x]*Log[1 - I/E^ArcCosh[c + d*x]] + (8*I)*Pi^3*Log[1
+ I/E^ArcCosh[c + d*x]] + 384*ArcCosh[c + d*x]*Log[1 + I/E^ArcCosh[c + d*x]] + 48*Pi^2*ArcCosh[c + d*x]*Log[1
+ I/E^ArcCosh[c + d*x]] - (96*I)*Pi*ArcCosh[c + d*x]^2*Log[1 + I/E^ArcCosh[c + d*x]] - 64*ArcCosh[c + d*x]^3*
Log[1 + I/E^ArcCosh[c + d*x]] - 48*Pi^2*ArcCosh[c + d*x]*Log[1 - I*E^ArcCosh[c + d*x]] + (96*I)*Pi*ArcCosh[c +
d*x]^2*Log[1 - I*E^ArcCosh[c + d*x]] - (8*I)*Pi^3*Log[1 + I*E^ArcCosh[c + d*x]] + 64*ArcCosh[c + d*x]^3*Log[1
+ I*E^ArcCosh[c + d*x]] + (8*I)*Pi^3*Log[Tan[(Pi + (2*I)*ArcCosh[c + d*x])/4]] + 384*PolyLog[2, I/E^ArcCosh[c
+ d*x]] + 192*ArcCosh[c + d*x]^2*PolyLog[2, (-I)*E^ArcCosh[c + d*x]] - 48*Pi^2*PolyLog[2, I*E^ArcCosh[c + d*x
]] + (192*I)*Pi*ArcCosh[c + d*x]*PolyLog[2, I*E^ArcCosh[c + d*x]] + (192*I)*Pi*PolyLog[3, (-I)/E^ArcCosh[c + d
*x]] + 384*ArcCosh[c + d*x]*PolyLog[3, (-I)/E^ArcCosh[c + d*x]] - 384*ArcCosh[c + d*x]*PolyLog[3, (-I)*E^ArcCo
sh[c + d*x]] - (192*I)*Pi*PolyLog[3, I*E^ArcCosh[c + d*x]] + 384*PolyLog[4, (-I)/E^ArcCosh[c + d*x]] + 384*Pol
yLog[4, (-I)*E^ArcCosh[c + d*x]])))/(d*e^4*Sqrt[(-1 + c + d*x)/(1 + c + d*x)]*Sqrt[1 + c + d*x])

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fricas [F]  time = 0.68, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{4} \operatorname {arcosh}\left (d x + c\right )^{4} + 4 \, a b^{3} \operatorname {arcosh}\left (d x + c\right )^{3} + 6 \, a^{2} b^{2} \operatorname {arcosh}\left (d x + c\right )^{2} + 4 \, a^{3} b \operatorname {arcosh}\left (d x + c\right ) + a^{4}}{d^{4} e^{4} x^{4} + 4 \, c d^{3} e^{4} x^{3} + 6 \, c^{2} d^{2} e^{4} x^{2} + 4 \, c^{3} d e^{4} x + c^{4} e^{4}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{4}}{{\left (d e x + c e\right )}^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.48, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +b \,\mathrm {arccosh}\left (d x +c \right )\right )^{4}}{\left (d e x +c e \right )^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {b^{4} \log \left (d x + \sqrt {d x + c + 1} \sqrt {d x + c - 1} + c\right )^{4}}{3 \, {\left (d^{4} e^{4} x^{3} + 3 \, c d^{3} e^{4} x^{2} + 3 \, c^{2} d^{2} e^{4} x + c^{3} d e^{4}\right )}} - \frac {a^{4}}{3 \, {\left (d^{4} e^{4} x^{3} + 3 \, c d^{3} e^{4} x^{2} + 3 \, c^{2} d^{2} e^{4} x + c^{3} d e^{4}\right )}} + \int \frac {2 \, {\left (2 \, {\left (3 \, {\left (c^{3} - c\right )} a b^{3} + {\left (c^{3} - c\right )} b^{4} + {\left (3 \, a b^{3} d^{3} + b^{4} d^{3}\right )} x^{3} + 3 \, {\left (3 \, a b^{3} c d^{2} + b^{4} c d^{2}\right )} x^{2} + {\left (b^{4} c^{2} + 3 \, {\left (c^{2} - 1\right )} a b^{3} + {\left (3 \, a b^{3} d^{2} + b^{4} d^{2}\right )} x^{2} + 2 \, {\left (3 \, a b^{3} c d + b^{4} c d\right )} x\right )} \sqrt {d x + c + 1} \sqrt {d x + c - 1} + {\left (3 \, {\left (3 \, c^{2} d - d\right )} a b^{3} + {\left (3 \, c^{2} d - d\right )} b^{4}\right )} x\right )} \log \left (d x + \sqrt {d x + c + 1} \sqrt {d x + c - 1} + c\right )^{3} + 9 \, {\left (a^{2} b^{2} d^{3} x^{3} + 3 \, a^{2} b^{2} c d^{2} x^{2} + {\left (3 \, c^{2} d - d\right )} a^{2} b^{2} x + {\left (c^{3} - c\right )} a^{2} b^{2} + {\left (a^{2} b^{2} d^{2} x^{2} + 2 \, a^{2} b^{2} c d x + {\left (c^{2} - 1\right )} a^{2} b^{2}\right )} \sqrt {d x + c + 1} \sqrt {d x + c - 1}\right )} \log \left (d x + \sqrt {d x + c + 1} \sqrt {d x + c - 1} + c\right )^{2} + 6 \, {\left (a^{3} b d^{3} x^{3} + 3 \, a^{3} b c d^{2} x^{2} + {\left (3 \, c^{2} d - d\right )} a^{3} b x + {\left (c^{3} - c\right )} a^{3} b + {\left (a^{3} b d^{2} x^{2} + 2 \, a^{3} b c d x + {\left (c^{2} - 1\right )} a^{3} b\right )} \sqrt {d x + c + 1} \sqrt {d x + c - 1}\right )} \log \left (d x + \sqrt {d x + c + 1} \sqrt {d x + c - 1} + c\right )\right )}}{3 \, {\left (d^{7} e^{4} x^{7} + 7 \, c d^{6} e^{4} x^{6} + c^{7} e^{4} - c^{5} e^{4} + {\left (21 \, c^{2} d^{5} e^{4} - d^{5} e^{4}\right )} x^{5} + 5 \, {\left (7 \, c^{3} d^{4} e^{4} - c d^{4} e^{4}\right )} x^{4} + 5 \, {\left (7 \, c^{4} d^{3} e^{4} - 2 \, c^{2} d^{3} e^{4}\right )} x^{3} + {\left (21 \, c^{5} d^{2} e^{4} - 10 \, c^{3} d^{2} e^{4}\right )} x^{2} + {\left (d^{6} e^{4} x^{6} + 6 \, c d^{5} e^{4} x^{5} + c^{6} e^{4} - c^{4} e^{4} + {\left (15 \, c^{2} d^{4} e^{4} - d^{4} e^{4}\right )} x^{4} + 4 \, {\left (5 \, c^{3} d^{3} e^{4} - c d^{3} e^{4}\right )} x^{3} + 3 \, {\left (5 \, c^{4} d^{2} e^{4} - 2 \, c^{2} d^{2} e^{4}\right )} x^{2} + 2 \, {\left (3 \, c^{5} d e^{4} - 2 \, c^{3} d e^{4}\right )} x\right )} \sqrt {d x + c + 1} \sqrt {d x + c - 1} + {\left (7 \, c^{6} d e^{4} - 5 \, c^{4} d e^{4}\right )} x\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*b^4*log(d*x + sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + c)^4/(d^4*e^4*x^3 + 3*c*d^3*e^4*x^2 + 3*c^2*d^2*e^4*x
+ c^3*d*e^4) - 1/3*a^4/(d^4*e^4*x^3 + 3*c*d^3*e^4*x^2 + 3*c^2*d^2*e^4*x + c^3*d*e^4) + integrate(2/3*(2*(3*(c
^3 - c)*a*b^3 + (c^3 - c)*b^4 + (3*a*b^3*d^3 + b^4*d^3)*x^3 + 3*(3*a*b^3*c*d^2 + b^4*c*d^2)*x^2 + (b^4*c^2 + 3
*(c^2 - 1)*a*b^3 + (3*a*b^3*d^2 + b^4*d^2)*x^2 + 2*(3*a*b^3*c*d + b^4*c*d)*x)*sqrt(d*x + c + 1)*sqrt(d*x + c -
1) + (3*(3*c^2*d - d)*a*b^3 + (3*c^2*d - d)*b^4)*x)*log(d*x + sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + c)^3 + 9*
(a^2*b^2*d^3*x^3 + 3*a^2*b^2*c*d^2*x^2 + (3*c^2*d - d)*a^2*b^2*x + (c^3 - c)*a^2*b^2 + (a^2*b^2*d^2*x^2 + 2*a^
2*b^2*c*d*x + (c^2 - 1)*a^2*b^2)*sqrt(d*x + c + 1)*sqrt(d*x + c - 1))*log(d*x + sqrt(d*x + c + 1)*sqrt(d*x + c
- 1) + c)^2 + 6*(a^3*b*d^3*x^3 + 3*a^3*b*c*d^2*x^2 + (3*c^2*d - d)*a^3*b*x + (c^3 - c)*a^3*b + (a^3*b*d^2*x^2
+ 2*a^3*b*c*d*x + (c^2 - 1)*a^3*b)*sqrt(d*x + c + 1)*sqrt(d*x + c - 1))*log(d*x + sqrt(d*x + c + 1)*sqrt(d*x
+ c - 1) + c))/(d^7*e^4*x^7 + 7*c*d^6*e^4*x^6 + c^7*e^4 - c^5*e^4 + (21*c^2*d^5*e^4 - d^5*e^4)*x^5 + 5*(7*c^3*
d^4*e^4 - c*d^4*e^4)*x^4 + 5*(7*c^4*d^3*e^4 - 2*c^2*d^3*e^4)*x^3 + (21*c^5*d^2*e^4 - 10*c^3*d^2*e^4)*x^2 + (d^
6*e^4*x^6 + 6*c*d^5*e^4*x^5 + c^6*e^4 - c^4*e^4 + (15*c^2*d^4*e^4 - d^4*e^4)*x^4 + 4*(5*c^3*d^3*e^4 - c*d^3*e^
4)*x^3 + 3*(5*c^4*d^2*e^4 - 2*c^2*d^2*e^4)*x^2 + 2*(3*c^5*d*e^4 - 2*c^3*d*e^4)*x)*sqrt(d*x + c + 1)*sqrt(d*x +
c - 1) + (7*c^6*d*e^4 - 5*c^4*d*e^4)*x), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right )}^4}{{\left (c\,e+d\,e\,x\right )}^4} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {a^{4}}{c^{4} + 4 c^{3} d x + 6 c^{2} d^{2} x^{2} + 4 c d^{3} x^{3} + d^{4} x^{4}}\, dx + \int \frac {b^{4} \operatorname {acosh}^{4}{\left (c + d x \right )}}{c^{4} + 4 c^{3} d x + 6 c^{2} d^{2} x^{2} + 4 c d^{3} x^{3} + d^{4} x^{4}}\, dx + \int \frac {4 a b^{3} \operatorname {acosh}^{3}{\left (c + d x \right )}}{c^{4} + 4 c^{3} d x + 6 c^{2} d^{2} x^{2} + 4 c d^{3} x^{3} + d^{4} x^{4}}\, dx + \int \frac {6 a^{2} b^{2} \operatorname {acosh}^{2}{\left (c + d x \right )}}{c^{4} + 4 c^{3} d x + 6 c^{2} d^{2} x^{2} + 4 c d^{3} x^{3} + d^{4} x^{4}}\, dx + \int \frac {4 a^{3} b \operatorname {acosh}{\left (c + d x \right )}}{c^{4} + 4 c^{3} d x + 6 c^{2} d^{2} x^{2} + 4 c d^{3} x^{3} + d^{4} x^{4}}\, dx}{e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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