3.2.43 \(\int \frac {(c e+d e x)^3}{(a+b \cosh ^{-1}(c+d x))^3} \, dx\) [143]

Optimal. Leaf size=254 \[ -\frac {e^3 \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x}}{2 b d \left (a+b \cosh ^{-1}(c+d x)\right )^2}+\frac {3 e^3 (c+d x)^2}{2 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac {2 e^3 (c+d x)^4}{b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac {e^3 \text {Chi}\left (\frac {2 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right ) \sinh \left (\frac {2 a}{b}\right )}{2 b^3 d}-\frac {e^3 \text {Chi}\left (\frac {4 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right ) \sinh \left (\frac {4 a}{b}\right )}{b^3 d}+\frac {e^3 \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )}{2 b^3 d}+\frac {e^3 \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )}{b^3 d} \]

[Out]

3/2*e^3*(d*x+c)^2/b^2/d/(a+b*arccosh(d*x+c))-2*e^3*(d*x+c)^4/b^2/d/(a+b*arccosh(d*x+c))+1/2*e^3*cosh(2*a/b)*Sh
i(2*(a+b*arccosh(d*x+c))/b)/b^3/d+e^3*cosh(4*a/b)*Shi(4*(a+b*arccosh(d*x+c))/b)/b^3/d-1/2*e^3*Chi(2*(a+b*arcco
sh(d*x+c))/b)*sinh(2*a/b)/b^3/d-e^3*Chi(4*(a+b*arccosh(d*x+c))/b)*sinh(4*a/b)/b^3/d-1/2*e^3*(d*x+c)^3*(d*x+c-1
)^(1/2)*(d*x+c+1)^(1/2)/b/d/(a+b*arccosh(d*x+c))^2

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Rubi [A]
time = 0.58, antiderivative size = 254, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 9, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {5996, 12, 5886, 5951, 5887, 5556, 3384, 3379, 3382} \begin {gather*} -\frac {e^3 \sinh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )}{2 b^3 d}-\frac {e^3 \sinh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )}{b^3 d}+\frac {e^3 \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )}{2 b^3 d}+\frac {e^3 \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )}{b^3 d}-\frac {2 e^3 (c+d x)^4}{b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}+\frac {3 e^3 (c+d x)^2}{2 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac {e^3 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^3}{2 b d \left (a+b \cosh ^{-1}(c+d x)\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*(e^3*Sqrt[-1 + c + d*x]*(c + d*x)^3*Sqrt[1 + c + d*x])/(b*d*(a + b*ArcCosh[c + d*x])^2) + (3*e^3*(c + d*x
)^2)/(2*b^2*d*(a + b*ArcCosh[c + d*x])) - (2*e^3*(c + d*x)^4)/(b^2*d*(a + b*ArcCosh[c + d*x])) - (e^3*CoshInte
gral[(2*(a + b*ArcCosh[c + d*x]))/b]*Sinh[(2*a)/b])/(2*b^3*d) - (e^3*CoshIntegral[(4*(a + b*ArcCosh[c + d*x]))
/b]*Sinh[(4*a)/b])/(b^3*d) + (e^3*Cosh[(2*a)/b]*SinhIntegral[(2*(a + b*ArcCosh[c + d*x]))/b])/(2*b^3*d) + (e^3
*Cosh[(4*a)/b]*SinhIntegral[(4*(a + b*ArcCosh[c + d*x]))/b])/(b^3*d)

Rule 12

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

Rule 3379

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[I*(SinhIntegral[c*f*(fz/
d) + f*fz*x]/d), x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*e - c*f*fz*I, 0]

Rule 3382

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CoshIntegral[c*f*(fz/d)
+ f*fz*x]/d, x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*(e - Pi/2) - c*f*fz*I, 0]

Rule 3384

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[c*(f/d) + f*x]
/(c + d*x), x], x] + Dist[Sin[(d*e - c*f)/d], Int[Cos[c*(f/d) + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f},
x] && NeQ[d*e - c*f, 0]

Rule 5556

Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int
[ExpandTrigReduce[(c + d*x)^m, Sinh[a + b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n,
0] && IGtQ[p, 0]

Rule 5886

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[x^m*Sqrt[1 + c*x]*Sqrt[-1 + c*x]*((
a + b*ArcCosh[c*x])^(n + 1)/(b*c*(n + 1))), x] + (-Dist[c*((m + 1)/(b*(n + 1))), Int[x^(m + 1)*((a + b*ArcCosh
[c*x])^(n + 1)/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])), x], x] + Dist[m/(b*c*(n + 1)), Int[x^(m - 1)*((a + b*ArcCosh[c
*x])^(n + 1)/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])), x], x]) /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && LtQ[n, -2]

Rule 5887

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Dist[1/(b*c^(m + 1)), Subst[Int[x^n*Cosh
[-a/b + x/b]^m*Sinh[-a/b + x/b], x], x, a + b*ArcCosh[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[m, 0]

Rule 5951

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*((f_.)*(x_))^(m_.))/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_
.)*(x_)]), x_Symbol] :> Simp[(f*x)^m*((a + b*ArcCosh[c*x])^(n + 1)/(b*c*(n + 1)))*Simp[Sqrt[1 + c*x]/Sqrt[d1 +
e1*x]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]], x] - Dist[f*(m/(b*c*(n + 1)))*Simp[Sqrt[1 + c*x]/Sqrt[d1 + e1*x]
]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]], Int[(f*x)^(m - 1)*(a + b*ArcCosh[c*x])^(n + 1), x], x] /; FreeQ[{a, b,
c, d1, e1, d2, e2, f, m}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && LtQ[n, -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
]

Rubi steps

\begin {align*} \int \frac {(c e+d e x)^3}{\left (a+b \cosh ^{-1}(c+d x)\right )^3} \, dx &=\frac {\text {Subst}\left (\int \frac {e^3 x^3}{\left (a+b \cosh ^{-1}(x)\right )^3} \, dx,x,c+d x\right )}{d}\\ &=\frac {e^3 \text {Subst}\left (\int \frac {x^3}{\left (a+b \cosh ^{-1}(x)\right )^3} \, dx,x,c+d x\right )}{d}\\ &=-\frac {e^3 \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x}}{2 b d \left (a+b \cosh ^{-1}(c+d x)\right )^2}-\frac {\left (3 e^3\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {-1+x} \sqrt {1+x} \left (a+b \cosh ^{-1}(x)\right )^2} \, dx,x,c+d x\right )}{2 b d}+\frac {\left (2 e^3\right ) \text {Subst}\left (\int \frac {x^4}{\sqrt {-1+x} \sqrt {1+x} \left (a+b \cosh ^{-1}(x)\right )^2} \, dx,x,c+d x\right )}{b d}\\ &=-\frac {e^3 \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x}}{2 b d \left (a+b \cosh ^{-1}(c+d x)\right )^2}+\frac {3 e^3 (c+d x)^2}{2 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac {2 e^3 (c+d x)^4}{b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac {\left (3 e^3\right ) \text {Subst}\left (\int \frac {x}{a+b \cosh ^{-1}(x)} \, dx,x,c+d x\right )}{b^2 d}+\frac {\left (8 e^3\right ) \text {Subst}\left (\int \frac {x^3}{a+b \cosh ^{-1}(x)} \, dx,x,c+d x\right )}{b^2 d}\\ &=-\frac {e^3 \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x}}{2 b d \left (a+b \cosh ^{-1}(c+d x)\right )^2}+\frac {3 e^3 (c+d x)^2}{2 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac {2 e^3 (c+d x)^4}{b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac {\left (3 e^3\right ) \text {Subst}\left (\int \frac {\cosh (x) \sinh (x)}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{b^2 d}+\frac {\left (8 e^3\right ) \text {Subst}\left (\int \frac {\cosh ^3(x) \sinh (x)}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{b^2 d}\\ &=-\frac {e^3 \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x}}{2 b d \left (a+b \cosh ^{-1}(c+d x)\right )^2}+\frac {3 e^3 (c+d x)^2}{2 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac {2 e^3 (c+d x)^4}{b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac {\left (3 e^3\right ) \text {Subst}\left (\int \frac {\sinh (2 x)}{2 (a+b x)} \, dx,x,\cosh ^{-1}(c+d x)\right )}{b^2 d}+\frac {\left (8 e^3\right ) \text {Subst}\left (\int \left (\frac {\sinh (2 x)}{4 (a+b x)}+\frac {\sinh (4 x)}{8 (a+b x)}\right ) \, dx,x,\cosh ^{-1}(c+d x)\right )}{b^2 d}\\ &=-\frac {e^3 \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x}}{2 b d \left (a+b \cosh ^{-1}(c+d x)\right )^2}+\frac {3 e^3 (c+d x)^2}{2 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac {2 e^3 (c+d x)^4}{b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}+\frac {e^3 \text {Subst}\left (\int \frac {\sinh (4 x)}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{b^2 d}-\frac {\left (3 e^3\right ) \text {Subst}\left (\int \frac {\sinh (2 x)}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{2 b^2 d}+\frac {\left (2 e^3\right ) \text {Subst}\left (\int \frac {\sinh (2 x)}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{b^2 d}\\ &=-\frac {e^3 \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x}}{2 b d \left (a+b \cosh ^{-1}(c+d x)\right )^2}+\frac {3 e^3 (c+d x)^2}{2 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac {2 e^3 (c+d x)^4}{b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac {\left (3 e^3 \cosh \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {2 a}{b}+2 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{2 b^2 d}+\frac {\left (2 e^3 \cosh \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {2 a}{b}+2 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{b^2 d}+\frac {\left (e^3 \cosh \left (\frac {4 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {4 a}{b}+4 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{b^2 d}+\frac {\left (3 e^3 \sinh \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {2 a}{b}+2 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{2 b^2 d}-\frac {\left (2 e^3 \sinh \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {2 a}{b}+2 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{b^2 d}-\frac {\left (e^3 \sinh \left (\frac {4 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {4 a}{b}+4 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{b^2 d}\\ &=-\frac {e^3 \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x}}{2 b d \left (a+b \cosh ^{-1}(c+d x)\right )^2}+\frac {3 e^3 (c+d x)^2}{2 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac {2 e^3 (c+d x)^4}{b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac {e^3 \text {Chi}\left (\frac {2 a}{b}+2 \cosh ^{-1}(c+d x)\right ) \sinh \left (\frac {2 a}{b}\right )}{2 b^3 d}-\frac {e^3 \text {Chi}\left (\frac {4 a}{b}+4 \cosh ^{-1}(c+d x)\right ) \sinh \left (\frac {4 a}{b}\right )}{b^3 d}+\frac {e^3 \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 a}{b}+2 \cosh ^{-1}(c+d x)\right )}{2 b^3 d}+\frac {e^3 \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 a}{b}+4 \cosh ^{-1}(c+d x)\right )}{b^3 d}\\ \end {align*}

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Mathematica [A]
time = 0.50, size = 186, normalized size = 0.73 \begin {gather*} \frac {e^3 \left (-\frac {b^2 \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x}}{\left (a+b \cosh ^{-1}(c+d x)\right )^2}+\frac {b \left (3 (c+d x)^2-4 (c+d x)^4\right )}{a+b \cosh ^{-1}(c+d x)}-\text {Chi}\left (2 \left (\frac {a}{b}+\cosh ^{-1}(c+d x)\right )\right ) \sinh \left (\frac {2 a}{b}\right )-2 \text {Chi}\left (4 \left (\frac {a}{b}+\cosh ^{-1}(c+d x)\right )\right ) \sinh \left (\frac {4 a}{b}\right )+\cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\cosh ^{-1}(c+d x)\right )\right )+2 \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (4 \left (\frac {a}{b}+\cosh ^{-1}(c+d x)\right )\right )\right )}{2 b^3 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(e^3*(-((b^2*Sqrt[-1 + c + d*x]*(c + d*x)^3*Sqrt[1 + c + d*x])/(a + b*ArcCosh[c + d*x])^2) + (b*(3*(c + d*x)^2
- 4*(c + d*x)^4))/(a + b*ArcCosh[c + d*x]) - CoshIntegral[2*(a/b + ArcCosh[c + d*x])]*Sinh[(2*a)/b] - 2*CoshI
ntegral[4*(a/b + ArcCosh[c + d*x])]*Sinh[(4*a)/b] + Cosh[(2*a)/b]*SinhIntegral[2*(a/b + ArcCosh[c + d*x])] + 2
*Cosh[(4*a)/b]*SinhIntegral[4*(a/b + ArcCosh[c + d*x])]))/(2*b^3*d)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(623\) vs. \(2(242)=484\).
time = 0.18, size = 624, normalized size = 2.46

method	result	size
derivativedivides	\(\frac {-\frac {\left (-8 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, \left (d x +c \right )^{3}+4 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, \left (d x +c \right )+8 \left (d x +c \right )^{4}-8 \left (d x +c \right )^{2}+1\right ) e^{3} \left (4 b \,\mathrm {arccosh}\left (d x +c \right )+4 a -b \right )}{32 b^{2} \left (b^{2} \mathrm {arccosh}\left (d x +c \right )^{2}+2 a b \,\mathrm {arccosh}\left (d x +c \right )+a^{2}\right )}+\frac {e^{3} {\mathrm e}^{\frac {4 a}{b}} \expIntegral \left (1, 4 \,\mathrm {arccosh}\left (d x +c \right )+\frac {4 a}{b}\right )}{2 b^{3}}-\frac {\left (-2 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, \left (d x +c \right )+2 \left (d x +c \right )^{2}-1\right ) e^{3} \left (2 b \,\mathrm {arccosh}\left (d x +c \right )+2 a -b \right )}{16 b^{2} \left (b^{2} \mathrm {arccosh}\left (d x +c \right )^{2}+2 a b \,\mathrm {arccosh}\left (d x +c \right )+a^{2}\right )}+\frac {e^{3} {\mathrm e}^{\frac {2 a}{b}} \expIntegral \left (1, 2 \,\mathrm {arccosh}\left (d x +c \right )+\frac {2 a}{b}\right )}{4 b^{3}}-\frac {e^{3} \left (2 \left (d x +c \right )^{2}-1+2 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, \left (d x +c \right )\right )}{16 b \left (a +b \,\mathrm {arccosh}\left (d x +c \right )\right )^{2}}-\frac {e^{3} \left (2 \left (d x +c \right )^{2}-1+2 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, \left (d x +c \right )\right )}{8 b^{2} \left (a +b \,\mathrm {arccosh}\left (d x +c \right )\right )}-\frac {e^{3} {\mathrm e}^{-\frac {2 a}{b}} \expIntegral \left (1, -2 \,\mathrm {arccosh}\left (d x +c \right )-\frac {2 a}{b}\right )}{4 b^{3}}-\frac {e^{3} \left (8 \left (d x +c \right )^{4}-8 \left (d x +c \right )^{2}+8 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, \left (d x +c \right )^{3}-4 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, \left (d x +c \right )+1\right )}{32 b \left (a +b \,\mathrm {arccosh}\left (d x +c \right )\right )^{2}}-\frac {e^{3} \left (8 \left (d x +c \right )^{4}-8 \left (d x +c \right )^{2}+8 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, \left (d x +c \right )^{3}-4 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, \left (d x +c \right )+1\right )}{8 b^{2} \left (a +b \,\mathrm {arccosh}\left (d x +c \right )\right )}-\frac {e^{3} {\mathrm e}^{-\frac {4 a}{b}} \expIntegral \left (1, -4 \,\mathrm {arccosh}\left (d x +c \right )-\frac {4 a}{b}\right )}{2 b^{3}}}{d}\)	\(624\)
default	\(\frac {-\frac {\left (-8 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, \left (d x +c \right )^{3}+4 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, \left (d x +c \right )+8 \left (d x +c \right )^{4}-8 \left (d x +c \right )^{2}+1\right ) e^{3} \left (4 b \,\mathrm {arccosh}\left (d x +c \right )+4 a -b \right )}{32 b^{2} \left (b^{2} \mathrm {arccosh}\left (d x +c \right )^{2}+2 a b \,\mathrm {arccosh}\left (d x +c \right )+a^{2}\right )}+\frac {e^{3} {\mathrm e}^{\frac {4 a}{b}} \expIntegral \left (1, 4 \,\mathrm {arccosh}\left (d x +c \right )+\frac {4 a}{b}\right )}{2 b^{3}}-\frac {\left (-2 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, \left (d x +c \right )+2 \left (d x +c \right )^{2}-1\right ) e^{3} \left (2 b \,\mathrm {arccosh}\left (d x +c \right )+2 a -b \right )}{16 b^{2} \left (b^{2} \mathrm {arccosh}\left (d x +c \right )^{2}+2 a b \,\mathrm {arccosh}\left (d x +c \right )+a^{2}\right )}+\frac {e^{3} {\mathrm e}^{\frac {2 a}{b}} \expIntegral \left (1, 2 \,\mathrm {arccosh}\left (d x +c \right )+\frac {2 a}{b}\right )}{4 b^{3}}-\frac {e^{3} \left (2 \left (d x +c \right )^{2}-1+2 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, \left (d x +c \right )\right )}{16 b \left (a +b \,\mathrm {arccosh}\left (d x +c \right )\right )^{2}}-\frac {e^{3} \left (2 \left (d x +c \right )^{2}-1+2 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, \left (d x +c \right )\right )}{8 b^{2} \left (a +b \,\mathrm {arccosh}\left (d x +c \right )\right )}-\frac {e^{3} {\mathrm e}^{-\frac {2 a}{b}} \expIntegral \left (1, -2 \,\mathrm {arccosh}\left (d x +c \right )-\frac {2 a}{b}\right )}{4 b^{3}}-\frac {e^{3} \left (8 \left (d x +c \right )^{4}-8 \left (d x +c \right )^{2}+8 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, \left (d x +c \right )^{3}-4 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, \left (d x +c \right )+1\right )}{32 b \left (a +b \,\mathrm {arccosh}\left (d x +c \right )\right )^{2}}-\frac {e^{3} \left (8 \left (d x +c \right )^{4}-8 \left (d x +c \right )^{2}+8 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, \left (d x +c \right )^{3}-4 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, \left (d x +c \right )+1\right )}{8 b^{2} \left (a +b \,\mathrm {arccosh}\left (d x +c \right )\right )}-\frac {e^{3} {\mathrm e}^{-\frac {4 a}{b}} \expIntegral \left (1, -4 \,\mathrm {arccosh}\left (d x +c \right )-\frac {4 a}{b}\right )}{2 b^{3}}}{d}\)	\(624\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*e*x+c*e)^3/(a+b*arccosh(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

1/d*(-1/32*(-8*(d*x+c+1)^(1/2)*(d*x+c-1)^(1/2)*(d*x+c)^3+4*(d*x+c+1)^(1/2)*(d*x+c-1)^(1/2)*(d*x+c)+8*(d*x+c)^4
-8*(d*x+c)^2+1)*e^3*(4*b*arccosh(d*x+c)+4*a-b)/b^2/(b^2*arccosh(d*x+c)^2+2*a*b*arccosh(d*x+c)+a^2)+1/2*e^3/b^3
*exp(4*a/b)*Ei(1,4*arccosh(d*x+c)+4*a/b)-1/16*(-2*(d*x+c+1)^(1/2)*(d*x+c-1)^(1/2)*(d*x+c)+2*(d*x+c)^2-1)*e^3*(
2*b*arccosh(d*x+c)+2*a-b)/b^2/(b^2*arccosh(d*x+c)^2+2*a*b*arccosh(d*x+c)+a^2)+1/4*e^3/b^3*exp(2*a/b)*Ei(1,2*ar
ccosh(d*x+c)+2*a/b)-1/16/b*e^3*(2*(d*x+c)^2-1+2*(d*x+c+1)^(1/2)*(d*x+c-1)^(1/2)*(d*x+c))/(a+b*arccosh(d*x+c))^
2-1/8/b^2*e^3*(2*(d*x+c)^2-1+2*(d*x+c+1)^(1/2)*(d*x+c-1)^(1/2)*(d*x+c))/(a+b*arccosh(d*x+c))-1/4/b^3*e^3*exp(-
2*a/b)*Ei(1,-2*arccosh(d*x+c)-2*a/b)-1/32/b*e^3*(8*(d*x+c)^4-8*(d*x+c)^2+8*(d*x+c+1)^(1/2)*(d*x+c-1)^(1/2)*(d*
x+c)^3-4*(d*x+c+1)^(1/2)*(d*x+c-1)^(1/2)*(d*x+c)+1)/(a+b*arccosh(d*x+c))^2-1/8/b^2*e^3*(8*(d*x+c)^4-8*(d*x+c)^
2+8*(d*x+c+1)^(1/2)*(d*x+c-1)^(1/2)*(d*x+c)^3-4*(d*x+c+1)^(1/2)*(d*x+c-1)^(1/2)*(d*x+c)+1)/(a+b*arccosh(d*x+c)
)-1/2/b^3*e^3*exp(-4*a/b)*Ei(1,-4*arccosh(d*x+c)-4*a/b))

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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((d*e*x+c*e)^3/(a+b*arccosh(d*x+c))^3,x, algorithm="maxima")

[Out]

-1/2*((4*a*d^10 + b*d^10)*x^10*e^3 + 10*(4*a*c*d^9 + b*c*d^9)*x^9*e^3 + 3*(4*(15*c^2*d^8 - d^8)*a + (15*c^2*d^
8 - d^8)*b)*x^8*e^3 + 24*(4*(5*c^3*d^7 - c*d^7)*a + (5*c^3*d^7 - c*d^7)*b)*x^7*e^3 + 3*(4*(70*c^4*d^6 - 28*c^2
*d^6 + d^6)*a + (70*c^4*d^6 - 28*c^2*d^6 + d^6)*b)*x^6*e^3 + 6*(4*(42*c^5*d^5 - 28*c^3*d^5 + 3*c*d^5)*a + (42*
c^5*d^5 - 28*c^3*d^5 + 3*c*d^5)*b)*x^5*e^3 + (4*(210*c^6*d^4 - 210*c^4*d^4 + 45*c^2*d^4 - d^4)*a + (210*c^6*d^
4 - 210*c^4*d^4 + 45*c^2*d^4 - d^4)*b)*x^4*e^3 + 4*(4*(30*c^7*d^3 - 42*c^5*d^3 + 15*c^3*d^3 - c*d^3)*a + (30*c
^7*d^3 - 42*c^5*d^3 + 15*c^3*d^3 - c*d^3)*b)*x^3*e^3 + ((4*a*d^7 + b*d^7)*x^7*e^3 + 7*(4*a*c*d^6 + b*c*d^6)*x^
6*e^3 + (6*(14*c^2*d^5 - d^5)*a + (21*c^2*d^5 - d^5)*b)*x^5*e^3 + 5*(2*(14*c^3*d^4 - 3*c*d^4)*a + (7*c^3*d^4 -
c*d^4)*b)*x^4*e^3 + (2*(70*c^4*d^3 - 30*c^2*d^3 + d^3)*a + 5*(7*c^4*d^3 - 2*c^2*d^3)*b)*x^3*e^3 + (6*(14*c^5*
d^2 - 10*c^3*d^2 + c*d^2)*a + (21*c^5*d^2 - 10*c^3*d^2)*b)*x^2*e^3 + (2*(14*c^6*d - 15*c^4*d + 3*c^2*d)*a + (7
*c^6*d - 5*c^4*d)*b)*x*e^3 + (2*(2*c^7 - 3*c^5 + c^3)*a + (c^7 - c^5)*b)*e^3)*(d*x + c + 1)^(3/2)*(d*x + c - 1
)^(3/2) + 3*(4*(15*c^8*d^2 - 28*c^6*d^2 + 15*c^4*d^2 - 2*c^2*d^2)*a + (15*c^8*d^2 - 28*c^6*d^2 + 15*c^4*d^2 -
2*c^2*d^2)*b)*x^2*e^3 + (3*(4*a*d^8 + b*d^8)*x^8*e^3 + 24*(4*a*c*d^7 + b*c*d^7)*x^7*e^3 + (24*(14*c^2*d^6 - d^
6)*a + (84*c^2*d^6 - 5*d^6)*b)*x^6*e^3 + 6*(8*(14*c^3*d^5 - 3*c*d^5)*a + (28*c^3*d^5 - 5*c*d^5)*b)*x^5*e^3 + (
15*(56*c^4*d^4 - 24*c^2*d^4 + d^4)*a + (210*c^4*d^4 - 75*c^2*d^4 + 2*d^4)*b)*x^4*e^3 + 4*(3*(56*c^5*d^3 - 40*c
^3*d^3 + 5*c*d^3)*a + (42*c^5*d^3 - 25*c^3*d^3 + 2*c*d^3)*b)*x^3*e^3 + 3*((112*c^6*d^2 - 120*c^4*d^2 + 30*c^2*
d^2 - d^2)*a + (28*c^6*d^2 - 25*c^4*d^2 + 4*c^2*d^2)*b)*x^2*e^3 + 2*(3*(16*c^7*d - 24*c^5*d + 10*c^3*d - c*d)*
a + (12*c^7*d - 15*c^5*d + 4*c^3*d)*b)*x*e^3 + (3*(4*c^8 - 8*c^6 + 5*c^4 - c^2)*a + (3*c^8 - 5*c^6 + 2*c^4)*b)
*e^3)*(d*x + c + 1)*(d*x + c - 1) + 2*(4*(5*c^9*d - 12*c^7*d + 9*c^5*d - 2*c^3*d)*a + (5*c^9*d - 12*c^7*d + 9*
c^5*d - 2*c^3*d)*b)*x*e^3 + (3*(4*a*d^9 + b*d^9)*x^9*e^3 + 27*(4*a*c*d^8 + b*c*d^8)*x^8*e^3 + (6*(72*c^2*d^7 -
5*d^7)*a + (108*c^2*d^7 - 7*d^7)*b)*x^7*e^3 + 7*(6*(24*c^3*d^6 - 5*c*d^6)*a + (36*c^3*d^6 - 7*c*d^6)*b)*x^6*e
^3 + ((1512*c^4*d^5 - 630*c^2*d^5 + 25*d^5)*a + (378*c^4*d^5 - 147*c^2*d^5 + 5*d^5)*b)*x^5*e^3 + ((1512*c^5*d^
4 - 1050*c^3*d^4 + 125*c*d^4)*a + (378*c^5*d^4 - 245*c^3*d^4 + 25*c*d^4)*b)*x^4*e^3 + ((1008*c^6*d^3 - 1050*c^
4*d^3 + 250*c^2*d^3 - 7*d^3)*a + (252*c^6*d^3 - 245*c^4*d^3 + 50*c^2*d^3 - d^3)*b)*x^3*e^3 + ((432*c^7*d^2 - 6
30*c^5*d^2 + 250*c^3*d^2 - 21*c*d^2)*a + (108*c^7*d^2 - 147*c^5*d^2 + 50*c^3*d^2 - 3*c*d^2)*b)*x^2*e^3 + ((108
*c^8*d - 210*c^6*d + 125*c^4*d - 21*c^2*d)*a + (27*c^8*d - 49*c^6*d + 25*c^4*d - 3*c^2*d)*b)*x*e^3 + ((12*c^9
- 30*c^7 + 25*c^5 - 7*c^3)*a + (3*c^9 - 7*c^7 + 5*c^5 - c^3)*b)*e^3)*sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + (4*
(c^10 - 3*c^8 + 3*c^6 - c^4)*a + (c^10 - 3*c^8 + 3*c^6 - c^4)*b)*e^3 + (4*b*d^10*x^10*e^3 + 40*b*c*d^9*x^9*e^3
+ 12*(15*c^2*d^8 - d^8)*b*x^8*e^3 + 96*(5*c^3*d^7 - c*d^7)*b*x^7*e^3 + 12*(70*c^4*d^6 - 28*c^2*d^6 + d^6)*b*x
^6*e^3 + 24*(42*c^5*d^5 - 28*c^3*d^5 + 3*c*d^5)*b*x^5*e^3 + 4*(210*c^6*d^4 - 210*c^4*d^4 + 45*c^2*d^4 - d^4)*b
*x^4*e^3 + 16*(30*c^7*d^3 - 42*c^5*d^3 + 15*c^3*d^3 - c*d^3)*b*x^3*e^3 + 12*(15*c^8*d^2 - 28*c^6*d^2 + 15*c^4*
d^2 - 2*c^2*d^2)*b*x^2*e^3 + 2*(2*b*d^7*x^7*e^3 + 14*b*c*d^6*x^6*e^3 + 3*(14*c^2*d^5 - d^5)*b*x^5*e^3 + 5*(14*
c^3*d^4 - 3*c*d^4)*b*x^4*e^3 + (70*c^4*d^3 - 30*c^2*d^3 + d^3)*b*x^3*e^3 + 3*(14*c^5*d^2 - 10*c^3*d^2 + c*d^2)
*b*x^2*e^3 + (14*c^6*d - 15*c^4*d + 3*c^2*d)*b*x*e^3 + (2*c^7 - 3*c^5 + c^3)*b*e^3)*(d*x + c + 1)^(3/2)*(d*x +
c - 1)^(3/2) + 8*(5*c^9*d - 12*c^7*d + 9*c^5*d - 2*c^3*d)*b*x*e^3 + 3*(4*b*d^8*x^8*e^3 + 32*b*c*d^7*x^7*e^3 +
8*(14*c^2*d^6 - d^6)*b*x^6*e^3 + 16*(14*c^3*d^5 - 3*c*d^5)*b*x^5*e^3 + 5*(56*c^4*d^4 - 24*c^2*d^4 + d^4)*b*x^
4*e^3 + 4*(56*c^5*d^3 - 40*c^3*d^3 + 5*c*d^3)*b*x^3*e^3 + (112*c^6*d^2 - 120*c^4*d^2 + 30*c^2*d^2 - d^2)*b*x^2
*e^3 + 2*(16*c^7*d - 24*c^5*d + 10*c^3*d - c*d)*b*x*e^3 + (4*c^8 - 8*c^6 + 5*c^4 - c^2)*b*e^3)*(d*x + c + 1)*(
d*x + c - 1) + 4*(c^10 - 3*c^8 + 3*c^6 - c^4)*b*e^3 + (12*b*d^9*x^9*e^3 + 108*b*c*d^8*x^8*e^3 + 6*(72*c^2*d^7
- 5*d^7)*b*x^7*e^3 + 42*(24*c^3*d^6 - 5*c*d^6)*b*x^6*e^3 + (1512*c^4*d^5 - 630*c^2*d^5 + 25*d^5)*b*x^5*e^3 + (
1512*c^5*d^4 - 1050*c^3*d^4 + 125*c*d^4)*b*x^4*e^3 + (1008*c^6*d^3 - 1050*c^4*d^3 + 250*c^2*d^3 - 7*d^3)*b*x^3
*e^3 + (432*c^7*d^2 - 630*c^5*d^2 + 250*c^3*d^2 - 21*c*d^2)*b*x^2*e^3 + (108*c^8*d - 210*c^6*d + 125*c^4*d - 2
1*c^2*d)*b*x*e^3 + (12*c^9 - 30*c^7 + 25*c^5 - 7*c^3)*b*e^3)*sqrt(d*x + c + 1)*sqrt(d*x + c - 1))*log(d*x + sq
rt(d*x + c + 1)*sqrt(d*x + c - 1) + c))/(a^2*b^2*d^7*x^6 + 6*a^2*b^2*c*d^6*x^5 + 3*(5*c^2*d^5 - d^5)*a^2*b^2*x
^4 + 4*(5*c^3*d^4 - 3*c*d^4)*a^2*b^2*x^3 + 3*(5*c^4*d^3 - 6*c^2*d^3 + d^3)*a^2*b^2*x^2 + 6*(c^5*d^2 - 2*c^3*d^
2 + c*d^2)*a^2*b^2*x + (c^6*d - 3*c^4*d + 3*c^2*d - d)*a^2*b^2 + (a^2*b^2*d^4*x^3 + 3*a^2*b^2*c*d^3*x^2 + 3*a^
2*b^2*c^2*d^2*x + a^2*b^2*c^3*d)*(d*x + c + 1)^...

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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((d*e*x+c*e)^3/(a+b*arccosh(d*x+c))^3,x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)**3/(a+b*acosh(d*x+c))**3,x)

[Out]

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

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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((d*e*x+c*e)^3/(a+b*arccosh(d*x+c))^3,x, algorithm="giac")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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