3.2.4 \(\int (c e+d e x)^4 (a+b \cosh ^{-1}(c+d x))^2 \, dx\) [104]

Optimal. Leaf size=218 \[ \frac {16}{75} b^2 e^4 x+\frac {8 b^2 e^4 (c+d x)^3}{225 d}+\frac {2 b^2 e^4 (c+d x)^5}{125 d}-\frac {16 b e^4 \sqrt {-1+c+d x} \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )}{75 d}-\frac {8 b e^4 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )}{75 d}-\frac {2 b e^4 \sqrt {-1+c+d x} (c+d x)^4 \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )}{25 d}+\frac {e^4 (c+d x)^5 \left (a+b \cosh ^{-1}(c+d x)\right )^2}{5 d} \]

[Out]

16/75*b^2*e^4*x+8/225*b^2*e^4*(d*x+c)^3/d+2/125*b^2*e^4*(d*x+c)^5/d+1/5*e^4*(d*x+c)^5*(a+b*arccosh(d*x+c))^2/d
-16/75*b*e^4*(a+b*arccosh(d*x+c))*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)/d-8/75*b*e^4*(d*x+c)^2*(a+b*arccosh(d*x+c))*
(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)/d-2/25*b*e^4*(d*x+c)^4*(a+b*arccosh(d*x+c))*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)/d

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Rubi [A]
time = 0.35, antiderivative size = 218, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {5996, 12, 5883, 5939, 5915, 8, 30} \begin {gather*} \frac {e^4 (c+d x)^5 \left (a+b \cosh ^{-1}(c+d x)\right )^2}{5 d}-\frac {2 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 )}{25 d}-\frac {8 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 )}{75 d}-\frac {16 b e^4 \sqrt {c+d x-1} \sqrt {c+d x+1} \left (a+b \cosh ^{-1}(c+d x)\right )}{75 d}+\frac {2 b^2 e^4 (c+d x)^5}{125 d}+\frac {8 b^2 e^4 (c+d x)^3}{225 d}+\frac {16}{75} b^2 e^4 x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c*e + d*e*x)^4*(a + b*ArcCosh[c + d*x])^2,x]

[Out]

(16*b^2*e^4*x)/75 + (8*b^2*e^4*(c + d*x)^3)/(225*d) + (2*b^2*e^4*(c + d*x)^5)/(125*d) - (16*b*e^4*Sqrt[-1 + c
+ d*x]*Sqrt[1 + c + d*x]*(a + b*ArcCosh[c + d*x]))/(75*d) - (8*b*e^4*Sqrt[-1 + c + d*x]*(c + d*x)^2*Sqrt[1 + c
+ d*x]*(a + b*ArcCosh[c + d*x]))/(75*d) - (2*b*e^4*Sqrt[-1 + c + d*x]*(c + d*x)^4*Sqrt[1 + c + d*x]*(a + b*Ar
cCosh[c + d*x]))/(25*d) + (e^4*(c + d*x)^5*(a + b*ArcCosh[c + d*x])^2)/(5*d)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 12

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

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

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

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Mathematica [A]
time = 0.20, size = 220, normalized size = 1.01 \begin {gather*} \frac {e^4 \left (240 b^2 (c+d x)+40 b^2 (c+d x)^3+9 \left (25 a^2+2 b^2\right ) (c+d x)^5+30 a b \sqrt {-1+c+d x} \sqrt {1+c+d x} \left (-8-4 (c+d x)^2-3 (c+d x)^4\right )+30 b \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)+225 b^2 (c+d x)^5 \cosh ^{-1}(c+d x)^2\right )}{1125 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c*e + d*e*x)^4*(a + b*ArcCosh[c + d*x])^2,x]

[Out]

(e^4*(240*b^2*(c + d*x) + 40*b^2*(c + d*x)^3 + 9*(25*a^2 + 2*b^2)*(c + d*x)^5 + 30*a*b*Sqrt[-1 + c + d*x]*Sqrt
[1 + c + d*x]*(-8 - 4*(c + d*x)^2 - 3*(c + d*x)^4) + 30*b*(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])*ArcCosh[c + d*x] + 225*b^2*(c + d*x)^5*ArcCosh[c + d*x]^2))/(1125*d)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(539\) vs. \(2(192)=384\).
time = 10.98, size = 540, normalized size = 2.48

method	result	size
default	\(\frac {e^{4} \left (d x +c \right )^{5} a^{2}}{5 d}+\frac {e^{4} b^{2} \left (225 \mathrm {arccosh}\left (d x +c \right )^{2} c^{5}+18 x^{5} d^{5}+240 c -90 \,\mathrm {arccosh}\left (d x +c \right ) \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, c^{4}-120 \,\mathrm {arccosh}\left (d x +c \right ) \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, c^{2}+1125 \mathrm {arccosh}\left (d x +c \right )^{2} x^{4} c \,d^{4}+2250 \mathrm {arccosh}\left (d x +c \right )^{2} x^{3} c^{2} d^{3}+2250 \mathrm {arccosh}\left (d x +c \right )^{2} x^{2} c^{3} d^{2}+1125 \mathrm {arccosh}\left (d x +c \right )^{2} x \,c^{4} d +40 c^{3}+40 d^{3} x^{3}+18 c^{5}+240 d x -90 \,\mathrm {arccosh}\left (d x +c \right ) \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, x^{4} d^{4}-120 \,\mathrm {arccosh}\left (d x +c \right ) \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, x^{2} d^{2}-540 \,\mathrm {arccosh}\left (d x +c \right ) \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, x^{2} c^{2} d^{2}-360 \,\mathrm {arccosh}\left (d x +c \right ) \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, x \,c^{3} d -240 \,\mathrm {arccosh}\left (d x +c \right ) \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, x c d -360 \,\mathrm {arccosh}\left (d x +c \right ) \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, x^{3} c \,d^{3}+120 c^{2} d x +90 x \,c^{4} d +180 x^{3} c^{2} d^{3}+90 x^{4} c \,d^{4}+180 x^{2} c^{3} d^{2}+120 x^{2} c \,d^{2}+225 \mathrm {arccosh}\left (d x +c \right )^{2} x^{5} d^{5}-240 \,\mathrm {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )}{1125 d}+\frac {2 e^{4} a b \left (\frac {\left (d x +c \right )^{5} \mathrm {arccosh}\left (d x +c \right )}{5}-\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (3 \left (d x +c \right )^{4}+4 \left (d x +c \right )^{2}+8\right )}{75}\right )}{d}\)	\(540\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*e*x+c*e)^4*(a+b*arccosh(d*x+c))^2,x,method=_RETURNVERBOSE)

[Out]

1/5*e^4*(d*x+c)^5*a^2/d+1/1125*e^4*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*arccosh(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*arccos
h(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+2*e^
4*a*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))^2,x, algorithm="maxima")

[Out]

1/5*a^2*d^4*x^5*e^4 + a^2*c*d^3*x^4*e^4 + 2*a^2*c^2*d^2*x^3*e^4 + 2*a^2*c^3*d*x^2*e^4 + 2*(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*b*c^3*d*e^4 + 2/3*(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*b*c^2*d^2*e^4 + 1/
12*(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*b*c*d^3*e^4 + 1/300*(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*b*d^4*e^4 + a^2*c^4*x*e^4 + 2*((d*x + c)*arccosh(d*x + c) - sqrt((d*x + c)^2 - 1)
)*a*b*c^4*e^4/d + 1/5*(b^2*d^4*x^5*e^4 + 5*b^2*c*d^3*x^4*e^4 + 10*b^2*c^2*d^2*x^3*e^4 + 10*b^2*c^3*d*x^2*e^4 +
5*b^2*c^4*x*e^4)*log(d*x + sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + c)^2 - integrate(2/5*(b^2*d^7*x^7*e^4 + 7*b^
2*c*d^6*x^6*e^4 + (21*c^2*d^5 - d^5)*b^2*x^5*e^4 + 5*(7*c^3*d^4 - c*d^4)*b^2*x^4*e^4 + 5*(7*c^4*d^3 - 2*c^2*d^
3)*b^2*x^3*e^4 + 10*(2*c^5*d^2 - c^3*d^2)*b^2*x^2*e^4 + 5*(c^6*d - c^4*d)*b^2*x*e^4 + (b^2*d^6*x^6*e^4 + 6*b^2
*c*d^5*x^5*e^4 + 15*b^2*c^2*d^4*x^4*e^4 + 20*b^2*c^3*d^3*x^3*e^4 + 15*b^2*c^4*d^2*x^2*e^4 + 5*b^2*c^5*d*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)/(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. 2430 vs. \(2 (185) = 370\).
time = 0.45, size = 2430, normalized size = 11.15 \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))^2,x, algorithm="fricas")

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1268 vs. \(2 (209) = 418\).
time = 0.82, size = 1268, normalized size = 5.82 \begin {gather*} \begin {cases} a^{2} c^{4} e^{4} x + 2 a^{2} c^{3} d e^{4} x^{2} + 2 a^{2} c^{2} d^{2} e^{4} x^{3} + a^{2} c d^{3} e^{4} x^{4} + \frac {a^{2} d^{4} e^{4} x^{5}}{5} + \frac {2 a b c^{5} e^{4} \operatorname {acosh}{\left (c + d x \right )}}{5 d} + 2 a b c^{4} e^{4} x \operatorname {acosh}{\left (c + d x \right )} - \frac {2 a b c^{4} e^{4} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1}}{25 d} + 4 a b c^{3} d e^{4} x^{2} \operatorname {acosh}{\left (c + d x \right )} - \frac {8 a b c^{3} e^{4} x \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1}}{25} + 4 a b c^{2} d^{2} e^{4} x^{3} \operatorname {acosh}{\left (c + d x \right )} - \frac {12 a b c^{2} d e^{4} x^{2} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1}}{25} - \frac {8 a b c^{2} e^{4} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1}}{75 d} + 2 a b c d^{3} e^{4} x^{4} \operatorname {acosh}{\left (c + d x \right )} - \frac {8 a b c d^{2} e^{4} x^{3} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1}}{25} - \frac {16 a b c e^{4} x \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1}}{75} + \frac {2 a b d^{4} e^{4} x^{5} \operatorname {acosh}{\left (c + d x \right )}}{5} - \frac {2 a b d^{3} e^{4} x^{4} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1}}{25} - \frac {8 a b d e^{4} x^{2} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1}}{75} - \frac {16 a b e^{4} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1}}{75 d} + \frac {b^{2} c^{5} e^{4} \operatorname {acosh}^{2}{\left (c + d x \right )}}{5 d} + b^{2} c^{4} e^{4} x \operatorname {acosh}^{2}{\left (c + d x \right )} + \frac {2 b^{2} c^{4} e^{4} x}{25} - \frac {2 b^{2} c^{4} e^{4} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1} \operatorname {acosh}{\left (c + d x \right )}}{25 d} + 2 b^{2} c^{3} d e^{4} x^{2} \operatorname {acosh}^{2}{\left (c + d x \right )} + \frac {4 b^{2} c^{3} d e^{4} x^{2}}{25} - \frac {8 b^{2} c^{3} e^{4} x \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1} \operatorname {acosh}{\left (c + d x \right )}}{25} + 2 b^{2} c^{2} d^{2} e^{4} x^{3} \operatorname {acosh}^{2}{\left (c + d x \right )} + \frac {4 b^{2} c^{2} d^{2} e^{4} x^{3}}{25} - \frac {12 b^{2} c^{2} d e^{4} x^{2} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1} \operatorname {acosh}{\left (c + d x \right )}}{25} + \frac {8 b^{2} c^{2} e^{4} x}{75} - \frac {8 b^{2} c^{2} e^{4} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1} \operatorname {acosh}{\left (c + d x \right )}}{75 d} + b^{2} c d^{3} e^{4} x^{4} \operatorname {acosh}^{2}{\left (c + d x \right )} + \frac {2 b^{2} c d^{3} e^{4} x^{4}}{25} - \frac {8 b^{2} c d^{2} e^{4} x^{3} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1} \operatorname {acosh}{\left (c + d x \right )}}{25} + \frac {8 b^{2} c d e^{4} x^{2}}{75} - \frac {16 b^{2} c e^{4} x \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1} \operatorname {acosh}{\left (c + d x \right )}}{75} + \frac {b^{2} d^{4} e^{4} x^{5} \operatorname {acosh}^{2}{\left (c + d x \right )}}{5} + \frac {2 b^{2} d^{4} e^{4} x^{5}}{125} - \frac {2 b^{2} d^{3} e^{4} x^{4} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1} \operatorname {acosh}{\left (c + d x \right )}}{25} + \frac {8 b^{2} d^{2} e^{4} x^{3}}{225} - \frac {8 b^{2} d e^{4} x^{2} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1} \operatorname {acosh}{\left (c + d x \right )}}{75} + \frac {16 b^{2} e^{4} x}{75} - \frac {16 b^{2} e^{4} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1} \operatorname {acosh}{\left (c + d x \right )}}{75 d} & \text {for}\: d \neq 0 \\c^{4} e^{4} x \left (a + b \operatorname {acosh}{\left (c \right )}\right )^{2} & \text {otherwise} \end {cases} \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))**2,x)

[Out]

Piecewise((a**2*c**4*e**4*x + 2*a**2*c**3*d*e**4*x**2 + 2*a**2*c**2*d**2*e**4*x**3 + a**2*c*d**3*e**4*x**4 + a
**2*d**4*e**4*x**5/5 + 2*a*b*c**5*e**4*acosh(c + d*x)/(5*d) + 2*a*b*c**4*e**4*x*acosh(c + d*x) - 2*a*b*c**4*e*
*4*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)/(25*d) + 4*a*b*c**3*d*e**4*x**2*acosh(c + d*x) - 8*a*b*c**3*e**4*x*sqr
t(c**2 + 2*c*d*x + d**2*x**2 - 1)/25 + 4*a*b*c**2*d**2*e**4*x**3*acosh(c + d*x) - 12*a*b*c**2*d*e**4*x**2*sqrt
(c**2 + 2*c*d*x + d**2*x**2 - 1)/25 - 8*a*b*c**2*e**4*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)/(75*d) + 2*a*b*c*d*
*3*e**4*x**4*acosh(c + d*x) - 8*a*b*c*d**2*e**4*x**3*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)/25 - 16*a*b*c*e**4*x
*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)/75 + 2*a*b*d**4*e**4*x**5*acosh(c + d*x)/5 - 2*a*b*d**3*e**4*x**4*sqrt(c
**2 + 2*c*d*x + d**2*x**2 - 1)/25 - 8*a*b*d*e**4*x**2*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)/75 - 16*a*b*e**4*sq
rt(c**2 + 2*c*d*x + d**2*x**2 - 1)/(75*d) + b**2*c**5*e**4*acosh(c + d*x)**2/(5*d) + b**2*c**4*e**4*x*acosh(c
+ d*x)**2 + 2*b**2*c**4*e**4*x/25 - 2*b**2*c**4*e**4*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)*acosh(c + d*x)/(25*d
) + 2*b**2*c**3*d*e**4*x**2*acosh(c + d*x)**2 + 4*b**2*c**3*d*e**4*x**2/25 - 8*b**2*c**3*e**4*x*sqrt(c**2 + 2*
c*d*x + d**2*x**2 - 1)*acosh(c + d*x)/25 + 2*b**2*c**2*d**2*e**4*x**3*acosh(c + d*x)**2 + 4*b**2*c**2*d**2*e**
4*x**3/25 - 12*b**2*c**2*d*e**4*x**2*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)*acosh(c + d*x)/25 + 8*b**2*c**2*e**4
*x/75 - 8*b**2*c**2*e**4*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)*acosh(c + d*x)/(75*d) + b**2*c*d**3*e**4*x**4*ac
osh(c + d*x)**2 + 2*b**2*c*d**3*e**4*x**4/25 - 8*b**2*c*d**2*e**4*x**3*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)*ac
osh(c + d*x)/25 + 8*b**2*c*d*e**4*x**2/75 - 16*b**2*c*e**4*x*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)*acosh(c + d*
x)/75 + b**2*d**4*e**4*x**5*acosh(c + d*x)**2/5 + 2*b**2*d**4*e**4*x**5/125 - 2*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*b**2*d**2*e**4*x**3/225 - 8*b**2*d*e**4*x**2*sqrt(c**2 + 2*c*
d*x + d**2*x**2 - 1)*acosh(c + d*x)/75 + 16*b**2*e**4*x/75 - 16*b**2*e**4*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)
*acosh(c + d*x)/(75*d), Ne(d, 0)), (c**4*e**4*x*(a + b*acosh(c))**2, 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))^2,x, algorithm="giac")

[Out]

integrate((d*e*x + c*e)^4*(b*arccosh(d*x + c) + a)^2, 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 )}^2 \,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))^2,x)

[Out]

int((c*e + d*e*x)^4*(a + b*acosh(c + d*x))^2, x)

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