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

Optimal. Leaf size=264 \[ \frac {24 i b^3 \text {Li}_3\left (-i e^{\cosh ^{-1}(c+d x)}\right ) \left (a+b \cosh ^{-1}(c+d x)\right )}{d e^2}-\frac {24 i b^3 \text {Li}_3\left (i e^{\cosh ^{-1}(c+d x)}\right ) \left (a+b \cosh ^{-1}(c+d x)\right )}{d e^2}-\frac {12 i b^2 \text {Li}_2\left (-i e^{\cosh ^{-1}(c+d x)}\right ) \left (a+b \cosh ^{-1}(c+d x)\right )^2}{d e^2}+\frac {12 i b^2 \text {Li}_2\left (i e^{\cosh ^{-1}(c+d x)}\right ) \left (a+b \cosh ^{-1}(c+d x)\right )^2}{d e^2}-\frac {\left (a+b \cosh ^{-1}(c+d x)\right )^4}{d e^2 (c+d x)}+\frac {8 b \tan ^{-1}\left (e^{\cosh ^{-1}(c+d x)}\right ) \left (a+b \cosh ^{-1}(c+d x)\right )^3}{d e^2}-\frac {24 i b^4 \text {Li}_4\left (-i e^{\cosh ^{-1}(c+d x)}\right )}{d e^2}+\frac {24 i b^4 \text {Li}_4\left (i e^{\cosh ^{-1}(c+d x)}\right )}{d e^2} \]

[Out]

-(a+b*arccosh(d*x+c))^4/d/e^2/(d*x+c)+8*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^2-12*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^2+12*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^2+24*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^2-24*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^2-24*I*b^4*polylog(4,-I*(d*x+c+(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)))/d/e^2+24
*I*b^4*polylog(4,I*(d*x+c+(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)))/d/e^2

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Rubi [A]  time = 0.41, antiderivative size = 264, normalized size of antiderivative = 1.00, 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 12

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

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 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 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)^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 = 2.52, size = 872, normalized size = 3.30 \[ \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 {Li}_2\left (-i e^{-\cosh ^{-1}(c+d x)}\right )-2 \text {Li}_2\left (i e^{-\cosh ^{-1}(c+d x)}\right )\right ) a^2+4 b^3 \left (3 i \left (-\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\right )-2 \left (\text {Li}_2\left (-i e^{-\cosh ^{-1}(c+d x)}\right )-\text {Li}_2\left (i e^{-\cosh ^{-1}(c+d x)}\right )\right ) \cosh ^{-1}(c+d x)-2 \text {Li}_3\left (-i e^{-\cosh ^{-1}(c+d x)}\right )+2 \text {Li}_3\left (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 {Li}_2\left (-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 {Li}_2\left (i e^{\cosh ^{-1}(c+d x)}\right ) \cosh ^{-1}(c+d x)-24 i \text {Li}_3\left (-i e^{-\cosh ^{-1}(c+d x)}\right ) \cosh ^{-1}(c+d x)+24 i \text {Li}_3\left (-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 {Li}_2\left (-i e^{-\cosh ^{-1}(c+d x)}\right )+3 i \pi ^2 \text {Li}_2\left (i e^{\cosh ^{-1}(c+d x)}\right )+12 \pi \text {Li}_3\left (-i e^{-\cosh ^{-1}(c+d x)}\right )-12 \pi \text {Li}_3\left (i e^{\cosh ^{-1}(c+d x)}\right )-24 i \text {Li}_4\left (-i e^{-\cosh ^{-1}(c+d x)}\right )-24 i \text {Li}_4\left (-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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fricas [F]  time = 0.75, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{4} \operatorname {arcosh}\left (d x + c\right )^{4} + 4 \, a b^{3} \operatorname {arcosh}\left (d x + c\right )^{3} + 6 \, a^{2} b^{2} \operatorname {arcosh}\left (d x + c\right )^{2} + 4 \, a^{3} b \operatorname {arcosh}\left (d x + c\right ) + a^{4}}{d^{2} e^{2} x^{2} + 2 \, c d e^{2} x + c^{2} e^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccosh(d*x+c))^4/(d*e*x+c*e)^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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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

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 >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Warn
ing, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Check [ab
s(t_nostep)]Evaluation time: 1.52sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad
Argument Value

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maple [F]  time = 0.22, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +b \,\mathrm {arccosh}\left (d x +c \right )\right )^{4}}{\left (d e x +c e \right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-b^4*log(d*x + sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + c)^4/(d^2*e^2*x + c*d*e^2) - 4*a^3*b*(arccosh(d*x + c)/(d
^2*e^2*x + c*d*e^2) + arcsin(d*e^2/abs(d^2*e^2*x + c*d*e^2))/(d*e^2)) - a^4/(d^2*e^2*x + c*d*e^2) + integrate(
2*(2*((c^3 - c)*a*b^3 + (c^3 - c)*b^4 + (a*b^3*d^3 + b^4*d^3)*x^3 + 3*(a*b^3*c*d^2 + b^4*c*d^2)*x^2 + (b^4*c^2
+ (c^2 - 1)*a*b^3 + (a*b^3*d^2 + b^4*d^2)*x^2 + 2*(a*b^3*c*d + b^4*c*d)*x)*sqrt(d*x + c + 1)*sqrt(d*x + c - 1
) + ((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 + 3*(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)/(d^5*e^2*x^5 + 5*c*d^4*e^2*x^4 + c^5*e^2 - c^3*e^2 + (10*c^2*d^3*e^2 - d^3*e^2)*x^3 + (10*c^3*d^2*e^
2 - 3*c*d^2*e^2)*x^2 + (d^4*e^2*x^4 + 4*c*d^3*e^2*x^3 + c^4*e^2 - c^2*e^2 + (6*c^2*d^2*e^2 - d^2*e^2)*x^2 + 2*
(2*c^3*d*e^2 - c*d*e^2)*x)*sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + (5*c^4*d*e^2 - 3*c^2*d*e^2)*x), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right )}^4}{{\left (c\,e+d\,e\,x\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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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