3.128 \(\int \frac{(a+b \cosh ^{-1}(c+d x))^4}{(c e+d e x)^3} \, dx\)

Optimal. Leaf size=195 \[ \frac{6 b^3 \text{PolyLog}\left (2,-e^{-2 \cosh ^{-1}(c+d x)}\right ) \left (a+b \cosh ^{-1}(c+d x)\right )}{d e^3}+\frac{3 b^4 \text{PolyLog}\left (3,-e^{-2 \cosh ^{-1}(c+d x)}\right )}{d e^3}-\frac{6 b^2 \log \left (e^{-2 \cosh ^{-1}(c+d x)}+1\right ) \left (a+b \cosh ^{-1}(c+d x)\right )^2}{d e^3}+\frac{2 b \sqrt{c+d x-1} \sqrt{c+d x+1} \left (a+b \cosh ^{-1}(c+d x)\right )^3}{d e^3 (c+d x)}-\frac{2 b \left (a+b \cosh ^{-1}(c+d x)\right )^3}{d e^3}-\frac{\left (a+b \cosh ^{-1}(c+d x)\right )^4}{2 d e^3 (c+d x)^2} \]

[Out]

(-2*b*(a + b*ArcCosh[c + d*x])^3)/(d*e^3) + (2*b*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x]*(a + b*ArcCosh[c + d*x])
^3)/(d*e^3*(c + d*x)) - (a + b*ArcCosh[c + d*x])^4/(2*d*e^3*(c + d*x)^2) - (6*b^2*(a + b*ArcCosh[c + d*x])^2*L
og[1 + E^(-2*ArcCosh[c + d*x])])/(d*e^3) + (6*b^3*(a + b*ArcCosh[c + d*x])*PolyLog[2, -E^(-2*ArcCosh[c + d*x])
])/(d*e^3) + (3*b^4*PolyLog[3, -E^(-2*ArcCosh[c + d*x])])/(d*e^3)

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Rubi [A]  time = 0.426362, antiderivative size = 195, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 10, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.435, Rules used = {5866, 12, 5662, 5724, 5660, 3718, 2190, 2531, 2282, 6589} \[ -\frac{6 b^3 \text{PolyLog}\left (2,-e^{2 \cosh ^{-1}(c+d x)}\right ) \left (a+b \cosh ^{-1}(c+d x)\right )}{d e^3}+\frac{3 b^4 \text{PolyLog}\left (3,-e^{2 \cosh ^{-1}(c+d x)}\right )}{d e^3}-\frac{6 b^2 \log \left (e^{2 \cosh ^{-1}(c+d x)}+1\right ) \left (a+b \cosh ^{-1}(c+d x)\right )^2}{d e^3}+\frac{2 b \sqrt{c+d x-1} \sqrt{c+d x+1} \left (a+b \cosh ^{-1}(c+d x)\right )^3}{d e^3 (c+d x)}+\frac{2 b \left (a+b \cosh ^{-1}(c+d x)\right )^3}{d e^3}-\frac{\left (a+b \cosh ^{-1}(c+d x)\right )^4}{2 d e^3 (c+d x)^2} \]

Warning: Unable to verify antiderivative.

[In]

Int[(a + b*ArcCosh[c + d*x])^4/(c*e + d*e*x)^3,x]

[Out]

(2*b*(a + b*ArcCosh[c + d*x])^3)/(d*e^3) + (2*b*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x]*(a + b*ArcCosh[c + d*x])^
3)/(d*e^3*(c + d*x)) - (a + b*ArcCosh[c + d*x])^4/(2*d*e^3*(c + d*x)^2) - (6*b^2*(a + b*ArcCosh[c + d*x])^2*Lo
g[1 + E^(2*ArcCosh[c + d*x])])/(d*e^3) - (6*b^3*(a + b*ArcCosh[c + d*x])*PolyLog[2, -E^(2*ArcCosh[c + d*x])])/
(d*e^3) + (3*b^4*PolyLog[3, -E^(2*ArcCosh[c + d*x])])/(d*e^3)

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 5724

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)/(d
1*d2*f*(m + 1)), 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, m, p}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2,
0] && GtQ[n, 0] && EqQ[m + 2*p + 3, 0] && NeQ[m, -1] && IntegerQ[p + 1/2]

Rule 5660

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> Subst[Int[(a + b*x)^n/Coth[x], x], x, ArcCosh
[c*x]] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0]

Rule 3718

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (Complex[0, fz_])*(f_.)*(x_)], x_Symbol] :> -Simp[(I*(c + d*x)^(m +
1))/(d*(m + 1)), x] + Dist[2*I, Int[((c + d*x)^m*E^(2*(-(I*e) + f*fz*x)))/(1 + E^(2*(-(I*e) + f*fz*x))), x],
x] /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]

Rule 2190

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

Rule 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 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)^3} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (a+b \cosh ^{-1}(x)\right )^4}{e^3 x^3} \, dx,x,c+d x\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\left (a+b \cosh ^{-1}(x)\right )^4}{x^3} \, dx,x,c+d x\right )}{d e^3}\\ &=-\frac{\left (a+b \cosh ^{-1}(c+d x)\right )^4}{2 d e^3 (c+d x)^2}+\frac{(2 b) \operatorname{Subst}\left (\int \frac{\left (a+b \cosh ^{-1}(x)\right )^3}{\sqrt{-1+x} x^2 \sqrt{1+x}} \, dx,x,c+d x\right )}{d e^3}\\ &=\frac{2 b \sqrt{-1+c+d x} \sqrt{1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^3}{d e^3 (c+d x)}-\frac{\left (a+b \cosh ^{-1}(c+d x)\right )^4}{2 d e^3 (c+d x)^2}-\frac{\left (6 b^2\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \cosh ^{-1}(x)\right )^2}{x} \, dx,x,c+d x\right )}{d e^3}\\ &=\frac{2 b \sqrt{-1+c+d x} \sqrt{1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^3}{d e^3 (c+d x)}-\frac{\left (a+b \cosh ^{-1}(c+d x)\right )^4}{2 d e^3 (c+d x)^2}-\frac{\left (6 b^2\right ) \operatorname{Subst}\left (\int (a+b x)^2 \tanh (x) \, dx,x,\cosh ^{-1}(c+d x)\right )}{d e^3}\\ &=\frac{2 b \left (a+b \cosh ^{-1}(c+d x)\right )^3}{d e^3}+\frac{2 b \sqrt{-1+c+d x} \sqrt{1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^3}{d e^3 (c+d x)}-\frac{\left (a+b \cosh ^{-1}(c+d x)\right )^4}{2 d e^3 (c+d x)^2}-\frac{\left (12 b^2\right ) \operatorname{Subst}\left (\int \frac{e^{2 x} (a+b x)^2}{1+e^{2 x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{d e^3}\\ &=\frac{2 b \left (a+b \cosh ^{-1}(c+d x)\right )^3}{d e^3}+\frac{2 b \sqrt{-1+c+d x} \sqrt{1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^3}{d e^3 (c+d x)}-\frac{\left (a+b \cosh ^{-1}(c+d x)\right )^4}{2 d e^3 (c+d x)^2}-\frac{6 b^2 \left (a+b \cosh ^{-1}(c+d x)\right )^2 \log \left (1+e^{2 \cosh ^{-1}(c+d x)}\right )}{d e^3}+\frac{\left (12 b^3\right ) \operatorname{Subst}\left (\int (a+b x) \log \left (1+e^{2 x}\right ) \, dx,x,\cosh ^{-1}(c+d x)\right )}{d e^3}\\ &=\frac{2 b \left (a+b \cosh ^{-1}(c+d x)\right )^3}{d e^3}+\frac{2 b \sqrt{-1+c+d x} \sqrt{1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^3}{d e^3 (c+d x)}-\frac{\left (a+b \cosh ^{-1}(c+d x)\right )^4}{2 d e^3 (c+d x)^2}-\frac{6 b^2 \left (a+b \cosh ^{-1}(c+d x)\right )^2 \log \left (1+e^{2 \cosh ^{-1}(c+d x)}\right )}{d e^3}-\frac{6 b^3 \left (a+b \cosh ^{-1}(c+d x)\right ) \text{Li}_2\left (-e^{2 \cosh ^{-1}(c+d x)}\right )}{d e^3}+\frac{\left (6 b^4\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (-e^{2 x}\right ) \, dx,x,\cosh ^{-1}(c+d x)\right )}{d e^3}\\ &=\frac{2 b \left (a+b \cosh ^{-1}(c+d x)\right )^3}{d e^3}+\frac{2 b \sqrt{-1+c+d x} \sqrt{1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^3}{d e^3 (c+d x)}-\frac{\left (a+b \cosh ^{-1}(c+d x)\right )^4}{2 d e^3 (c+d x)^2}-\frac{6 b^2 \left (a+b \cosh ^{-1}(c+d x)\right )^2 \log \left (1+e^{2 \cosh ^{-1}(c+d x)}\right )}{d e^3}-\frac{6 b^3 \left (a+b \cosh ^{-1}(c+d x)\right ) \text{Li}_2\left (-e^{2 \cosh ^{-1}(c+d x)}\right )}{d e^3}+\frac{\left (3 b^4\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-x)}{x} \, dx,x,e^{2 \cosh ^{-1}(c+d x)}\right )}{d e^3}\\ &=\frac{2 b \left (a+b \cosh ^{-1}(c+d x)\right )^3}{d e^3}+\frac{2 b \sqrt{-1+c+d x} \sqrt{1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^3}{d e^3 (c+d x)}-\frac{\left (a+b \cosh ^{-1}(c+d x)\right )^4}{2 d e^3 (c+d x)^2}-\frac{6 b^2 \left (a+b \cosh ^{-1}(c+d x)\right )^2 \log \left (1+e^{2 \cosh ^{-1}(c+d x)}\right )}{d e^3}-\frac{6 b^3 \left (a+b \cosh ^{-1}(c+d x)\right ) \text{Li}_2\left (-e^{2 \cosh ^{-1}(c+d x)}\right )}{d e^3}+\frac{3 b^4 \text{Li}_3\left (-e^{2 \cosh ^{-1}(c+d x)}\right )}{d e^3}\\ \end{align*}

Mathematica [B]  time = 2.21836, size = 398, normalized size = 2.04 \[ \frac{4 a b^3 \left (3 \text{PolyLog}\left (2,-e^{-2 \cosh ^{-1}(c+d x)}\right )-\cosh ^{-1}(c+d x) \left (\frac{\cosh ^{-1}(c+d x)^2}{(c+d x)^2}-\frac{3 \sqrt{\frac{c+d x-1}{c+d x+1}} (c+d x+1) \cosh ^{-1}(c+d x)}{c+d x}+3 \cosh ^{-1}(c+d x)+6 \log \left (e^{-2 \cosh ^{-1}(c+d x)}+1\right )\right )\right )+2 b^4 \left (6 \cosh ^{-1}(c+d x) \text{PolyLog}\left (2,-e^{-2 \cosh ^{-1}(c+d x)}\right )+3 \text{PolyLog}\left (3,-e^{-2 \cosh ^{-1}(c+d x)}\right )+2 \cosh ^{-1}(c+d x)^2 \left (\frac{\sqrt{\frac{c+d x-1}{c+d x+1}} (c+d x+1) \cosh ^{-1}(c+d x)}{c+d x}-\cosh ^{-1}(c+d x)-3 \log \left (e^{-2 \cosh ^{-1}(c+d x)}+1\right )\right )\right )+12 a^2 b^2 \left (-\log (c+d x)-\frac{\cosh ^{-1}(c+d x)^2}{2 (c+d x)^2}+\frac{\sqrt{\frac{c+d x-1}{c+d x+1}} (c+d x+1) \cosh ^{-1}(c+d x)}{c+d x}\right )+\frac{4 a^3 b \sqrt{c+d x-1} \sqrt{c+d x+1}}{c+d x}-\frac{4 a^3 b \cosh ^{-1}(c+d x)}{(c+d x)^2}-\frac{a^4}{(c+d x)^2}-\frac{b^4 \cosh ^{-1}(c+d x)^4}{(c+d x)^2}}{2 d e^3} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + b*ArcCosh[c + d*x])^4/(c*e + d*e*x)^3,x]

[Out]

(-(a^4/(c + d*x)^2) + (4*a^3*b*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x])/(c + d*x) - (4*a^3*b*ArcCosh[c + d*x])/(c
+ d*x)^2 - (b^4*ArcCosh[c + d*x]^4)/(c + d*x)^2 + 12*a^2*b^2*((Sqrt[(-1 + c + d*x)/(1 + c + d*x)]*(1 + c + d*
x)*ArcCosh[c + d*x])/(c + d*x) - ArcCosh[c + d*x]^2/(2*(c + d*x)^2) - Log[c + d*x]) + 4*a*b^3*(-(ArcCosh[c + d
*x]*(3*ArcCosh[c + d*x] - (3*Sqrt[(-1 + c + d*x)/(1 + c + d*x)]*(1 + c + d*x)*ArcCosh[c + d*x])/(c + d*x) + Ar
cCosh[c + d*x]^2/(c + d*x)^2 + 6*Log[1 + E^(-2*ArcCosh[c + d*x])])) + 3*PolyLog[2, -E^(-2*ArcCosh[c + d*x])])
+ 2*b^4*(2*ArcCosh[c + d*x]^2*(-ArcCosh[c + d*x] + (Sqrt[(-1 + c + d*x)/(1 + c + d*x)]*(1 + c + d*x)*ArcCosh[c
+ d*x])/(c + d*x) - 3*Log[1 + E^(-2*ArcCosh[c + d*x])]) + 6*ArcCosh[c + d*x]*PolyLog[2, -E^(-2*ArcCosh[c + d*
x])] + 3*PolyLog[3, -E^(-2*ArcCosh[c + d*x])]))/(2*d*e^3)

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Maple [B]  time = 0.066, size = 605, normalized size = 3.1 \begin{align*} -{\frac{{a}^{4}}{2\,d{e}^{3} \left ( dx+c \right ) ^{2}}}+2\,{\frac{{b}^{4} \left ({\rm arccosh} \left (dx+c\right ) \right ) ^{3}\sqrt{dx+c+1}\sqrt{dx+c-1}}{d{e}^{3} \left ( dx+c \right ) }}+2\,{\frac{{b}^{4} \left ({\rm arccosh} \left (dx+c\right ) \right ) ^{3}}{d{e}^{3}}}-{\frac{{b}^{4} \left ({\rm arccosh} \left (dx+c\right ) \right ) ^{4}}{2\,d{e}^{3} \left ( dx+c \right ) ^{2}}}-6\,{\frac{{b}^{4} \left ({\rm arccosh} \left (dx+c\right ) \right ) ^{2}\ln \left ( \left ( dx+c+\sqrt{dx+c-1}\sqrt{dx+c+1} \right ) ^{2}+1 \right ) }{d{e}^{3}}}-6\,{\frac{{b}^{4}{\rm arccosh} \left (dx+c\right ){\it polylog} \left ( 2,- \left ( dx+c+\sqrt{dx+c-1}\sqrt{dx+c+1} \right ) ^{2} \right ) }{d{e}^{3}}}+3\,{\frac{{b}^{4}{\it polylog} \left ( 3,- \left ( dx+c+\sqrt{dx+c-1}\sqrt{dx+c+1} \right ) ^{2} \right ) }{d{e}^{3}}}+6\,{\frac{a{b}^{3} \left ({\rm arccosh} \left (dx+c\right ) \right ) ^{2}\sqrt{dx+c+1}\sqrt{dx+c-1}}{d{e}^{3} \left ( dx+c \right ) }}+6\,{\frac{a{b}^{3} \left ({\rm arccosh} \left (dx+c\right ) \right ) ^{2}}{d{e}^{3}}}-2\,{\frac{a{b}^{3} \left ({\rm arccosh} \left (dx+c\right ) \right ) ^{3}}{d{e}^{3} \left ( dx+c \right ) ^{2}}}-12\,{\frac{a{b}^{3}{\rm arccosh} \left (dx+c\right )\ln \left ( \left ( dx+c+\sqrt{dx+c-1}\sqrt{dx+c+1} \right ) ^{2}+1 \right ) }{d{e}^{3}}}-6\,{\frac{a{b}^{3}{\it polylog} \left ( 2,- \left ( dx+c+\sqrt{dx+c-1}\sqrt{dx+c+1} \right ) ^{2} \right ) }{d{e}^{3}}}+6\,{\frac{{a}^{2}{b}^{2}{\rm arccosh} \left (dx+c\right )}{d{e}^{3}}}+6\,{\frac{{a}^{2}{b}^{2}{\rm arccosh} \left (dx+c\right )\sqrt{dx+c+1}\sqrt{dx+c-1}}{d{e}^{3} \left ( dx+c \right ) }}-3\,{\frac{{a}^{2}{b}^{2} \left ({\rm arccosh} \left (dx+c\right ) \right ) ^{2}}{d{e}^{3} \left ( dx+c \right ) ^{2}}}-6\,{\frac{{a}^{2}{b}^{2}\ln \left ( \left ( dx+c+\sqrt{dx+c-1}\sqrt{dx+c+1} \right ) ^{2}+1 \right ) }{d{e}^{3}}}-2\,{\frac{{a}^{3}b{\rm arccosh} \left (dx+c\right )}{d{e}^{3} \left ( dx+c \right ) ^{2}}}+2\,{\frac{{a}^{3}b\sqrt{dx+c-1}\sqrt{dx+c+1}}{d{e}^{3} \left ( dx+c \right ) }} \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)^3,x)

[Out]

-1/2/d*a^4/e^3/(d*x+c)^2+2/d*b^4/e^3*arccosh(d*x+c)^3/(d*x+c)*(d*x+c+1)^(1/2)*(d*x+c-1)^(1/2)+2/d*b^4/e^3*arcc
osh(d*x+c)^3-1/2/d*b^4/e^3*arccosh(d*x+c)^4/(d*x+c)^2-6/d*b^4/e^3*arccosh(d*x+c)^2*ln((d*x+c+(d*x+c-1)^(1/2)*(
d*x+c+1)^(1/2))^2+1)-6/d*b^4/e^3*arccosh(d*x+c)*polylog(2,-(d*x+c+(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2))^2)+3/d*b^4/
e^3*polylog(3,-(d*x+c+(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2))^2)+6/d*a*b^3/e^3*arccosh(d*x+c)^2/(d*x+c)*(d*x+c+1)^(1/
2)*(d*x+c-1)^(1/2)+6/d*a*b^3/e^3*arccosh(d*x+c)^2-2/d*a*b^3/e^3*arccosh(d*x+c)^3/(d*x+c)^2-12/d*a*b^3/e^3*arcc
osh(d*x+c)*ln((d*x+c+(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2))^2+1)-6/d*a*b^3/e^3*polylog(2,-(d*x+c+(d*x+c-1)^(1/2)*(d*
x+c+1)^(1/2))^2)+6/d*a^2*b^2/e^3*arccosh(d*x+c)+6/d*a^2*b^2/e^3*arccosh(d*x+c)/(d*x+c)*(d*x+c+1)^(1/2)*(d*x+c-
1)^(1/2)-3/d*a^2*b^2/e^3*arccosh(d*x+c)^2/(d*x+c)^2-6/d*a^2*b^2/e^3*ln((d*x+c+(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2))
^2+1)-2/d*a^3*b/e^3/(d*x+c)^2*arccosh(d*x+c)+2/d*a^3*b/e^3*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)/(d*x+c)

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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)^3,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^{3} e^{3} x^{3} + 3 \, c d^{2} e^{3} x^{2} + 3 \, c^{2} d e^{3} x + c^{3} e^{3}}, 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)^3,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^3*e^3*x^3 + 3*c*d^2*e^3*x^2 + 3*c^2*d*e^3*x + c^3*e^3), x)

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \operatorname{arcosh}\left (d x + c\right ) + a\right )}^{4}}{{\left (d e x + c e\right )}^{3}}\,{d x} \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)^3,x, algorithm="giac")

[Out]

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