3.2.13 \(\int (c e+d e x)^4 (a+b \cosh ^{-1}(c+d x))^3 \, dx\) [113]

Optimal. Leaf size=382 \[ \frac {16}{25} a b^2 e^4 x-\frac {4144 b^3 e^4 \sqrt {-1+c+d x} \sqrt {1+c+d x}}{5625 d}-\frac {272 b^3 e^4 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{5625 d}-\frac {6 b^3 e^4 \sqrt {-1+c+d x} (c+d x)^4 \sqrt {1+c+d x}}{625 d}+\frac {16 b^3 e^4 (c+d x) \cosh ^{-1}(c+d x)}{25 d}+\frac {8 b^2 e^4 (c+d x)^3 \left (a+b \cosh ^{-1}(c+d x)\right )}{75 d}+\frac {6 b^2 e^4 (c+d x)^5 \left (a+b \cosh ^{-1}(c+d x)\right )}{125 d}-\frac {8 b e^4 \sqrt {-1+c+d x} \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{25 d}-\frac {4 b e^4 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{25 d}-\frac {3 b e^4 \sqrt {-1+c+d x} (c+d x)^4 \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{25 d}+\frac {e^4 (c+d x)^5 \left (a+b \cosh ^{-1}(c+d x)\right )^3}{5 d} \]

[Out]

16/25*a*b^2*e^4*x+16/25*b^3*e^4*(d*x+c)*arccosh(d*x+c)/d+8/75*b^2*e^4*(d*x+c)^3*(a+b*arccosh(d*x+c))/d+6/125*b
^2*e^4*(d*x+c)^5*(a+b*arccosh(d*x+c))/d+1/5*e^4*(d*x+c)^5*(a+b*arccosh(d*x+c))^3/d-4144/5625*b^3*e^4*(d*x+c-1)
^(1/2)*(d*x+c+1)^(1/2)/d-272/5625*b^3*e^4*(d*x+c)^2*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)/d-6/625*b^3*e^4*(d*x+c)^4*
(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)/d-8/25*b*e^4*(a+b*arccosh(d*x+c))^2*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)/d-4/25*b*e
^4*(d*x+c)^2*(a+b*arccosh(d*x+c))^2*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)/d-3/25*b*e^4*(d*x+c)^4*(a+b*arccosh(d*x+c)
)^2*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)/d

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Rubi [A]
time = 0.48, antiderivative size = 382, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 8, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {5996, 12, 5883, 5939, 5915, 5879, 75, 102} \begin {gather*} \frac {6 b^2 e^4 (c+d x)^5 \left (a+b \cosh ^{-1}(c+d x)\right )}{125 d}+\frac {8 b^2 e^4 (c+d x)^3 \left (a+b \cosh ^{-1}(c+d x)\right )}{75 d}+\frac {16}{25} a b^2 e^4 x+\frac {e^4 (c+d x)^5 \left (a+b \cosh ^{-1}(c+d x)\right )^3}{5 d}-\frac {3 b e^4 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^4 \left (a+b \cosh ^{-1}(c+d x)\right )^2}{25 d}-\frac {4 b e^4 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^2}{25 d}-\frac {8 b e^4 \sqrt {c+d x-1} \sqrt {c+d x+1} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{25 d}-\frac {6 b^3 e^4 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^4}{625 d}-\frac {272 b^3 e^4 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2}{5625 d}-\frac {4144 b^3 e^4 \sqrt {c+d x-1} \sqrt {c+d x+1}}{5625 d}+\frac {16 b^3 e^4 (c+d x) \cosh ^{-1}(c+d x)}{25 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(16*a*b^2*e^4*x)/25 - (4144*b^3*e^4*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x])/(5625*d) - (272*b^3*e^4*Sqrt[-1 + c
+ d*x]*(c + d*x)^2*Sqrt[1 + c + d*x])/(5625*d) - (6*b^3*e^4*Sqrt[-1 + c + d*x]*(c + d*x)^4*Sqrt[1 + c + d*x])/
(625*d) + (16*b^3*e^4*(c + d*x)*ArcCosh[c + d*x])/(25*d) + (8*b^2*e^4*(c + d*x)^3*(a + b*ArcCosh[c + d*x]))/(7
5*d) + (6*b^2*e^4*(c + d*x)^5*(a + b*ArcCosh[c + d*x]))/(125*d) - (8*b*e^4*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x
]*(a + b*ArcCosh[c + d*x])^2)/(25*d) - (4*b*e^4*Sqrt[-1 + c + d*x]*(c + d*x)^2*Sqrt[1 + c + d*x]*(a + b*ArcCos
h[c + d*x])^2)/(25*d) - (3*b*e^4*Sqrt[-1 + c + d*x]*(c + d*x)^4*Sqrt[1 + c + d*x]*(a + b*ArcCosh[c + d*x])^2)/
(25*d) + (e^4*(c + d*x)^5*(a + b*ArcCosh[c + d*x])^3)/(5*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 75

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(c + d*x)^
(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0] &
& EqQ[a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)), 0]

Rule 102

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(a +
b*x)^(m - 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(m + n + p + 1))), x] + Dist[1/(d*f*(m + n + p + 1)), I
nt[(a + b*x)^(m - 2)*(c + d*x)^n*(e + f*x)^p*Simp[a^2*d*f*(m + n + p + 1) - b*(b*c*e*(m - 1) + a*(d*e*(n + 1)
+ c*f*(p + 1))) + b*(a*d*f*(2*m + n + p) - b*(d*e*(m + n) + c*f*(m + p)))*x, x], x], x] /; FreeQ[{a, b, c, d,
e, f, n, p}, x] && GtQ[m, 1] && NeQ[m + n + p + 1, 0] && IntegerQ[m]

Rule 5879

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*ArcCosh[c*x])^n, x] - Dist[b*c*n, In
t[x*((a + b*ArcCosh[c*x])^(n - 1)/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])), x], x] /; FreeQ[{a, b, c}, x] && GtQ[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 5915

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d1_) + (e1_.)*(x_))^(p_)*((d2_) + (e2_.)*(x_))^(p_), x_Sy
mbol] :> Simp[(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*((a + b*ArcCosh[c*x])^n/(2*e1*e2*(p + 1))), x] - Dist[b*
(n/(2*c*(p + 1)))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*x)^p/(-1 + c*x)^p], Int[(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, p}, x] && EqQ[e1, c
*d1] && EqQ[e2, (-c)*d2] && GtQ[n, 0] && NeQ[p, -1]

Rule 5939

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d1_) + (e1_.)*(x_))^(p_)*((d2_) + (e2_.)*(x_
))^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 1)*(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*((a + b*ArcCosh[c*x])^n/(e1
*e2*(m + 2*p + 1))), x] + (Dist[f^2*((m - 1)/(c^2*(m + 2*p + 1))), Int[(f*x)^(m - 2)*(d1 + e1*x)^p*(d2 + e2*x)
^p*(a + b*ArcCosh[c*x])^n, x], x] - Dist[b*f*(n/(c*(m + 2*p + 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, p}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && GtQ[n, 0] && IG
tQ[m, 1] && NeQ[m + 2*p + 1, 0]

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 (c e+d e x)^4 \left (a+b \cosh ^{-1}(c+d x)\right )^3 \, dx &=\frac {\text {Subst}\left (\int 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 x^4 \left (a+b \cosh ^{-1}(x)\right )^3 \, dx,x,c+d x\right )}{d}\\ &=\frac {e^4 (c+d x)^5 \left (a+b \cosh ^{-1}(c+d x)\right )^3}{5 d}-\frac {\left (3 b e^4\right ) \text {Subst}\left (\int \frac {x^5 \left (a+b \cosh ^{-1}(x)\right )^2}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{5 d}\\ &=-\frac {3 b e^4 \sqrt {-1+c+d x} (c+d x)^4 \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{25 d}+\frac {e^4 (c+d x)^5 \left (a+b \cosh ^{-1}(c+d x)\right )^3}{5 d}-\frac {\left (12 b e^4\right ) \text {Subst}\left (\int \frac {x^3 \left (a+b \cosh ^{-1}(x)\right )^2}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{25 d}+\frac {\left (6 b^2 e^4\right ) \text {Subst}\left (\int x^4 \left (a+b \cosh ^{-1}(x)\right ) \, dx,x,c+d x\right )}{25 d}\\ &=\frac {6 b^2 e^4 (c+d x)^5 \left (a+b \cosh ^{-1}(c+d x)\right )}{125 d}-\frac {4 b e^4 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{25 d}-\frac {3 b e^4 \sqrt {-1+c+d x} (c+d x)^4 \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{25 d}+\frac {e^4 (c+d x)^5 \left (a+b \cosh ^{-1}(c+d x)\right )^3}{5 d}-\frac {\left (8 b e^4\right ) \text {Subst}\left (\int \frac {x \left (a+b \cosh ^{-1}(x)\right )^2}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{25 d}+\frac {\left (8 b^2 e^4\right ) \text {Subst}\left (\int x^2 \left (a+b \cosh ^{-1}(x)\right ) \, dx,x,c+d x\right )}{25 d}-\frac {\left (6 b^3 e^4\right ) \text {Subst}\left (\int \frac {x^5}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{125 d}\\ &=-\frac {6 b^3 e^4 \sqrt {-1+c+d x} (c+d x)^4 \sqrt {1+c+d x}}{625 d}+\frac {8 b^2 e^4 (c+d x)^3 \left (a+b \cosh ^{-1}(c+d x)\right )}{75 d}+\frac {6 b^2 e^4 (c+d x)^5 \left (a+b \cosh ^{-1}(c+d x)\right )}{125 d}-\frac {8 b e^4 \sqrt {-1+c+d x} \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{25 d}-\frac {4 b e^4 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{25 d}-\frac {3 b e^4 \sqrt {-1+c+d x} (c+d x)^4 \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{25 d}+\frac {e^4 (c+d x)^5 \left (a+b \cosh ^{-1}(c+d x)\right )^3}{5 d}+\frac {\left (16 b^2 e^4\right ) \text {Subst}\left (\int \left (a+b \cosh ^{-1}(x)\right ) \, dx,x,c+d x\right )}{25 d}-\frac {\left (6 b^3 e^4\right ) \text {Subst}\left (\int \frac {4 x^3}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{625 d}-\frac {\left (8 b^3 e^4\right ) \text {Subst}\left (\int \frac {x^3}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{75 d}\\ &=\frac {16}{25} a b^2 e^4 x-\frac {8 b^3 e^4 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{225 d}-\frac {6 b^3 e^4 \sqrt {-1+c+d x} (c+d x)^4 \sqrt {1+c+d x}}{625 d}+\frac {8 b^2 e^4 (c+d x)^3 \left (a+b \cosh ^{-1}(c+d x)\right )}{75 d}+\frac {6 b^2 e^4 (c+d x)^5 \left (a+b \cosh ^{-1}(c+d x)\right )}{125 d}-\frac {8 b e^4 \sqrt {-1+c+d x} \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{25 d}-\frac {4 b e^4 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{25 d}-\frac {3 b e^4 \sqrt {-1+c+d x} (c+d x)^4 \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{25 d}+\frac {e^4 (c+d x)^5 \left (a+b \cosh ^{-1}(c+d x)\right )^3}{5 d}-\frac {\left (8 b^3 e^4\right ) \text {Subst}\left (\int \frac {2 x}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{225 d}-\frac {\left (24 b^3 e^4\right ) \text {Subst}\left (\int \frac {x^3}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{625 d}+\frac {\left (16 b^3 e^4\right ) \text {Subst}\left (\int \cosh ^{-1}(x) \, dx,x,c+d x\right )}{25 d}\\ &=\frac {16}{25} a b^2 e^4 x-\frac {272 b^3 e^4 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{5625 d}-\frac {6 b^3 e^4 \sqrt {-1+c+d x} (c+d x)^4 \sqrt {1+c+d x}}{625 d}+\frac {16 b^3 e^4 (c+d x) \cosh ^{-1}(c+d x)}{25 d}+\frac {8 b^2 e^4 (c+d x)^3 \left (a+b \cosh ^{-1}(c+d x)\right )}{75 d}+\frac {6 b^2 e^4 (c+d x)^5 \left (a+b \cosh ^{-1}(c+d x)\right )}{125 d}-\frac {8 b e^4 \sqrt {-1+c+d x} \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{25 d}-\frac {4 b e^4 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{25 d}-\frac {3 b e^4 \sqrt {-1+c+d x} (c+d x)^4 \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{25 d}+\frac {e^4 (c+d x)^5 \left (a+b \cosh ^{-1}(c+d x)\right )^3}{5 d}-\frac {\left (8 b^3 e^4\right ) \text {Subst}\left (\int \frac {2 x}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{625 d}-\frac {\left (16 b^3 e^4\right ) \text {Subst}\left (\int \frac {x}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{225 d}-\frac {\left (16 b^3 e^4\right ) \text {Subst}\left (\int \frac {x}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{25 d}\\ &=\frac {16}{25} a b^2 e^4 x-\frac {32 b^3 e^4 \sqrt {-1+c+d x} \sqrt {1+c+d x}}{45 d}-\frac {272 b^3 e^4 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{5625 d}-\frac {6 b^3 e^4 \sqrt {-1+c+d x} (c+d x)^4 \sqrt {1+c+d x}}{625 d}+\frac {16 b^3 e^4 (c+d x) \cosh ^{-1}(c+d x)}{25 d}+\frac {8 b^2 e^4 (c+d x)^3 \left (a+b \cosh ^{-1}(c+d x)\right )}{75 d}+\frac {6 b^2 e^4 (c+d x)^5 \left (a+b \cosh ^{-1}(c+d x)\right )}{125 d}-\frac {8 b e^4 \sqrt {-1+c+d x} \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{25 d}-\frac {4 b e^4 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{25 d}-\frac {3 b e^4 \sqrt {-1+c+d x} (c+d x)^4 \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{25 d}+\frac {e^4 (c+d x)^5 \left (a+b \cosh ^{-1}(c+d x)\right )^3}{5 d}-\frac {\left (16 b^3 e^4\right ) \text {Subst}\left (\int \frac {x}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{625 d}\\ &=\frac {16}{25} a b^2 e^4 x-\frac {4144 b^3 e^4 \sqrt {-1+c+d x} \sqrt {1+c+d x}}{5625 d}-\frac {272 b^3 e^4 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{5625 d}-\frac {6 b^3 e^4 \sqrt {-1+c+d x} (c+d x)^4 \sqrt {1+c+d x}}{625 d}+\frac {16 b^3 e^4 (c+d x) \cosh ^{-1}(c+d x)}{25 d}+\frac {8 b^2 e^4 (c+d x)^3 \left (a+b \cosh ^{-1}(c+d x)\right )}{75 d}+\frac {6 b^2 e^4 (c+d x)^5 \left (a+b \cosh ^{-1}(c+d x)\right )}{125 d}-\frac {8 b e^4 \sqrt {-1+c+d x} \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{25 d}-\frac {4 b e^4 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{25 d}-\frac {3 b e^4 \sqrt {-1+c+d x} (c+d x)^4 \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{25 d}+\frac {e^4 (c+d x)^5 \left (a+b \cosh ^{-1}(c+d x)\right )^3}{5 d}\\ \end {align*}

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Mathematica [A]
time = 0.37, size = 404, normalized size = 1.06 \begin {gather*} \frac {e^4 \left (240 a b^2 (c+d x)+40 a b^2 (c+d x)^3+3 a \left (25 a^2+6 b^2\right ) (c+d x)^5+\frac {1}{15} b \sqrt {-1+c+d x} \sqrt {1+c+d x} \left (-8 \left (225 a^2+518 b^2\right )-4 \left (225 a^2+68 b^2\right ) (c+d x)^2-27 \left (25 a^2+2 b^2\right ) (c+d x)^4\right )-b \left (-240 b^2 (c+d x)-40 b^2 (c+d x)^3-225 a^2 (c+d x)^5-18 b^2 (c+d x)^5+240 a b \sqrt {-1+c+d x} \sqrt {1+c+d x}+120 a b \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}+90 a b \sqrt {-1+c+d x} (c+d x)^4 \sqrt {1+c+d x}\right ) \cosh ^{-1}(c+d x)-15 b^2 \left (-15 a (c+d x)^5+8 b \sqrt {-1+c+d x} \sqrt {1+c+d x}+4 b \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}+3 b \sqrt {-1+c+d x} (c+d x)^4 \sqrt {1+c+d x}\right ) \cosh ^{-1}(c+d x)^2+75 b^3 (c+d x)^5 \cosh ^{-1}(c+d x)^3\right )}{375 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*(240*a*b^2*(c + d*x) + 40*a*b^2*(c + d*x)^3 + 3*a*(25*a^2 + 6*b^2)*(c + d*x)^5 + (b*Sqrt[-1 + c + d*x]*Sq
rt[1 + c + d*x]*(-8*(225*a^2 + 518*b^2) - 4*(225*a^2 + 68*b^2)*(c + d*x)^2 - 27*(25*a^2 + 2*b^2)*(c + d*x)^4))
/15 - b*(-240*b^2*(c + d*x) - 40*b^2*(c + d*x)^3 - 225*a^2*(c + d*x)^5 - 18*b^2*(c + d*x)^5 + 240*a*b*Sqrt[-1
+ c + d*x]*Sqrt[1 + c + d*x] + 120*a*b*Sqrt[-1 + c + d*x]*(c + d*x)^2*Sqrt[1 + c + d*x] + 90*a*b*Sqrt[-1 + c +
d*x]*(c + d*x)^4*Sqrt[1 + c + d*x])*ArcCosh[c + d*x] - 15*b^2*(-15*a*(c + d*x)^5 + 8*b*Sqrt[-1 + c + d*x]*Sqr
t[1 + c + d*x] + 4*b*Sqrt[-1 + c + d*x]*(c + d*x)^2*Sqrt[1 + c + d*x] + 3*b*Sqrt[-1 + c + d*x]*(c + d*x)^4*Sqr
t[1 + c + d*x])*ArcCosh[c + d*x]^2 + 75*b^3*(c + d*x)^5*ArcCosh[c + d*x]^3))/(375*d)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1292\) vs. \(2(336)=672\).
time = 31.69, size = 1293, normalized size = 3.38

method	result	size
default	\(\text {Expression too large to display}\)	\(1293\)

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]

1/5*e^4*(d*x+c)^5*a^3/d+1/5625*e^4*b^3*(-4144*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)-2700*arccosh(d*x+c)^2*(d*x+c+1)^
(1/2)*(d*x+c-1)^(1/2)*x^3*c*d^3-272*(d*x+c+1)^(1/2)*(d*x+c-1)^(1/2)*x^2*d^2+5625*arccosh(d*x+c)^3*x^4*c*d^4+11
250*arccosh(d*x+c)^3*x^3*c^2*d^3+11250*arccosh(d*x+c)^3*x^2*c^3*d^2+1350*arccosh(d*x+c)*x^4*c*d^4+5625*arccosh
(d*x+c)^3*x*c^4*d+2700*arccosh(d*x+c)*x^3*c^2*d^3+2700*arccosh(d*x+c)*x^2*c^3*d^2+1350*arccosh(d*x+c)*x*c^4*d+
1800*arccosh(d*x+c)*x^2*c*d^2+1800*arccosh(d*x+c)*x*c^2*d-675*arccosh(d*x+c)^2*(d*x+c+1)^(1/2)*(d*x+c-1)^(1/2)
*c^4-900*arccosh(d*x+c)^2*(d*x+c+1)^(1/2)*(d*x+c-1)^(1/2)*c^2-216*(d*x+c+1)^(1/2)*(d*x+c-1)^(1/2)*x^3*c*d^3-32
4*(d*x+c+1)^(1/2)*(d*x+c-1)^(1/2)*x^2*c^2*d^2-216*(d*x+c+1)^(1/2)*(d*x+c-1)^(1/2)*x*c^3*d-544*(d*x+c+1)^(1/2)*
(d*x+c-1)^(1/2)*x*c*d-675*arccosh(d*x+c)^2*(d*x+c+1)^(1/2)*(d*x+c-1)^(1/2)*x^4*d^4-900*arccosh(d*x+c)^2*(d*x+c
+1)^(1/2)*(d*x+c-1)^(1/2)*x^2*d^2-4050*arccosh(d*x+c)^2*(d*x+c+1)^(1/2)*(d*x+c-1)^(1/2)*x^2*c^2*d^2-2700*arcco
sh(d*x+c)^2*(d*x+c+1)^(1/2)*(d*x+c-1)^(1/2)*x*c^3*d-1800*arccosh(d*x+c)^2*(d*x+c+1)^(1/2)*(d*x+c-1)^(1/2)*x*c*
d-54*(d*x+c+1)^(1/2)*(d*x+c-1)^(1/2)*c^4-54*(d*x+c+1)^(1/2)*(d*x+c-1)^(1/2)*x^4*d^4+1125*arccosh(d*x+c)^3*c^5+
270*arccosh(d*x+c)*c^5+600*arccosh(d*x+c)*c^3+3600*arccosh(d*x+c)*c-272*(d*x+c+1)^(1/2)*(d*x+c-1)^(1/2)*c^2+11
25*arccosh(d*x+c)^3*x^5*d^5+270*arccosh(d*x+c)*x^5*d^5+600*arccosh(d*x+c)*x^3*d^3+3600*arccosh(d*x+c)*x*d-1800
*arccosh(d*x+c)^2*(d*x+c+1)^(1/2)*(d*x+c-1)^(1/2))/d+1/375*e^4*a*b^2*(225*arccosh(d*x+c)^2*c^5+18*x^5*d^5+240*
c-90*arccosh(d*x+c)*(d*x+c+1)^(1/2)*(d*x+c-1)^(1/2)*c^4-120*arccosh(d*x+c)*(d*x+c+1)^(1/2)*(d*x+c-1)^(1/2)*c^2
+1125*arccosh(d*x+c)^2*x^4*c*d^4+2250*arccosh(d*x+c)^2*x^3*c^2*d^3+2250*arccosh(d*x+c)^2*x^2*c^3*d^2+1125*arcc
osh(d*x+c)^2*x*c^4*d+40*c^3+40*d^3*x^3+18*c^5+240*d*x-90*arccosh(d*x+c)*(d*x+c+1)^(1/2)*(d*x+c-1)^(1/2)*x^4*d^
4-120*arccosh(d*x+c)*(d*x+c+1)^(1/2)*(d*x+c-1)^(1/2)*x^2*d^2-540*arccosh(d*x+c)*(d*x+c+1)^(1/2)*(d*x+c-1)^(1/2
)*x^2*c^2*d^2-360*arccosh(d*x+c)*(d*x+c+1)^(1/2)*(d*x+c-1)^(1/2)*x*c^3*d-240*arccosh(d*x+c)*(d*x+c+1)^(1/2)*(d
*x+c-1)^(1/2)*x*c*d-360*arccosh(d*x+c)*(d*x+c+1)^(1/2)*(d*x+c-1)^(1/2)*x^3*c*d^3+120*c^2*d*x+90*x*c^4*d+180*x^
3*c^2*d^3+90*x^4*c*d^4+180*x^2*c^3*d^2+120*x^2*c*d^2+225*arccosh(d*x+c)^2*x^5*d^5-240*arccosh(d*x+c)*(d*x+c-1)
^(1/2)*(d*x+c+1)^(1/2))/d+3*e^4*a^2*b/d*(1/5*(d*x+c)^5*arccosh(d*x+c)-1/75*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)*(3*
(d*x+c)^4+4*(d*x+c)^2+8))

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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]

1/5*a^3*d^4*x^5*e^4 + a^3*c*d^3*x^4*e^4 + 2*a^3*c^2*d^2*x^3*e^4 + 2*a^3*c^3*d*x^2*e^4 + 3*(2*x^2*arccosh(d*x +
c) - d*(3*c^2*log(2*d^2*x + 2*c*d + 2*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*d)/d^3 + sqrt(d^2*x^2 + 2*c*d*x + c^2
- 1)*x/d^2 - (c^2 - 1)*log(2*d^2*x + 2*c*d + 2*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*d)/d^3 - 3*sqrt(d^2*x^2 + 2*
c*d*x + c^2 - 1)*c/d^3))*a^2*b*c^3*d*e^4 + (6*x^3*arccosh(d*x + c) - d*(2*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*x^
2/d^2 - 15*c^3*log(2*d^2*x + 2*c*d + 2*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*d)/d^4 - 5*sqrt(d^2*x^2 + 2*c*d*x + c
^2 - 1)*c*x/d^3 + 9*(c^2 - 1)*c*log(2*d^2*x + 2*c*d + 2*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*d)/d^4 + 15*sqrt(d^2
*x^2 + 2*c*d*x + c^2 - 1)*c^2/d^4 - 4*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*(c^2 - 1)/d^4))*a^2*b*c^2*d^2*e^4 + 1/
8*(24*x^4*arccosh(d*x + c) - (6*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*x^3/d^2 - 14*sqrt(d^2*x^2 + 2*c*d*x + c^2 -
1)*c*x^2/d^3 + 105*c^4*log(2*d^2*x + 2*c*d + 2*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*d)/d^5 + 35*sqrt(d^2*x^2 + 2*
c*d*x + c^2 - 1)*c^2*x/d^4 - 90*(c^2 - 1)*c^2*log(2*d^2*x + 2*c*d + 2*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*d)/d^5
- 105*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*c^3/d^5 - 9*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*(c^2 - 1)*x/d^4 + 9*(c^
2 - 1)^2*log(2*d^2*x + 2*c*d + 2*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*d)/d^5 + 55*sqrt(d^2*x^2 + 2*c*d*x + c^2 -
1)*(c^2 - 1)*c/d^5)*d)*a^2*b*c*d^3*e^4 + 1/200*(120*x^5*arccosh(d*x + c) - (24*sqrt(d^2*x^2 + 2*c*d*x + c^2 -
1)*x^4/d^2 - 54*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*c*x^3/d^3 + 126*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*c^2*x^2/d^
4 - 945*c^5*log(2*d^2*x + 2*c*d + 2*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*d)/d^6 - 315*sqrt(d^2*x^2 + 2*c*d*x + c^
2 - 1)*c^3*x/d^5 - 32*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*(c^2 - 1)*x^2/d^4 + 1050*(c^2 - 1)*c^3*log(2*d^2*x + 2
*c*d + 2*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*d)/d^6 + 945*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*c^4/d^6 + 161*sqrt(d
^2*x^2 + 2*c*d*x + c^2 - 1)*(c^2 - 1)*c*x/d^5 - 225*(c^2 - 1)^2*c*log(2*d^2*x + 2*c*d + 2*sqrt(d^2*x^2 + 2*c*d
*x + c^2 - 1)*d)/d^6 - 735*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*(c^2 - 1)*c^2/d^6 + 64*sqrt(d^2*x^2 + 2*c*d*x + c
^2 - 1)*(c^2 - 1)^2/d^6)*d)*a^2*b*d^4*e^4 + a^3*c^4*x*e^4 + 3*((d*x + c)*arccosh(d*x + c) - sqrt((d*x + c)^2 -
1))*a^2*b*c^4*e^4/d + 1/5*(b^3*d^4*x^5*e^4 + 5*b^3*c*d^3*x^4*e^4 + 10*b^3*c^2*d^2*x^3*e^4 + 10*b^3*c^3*d*x^2*
e^4 + 5*b^3*c^4*x*e^4)*log(d*x + sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + c)^3 + integrate(3/5*((5*a*b^2*d^7 - b^
3*d^7)*x^7*e^4 + 7*(5*a*b^2*c*d^6 - b^3*c*d^6)*x^6*e^4 + (5*(21*c^2*d^5 - d^5)*a*b^2 - (21*c^2*d^5 - d^5)*b^3)
*x^5*e^4 + 5*(5*(7*c^3*d^4 - c*d^4)*a*b^2 - (7*c^3*d^4 - c*d^4)*b^3)*x^4*e^4 + 5*(c^7 - c^5)*a*b^2*e^4 + 5*(5*
(7*c^4*d^3 - 2*c^2*d^3)*a*b^2 - (7*c^4*d^3 - 2*c^2*d^3)*b^3)*x^3*e^4 + 5*((21*c^5*d^2 - 10*c^3*d^2)*a*b^2 - 2*
(2*c^5*d^2 - c^3*d^2)*b^3)*x^2*e^4 + 5*((7*c^6*d - 5*c^4*d)*a*b^2 - (c^6*d - c^4*d)*b^3)*x*e^4 + ((5*a*b^2*d^6
- b^3*d^6)*x^6*e^4 + 6*(5*a*b^2*c*d^5 - b^3*c*d^5)*x^5*e^4 - 5*(3*b^3*c^2*d^4 - (15*c^2*d^4 - d^4)*a*b^2)*x^4
*e^4 + 5*(c^6 - c^4)*a*b^2*e^4 - 20*(b^3*c^3*d^3 - (5*c^3*d^3 - c*d^3)*a*b^2)*x^3*e^4 - 15*(b^3*c^4*d^2 - (5*c
^4*d^2 - 2*c^2*d^2)*a*b^2)*x^2*e^4 - 5*(b^3*c^5*d - 2*(3*c^5*d - 2*c^3*d)*a*b^2)*x*e^4)*sqrt(d*x + c + 1)*sqrt
(d*x + c - 1))*log(d*x + sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + c)^2/(d^3*x^3 + 3*c*d^2*x^2 + c^3 + (d^2*x^2 +
2*c*d*x + c^2 - 1)*sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + (3*c^2*d - d)*x - c), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 4401 vs. \(2 (325) = 650\).
time = 0.43, size = 4401, normalized size = 11.52 \begin {gather*} \text {Too large to display} \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]

1/5625*(15*(3*(25*a^3 + 6*a*b^2)*d^5*x^5 + 15*(25*a^3 + 6*a*b^2)*c*d^4*x^4 + 10*(4*a*b^2 + 3*(25*a^3 + 6*a*b^2
)*c^2)*d^3*x^3 + 30*(4*a*b^2*c + (25*a^3 + 6*a*b^2)*c^3)*d^2*x^2 + 15*(8*a*b^2*c^2 + (25*a^3 + 6*a*b^2)*c^4 +
16*a*b^2)*d*x)*cosh(1)^4 + 60*(3*(25*a^3 + 6*a*b^2)*d^5*x^5 + 15*(25*a^3 + 6*a*b^2)*c*d^4*x^4 + 10*(4*a*b^2 +
3*(25*a^3 + 6*a*b^2)*c^2)*d^3*x^3 + 30*(4*a*b^2*c + (25*a^3 + 6*a*b^2)*c^3)*d^2*x^2 + 15*(8*a*b^2*c^2 + (25*a^
3 + 6*a*b^2)*c^4 + 16*a*b^2)*d*x)*cosh(1)^3*sinh(1) + 90*(3*(25*a^3 + 6*a*b^2)*d^5*x^5 + 15*(25*a^3 + 6*a*b^2)
*c*d^4*x^4 + 10*(4*a*b^2 + 3*(25*a^3 + 6*a*b^2)*c^2)*d^3*x^3 + 30*(4*a*b^2*c + (25*a^3 + 6*a*b^2)*c^3)*d^2*x^2
+ 15*(8*a*b^2*c^2 + (25*a^3 + 6*a*b^2)*c^4 + 16*a*b^2)*d*x)*cosh(1)^2*sinh(1)^2 + 60*(3*(25*a^3 + 6*a*b^2)*d^
5*x^5 + 15*(25*a^3 + 6*a*b^2)*c*d^4*x^4 + 10*(4*a*b^2 + 3*(25*a^3 + 6*a*b^2)*c^2)*d^3*x^3 + 30*(4*a*b^2*c + (2
5*a^3 + 6*a*b^2)*c^3)*d^2*x^2 + 15*(8*a*b^2*c^2 + (25*a^3 + 6*a*b^2)*c^4 + 16*a*b^2)*d*x)*cosh(1)*sinh(1)^3 +
15*(3*(25*a^3 + 6*a*b^2)*d^5*x^5 + 15*(25*a^3 + 6*a*b^2)*c*d^4*x^4 + 10*(4*a*b^2 + 3*(25*a^3 + 6*a*b^2)*c^2)*d
^3*x^3 + 30*(4*a*b^2*c + (25*a^3 + 6*a*b^2)*c^3)*d^2*x^2 + 15*(8*a*b^2*c^2 + (25*a^3 + 6*a*b^2)*c^4 + 16*a*b^2
)*d*x)*sinh(1)^4 + 1125*((b^3*d^5*x^5 + 5*b^3*c*d^4*x^4 + 10*b^3*c^2*d^3*x^3 + 10*b^3*c^3*d^2*x^2 + 5*b^3*c^4*
d*x + b^3*c^5)*cosh(1)^4 + 4*(b^3*d^5*x^5 + 5*b^3*c*d^4*x^4 + 10*b^3*c^2*d^3*x^3 + 10*b^3*c^3*d^2*x^2 + 5*b^3*
c^4*d*x + b^3*c^5)*cosh(1)^3*sinh(1) + 6*(b^3*d^5*x^5 + 5*b^3*c*d^4*x^4 + 10*b^3*c^2*d^3*x^3 + 10*b^3*c^3*d^2*
x^2 + 5*b^3*c^4*d*x + b^3*c^5)*cosh(1)^2*sinh(1)^2 + 4*(b^3*d^5*x^5 + 5*b^3*c*d^4*x^4 + 10*b^3*c^2*d^3*x^3 + 1
0*b^3*c^3*d^2*x^2 + 5*b^3*c^4*d*x + b^3*c^5)*cosh(1)*sinh(1)^3 + (b^3*d^5*x^5 + 5*b^3*c*d^4*x^4 + 10*b^3*c^2*d
^3*x^3 + 10*b^3*c^3*d^2*x^2 + 5*b^3*c^4*d*x + b^3*c^5)*sinh(1)^4)*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 -
1))^3 + 225*(15*(a*b^2*d^5*x^5 + 5*a*b^2*c*d^4*x^4 + 10*a*b^2*c^2*d^3*x^3 + 10*a*b^2*c^3*d^2*x^2 + 5*a*b^2*c^
4*d*x + a*b^2*c^5)*cosh(1)^4 + 60*(a*b^2*d^5*x^5 + 5*a*b^2*c*d^4*x^4 + 10*a*b^2*c^2*d^3*x^3 + 10*a*b^2*c^3*d^2
*x^2 + 5*a*b^2*c^4*d*x + a*b^2*c^5)*cosh(1)^3*sinh(1) + 90*(a*b^2*d^5*x^5 + 5*a*b^2*c*d^4*x^4 + 10*a*b^2*c^2*d
^3*x^3 + 10*a*b^2*c^3*d^2*x^2 + 5*a*b^2*c^4*d*x + a*b^2*c^5)*cosh(1)^2*sinh(1)^2 + 60*(a*b^2*d^5*x^5 + 5*a*b^2
*c*d^4*x^4 + 10*a*b^2*c^2*d^3*x^3 + 10*a*b^2*c^3*d^2*x^2 + 5*a*b^2*c^4*d*x + a*b^2*c^5)*cosh(1)*sinh(1)^3 + 15
*(a*b^2*d^5*x^5 + 5*a*b^2*c*d^4*x^4 + 10*a*b^2*c^2*d^3*x^3 + 10*a*b^2*c^3*d^2*x^2 + 5*a*b^2*c^4*d*x + a*b^2*c^
5)*sinh(1)^4 - ((3*b^3*d^4*x^4 + 12*b^3*c*d^3*x^3 + 3*b^3*c^4 + 4*b^3*c^2 + 2*(9*b^3*c^2 + 2*b^3)*d^2*x^2 + 8*
b^3 + 4*(3*b^3*c^3 + 2*b^3*c)*d*x)*cosh(1)^4 + 4*(3*b^3*d^4*x^4 + 12*b^3*c*d^3*x^3 + 3*b^3*c^4 + 4*b^3*c^2 + 2
*(9*b^3*c^2 + 2*b^3)*d^2*x^2 + 8*b^3 + 4*(3*b^3*c^3 + 2*b^3*c)*d*x)*cosh(1)^3*sinh(1) + 6*(3*b^3*d^4*x^4 + 12*
b^3*c*d^3*x^3 + 3*b^3*c^4 + 4*b^3*c^2 + 2*(9*b^3*c^2 + 2*b^3)*d^2*x^2 + 8*b^3 + 4*(3*b^3*c^3 + 2*b^3*c)*d*x)*c
osh(1)^2*sinh(1)^2 + 4*(3*b^3*d^4*x^4 + 12*b^3*c*d^3*x^3 + 3*b^3*c^4 + 4*b^3*c^2 + 2*(9*b^3*c^2 + 2*b^3)*d^2*x
^2 + 8*b^3 + 4*(3*b^3*c^3 + 2*b^3*c)*d*x)*cosh(1)*sinh(1)^3 + (3*b^3*d^4*x^4 + 12*b^3*c*d^3*x^3 + 3*b^3*c^4 +
4*b^3*c^2 + 2*(9*b^3*c^2 + 2*b^3)*d^2*x^2 + 8*b^3 + 4*(3*b^3*c^3 + 2*b^3*c)*d*x)*sinh(1)^4)*sqrt(d^2*x^2 + 2*c
*d*x + c^2 - 1))*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1))^2 + 15*((9*(25*a^2*b + 2*b^3)*d^5*x^5 + 45*(
25*a^2*b + 2*b^3)*c*d^4*x^4 + 10*(4*b^3 + 9*(25*a^2*b + 2*b^3)*c^2)*d^3*x^3 + 40*b^3*c^3 + 9*(25*a^2*b + 2*b^3
)*c^5 + 30*(4*b^3*c + 3*(25*a^2*b + 2*b^3)*c^3)*d^2*x^2 + 240*b^3*c + 15*(8*b^3*c^2 + 3*(25*a^2*b + 2*b^3)*c^4
+ 16*b^3)*d*x)*cosh(1)^4 + 4*(9*(25*a^2*b + 2*b^3)*d^5*x^5 + 45*(25*a^2*b + 2*b^3)*c*d^4*x^4 + 10*(4*b^3 + 9*
(25*a^2*b + 2*b^3)*c^2)*d^3*x^3 + 40*b^3*c^3 + 9*(25*a^2*b + 2*b^3)*c^5 + 30*(4*b^3*c + 3*(25*a^2*b + 2*b^3)*c
^3)*d^2*x^2 + 240*b^3*c + 15*(8*b^3*c^2 + 3*(25*a^2*b + 2*b^3)*c^4 + 16*b^3)*d*x)*cosh(1)^3*sinh(1) + 6*(9*(25
*a^2*b + 2*b^3)*d^5*x^5 + 45*(25*a^2*b + 2*b^3)*c*d^4*x^4 + 10*(4*b^3 + 9*(25*a^2*b + 2*b^3)*c^2)*d^3*x^3 + 40
*b^3*c^3 + 9*(25*a^2*b + 2*b^3)*c^5 + 30*(4*b^3*c + 3*(25*a^2*b + 2*b^3)*c^3)*d^2*x^2 + 240*b^3*c + 15*(8*b^3*
c^2 + 3*(25*a^2*b + 2*b^3)*c^4 + 16*b^3)*d*x)*cosh(1)^2*sinh(1)^2 + 4*(9*(25*a^2*b + 2*b^3)*d^5*x^5 + 45*(25*a
^2*b + 2*b^3)*c*d^4*x^4 + 10*(4*b^3 + 9*(25*a^2*b + 2*b^3)*c^2)*d^3*x^3 + 40*b^3*c^3 + 9*(25*a^2*b + 2*b^3)*c^
5 + 30*(4*b^3*c + 3*(25*a^2*b + 2*b^3)*c^3)*d^2*x^2 + 240*b^3*c + 15*(8*b^3*c^2 + 3*(25*a^2*b + 2*b^3)*c^4 + 1
6*b^3)*d*x)*cosh(1)*sinh(1)^3 + (9*(25*a^2*b + 2*b^3)*d^5*x^5 + 45*(25*a^2*b + 2*b^3)*c*d^4*x^4 + 10*(4*b^3 +
9*(25*a^2*b + 2*b^3)*c^2)*d^3*x^3 + 40*b^3*c^3 + 9*(25*a^2*b + 2*b^3)*c^5 + 30*(4*b^3*c + 3*(25*a^2*b + 2*b^3)
*c^3)*d^2*x^2 + 240*b^3*c + 15*(8*b^3*c^2 + 3*(25*a^2*b + 2*b^3)*c^4 + 16*b^3)*d*x)*sinh(1)^4 - 30*((3*a*b^2*d
^4*x^4 + 12*a*b^2*c*d^3*x^3 + 3*a*b^2*c^4 + 4*a*b^2*c^2 + 2*(9*a*b^2*c^2 + 2*a*b^2)*d^2*x^2 + 8*a*b^2 + 4*(3*a
*b^2*c^3 + 2*a*b^2*c)*d*x)*cosh(1)^4 + 4*(3*a*b...

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 2518 vs. \(2 (371) = 742\).
time = 1.38, size = 2518, normalized size = 6.59 \begin {gather*} \text {Too large to display} \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]

Piecewise((a**3*c**4*e**4*x + 2*a**3*c**3*d*e**4*x**2 + 2*a**3*c**2*d**2*e**4*x**3 + a**3*c*d**3*e**4*x**4 + a
**3*d**4*e**4*x**5/5 + 3*a**2*b*c**5*e**4*acosh(c + d*x)/(5*d) + 3*a**2*b*c**4*e**4*x*acosh(c + d*x) - 3*a**2*
b*c**4*e**4*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)/(25*d) + 6*a**2*b*c**3*d*e**4*x**2*acosh(c + d*x) - 12*a**2*b
*c**3*e**4*x*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)/25 + 6*a**2*b*c**2*d**2*e**4*x**3*acosh(c + d*x) - 18*a**2*b
*c**2*d*e**4*x**2*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)/25 - 4*a**2*b*c**2*e**4*sqrt(c**2 + 2*c*d*x + d**2*x**2
- 1)/(25*d) + 3*a**2*b*c*d**3*e**4*x**4*acosh(c + d*x) - 12*a**2*b*c*d**2*e**4*x**3*sqrt(c**2 + 2*c*d*x + d**
2*x**2 - 1)/25 - 8*a**2*b*c*e**4*x*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)/25 + 3*a**2*b*d**4*e**4*x**5*acosh(c +
d*x)/5 - 3*a**2*b*d**3*e**4*x**4*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)/25 - 4*a**2*b*d*e**4*x**2*sqrt(c**2 + 2
*c*d*x + d**2*x**2 - 1)/25 - 8*a**2*b*e**4*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)/(25*d) + 3*a*b**2*c**5*e**4*ac
osh(c + d*x)**2/(5*d) + 3*a*b**2*c**4*e**4*x*acosh(c + d*x)**2 + 6*a*b**2*c**4*e**4*x/25 - 6*a*b**2*c**4*e**4*
sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)*acosh(c + d*x)/(25*d) + 6*a*b**2*c**3*d*e**4*x**2*acosh(c + d*x)**2 + 12*
a*b**2*c**3*d*e**4*x**2/25 - 24*a*b**2*c**3*e**4*x*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)*acosh(c + d*x)/25 + 6*
a*b**2*c**2*d**2*e**4*x**3*acosh(c + d*x)**2 + 12*a*b**2*c**2*d**2*e**4*x**3/25 - 36*a*b**2*c**2*d*e**4*x**2*s
qrt(c**2 + 2*c*d*x + d**2*x**2 - 1)*acosh(c + d*x)/25 + 8*a*b**2*c**2*e**4*x/25 - 8*a*b**2*c**2*e**4*sqrt(c**2
+ 2*c*d*x + d**2*x**2 - 1)*acosh(c + d*x)/(25*d) + 3*a*b**2*c*d**3*e**4*x**4*acosh(c + d*x)**2 + 6*a*b**2*c*d
**3*e**4*x**4/25 - 24*a*b**2*c*d**2*e**4*x**3*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)*acosh(c + d*x)/25 + 8*a*b**
2*c*d*e**4*x**2/25 - 16*a*b**2*c*e**4*x*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)*acosh(c + d*x)/25 + 3*a*b**2*d**4
*e**4*x**5*acosh(c + d*x)**2/5 + 6*a*b**2*d**4*e**4*x**5/125 - 6*a*b**2*d**3*e**4*x**4*sqrt(c**2 + 2*c*d*x + d
**2*x**2 - 1)*acosh(c + d*x)/25 + 8*a*b**2*d**2*e**4*x**3/75 - 8*a*b**2*d*e**4*x**2*sqrt(c**2 + 2*c*d*x + d**2
*x**2 - 1)*acosh(c + d*x)/25 + 16*a*b**2*e**4*x/25 - 16*a*b**2*e**4*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)*acosh
(c + d*x)/(25*d) + b**3*c**5*e**4*acosh(c + d*x)**3/(5*d) + 6*b**3*c**5*e**4*acosh(c + d*x)/(125*d) + b**3*c**
4*e**4*x*acosh(c + d*x)**3 + 6*b**3*c**4*e**4*x*acosh(c + d*x)/25 - 3*b**3*c**4*e**4*sqrt(c**2 + 2*c*d*x + d**
2*x**2 - 1)*acosh(c + d*x)**2/(25*d) - 6*b**3*c**4*e**4*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)/(625*d) + 2*b**3*
c**3*d*e**4*x**2*acosh(c + d*x)**3 + 12*b**3*c**3*d*e**4*x**2*acosh(c + d*x)/25 - 12*b**3*c**3*e**4*x*sqrt(c**
2 + 2*c*d*x + d**2*x**2 - 1)*acosh(c + d*x)**2/25 - 24*b**3*c**3*e**4*x*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)/6
25 + 8*b**3*c**3*e**4*acosh(c + d*x)/(75*d) + 2*b**3*c**2*d**2*e**4*x**3*acosh(c + d*x)**3 + 12*b**3*c**2*d**2
*e**4*x**3*acosh(c + d*x)/25 - 18*b**3*c**2*d*e**4*x**2*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)*acosh(c + d*x)**2
/25 - 36*b**3*c**2*d*e**4*x**2*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)/625 + 8*b**3*c**2*e**4*x*acosh(c + d*x)/25
- 4*b**3*c**2*e**4*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)*acosh(c + d*x)**2/(25*d) - 272*b**3*c**2*e**4*sqrt(c*
*2 + 2*c*d*x + d**2*x**2 - 1)/(5625*d) + b**3*c*d**3*e**4*x**4*acosh(c + d*x)**3 + 6*b**3*c*d**3*e**4*x**4*aco
sh(c + d*x)/25 - 12*b**3*c*d**2*e**4*x**3*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)*acosh(c + d*x)**2/25 - 24*b**3*
c*d**2*e**4*x**3*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)/625 + 8*b**3*c*d*e**4*x**2*acosh(c + d*x)/25 - 8*b**3*c*
e**4*x*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)*acosh(c + d*x)**2/25 - 544*b**3*c*e**4*x*sqrt(c**2 + 2*c*d*x + d**
2*x**2 - 1)/5625 + 16*b**3*c*e**4*acosh(c + d*x)/(25*d) + b**3*d**4*e**4*x**5*acosh(c + d*x)**3/5 + 6*b**3*d**
4*e**4*x**5*acosh(c + d*x)/125 - 3*b**3*d**3*e**4*x**4*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)*acosh(c + d*x)**2/
25 - 6*b**3*d**3*e**4*x**4*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)/625 + 8*b**3*d**2*e**4*x**3*acosh(c + d*x)/75
- 4*b**3*d*e**4*x**2*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)*acosh(c + d*x)**2/25 - 272*b**3*d*e**4*x**2*sqrt(c**
2 + 2*c*d*x + d**2*x**2 - 1)/5625 + 16*b**3*e**4*x*acosh(c + d*x)/25 - 8*b**3*e**4*sqrt(c**2 + 2*c*d*x + d**2*
x**2 - 1)*acosh(c + d*x)**2/(25*d) - 4144*b**3*e**4*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)/(5625*d), Ne(d, 0)),
(c**4*e**4*x*(a + b*acosh(c))**3, True))

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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 {\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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