Integrand size = 23, antiderivative size = 262 \[ \int (c e+d e x)^2 (a+b \text {arccosh}(c+d x))^3 \, dx=\frac {4}{3} a b^2 e^2 x-\frac {40 b^3 e^2 \sqrt {-1+c+d x} \sqrt {1+c+d x}}{27 d}-\frac {2 b^3 e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{27 d}+\frac {4 b^3 e^2 (c+d x) \text {arccosh}(c+d x)}{3 d}+\frac {2 b^2 e^2 (c+d x)^3 (a+b \text {arccosh}(c+d x))}{9 d}-\frac {2 b e^2 \sqrt {-1+c+d x} \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))^2}{3 d}-\frac {b e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))^2}{3 d}+\frac {e^2 (c+d x)^3 (a+b \text {arccosh}(c+d x))^3}{3 d} \]
4/3*a*b^2*e^2*x+4/3*b^3*e^2*(d*x+c)*arccosh(d*x+c)/d+2/9*b^2*e^2*(d*x+c)^3 *(a+b*arccosh(d*x+c))/d+1/3*e^2*(d*x+c)^3*(a+b*arccosh(d*x+c))^3/d-40/27*b ^3*e^2*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)/d-2/27*b^3*e^2*(d*x+c)^2*(d*x+c-1)^ (1/2)*(d*x+c+1)^(1/2)/d-2/3*b*e^2*(a+b*arccosh(d*x+c))^2*(d*x+c-1)^(1/2)*( d*x+c+1)^(1/2)/d-1/3*b*e^2*(d*x+c)^2*(a+b*arccosh(d*x+c))^2*(d*x+c-1)^(1/2 )*(d*x+c+1)^(1/2)/d
Time = 0.50 (sec) , antiderivative size = 296, normalized size of antiderivative = 1.13 \[ \int (c e+d e x)^2 (a+b \text {arccosh}(c+d x))^3 \, dx=\frac {e^2 \left (12 a b^2 (c+d x)+a \left (3 a^2+2 b^2\right ) (c+d x)^3+\frac {1}{3} b \sqrt {-1+c+d x} \sqrt {1+c+d x} \left (-2 \left (9 a^2+20 b^2\right )-\left (9 a^2+2 b^2\right ) (c+d x)^2\right )-b \left (-12 b^2 (c+d x)-9 a^2 (c+d x)^3-2 b^2 (c+d x)^3+12 a b \sqrt {-1+c+d x} \sqrt {1+c+d x}+6 a b \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}\right ) \text {arccosh}(c+d x)-3 b^2 \left (-3 a (c+d x)^3+2 b \sqrt {-1+c+d x} \sqrt {1+c+d x}+b \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}\right ) \text {arccosh}(c+d x)^2+3 b^3 (c+d x)^3 \text {arccosh}(c+d x)^3\right )}{9 d} \]
(e^2*(12*a*b^2*(c + d*x) + a*(3*a^2 + 2*b^2)*(c + d*x)^3 + (b*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x]*(-2*(9*a^2 + 20*b^2) - (9*a^2 + 2*b^2)*(c + d*x)^ 2))/3 - b*(-12*b^2*(c + d*x) - 9*a^2*(c + d*x)^3 - 2*b^2*(c + d*x)^3 + 12* a*b*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x] + 6*a*b*Sqrt[-1 + c + d*x]*(c + d *x)^2*Sqrt[1 + c + d*x])*ArcCosh[c + d*x] - 3*b^2*(-3*a*(c + d*x)^3 + 2*b* Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x] + b*Sqrt[-1 + c + d*x]*(c + d*x)^2*Sq rt[1 + c + d*x])*ArcCosh[c + d*x]^2 + 3*b^3*(c + d*x)^3*ArcCosh[c + d*x]^3 ))/(9*d)
Time = 1.07 (sec) , antiderivative size = 251, normalized size of antiderivative = 0.96, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {6411, 27, 6298, 6354, 6298, 111, 27, 83, 6330, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int (c e+d e x)^2 (a+b \text {arccosh}(c+d x))^3 \, dx\) |
| \(\Big \downarrow \) 6411 |
| \(\displaystyle \frac {\int e^2 (c+d x)^2 (a+b \text {arccosh}(c+d x))^3d(c+d x)}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e^2 \int (c+d x)^2 (a+b \text {arccosh}(c+d x))^3d(c+d x)}{d}\) |
| \(\Big \downarrow \) 6298 |
| \(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \text {arccosh}(c+d x))^3-b \int \frac {(c+d x)^3 (a+b \text {arccosh}(c+d x))^2}{\sqrt {c+d x-1} \sqrt {c+d x+1}}d(c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 6354 |
| \(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \text {arccosh}(c+d x))^3-b \left (-\frac {2}{3} b \int (c+d x)^2 (a+b \text {arccosh}(c+d x))d(c+d x)+\frac {2}{3} \int \frac {(c+d x) (a+b \text {arccosh}(c+d x))^2}{\sqrt {c+d x-1} \sqrt {c+d x+1}}d(c+d x)+\frac {1}{3} \sqrt {c+d x-1} (c+d x)^2 \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^2\right )\right )}{d}\) |
| \(\Big \downarrow \) 6298 |
| \(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \text {arccosh}(c+d x))^3-b \left (-\frac {2}{3} b \left (\frac {1}{3} (c+d x)^3 (a+b \text {arccosh}(c+d x))-\frac {1}{3} b \int \frac {(c+d x)^3}{\sqrt {c+d x-1} \sqrt {c+d x+1}}d(c+d x)\right )+\frac {2}{3} \int \frac {(c+d x) (a+b \text {arccosh}(c+d x))^2}{\sqrt {c+d x-1} \sqrt {c+d x+1}}d(c+d x)+\frac {1}{3} \sqrt {c+d x-1} (c+d x)^2 \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^2\right )\right )}{d}\) |
| \(\Big \downarrow \) 111 |
| \(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \text {arccosh}(c+d x))^3-b \left (-\frac {2}{3} b \left (\frac {1}{3} (c+d x)^3 (a+b \text {arccosh}(c+d x))-\frac {1}{3} b \left (\frac {1}{3} \int \frac {2 (c+d x)}{\sqrt {c+d x-1} \sqrt {c+d x+1}}d(c+d x)+\frac {1}{3} \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2\right )\right )+\frac {2}{3} \int \frac {(c+d x) (a+b \text {arccosh}(c+d x))^2}{\sqrt {c+d x-1} \sqrt {c+d x+1}}d(c+d x)+\frac {1}{3} \sqrt {c+d x-1} (c+d x)^2 \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^2\right )\right )}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \text {arccosh}(c+d x))^3-b \left (-\frac {2}{3} b \left (\frac {1}{3} (c+d x)^3 (a+b \text {arccosh}(c+d x))-\frac {1}{3} b \left (\frac {2}{3} \int \frac {c+d x}{\sqrt {c+d x-1} \sqrt {c+d x+1}}d(c+d x)+\frac {1}{3} \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2\right )\right )+\frac {2}{3} \int \frac {(c+d x) (a+b \text {arccosh}(c+d x))^2}{\sqrt {c+d x-1} \sqrt {c+d x+1}}d(c+d x)+\frac {1}{3} \sqrt {c+d x-1} (c+d x)^2 \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^2\right )\right )}{d}\) |
| \(\Big \downarrow \) 83 |
| \(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \text {arccosh}(c+d x))^3-b \left (\frac {2}{3} \int \frac {(c+d x) (a+b \text {arccosh}(c+d x))^2}{\sqrt {c+d x-1} \sqrt {c+d x+1}}d(c+d x)+\frac {1}{3} \sqrt {c+d x-1} (c+d x)^2 \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^2-\frac {2}{3} b \left (\frac {1}{3} (c+d x)^3 (a+b \text {arccosh}(c+d x))-\frac {1}{3} b \left (\frac {1}{3} \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2+\frac {2}{3} \sqrt {c+d x-1} \sqrt {c+d x+1}\right )\right )\right )\right )}{d}\) |
| \(\Big \downarrow \) 6330 |
| \(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \text {arccosh}(c+d x))^3-b \left (\frac {2}{3} \left (\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^2-2 b \int (a+b \text {arccosh}(c+d x))d(c+d x)\right )+\frac {1}{3} \sqrt {c+d x-1} (c+d x)^2 \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^2-\frac {2}{3} b \left (\frac {1}{3} (c+d x)^3 (a+b \text {arccosh}(c+d x))-\frac {1}{3} b \left (\frac {1}{3} \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2+\frac {2}{3} \sqrt {c+d x-1} \sqrt {c+d x+1}\right )\right )\right )\right )}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \text {arccosh}(c+d x))^3-b \left (\frac {1}{3} \sqrt {c+d x-1} (c+d x)^2 \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^2-\frac {2}{3} b \left (\frac {1}{3} (c+d x)^3 (a+b \text {arccosh}(c+d x))-\frac {1}{3} b \left (\frac {1}{3} \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2+\frac {2}{3} \sqrt {c+d x-1} \sqrt {c+d x+1}\right )\right )+\frac {2}{3} \left (\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^2-2 b \left (a (c+d x)+b (c+d x) \text {arccosh}(c+d x)-b \sqrt {c+d x-1} \sqrt {c+d x+1}\right )\right )\right )\right )}{d}\) |
(e^2*(((c + d*x)^3*(a + b*ArcCosh[c + d*x])^3)/3 - b*((Sqrt[-1 + c + d*x]* (c + d*x)^2*Sqrt[1 + c + d*x]*(a + b*ArcCosh[c + d*x])^2)/3 - (2*b*(-1/3*( b*((2*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x])/3 + (Sqrt[-1 + c + d*x]*(c + d *x)^2*Sqrt[1 + c + d*x])/3)) + ((c + d*x)^3*(a + b*ArcCosh[c + d*x]))/3))/ 3 + (2*(Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x]*(a + b*ArcCosh[c + d*x])^2 - 2*b*(a*(c + d*x) - b*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x] + b*(c + d*x)*Ar cCosh[c + d*x])))/3)))/d
3.2.15.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> 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]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[b*(a + b*x)^(m - 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 )/(d*f*(m + n + p + 1))), x] + Simp[1/(d*f*(m + n + p + 1)) Int[(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]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcCosh[c*x])^n/(d*(m + 1))), x] - Simp[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]
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] - Simp[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] && E qQ[e2, (-c)*d2] && GtQ[n, 0] && NeQ[p, -1]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d1_) + (e 1_.)*(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] + (Simp[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] - Simp[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] && IGtQ[m, 1] && N eQ[m + 2*p + 1, 0]
Int[((a_.) + ArcCosh[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^( m_.), x_Symbol] :> Simp[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]
Time = 0.67 (sec) , antiderivative size = 326, normalized size of antiderivative = 1.24
| method | result | size |
| derivativedivides | \(\frac {\frac {e^{2} a^{3} \left (d x +c \right )^{3}}{3}+e^{2} b^{3} \left (\frac {\left (d x +c \right )^{3} \operatorname {arccosh}\left (d x +c \right )^{3}}{3}-\frac {2 \operatorname {arccosh}\left (d x +c \right )^{2} \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{3}-\frac {\left (d x +c \right )^{2} \operatorname {arccosh}\left (d x +c \right )^{2} \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{3}+\frac {4 \left (d x +c \right ) \operatorname {arccosh}\left (d x +c \right )}{3}-\frac {40 \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{27}+\frac {2 \left (d x +c \right )^{3} \operatorname {arccosh}\left (d x +c \right )}{9}-\frac {2 \left (d x +c \right )^{2} \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{27}\right )+3 e^{2} a \,b^{2} \left (\frac {\left (d x +c \right )^{3} \operatorname {arccosh}\left (d x +c \right )^{2}}{3}-\frac {4 \,\operatorname {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{9}-\frac {2 \left (d x +c \right )^{2} \operatorname {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{9}+\frac {4 d x}{9}+\frac {4 c}{9}+\frac {2 \left (d x +c \right )^{3}}{27}\right )+3 e^{2} a^{2} b \left (\frac {\left (d x +c \right )^{3} \operatorname {arccosh}\left (d x +c \right )}{3}-\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (\left (d x +c \right )^{2}+2\right )}{9}\right )}{d}\) | \(326\) |
| default | \(\frac {\frac {e^{2} a^{3} \left (d x +c \right )^{3}}{3}+e^{2} b^{3} \left (\frac {\left (d x +c \right )^{3} \operatorname {arccosh}\left (d x +c \right )^{3}}{3}-\frac {2 \operatorname {arccosh}\left (d x +c \right )^{2} \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{3}-\frac {\left (d x +c \right )^{2} \operatorname {arccosh}\left (d x +c \right )^{2} \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{3}+\frac {4 \left (d x +c \right ) \operatorname {arccosh}\left (d x +c \right )}{3}-\frac {40 \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{27}+\frac {2 \left (d x +c \right )^{3} \operatorname {arccosh}\left (d x +c \right )}{9}-\frac {2 \left (d x +c \right )^{2} \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{27}\right )+3 e^{2} a \,b^{2} \left (\frac {\left (d x +c \right )^{3} \operatorname {arccosh}\left (d x +c \right )^{2}}{3}-\frac {4 \,\operatorname {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{9}-\frac {2 \left (d x +c \right )^{2} \operatorname {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{9}+\frac {4 d x}{9}+\frac {4 c}{9}+\frac {2 \left (d x +c \right )^{3}}{27}\right )+3 e^{2} a^{2} b \left (\frac {\left (d x +c \right )^{3} \operatorname {arccosh}\left (d x +c \right )}{3}-\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (\left (d x +c \right )^{2}+2\right )}{9}\right )}{d}\) | \(326\) |
| parts | \(\frac {e^{2} a^{3} \left (d x +c \right )^{3}}{3 d}+\frac {e^{2} b^{3} \left (\frac {\left (d x +c \right )^{3} \operatorname {arccosh}\left (d x +c \right )^{3}}{3}-\frac {2 \operatorname {arccosh}\left (d x +c \right )^{2} \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{3}-\frac {\left (d x +c \right )^{2} \operatorname {arccosh}\left (d x +c \right )^{2} \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{3}+\frac {4 \left (d x +c \right ) \operatorname {arccosh}\left (d x +c \right )}{3}-\frac {40 \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{27}+\frac {2 \left (d x +c \right )^{3} \operatorname {arccosh}\left (d x +c \right )}{9}-\frac {2 \left (d x +c \right )^{2} \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{27}\right )}{d}+\frac {3 e^{2} a \,b^{2} \left (\frac {\left (d x +c \right )^{3} \operatorname {arccosh}\left (d x +c \right )^{2}}{3}-\frac {4 \,\operatorname {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{9}-\frac {2 \left (d x +c \right )^{2} \operatorname {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{9}+\frac {4 d x}{9}+\frac {4 c}{9}+\frac {2 \left (d x +c \right )^{3}}{27}\right )}{d}+\frac {3 e^{2} a^{2} b \left (\frac {\left (d x +c \right )^{3} \operatorname {arccosh}\left (d x +c \right )}{3}-\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (\left (d x +c \right )^{2}+2\right )}{9}\right )}{d}\) | \(334\) |
1/d*(1/3*e^2*a^3*(d*x+c)^3+e^2*b^3*(1/3*(d*x+c)^3*arccosh(d*x+c)^3-2/3*arc cosh(d*x+c)^2*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)-1/3*(d*x+c)^2*arccosh(d*x+c) ^2*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)+4/3*(d*x+c)*arccosh(d*x+c)-40/27*(d*x+c -1)^(1/2)*(d*x+c+1)^(1/2)+2/9*(d*x+c)^3*arccosh(d*x+c)-2/27*(d*x+c)^2*(d*x +c-1)^(1/2)*(d*x+c+1)^(1/2))+3*e^2*a*b^2*(1/3*(d*x+c)^3*arccosh(d*x+c)^2-4 /9*arccosh(d*x+c)*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)-2/9*(d*x+c)^2*arccosh(d* x+c)*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)+4/9*d*x+4/9*c+2/27*(d*x+c)^3)+3*e^2*a ^2*b*(1/3*(d*x+c)^3*arccosh(d*x+c)-1/9*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)*((d *x+c)^2+2)))
Leaf count of result is larger than twice the leaf count of optimal. 607 vs. \(2 (230) = 460\).
Time = 0.28 (sec) , antiderivative size = 607, normalized size of antiderivative = 2.32 \[ \int (c e+d e x)^2 (a+b \text {arccosh}(c+d x))^3 \, dx=\frac {3 \, {\left (3 \, a^{3} + 2 \, a b^{2}\right )} d^{3} e^{2} x^{3} + 9 \, {\left (3 \, a^{3} + 2 \, a b^{2}\right )} c d^{2} e^{2} x^{2} + 9 \, {\left (4 \, a b^{2} + {\left (3 \, a^{3} + 2 \, a b^{2}\right )} c^{2}\right )} d e^{2} x + 9 \, {\left (b^{3} d^{3} e^{2} x^{3} + 3 \, b^{3} c d^{2} e^{2} x^{2} + 3 \, b^{3} c^{2} d e^{2} x + b^{3} c^{3} e^{2}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )^{3} + 9 \, {\left (3 \, a b^{2} d^{3} e^{2} x^{3} + 9 \, a b^{2} c d^{2} e^{2} x^{2} + 9 \, a b^{2} c^{2} d e^{2} x + 3 \, a b^{2} c^{3} e^{2} - {\left (b^{3} d^{2} e^{2} x^{2} + 2 \, b^{3} c d e^{2} x + {\left (b^{3} c^{2} + 2 \, b^{3}\right )} e^{2}\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )^{2} + 3 \, {\left ({\left (9 \, a^{2} b + 2 \, b^{3}\right )} d^{3} e^{2} x^{3} + 3 \, {\left (9 \, a^{2} b + 2 \, b^{3}\right )} c d^{2} e^{2} x^{2} + 3 \, {\left (4 \, b^{3} + {\left (9 \, a^{2} b + 2 \, b^{3}\right )} c^{2}\right )} d e^{2} x + {\left (12 \, b^{3} c + {\left (9 \, a^{2} b + 2 \, b^{3}\right )} c^{3}\right )} e^{2} - 6 \, {\left (a b^{2} d^{2} e^{2} x^{2} + 2 \, a b^{2} c d e^{2} x + {\left (a b^{2} c^{2} + 2 \, a b^{2}\right )} e^{2}\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) - {\left ({\left (9 \, a^{2} b + 2 \, b^{3}\right )} d^{2} e^{2} x^{2} + 2 \, {\left (9 \, a^{2} b + 2 \, b^{3}\right )} c d e^{2} x + {\left (18 \, a^{2} b + 40 \, b^{3} + {\left (9 \, a^{2} b + 2 \, b^{3}\right )} c^{2}\right )} e^{2}\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}}{27 \, d} \]
1/27*(3*(3*a^3 + 2*a*b^2)*d^3*e^2*x^3 + 9*(3*a^3 + 2*a*b^2)*c*d^2*e^2*x^2 + 9*(4*a*b^2 + (3*a^3 + 2*a*b^2)*c^2)*d*e^2*x + 9*(b^3*d^3*e^2*x^3 + 3*b^3 *c*d^2*e^2*x^2 + 3*b^3*c^2*d*e^2*x + b^3*c^3*e^2)*log(d*x + c + sqrt(d^2*x ^2 + 2*c*d*x + c^2 - 1))^3 + 9*(3*a*b^2*d^3*e^2*x^3 + 9*a*b^2*c*d^2*e^2*x^ 2 + 9*a*b^2*c^2*d*e^2*x + 3*a*b^2*c^3*e^2 - (b^3*d^2*e^2*x^2 + 2*b^3*c*d*e ^2*x + (b^3*c^2 + 2*b^3)*e^2)*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))^2 + 3*((9*a^2*b + 2*b^3)*d^3*e^2*x ^3 + 3*(9*a^2*b + 2*b^3)*c*d^2*e^2*x^2 + 3*(4*b^3 + (9*a^2*b + 2*b^3)*c^2) *d*e^2*x + (12*b^3*c + (9*a^2*b + 2*b^3)*c^3)*e^2 - 6*(a*b^2*d^2*e^2*x^2 + 2*a*b^2*c*d*e^2*x + (a*b^2*c^2 + 2*a*b^2)*e^2)*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)) - ((9*a^2*b + 2* b^3)*d^2*e^2*x^2 + 2*(9*a^2*b + 2*b^3)*c*d*e^2*x + (18*a^2*b + 40*b^3 + (9 *a^2*b + 2*b^3)*c^2)*e^2)*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1))/d
\[ \int (c e+d e x)^2 (a+b \text {arccosh}(c+d x))^3 \, dx=e^{2} \left (\int a^{3} c^{2}\, dx + \int a^{3} d^{2} x^{2}\, dx + \int b^{3} c^{2} \operatorname {acosh}^{3}{\left (c + d x \right )}\, dx + \int 3 a b^{2} c^{2} \operatorname {acosh}^{2}{\left (c + d x \right )}\, dx + \int 3 a^{2} b c^{2} \operatorname {acosh}{\left (c + d x \right )}\, dx + \int 2 a^{3} c d x\, dx + \int b^{3} d^{2} x^{2} \operatorname {acosh}^{3}{\left (c + d x \right )}\, dx + \int 3 a b^{2} d^{2} x^{2} \operatorname {acosh}^{2}{\left (c + d x \right )}\, dx + \int 3 a^{2} b d^{2} x^{2} \operatorname {acosh}{\left (c + d x \right )}\, dx + \int 2 b^{3} c d x \operatorname {acosh}^{3}{\left (c + d x \right )}\, dx + \int 6 a b^{2} c d x \operatorname {acosh}^{2}{\left (c + d x \right )}\, dx + \int 6 a^{2} b c d x \operatorname {acosh}{\left (c + d x \right )}\, dx\right ) \]
e**2*(Integral(a**3*c**2, x) + Integral(a**3*d**2*x**2, x) + Integral(b**3 *c**2*acosh(c + d*x)**3, x) + Integral(3*a*b**2*c**2*acosh(c + d*x)**2, x) + Integral(3*a**2*b*c**2*acosh(c + d*x), x) + Integral(2*a**3*c*d*x, x) + Integral(b**3*d**2*x**2*acosh(c + d*x)**3, x) + Integral(3*a*b**2*d**2*x* *2*acosh(c + d*x)**2, x) + Integral(3*a**2*b*d**2*x**2*acosh(c + d*x), x) + Integral(2*b**3*c*d*x*acosh(c + d*x)**3, x) + Integral(6*a*b**2*c*d*x*ac osh(c + d*x)**2, x) + Integral(6*a**2*b*c*d*x*acosh(c + d*x), x))
\[ \int (c e+d e x)^2 (a+b \text {arccosh}(c+d x))^3 \, dx=\int { {\left (d e x + c e\right )}^{2} {\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{3} \,d x } \]
1/3*a^3*d^2*e^2*x^3 + a^3*c*d*e^2*x^2 + 3/2*(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 + sq rt(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^2*b*c*d*e^2 + 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))*a^2*b*d^2*e^2 + a^3*c^2*e^2*x + 3*(( d*x + c)*arccosh(d*x + c) - sqrt((d*x + c)^2 - 1))*a^2*b*c^2*e^2/d + 1/3*( b^3*d^2*e^2*x^3 + 3*b^3*c*d*e^2*x^2 + 3*b^3*c^2*e^2*x)*log(d*x + sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + c)^3 + integrate(((3*a*b^2*d^5*e^2 - b^3*d^5* e^2)*x^5 + 5*(3*a*b^2*c*d^4*e^2 - b^3*c*d^4*e^2)*x^4 + 3*(c^5*e^2 - c^3*e^ 2)*a*b^2 + (3*(10*c^2*d^3*e^2 - d^3*e^2)*a*b^2 - (10*c^2*d^3*e^2 - d^3*e^2 )*b^3)*x^3 + 3*((10*c^3*d^2*e^2 - 3*c*d^2*e^2)*a*b^2 - (3*c^3*d^2*e^2 - c* d^2*e^2)*b^3)*x^2 + ((3*a*b^2*d^4*e^2 - b^3*d^4*e^2)*x^4 + 3*(c^4*e^2 - c^ 2*e^2)*a*b^2 + 4*(3*a*b^2*c*d^3*e^2 - b^3*c*d^3*e^2)*x^3 - 3*(2*b^3*c^2*d^ 2*e^2 - (6*c^2*d^2*e^2 - d^2*e^2)*a*b^2)*x^2 - 3*(b^3*c^3*d*e^2 - 2*(2*c^3 *d*e^2 - c*d*e^2)*a*b^2)*x)*sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + 3*((5...
\[ \int (c e+d e x)^2 (a+b \text {arccosh}(c+d x))^3 \, dx=\int { {\left (d e x + c e\right )}^{2} {\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{3} \,d x } \]
Timed out. \[ \int (c e+d e x)^2 (a+b \text {arccosh}(c+d x))^3 \, dx=\int {\left (c\,e+d\,e\,x\right )}^2\,{\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right )}^3 \,d x \]