3.2.44 \(\int \frac {(c e+d e x)^2}{(a+b \cosh ^{-1}(c+d x))^3} \, dx\) [144]

Optimal. Leaf size=252 \[ -\frac {e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{2 b d \left (a+b \cosh ^{-1}(c+d x)\right )^2}+\frac {e^2 (c+d x)}{b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac {3 e^2 (c+d x)^3}{2 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac {e^2 \text {Chi}\left (\frac {a+b \cosh ^{-1}(c+d x)}{b}\right ) \sinh \left (\frac {a}{b}\right )}{8 b^3 d}-\frac {9 e^2 \text {Chi}\left (\frac {3 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right ) \sinh \left (\frac {3 a}{b}\right )}{8 b^3 d}+\frac {e^2 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \cosh ^{-1}(c+d x)}{b}\right )}{8 b^3 d}+\frac {9 e^2 \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )}{8 b^3 d} \]

[Out]

e^2*(d*x+c)/b^2/d/(a+b*arccosh(d*x+c))-3/2*e^2*(d*x+c)^3/b^2/d/(a+b*arccosh(d*x+c))+1/8*e^2*cosh(a/b)*Shi((a+b
*arccosh(d*x+c))/b)/b^3/d+9/8*e^2*cosh(3*a/b)*Shi(3*(a+b*arccosh(d*x+c))/b)/b^3/d-1/8*e^2*Chi((a+b*arccosh(d*x
+c))/b)*sinh(a/b)/b^3/d-9/8*e^2*Chi(3*(a+b*arccosh(d*x+c))/b)*sinh(3*a/b)/b^3/d-1/2*e^2*(d*x+c)^2*(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 = 0.53, antiderivative size = 252, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 10, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {5996, 12, 5886, 5951, 5887, 5556, 3384, 3379, 3382, 5881} \begin {gather*} -\frac {e^2 \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \cosh ^{-1}(c+d x)}{b}\right )}{8 b^3 d}-\frac {9 e^2 \sinh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )}{8 b^3 d}+\frac {e^2 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \cosh ^{-1}(c+d x)}{b}\right )}{8 b^3 d}+\frac {9 e^2 \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )}{8 b^3 d}-\frac {3 e^2 (c+d x)^3}{2 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}+\frac {e^2 (c+d x)}{b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac {e^2 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2}{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)^2/(a + b*ArcCosh[c + d*x])^3,x]

[Out]

-1/2*(e^2*Sqrt[-1 + c + d*x]*(c + d*x)^2*Sqrt[1 + c + d*x])/(b*d*(a + b*ArcCosh[c + d*x])^2) + (e^2*(c + d*x))
/(b^2*d*(a + b*ArcCosh[c + d*x])) - (3*e^2*(c + d*x)^3)/(2*b^2*d*(a + b*ArcCosh[c + d*x])) - (e^2*CoshIntegral
[(a + b*ArcCosh[c + d*x])/b]*Sinh[a/b])/(8*b^3*d) - (9*e^2*CoshIntegral[(3*(a + b*ArcCosh[c + d*x]))/b]*Sinh[(
3*a)/b])/(8*b^3*d) + (e^2*Cosh[a/b]*SinhIntegral[(a + b*ArcCosh[c + d*x])/b])/(8*b^3*d) + (9*e^2*Cosh[(3*a)/b]
*SinhIntegral[(3*(a + b*ArcCosh[c + d*x]))/b])/(8*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 5881

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

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)^2}{\left (a+b \cosh ^{-1}(c+d x)\right )^3} \, dx &=\frac {\text {Subst}\left (\int \frac {e^2 x^2}{\left (a+b \cosh ^{-1}(x)\right )^3} \, dx,x,c+d x\right )}{d}\\ &=\frac {e^2 \text {Subst}\left (\int \frac {x^2}{\left (a+b \cosh ^{-1}(x)\right )^3} \, dx,x,c+d x\right )}{d}\\ &=-\frac {e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{2 b d \left (a+b \cosh ^{-1}(c+d x)\right )^2}-\frac {e^2 \text {Subst}\left (\int \frac {x}{\sqrt {-1+x} \sqrt {1+x} \left (a+b \cosh ^{-1}(x)\right )^2} \, dx,x,c+d x\right )}{b d}+\frac {\left (3 e^2\right ) \text {Subst}\left (\int \frac {x^3}{\sqrt {-1+x} \sqrt {1+x} \left (a+b \cosh ^{-1}(x)\right )^2} \, dx,x,c+d x\right )}{2 b d}\\ &=-\frac {e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{2 b d \left (a+b \cosh ^{-1}(c+d x)\right )^2}+\frac {e^2 (c+d x)}{b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac {3 e^2 (c+d x)^3}{2 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac {e^2 \text {Subst}\left (\int \frac {1}{a+b \cosh ^{-1}(x)} \, dx,x,c+d x\right )}{b^2 d}+\frac {\left (9 e^2\right ) \text {Subst}\left (\int \frac {x^2}{a+b \cosh ^{-1}(x)} \, dx,x,c+d x\right )}{2 b^2 d}\\ &=-\frac {e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{2 b d \left (a+b \cosh ^{-1}(c+d x)\right )^2}+\frac {e^2 (c+d x)}{b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac {3 e^2 (c+d x)^3}{2 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}+\frac {e^2 \text {Subst}\left (\int \frac {\sinh \left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \cosh ^{-1}(c+d x)\right )}{b^3 d}+\frac {\left (9 e^2\right ) \text {Subst}\left (\int \frac {\cosh ^2(x) \sinh (x)}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{2 b^2 d}\\ &=-\frac {e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{2 b d \left (a+b \cosh ^{-1}(c+d x)\right )^2}+\frac {e^2 (c+d x)}{b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac {3 e^2 (c+d x)^3}{2 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}+\frac {\left (9 e^2\right ) \text {Subst}\left (\int \left (\frac {\sinh (x)}{4 (a+b x)}+\frac {\sinh (3 x)}{4 (a+b x)}\right ) \, dx,x,\cosh ^{-1}(c+d x)\right )}{2 b^2 d}-\frac {\left (e^2 \cosh \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \cosh ^{-1}(c+d x)\right )}{b^3 d}+\frac {\left (e^2 \sinh \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \cosh ^{-1}(c+d x)\right )}{b^3 d}\\ &=-\frac {e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{2 b d \left (a+b \cosh ^{-1}(c+d x)\right )^2}+\frac {e^2 (c+d x)}{b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac {3 e^2 (c+d x)^3}{2 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}+\frac {e^2 \text {Chi}\left (\frac {a+b \cosh ^{-1}(c+d x)}{b}\right ) \sinh \left (\frac {a}{b}\right )}{b^3 d}-\frac {e^2 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \cosh ^{-1}(c+d x)}{b}\right )}{b^3 d}+\frac {\left (9 e^2\right ) \text {Subst}\left (\int \frac {\sinh (x)}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{8 b^2 d}+\frac {\left (9 e^2\right ) \text {Subst}\left (\int \frac {\sinh (3 x)}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{8 b^2 d}\\ &=-\frac {e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{2 b d \left (a+b \cosh ^{-1}(c+d x)\right )^2}+\frac {e^2 (c+d x)}{b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac {3 e^2 (c+d x)^3}{2 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}+\frac {e^2 \text {Chi}\left (\frac {a+b \cosh ^{-1}(c+d x)}{b}\right ) \sinh \left (\frac {a}{b}\right )}{b^3 d}-\frac {e^2 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \cosh ^{-1}(c+d x)}{b}\right )}{b^3 d}+\frac {\left (9 e^2 \cosh \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{8 b^2 d}+\frac {\left (9 e^2 \cosh \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {3 a}{b}+3 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{8 b^2 d}-\frac {\left (9 e^2 \sinh \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{8 b^2 d}-\frac {\left (9 e^2 \sinh \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {3 a}{b}+3 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{8 b^2 d}\\ &=-\frac {e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{2 b d \left (a+b \cosh ^{-1}(c+d x)\right )^2}+\frac {e^2 (c+d x)}{b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac {3 e^2 (c+d x)^3}{2 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac {9 e^2 \text {Chi}\left (\frac {a}{b}+\cosh ^{-1}(c+d x)\right ) \sinh \left (\frac {a}{b}\right )}{8 b^3 d}+\frac {e^2 \text {Chi}\left (\frac {a+b \cosh ^{-1}(c+d x)}{b}\right ) \sinh \left (\frac {a}{b}\right )}{b^3 d}-\frac {9 e^2 \text {Chi}\left (\frac {3 a}{b}+3 \cosh ^{-1}(c+d x)\right ) \sinh \left (\frac {3 a}{b}\right )}{8 b^3 d}+\frac {9 e^2 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\cosh ^{-1}(c+d x)\right )}{8 b^3 d}+\frac {9 e^2 \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 a}{b}+3 \cosh ^{-1}(c+d x)\right )}{8 b^3 d}-\frac {e^2 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \cosh ^{-1}(c+d x)}{b}\right )}{b^3 d}\\ \end {align*}

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Mathematica [A]
time = 0.51, size = 223, normalized size = 0.88 \begin {gather*} \frac {e^2 \left (-\frac {4 b^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{\left (a+b \cosh ^{-1}(c+d x)\right )^2}+\frac {4 b \left (2 (c+d x)-3 (c+d x)^3\right )}{a+b \cosh ^{-1}(c+d x)}+8 \text {Chi}\left (\frac {a}{b}+\cosh ^{-1}(c+d x)\right ) \sinh \left (\frac {a}{b}\right )-8 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\cosh ^{-1}(c+d x)\right )+9 \left (-\text {Chi}\left (\frac {a}{b}+\cosh ^{-1}(c+d x)\right ) \sinh \left (\frac {a}{b}\right )-\text {Chi}\left (3 \left (\frac {a}{b}+\cosh ^{-1}(c+d x)\right )\right ) \sinh \left (\frac {3 a}{b}\right )+\cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\cosh ^{-1}(c+d x)\right )+\cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\cosh ^{-1}(c+d x)\right )\right )\right )\right )}{8 b^3 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c*e + d*e*x)^2/(a + b*ArcCosh[c + d*x])^3,x]

[Out]

(e^2*((-4*b^2*Sqrt[-1 + c + d*x]*(c + d*x)^2*Sqrt[1 + c + d*x])/(a + b*ArcCosh[c + d*x])^2 + (4*b*(2*(c + d*x)
- 3*(c + d*x)^3))/(a + b*ArcCosh[c + d*x]) + 8*CoshIntegral[a/b + ArcCosh[c + d*x]]*Sinh[a/b] - 8*Cosh[a/b]*S
inhIntegral[a/b + ArcCosh[c + d*x]] + 9*(-(CoshIntegral[a/b + ArcCosh[c + d*x]]*Sinh[a/b]) - CoshIntegral[3*(a
/b + ArcCosh[c + d*x])]*Sinh[(3*a)/b] + Cosh[a/b]*SinhIntegral[a/b + ArcCosh[c + d*x]] + Cosh[(3*a)/b]*SinhInt
egral[3*(a/b + ArcCosh[c + d*x])])))/(8*b^3*d)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(556\) vs. \(2(236)=472\).
time = 0.11, size = 557, normalized size = 2.21

method	result	size
derivativedivides	\(\frac {-\frac {\left (-4 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, \left (d x +c \right )^{2}+\sqrt {d x +c -1}\, \sqrt {d x +c +1}+4 \left (d x +c \right )^{3}-3 d x -3 c \right ) e^{2} \left (3 b \,\mathrm {arccosh}\left (d x +c \right )+3 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 {9 e^{2} {\mathrm e}^{\frac {3 a}{b}} \expIntegral \left (1, 3 \,\mathrm {arccosh}\left (d x +c \right )+\frac {3 a}{b}\right )}{16 b^{3}}-\frac {\left (-\sqrt {d x +c -1}\, \sqrt {d x +c +1}+d x +c \right ) e^{2} \left (b \,\mathrm {arccosh}\left (d x +c \right )+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^{2} {\mathrm e}^{\frac {a}{b}} \expIntegral \left (1, \mathrm {arccosh}\left (d x +c \right )+\frac {a}{b}\right )}{16 b^{3}}-\frac {e^{2} \left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )}{16 b \left (a +b \,\mathrm {arccosh}\left (d x +c \right )\right )^{2}}-\frac {e^{2} \left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )}{16 b^{2} \left (a +b \,\mathrm {arccosh}\left (d x +c \right )\right )}-\frac {e^{2} {\mathrm e}^{-\frac {a}{b}} \expIntegral \left (1, -\mathrm {arccosh}\left (d x +c \right )-\frac {a}{b}\right )}{16 b^{3}}-\frac {e^{2} \left (4 \left (d x +c \right )^{3}-3 d x -3 c +4 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, \left (d x +c \right )^{2}-\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )}{16 b \left (a +b \,\mathrm {arccosh}\left (d x +c \right )\right )^{2}}-\frac {3 e^{2} \left (4 \left (d x +c \right )^{3}-3 d x -3 c +4 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, \left (d x +c \right )^{2}-\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )}{16 b^{2} \left (a +b \,\mathrm {arccosh}\left (d x +c \right )\right )}-\frac {9 e^{2} {\mathrm e}^{-\frac {3 a}{b}} \expIntegral \left (1, -3 \,\mathrm {arccosh}\left (d x +c \right )-\frac {3 a}{b}\right )}{16 b^{3}}}{d}\)	\(557\)
default	\(\frac {-\frac {\left (-4 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, \left (d x +c \right )^{2}+\sqrt {d x +c -1}\, \sqrt {d x +c +1}+4 \left (d x +c \right )^{3}-3 d x -3 c \right ) e^{2} \left (3 b \,\mathrm {arccosh}\left (d x +c \right )+3 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 {9 e^{2} {\mathrm e}^{\frac {3 a}{b}} \expIntegral \left (1, 3 \,\mathrm {arccosh}\left (d x +c \right )+\frac {3 a}{b}\right )}{16 b^{3}}-\frac {\left (-\sqrt {d x +c -1}\, \sqrt {d x +c +1}+d x +c \right ) e^{2} \left (b \,\mathrm {arccosh}\left (d x +c \right )+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^{2} {\mathrm e}^{\frac {a}{b}} \expIntegral \left (1, \mathrm {arccosh}\left (d x +c \right )+\frac {a}{b}\right )}{16 b^{3}}-\frac {e^{2} \left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )}{16 b \left (a +b \,\mathrm {arccosh}\left (d x +c \right )\right )^{2}}-\frac {e^{2} \left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )}{16 b^{2} \left (a +b \,\mathrm {arccosh}\left (d x +c \right )\right )}-\frac {e^{2} {\mathrm e}^{-\frac {a}{b}} \expIntegral \left (1, -\mathrm {arccosh}\left (d x +c \right )-\frac {a}{b}\right )}{16 b^{3}}-\frac {e^{2} \left (4 \left (d x +c \right )^{3}-3 d x -3 c +4 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, \left (d x +c \right )^{2}-\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )}{16 b \left (a +b \,\mathrm {arccosh}\left (d x +c \right )\right )^{2}}-\frac {3 e^{2} \left (4 \left (d x +c \right )^{3}-3 d x -3 c +4 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, \left (d x +c \right )^{2}-\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )}{16 b^{2} \left (a +b \,\mathrm {arccosh}\left (d x +c \right )\right )}-\frac {9 e^{2} {\mathrm e}^{-\frac {3 a}{b}} \expIntegral \left (1, -3 \,\mathrm {arccosh}\left (d x +c \right )-\frac {3 a}{b}\right )}{16 b^{3}}}{d}\)	\(557\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*e*x+c*e)^2/(a+b*arccosh(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

1/d*(-1/16*(-4*(d*x+c+1)^(1/2)*(d*x+c-1)^(1/2)*(d*x+c)^2+(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)+4*(d*x+c)^3-3*d*x-3*c
)*e^2*(3*b*arccosh(d*x+c)+3*a-b)/b^2/(b^2*arccosh(d*x+c)^2+2*a*b*arccosh(d*x+c)+a^2)+9/16*e^2/b^3*exp(3*a/b)*E
i(1,3*arccosh(d*x+c)+3*a/b)-1/16*(-(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)+d*x+c)*e^2*(b*arccosh(d*x+c)+a-b)/b^2/(b^2*
arccosh(d*x+c)^2+2*a*b*arccosh(d*x+c)+a^2)+1/16*e^2/b^3*exp(a/b)*Ei(1,arccosh(d*x+c)+a/b)-1/16/b*e^2*(d*x+c+(d
*x+c-1)^(1/2)*(d*x+c+1)^(1/2))/(a+b*arccosh(d*x+c))^2-1/16/b^2*e^2*(d*x+c+(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2))/(a+
b*arccosh(d*x+c))-1/16/b^3*e^2*exp(-a/b)*Ei(1,-arccosh(d*x+c)-a/b)-1/16/b*e^2*(4*(d*x+c)^3-3*d*x-3*c+4*(d*x+c+
1)^(1/2)*(d*x+c-1)^(1/2)*(d*x+c)^2-(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2))/(a+b*arccosh(d*x+c))^2-3/16/b^2*e^2*(4*(d*
x+c)^3-3*d*x-3*c+4*(d*x+c+1)^(1/2)*(d*x+c-1)^(1/2)*(d*x+c)^2-(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2))/(a+b*arccosh(d*x
+c))-9/16/b^3*e^2*exp(-3*a/b)*Ei(1,-3*arccosh(d*x+c)-3*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)^2/(a+b*arccosh(d*x+c))^3,x, algorithm="maxima")

[Out]

-1/2*((3*a*d^9 + b*d^9)*x^9*e^2 + 9*(3*a*c*d^8 + b*c*d^8)*x^8*e^2 + 3*(3*(12*c^2*d^7 - d^7)*a + (12*c^2*d^7 -
d^7)*b)*x^7*e^2 + 21*(3*(4*c^3*d^6 - c*d^6)*a + (4*c^3*d^6 - c*d^6)*b)*x^6*e^2 + 3*(3*(42*c^4*d^5 - 21*c^2*d^5
+ d^5)*a + (42*c^4*d^5 - 21*c^2*d^5 + d^5)*b)*x^5*e^2 + 3*(3*(42*c^5*d^4 - 35*c^3*d^4 + 5*c*d^4)*a + (42*c^5*
d^4 - 35*c^3*d^4 + 5*c*d^4)*b)*x^4*e^2 + (3*(84*c^6*d^3 - 105*c^4*d^3 + 30*c^2*d^3 - d^3)*a + (84*c^6*d^3 - 10
5*c^4*d^3 + 30*c^2*d^3 - d^3)*b)*x^3*e^2 + ((3*a*d^6 + b*d^6)*x^6*e^2 + 6*(3*a*c*d^5 + b*c*d^5)*x^5*e^2 + ((45
*c^2*d^4 - 4*d^4)*a + (15*c^2*d^4 - d^4)*b)*x^4*e^2 + 4*((15*c^3*d^3 - 4*c*d^3)*a + (5*c^3*d^3 - c*d^3)*b)*x^3
*e^2 + ((45*c^4*d^2 - 24*c^2*d^2 + d^2)*a + 3*(5*c^4*d^2 - 2*c^2*d^2)*b)*x^2*e^2 + 2*((9*c^5*d - 8*c^3*d + c*d
)*a + (3*c^5*d - 2*c^3*d)*b)*x*e^2 + ((3*c^6 - 4*c^4 + c^2)*a + (c^6 - c^4)*b)*e^2)*(d*x + c + 1)^(3/2)*(d*x +
c - 1)^(3/2) + 3*(3*(12*c^7*d^2 - 21*c^5*d^2 + 10*c^3*d^2 - c*d^2)*a + (12*c^7*d^2 - 21*c^5*d^2 + 10*c^3*d^2
- c*d^2)*b)*x^2*e^2 + (3*(3*a*d^7 + b*d^7)*x^7*e^2 + 21*(3*a*c*d^6 + b*c*d^6)*x^6*e^2 + ((189*c^2*d^5 - 17*d^5
)*a + (63*c^2*d^5 - 5*d^5)*b)*x^5*e^2 + 5*((63*c^3*d^4 - 17*c*d^4)*a + (21*c^3*d^4 - 5*c*d^4)*b)*x^4*e^2 + (5*
(63*c^4*d^3 - 34*c^2*d^3 + 2*d^3)*a + (105*c^4*d^3 - 50*c^2*d^3 + 2*d^3)*b)*x^3*e^2 + ((189*c^5*d^2 - 170*c^3*
d^2 + 30*c*d^2)*a + (63*c^5*d^2 - 50*c^3*d^2 + 6*c*d^2)*b)*x^2*e^2 + ((63*c^6*d - 85*c^4*d + 30*c^2*d - 2*d)*a
+ (21*c^6*d - 25*c^4*d + 6*c^2*d)*b)*x*e^2 + ((9*c^7 - 17*c^5 + 10*c^3 - 2*c)*a + (3*c^7 - 5*c^5 + 2*c^3)*b)*
e^2)*(d*x + c + 1)*(d*x + c - 1) + 3*(3*(3*c^8*d - 7*c^6*d + 5*c^4*d - c^2*d)*a + (3*c^8*d - 7*c^6*d + 5*c^4*d
- c^2*d)*b)*x*e^2 + (3*(3*a*d^8 + b*d^8)*x^8*e^2 + 24*(3*a*c*d^7 + b*c*d^7)*x^7*e^2 + (2*(126*c^2*d^6 - 11*d^
6)*a + 7*(12*c^2*d^6 - d^6)*b)*x^6*e^2 + 6*(2*(42*c^3*d^5 - 11*c*d^5)*a + 7*(4*c^3*d^5 - c*d^5)*b)*x^5*e^2 + (
6*(105*c^4*d^4 - 55*c^2*d^4 + 3*d^4)*a + 5*(42*c^4*d^4 - 21*c^2*d^4 + d^4)*b)*x^4*e^2 + 4*(2*(63*c^5*d^3 - 55*
c^3*d^3 + 9*c*d^3)*a + (42*c^5*d^3 - 35*c^3*d^3 + 5*c*d^3)*b)*x^3*e^2 + ((252*c^6*d^2 - 330*c^4*d^2 + 108*c^2*
d^2 - 5*d^2)*a + (84*c^6*d^2 - 105*c^4*d^2 + 30*c^2*d^2 - d^2)*b)*x^2*e^2 + 2*((36*c^7*d - 66*c^5*d + 36*c^3*d
- 5*c*d)*a + (12*c^7*d - 21*c^5*d + 10*c^3*d - c*d)*b)*x*e^2 + ((9*c^8 - 22*c^6 + 18*c^4 - 5*c^2)*a + (3*c^8
- 7*c^6 + 5*c^4 - c^2)*b)*e^2)*sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + (3*(c^9 - 3*c^7 + 3*c^5 - c^3)*a + (c^9 -
3*c^7 + 3*c^5 - c^3)*b)*e^2 + (3*b*d^9*x^9*e^2 + 27*b*c*d^8*x^8*e^2 + 9*(12*c^2*d^7 - d^7)*b*x^7*e^2 + 63*(4*
c^3*d^6 - c*d^6)*b*x^6*e^2 + 9*(42*c^4*d^5 - 21*c^2*d^5 + d^5)*b*x^5*e^2 + 9*(42*c^5*d^4 - 35*c^3*d^4 + 5*c*d^
4)*b*x^4*e^2 + 3*(84*c^6*d^3 - 105*c^4*d^3 + 30*c^2*d^3 - d^3)*b*x^3*e^2 + 9*(12*c^7*d^2 - 21*c^5*d^2 + 10*c^3
*d^2 - c*d^2)*b*x^2*e^2 + (3*b*d^6*x^6*e^2 + 18*b*c*d^5*x^5*e^2 + (45*c^2*d^4 - 4*d^4)*b*x^4*e^2 + 4*(15*c^3*d
^3 - 4*c*d^3)*b*x^3*e^2 + (45*c^4*d^2 - 24*c^2*d^2 + d^2)*b*x^2*e^2 + 2*(9*c^5*d - 8*c^3*d + c*d)*b*x*e^2 + (3
*c^6 - 4*c^4 + c^2)*b*e^2)*(d*x + c + 1)^(3/2)*(d*x + c - 1)^(3/2) + 9*(3*c^8*d - 7*c^6*d + 5*c^4*d - c^2*d)*b
*x*e^2 + (9*b*d^7*x^7*e^2 + 63*b*c*d^6*x^6*e^2 + (189*c^2*d^5 - 17*d^5)*b*x^5*e^2 + 5*(63*c^3*d^4 - 17*c*d^4)*
b*x^4*e^2 + 5*(63*c^4*d^3 - 34*c^2*d^3 + 2*d^3)*b*x^3*e^2 + (189*c^5*d^2 - 170*c^3*d^2 + 30*c*d^2)*b*x^2*e^2 +
(63*c^6*d - 85*c^4*d + 30*c^2*d - 2*d)*b*x*e^2 + (9*c^7 - 17*c^5 + 10*c^3 - 2*c)*b*e^2)*(d*x + c + 1)*(d*x +
c - 1) + 3*(c^9 - 3*c^7 + 3*c^5 - c^3)*b*e^2 + (9*b*d^8*x^8*e^2 + 72*b*c*d^7*x^7*e^2 + 2*(126*c^2*d^6 - 11*d^6
)*b*x^6*e^2 + 12*(42*c^3*d^5 - 11*c*d^5)*b*x^5*e^2 + 6*(105*c^4*d^4 - 55*c^2*d^4 + 3*d^4)*b*x^4*e^2 + 8*(63*c^
5*d^3 - 55*c^3*d^3 + 9*c*d^3)*b*x^3*e^2 + (252*c^6*d^2 - 330*c^4*d^2 + 108*c^2*d^2 - 5*d^2)*b*x^2*e^2 + 2*(36*
c^7*d - 66*c^5*d + 36*c^3*d - 5*c*d)*b*x*e^2 + (9*c^8 - 22*c^6 + 18*c^4 - 5*c^2)*b*e^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))/(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)^(3/2)*(d*x + c - 1)^(3/2) + 3*(a^2*b
^2*d^5*x^4 + 4*a^2*b^2*c*d^4*x^3 + (6*c^2*d^3 - d^3)*a^2*b^2*x^2 + 2*(2*c^3*d^2 - c*d^2)*a^2*b^2*x + (c^4*d -
c^2*d)*a^2*b^2)*(d*x + c + 1)*(d*x + c - 1) + (b^4*d^7*x^6 + 6*b^4*c*d^6*x^5 + 3*(5*c^2*d^5 - d^5)*b^4*x^4 + 4
*(5*c^3*d^4 - 3*c*d^4)*b^4*x^3 + 3*(5*c^4*d^3 - 6*c^2*d^3 + d^3)*b^4*x^2 + 6*(c^5*d^2 - 2*c^3*d^2 + c*d^2)*b^4
*x + (c^6*d - 3*c^4*d + 3*c^2*d - d)*b^4 + (b^4*d^4*x^3 + 3*b^4*c*d^3*x^2 + 3*b^4*c^2*d^2*x + b^4*c^3*d)*(d*x
+ c + 1)^(3/2)*(d*x + c - 1)^(3/2) + 3*(b^4*d^5*x^4 + 4*b^4*c*d^4*x^3 + (6*c^2*d^3 - d^3)*b^4*x^2 + 2*(2*c^3*d
^2 - c*d^2)*b^4*x + (c^4*d - c^2*d)*b^4)*(d*x + c + 1)*(d*x + c - 1) + 3*(b^4*d^6*x^5 + 5*b^4*c*d^5*x^4 + 2*(5
*c^2*d^4 - d^4)*b^4*x^3 + 2*(5*c^3*d^3 - 3*c*d^...

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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)^2/(a+b*arccosh(d*x+c))^3,x, algorithm="fricas")

[Out]

integral((d^2*x^2 + 2*c*d*x + c^2)*e^2/(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^{2} \left (\int \frac {c^{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 {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 {2 c 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)**2/(a+b*acosh(d*x+c))**3,x)

[Out]

e**2*(Integral(c**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(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(2*c*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)^2/(a+b*arccosh(d*x+c))^3,x, algorithm="giac")

[Out]

integrate((d*e*x + c*e)^2/(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 )}^2}{{\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)^2/(a + b*acosh(c + d*x))^3,x)

[Out]

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

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