Integrand size = 10, antiderivative size = 104 \[ \int x^2 \text {arccosh}(a+b x) \, dx=-\frac {x^2 \sqrt {-1+a+b x} \sqrt {1+a+b x}}{9 b}-\frac {\sqrt {-1+a+b x} \sqrt {1+a+b x} \left (4+11 a^2-5 a b x\right )}{18 b^3}+\frac {a \left (3+2 a^2\right ) \text {arccosh}(a+b x)}{6 b^3}+\frac {1}{3} x^3 \text {arccosh}(a+b x) \]
1/6*a*(2*a^2+3)*arccosh(b*x+a)/b^3+1/3*x^3*arccosh(b*x+a)-1/9*x^2*(b*x+a-1 )^(1/2)*(b*x+a+1)^(1/2)/b-1/18*(-5*a*b*x+11*a^2+4)*(b*x+a-1)^(1/2)*(b*x+a+ 1)^(1/2)/b^3
Time = 0.14 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.97 \[ \int x^2 \text {arccosh}(a+b x) \, dx=\frac {-\sqrt {-1+a+b x} \sqrt {1+a+b x} \left (4+11 a^2-5 a b x+2 b^2 x^2\right )+6 b^3 x^3 \text {arccosh}(a+b x)+\left (9 a+6 a^3\right ) \log \left (a+b x+\sqrt {-1+a+b x} \sqrt {1+a+b x}\right )}{18 b^3} \]
(-(Sqrt[-1 + a + b*x]*Sqrt[1 + a + b*x]*(4 + 11*a^2 - 5*a*b*x + 2*b^2*x^2) ) + 6*b^3*x^3*ArcCosh[a + b*x] + (9*a + 6*a^3)*Log[a + b*x + Sqrt[-1 + a + b*x]*Sqrt[1 + a + b*x]])/(18*b^3)
Time = 0.31 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.16, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {6411, 27, 6378, 111, 164, 43}
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.
\(\displaystyle \int x^2 \text {arccosh}(a+b x) \, dx\) |
\(\Big \downarrow \) 6411 |
\(\displaystyle \frac {\int x^2 \text {arccosh}(a+b x)d(a+b x)}{b}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int b^2 x^2 \text {arccosh}(a+b x)d(a+b x)}{b^3}\) |
\(\Big \downarrow \) 6378 |
\(\displaystyle \frac {\frac {1}{3} \int -\frac {b^3 x^3}{\sqrt {a+b x-1} \sqrt {a+b x+1}}d(a+b x)+\frac {1}{3} b^3 x^3 \text {arccosh}(a+b x)}{b^3}\) |
\(\Big \downarrow \) 111 |
\(\displaystyle \frac {\frac {1}{3} \left (\frac {1}{3} \int -\frac {b x \left (3 a^2-5 (a+b x) a+2\right )}{\sqrt {a+b x-1} \sqrt {a+b x+1}}d(a+b x)-\frac {1}{3} b^2 x^2 \sqrt {a+b x-1} \sqrt {a+b x+1}\right )+\frac {1}{3} b^3 x^3 \text {arccosh}(a+b x)}{b^3}\) |
\(\Big \downarrow \) 164 |
\(\displaystyle \frac {\frac {1}{3} \left (\frac {1}{3} \left (\frac {3}{2} a \left (2 a^2+3\right ) \int \frac {1}{\sqrt {a+b x-1} \sqrt {a+b x+1}}d(a+b x)-\frac {1}{2} \sqrt {a+b x-1} \sqrt {a+b x+1} \left (4 \left (4 a^2+1\right )-5 a (a+b x)\right )\right )-\frac {1}{3} b^2 x^2 \sqrt {a+b x-1} \sqrt {a+b x+1}\right )+\frac {1}{3} b^3 x^3 \text {arccosh}(a+b x)}{b^3}\) |
\(\Big \downarrow \) 43 |
\(\displaystyle \frac {\frac {1}{3} \left (\frac {1}{3} \left (\frac {3}{2} a \left (2 a^2+3\right ) \text {arccosh}(a+b x)-\frac {1}{2} \sqrt {a+b x-1} \sqrt {a+b x+1} \left (4 \left (4 a^2+1\right )-5 a (a+b x)\right )\right )-\frac {1}{3} b^2 x^2 \sqrt {a+b x-1} \sqrt {a+b x+1}\right )+\frac {1}{3} b^3 x^3 \text {arccosh}(a+b x)}{b^3}\) |
((b^3*x^3*ArcCosh[a + b*x])/3 + (-1/3*(b^2*x^2*Sqrt[-1 + a + b*x]*Sqrt[1 + a + b*x]) + (-1/2*(Sqrt[-1 + a + b*x]*Sqrt[1 + a + b*x]*(4*(1 + 4*a^2) - 5*a*(a + b*x))) + (3*a*(3 + 2*a^2)*ArcCosh[a + b*x])/2)/3)/3)/b^3
3.1.84.3.1 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[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ ArcCosh[b*(x/a)]/(b*Sqrt[d/b]), x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c + a *d, 0] && GtQ[a, 0] && GtQ[d/b, 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_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_ ))*((g_.) + (h_.)*(x_)), x_] :> Simp[(-(a*d*f*h*(n + 2) + b*c*f*h*(m + 2) - b*d*(f*g + e*h)*(m + n + 3) - b*d*f*h*(m + n + 2)*x))*(a + b*x)^(m + 1)*(( c + d*x)^(n + 1)/(b^2*d^2*(m + n + 2)*(m + n + 3))), x] + Simp[(a^2*d^2*f*h *(n + 1)*(n + 2) + a*b*d*(n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c*d*(f*g + e*h)*(m + 1)*(m + n + 3) + d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d^2*(m + n + 2)*(m + n + 3)) Int[( a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && NeQ[m + n + 2, 0] && NeQ[m + n + 3, 0]
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] - Simp[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]
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]
Leaf count of result is larger than twice the leaf count of optimal. \(202\) vs. \(2(88)=176\).
Time = 0.03 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.95
method | result | size |
derivativedivides | \(\frac {-\frac {\operatorname {arccosh}\left (b x +a \right ) a^{3}}{3}+\operatorname {arccosh}\left (b x +a \right ) a^{2} \left (b x +a \right )-\operatorname {arccosh}\left (b x +a \right ) a \left (b x +a \right )^{2}+\frac {\operatorname {arccosh}\left (b x +a \right ) \left (b x +a \right )^{3}}{3}+\frac {\sqrt {b x +a -1}\, \sqrt {b x +a +1}\, \left (6 a^{3} \ln \left (b x +a +\sqrt {\left (b x +a \right )^{2}-1}\right )-18 a^{2} \sqrt {\left (b x +a \right )^{2}-1}+9 a \left (b x +a \right ) \sqrt {\left (b x +a \right )^{2}-1}-2 \left (b x +a \right )^{2} \sqrt {\left (b x +a \right )^{2}-1}+9 a \ln \left (b x +a +\sqrt {\left (b x +a \right )^{2}-1}\right )-4 \sqrt {\left (b x +a \right )^{2}-1}\right )}{18 \sqrt {\left (b x +a \right )^{2}-1}}}{b^{3}}\) | \(203\) |
default | \(\frac {-\frac {\operatorname {arccosh}\left (b x +a \right ) a^{3}}{3}+\operatorname {arccosh}\left (b x +a \right ) a^{2} \left (b x +a \right )-\operatorname {arccosh}\left (b x +a \right ) a \left (b x +a \right )^{2}+\frac {\operatorname {arccosh}\left (b x +a \right ) \left (b x +a \right )^{3}}{3}+\frac {\sqrt {b x +a -1}\, \sqrt {b x +a +1}\, \left (6 a^{3} \ln \left (b x +a +\sqrt {\left (b x +a \right )^{2}-1}\right )-18 a^{2} \sqrt {\left (b x +a \right )^{2}-1}+9 a \left (b x +a \right ) \sqrt {\left (b x +a \right )^{2}-1}-2 \left (b x +a \right )^{2} \sqrt {\left (b x +a \right )^{2}-1}+9 a \ln \left (b x +a +\sqrt {\left (b x +a \right )^{2}-1}\right )-4 \sqrt {\left (b x +a \right )^{2}-1}\right )}{18 \sqrt {\left (b x +a \right )^{2}-1}}}{b^{3}}\) | \(203\) |
parts | \(\frac {x^{3} \operatorname {arccosh}\left (b x +a \right )}{3}-\frac {\sqrt {b x +a -1}\, \sqrt {b x +a +1}\, \left (2 \,\operatorname {csgn}\left (b \right ) b^{2} x^{2} \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}-5 \,\operatorname {csgn}\left (b \right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, a b x +11 \,\operatorname {csgn}\left (b \right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, a^{2}-6 \ln \left (\left (\sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, \operatorname {csgn}\left (b \right )+b x +a \right ) \operatorname {csgn}\left (b \right )\right ) a^{3}+4 \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, \operatorname {csgn}\left (b \right )-9 \ln \left (\left (\sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, \operatorname {csgn}\left (b \right )+b x +a \right ) \operatorname {csgn}\left (b \right )\right ) a \right ) \operatorname {csgn}\left (b \right )}{18 b^{3} \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}}\) | \(230\) |
1/b^3*(-1/3*arccosh(b*x+a)*a^3+arccosh(b*x+a)*a^2*(b*x+a)-arccosh(b*x+a)*a *(b*x+a)^2+1/3*arccosh(b*x+a)*(b*x+a)^3+1/18*(b*x+a-1)^(1/2)*(b*x+a+1)^(1/ 2)*(6*a^3*ln(b*x+a+((b*x+a)^2-1)^(1/2))-18*a^2*((b*x+a)^2-1)^(1/2)+9*a*(b* x+a)*((b*x+a)^2-1)^(1/2)-2*(b*x+a)^2*((b*x+a)^2-1)^(1/2)+9*a*ln(b*x+a+((b* x+a)^2-1)^(1/2))-4*((b*x+a)^2-1)^(1/2))/((b*x+a)^2-1)^(1/2))
Time = 0.27 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.88 \[ \int x^2 \text {arccosh}(a+b x) \, dx=\frac {3 \, {\left (2 \, b^{3} x^{3} + 2 \, a^{3} + 3 \, a\right )} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) - {\left (2 \, b^{2} x^{2} - 5 \, a b x + 11 \, a^{2} + 4\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}}{18 \, b^{3}} \]
1/18*(3*(2*b^3*x^3 + 2*a^3 + 3*a)*log(b*x + a + sqrt(b^2*x^2 + 2*a*b*x + a ^2 - 1)) - (2*b^2*x^2 - 5*a*b*x + 11*a^2 + 4)*sqrt(b^2*x^2 + 2*a*b*x + a^2 - 1))/b^3
\[ \int x^2 \text {arccosh}(a+b x) \, dx=\int x^{2} \operatorname {acosh}{\left (a + b x \right )}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 212 vs. \(2 (88) = 176\).
Time = 0.21 (sec) , antiderivative size = 212, normalized size of antiderivative = 2.04 \[ \int x^2 \text {arccosh}(a+b x) \, dx=\frac {1}{3} \, x^{3} \operatorname {arcosh}\left (b x + a\right ) - \frac {1}{18} \, b {\left (\frac {2 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} x^{2}}{b^{2}} - \frac {15 \, a^{3} \log \left (2 \, b^{2} x + 2 \, a b + 2 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} b\right )}{b^{4}} - \frac {5 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} a x}{b^{3}} + \frac {9 \, {\left (a^{2} - 1\right )} a \log \left (2 \, b^{2} x + 2 \, a b + 2 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} b\right )}{b^{4}} + \frac {15 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} a^{2}}{b^{4}} - \frac {4 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} {\left (a^{2} - 1\right )}}{b^{4}}\right )} \]
1/3*x^3*arccosh(b*x + a) - 1/18*b*(2*sqrt(b^2*x^2 + 2*a*b*x + a^2 - 1)*x^2 /b^2 - 15*a^3*log(2*b^2*x + 2*a*b + 2*sqrt(b^2*x^2 + 2*a*b*x + a^2 - 1)*b) /b^4 - 5*sqrt(b^2*x^2 + 2*a*b*x + a^2 - 1)*a*x/b^3 + 9*(a^2 - 1)*a*log(2*b ^2*x + 2*a*b + 2*sqrt(b^2*x^2 + 2*a*b*x + a^2 - 1)*b)/b^4 + 15*sqrt(b^2*x^ 2 + 2*a*b*x + a^2 - 1)*a^2/b^4 - 4*sqrt(b^2*x^2 + 2*a*b*x + a^2 - 1)*(a^2 - 1)/b^4)
Time = 0.31 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.27 \[ \int x^2 \text {arccosh}(a+b x) \, dx=\frac {1}{3} \, x^{3} \log \left (b x + a + \sqrt {{\left (b x + a\right )}^{2} - 1}\right ) - \frac {1}{18} \, {\left (\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} {\left (x {\left (\frac {2 \, x}{b^{2}} - \frac {5 \, a}{b^{3}}\right )} + \frac {11 \, a^{2} b + 4 \, b}{b^{5}}\right )} + \frac {3 \, {\left (2 \, a^{3} + 3 \, a\right )} \log \left ({\left | -a b - {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right )} {\left | b \right |} \right |}\right )}{b^{3} {\left | b \right |}}\right )} b \]
1/3*x^3*log(b*x + a + sqrt((b*x + a)^2 - 1)) - 1/18*(sqrt(b^2*x^2 + 2*a*b* x + a^2 - 1)*(x*(2*x/b^2 - 5*a/b^3) + (11*a^2*b + 4*b)/b^5) + 3*(2*a^3 + 3 *a)*log(abs(-a*b - (x*abs(b) - sqrt(b^2*x^2 + 2*a*b*x + a^2 - 1))*abs(b))) /(b^3*abs(b)))*b
Timed out. \[ \int x^2 \text {arccosh}(a+b x) \, dx=\int x^2\,\mathrm {acosh}\left (a+b\,x\right ) \,d x \]