Integrand size = 16, antiderivative size = 213 \[ \int x^2 \sqrt {a+b \text {arccosh}(c x)} \, dx=\frac {1}{3} x^3 \sqrt {a+b \text {arccosh}(c x)}-\frac {\sqrt {b} e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{16 c^3}-\frac {\sqrt {b} e^{\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{48 c^3}-\frac {\sqrt {b} e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{16 c^3}-\frac {\sqrt {b} e^{-\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{48 c^3} \]
-1/144*exp(3*a/b)*erf(3^(1/2)*(a+b*arccosh(c*x))^(1/2)/b^(1/2))*b^(1/2)*3^ (1/2)*Pi^(1/2)/c^3-1/144*erfi(3^(1/2)*(a+b*arccosh(c*x))^(1/2)/b^(1/2))*b^ (1/2)*3^(1/2)*Pi^(1/2)/c^3/exp(3*a/b)-1/16*exp(a/b)*erf((a+b*arccosh(c*x)) ^(1/2)/b^(1/2))*b^(1/2)*Pi^(1/2)/c^3-1/16*erfi((a+b*arccosh(c*x))^(1/2)/b^ (1/2))*b^(1/2)*Pi^(1/2)/c^3/exp(a/b)+1/3*x^3*(a+b*arccosh(c*x))^(1/2)
Time = 0.36 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.00 \[ \int x^2 \sqrt {a+b \text {arccosh}(c x)} \, dx=\frac {e^{-\frac {3 a}{b}} \sqrt {a+b \text {arccosh}(c x)} \left (9 e^{\frac {4 a}{b}} \sqrt {-\frac {a+b \text {arccosh}(c x)}{b}} \Gamma \left (\frac {3}{2},\frac {a}{b}+\text {arccosh}(c x)\right )+\sqrt {3} \sqrt {\frac {a}{b}+\text {arccosh}(c x)} \Gamma \left (\frac {3}{2},-\frac {3 (a+b \text {arccosh}(c x))}{b}\right )+9 e^{\frac {2 a}{b}} \sqrt {\frac {a}{b}+\text {arccosh}(c x)} \Gamma \left (\frac {3}{2},-\frac {a+b \text {arccosh}(c x)}{b}\right )+\sqrt {3} e^{\frac {6 a}{b}} \sqrt {-\frac {a+b \text {arccosh}(c x)}{b}} \Gamma \left (\frac {3}{2},\frac {3 (a+b \text {arccosh}(c x))}{b}\right )\right )}{72 c^3 \sqrt {-\frac {(a+b \text {arccosh}(c x))^2}{b^2}}} \]
(Sqrt[a + b*ArcCosh[c*x]]*(9*E^((4*a)/b)*Sqrt[-((a + b*ArcCosh[c*x])/b)]*G amma[3/2, a/b + ArcCosh[c*x]] + Sqrt[3]*Sqrt[a/b + ArcCosh[c*x]]*Gamma[3/2 , (-3*(a + b*ArcCosh[c*x]))/b] + 9*E^((2*a)/b)*Sqrt[a/b + ArcCosh[c*x]]*Ga mma[3/2, -((a + b*ArcCosh[c*x])/b)] + Sqrt[3]*E^((6*a)/b)*Sqrt[-((a + b*Ar cCosh[c*x])/b)]*Gamma[3/2, (3*(a + b*ArcCosh[c*x]))/b]))/(72*c^3*E^((3*a)/ b)*Sqrt[-((a + b*ArcCosh[c*x])^2/b^2)])
Time = 1.07 (sec) , antiderivative size = 209, normalized size of antiderivative = 0.98, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {6299, 6368, 3042, 3793, 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.
\(\displaystyle \int x^2 \sqrt {a+b \text {arccosh}(c x)} \, dx\) |
\(\Big \downarrow \) 6299 |
\(\displaystyle \frac {1}{3} x^3 \sqrt {a+b \text {arccosh}(c x)}-\frac {1}{6} b c \int \frac {x^3}{\sqrt {c x-1} \sqrt {c x+1} \sqrt {a+b \text {arccosh}(c x)}}dx\) |
\(\Big \downarrow \) 6368 |
\(\displaystyle \frac {1}{3} x^3 \sqrt {a+b \text {arccosh}(c x)}-\frac {\int \frac {\cosh ^3\left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}{\sqrt {a+b \text {arccosh}(c x)}}d(a+b \text {arccosh}(c x))}{6 c^3}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{3} x^3 \sqrt {a+b \text {arccosh}(c x)}-\frac {\int \frac {\sin \left (\frac {i a}{b}-\frac {i (a+b \text {arccosh}(c x))}{b}+\frac {\pi }{2}\right )^3}{\sqrt {a+b \text {arccosh}(c x)}}d(a+b \text {arccosh}(c x))}{6 c^3}\) |
\(\Big \downarrow \) 3793 |
\(\displaystyle \frac {1}{3} x^3 \sqrt {a+b \text {arccosh}(c x)}-\frac {\int \left (\frac {\cosh \left (\frac {3 a}{b}-\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{4 \sqrt {a+b \text {arccosh}(c x)}}+\frac {3 \cosh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}{4 \sqrt {a+b \text {arccosh}(c x)}}\right )d(a+b \text {arccosh}(c x))}{6 c^3}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{3} x^3 \sqrt {a+b \text {arccosh}(c x)}-\frac {\frac {3}{8} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \sqrt {b} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )+\frac {3}{8} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \sqrt {b} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{6 c^3}\) |
(x^3*Sqrt[a + b*ArcCosh[c*x]])/3 - ((3*Sqrt[b]*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/8 + (Sqrt[b]*E^((3*a)/b)*Sqrt[Pi/3]*Erf[(Sqrt [3]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/8 + (3*Sqrt[b]*Sqrt[Pi]*Erfi[Sqrt[ a + b*ArcCosh[c*x]]/Sqrt[b]])/(8*E^(a/b)) + (Sqrt[b]*Sqrt[Pi/3]*Erfi[(Sqrt [3]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(8*E^((3*a)/b)))/(6*c^3)
3.2.42.3.1 Defintions of rubi rules used
Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> In t[ExpandTrigReduce[(c + d*x)^m, Sin[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f , m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1]))
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[ x^(m + 1)*((a + b*ArcCosh[c*x])^n/(m + 1)), x] - Simp[b*c*(n/(m + 1)) Int [x^(m + 1)*((a + b*ArcCosh[c*x])^(n - 1)/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])), x ], x] /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && GtQ[n, 0]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d1_) + (e1_.)*(x _))^(p_.)*((d2_) + (e2_.)*(x_))^(p_.), x_Symbol] :> Simp[(1/(b*c^(m + 1)))* Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*x)^p/(-1 + c*x)^p] Subst[In t[x^n*Cosh[-a/b + x/b]^m*Sinh[-a/b + x/b]^(2*p + 1), x], x, a + b*ArcCosh[c *x]], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[e1, c*d1] && EqQ[ e2, (-c)*d2] && IGtQ[p + 3/2, 0] && IGtQ[m, 0]
\[\int x^{2} \sqrt {a +b \,\operatorname {arccosh}\left (c x \right )}d x\]
Exception generated. \[ \int x^2 \sqrt {a+b \text {arccosh}(c x)} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int x^2 \sqrt {a+b \text {arccosh}(c x)} \, dx=\int x^{2} \sqrt {a + b \operatorname {acosh}{\left (c x \right )}}\, dx \]
\[ \int x^2 \sqrt {a+b \text {arccosh}(c x)} \, dx=\int { \sqrt {b \operatorname {arcosh}\left (c x\right ) + a} x^{2} \,d x } \]
\[ \int x^2 \sqrt {a+b \text {arccosh}(c x)} \, dx=\int { \sqrt {b \operatorname {arcosh}\left (c x\right ) + a} x^{2} \,d x } \]
Timed out. \[ \int x^2 \sqrt {a+b \text {arccosh}(c x)} \, dx=\int x^2\,\sqrt {a+b\,\mathrm {acosh}\left (c\,x\right )} \,d x \]