Integrand size = 8, antiderivative size = 90 \[ \int x \text {arccosh}(a+b x) \, dx=\frac {3 a \sqrt {-1+a+b x} \sqrt {1+a+b x}}{4 b^2}-\frac {x \sqrt {-1+a+b x} \sqrt {1+a+b x}}{4 b}-\frac {\left (1+2 a^2\right ) \text {arccosh}(a+b x)}{4 b^2}+\frac {1}{2} x^2 \text {arccosh}(a+b x) \]
-1/4*(2*a^2+1)*arccosh(b*x+a)/b^2+1/2*x^2*arccosh(b*x+a)+3/4*a*(b*x+a-1)^( 1/2)*(b*x+a+1)^(1/2)/b^2-1/4*x*(b*x+a-1)^(1/2)*(b*x+a+1)^(1/2)/b
Time = 0.08 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.97 \[ \int x \text {arccosh}(a+b x) \, dx=\frac {(3 a-b x) \sqrt {-1+a+b x} \sqrt {1+a+b x}+2 b^2 x^2 \text {arccosh}(a+b x)-\left (1+2 a^2\right ) \log \left (a+b x+\sqrt {-1+a+b x} \sqrt {1+a+b x}\right )}{4 b^2} \]
((3*a - b*x)*Sqrt[-1 + a + b*x]*Sqrt[1 + a + b*x] + 2*b^2*x^2*ArcCosh[a + b*x] - (1 + 2*a^2)*Log[a + b*x + Sqrt[-1 + a + b*x]*Sqrt[1 + a + b*x]])/(4 *b^2)
Time = 0.27 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.06, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.875, Rules used = {6411, 25, 27, 6378, 101, 90, 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.
\(\displaystyle \int x \text {arccosh}(a+b x) \, dx\) |
\(\Big \downarrow \) 6411 |
\(\displaystyle \frac {\int x \text {arccosh}(a+b x)d(a+b x)}{b}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\int -x \text {arccosh}(a+b x)d(a+b x)}{b}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\int -b x \text {arccosh}(a+b x)d(a+b x)}{b^2}\) |
\(\Big \downarrow \) 6378 |
\(\displaystyle -\frac {\frac {1}{2} \int \frac {b^2 x^2}{\sqrt {a+b x-1} \sqrt {a+b x+1}}d(a+b x)-\frac {1}{2} b^2 x^2 \text {arccosh}(a+b x)}{b^2}\) |
\(\Big \downarrow \) 101 |
\(\displaystyle -\frac {\frac {1}{2} \left (\frac {1}{2} \int \frac {2 a^2-3 (a+b x) a+1}{\sqrt {a+b x-1} \sqrt {a+b x+1}}d(a+b x)+\frac {1}{2} b x \sqrt {a+b x-1} \sqrt {a+b x+1}\right )-\frac {1}{2} b^2 x^2 \text {arccosh}(a+b x)}{b^2}\) |
\(\Big \downarrow \) 90 |
\(\displaystyle -\frac {\frac {1}{2} \left (\frac {1}{2} \left (\left (2 a^2+1\right ) \int \frac {1}{\sqrt {a+b x-1} \sqrt {a+b x+1}}d(a+b x)-3 a \sqrt {a+b x-1} \sqrt {a+b x+1}\right )+\frac {1}{2} b x \sqrt {a+b x-1} \sqrt {a+b x+1}\right )-\frac {1}{2} b^2 x^2 \text {arccosh}(a+b x)}{b^2}\) |
\(\Big \downarrow \) 43 |
\(\displaystyle -\frac {\frac {1}{2} \left (\frac {1}{2} \left (\left (2 a^2+1\right ) \text {arccosh}(a+b x)-3 a \sqrt {a+b x-1} \sqrt {a+b x+1}\right )+\frac {1}{2} b x \sqrt {a+b x-1} \sqrt {a+b x+1}\right )-\frac {1}{2} b^2 x^2 \text {arccosh}(a+b x)}{b^2}\) |
-((-1/2*(b^2*x^2*ArcCosh[a + b*x]) + ((b*x*Sqrt[-1 + a + b*x]*Sqrt[1 + a + b*x])/2 + (-3*a*Sqrt[-1 + a + b*x]*Sqrt[1 + a + b*x] + (1 + 2*a^2)*ArcCos h[a + b*x])/2)/2)/b^2)
3.1.85.3.1 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_))*((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] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p + 2)) Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0]
Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( p_), x_] :> Simp[b*(a + b*x)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 3))), x] + Simp[1/(d*f*(n + p + 3)) Int[(c + d*x)^n*(e + f*x)^p*Simp [a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f *(n + p + 4) - b*(d*e*(n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 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]
Time = 0.03 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.26
method | result | size |
derivativedivides | \(\frac {\frac {\operatorname {arccosh}\left (b x +a \right ) \left (b x +a \right )^{2}}{2}-\operatorname {arccosh}\left (b x +a \right ) a \left (b x +a \right )-\frac {\sqrt {b x +a -1}\, \sqrt {b x +a +1}\, \left (-4 a \sqrt {\left (b x +a \right )^{2}-1}+\left (b x +a \right ) \sqrt {\left (b x +a \right )^{2}-1}+\ln \left (b x +a +\sqrt {\left (b x +a \right )^{2}-1}\right )\right )}{4 \sqrt {\left (b x +a \right )^{2}-1}}}{b^{2}}\) | \(113\) |
default | \(\frac {\frac {\operatorname {arccosh}\left (b x +a \right ) \left (b x +a \right )^{2}}{2}-\operatorname {arccosh}\left (b x +a \right ) a \left (b x +a \right )-\frac {\sqrt {b x +a -1}\, \sqrt {b x +a +1}\, \left (-4 a \sqrt {\left (b x +a \right )^{2}-1}+\left (b x +a \right ) \sqrt {\left (b x +a \right )^{2}-1}+\ln \left (b x +a +\sqrt {\left (b x +a \right )^{2}-1}\right )\right )}{4 \sqrt {\left (b x +a \right )^{2}-1}}}{b^{2}}\) | \(113\) |
parts | \(\frac {x^{2} \operatorname {arccosh}\left (b x +a \right )}{2}+\frac {\sqrt {b x +a -1}\, \sqrt {b x +a +1}\, \left (-\sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, \operatorname {csgn}\left (b \right ) b x +3 \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, \operatorname {csgn}\left (b \right ) a -2 \ln \left (\left (\sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, \operatorname {csgn}\left (b \right )+b x +a \right ) \operatorname {csgn}\left (b \right )\right ) a^{2}-\ln \left (\left (\sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, \operatorname {csgn}\left (b \right )+b x +a \right ) \operatorname {csgn}\left (b \right )\right )\right ) \operatorname {csgn}\left (b \right )}{4 b^{2} \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}}\) | \(174\) |
1/b^2*(1/2*arccosh(b*x+a)*(b*x+a)^2-arccosh(b*x+a)*a*(b*x+a)-1/4*(b*x+a-1) ^(1/2)*(b*x+a+1)^(1/2)*(-4*a*((b*x+a)^2-1)^(1/2)+(b*x+a)*((b*x+a)^2-1)^(1/ 2)+ln(b*x+a+((b*x+a)^2-1)^(1/2)))/((b*x+a)^2-1)^(1/2))
Time = 0.25 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.83 \[ \int x \text {arccosh}(a+b x) \, dx=\frac {{\left (2 \, b^{2} x^{2} - 2 \, a^{2} - 1\right )} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} {\left (b x - 3 \, a\right )}}{4 \, b^{2}} \]
1/4*((2*b^2*x^2 - 2*a^2 - 1)*log(b*x + a + sqrt(b^2*x^2 + 2*a*b*x + a^2 - 1)) - sqrt(b^2*x^2 + 2*a*b*x + a^2 - 1)*(b*x - 3*a))/b^2
\[ \int x \text {arccosh}(a+b x) \, dx=\int x \operatorname {acosh}{\left (a + b x \right )}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 151 vs. \(2 (74) = 148\).
Time = 0.21 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.68 \[ \int x \text {arccosh}(a+b x) \, dx=\frac {1}{2} \, x^{2} \operatorname {arcosh}\left (b x + a\right ) - \frac {1}{4} \, b {\left (\frac {3 \, a^{2} \log \left (2 \, b^{2} x + 2 \, a b + 2 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} b\right )}{b^{3}} + \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} x}{b^{2}} - \frac {{\left (a^{2} - 1\right )} \log \left (2 \, b^{2} x + 2 \, a b + 2 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} b\right )}{b^{3}} - \frac {3 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} a}{b^{3}}\right )} \]
1/2*x^2*arccosh(b*x + a) - 1/4*b*(3*a^2*log(2*b^2*x + 2*a*b + 2*sqrt(b^2*x ^2 + 2*a*b*x + a^2 - 1)*b)/b^3 + sqrt(b^2*x^2 + 2*a*b*x + a^2 - 1)*x/b^2 - (a^2 - 1)*log(2*b^2*x + 2*a*b + 2*sqrt(b^2*x^2 + 2*a*b*x + a^2 - 1)*b)/b^ 3 - 3*sqrt(b^2*x^2 + 2*a*b*x + a^2 - 1)*a/b^3)
Time = 0.32 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.24 \[ \int x \text {arccosh}(a+b x) \, dx=\frac {1}{2} \, x^{2} \log \left (b x + a + \sqrt {{\left (b x + a\right )}^{2} - 1}\right ) - \frac {1}{4} \, {\left (\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} {\left (\frac {x}{b^{2}} - \frac {3 \, a}{b^{3}}\right )} - \frac {{\left (2 \, a^{2} + 1\right )} \log \left ({\left | -a b - {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right )} {\left | b \right |} \right |}\right )}{b^{2} {\left | b \right |}}\right )} b \]
1/2*x^2*log(b*x + a + sqrt((b*x + a)^2 - 1)) - 1/4*(sqrt(b^2*x^2 + 2*a*b*x + a^2 - 1)*(x/b^2 - 3*a/b^3) - (2*a^2 + 1)*log(abs(-a*b - (x*abs(b) - sqr t(b^2*x^2 + 2*a*b*x + a^2 - 1))*abs(b)))/(b^2*abs(b)))*b
Timed out. \[ \int x \text {arccosh}(a+b x) \, dx=\int x\,\mathrm {acosh}\left (a+b\,x\right ) \,d x \]