3.1.93 \(\int (c e+d e x)^4 (a+b \cosh ^{-1}(c+d x)) \, dx\) [93]

Optimal. Leaf size=135 \[ -\frac {8 b e^4 \sqrt {-1+c+d x} \sqrt {1+c+d x}}{75 d}-\frac {4 b e^4 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{75 d}-\frac {b e^4 \sqrt {-1+c+d x} (c+d x)^4 \sqrt {1+c+d x}}{25 d}+\frac {e^4 (c+d x)^5 \left (a+b \cosh ^{-1}(c+d x)\right )}{5 d} \]

[Out]

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

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Rubi [A]
time = 0.06, antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {5996, 12, 5883, 102, 75} \begin {gather*} \frac {e^4 (c+d x)^5 \left (a+b \cosh ^{-1}(c+d x)\right )}{5 d}-\frac {b e^4 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^4}{25 d}-\frac {4 b e^4 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2}{75 d}-\frac {8 b e^4 \sqrt {c+d x-1} \sqrt {c+d x+1}}{75 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]
time = 0.09, size = 103, normalized size = 0.76 \begin {gather*} \frac {e^4 \left (-\frac {1}{5} b \sqrt {-1+c+d x} (c+d x)^4 \sqrt {1+c+d x}-\frac {4}{15} b \sqrt {-1+c+d x} \sqrt {1+c+d x} \left (2+c^2+2 c d x+d^2 x^2\right )+(c+d x)^5 \left (a+b \cosh ^{-1}(c+d x)\right )\right )}{5 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(e^4*(-1/5*(b*Sqrt[-1 + c + d*x]*(c + d*x)^4*Sqrt[1 + c + d*x]) - (4*b*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x]*(2
+ c^2 + 2*c*d*x + d^2*x^2))/15 + (c + d*x)^5*(a + b*ArcCosh[c + d*x])))/(5*d)

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Maple [A]
time = 3.90, size = 78, normalized size = 0.58

method	result	size
derivativedivides	\(\frac {\frac {e^{4} \left (d x +c \right )^{5} a}{5}+e^{4} b \left (\frac {\left (d x +c \right )^{5} \mathrm {arccosh}\left (d x +c \right )}{5}-\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (3 \left (d x +c \right )^{4}+4 \left (d x +c \right )^{2}+8\right )}{75}\right )}{d}\)	\(78\)
default	\(\frac {\frac {e^{4} \left (d x +c \right )^{5} a}{5}+e^{4} b \left (\frac {\left (d x +c \right )^{5} \mathrm {arccosh}\left (d x +c \right )}{5}-\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (3 \left (d x +c \right )^{4}+4 \left (d x +c \right )^{2}+8\right )}{75}\right )}{d}\)	\(78\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(1/5*e^4*(d*x+c)^5*a+e^4*b*(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 [B] Leaf count of result is larger than twice the leaf count of optimal. 1231 vs. \(2 (111) = 222\).
time = 0.28, size = 1231, normalized size = 9.12 \begin {gather*} \frac {1}{5} \, a d^{4} x^{5} e^{4} + a c d^{3} x^{4} e^{4} + 2 \, a c^{2} d^{2} x^{3} e^{4} + 2 \, a c^{3} d x^{2} e^{4} + {\left (2 \, x^{2} \operatorname {arcosh}\left (d x + c\right ) - d {\left (\frac {3 \, c^{2} \log \left (2 \, d^{2} x + 2 \, c d + 2 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} d\right )}{d^{3}} + \frac {\sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} x}{d^{2}} - \frac {{\left (c^{2} - 1\right )} \log \left (2 \, d^{2} x + 2 \, c d + 2 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} d\right )}{d^{3}} - \frac {3 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} c}{d^{3}}\right )}\right )} b c^{3} d e^{4} + \frac {1}{3} \, {\left (6 \, x^{3} \operatorname {arcosh}\left (d x + c\right ) - d {\left (\frac {2 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} x^{2}}{d^{2}} - \frac {15 \, c^{3} \log \left (2 \, d^{2} x + 2 \, c d + 2 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} d\right )}{d^{4}} - \frac {5 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} c x}{d^{3}} + \frac {9 \, {\left (c^{2} - 1\right )} c \log \left (2 \, d^{2} x + 2 \, c d + 2 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} d\right )}{d^{4}} + \frac {15 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} c^{2}}{d^{4}} - \frac {4 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} {\left (c^{2} - 1\right )}}{d^{4}}\right )}\right )} b c^{2} d^{2} e^{4} + \frac {1}{24} \, {\left (24 \, x^{4} \operatorname {arcosh}\left (d x + c\right ) - {\left (\frac {6 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} x^{3}}{d^{2}} - \frac {14 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} c x^{2}}{d^{3}} + \frac {105 \, c^{4} \log \left (2 \, d^{2} x + 2 \, c d + 2 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} d\right )}{d^{5}} + \frac {35 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} c^{2} x}{d^{4}} - \frac {90 \, {\left (c^{2} - 1\right )} c^{2} \log \left (2 \, d^{2} x + 2 \, c d + 2 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} d\right )}{d^{5}} - \frac {105 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} c^{3}}{d^{5}} - \frac {9 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} {\left (c^{2} - 1\right )} x}{d^{4}} + \frac {9 \, {\left (c^{2} - 1\right )}^{2} \log \left (2 \, d^{2} x + 2 \, c d + 2 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} d\right )}{d^{5}} + \frac {55 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} {\left (c^{2} - 1\right )} c}{d^{5}}\right )} d\right )} b c d^{3} e^{4} + \frac {1}{600} \, {\left (120 \, x^{5} \operatorname {arcosh}\left (d x + c\right ) - {\left (\frac {24 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} x^{4}}{d^{2}} - \frac {54 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} c x^{3}}{d^{3}} + \frac {126 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} c^{2} x^{2}}{d^{4}} - \frac {945 \, c^{5} \log \left (2 \, d^{2} x + 2 \, c d + 2 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} d\right )}{d^{6}} - \frac {315 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} c^{3} x}{d^{5}} - \frac {32 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} {\left (c^{2} - 1\right )} x^{2}}{d^{4}} + \frac {1050 \, {\left (c^{2} - 1\right )} c^{3} \log \left (2 \, d^{2} x + 2 \, c d + 2 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} d\right )}{d^{6}} + \frac {945 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} c^{4}}{d^{6}} + \frac {161 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} {\left (c^{2} - 1\right )} c x}{d^{5}} - \frac {225 \, {\left (c^{2} - 1\right )}^{2} c \log \left (2 \, d^{2} x + 2 \, c d + 2 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} d\right )}{d^{6}} - \frac {735 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} {\left (c^{2} - 1\right )} c^{2}}{d^{6}} + \frac {64 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} {\left (c^{2} - 1\right )}^{2}}{d^{6}}\right )} d\right )} b d^{4} e^{4} + a c^{4} x e^{4} + \frac {{\left ({\left (d x + c\right )} \operatorname {arcosh}\left (d x + c\right ) - \sqrt {{\left (d x + c\right )}^{2} - 1}\right )} b c^{4} e^{4}}{d} \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)),x, algorithm="maxima")

[Out]

1/5*a*d^4*x^5*e^4 + a*c*d^3*x^4*e^4 + 2*a*c^2*d^2*x^3*e^4 + 2*a*c^3*d*x^2*e^4 + (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))*b*c^3*d*e^4 + 1/3*(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))*b*c^2*d^2*e^4 + 1/24*(24*x^4*arc
cosh(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)*b*c*d^3*e^4 + 1/600*(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*s
qrt(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)*b*d^4*e^4 + a*c^4*x*e^4 + ((d*x + c)*arccosh(d*x + c) - sqrt((d*x + c)^2 - 1))*b*c^4*e^4/d

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1041 vs. \(2 (111) = 222\).
time = 0.36, size = 1041, normalized size = 7.71 \begin {gather*} \frac {15 \, {\left (a d^{5} x^{5} + 5 \, a c d^{4} x^{4} + 10 \, a c^{2} d^{3} x^{3} + 10 \, a c^{3} d^{2} x^{2} + 5 \, a c^{4} d x\right )} \cosh \left (1\right )^{4} + 60 \, {\left (a d^{5} x^{5} + 5 \, a c d^{4} x^{4} + 10 \, a c^{2} d^{3} x^{3} + 10 \, a c^{3} d^{2} x^{2} + 5 \, a c^{4} d x\right )} \cosh \left (1\right )^{3} \sinh \left (1\right ) + 90 \, {\left (a d^{5} x^{5} + 5 \, a c d^{4} x^{4} + 10 \, a c^{2} d^{3} x^{3} + 10 \, a c^{3} d^{2} x^{2} + 5 \, a c^{4} d x\right )} \cosh \left (1\right )^{2} \sinh \left (1\right )^{2} + 60 \, {\left (a d^{5} x^{5} + 5 \, a c d^{4} x^{4} + 10 \, a c^{2} d^{3} x^{3} + 10 \, a c^{3} d^{2} x^{2} + 5 \, a c^{4} d x\right )} \cosh \left (1\right ) \sinh \left (1\right )^{3} + 15 \, {\left (a d^{5} x^{5} + 5 \, a c d^{4} x^{4} + 10 \, a c^{2} d^{3} x^{3} + 10 \, a c^{3} d^{2} x^{2} + 5 \, a c^{4} d x\right )} \sinh \left (1\right )^{4} + 15 \, {\left ({\left (b d^{5} x^{5} + 5 \, b c d^{4} x^{4} + 10 \, b c^{2} d^{3} x^{3} + 10 \, b c^{3} d^{2} x^{2} + 5 \, b c^{4} d x + b c^{5}\right )} \cosh \left (1\right )^{4} + 4 \, {\left (b d^{5} x^{5} + 5 \, b c d^{4} x^{4} + 10 \, b c^{2} d^{3} x^{3} + 10 \, b c^{3} d^{2} x^{2} + 5 \, b c^{4} d x + b c^{5}\right )} \cosh \left (1\right )^{3} \sinh \left (1\right ) + 6 \, {\left (b d^{5} x^{5} + 5 \, b c d^{4} x^{4} + 10 \, b c^{2} d^{3} x^{3} + 10 \, b c^{3} d^{2} x^{2} + 5 \, b c^{4} d x + b c^{5}\right )} \cosh \left (1\right )^{2} \sinh \left (1\right )^{2} + 4 \, {\left (b d^{5} x^{5} + 5 \, b c d^{4} x^{4} + 10 \, b c^{2} d^{3} x^{3} + 10 \, b c^{3} d^{2} x^{2} + 5 \, b c^{4} d x + b c^{5}\right )} \cosh \left (1\right ) \sinh \left (1\right )^{3} + {\left (b d^{5} x^{5} + 5 \, b c d^{4} x^{4} + 10 \, b c^{2} d^{3} x^{3} + 10 \, b c^{3} d^{2} x^{2} + 5 \, b c^{4} d x + b c^{5}\right )} \sinh \left (1\right )^{4}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) - {\left ({\left (3 \, b d^{4} x^{4} + 12 \, b c d^{3} x^{3} + 3 \, b c^{4} + 2 \, {\left (9 \, b c^{2} + 2 \, b\right )} d^{2} x^{2} + 4 \, b c^{2} + 4 \, {\left (3 \, b c^{3} + 2 \, b c\right )} d x + 8 \, b\right )} \cosh \left (1\right )^{4} + 4 \, {\left (3 \, b d^{4} x^{4} + 12 \, b c d^{3} x^{3} + 3 \, b c^{4} + 2 \, {\left (9 \, b c^{2} + 2 \, b\right )} d^{2} x^{2} + 4 \, b c^{2} + 4 \, {\left (3 \, b c^{3} + 2 \, b c\right )} d x + 8 \, b\right )} \cosh \left (1\right )^{3} \sinh \left (1\right ) + 6 \, {\left (3 \, b d^{4} x^{4} + 12 \, b c d^{3} x^{3} + 3 \, b c^{4} + 2 \, {\left (9 \, b c^{2} + 2 \, b\right )} d^{2} x^{2} + 4 \, b c^{2} + 4 \, {\left (3 \, b c^{3} + 2 \, b c\right )} d x + 8 \, b\right )} \cosh \left (1\right )^{2} \sinh \left (1\right )^{2} + 4 \, {\left (3 \, b d^{4} x^{4} + 12 \, b c d^{3} x^{3} + 3 \, b c^{4} + 2 \, {\left (9 \, b c^{2} + 2 \, b\right )} d^{2} x^{2} + 4 \, b c^{2} + 4 \, {\left (3 \, b c^{3} + 2 \, b c\right )} d x + 8 \, b\right )} \cosh \left (1\right ) \sinh \left (1\right )^{3} + {\left (3 \, b d^{4} x^{4} + 12 \, b c d^{3} x^{3} + 3 \, b c^{4} + 2 \, {\left (9 \, b c^{2} + 2 \, b\right )} d^{2} x^{2} + 4 \, b c^{2} + 4 \, {\left (3 \, b c^{3} + 2 \, b c\right )} d x + 8 \, b\right )} \sinh \left (1\right )^{4}\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}}{75 \, d} \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)),x, algorithm="fricas")

[Out]

1/75*(15*(a*d^5*x^5 + 5*a*c*d^4*x^4 + 10*a*c^2*d^3*x^3 + 10*a*c^3*d^2*x^2 + 5*a*c^4*d*x)*cosh(1)^4 + 60*(a*d^5
*x^5 + 5*a*c*d^4*x^4 + 10*a*c^2*d^3*x^3 + 10*a*c^3*d^2*x^2 + 5*a*c^4*d*x)*cosh(1)^3*sinh(1) + 90*(a*d^5*x^5 +
5*a*c*d^4*x^4 + 10*a*c^2*d^3*x^3 + 10*a*c^3*d^2*x^2 + 5*a*c^4*d*x)*cosh(1)^2*sinh(1)^2 + 60*(a*d^5*x^5 + 5*a*c
*d^4*x^4 + 10*a*c^2*d^3*x^3 + 10*a*c^3*d^2*x^2 + 5*a*c^4*d*x)*cosh(1)*sinh(1)^3 + 15*(a*d^5*x^5 + 5*a*c*d^4*x^
4 + 10*a*c^2*d^3*x^3 + 10*a*c^3*d^2*x^2 + 5*a*c^4*d*x)*sinh(1)^4 + 15*((b*d^5*x^5 + 5*b*c*d^4*x^4 + 10*b*c^2*d
^3*x^3 + 10*b*c^3*d^2*x^2 + 5*b*c^4*d*x + b*c^5)*cosh(1)^4 + 4*(b*d^5*x^5 + 5*b*c*d^4*x^4 + 10*b*c^2*d^3*x^3 +
10*b*c^3*d^2*x^2 + 5*b*c^4*d*x + b*c^5)*cosh(1)^3*sinh(1) + 6*(b*d^5*x^5 + 5*b*c*d^4*x^4 + 10*b*c^2*d^3*x^3 +
10*b*c^3*d^2*x^2 + 5*b*c^4*d*x + b*c^5)*cosh(1)^2*sinh(1)^2 + 4*(b*d^5*x^5 + 5*b*c*d^4*x^4 + 10*b*c^2*d^3*x^3
+ 10*b*c^3*d^2*x^2 + 5*b*c^4*d*x + b*c^5)*cosh(1)*sinh(1)^3 + (b*d^5*x^5 + 5*b*c*d^4*x^4 + 10*b*c^2*d^3*x^3 +
10*b*c^3*d^2*x^2 + 5*b*c^4*d*x + b*c^5)*sinh(1)^4)*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)) - ((3*b*d
^4*x^4 + 12*b*c*d^3*x^3 + 3*b*c^4 + 2*(9*b*c^2 + 2*b)*d^2*x^2 + 4*b*c^2 + 4*(3*b*c^3 + 2*b*c)*d*x + 8*b)*cosh(
1)^4 + 4*(3*b*d^4*x^4 + 12*b*c*d^3*x^3 + 3*b*c^4 + 2*(9*b*c^2 + 2*b)*d^2*x^2 + 4*b*c^2 + 4*(3*b*c^3 + 2*b*c)*d
*x + 8*b)*cosh(1)^3*sinh(1) + 6*(3*b*d^4*x^4 + 12*b*c*d^3*x^3 + 3*b*c^4 + 2*(9*b*c^2 + 2*b)*d^2*x^2 + 4*b*c^2
+ 4*(3*b*c^3 + 2*b*c)*d*x + 8*b)*cosh(1)^2*sinh(1)^2 + 4*(3*b*d^4*x^4 + 12*b*c*d^3*x^3 + 3*b*c^4 + 2*(9*b*c^2
+ 2*b)*d^2*x^2 + 4*b*c^2 + 4*(3*b*c^3 + 2*b*c)*d*x + 8*b)*cosh(1)*sinh(1)^3 + (3*b*d^4*x^4 + 12*b*c*d^3*x^3 +
3*b*c^4 + 2*(9*b*c^2 + 2*b)*d^2*x^2 + 4*b*c^2 + 4*(3*b*c^3 + 2*b*c)*d*x + 8*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. 527 vs. \(2 (126) = 252\).
time = 0.50, size = 527, normalized size = 3.90 \begin {gather*} \begin {cases} a c^{4} e^{4} x + 2 a c^{3} d e^{4} x^{2} + 2 a c^{2} d^{2} e^{4} x^{3} + a c d^{3} e^{4} x^{4} + \frac {a d^{4} e^{4} x^{5}}{5} + \frac {b c^{5} e^{4} \operatorname {acosh}{\left (c + d x \right )}}{5 d} + b c^{4} e^{4} x \operatorname {acosh}{\left (c + d x \right )} - \frac {b c^{4} e^{4} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1}}{25 d} + 2 b c^{3} d e^{4} x^{2} \operatorname {acosh}{\left (c + d x \right )} - \frac {4 b c^{3} e^{4} x \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1}}{25} + 2 b c^{2} d^{2} e^{4} x^{3} \operatorname {acosh}{\left (c + d x \right )} - \frac {6 b c^{2} d e^{4} x^{2} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1}}{25} - \frac {4 b c^{2} e^{4} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1}}{75 d} + b c d^{3} e^{4} x^{4} \operatorname {acosh}{\left (c + d x \right )} - \frac {4 b c d^{2} e^{4} x^{3} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1}}{25} - \frac {8 b c e^{4} x \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1}}{75} + \frac {b d^{4} e^{4} x^{5} \operatorname {acosh}{\left (c + d x \right )}}{5} - \frac {b d^{3} e^{4} x^{4} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1}}{25} - \frac {4 b d e^{4} x^{2} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1}}{75} - \frac {8 b e^{4} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1}}{75 d} & \text {for}\: d \neq 0 \\c^{4} e^{4} x \left (a + b \operatorname {acosh}{\left (c \right )}\right ) & \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)),x)

[Out]

Piecewise((a*c**4*e**4*x + 2*a*c**3*d*e**4*x**2 + 2*a*c**2*d**2*e**4*x**3 + a*c*d**3*e**4*x**4 + a*d**4*e**4*x
**5/5 + b*c**5*e**4*acosh(c + d*x)/(5*d) + b*c**4*e**4*x*acosh(c + d*x) - b*c**4*e**4*sqrt(c**2 + 2*c*d*x + d*
*2*x**2 - 1)/(25*d) + 2*b*c**3*d*e**4*x**2*acosh(c + d*x) - 4*b*c**3*e**4*x*sqrt(c**2 + 2*c*d*x + d**2*x**2 -
1)/25 + 2*b*c**2*d**2*e**4*x**3*acosh(c + d*x) - 6*b*c**2*d*e**4*x**2*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)/25
- 4*b*c**2*e**4*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)/(75*d) + b*c*d**3*e**4*x**4*acosh(c + d*x) - 4*b*c*d**2*e
**4*x**3*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)/25 - 8*b*c*e**4*x*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)/75 + b*d*
*4*e**4*x**5*acosh(c + d*x)/5 - b*d**3*e**4*x**4*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)/25 - 4*b*d*e**4*x**2*sqr
t(c**2 + 2*c*d*x + d**2*x**2 - 1)/75 - 8*b*e**4*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)/(75*d), Ne(d, 0)), (c**4*
e**4*x*(a + b*acosh(c)), True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 846 vs. \(2 (115) = 230\).
time = 1.06, size = 846, normalized size = 6.27 \begin {gather*} \frac {1}{5} \, a d^{4} e^{4} x^{5} + a c d^{3} e^{4} x^{4} + 2 \, a c^{2} d^{2} e^{4} x^{3} + 2 \, a c^{3} d e^{4} x^{2} - {\left (d {\left (\frac {c \log \left ({\left | -c d - {\left (x {\left | d \right |} - \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )} {\left | d \right |} \right |}\right )}{d {\left | d \right |}} + \frac {\sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}}{d^{2}}\right )} - x \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )\right )} b c^{4} e^{4} + {\left (2 \, x^{2} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) - {\left (\sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} {\left (\frac {x}{d^{2}} - \frac {3 \, c}{d^{3}}\right )} - \frac {{\left (2 \, c^{2} + 1\right )} \log \left ({\left | -c d - {\left (x {\left | d \right |} - \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )} {\left | d \right |} \right |}\right )}{d^{2} {\left | d \right |}}\right )} d\right )} b c^{3} d e^{4} + \frac {1}{3} \, {\left (6 \, x^{3} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) - {\left (\sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} {\left (x {\left (\frac {2 \, x}{d^{2}} - \frac {5 \, c}{d^{3}}\right )} + \frac {11 \, c^{2} d + 4 \, d}{d^{5}}\right )} + \frac {3 \, {\left (2 \, c^{3} + 3 \, c\right )} \log \left ({\left | -c d - {\left (x {\left | d \right |} - \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )} {\left | d \right |} \right |}\right )}{d^{3} {\left | d \right |}}\right )} d\right )} b c^{2} d^{2} e^{4} + \frac {1}{24} \, {\left (24 \, x^{4} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) - {\left (\sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} {\left ({\left (2 \, x {\left (\frac {3 \, x}{d^{2}} - \frac {7 \, c}{d^{3}}\right )} + \frac {26 \, c^{2} d^{3} + 9 \, d^{3}}{d^{7}}\right )} x - \frac {5 \, {\left (10 \, c^{3} d^{2} + 11 \, c d^{2}\right )}}{d^{7}}\right )} - \frac {3 \, {\left (8 \, c^{4} + 24 \, c^{2} + 3\right )} \log \left ({\left | -c d - {\left (x {\left | d \right |} - \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )} {\left | d \right |} \right |}\right )}{d^{4} {\left | d \right |}}\right )} d\right )} b c d^{3} e^{4} + \frac {1}{600} \, {\left (120 \, x^{5} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) - {\left (\sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} {\left ({\left (2 \, {\left (3 \, x {\left (\frac {4 \, x}{d^{2}} - \frac {9 \, c}{d^{3}}\right )} + \frac {47 \, c^{2} d^{5} + 16 \, d^{5}}{d^{9}}\right )} x - \frac {7 \, {\left (22 \, c^{3} d^{4} + 23 \, c d^{4}\right )}}{d^{9}}\right )} x + \frac {274 \, c^{4} d^{3} + 607 \, c^{2} d^{3} + 64 \, d^{3}}{d^{9}}\right )} + \frac {15 \, {\left (8 \, c^{5} + 40 \, c^{3} + 15 \, c\right )} \log \left ({\left | -c d - {\left (x {\left | d \right |} - \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )} {\left | d \right |} \right |}\right )}{d^{5} {\left | d \right |}}\right )} d\right )} b d^{4} e^{4} + a c^{4} e^{4} x \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)),x, algorithm="giac")

[Out]

1/5*a*d^4*e^4*x^5 + a*c*d^3*e^4*x^4 + 2*a*c^2*d^2*e^4*x^3 + 2*a*c^3*d*e^4*x^2 - (d*(c*log(abs(-c*d - (x*abs(d)
- sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1))*abs(d)))/(d*abs(d)) + sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)/d^2) - x*log(d*x
+ c + sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)))*b*c^4*e^4 + (2*x^2*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1))
- (sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*(x/d^2 - 3*c/d^3) - (2*c^2 + 1)*log(abs(-c*d - (x*abs(d) - sqrt(d^2*x^2
+ 2*c*d*x + c^2 - 1))*abs(d)))/(d^2*abs(d)))*d)*b*c^3*d*e^4 + 1/3*(6*x^3*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x
+ c^2 - 1)) - (sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*(x*(2*x/d^2 - 5*c/d^3) + (11*c^2*d + 4*d)/d^5) + 3*(2*c^3 + 3
*c)*log(abs(-c*d - (x*abs(d) - sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1))*abs(d)))/(d^3*abs(d)))*d)*b*c^2*d^2*e^4 + 1/
24*(24*x^4*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)) - (sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*((2*x*(3*x/d^
2 - 7*c/d^3) + (26*c^2*d^3 + 9*d^3)/d^7)*x - 5*(10*c^3*d^2 + 11*c*d^2)/d^7) - 3*(8*c^4 + 24*c^2 + 3)*log(abs(-
c*d - (x*abs(d) - sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1))*abs(d)))/(d^4*abs(d)))*d)*b*c*d^3*e^4 + 1/600*(120*x^5*lo
g(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)) - (sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*((2*(3*x*(4*x/d^2 - 9*c/d^
3) + (47*c^2*d^5 + 16*d^5)/d^9)*x - 7*(22*c^3*d^4 + 23*c*d^4)/d^9)*x + (274*c^4*d^3 + 607*c^2*d^3 + 64*d^3)/d^
9) + 15*(8*c^5 + 40*c^3 + 15*c)*log(abs(-c*d - (x*abs(d) - sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1))*abs(d)))/(d^5*ab
s(d)))*d)*b*d^4*e^4 + a*c^4*e^4*x

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (c\,e+d\,e\,x\right )}^4\,\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right ) \,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)),x)

[Out]

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

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