Integrand size = 10, antiderivative size = 154 \[ \int x^m \text {arccosh}(a x)^2 \, dx=\frac {x^{1+m} \text {arccosh}(a x)^2}{1+m}-\frac {2 a x^{2+m} \sqrt {1-a x} \text {arccosh}(a x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},a^2 x^2\right )}{\left (2+3 m+m^2\right ) \sqrt {-1+a x}}-\frac {2 a^2 x^{3+m} \, _3F_2\left (1,\frac {3}{2}+\frac {m}{2},\frac {3}{2}+\frac {m}{2};2+\frac {m}{2},\frac {5}{2}+\frac {m}{2};a^2 x^2\right )}{6+11 m+6 m^2+m^3} \]
x^(1+m)*arccosh(a*x)^2/(1+m)-2*a^2*x^(3+m)*hypergeom([1, 3/2+1/2*m, 3/2+1/ 2*m],[2+1/2*m, 5/2+1/2*m],a^2*x^2)/(m^3+6*m^2+11*m+6)-2*a*x^(2+m)*arccosh( a*x)*hypergeom([1/2, 1+1/2*m],[2+1/2*m],a^2*x^2)*(-a*x+1)^(1/2)/(m^2+3*m+2 )/(a*x-1)^(1/2)
Time = 0.23 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.93 \[ \int x^m \text {arccosh}(a x)^2 \, dx=\frac {x^{1+m} \left (\text {arccosh}(a x)^2-\frac {2 a x \left (\frac {\sqrt {1-a^2 x^2} \text {arccosh}(a x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},a^2 x^2\right )}{\sqrt {-1+a x} \sqrt {1+a x}}+\frac {a x \, _3F_2\left (1,\frac {3}{2}+\frac {m}{2},\frac {3}{2}+\frac {m}{2};2+\frac {m}{2},\frac {5}{2}+\frac {m}{2};a^2 x^2\right )}{3+m}\right )}{2+m}\right )}{1+m} \]
(x^(1 + m)*(ArcCosh[a*x]^2 - (2*a*x*((Sqrt[1 - a^2*x^2]*ArcCosh[a*x]*Hyper geometric2F1[1/2, (2 + m)/2, (4 + m)/2, a^2*x^2])/(Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (a*x*HypergeometricPFQ[{1, 3/2 + m/2, 3/2 + m/2}, {2 + m/2, 5/2 + m/2}, a^2*x^2])/(3 + m)))/(2 + m)))/(1 + m)
Time = 0.47 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.96, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6298, 6364}
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.
\(\displaystyle \int x^m \text {arccosh}(a x)^2 \, dx\) |
\(\Big \downarrow \) 6298 |
\(\displaystyle \frac {x^{m+1} \text {arccosh}(a x)^2}{m+1}-\frac {2 a \int \frac {x^{m+1} \text {arccosh}(a x)}{\sqrt {a x-1} \sqrt {a x+1}}dx}{m+1}\) |
\(\Big \downarrow \) 6364 |
\(\displaystyle \frac {x^{m+1} \text {arccosh}(a x)^2}{m+1}-\frac {2 a \left (\frac {a x^{m+3} \, _3F_2\left (1,\frac {m}{2}+\frac {3}{2},\frac {m}{2}+\frac {3}{2};\frac {m}{2}+2,\frac {m}{2}+\frac {5}{2};a^2 x^2\right )}{m^2+5 m+6}+\frac {\sqrt {1-a x} x^{m+2} \text {arccosh}(a x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+2}{2},\frac {m+4}{2},a^2 x^2\right )}{(m+2) \sqrt {a x-1}}\right )}{m+1}\) |
(x^(1 + m)*ArcCosh[a*x]^2)/(1 + m) - (2*a*((x^(2 + m)*Sqrt[1 - a*x]*ArcCos h[a*x]*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, a^2*x^2])/((2 + m)*Sqr t[-1 + a*x]) + (a*x^(3 + m)*HypergeometricPFQ[{1, 3/2 + m/2, 3/2 + m/2}, { 2 + m/2, 5/2 + m/2}, a^2*x^2])/(6 + 5*m + m^2)))/(1 + m)
3.2.17.3.1 Defintions of rubi rules used
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_.))*((f_.)*(x_))^(m_))/(Sqrt[(d1_) + ( e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)]), x_Symbol] :> Simp[((f*x)^(m + 1)/(f *(m + 1)))*Simp[Sqrt[1 - c^2*x^2]/(Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x])]*(a + b *ArcCosh[c*x])*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, c^2*x^2], x] + Simp[b*c*((f*x)^(m + 2)/(f^2*(m + 1)*(m + 2)))*Simp[Sqrt[1 + c*x]/Sqrt[d1 + e1*x]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]]*HypergeometricPFQ[{1, 1 + m/2, 1 + m/2}, {3/2 + m/2, 2 + m/2}, c^2*x^2], x] /; FreeQ[{a, b, c, d1, e1, d2 , e2, f, m}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && !IntegerQ[m]
\[\int x^{m} \operatorname {arccosh}\left (a x \right )^{2}d x\]
\[ \int x^m \text {arccosh}(a x)^2 \, dx=\int { x^{m} \operatorname {arcosh}\left (a x\right )^{2} \,d x } \]
\[ \int x^m \text {arccosh}(a x)^2 \, dx=\int x^{m} \operatorname {acosh}^{2}{\left (a x \right )}\, dx \]
\[ \int x^m \text {arccosh}(a x)^2 \, dx=\int { x^{m} \operatorname {arcosh}\left (a x\right )^{2} \,d x } \]
x*x^m*log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1))^2/(m + 1) - integrate(2*(sqrt (a*x + 1)*sqrt(a*x - 1)*a^2*x^2*x^m + (a^3*x^3 - a*x)*x^m)*log(a*x + sqrt( a*x + 1)*sqrt(a*x - 1))/(a^3*(m + 1)*x^3 - a*(m + 1)*x + (a^2*(m + 1)*x^2 - m - 1)*sqrt(a*x + 1)*sqrt(a*x - 1)), x)
\[ \int x^m \text {arccosh}(a x)^2 \, dx=\int { x^{m} \operatorname {arcosh}\left (a x\right )^{2} \,d x } \]
Timed out. \[ \int x^m \text {arccosh}(a x)^2 \, dx=\int x^m\,{\mathrm {acosh}\left (a\,x\right )}^2 \,d x \]