3.94 \(\int (c e+d e x)^3 (a+b \cosh ^{-1}(c+d x)) \, dx\)

Optimal. Leaf size=119 \[ \frac {e^3 (c+d x)^4 \left (a+b \cosh ^{-1}(c+d x)\right )}{4 d}-\frac {b e^3 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^3}{16 d}-\frac {3 b e^3 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)}{32 d}-\frac {3 b e^3 \cosh ^{-1}(c+d x)}{32 d} \]

[Out]

-3/32*b*e^3*arccosh(d*x+c)/d+1/4*e^3*(d*x+c)^4*(a+b*arccosh(d*x+c))/d-3/32*b*e^3*(d*x+c)*(d*x+c-1)^(1/2)*(d*x+
c+1)^(1/2)/d-1/16*b*e^3*(d*x+c)^3*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)/d

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Rubi [A]  time = 0.07, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {5866, 12, 5662, 100, 90, 52} \[ \frac {e^3 (c+d x)^4 \left (a+b \cosh ^{-1}(c+d x)\right )}{4 d}-\frac {b e^3 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^3}{16 d}-\frac {3 b e^3 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)}{32 d}-\frac {3 b e^3 \cosh ^{-1}(c+d x)}{32 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*b*e^3*Sqrt[-1 + c + d*x]*(c + d*x)*Sqrt[1 + c + d*x])/(32*d) - (b*e^3*Sqrt[-1 + c + d*x]*(c + d*x)^3*Sqrt[
1 + c + d*x])/(16*d) - (3*b*e^3*ArcCosh[c + d*x])/(32*d) + (e^3*(c + d*x)^4*(a + b*ArcCosh[c + d*x]))/(4*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 52

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ArcCosh[(b*x)/a]/b, x] /; FreeQ[{a,
b, c, d}, x] && EqQ[a + c, 0] && EqQ[b - d, 0] && GtQ[a, 0]

Rule 90

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a + b*
x)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 3)), x] + Dist[1/(d*f*(n + p + 3)), Int[(c + d*x)^n*(e +
f*x)^p*Simp[a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(n + p + 4) - b*(d*e*(
n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 3, 0]

Rule 100

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 5662

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))/(Sqr
t[-1 + c*x]*Sqrt[1 + c*x]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rule 5866

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)^3 \left (a+b \cosh ^{-1}(c+d x)\right ) \, dx &=\frac {\operatorname {Subst}\left (\int e^3 x^3 \left (a+b \cosh ^{-1}(x)\right ) \, dx,x,c+d x\right )}{d}\\ &=\frac {e^3 \operatorname {Subst}\left (\int x^3 \left (a+b \cosh ^{-1}(x)\right ) \, dx,x,c+d x\right )}{d}\\ &=\frac {e^3 (c+d x)^4 \left (a+b \cosh ^{-1}(c+d x)\right )}{4 d}-\frac {\left (b e^3\right ) \operatorname {Subst}\left (\int \frac {x^4}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{4 d}\\ &=-\frac {b e^3 \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x}}{16 d}+\frac {e^3 (c+d x)^4 \left (a+b \cosh ^{-1}(c+d x)\right )}{4 d}-\frac {\left (b e^3\right ) \operatorname {Subst}\left (\int \frac {3 x^2}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{16 d}\\ &=-\frac {b e^3 \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x}}{16 d}+\frac {e^3 (c+d x)^4 \left (a+b \cosh ^{-1}(c+d x)\right )}{4 d}-\frac {\left (3 b e^3\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{16 d}\\ &=-\frac {3 b e^3 \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}}{32 d}-\frac {b e^3 \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x}}{16 d}+\frac {e^3 (c+d x)^4 \left (a+b \cosh ^{-1}(c+d x)\right )}{4 d}-\frac {\left (3 b e^3\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{32 d}\\ &=-\frac {3 b e^3 \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}}{32 d}-\frac {b e^3 \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x}}{16 d}-\frac {3 b e^3 \cosh ^{-1}(c+d x)}{32 d}+\frac {e^3 (c+d x)^4 \left (a+b \cosh ^{-1}(c+d x)\right )}{4 d}\\ \end {align*}

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Mathematica [A]  time = 0.13, size = 115, normalized size = 0.97 \[ \frac {e^3 \left ((c+d x)^4 \left (a+b \cosh ^{-1}(c+d x)\right )-\frac {1}{4} b \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^3-\frac {3}{8} b \left (\sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)+2 \tanh ^{-1}\left (\sqrt {\frac {c+d x-1}{c+d x+1}}\right )\right )\right )}{4 d} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(e^3*(-1/4*(b*Sqrt[-1 + c + d*x]*(c + d*x)^3*Sqrt[1 + c + d*x]) + (c + d*x)^4*(a + b*ArcCosh[c + d*x]) - (3*b*
(Sqrt[-1 + c + d*x]*(c + d*x)*Sqrt[1 + c + d*x] + 2*ArcTanh[Sqrt[(-1 + c + d*x)/(1 + c + d*x)]]))/8))/(4*d)

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fricas [B]  time = 0.87, size = 226, normalized size = 1.90 \[ \frac {8 \, a d^{4} e^{3} x^{4} + 32 \, a c d^{3} e^{3} x^{3} + 48 \, a c^{2} d^{2} e^{3} x^{2} + 32 \, a c^{3} d e^{3} x + {\left (8 \, b d^{4} e^{3} x^{4} + 32 \, b c d^{3} e^{3} x^{3} + 48 \, b c^{2} d^{2} e^{3} x^{2} + 32 \, b c^{3} d e^{3} x + {\left (8 \, b c^{4} - 3 \, b\right )} e^{3}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) - {\left (2 \, b d^{3} e^{3} x^{3} + 6 \, b c d^{2} e^{3} x^{2} + 3 \, {\left (2 \, b c^{2} + b\right )} d e^{3} x + {\left (2 \, b c^{3} + 3 \, b c\right )} e^{3}\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}}{32 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/32*(8*a*d^4*e^3*x^4 + 32*a*c*d^3*e^3*x^3 + 48*a*c^2*d^2*e^3*x^2 + 32*a*c^3*d*e^3*x + (8*b*d^4*e^3*x^4 + 32*b
*c*d^3*e^3*x^3 + 48*b*c^2*d^2*e^3*x^2 + 32*b*c^3*d*e^3*x + (8*b*c^4 - 3*b)*e^3)*log(d*x + c + sqrt(d^2*x^2 + 2
*c*d*x + c^2 - 1)) - (2*b*d^3*e^3*x^3 + 6*b*c*d^2*e^3*x^2 + 3*(2*b*c^2 + b)*d*e^3*x + (2*b*c^3 + 3*b*c)*e^3)*s
qrt(d^2*x^2 + 2*c*d*x + c^2 - 1))/d

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giac [B]  time = 4.27, size = 598, normalized size = 5.03 \[ \frac {1}{96} \, {\left (24 \, a d^{3} x^{4} + 96 \, a c d^{2} x^{3} + 144 \, a c^{2} d x^{2} - 96 \, {\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^{3} + 72 \, {\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^{2} d + 16 \, {\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 d^{2} + {\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 d^{3} + 96 \, a c^{3} x\right )} e^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/96*(24*a*d^3*x^4 + 96*a*c*d^2*x^3 + 144*a*c^2*d*x^2 - 96*(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^3 + 72*(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^2*d + 16*(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*d^2 + (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*d^3 + 96*a*c^3*x)*e^3

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maple [B]  time = 0.01, size = 359, normalized size = 3.02 \[ \frac {d^{3} x^{4} a \,e^{3}}{4}+d^{2} x^{3} a c \,e^{3}+\frac {3 d \,x^{2} a \,c^{2} e^{3}}{2}+x a \,c^{3} e^{3}+\frac {a \,c^{4} e^{3}}{4 d}+\frac {d^{3} \mathrm {arccosh}\left (d x +c \right ) x^{4} b \,e^{3}}{4}+d^{2} \mathrm {arccosh}\left (d x +c \right ) x^{3} b c \,e^{3}+\frac {3 d \,\mathrm {arccosh}\left (d x +c \right ) x^{2} b \,c^{2} e^{3}}{2}+\mathrm {arccosh}\left (d x +c \right ) x b \,c^{3} e^{3}+\frac {\mathrm {arccosh}\left (d x +c \right ) b \,c^{4} e^{3}}{4 d}-\frac {d^{2} \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, x^{3} b \,e^{3}}{16}-\frac {3 d \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, x^{2} b c \,e^{3}}{16}-\frac {3 \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, x b \,c^{2} e^{3}}{16}-\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}\, b \,c^{3} e^{3}}{16 d}-\frac {3 \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, x b \,e^{3}}{32}-\frac {3 \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, b c \,e^{3}}{32 d}-\frac {3 e^{3} b \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \ln \left (d x +c +\sqrt {\left (d x +c \right )^{2}-1}\right )}{32 d \sqrt {\left (d x +c \right )^{2}-1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*e*x+c*e)^3*(a+b*arccosh(d*x+c)),x)

[Out]

1/4*d^3*x^4*a*e^3+d^2*x^3*a*c*e^3+3/2*d*x^2*a*c^2*e^3+x*a*c^3*e^3+1/4/d*a*c^4*e^3+1/4*d^3*arccosh(d*x+c)*x^4*b
*e^3+d^2*arccosh(d*x+c)*x^3*b*c*e^3+3/2*d*arccosh(d*x+c)*x^2*b*c^2*e^3+arccosh(d*x+c)*x*b*c^3*e^3+1/4/d*arccos
h(d*x+c)*b*c^4*e^3-1/16*d^2*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)*x^3*b*e^3-3/16*d*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)*x
^2*b*c*e^3-3/16*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)*x*b*c^2*e^3-1/16/d*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)*b*c^3*e^3-3
/32*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)*x*b*e^3-3/32/d*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)*b*c*e^3-3/32/d*e^3*b*(d*x+c
-1)^(1/2)*(d*x+c+1)^(1/2)/((d*x+c)^2-1)^(1/2)*ln(d*x+c+((d*x+c)^2-1)^(1/2))

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maxima [B]  time = 0.44, size = 797, normalized size = 6.70 \[ \frac {1}{4} \, a d^{3} e^{3} x^{4} + a c d^{2} e^{3} x^{3} + \frac {3}{2} \, a c^{2} d e^{3} x^{2} + \frac {3}{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^{2} d e^{3} + \frac {1}{6} \, {\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 d^{2} e^{3} + \frac {1}{96} \, {\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 d^{3} e^{3} + a c^{3} e^{3} x + \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^{3} e^{3}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)^3*(a+b*arccosh(d*x+c)),x, algorithm="maxima")

[Out]

1/4*a*d^3*e^3*x^4 + a*c*d^2*e^3*x^3 + 3/2*a*c^2*d*e^3*x^2 + 3/4*(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)*lo
g(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^2*d*e^3 + 1/6*(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*d^2*e^3 + 1/96*(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)*b*d^3*e^
3 + a*c^3*e^3*x + ((d*x + c)*arccosh(d*x + c) - sqrt((d*x + c)^2 - 1))*b*c^3*e^3/d

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (c\,e+d\,e\,x\right )}^3\,\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right ) \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 1.60, size = 394, normalized size = 3.31 \[ \begin {cases} a c^{3} e^{3} x + \frac {3 a c^{2} d e^{3} x^{2}}{2} + a c d^{2} e^{3} x^{3} + \frac {a d^{3} e^{3} x^{4}}{4} + \frac {b c^{4} e^{3} \operatorname {acosh}{\left (c + d x \right )}}{4 d} + b c^{3} e^{3} x \operatorname {acosh}{\left (c + d x \right )} - \frac {b c^{3} e^{3} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1}}{16 d} + \frac {3 b c^{2} d e^{3} x^{2} \operatorname {acosh}{\left (c + d x \right )}}{2} - \frac {3 b c^{2} e^{3} x \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1}}{16} + b c d^{2} e^{3} x^{3} \operatorname {acosh}{\left (c + d x \right )} - \frac {3 b c d e^{3} x^{2} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1}}{16} - \frac {3 b c e^{3} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1}}{32 d} + \frac {b d^{3} e^{3} x^{4} \operatorname {acosh}{\left (c + d x \right )}}{4} - \frac {b d^{2} e^{3} x^{3} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1}}{16} - \frac {3 b e^{3} x \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1}}{32} - \frac {3 b e^{3} \operatorname {acosh}{\left (c + d x \right )}}{32 d} & \text {for}\: d \neq 0 \\c^{3} e^{3} x \left (a + b \operatorname {acosh}{\relax (c )}\right ) & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)**3*(a+b*acosh(d*x+c)),x)

[Out]

Piecewise((a*c**3*e**3*x + 3*a*c**2*d*e**3*x**2/2 + a*c*d**2*e**3*x**3 + a*d**3*e**3*x**4/4 + b*c**4*e**3*acos
h(c + d*x)/(4*d) + b*c**3*e**3*x*acosh(c + d*x) - b*c**3*e**3*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)/(16*d) + 3*
b*c**2*d*e**3*x**2*acosh(c + d*x)/2 - 3*b*c**2*e**3*x*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)/16 + b*c*d**2*e**3*
x**3*acosh(c + d*x) - 3*b*c*d*e**3*x**2*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)/16 - 3*b*c*e**3*sqrt(c**2 + 2*c*d
*x + d**2*x**2 - 1)/(32*d) + b*d**3*e**3*x**4*acosh(c + d*x)/4 - b*d**2*e**3*x**3*sqrt(c**2 + 2*c*d*x + d**2*x
**2 - 1)/16 - 3*b*e**3*x*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)/32 - 3*b*e**3*acosh(c + d*x)/(32*d), Ne(d, 0)),
(c**3*e**3*x*(a + b*acosh(c)), True))

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