3.2.28 \(\int \frac {(a+b \cosh ^{-1}(c+d x))^4}{(c e+d e x)^3} \, dx\) [128]

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

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.32, antiderivative size = 195, normalized size of antiderivative = 1.00, 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 = {5996, 12, 5883, 5918, 5882, 3799, 2221, 2611, 2320, 6724} \begin {gather*} \frac {6 b^3 \text {Li}_2\left (-e^{-2 \cosh ^{-1}(c+d x)}\right ) \left (a+b \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}+\frac {3 b^4 \text {Li}_3\left (-e^{-2 \cosh ^{-1}(c+d x)}\right )}{d e^3} \end {gather*}

Antiderivative was successfully verified.

[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*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)

Rule 12

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

Rule 2221

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/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], 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 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 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 3799

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 5882

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> Dist[1/b, Subst[Int[x^n*Tanh[-a/b + x/b], x],
x, a + b*ArcCosh[c*x]], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 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 5918

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

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]

Rubi steps

\begin {align*} \int \frac {\left (a+b \cosh ^{-1}(c+d x)\right )^4}{(c e+d e x)^3} \, dx &=\frac {\text {Subst}\left (\int \frac {\left (a+b \cosh ^{-1}(x)\right )^4}{e^3 x^3} \, dx,x,c+d x\right )}{d}\\ &=\frac {\text {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) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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] Leaf count is larger than twice the leaf count of optimal. \(398\) vs. \(2(195)=390\).
time = 1.54, size = 398, normalized size = 2.04 \begin {gather*} \frac {-\frac {a^4}{(c+d x)^2}+\frac {4 a^3 b \sqrt {-1+c+d x} \sqrt {1+c+d x}}{c+d x}-\frac {4 a^3 b \cosh ^{-1}(c+d x)}{(c+d x)^2}-\frac {b^4 \cosh ^{-1}(c+d x)^4}{(c+d x)^2}+12 a^2 b^2 \left (\frac {\sqrt {\frac {-1+c+d x}{1+c+d x}} (1+c+d x) \cosh ^{-1}(c+d x)}{c+d x}-\frac {\cosh ^{-1}(c+d x)^2}{2 (c+d x)^2}-\log (c+d x)\right )+4 a b^3 \left (-\cosh ^{-1}(c+d x) \left (3 \cosh ^{-1}(c+d x)-\frac {3 \sqrt {\frac {-1+c+d x}{1+c+d x}} (1+c+d x) \cosh ^{-1}(c+d x)}{c+d x}+\frac {\cosh ^{-1}(c+d x)^2}{(c+d x)^2}+6 \log \left (1+e^{-2 \cosh ^{-1}(c+d x)}\right )\right )+3 \text {PolyLog}\left (2,-e^{-2 \cosh ^{-1}(c+d x)}\right )\right )+2 b^4 \left (2 \cosh ^{-1}(c+d x)^2 \left (-\cosh ^{-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}-3 \log \left (1+e^{-2 \cosh ^{-1}(c+d x)}\right )\right )+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 )\right )}{2 d e^3} \end {gather*}

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)

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(550\) vs. \(2(231)=462\).
time = 56.55, size = 551, normalized size = 2.83

method	result	size
derivativedivides	\(\frac {-\frac {a^{4}}{2 e^{3} \left (d x +c \right )^{2}}+\frac {2 b^{4} \mathrm {arccosh}\left (d x +c \right )^{3} \sqrt {d x +c +1}\, \sqrt {d x +c -1}}{e^{3} \left (d x +c \right )}+\frac {2 b^{4} \mathrm {arccosh}\left (d x +c \right )^{3}}{e^{3}}-\frac {b^{4} \mathrm {arccosh}\left (d x +c \right )^{4}}{2 e^{3} \left (d x +c \right )^{2}}-\frac {6 b^{4} \mathrm {arccosh}\left (d x +c \right )^{2} \ln \left (1+\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )}{e^{3}}-\frac {6 b^{4} \mathrm {arccosh}\left (d x +c \right ) \polylog \left (2, -\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )}{e^{3}}+\frac {3 b^{4} \polylog \left (3, -\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )}{e^{3}}+\frac {6 a \,b^{3} \mathrm {arccosh}\left (d x +c \right )^{2} \sqrt {d x +c +1}\, \sqrt {d x +c -1}}{e^{3} \left (d x +c \right )}+\frac {6 a \,b^{3} \mathrm {arccosh}\left (d x +c \right )^{2}}{e^{3}}-\frac {2 a \,b^{3} \mathrm {arccosh}\left (d x +c \right )^{3}}{e^{3} \left (d x +c \right )^{2}}-\frac {12 a \,b^{3} \mathrm {arccosh}\left (d x +c \right ) \ln \left (1+\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )}{e^{3}}-\frac {6 a \,b^{3} \polylog \left (2, -\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )}{e^{3}}+\frac {6 a^{2} b^{2} \mathrm {arccosh}\left (d x +c \right )}{e^{3}}+\frac {6 a^{2} b^{2} \mathrm {arccosh}\left (d x +c \right ) \sqrt {d x +c +1}\, \sqrt {d x +c -1}}{e^{3} \left (d x +c \right )}-\frac {3 a^{2} b^{2} \mathrm {arccosh}\left (d x +c \right )^{2}}{e^{3} \left (d x +c \right )^{2}}-\frac {6 a^{2} b^{2} \ln \left (1+\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )}{e^{3}}+\frac {4 a^{3} b \left (-\frac {\mathrm {arccosh}\left (d x +c \right )}{2 \left (d x +c \right )^{2}}+\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}}{2 d x +2 c}\right )}{e^{3}}}{d}\)	\(551\)
default	\(\frac {-\frac {a^{4}}{2 e^{3} \left (d x +c \right )^{2}}+\frac {2 b^{4} \mathrm {arccosh}\left (d x +c \right )^{3} \sqrt {d x +c +1}\, \sqrt {d x +c -1}}{e^{3} \left (d x +c \right )}+\frac {2 b^{4} \mathrm {arccosh}\left (d x +c \right )^{3}}{e^{3}}-\frac {b^{4} \mathrm {arccosh}\left (d x +c \right )^{4}}{2 e^{3} \left (d x +c \right )^{2}}-\frac {6 b^{4} \mathrm {arccosh}\left (d x +c \right )^{2} \ln \left (1+\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )}{e^{3}}-\frac {6 b^{4} \mathrm {arccosh}\left (d x +c \right ) \polylog \left (2, -\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )}{e^{3}}+\frac {3 b^{4} \polylog \left (3, -\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )}{e^{3}}+\frac {6 a \,b^{3} \mathrm {arccosh}\left (d x +c \right )^{2} \sqrt {d x +c +1}\, \sqrt {d x +c -1}}{e^{3} \left (d x +c \right )}+\frac {6 a \,b^{3} \mathrm {arccosh}\left (d x +c \right )^{2}}{e^{3}}-\frac {2 a \,b^{3} \mathrm {arccosh}\left (d x +c \right )^{3}}{e^{3} \left (d x +c \right )^{2}}-\frac {12 a \,b^{3} \mathrm {arccosh}\left (d x +c \right ) \ln \left (1+\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )}{e^{3}}-\frac {6 a \,b^{3} \polylog \left (2, -\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )}{e^{3}}+\frac {6 a^{2} b^{2} \mathrm {arccosh}\left (d x +c \right )}{e^{3}}+\frac {6 a^{2} b^{2} \mathrm {arccosh}\left (d x +c \right ) \sqrt {d x +c +1}\, \sqrt {d x +c -1}}{e^{3} \left (d x +c \right )}-\frac {3 a^{2} b^{2} \mathrm {arccosh}\left (d x +c \right )^{2}}{e^{3} \left (d x +c \right )^{2}}-\frac {6 a^{2} b^{2} \ln \left (1+\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )}{e^{3}}+\frac {4 a^{3} b \left (-\frac {\mathrm {arccosh}\left (d x +c \right )}{2 \left (d x +c \right )^{2}}+\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}}{2 d x +2 c}\right )}{e^{3}}}{d}\)	\(551\)

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,method=_RETURNVERBOSE)

[Out]

1/d*(-1/2*a^4/e^3/(d*x+c)^2+2*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*b^4/e^3*arcco
sh(d*x+c)^3-1/2*b^4/e^3*arccosh(d*x+c)^4/(d*x+c)^2-6*b^4/e^3*arccosh(d*x+c)^2*ln(1+(d*x+c+(d*x+c-1)^(1/2)*(d*x
+c+1)^(1/2))^2)-6*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*b^4/e^3*polyl
og(3,-(d*x+c+(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2))^2)+6*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*a*b^3/e^3*arccosh(d*x+c)^2-2*a*b^3/e^3*arccosh(d*x+c)^3/(d*x+c)^2-12*a*b^3/e^3*arccosh(d*x+c)*ln(1+(
d*x+c+(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2))^2)-6*a*b^3/e^3*polylog(2,-(d*x+c+(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2))^2)+6*
a^2*b^2/e^3*arccosh(d*x+c)+6*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*a^2*b^2/e^3*
arccosh(d*x+c)^2/(d*x+c)^2-6*a^2*b^2/e^3*ln(1+(d*x+c+(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2))^2)+4*a^3*b/e^3*(-1/2/(d*
x+c)^2*arccosh(d*x+c)+1/2*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)/(d*x+c)))

________________________________________________________________________________________

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)^3,x, algorithm="maxima")

[Out]

-1/2*b^4*log(d*x + sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + c)^4/(d^3*x^2*e^3 + 2*c*d^2*x*e^3 + c^2*d*e^3) + 6*(s
qrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*d*arccosh(d*x + c)/(d^3*x*e^3 + c*d^2*e^3) - e^(-3)*log(d*x + c)/d)*a^2*b^2 +
2*a^3*b*(sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*d/(d^3*x*e^3 + c*d^2*e^3) - arccosh(d*x + c)/(d^3*x^2*e^3 + 2*c*d^
2*x*e^3 + c^2*d*e^3)) - 3*a^2*b^2*arccosh(d*x + c)^2/(d^3*x^2*e^3 + 2*c*d^2*x*e^3 + c^2*d*e^3) - 1/2*a^4/(d^3*
x^2*e^3 + 2*c*d^2*x*e^3 + c^2*d*e^3) + integrate(2*(2*(c^3 - c)*a*b^3 + (c^3 - c)*b^4 + (2*a*b^3*d^3 + b^4*d^3
)*x^3 + 3*(2*a*b^3*c*d^2 + b^4*c*d^2)*x^2 + (b^4*c^2 + 2*(c^2 - 1)*a*b^3 + (2*a*b^3*d^2 + b^4*d^2)*x^2 + 2*(2*
a*b^3*c*d + b^4*c*d)*x)*sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + (2*(3*c^2*d - d)*a*b^3 + (3*c^2*d - d)*b^4)*x)*l
og(d*x + sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + c)^3/(d^6*x^6*e^3 + 6*c*d^5*x^5*e^3 + (15*c^2*d^4 - d^4)*x^4*e^
3 + 4*(5*c^3*d^3 - c*d^3)*x^3*e^3 + 3*(5*c^4*d^2 - 2*c^2*d^2)*x^2*e^3 + 2*(3*c^5*d - 2*c^3*d)*x*e^3 + (d^5*x^5
*e^3 + 5*c*d^4*x^4*e^3 + (10*c^2*d^3 - d^3)*x^3*e^3 + (10*c^3*d^2 - 3*c*d^2)*x^2*e^3 + (5*c^4*d - 3*c^2*d)*x*e
^3 + (c^5 - c^3)*e^3)*sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + (c^6 - c^4)*e^3), x)

________________________________________________________________________________________

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

________________________________________________________________________________________

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

________________________________________________________________________________________

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)^3,x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right )}^4}{{\left (c\,e+d\,e\,x\right )}^3} \,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)^3,x)

[Out]

int((a + b*acosh(c + d*x))^4/(c*e + d*e*x)^3, x)

________________________________________________________________________________________