Integrand size = 12, antiderivative size = 114 \[ \int (a+b \text {arccosh}(c+d x))^3 \, dx=6 a b^2 x-\frac {6 b^3 \sqrt {-1+c+d x} \sqrt {1+c+d x}}{d}+\frac {6 b^3 (c+d x) \text {arccosh}(c+d x)}{d}-\frac {3 b \sqrt {-1+c+d x} \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))^2}{d}+\frac {(c+d x) (a+b \text {arccosh}(c+d x))^3}{d} \]
6*a*b^2*x+6*b^3*(d*x+c)*arccosh(d*x+c)/d+(d*x+c)*(a+b*arccosh(d*x+c))^3/d- 6*b^3*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)/d-3*b*(a+b*arccosh(d*x+c))^2*(d*x+c- 1)^(1/2)*(d*x+c+1)^(1/2)/d
Time = 0.17 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.47 \[ \int (a+b \text {arccosh}(c+d x))^3 \, dx=\frac {a \left (a^2+6 b^2\right ) (c+d x)-3 b \left (a^2+2 b^2\right ) \sqrt {-1+c+d x} \sqrt {1+c+d x}-3 b \left (-a^2 (c+d x)-2 b^2 (c+d x)+2 a b \sqrt {-1+c+d x} \sqrt {1+c+d x}\right ) \text {arccosh}(c+d x)-3 b^2 \left (-a (c+d x)+b \sqrt {-1+c+d x} \sqrt {1+c+d x}\right ) \text {arccosh}(c+d x)^2+b^3 (c+d x) \text {arccosh}(c+d x)^3}{d} \]
(a*(a^2 + 6*b^2)*(c + d*x) - 3*b*(a^2 + 2*b^2)*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x] - 3*b*(-(a^2*(c + d*x)) - 2*b^2*(c + d*x) + 2*a*b*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x])*ArcCosh[c + d*x] - 3*b^2*(-(a*(c + d*x)) + b*Sqrt[ -1 + c + d*x]*Sqrt[1 + c + d*x])*ArcCosh[c + d*x]^2 + b^3*(c + d*x)*ArcCos h[c + d*x]^3)/d
Time = 0.45 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.94, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6410, 6294, 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.
| \(\displaystyle \int (a+b \text {arccosh}(c+d x))^3 \, dx\) |
| \(\Big \downarrow \) 6410 |
| \(\displaystyle \frac {\int (a+b \text {arccosh}(c+d x))^3d(c+d x)}{d}\) |
| \(\Big \downarrow \) 6294 |
| \(\displaystyle \frac {(c+d x) (a+b \text {arccosh}(c+d x))^3-3 b \int \frac {(c+d x) (a+b \text {arccosh}(c+d x))^2}{\sqrt {c+d x-1} \sqrt {c+d x+1}}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 6330 |
| \(\displaystyle \frac {(c+d x) (a+b \text {arccosh}(c+d x))^3-3 b \left (\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^2-2 b \int (a+b \text {arccosh}(c+d x))d(c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {(c+d x) (a+b \text {arccosh}(c+d x))^3-3 b \left (\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^2-2 b \left (a (c+d x)+b (c+d x) \text {arccosh}(c+d x)-b \sqrt {c+d x-1} \sqrt {c+d x+1}\right )\right )}{d}\) |
((c + d*x)*(a + b*ArcCosh[c + d*x])^3 - 3*b*(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)*ArcCosh[c + d*x])))/d
3.2.17.3.1 Defintions of rubi rules used
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*A rcCosh[c*x])^n, x] - Simp[b*c*n Int[x*((a + b*ArcCosh[c*x])^(n - 1)/(Sqrt [1 + c*x]*Sqrt[-1 + c*x])), x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]
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_) + (d_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[1/d Subst[Int[(a + b*ArcCosh[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d , n}, x]
Time = 0.25 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.58
| method | result | size |
| derivativedivides | \(\frac {\left (d x +c \right ) a^{3}+b^{3} \left (\left (d x +c \right ) \operatorname {arccosh}\left (d x +c \right )^{3}-3 \operatorname {arccosh}\left (d x +c \right )^{2} \sqrt {d x +c -1}\, \sqrt {d x +c +1}+6 \left (d x +c \right ) \operatorname {arccosh}\left (d x +c \right )-6 \sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )+3 a \,b^{2} \left (\left (d x +c \right ) \operatorname {arccosh}\left (d x +c \right )^{2}-2 \,\operatorname {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}+2 d x +2 c \right )+3 a^{2} b \left (\left (d x +c \right ) \operatorname {arccosh}\left (d x +c \right )-\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )}{d}\) | \(180\) |
| default | \(\frac {\left (d x +c \right ) a^{3}+b^{3} \left (\left (d x +c \right ) \operatorname {arccosh}\left (d x +c \right )^{3}-3 \operatorname {arccosh}\left (d x +c \right )^{2} \sqrt {d x +c -1}\, \sqrt {d x +c +1}+6 \left (d x +c \right ) \operatorname {arccosh}\left (d x +c \right )-6 \sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )+3 a \,b^{2} \left (\left (d x +c \right ) \operatorname {arccosh}\left (d x +c \right )^{2}-2 \,\operatorname {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}+2 d x +2 c \right )+3 a^{2} b \left (\left (d x +c \right ) \operatorname {arccosh}\left (d x +c \right )-\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )}{d}\) | \(180\) |
| parts | \(a^{3} x +\frac {b^{3} \left (\left (d x +c \right ) \operatorname {arccosh}\left (d x +c \right )^{3}-3 \operatorname {arccosh}\left (d x +c \right )^{2} \sqrt {d x +c -1}\, \sqrt {d x +c +1}+6 \left (d x +c \right ) \operatorname {arccosh}\left (d x +c \right )-6 \sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )}{d}+\frac {3 a \,b^{2} \left (\left (d x +c \right ) \operatorname {arccosh}\left (d x +c \right )^{2}-2 \,\operatorname {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}+2 d x +2 c \right )}{d}+\frac {3 a^{2} b \left (\left (d x +c \right ) \operatorname {arccosh}\left (d x +c \right )-\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )}{d}\) | \(181\) |
1/d*((d*x+c)*a^3+b^3*((d*x+c)*arccosh(d*x+c)^3-3*arccosh(d*x+c)^2*(d*x+c-1 )^(1/2)*(d*x+c+1)^(1/2)+6*(d*x+c)*arccosh(d*x+c)-6*(d*x+c-1)^(1/2)*(d*x+c+ 1)^(1/2))+3*a*b^2*((d*x+c)*arccosh(d*x+c)^2-2*arccosh(d*x+c)*(d*x+c-1)^(1/ 2)*(d*x+c+1)^(1/2)+2*d*x+2*c)+3*a^2*b*((d*x+c)*arccosh(d*x+c)-(d*x+c-1)^(1 /2)*(d*x+c+1)^(1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 239 vs. \(2 (106) = 212\).
Time = 0.25 (sec) , antiderivative size = 239, normalized size of antiderivative = 2.10 \[ \int (a+b \text {arccosh}(c+d x))^3 \, dx=\frac {{\left (b^{3} d x + b^{3} c\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )^{3} + {\left (a^{3} + 6 \, a b^{2}\right )} d x + 3 \, {\left (a b^{2} d x + a b^{2} c - \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} b^{3}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )^{2} - 3 \, {\left (2 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} a b^{2} - {\left (a^{2} b + 2 \, b^{3}\right )} d x - {\left (a^{2} b + 2 \, b^{3}\right )} c\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) - 3 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} {\left (a^{2} b + 2 \, b^{3}\right )}}{d} \]
((b^3*d*x + b^3*c)*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1))^3 + (a ^3 + 6*a*b^2)*d*x + 3*(a*b^2*d*x + a*b^2*c - sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*b^3)*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1))^2 - 3*(2*sqrt(d ^2*x^2 + 2*c*d*x + c^2 - 1)*a*b^2 - (a^2*b + 2*b^3)*d*x - (a^2*b + 2*b^3)* c)*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)) - 3*sqrt(d^2*x^2 + 2*c *d*x + c^2 - 1)*(a^2*b + 2*b^3))/d
\[ \int (a+b \text {arccosh}(c+d x))^3 \, dx=\int \left (a + b \operatorname {acosh}{\left (c + d x \right )}\right )^{3}\, dx \]
\[ \int (a+b \text {arccosh}(c+d x))^3 \, dx=\int { {\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{3} \,d x } \]
b^3*x*log(d*x + sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + c)^3 + a^3*x + 3*((d *x + c)*arccosh(d*x + c) - sqrt((d*x + c)^2 - 1))*a^2*b/d + integrate(3*(( c^3 - c)*a*b^2 + (a*b^2*d^3 - b^3*d^3)*x^3 + (3*a*b^2*c*d^2 - 2*b^3*c*d^2) *x^2 + ((c^2 - 1)*a*b^2 + (a*b^2*d^2 - b^3*d^2)*x^2 + (2*a*b^2*c*d - b^3*c *d)*x)*sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + ((3*c^2*d - d)*a*b^2 - (c^2*d - d)*b^3)*x)*log(d*x + sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + c)^2/(d^3*x^ 3 + 3*c*d^2*x^2 + c^3 + (d^2*x^2 + 2*c*d*x + c^2 - 1)*sqrt(d*x + c + 1)*sq rt(d*x + c - 1) + (3*c^2*d - d)*x - c), x)
\[ \int (a+b \text {arccosh}(c+d x))^3 \, dx=\int { {\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{3} \,d x } \]
Timed out. \[ \int (a+b \text {arccosh}(c+d x))^3 \, dx=\int {\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right )}^3 \,d x \]