\(\int (d+e x)^3 (a+b \text {arccosh}(c x))^2 \, dx\) [21]

Optimal result

Integrand size = 18, antiderivative size = 398 \[ \int (d+e x)^3 (a+b \text {arccosh}(c x))^2 \, dx=2 b^2 d^3 x+\frac {4 b^2 d e^2 x}{3 c^2}+\frac {3}{4} b^2 d^2 e x^2+\frac {3 b^2 e^3 x^2}{32 c^2}+\frac {2}{9} b^2 d e^2 x^3+\frac {1}{32} b^2 e^3 x^4-\frac {2 b d^3 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{c}-\frac {4 b d e^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{3 c^3}-\frac {3 b d^2 e x \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{2 c}-\frac {3 b e^3 x \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{16 c^3}-\frac {2 b d e^2 x^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{3 c}-\frac {b e^3 x^3 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{8 c}-\frac {d^4 (a+b \text {arccosh}(c x))^2}{4 e}-\frac {3 d^2 e (a+b \text {arccosh}(c x))^2}{4 c^2}-\frac {3 e^3 (a+b \text {arccosh}(c x))^2}{32 c^4}+\frac {(d+e x)^4 (a+b \text {arccosh}(c x))^2}{4 e} \]

[Out]

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

Rubi [A] (verified)

Time = 1.16 (sec) , antiderivative size = 398, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {5963, 5975, 5893, 5915, 8, 5939, 30} \[ \int (d+e x)^3 (a+b \text {arccosh}(c x))^2 \, dx=-\frac {3 e^3 (a+b \text {arccosh}(c x))^2}{32 c^4}-\frac {4 b d e^2 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{3 c^3}-\frac {3 b e^3 x \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{16 c^3}-\frac {3 d^2 e (a+b \text {arccosh}(c x))^2}{4 c^2}-\frac {d^4 (a+b \text {arccosh}(c x))^2}{4 e}-\frac {2 b d^3 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{c}-\frac {3 b d^2 e x \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{2 c}-\frac {2 b d e^2 x^2 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{3 c}+\frac {(d+e x)^4 (a+b \text {arccosh}(c x))^2}{4 e}-\frac {b e^3 x^3 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{8 c}+\frac {4 b^2 d e^2 x}{3 c^2}+\frac {3 b^2 e^3 x^2}{32 c^2}+2 b^2 d^3 x+\frac {3}{4} b^2 d^2 e x^2+\frac {2}{9} b^2 d e^2 x^3+\frac {1}{32} b^2 e^3 x^4 \]

[In]

Int[(d + e*x)^3*(a + b*ArcCosh[c*x])^2,x]

[Out]

2*b^2*d^3*x + (4*b^2*d*e^2*x)/(3*c^2) + (3*b^2*d^2*e*x^2)/4 + (3*b^2*e^3*x^2)/(32*c^2) + (2*b^2*d*e^2*x^3)/9 +
(b^2*e^3*x^4)/32 - (2*b*d^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/c - (4*b*d*e^2*Sqrt[-1 + c*x]*
Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(3*c^3) - (3*b*d^2*e*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/
(2*c) - (3*b*e^3*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(16*c^3) - (2*b*d*e^2*x^2*Sqrt[-1 + c*x]
*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(3*c) - (b*e^3*x^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(8*
c) - (d^4*(a + b*ArcCosh[c*x])^2)/(4*e) - (3*d^2*e*(a + b*ArcCosh[c*x])^2)/(4*c^2) - (3*e^3*(a + b*ArcCosh[c*x
])^2)/(32*c^4) + ((d + e*x)^4*(a + b*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 5893

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)]), x_Symbol]
:> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 + c*x]/Sqrt[d1 + e1*x]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]]*(a + b*Arc
Cosh[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && NeQ[n
, -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 5963

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 5975

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

Mathematica [A] (verified)

Time = 0.40 (sec) , antiderivative size = 386, normalized size of antiderivative = 0.97 \[ \int (d+e x)^3 (a+b \text {arccosh}(c x))^2 \, dx=\frac {c \left (72 a^2 c^3 x \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )-6 a b \sqrt {-1+c x} \sqrt {1+c x} \left (e^2 (64 d+9 e x)+c^2 \left (96 d^3+72 d^2 e x+32 d e^2 x^2+6 e^3 x^3\right )\right )+b^2 c x \left (3 e^2 (128 d+9 e x)+c^2 \left (576 d^3+216 d^2 e x+64 d e^2 x^2+9 e^3 x^3\right )\right )\right )-6 b c \left (-24 a c^3 x \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )+b \sqrt {-1+c x} \sqrt {1+c x} \left (e^2 (64 d+9 e x)+c^2 \left (96 d^3+72 d^2 e x+32 d e^2 x^2+6 e^3 x^3\right )\right )\right ) \text {arccosh}(c x)+9 b^2 \left (-24 c^2 d^2 e-3 e^3+8 c^4 x \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )\right ) \text {arccosh}(c x)^2-54 a b e \left (8 c^2 d^2+e^2\right ) \log \left (c x+\sqrt {-1+c x} \sqrt {1+c x}\right )}{288 c^4} \]

[In]

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

[Out]

(c*(72*a^2*c^3*x*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3) - 6*a*b*Sqrt[-1 + c*x]*Sqrt[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)) + b^2*c*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*b*c*(-24*a*c^3*x*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3)
+ b*Sqrt[-1 + c*x]*Sqrt[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*b^2*(-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 - 54*a*b*e*(8*c^2*d^2 + e^2)*Log[c*x + Sqrt[-1 + c*x]*Sqrt[1 + c*x]])/(288*c^4)

Maple [A] (verified)

Time = 0.82 (sec) , antiderivative size = 616, normalized size of antiderivative = 1.55

method	result	size
derivativedivides	\(\frac {\frac {a^{2} \left (e c x +c d \right )^{4}}{4 c^{3} e}+\frac {b^{2} \left (\frac {e^{3} \left (8 \operatorname {arccosh}\left (c x \right )^{2} x^{4} c^{4}-4 \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, c^{3} x^{3}-6 \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, c x +c^{4} x^{4}-3 \operatorname {arccosh}\left (c x \right )^{2}+3 c^{2} x^{2}\right )}{32}+\frac {c d \,e^{2} \left (9 \operatorname {arccosh}\left (c x \right )^{2} x^{3} c^{3}-6 \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, c^{2} x^{2}-12 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}+2 c^{3} x^{3}+12 c x \right )}{9}+\frac {3 d^{2} e \,c^{2} \left (2 \operatorname {arccosh}\left (c x \right )^{2} x^{2} c^{2}-2 \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, c x -\operatorname {arccosh}\left (c x \right )^{2}+c^{2} x^{2}\right )}{4}+c^{3} d^{3} \left (\operatorname {arccosh}\left (c x \right )^{2} x c -2 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}+2 c x \right )\right )}{c^{3}}+\frac {2 a b \left (\frac {\operatorname {arccosh}\left (c x \right ) c^{4} d^{4}}{4 e}+\operatorname {arccosh}\left (c x \right ) c^{4} d^{3} x +\frac {3 e \,\operatorname {arccosh}\left (c x \right ) c^{4} d^{2} x^{2}}{2}+e^{2} \operatorname {arccosh}\left (c x \right ) c^{4} d \,x^{3}+\frac {\operatorname {arccosh}\left (c x \right ) e^{3} c^{4} x^{4}}{4}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (24 c^{4} d^{4} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+96 c^{3} d^{3} e \sqrt {c^{2} x^{2}-1}+72 c^{3} d^{2} e^{2} x \sqrt {c^{2} x^{2}-1}+32 c^{3} d \,e^{3} \sqrt {c^{2} x^{2}-1}\, x^{2}+6 e^{4} c^{3} x^{3} \sqrt {c^{2} x^{2}-1}+72 c^{2} d^{2} e^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+64 c d \,e^{3} \sqrt {c^{2} x^{2}-1}+9 e^{4} c x \sqrt {c^{2} x^{2}-1}+9 e^{4} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )\right )}{96 e \sqrt {c^{2} x^{2}-1}}\right )}{c^{3}}}{c}\)	\(616\)
default	\(\frac {\frac {a^{2} \left (e c x +c d \right )^{4}}{4 c^{3} e}+\frac {b^{2} \left (\frac {e^{3} \left (8 \operatorname {arccosh}\left (c x \right )^{2} x^{4} c^{4}-4 \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, c^{3} x^{3}-6 \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, c x +c^{4} x^{4}-3 \operatorname {arccosh}\left (c x \right )^{2}+3 c^{2} x^{2}\right )}{32}+\frac {c d \,e^{2} \left (9 \operatorname {arccosh}\left (c x \right )^{2} x^{3} c^{3}-6 \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, c^{2} x^{2}-12 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}+2 c^{3} x^{3}+12 c x \right )}{9}+\frac {3 d^{2} e \,c^{2} \left (2 \operatorname {arccosh}\left (c x \right )^{2} x^{2} c^{2}-2 \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, c x -\operatorname {arccosh}\left (c x \right )^{2}+c^{2} x^{2}\right )}{4}+c^{3} d^{3} \left (\operatorname {arccosh}\left (c x \right )^{2} x c -2 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}+2 c x \right )\right )}{c^{3}}+\frac {2 a b \left (\frac {\operatorname {arccosh}\left (c x \right ) c^{4} d^{4}}{4 e}+\operatorname {arccosh}\left (c x \right ) c^{4} d^{3} x +\frac {3 e \,\operatorname {arccosh}\left (c x \right ) c^{4} d^{2} x^{2}}{2}+e^{2} \operatorname {arccosh}\left (c x \right ) c^{4} d \,x^{3}+\frac {\operatorname {arccosh}\left (c x \right ) e^{3} c^{4} x^{4}}{4}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (24 c^{4} d^{4} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+96 c^{3} d^{3} e \sqrt {c^{2} x^{2}-1}+72 c^{3} d^{2} e^{2} x \sqrt {c^{2} x^{2}-1}+32 c^{3} d \,e^{3} \sqrt {c^{2} x^{2}-1}\, x^{2}+6 e^{4} c^{3} x^{3} \sqrt {c^{2} x^{2}-1}+72 c^{2} d^{2} e^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+64 c d \,e^{3} \sqrt {c^{2} x^{2}-1}+9 e^{4} c x \sqrt {c^{2} x^{2}-1}+9 e^{4} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )\right )}{96 e \sqrt {c^{2} x^{2}-1}}\right )}{c^{3}}}{c}\)	\(616\)
parts	\(\frac {a^{2} \left (e x +d \right )^{4}}{4 e}+\frac {b^{2} \left (288 \operatorname {arccosh}\left (c x \right )^{2} c^{4} d^{3} x +432 \operatorname {arccosh}\left (c x \right )^{2} c^{4} d^{2} e \,x^{2}+288 \operatorname {arccosh}\left (c x \right )^{2} c^{4} d \,e^{2} x^{3}+72 \operatorname {arccosh}\left (c x \right )^{2} e^{3} c^{4} x^{4}-576 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} d^{3}-432 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} d^{2} e x -192 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} d \,e^{2} x^{2}-36 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}\, e^{3} c^{3} x^{3}-216 \operatorname {arccosh}\left (c x \right )^{2} c^{2} d^{2} e -384 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}\, c d \,e^{2}-54 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}\, e^{3} c x +576 x \,c^{4} d^{3}+216 c^{4} x^{2} d^{2} e +64 c^{4} d \,e^{2} x^{3}+9 c^{4} x^{4} e^{3}-27 \operatorname {arccosh}\left (c x \right )^{2} e^{3}+384 c^{2} x d \,e^{2}+27 c^{2} x^{2} e^{3}\right )}{288 c^{4}}+\frac {2 a b \left (\frac {c \,\operatorname {arccosh}\left (c x \right ) d^{4}}{4 e}+\operatorname {arccosh}\left (c x \right ) c x \,d^{3}+\frac {3 c \,\operatorname {arccosh}\left (c x \right ) d^{2} e \,x^{2}}{2}+c \,e^{2} \operatorname {arccosh}\left (c x \right ) d \,x^{3}+\frac {c \,e^{3} \operatorname {arccosh}\left (c x \right ) x^{4}}{4}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (24 c^{4} d^{4} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+96 c^{3} d^{3} e \sqrt {c^{2} x^{2}-1}+72 c^{3} d^{2} e^{2} x \sqrt {c^{2} x^{2}-1}+32 c^{3} d \,e^{3} \sqrt {c^{2} x^{2}-1}\, x^{2}+6 e^{4} c^{3} x^{3} \sqrt {c^{2} x^{2}-1}+72 c^{2} d^{2} e^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+64 c d \,e^{3} \sqrt {c^{2} x^{2}-1}+9 e^{4} c x \sqrt {c^{2} x^{2}-1}+9 e^{4} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )\right )}{96 c^{3} e \sqrt {c^{2} x^{2}-1}}\right )}{c}\)	\(649\)

[In]

int((e*x+d)^3*(a+b*arccosh(c*x))^2,x,method=_RETURNVERBOSE)

[Out]

1/c*(1/4*a^2/c^3*(c*e*x+c*d)^4/e+b^2/c^3*(1/32*e^3*(8*arccosh(c*x)^2*x^4*c^4-4*(c*x+1)^(1/2)*arccosh(c*x)*(c*x
-1)^(1/2)*c^3*x^3-6*(c*x+1)^(1/2)*arccosh(c*x)*(c*x-1)^(1/2)*c*x+c^4*x^4-3*arccosh(c*x)^2+3*c^2*x^2)+1/9*c*d*e
^2*(9*arccosh(c*x)^2*x^3*c^3-6*(c*x+1)^(1/2)*arccosh(c*x)*(c*x-1)^(1/2)*c^2*x^2-12*arccosh(c*x)*(c*x-1)^(1/2)*
(c*x+1)^(1/2)+2*c^3*x^3+12*c*x)+3/4*d^2*e*c^2*(2*arccosh(c*x)^2*x^2*c^2-2*(c*x+1)^(1/2)*arccosh(c*x)*(c*x-1)^(
1/2)*c*x-arccosh(c*x)^2+c^2*x^2)+c^3*d^3*(arccosh(c*x)^2*x*c-2*arccosh(c*x)*(c*x-1)^(1/2)*(c*x+1)^(1/2)+2*c*x)
)+2*a*b/c^3*(1/4/e*arccosh(c*x)*c^4*d^4+arccosh(c*x)*c^4*d^3*x+3/2*e*arccosh(c*x)*c^4*d^2*x^2+e^2*arccosh(c*x)
*c^4*d*x^3+1/4*arccosh(c*x)*e^3*c^4*x^4-1/96/e*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(24*c^4*d^4*ln(c*x+(c^2*x^2-1)^(1/2
))+96*c^3*d^3*e*(c^2*x^2-1)^(1/2)+72*c^3*d^2*e^2*x*(c^2*x^2-1)^(1/2)+32*c^3*d*e^3*(c^2*x^2-1)^(1/2)*x^2+6*e^4*
c^3*x^3*(c^2*x^2-1)^(1/2)+72*c^2*d^2*e^2*ln(c*x+(c^2*x^2-1)^(1/2))+64*c*d*e^3*(c^2*x^2-1)^(1/2)+9*e^4*c*x*(c^2
*x^2-1)^(1/2)+9*e^4*ln(c*x+(c^2*x^2-1)^(1/2)))/(c^2*x^2-1)^(1/2)))

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 472, normalized size of antiderivative = 1.19 \[ \int (d+e x)^3 (a+b \text {arccosh}(c x))^2 \, dx=\frac {9 \, {\left (8 \, a^{2} + b^{2}\right )} c^{4} e^{3} x^{4} + 32 \, {\left (9 \, a^{2} + 2 \, b^{2}\right )} c^{4} d e^{2} x^{3} + 27 \, {\left (8 \, {\left (2 \, a^{2} + b^{2}\right )} c^{4} d^{2} e + b^{2} c^{2} e^{3}\right )} x^{2} + 9 \, {\left (8 \, b^{2} c^{4} e^{3} x^{4} + 32 \, b^{2} c^{4} d e^{2} x^{3} + 48 \, b^{2} c^{4} d^{2} e x^{2} + 32 \, b^{2} c^{4} d^{3} x - 24 \, b^{2} c^{2} d^{2} e - 3 \, b^{2} e^{3}\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right )^{2} + 96 \, {\left (3 \, {\left (a^{2} + 2 \, b^{2}\right )} c^{4} d^{3} + 4 \, b^{2} c^{2} d e^{2}\right )} x + 6 \, {\left (24 \, a b c^{4} e^{3} x^{4} + 96 \, a b c^{4} d e^{2} x^{3} + 144 \, a b c^{4} d^{2} e x^{2} + 96 \, a b c^{4} d^{3} x - 72 \, a b c^{2} d^{2} e - 9 \, a b e^{3} - {\left (6 \, b^{2} c^{3} e^{3} x^{3} + 32 \, b^{2} c^{3} d e^{2} x^{2} + 96 \, b^{2} c^{3} d^{3} + 64 \, b^{2} c d e^{2} + 9 \, {\left (8 \, b^{2} c^{3} d^{2} e + b^{2} c e^{3}\right )} x\right )} \sqrt {c^{2} x^{2} - 1}\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - 6 \, {\left (6 \, a b c^{3} e^{3} x^{3} + 32 \, a b c^{3} d e^{2} x^{2} + 96 \, a b c^{3} d^{3} + 64 \, a b c d e^{2} + 9 \, {\left (8 \, a b c^{3} d^{2} e + a b c e^{3}\right )} x\right )} \sqrt {c^{2} x^{2} - 1}}{288 \, c^{4}} \]

[In]

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

[Out]

1/288*(9*(8*a^2 + b^2)*c^4*e^3*x^4 + 32*(9*a^2 + 2*b^2)*c^4*d*e^2*x^3 + 27*(8*(2*a^2 + b^2)*c^4*d^2*e + b^2*c^
2*e^3)*x^2 + 9*(8*b^2*c^4*e^3*x^4 + 32*b^2*c^4*d*e^2*x^3 + 48*b^2*c^4*d^2*e*x^2 + 32*b^2*c^4*d^3*x - 24*b^2*c^
2*d^2*e - 3*b^2*e^3)*log(c*x + sqrt(c^2*x^2 - 1))^2 + 96*(3*(a^2 + 2*b^2)*c^4*d^3 + 4*b^2*c^2*d*e^2)*x + 6*(24
*a*b*c^4*e^3*x^4 + 96*a*b*c^4*d*e^2*x^3 + 144*a*b*c^4*d^2*e*x^2 + 96*a*b*c^4*d^3*x - 72*a*b*c^2*d^2*e - 9*a*b*
e^3 - (6*b^2*c^3*e^3*x^3 + 32*b^2*c^3*d*e^2*x^2 + 96*b^2*c^3*d^3 + 64*b^2*c*d*e^2 + 9*(8*b^2*c^3*d^2*e + b^2*c
*e^3)*x)*sqrt(c^2*x^2 - 1))*log(c*x + sqrt(c^2*x^2 - 1)) - 6*(6*a*b*c^3*e^3*x^3 + 32*a*b*c^3*d*e^2*x^2 + 96*a*
b*c^3*d^3 + 64*a*b*c*d*e^2 + 9*(8*a*b*c^3*d^2*e + a*b*c*e^3)*x)*sqrt(c^2*x^2 - 1))/c^4

Sympy [F]

\[ \int (d+e x)^3 (a+b \text {arccosh}(c x))^2 \, dx=\int \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2} \left (d + e x\right )^{3}\, dx \]

[In]

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

[Out]

Integral((a + b*acosh(c*x))**2*(d + e*x)**3, x)

Maxima [F]

\[ \int (d+e x)^3 (a+b \text {arccosh}(c x))^2 \, dx=\int { {\left (e x + d\right )}^{3} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2} \,d x } \]

[In]

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

[Out]

1/4*a^2*e^3*x^4 + a^2*d*e^2*x^3 + b^2*d^3*x*arccosh(c*x)^2 + 3/2*a^2*d^2*e*x^2 + 3/2*(2*x^2*arccosh(c*x) - c*(
sqrt(c^2*x^2 - 1)*x/c^2 + log(2*c^2*x + 2*sqrt(c^2*x^2 - 1)*c)/c^3))*a*b*d^2*e + 2/3*(3*x^3*arccosh(c*x) - c*(
sqrt(c^2*x^2 - 1)*x^2/c^2 + 2*sqrt(c^2*x^2 - 1)/c^4))*a*b*d*e^2 + 1/16*(8*x^4*arccosh(c*x) - (2*sqrt(c^2*x^2 -
1)*x^3/c^2 + 3*sqrt(c^2*x^2 - 1)*x/c^4 + 3*log(2*c^2*x + 2*sqrt(c^2*x^2 - 1)*c)/c^5)*c)*a*b*e^3 + 2*b^2*d^3*(
x - sqrt(c^2*x^2 - 1)*arccosh(c*x)/c) + a^2*d^3*x + 2*(c*x*arccosh(c*x) - sqrt(c^2*x^2 - 1))*a*b*d^3/c + 1/4*(
b^2*e^3*x^4 + 4*b^2*d*e^2*x^3 + 6*b^2*d^2*e*x^2)*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))^2 - integrate(1/2*(b^2
*c^3*e^3*x^6 + 4*b^2*c^3*d*e^2*x^5 - 4*b^2*c*d*e^2*x^3 - 6*b^2*c*d^2*e*x^2 + (6*c^3*d^2*e - c*e^3)*b^2*x^4 + (
b^2*c^2*e^3*x^5 + 4*b^2*c^2*d*e^2*x^4 + 6*b^2*c^2*d^2*e*x^3)*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)

Giac [F(-2)]

Exception generated. \[ \int (d+e x)^3 (a+b \text {arccosh}(c x))^2 \, dx=\text {Exception raised: RuntimeError} \]

[In]

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

[Out]

Exception raised: RuntimeError >> an error occurred running a Giac command:INPUT:sage2OUTPUT:sym2poly/r2sym(co
nst gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

Timed out. \[ \int (d+e x)^3 (a+b \text {arccosh}(c x))^2 \, dx=\int {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2\,{\left (d+e\,x\right )}^3 \,d x \]

[In]

int((a + b*acosh(c*x))^2*(d + e*x)^3,x)

[Out]

int((a + b*acosh(c*x))^2*(d + e*x)^3, x)