3.2.29 \(\int \frac {(a+b \cosh ^{-1}(c+d x))^4}{(c e+d e x)^4} \, dx\) [129]

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

[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

________________________________________________________________________________________

Rubi [A]
time = 0.55, 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 = {5996, 12, 5883, 5933, 5947, 4265, 2611, 6744, 2320, 6724, 2317, 2438} \begin {gather*} -\frac {8 b^3 \text {ArcTan}\left (e^{\cosh ^{-1}(c+d x)}\right ) \left (a+b \cosh ^{-1}(c+d x)\right )}{d e^4}+\frac {4 b \text {ArcTan}\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^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 {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 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} \end {gather*}

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 2317

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 2320

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 2438

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 2611

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 4265

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 5883

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

Rule 5933

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*d
2*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/(f*(m + 1)))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*x)^p/(-1
+ c*x)^p], Int[(f*x)^(m + 1)*(1 + c*x)^(p + 1/2)*(-1 + c*x)^(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] && EqQ[e2, (-c)*d2] && GtQ[n, 0] && ILtQ[m, -1]

Rule 5947

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_))/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)]
), x_Symbol] :> Dist[(1/c^(m + 1))*Simp[Sqrt[1 + c*x]/Sqrt[d1 + e1*x]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]], S
ubst[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*d
1] && EqQ[e2, (-c)*d2] && IGtQ[n, 0] && IntegerQ[m]

Rule 5996

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 6724

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 6744

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 {\text {Subst}\left (\int \frac {\left (a+b \cosh ^{-1}(x)\right )^4}{e^4 x^4} \, dx,x,c+d x\right )}{d}\\ &=\frac {\text {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) \text {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) \text {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 ) \text {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) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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] Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1309\) vs. \(2(432)=864\).
time = 7.42, size = 1309, normalized size = 3.03 \begin {gather*} \frac {-\frac {a^4}{(c+d x)^3}+2 a^3 b \left (\frac {\sqrt {\frac {-1+c+d x}{1+c+d x}} (1+c+d x)}{(c+d x)^2}-\frac {2 \cosh ^{-1}(c+d x)}{(c+d x)^3}+2 \text {ArcTan}\left (\tanh \left (\frac {1}{2} \cosh ^{-1}(c+d x)\right )\right )\right )+6 a^2 b^2 \left (\frac {1}{c+d x}+\frac {\sqrt {\frac {-1+c+d x}{1+c+d x}} (1+c+d x) \cosh ^{-1}(c+d x)}{(c+d x)^2}-\frac {\cosh ^{-1}(c+d x)^2}{(c+d x)^3}-i \cosh ^{-1}(c+d x) \log \left (1-i e^{-\cosh ^{-1}(c+d x)}\right )+i \cosh ^{-1}(c+d x) \log \left (1+i e^{-\cosh ^{-1}(c+d x)}\right )-i \text {PolyLog}\left (2,-i e^{-\cosh ^{-1}(c+d x)}\right )+i \text {PolyLog}\left (2,i e^{-\cosh ^{-1}(c+d x)}\right )\right )+12 a b^3 \left (\frac {\cosh ^{-1}(c+d x)}{c+d x}+\frac {\sqrt {\frac {-1+c+d x}{1+c+d x}} (1+c+d x) \cosh ^{-1}(c+d x)^2}{2 (c+d x)^2}-\frac {\cosh ^{-1}(c+d x)^3}{3 (c+d x)^3}-2 \text {ArcTan}\left (c+d x+\sqrt {\frac {-1+c+d x}{1+c+d x}} (1+c+d x)\right )+\cosh ^{-1}(c+d x)^2 \text {ArcTan}\left (c+d x+\sqrt {\frac {-1+c+d x}{1+c+d x}} (1+c+d x)\right )-i \cosh ^{-1}(c+d x) \text {PolyLog}\left (2,-i \left (c+d x+\sqrt {\frac {-1+c+d x}{1+c+d x}} (1+c+d x)\right )\right )+i \cosh ^{-1}(c+d x) \text {PolyLog}\left (2,i \left (c+d x+\sqrt {\frac {-1+c+d x}{1+c+d x}} (1+c+d x)\right )\right )+i \text {PolyLog}\left (3,-i \left (c+d x+\sqrt {\frac {-1+c+d x}{1+c+d x}} (1+c+d x)\right )\right )-i \text {PolyLog}\left (3,i \left (c+d x+\sqrt {\frac {-1+c+d x}{1+c+d x}} (1+c+d x)\right )\right )\right )+3 b^4 \left (-\frac {7 i \pi ^4}{96}+\frac {1}{12} \pi ^3 \cosh ^{-1}(c+d x)-\frac {1}{4} i \pi ^2 \cosh ^{-1}(c+d x)^2+\frac {2 \cosh ^{-1}(c+d x)^2}{c+d x}-\frac {1}{3} \pi \cosh ^{-1}(c+d x)^3+\frac {2 \sqrt {\frac {-1+c+d x}{1+c+d x}} (1+c+d x) \cosh ^{-1}(c+d x)^3}{3 (c+d x)^2}+\frac {1}{6} i \cosh ^{-1}(c+d x)^4-\frac {\cosh ^{-1}(c+d x)^4}{3 (c+d x)^3}+4 i \cosh ^{-1}(c+d x) \log \left (1-i e^{-\cosh ^{-1}(c+d x)}\right )+\frac {1}{12} \pi ^3 \log \left (1+i e^{-\cosh ^{-1}(c+d x)}\right )-4 i \cosh ^{-1}(c+d x) \log \left (1+i e^{-\cosh ^{-1}(c+d x)}\right )-\frac {1}{2} i \pi ^2 \cosh ^{-1}(c+d x) \log \left (1+i e^{-\cosh ^{-1}(c+d x)}\right )-\pi \cosh ^{-1}(c+d x)^2 \log \left (1+i e^{-\cosh ^{-1}(c+d x)}\right )+\frac {2}{3} i \cosh ^{-1}(c+d x)^3 \log \left (1+i e^{-\cosh ^{-1}(c+d x)}\right )+\frac {1}{2} i \pi ^2 \cosh ^{-1}(c+d x) \log \left (1-i e^{\cosh ^{-1}(c+d x)}\right )+\pi \cosh ^{-1}(c+d x)^2 \log \left (1-i e^{\cosh ^{-1}(c+d x)}\right )-\frac {1}{12} \pi ^3 \log \left (1+i e^{\cosh ^{-1}(c+d x)}\right )-\frac {2}{3} i \cosh ^{-1}(c+d x)^3 \log \left (1+i e^{\cosh ^{-1}(c+d x)}\right )+\frac {1}{12} \pi ^3 \log \left (\tan \left (\frac {1}{4} \left (\pi +2 i \cosh ^{-1}(c+d x)\right )\right )\right )+\frac {1}{2} i \left (8+\pi ^2-4 i \pi \cosh ^{-1}(c+d x)-4 \cosh ^{-1}(c+d x)^2\right ) \text {PolyLog}\left (2,-i e^{-\cosh ^{-1}(c+d x)}\right )-4 i \text {PolyLog}\left (2,i e^{-\cosh ^{-1}(c+d x)}\right )-2 i \cosh ^{-1}(c+d x)^2 \text {PolyLog}\left (2,-i e^{\cosh ^{-1}(c+d x)}\right )+\frac {1}{2} i \pi ^2 \text {PolyLog}\left (2,i e^{\cosh ^{-1}(c+d x)}\right )+2 \pi \cosh ^{-1}(c+d x) \text {PolyLog}\left (2,i e^{\cosh ^{-1}(c+d x)}\right )+2 \pi \text {PolyLog}\left (3,-i e^{-\cosh ^{-1}(c+d x)}\right )-4 i \cosh ^{-1}(c+d x) \text {PolyLog}\left (3,-i e^{-\cosh ^{-1}(c+d x)}\right )+4 i \cosh ^{-1}(c+d x) \text {PolyLog}\left (3,-i e^{\cosh ^{-1}(c+d x)}\right )-2 \pi \text {PolyLog}\left (3,i e^{\cosh ^{-1}(c+d x)}\right )-4 i \text {PolyLog}\left (4,-i e^{-\cosh ^{-1}(c+d x)}\right )-4 i \text {PolyLog}\left (4,-i e^{\cosh ^{-1}(c+d x)}\right )\right )}{3 d e^4} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-(a^4/(c + d*x)^3) + 2*a^3*b*((Sqrt[(-1 + c + d*x)/(1 + c + d*x)]*(1 + c + d*x))/(c + d*x)^2 - (2*ArcCosh[c +
d*x])/(c + d*x)^3 + 2*ArcTan[Tanh[ArcCosh[c + d*x]/2]]) + 6*a^2*b^2*((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^ArcCos
h[c + d*x]] + I*PolyLog[2, I/E^ArcCosh[c + d*x]]) + 12*a*b^3*(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) - 2*Ar
cTan[c + d*x + Sqrt[(-1 + c + d*x)/(1 + c + d*x)]*(1 + c + d*x)] + ArcCosh[c + d*x]^2*ArcTan[c + d*x + Sqrt[(-
1 + c + d*x)/(1 + c + d*x)]*(1 + c + d*x)] - I*ArcCosh[c + d*x]*PolyLog[2, (-I)*(c + d*x + Sqrt[(-1 + c + d*x)
/(1 + c + d*x)]*(1 + c + d*x))] + I*ArcCosh[c + d*x]*PolyLog[2, I*(c + d*x + Sqrt[(-1 + c + d*x)/(1 + c + d*x)
]*(1 + c + d*x))] + I*PolyLog[3, (-I)*(c + d*x + Sqrt[(-1 + c + d*x)/(1 + c + d*x)]*(1 + c + d*x))] - I*PolyLo
g[3, I*(c + d*x + Sqrt[(-1 + c + d*x)/(1 + c + d*x)]*(1 + c + d*x))]) + 3*b^4*(((-7*I)/96)*Pi^4 + (Pi^3*ArcCos
h[c + d*x])/12 - (I/4)*Pi^2*ArcCosh[c + d*x]^2 + (2*ArcCosh[c + d*x]^2)/(c + d*x) - (Pi*ArcCosh[c + d*x]^3)/3
+ (2*Sqrt[(-1 + c + d*x)/(1 + c + d*x)]*(1 + c + d*x)*ArcCosh[c + d*x]^3)/(3*(c + d*x)^2) + (I/6)*ArcCosh[c +
d*x]^4 - ArcCosh[c + d*x]^4/(3*(c + d*x)^3) + (4*I)*ArcCosh[c + d*x]*Log[1 - I/E^ArcCosh[c + d*x]] + (Pi^3*Log
[1 + I/E^ArcCosh[c + d*x]])/12 - (4*I)*ArcCosh[c + d*x]*Log[1 + I/E^ArcCosh[c + d*x]] - (I/2)*Pi^2*ArcCosh[c +
d*x]*Log[1 + I/E^ArcCosh[c + d*x]] - Pi*ArcCosh[c + d*x]^2*Log[1 + I/E^ArcCosh[c + d*x]] + ((2*I)/3)*ArcCosh[
c + d*x]^3*Log[1 + I/E^ArcCosh[c + d*x]] + (I/2)*Pi^2*ArcCosh[c + d*x]*Log[1 - I*E^ArcCosh[c + d*x]] + Pi*ArcC
osh[c + d*x]^2*Log[1 - I*E^ArcCosh[c + d*x]] - (Pi^3*Log[1 + I*E^ArcCosh[c + d*x]])/12 - ((2*I)/3)*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]])/12 + (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]] - (4*I)*PolyLog[2, I/E
^ArcCosh[c + d*x]] - (2*I)*ArcCosh[c + d*x]^2*PolyLog[2, (-I)*E^ArcCosh[c + d*x]] + (I/2)*Pi^2*PolyLog[2, I*E^
ArcCosh[c + d*x]] + 2*Pi*ArcCosh[c + d*x]*PolyLog[2, I*E^ArcCosh[c + d*x]] + 2*Pi*PolyLog[3, (-I)/E^ArcCosh[c
+ d*x]] - (4*I)*ArcCosh[c + d*x]*PolyLog[3, (-I)/E^ArcCosh[c + d*x]] + (4*I)*ArcCosh[c + d*x]*PolyLog[3, (-I)*
E^ArcCosh[c + d*x]] - 2*Pi*PolyLog[3, I*E^ArcCosh[c + d*x]] - (4*I)*PolyLog[4, (-I)/E^ArcCosh[c + d*x]] - (4*I
)*PolyLog[4, (-I)*E^ArcCosh[c + d*x]]))/(3*d*e^4)

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Maple [F]
time = 0.20, 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 \begin {gather*} \text {Failed to integrate} \end {gather*}

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

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)*e^(-4)/(d^4*x^4 + 4*c*d^3*x^3 + 6*c^2*d^2*x^2 + 4*c^3*d*x + c^4), x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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}} \end {gather*}

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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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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

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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