3.8 \(\int (d+e x)^3 \cosh ^{-1}(c x)^2 \, dx\)

Optimal. Leaf size=334 \[ -\frac {3 e^3 \cosh ^{-1}(c x)^2}{32 c^4}-\frac {4 d e^2 \sqrt {c x-1} \sqrt {c x+1} \cosh ^{-1}(c x)}{3 c^3}-\frac {3 e^3 x \sqrt {c x-1} \sqrt {c x+1} \cosh ^{-1}(c x)}{16 c^3}-\frac {3 d^2 e \cosh ^{-1}(c x)^2}{4 c^2}+\frac {4 d e^2 x}{3 c^2}+\frac {3 e^3 x^2}{32 c^2}-\frac {d^4 \cosh ^{-1}(c x)^2}{4 e}-\frac {2 d^3 \sqrt {c x-1} \sqrt {c x+1} \cosh ^{-1}(c x)}{c}-\frac {3 d^2 e x \sqrt {c x-1} \sqrt {c x+1} \cosh ^{-1}(c x)}{2 c}-\frac {2 d e^2 x^2 \sqrt {c x-1} \sqrt {c x+1} \cosh ^{-1}(c x)}{3 c}+\frac {\cosh ^{-1}(c x)^2 (d+e x)^4}{4 e}-\frac {e^3 x^3 \sqrt {c x-1} \sqrt {c x+1} \cosh ^{-1}(c x)}{8 c}+2 d^3 x+\frac {3}{4} d^2 e x^2+\frac {2}{9} d e^2 x^3+\frac {e^3 x^4}{32} \]

[Out]

2*d^3*x+4/3*d*e^2*x/c^2+3/4*d^2*e*x^2+3/32*e^3*x^2/c^2+2/9*d*e^2*x^3+1/32*e^3*x^4-1/4*d^4*arccosh(c*x)^2/e-3/4
*d^2*e*arccosh(c*x)^2/c^2-3/32*e^3*arccosh(c*x)^2/c^4+1/4*(e*x+d)^4*arccosh(c*x)^2/e-2*d^3*arccosh(c*x)*(c*x-1
)^(1/2)*(c*x+1)^(1/2)/c-4/3*d*e^2*arccosh(c*x)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c^3-3/2*d^2*e*x*arccosh(c*x)*(c*x-1
)^(1/2)*(c*x+1)^(1/2)/c-3/16*e^3*x*arccosh(c*x)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c^3-2/3*d*e^2*x^2*arccosh(c*x)*(c*
x-1)^(1/2)*(c*x+1)^(1/2)/c-1/8*e^3*x^3*arccosh(c*x)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c

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Rubi [A]  time = 1.46, antiderivative size = 334, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 7, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5802, 5822, 5676, 5718, 8, 5759, 30} \[ -\frac {3 d^2 e \cosh ^{-1}(c x)^2}{4 c^2}+\frac {4 d e^2 x}{3 c^2}-\frac {4 d e^2 \sqrt {c x-1} \sqrt {c x+1} \cosh ^{-1}(c x)}{3 c^3}+\frac {3 e^3 x^2}{32 c^2}-\frac {3 e^3 \cosh ^{-1}(c x)^2}{32 c^4}-\frac {3 e^3 x \sqrt {c x-1} \sqrt {c x+1} \cosh ^{-1}(c x)}{16 c^3}-\frac {d^4 \cosh ^{-1}(c x)^2}{4 e}-\frac {3 d^2 e x \sqrt {c x-1} \sqrt {c x+1} \cosh ^{-1}(c x)}{2 c}-\frac {2 d^3 \sqrt {c x-1} \sqrt {c x+1} \cosh ^{-1}(c x)}{c}-\frac {2 d e^2 x^2 \sqrt {c x-1} \sqrt {c x+1} \cosh ^{-1}(c x)}{3 c}+\frac {\cosh ^{-1}(c x)^2 (d+e x)^4}{4 e}-\frac {e^3 x^3 \sqrt {c x-1} \sqrt {c x+1} \cosh ^{-1}(c x)}{8 c}+\frac {3}{4} d^2 e x^2+2 d^3 x+\frac {2}{9} d e^2 x^3+\frac {e^3 x^4}{32} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^3*ArcCosh[c*x]^2,x]

[Out]

2*d^3*x + (4*d*e^2*x)/(3*c^2) + (3*d^2*e*x^2)/4 + (3*e^3*x^2)/(32*c^2) + (2*d*e^2*x^3)/9 + (e^3*x^4)/32 - (2*d
^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*ArcCosh[c*x])/c - (4*d*e^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*ArcCosh[c*x])/(3*c^3)
- (3*d^2*e*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*ArcCosh[c*x])/(2*c) - (3*e^3*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*ArcCosh[
c*x])/(16*c^3) - (2*d*e^2*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*ArcCosh[c*x])/(3*c) - (e^3*x^3*Sqrt[-1 + c*x]*Sqrt[
1 + c*x]*ArcCosh[c*x])/(8*c) - (d^4*ArcCosh[c*x]^2)/(4*e) - (3*d^2*e*ArcCosh[c*x]^2)/(4*c^2) - (3*e^3*ArcCosh[
c*x]^2)/(32*c^4) + ((d + e*x)^4*ArcCosh[c*x]^2)/(4*e)

Rule 8

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

Rule 30

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

Rule 5676

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)]), x_Symbol]
:> Simp[(a + b*ArcCosh[c*x])^(n + 1)/(b*c*Sqrt[-(d1*d2)]*(n + 1)), x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n},
x] && EqQ[e1, c*d1] && EqQ[e2, -(c*d2)] && GtQ[d1, 0] && LtQ[d2, 0] && NeQ[n, -1]

Rule 5718

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d1_) + (e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(x_))^(p_.), x_
Symbol] :> 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*(-(d1*d2))^IntPart[p]*(d1 + e1*x)^FracPart[p]*(d2 + e2*x)^FracPart[p])/(2*c*(p + 1)*(1 + c*x)^FracPart[p]
*(-1 + c*x)^FracPart[p]), Int[(-1 + c^2*x^2)^(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, 0] && EqQ[e2 + c*d2, 0] && GtQ[n, 0] && NeQ[p, -1] && IntegerQ[p + 1
/2]

Rule 5759

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

Rule 5802

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Simp[((d + e*x)^(m + 1)
*(a + b*ArcCosh[c*x])^n)/(e*(m + 1)), x] - Dist[(b*c*n)/(e*(m + 1)), Int[((d + e*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, e, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rule 5822

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d1_) + (e1_.)*(x_))^(p_)*((d2_) + (e2_.)*(x_))^(p_)*((f_) + (g
_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(d1 + e1*x)^p*(d2 + e2*x)^p*(a + b*ArcCosh[c*x])^n, (f + g*x
)^m, x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, f, g}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && IGtQ[m,
0] && IntegerQ[p + 1/2] && GtQ[d1, 0] && LtQ[d2, 0] && IGtQ[n, 0] && ((EqQ[n, 1] && GtQ[p, -1]) || GtQ[p, 0] |
| EqQ[m, 1] || (EqQ[m, 2] && LtQ[p, -2]))

Rubi steps

\begin {align*} \int (d+e x)^3 \cosh ^{-1}(c x)^2 \, dx &=\frac {(d+e x)^4 \cosh ^{-1}(c x)^2}{4 e}-\frac {c \int \frac {(d+e x)^4 \cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{2 e}\\ &=\frac {(d+e x)^4 \cosh ^{-1}(c x)^2}{4 e}-\frac {c \int \left (\frac {d^4 \cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {4 d^3 e x \cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {6 d^2 e^2 x^2 \cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {4 d e^3 x^3 \cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {e^4 x^4 \cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}}\right ) \, dx}{2 e}\\ &=\frac {(d+e x)^4 \cosh ^{-1}(c x)^2}{4 e}-\left (2 c d^3\right ) \int \frac {x \cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx-\frac {\left (c d^4\right ) \int \frac {\cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{2 e}-\left (3 c d^2 e\right ) \int \frac {x^2 \cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx-\left (2 c d e^2\right ) \int \frac {x^3 \cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx-\frac {1}{2} \left (c e^3\right ) \int \frac {x^4 \cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=-\frac {2 d^3 \sqrt {-1+c x} \sqrt {1+c x} \cosh ^{-1}(c x)}{c}-\frac {3 d^2 e x \sqrt {-1+c x} \sqrt {1+c x} \cosh ^{-1}(c x)}{2 c}-\frac {2 d e^2 x^2 \sqrt {-1+c x} \sqrt {1+c x} \cosh ^{-1}(c x)}{3 c}-\frac {e^3 x^3 \sqrt {-1+c x} \sqrt {1+c x} \cosh ^{-1}(c x)}{8 c}-\frac {d^4 \cosh ^{-1}(c x)^2}{4 e}+\frac {(d+e x)^4 \cosh ^{-1}(c x)^2}{4 e}+\left (2 d^3\right ) \int 1 \, dx+\frac {1}{2} \left (3 d^2 e\right ) \int x \, dx-\frac {\left (3 d^2 e\right ) \int \frac {\cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{2 c}+\frac {1}{3} \left (2 d e^2\right ) \int x^2 \, dx-\frac {\left (4 d e^2\right ) \int \frac {x \cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{3 c}+\frac {1}{8} e^3 \int x^3 \, dx-\frac {\left (3 e^3\right ) \int \frac {x^2 \cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{8 c}\\ &=2 d^3 x+\frac {3}{4} d^2 e x^2+\frac {2}{9} d e^2 x^3+\frac {e^3 x^4}{32}-\frac {2 d^3 \sqrt {-1+c x} \sqrt {1+c x} \cosh ^{-1}(c x)}{c}-\frac {4 d e^2 \sqrt {-1+c x} \sqrt {1+c x} \cosh ^{-1}(c x)}{3 c^3}-\frac {3 d^2 e x \sqrt {-1+c x} \sqrt {1+c x} \cosh ^{-1}(c x)}{2 c}-\frac {3 e^3 x \sqrt {-1+c x} \sqrt {1+c x} \cosh ^{-1}(c x)}{16 c^3}-\frac {2 d e^2 x^2 \sqrt {-1+c x} \sqrt {1+c x} \cosh ^{-1}(c x)}{3 c}-\frac {e^3 x^3 \sqrt {-1+c x} \sqrt {1+c x} \cosh ^{-1}(c x)}{8 c}-\frac {d^4 \cosh ^{-1}(c x)^2}{4 e}-\frac {3 d^2 e \cosh ^{-1}(c x)^2}{4 c^2}+\frac {(d+e x)^4 \cosh ^{-1}(c x)^2}{4 e}+\frac {\left (4 d e^2\right ) \int 1 \, dx}{3 c^2}-\frac {\left (3 e^3\right ) \int \frac {\cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{16 c^3}+\frac {\left (3 e^3\right ) \int x \, dx}{16 c^2}\\ &=2 d^3 x+\frac {4 d e^2 x}{3 c^2}+\frac {3}{4} d^2 e x^2+\frac {3 e^3 x^2}{32 c^2}+\frac {2}{9} d e^2 x^3+\frac {e^3 x^4}{32}-\frac {2 d^3 \sqrt {-1+c x} \sqrt {1+c x} \cosh ^{-1}(c x)}{c}-\frac {4 d e^2 \sqrt {-1+c x} \sqrt {1+c x} \cosh ^{-1}(c x)}{3 c^3}-\frac {3 d^2 e x \sqrt {-1+c x} \sqrt {1+c x} \cosh ^{-1}(c x)}{2 c}-\frac {3 e^3 x \sqrt {-1+c x} \sqrt {1+c x} \cosh ^{-1}(c x)}{16 c^3}-\frac {2 d e^2 x^2 \sqrt {-1+c x} \sqrt {1+c x} \cosh ^{-1}(c x)}{3 c}-\frac {e^3 x^3 \sqrt {-1+c x} \sqrt {1+c x} \cosh ^{-1}(c x)}{8 c}-\frac {d^4 \cosh ^{-1}(c x)^2}{4 e}-\frac {3 d^2 e \cosh ^{-1}(c x)^2}{4 c^2}-\frac {3 e^3 \cosh ^{-1}(c x)^2}{32 c^4}+\frac {(d+e x)^4 \cosh ^{-1}(c x)^2}{4 e}\\ \end {align*}

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Mathematica [A]  time = 0.27, size = 191, normalized size = 0.57 \[ \frac {c^2 x \left (c^2 \left (576 d^3+216 d^2 e x+64 d e^2 x^2+9 e^3 x^3\right )+3 e^2 (128 d+9 e x)\right )-6 c \sqrt {c x-1} \sqrt {c x+1} \cosh ^{-1}(c x) \left (c^2 \left (96 d^3+72 d^2 e x+32 d e^2 x^2+6 e^3 x^3\right )+e^2 (64 d+9 e x)\right )+9 \cosh ^{-1}(c x)^2 \left (8 c^4 x \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )-24 c^2 d^2 e-3 e^3\right )}{288 c^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^3*ArcCosh[c*x]^2,x]

[Out]

(c^2*x*(3*e^2*(128*d + 9*e*x) + c^2*(576*d^3 + 216*d^2*e*x + 64*d*e^2*x^2 + 9*e^3*x^3)) - 6*c*Sqrt[-1 + c*x]*S
qrt[1 + c*x]*(e^2*(64*d + 9*e*x) + c^2*(96*d^3 + 72*d^2*e*x + 32*d*e^2*x^2 + 6*e^3*x^3))*ArcCosh[c*x] + 9*(-24
*c^2*d^2*e - 3*e^3 + 8*c^4*x*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3))*ArcCosh[c*x]^2)/(288*c^4)

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fricas [A]  time = 0.74, size = 237, normalized size = 0.71 \[ \frac {9 \, c^{4} e^{3} x^{4} + 64 \, c^{4} d e^{2} x^{3} + 27 \, {\left (8 \, c^{4} d^{2} e + c^{2} e^{3}\right )} x^{2} + 9 \, {\left (8 \, c^{4} e^{3} x^{4} + 32 \, c^{4} d e^{2} x^{3} + 48 \, c^{4} d^{2} e x^{2} + 32 \, c^{4} d^{3} x - 24 \, c^{2} d^{2} e - 3 \, e^{3}\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right )^{2} - 6 \, {\left (6 \, c^{3} e^{3} x^{3} + 32 \, c^{3} d e^{2} x^{2} + 96 \, c^{3} d^{3} + 64 \, c d e^{2} + 9 \, {\left (8 \, c^{3} d^{2} e + c e^{3}\right )} x\right )} \sqrt {c^{2} x^{2} - 1} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) + 192 \, {\left (3 \, c^{4} d^{3} + 2 \, c^{2} d e^{2}\right )} x}{288 \, c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*arccosh(c*x)^2,x, algorithm="fricas")

[Out]

1/288*(9*c^4*e^3*x^4 + 64*c^4*d*e^2*x^3 + 27*(8*c^4*d^2*e + c^2*e^3)*x^2 + 9*(8*c^4*e^3*x^4 + 32*c^4*d*e^2*x^3
+ 48*c^4*d^2*e*x^2 + 32*c^4*d^3*x - 24*c^2*d^2*e - 3*e^3)*log(c*x + sqrt(c^2*x^2 - 1))^2 - 6*(6*c^3*e^3*x^3 +
32*c^3*d*e^2*x^2 + 96*c^3*d^3 + 64*c*d*e^2 + 9*(8*c^3*d^2*e + c*e^3)*x)*sqrt(c^2*x^2 - 1)*log(c*x + sqrt(c^2*
x^2 - 1)) + 192*(3*c^4*d^3 + 2*c^2*d*e^2)*x)/c^4

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*arccosh(c*x)^2,x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:sym2
poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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maple [A]  time = 0.13, size = 329, normalized size = 0.99 \[ \frac {72 \mathrm {arccosh}\left (c x \right )^{2} c^{4} x^{4} e^{3}+288 \mathrm {arccosh}\left (c x \right )^{2} c^{4} x^{3} d \,e^{2}+432 \mathrm {arccosh}\left (c x \right )^{2} c^{4} x^{2} d^{2} e +288 \mathrm {arccosh}\left (c x \right )^{2} c^{4} x \,d^{3}-36 \,\mathrm {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{3} e^{3}-192 \,\mathrm {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{2} d \,e^{2}-432 \,\mathrm {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x \,d^{2} e -576 \,\mathrm {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} d^{3}-216 \mathrm {arccosh}\left (c x \right )^{2} c^{2} d^{2} e -54 \,\mathrm {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}\, c x \,e^{3}-384 \,\mathrm {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}\, c d \,e^{2}+9 c^{4} e^{3} x^{4}+64 c^{4} d \,e^{2} x^{3}+216 c^{4} d^{2} e \,x^{2}+576 x \,c^{4} d^{3}-27 \mathrm {arccosh}\left (c x \right )^{2} e^{3}+27 c^{2} x^{2} e^{3}+384 x \,c^{2} d \,e^{2}}{288 c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^3*arccosh(c*x)^2,x)

[Out]

1/288/c^4*(72*arccosh(c*x)^2*c^4*x^4*e^3+288*arccosh(c*x)^2*c^4*x^3*d*e^2+432*arccosh(c*x)^2*c^4*x^2*d^2*e+288
*arccosh(c*x)^2*c^4*x*d^3-36*arccosh(c*x)*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^3*x^3*e^3-192*arccosh(c*x)*(c*x-1)^(1/
2)*(c*x+1)^(1/2)*c^3*x^2*d*e^2-432*arccosh(c*x)*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^3*x*d^2*e-576*arccosh(c*x)*(c*x-
1)^(1/2)*(c*x+1)^(1/2)*c^3*d^3-216*arccosh(c*x)^2*c^2*d^2*e-54*arccosh(c*x)*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x*e^
3-384*arccosh(c*x)*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c*d*e^2+9*c^4*e^3*x^4+64*c^4*d*e^2*x^3+216*c^4*d^2*e*x^2+576*x*
c^4*d^3-27*arccosh(c*x)^2*e^3+27*c^2*x^2*e^3+384*x*c^2*d*e^2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{4} \, {\left (e^{3} x^{4} + 4 \, d e^{2} x^{3} + 6 \, d^{2} e x^{2} + 4 \, d^{3} x\right )} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )^{2} - \int \frac {{\left (c^{3} e^{3} x^{6} + 4 \, c^{3} d e^{2} x^{5} - 6 \, c d^{2} e x^{2} - 4 \, c d^{3} x + {\left (6 \, c^{3} d^{2} e - c e^{3}\right )} x^{4} + 4 \, {\left (c^{3} d^{3} - c d e^{2}\right )} x^{3} + {\left (c^{2} e^{3} x^{5} + 4 \, c^{2} d e^{2} x^{4} + 6 \, c^{2} d^{2} e x^{3} + 4 \, c^{2} d^{3} x^{2}\right )} \sqrt {c x + 1} \sqrt {c x - 1}\right )} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )}{2 \, {\left (c^{3} x^{3} + {\left (c^{2} x^{2} - 1\right )} \sqrt {c x + 1} \sqrt {c x - 1} - c x\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*arccosh(c*x)^2,x, algorithm="maxima")

[Out]

1/4*(e^3*x^4 + 4*d*e^2*x^3 + 6*d^2*e*x^2 + 4*d^3*x)*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))^2 - integrate(1/2*(
c^3*e^3*x^6 + 4*c^3*d*e^2*x^5 - 6*c*d^2*e*x^2 - 4*c*d^3*x + (6*c^3*d^2*e - c*e^3)*x^4 + 4*(c^3*d^3 - c*d*e^2)*
x^3 + (c^2*e^3*x^5 + 4*c^2*d*e^2*x^4 + 6*c^2*d^2*e*x^3 + 4*c^2*d^3*x^2)*sqrt(c*x + 1)*sqrt(c*x - 1))*log(c*x +
sqrt(c*x + 1)*sqrt(c*x - 1))/(c^3*x^3 + (c^2*x^2 - 1)*sqrt(c*x + 1)*sqrt(c*x - 1) - c*x), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int {\mathrm {acosh}\left (c\,x\right )}^2\,{\left (d+e\,x\right )}^3 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(acosh(c*x)^2*(d + e*x)^3,x)

[Out]

int(acosh(c*x)^2*(d + e*x)^3, x)

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sympy [A]  time = 2.46, size = 371, normalized size = 1.11 \[ \begin {cases} d^{3} x \operatorname {acosh}^{2}{\left (c x \right )} + 2 d^{3} x + \frac {3 d^{2} e x^{2} \operatorname {acosh}^{2}{\left (c x \right )}}{2} + \frac {3 d^{2} e x^{2}}{4} + d e^{2} x^{3} \operatorname {acosh}^{2}{\left (c x \right )} + \frac {2 d e^{2} x^{3}}{9} + \frac {e^{3} x^{4} \operatorname {acosh}^{2}{\left (c x \right )}}{4} + \frac {e^{3} x^{4}}{32} - \frac {2 d^{3} \sqrt {c^{2} x^{2} - 1} \operatorname {acosh}{\left (c x \right )}}{c} - \frac {3 d^{2} e x \sqrt {c^{2} x^{2} - 1} \operatorname {acosh}{\left (c x \right )}}{2 c} - \frac {2 d e^{2} x^{2} \sqrt {c^{2} x^{2} - 1} \operatorname {acosh}{\left (c x \right )}}{3 c} - \frac {e^{3} x^{3} \sqrt {c^{2} x^{2} - 1} \operatorname {acosh}{\left (c x \right )}}{8 c} - \frac {3 d^{2} e \operatorname {acosh}^{2}{\left (c x \right )}}{4 c^{2}} + \frac {4 d e^{2} x}{3 c^{2}} + \frac {3 e^{3} x^{2}}{32 c^{2}} - \frac {4 d e^{2} \sqrt {c^{2} x^{2} - 1} \operatorname {acosh}{\left (c x \right )}}{3 c^{3}} - \frac {3 e^{3} x \sqrt {c^{2} x^{2} - 1} \operatorname {acosh}{\left (c x \right )}}{16 c^{3}} - \frac {3 e^{3} \operatorname {acosh}^{2}{\left (c x \right )}}{32 c^{4}} & \text {for}\: c \neq 0 \\- \frac {\pi ^{2} \left (d^{3} x + \frac {3 d^{2} e x^{2}}{2} + d e^{2} x^{3} + \frac {e^{3} x^{4}}{4}\right )}{4} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**3*acosh(c*x)**2,x)

[Out]

Piecewise((d**3*x*acosh(c*x)**2 + 2*d**3*x + 3*d**2*e*x**2*acosh(c*x)**2/2 + 3*d**2*e*x**2/4 + d*e**2*x**3*aco
sh(c*x)**2 + 2*d*e**2*x**3/9 + e**3*x**4*acosh(c*x)**2/4 + e**3*x**4/32 - 2*d**3*sqrt(c**2*x**2 - 1)*acosh(c*x
)/c - 3*d**2*e*x*sqrt(c**2*x**2 - 1)*acosh(c*x)/(2*c) - 2*d*e**2*x**2*sqrt(c**2*x**2 - 1)*acosh(c*x)/(3*c) - e
**3*x**3*sqrt(c**2*x**2 - 1)*acosh(c*x)/(8*c) - 3*d**2*e*acosh(c*x)**2/(4*c**2) + 4*d*e**2*x/(3*c**2) + 3*e**3
*x**2/(32*c**2) - 4*d*e**2*sqrt(c**2*x**2 - 1)*acosh(c*x)/(3*c**3) - 3*e**3*x*sqrt(c**2*x**2 - 1)*acosh(c*x)/(
16*c**3) - 3*e**3*acosh(c*x)**2/(32*c**4), Ne(c, 0)), (-pi**2*(d**3*x + 3*d**2*e*x**2/2 + d*e**2*x**3 + e**3*x
**4/4)/4, True))

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