3.2.42 \(\int \frac {(c e+d e x)^4}{(a+b \cosh ^{-1}(c+d x))^3} \, dx\) [142]

Optimal. Leaf size=327 \[ -\frac {e^4 \sqrt {-1+c+d x} (c+d x)^4 \sqrt {1+c+d x}}{2 b d \left (a+b \cosh ^{-1}(c+d x)\right )^2}+\frac {2 e^4 (c+d x)^3}{b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac {5 e^4 (c+d x)^5}{2 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac {e^4 \text {Chi}\left (\frac {a+b \cosh ^{-1}(c+d x)}{b}\right ) \sinh \left (\frac {a}{b}\right )}{16 b^3 d}-\frac {27 e^4 \text {Chi}\left (\frac {3 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right ) \sinh \left (\frac {3 a}{b}\right )}{32 b^3 d}-\frac {25 e^4 \text {Chi}\left (\frac {5 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right ) \sinh \left (\frac {5 a}{b}\right )}{32 b^3 d}+\frac {e^4 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \cosh ^{-1}(c+d x)}{b}\right )}{16 b^3 d}+\frac {27 e^4 \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )}{32 b^3 d}+\frac {25 e^4 \cosh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )}{32 b^3 d} \]

[Out]

2*e^4*(d*x+c)^3/b^2/d/(a+b*arccosh(d*x+c))-5/2*e^4*(d*x+c)^5/b^2/d/(a+b*arccosh(d*x+c))+1/16*e^4*cosh(a/b)*Shi
((a+b*arccosh(d*x+c))/b)/b^3/d+27/32*e^4*cosh(3*a/b)*Shi(3*(a+b*arccosh(d*x+c))/b)/b^3/d+25/32*e^4*cosh(5*a/b)
*Shi(5*(a+b*arccosh(d*x+c))/b)/b^3/d-1/16*e^4*Chi((a+b*arccosh(d*x+c))/b)*sinh(a/b)/b^3/d-27/32*e^4*Chi(3*(a+b
*arccosh(d*x+c))/b)*sinh(3*a/b)/b^3/d-25/32*e^4*Chi(5*(a+b*arccosh(d*x+c))/b)*sinh(5*a/b)/b^3/d-1/2*e^4*(d*x+c
)^4*(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 = 0.74, antiderivative size = 327, normalized size of antiderivative = 1.00, number of steps used = 26, 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^4 \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \cosh ^{-1}(c+d x)}{b}\right )}{16 b^3 d}-\frac {27 e^4 \sinh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )}{32 b^3 d}-\frac {25 e^4 \sinh \left (\frac {5 a}{b}\right ) \text {Chi}\left (\frac {5 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )}{32 b^3 d}+\frac {e^4 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \cosh ^{-1}(c+d x)}{b}\right )}{16 b^3 d}+\frac {27 e^4 \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )}{32 b^3 d}+\frac {25 e^4 \cosh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )}{32 b^3 d}-\frac {5 e^4 (c+d x)^5}{2 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}+\frac {2 e^4 (c+d x)^3}{b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac {e^4 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^4}{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)^4/(a + b*ArcCosh[c + d*x])^3,x]

[Out]

-1/2*(e^4*Sqrt[-1 + c + d*x]*(c + d*x)^4*Sqrt[1 + c + d*x])/(b*d*(a + b*ArcCosh[c + d*x])^2) + (2*e^4*(c + d*x
)^3)/(b^2*d*(a + b*ArcCosh[c + d*x])) - (5*e^4*(c + d*x)^5)/(2*b^2*d*(a + b*ArcCosh[c + d*x])) - (e^4*CoshInte
gral[(a + b*ArcCosh[c + d*x])/b]*Sinh[a/b])/(16*b^3*d) - (27*e^4*CoshIntegral[(3*(a + b*ArcCosh[c + d*x]))/b]*
Sinh[(3*a)/b])/(32*b^3*d) - (25*e^4*CoshIntegral[(5*(a + b*ArcCosh[c + d*x]))/b]*Sinh[(5*a)/b])/(32*b^3*d) + (
e^4*Cosh[a/b]*SinhIntegral[(a + b*ArcCosh[c + d*x])/b])/(16*b^3*d) + (27*e^4*Cosh[(3*a)/b]*SinhIntegral[(3*(a
+ b*ArcCosh[c + d*x]))/b])/(32*b^3*d) + (25*e^4*Cosh[(5*a)/b]*SinhIntegral[(5*(a + b*ArcCosh[c + d*x]))/b])/(3
2*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)^4}{\left (a+b \cosh ^{-1}(c+d x)\right )^3} \, dx &=\frac {\text {Subst}\left (\int \frac {e^4 x^4}{\left (a+b \cosh ^{-1}(x)\right )^3} \, dx,x,c+d x\right )}{d}\\ &=\frac {e^4 \text {Subst}\left (\int \frac {x^4}{\left (a+b \cosh ^{-1}(x)\right )^3} \, dx,x,c+d x\right )}{d}\\ &=-\frac {e^4 \sqrt {-1+c+d x} (c+d x)^4 \sqrt {1+c+d x}}{2 b d \left (a+b \cosh ^{-1}(c+d x)\right )^2}-\frac {\left (2 e^4\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 )}{b d}+\frac {\left (5 e^4\right ) \text {Subst}\left (\int \frac {x^5}{\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^4 \sqrt {-1+c+d x} (c+d x)^4 \sqrt {1+c+d x}}{2 b d \left (a+b \cosh ^{-1}(c+d x)\right )^2}+\frac {2 e^4 (c+d x)^3}{b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac {5 e^4 (c+d x)^5}{2 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac {\left (6 e^4\right ) \text {Subst}\left (\int \frac {x^2}{a+b \cosh ^{-1}(x)} \, dx,x,c+d x\right )}{b^2 d}+\frac {\left (25 e^4\right ) \text {Subst}\left (\int \frac {x^4}{a+b \cosh ^{-1}(x)} \, dx,x,c+d x\right )}{2 b^2 d}\\ &=-\frac {e^4 \sqrt {-1+c+d x} (c+d x)^4 \sqrt {1+c+d x}}{2 b d \left (a+b \cosh ^{-1}(c+d x)\right )^2}+\frac {2 e^4 (c+d x)^3}{b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac {5 e^4 (c+d x)^5}{2 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac {\left (6 e^4\right ) \text {Subst}\left (\int \frac {\cosh ^2(x) \sinh (x)}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{b^2 d}+\frac {\left (25 e^4\right ) \text {Subst}\left (\int \frac {\cosh ^4(x) \sinh (x)}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{2 b^2 d}\\ &=-\frac {e^4 \sqrt {-1+c+d x} (c+d x)^4 \sqrt {1+c+d x}}{2 b d \left (a+b \cosh ^{-1}(c+d x)\right )^2}+\frac {2 e^4 (c+d x)^3}{b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac {5 e^4 (c+d x)^5}{2 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac {\left (6 e^4\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 )}{b^2 d}+\frac {\left (25 e^4\right ) \text {Subst}\left (\int \left (\frac {\sinh (x)}{8 (a+b x)}+\frac {3 \sinh (3 x)}{16 (a+b x)}+\frac {\sinh (5 x)}{16 (a+b x)}\right ) \, dx,x,\cosh ^{-1}(c+d x)\right )}{2 b^2 d}\\ &=-\frac {e^4 \sqrt {-1+c+d x} (c+d x)^4 \sqrt {1+c+d x}}{2 b d \left (a+b \cosh ^{-1}(c+d x)\right )^2}+\frac {2 e^4 (c+d x)^3}{b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac {5 e^4 (c+d x)^5}{2 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}+\frac {\left (25 e^4\right ) \text {Subst}\left (\int \frac {\sinh (5 x)}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{32 b^2 d}-\frac {\left (3 e^4\right ) \text {Subst}\left (\int \frac {\sinh (x)}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{2 b^2 d}-\frac {\left (3 e^4\right ) \text {Subst}\left (\int \frac {\sinh (3 x)}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{2 b^2 d}+\frac {\left (25 e^4\right ) \text {Subst}\left (\int \frac {\sinh (x)}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{16 b^2 d}+\frac {\left (75 e^4\right ) \text {Subst}\left (\int \frac {\sinh (3 x)}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{32 b^2 d}\\ &=-\frac {e^4 \sqrt {-1+c+d x} (c+d x)^4 \sqrt {1+c+d x}}{2 b d \left (a+b \cosh ^{-1}(c+d x)\right )^2}+\frac {2 e^4 (c+d x)^3}{b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac {5 e^4 (c+d x)^5}{2 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac {\left (3 e^4 \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 )}{2 b^2 d}+\frac {\left (25 e^4 \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 )}{16 b^2 d}-\frac {\left (3 e^4 \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 )}{2 b^2 d}+\frac {\left (75 e^4 \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 )}{32 b^2 d}+\frac {\left (25 e^4 \cosh \left (\frac {5 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {5 a}{b}+5 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{32 b^2 d}+\frac {\left (3 e^4 \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 )}{2 b^2 d}-\frac {\left (25 e^4 \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 )}{16 b^2 d}+\frac {\left (3 e^4 \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 )}{2 b^2 d}-\frac {\left (75 e^4 \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 )}{32 b^2 d}-\frac {\left (25 e^4 \sinh \left (\frac {5 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {5 a}{b}+5 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{32 b^2 d}\\ &=-\frac {e^4 \sqrt {-1+c+d x} (c+d x)^4 \sqrt {1+c+d x}}{2 b d \left (a+b \cosh ^{-1}(c+d x)\right )^2}+\frac {2 e^4 (c+d x)^3}{b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac {5 e^4 (c+d x)^5}{2 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac {e^4 \text {Chi}\left (\frac {a}{b}+\cosh ^{-1}(c+d x)\right ) \sinh \left (\frac {a}{b}\right )}{16 b^3 d}-\frac {27 e^4 \text {Chi}\left (\frac {3 a}{b}+3 \cosh ^{-1}(c+d x)\right ) \sinh \left (\frac {3 a}{b}\right )}{32 b^3 d}-\frac {25 e^4 \text {Chi}\left (\frac {5 a}{b}+5 \cosh ^{-1}(c+d x)\right ) \sinh \left (\frac {5 a}{b}\right )}{32 b^3 d}+\frac {e^4 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\cosh ^{-1}(c+d x)\right )}{16 b^3 d}+\frac {27 e^4 \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 a}{b}+3 \cosh ^{-1}(c+d x)\right )}{32 b^3 d}+\frac {25 e^4 \cosh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 a}{b}+5 \cosh ^{-1}(c+d x)\right )}{32 b^3 d}\\ \end {align*}

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Mathematica [A]
time = 1.12, size = 323, normalized size = 0.99 \begin {gather*} \frac {e^4 \left (-\frac {16 b^2 \sqrt {-1+c+d x} (c+d x)^4 \sqrt {1+c+d x}}{\left (a+b \cosh ^{-1}(c+d x)\right )^2}+\frac {16 b \left (4 (c+d x)^3-5 (c+d x)^5\right )}{a+b \cosh ^{-1}(c+d x)}+48 \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 )+25 \left (-2 \text {Chi}\left (\frac {a}{b}+\cosh ^{-1}(c+d x)\right ) \sinh \left (\frac {a}{b}\right )-3 \text {Chi}\left (3 \left (\frac {a}{b}+\cosh ^{-1}(c+d x)\right )\right ) \sinh \left (\frac {3 a}{b}\right )-\text {Chi}\left (5 \left (\frac {a}{b}+\cosh ^{-1}(c+d x)\right )\right ) \sinh \left (\frac {5 a}{b}\right )+2 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\cosh ^{-1}(c+d x)\right )+3 \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\cosh ^{-1}(c+d x)\right )\right )+\cosh \left (\frac {5 a}{b}\right ) \text {Shi}\left (5 \left (\frac {a}{b}+\cosh ^{-1}(c+d x)\right )\right )\right )\right )}{32 b^3 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(e^4*((-16*b^2*Sqrt[-1 + c + d*x]*(c + d*x)^4*Sqrt[1 + c + d*x])/(a + b*ArcCosh[c + d*x])^2 + (16*b*(4*(c + d*
x)^3 - 5*(c + d*x)^5))/(a + b*ArcCosh[c + d*x]) + 48*(CoshIntegral[a/b + ArcCosh[c + d*x]]*Sinh[a/b] + CoshInt
egral[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]*SinhIntegral[3*(a/b + ArcCosh[c + d*x])]) + 25*(-2*CoshIntegral[a/b + ArcCosh[c + d*x]]*Sinh[a/b] - 3*CoshI
ntegral[3*(a/b + ArcCosh[c + d*x])]*Sinh[(3*a)/b] - CoshIntegral[5*(a/b + ArcCosh[c + d*x])]*Sinh[(5*a)/b] + 2
*Cosh[a/b]*SinhIntegral[a/b + ArcCosh[c + d*x]] + 3*Cosh[(3*a)/b]*SinhIntegral[3*(a/b + ArcCosh[c + d*x])] + C
osh[(5*a)/b]*SinhIntegral[5*(a/b + ArcCosh[c + d*x])])))/(32*b^3*d)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(992\) vs. \(2(307)=614\).
time = 0.53, size = 993, normalized size = 3.04 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*e*x+c*e)^4/(a+b*arccosh(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

1/d*(-1/64*(-16*(d*x+c+1)^(1/2)*(d*x+c-1)^(1/2)*(d*x+c)^4+12*(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)+16*(d*x+c)^5-20*(d*x+c)^3+5*d*x+5*c)*e^4*(5*b*arccosh(d*x+c)+5*a-b)/b^2/(b^2*arccosh(
d*x+c)^2+2*a*b*arccosh(d*x+c)+a^2)+25/64*e^4/b^3*exp(5*a/b)*Ei(1,5*arccosh(d*x+c)+5*a/b)-3/64*(-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^4*(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)+27/64*e^4/b^3*exp(3*a/b)*Ei(1,3*arccosh(d*x+c)+3*a/b
)-1/32*(-(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)+d*x+c)*e^4*(b*arccosh(d*x+c)+a-b)/b^2/(b^2*arccosh(d*x+c)^2+2*a*b*arc
cosh(d*x+c)+a^2)+1/32*e^4/b^3*exp(a/b)*Ei(1,arccosh(d*x+c)+a/b)-1/32/b*e^4*(d*x+c+(d*x+c-1)^(1/2)*(d*x+c+1)^(1
/2))/(a+b*arccosh(d*x+c))^2-1/32/b^2*e^4*(d*x+c+(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2))/(a+b*arccosh(d*x+c))-1/32/b^3
*e^4*exp(-a/b)*Ei(1,-arccosh(d*x+c)-a/b)-3/64/b*e^4*(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))^2-9/64/b^2*e^4*(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))-27/64/b^3*e^4*exp(-3*
a/b)*Ei(1,-3*arccosh(d*x+c)-3*a/b)-1/64/b*e^4*(16*(d*x+c)^5-20*(d*x+c)^3+16*(d*x+c+1)^(1/2)*(d*x+c-1)^(1/2)*(d
*x+c)^4+5*d*x+5*c-12*(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))^2-5/64/b^2*e^4*(16*(d*x+c)^5-20*(d*x+c)^3+16*(d*x+c+1)^(1/2)*(d*x+c-1)^(1/2)*(d*x+c)^4+5*d*x+5*c-12*(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))-25/64/b^3*e^4*exp
(-5*a/b)*Ei(1,-5*arccosh(d*x+c)-5*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)^4/(a+b*arccosh(d*x+c))^3,x, algorithm="maxima")

[Out]

-1/2*((5*a*d^11 + b*d^11)*x^11*e^4 + 11*(5*a*c*d^10 + b*c*d^10)*x^10*e^4 + (5*(55*c^2*d^9 - 3*d^9)*a + (55*c^2
*d^9 - 3*d^9)*b)*x^9*e^4 + 3*(5*(55*c^3*d^8 - 9*c*d^8)*a + (55*c^3*d^8 - 9*c*d^8)*b)*x^8*e^4 + 3*(5*(110*c^4*d
^7 - 36*c^2*d^7 + d^7)*a + (110*c^4*d^7 - 36*c^2*d^7 + d^7)*b)*x^7*e^4 + 21*(5*(22*c^5*d^6 - 12*c^3*d^6 + c*d^
6)*a + (22*c^5*d^6 - 12*c^3*d^6 + c*d^6)*b)*x^6*e^4 + (5*(462*c^6*d^5 - 378*c^4*d^5 + 63*c^2*d^5 - d^5)*a + (4
62*c^6*d^5 - 378*c^4*d^5 + 63*c^2*d^5 - d^5)*b)*x^5*e^4 + (5*(330*c^7*d^4 - 378*c^5*d^4 + 105*c^3*d^4 - 5*c*d^
4)*a + (330*c^7*d^4 - 378*c^5*d^4 + 105*c^3*d^4 - 5*c*d^4)*b)*x^4*e^4 + (5*(165*c^8*d^3 - 252*c^6*d^3 + 105*c^
4*d^3 - 10*c^2*d^3)*a + (165*c^8*d^3 - 252*c^6*d^3 + 105*c^4*d^3 - 10*c^2*d^3)*b)*x^3*e^4 + ((5*a*d^8 + b*d^8)
*x^8*e^4 + 8*(5*a*c*d^7 + b*c*d^7)*x^7*e^4 + (4*(35*c^2*d^6 - 2*d^6)*a + (28*c^2*d^6 - d^6)*b)*x^6*e^4 + 2*(4*
(35*c^3*d^5 - 6*c*d^5)*a + (28*c^3*d^5 - 3*c*d^5)*b)*x^5*e^4 + ((350*c^4*d^4 - 120*c^2*d^4 + 3*d^4)*a + 5*(14*
c^4*d^4 - 3*c^2*d^4)*b)*x^4*e^4 + 4*((70*c^5*d^3 - 40*c^3*d^3 + 3*c*d^3)*a + (14*c^5*d^3 - 5*c^3*d^3)*b)*x^3*e
^4 + (2*(70*c^6*d^2 - 60*c^4*d^2 + 9*c^2*d^2)*a + (28*c^6*d^2 - 15*c^4*d^2)*b)*x^2*e^4 + 2*(2*(10*c^7*d - 12*c
^5*d + 3*c^3*d)*a + (4*c^7*d - 3*c^5*d)*b)*x*e^4 + ((5*c^8 - 8*c^6 + 3*c^4)*a + (c^8 - c^6)*b)*e^4)*(d*x + c +
1)^(3/2)*(d*x + c - 1)^(3/2) + (5*(55*c^9*d^2 - 108*c^7*d^2 + 63*c^5*d^2 - 10*c^3*d^2)*a + (55*c^9*d^2 - 108*
c^7*d^2 + 63*c^5*d^2 - 10*c^3*d^2)*b)*x^2*e^4 + (3*(5*a*d^9 + b*d^9)*x^9*e^4 + 27*(5*a*c*d^8 + b*c*d^8)*x^8*e^
4 + ((540*c^2*d^7 - 31*d^7)*a + (108*c^2*d^7 - 5*d^7)*b)*x^7*e^4 + 7*((180*c^3*d^6 - 31*c*d^6)*a + (36*c^3*d^6
- 5*c*d^6)*b)*x^6*e^4 + ((1890*c^4*d^5 - 651*c^2*d^5 + 20*d^5)*a + (378*c^4*d^5 - 105*c^2*d^5 + 2*d^5)*b)*x^5
*e^4 + (5*(378*c^5*d^4 - 217*c^3*d^4 + 20*c*d^4)*a + (378*c^5*d^4 - 175*c^3*d^4 + 10*c*d^4)*b)*x^4*e^4 + ((126
0*c^6*d^3 - 1085*c^4*d^3 + 200*c^2*d^3 - 4*d^3)*a + (252*c^6*d^3 - 175*c^4*d^3 + 20*c^2*d^3)*b)*x^3*e^4 + ((54
0*c^7*d^2 - 651*c^5*d^2 + 200*c^3*d^2 - 12*c*d^2)*a + (108*c^7*d^2 - 105*c^5*d^2 + 20*c^3*d^2)*b)*x^2*e^4 + ((
135*c^8*d - 217*c^6*d + 100*c^4*d - 12*c^2*d)*a + (27*c^8*d - 35*c^6*d + 10*c^4*d)*b)*x*e^4 + ((15*c^9 - 31*c^
7 + 20*c^5 - 4*c^3)*a + (3*c^9 - 5*c^7 + 2*c^5)*b)*e^4)*(d*x + c + 1)*(d*x + c - 1) + (5*(11*c^10*d - 27*c^8*d
+ 21*c^6*d - 5*c^4*d)*a + (11*c^10*d - 27*c^8*d + 21*c^6*d - 5*c^4*d)*b)*x*e^4 + (3*(5*a*d^10 + b*d^10)*x^10*
e^4 + 30*(5*a*c*d^9 + b*c*d^9)*x^9*e^4 + ((675*c^2*d^8 - 38*d^8)*a + (135*c^2*d^8 - 7*d^8)*b)*x^8*e^4 + 8*((22
5*c^3*d^7 - 38*c*d^7)*a + (45*c^3*d^7 - 7*c*d^7)*b)*x^7*e^4 + (2*(1575*c^4*d^6 - 532*c^2*d^6 + 16*d^6)*a + (63
0*c^4*d^6 - 196*c^2*d^6 + 5*d^6)*b)*x^6*e^4 + 2*(2*(945*c^5*d^5 - 532*c^3*d^5 + 48*c*d^5)*a + (378*c^5*d^5 - 1
96*c^3*d^5 + 15*c*d^5)*b)*x^5*e^4 + ((3150*c^6*d^4 - 2660*c^4*d^4 + 480*c^2*d^4 - 9*d^4)*a + (630*c^6*d^4 - 49
0*c^4*d^4 + 75*c^2*d^4 - d^4)*b)*x^4*e^4 + 4*((450*c^7*d^3 - 532*c^5*d^3 + 160*c^3*d^3 - 9*c*d^3)*a + (90*c^7*
d^3 - 98*c^5*d^3 + 25*c^3*d^3 - c*d^3)*b)*x^3*e^4 + ((675*c^8*d^2 - 1064*c^6*d^2 + 480*c^4*d^2 - 54*c^2*d^2)*a
+ (135*c^8*d^2 - 196*c^6*d^2 + 75*c^4*d^2 - 6*c^2*d^2)*b)*x^2*e^4 + 2*((75*c^9*d - 152*c^7*d + 96*c^5*d - 18*
c^3*d)*a + (15*c^9*d - 28*c^7*d + 15*c^5*d - 2*c^3*d)*b)*x*e^4 + ((15*c^10 - 38*c^8 + 32*c^6 - 9*c^4)*a + (3*c
^10 - 7*c^8 + 5*c^6 - c^4)*b)*e^4)*sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + (5*(c^11 - 3*c^9 + 3*c^7 - c^5)*a + (
c^11 - 3*c^9 + 3*c^7 - c^5)*b)*e^4 + (5*b*d^11*x^11*e^4 + 55*b*c*d^10*x^10*e^4 + 5*(55*c^2*d^9 - 3*d^9)*b*x^9*
e^4 + 15*(55*c^3*d^8 - 9*c*d^8)*b*x^8*e^4 + 15*(110*c^4*d^7 - 36*c^2*d^7 + d^7)*b*x^7*e^4 + 105*(22*c^5*d^6 -
12*c^3*d^6 + c*d^6)*b*x^6*e^4 + 5*(462*c^6*d^5 - 378*c^4*d^5 + 63*c^2*d^5 - d^5)*b*x^5*e^4 + 5*(330*c^7*d^4 -
378*c^5*d^4 + 105*c^3*d^4 - 5*c*d^4)*b*x^4*e^4 + 5*(165*c^8*d^3 - 252*c^6*d^3 + 105*c^4*d^3 - 10*c^2*d^3)*b*x^
3*e^4 + 5*(55*c^9*d^2 - 108*c^7*d^2 + 63*c^5*d^2 - 10*c^3*d^2)*b*x^2*e^4 + (5*b*d^8*x^8*e^4 + 40*b*c*d^7*x^7*e
^4 + 4*(35*c^2*d^6 - 2*d^6)*b*x^6*e^4 + 8*(35*c^3*d^5 - 6*c*d^5)*b*x^5*e^4 + (350*c^4*d^4 - 120*c^2*d^4 + 3*d^
4)*b*x^4*e^4 + 4*(70*c^5*d^3 - 40*c^3*d^3 + 3*c*d^3)*b*x^3*e^4 + 2*(70*c^6*d^2 - 60*c^4*d^2 + 9*c^2*d^2)*b*x^2
*e^4 + 4*(10*c^7*d - 12*c^5*d + 3*c^3*d)*b*x*e^4 + (5*c^8 - 8*c^6 + 3*c^4)*b*e^4)*(d*x + c + 1)^(3/2)*(d*x + c
- 1)^(3/2) + 5*(11*c^10*d - 27*c^8*d + 21*c^6*d - 5*c^4*d)*b*x*e^4 + (15*b*d^9*x^9*e^4 + 135*b*c*d^8*x^8*e^4
+ (540*c^2*d^7 - 31*d^7)*b*x^7*e^4 + 7*(180*c^3*d^6 - 31*c*d^6)*b*x^6*e^4 + (1890*c^4*d^5 - 651*c^2*d^5 + 20*d
^5)*b*x^5*e^4 + 5*(378*c^5*d^4 - 217*c^3*d^4 + 20*c*d^4)*b*x^4*e^4 + (1260*c^6*d^3 - 1085*c^4*d^3 + 200*c^2*d^
3 - 4*d^3)*b*x^3*e^4 + (540*c^7*d^2 - 651*c^5*d^2 + 200*c^3*d^2 - 12*c*d^2)*b*x^2*e^4 + (135*c^8*d - 217*c^6*d
+ 100*c^4*d - 12*c^2*d)*b*x*e^4 + (15*c^9 - 31*c^7 + 20*c^5 - 4*c^3)*b*e^4)*(d*x + c + 1)*(d*x + c - 1) + 5*(
c^11 - 3*c^9 + 3*c^7 - c^5)*b*e^4 + (15*b*d^10*x^10*e^4 + 150*b*c*d^9*x^9*e^4 + (675*c^2*d^8 - 38*d^8)*b*x^8*e
^4 + 8*(225*c^3*d^7 - 38*c*d^7)*b*x^7*e^4 + 2*(...

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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)^4/(a+b*arccosh(d*x+c))^3,x, algorithm="fricas")

[Out]

integral((d^4*x^4 + 4*c*d^3*x^3 + 6*c^2*d^2*x^2 + 4*c^3*d*x + c^4)*e^4/(b^3*arccosh(d*x + c)^3 + 3*a*b^2*arcco
sh(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^{4} \left (\int \frac {c^{4}}{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^{4} x^{4}}{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 {4 c 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 {6 c^{2} 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 {4 c^{3} 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)**4/(a+b*acosh(d*x+c))**3,x)

[Out]

e**4*(Integral(c**4/(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**4*x**4/(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(4*c*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(6*c**2*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(4*c**3*d*x/(a**3 + 3*a**2*b*acosh(c + d*x) + 3*a*b**2*acosh(c + d*x)**2 + b**3*ac
osh(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)^4/(a+b*arccosh(d*x+c))^3,x, algorithm="giac")

[Out]

integrate((d*e*x + c*e)^4/(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 )}^4}{{\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)^4/(a + b*acosh(c + d*x))^3,x)

[Out]

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

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