Integrand size = 16, antiderivative size = 276 \[ \int \frac {x^2}{(a+b \text {arccosh}(c x))^{5/2}} \, dx=-\frac {2 x^2 \sqrt {-1+c x} \sqrt {1+c x}}{3 b c (a+b \text {arccosh}(c x))^{3/2}}+\frac {8 x}{3 b^2 c^2 \sqrt {a+b \text {arccosh}(c x)}}-\frac {4 x^3}{b^2 \sqrt {a+b \text {arccosh}(c x)}}-\frac {e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{6 b^{5/2} c^3}-\frac {e^{\frac {3 a}{b}} \sqrt {3 \pi } \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{2 b^{5/2} c^3}+\frac {e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{6 b^{5/2} c^3}+\frac {e^{-\frac {3 a}{b}} \sqrt {3 \pi } \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{2 b^{5/2} c^3} \]
-1/6*exp(a/b)*erf((a+b*arccosh(c*x))^(1/2)/b^(1/2))*Pi^(1/2)/b^(5/2)/c^3+1 /6*erfi((a+b*arccosh(c*x))^(1/2)/b^(1/2))*Pi^(1/2)/b^(5/2)/c^3/exp(a/b)-1/ 2*exp(3*a/b)*erf(3^(1/2)*(a+b*arccosh(c*x))^(1/2)/b^(1/2))*3^(1/2)*Pi^(1/2 )/b^(5/2)/c^3+1/2*erfi(3^(1/2)*(a+b*arccosh(c*x))^(1/2)/b^(1/2))*3^(1/2)*P i^(1/2)/b^(5/2)/c^3/exp(3*a/b)-2/3*x^2*(c*x-1)^(1/2)*(c*x+1)^(1/2)/b/c/(a+ b*arccosh(c*x))^(3/2)+8/3*x/b^2/c^2/(a+b*arccosh(c*x))^(1/2)-4*x^3/b^2/(a+ b*arccosh(c*x))^(1/2)
Time = 1.50 (sec) , antiderivative size = 340, normalized size of antiderivative = 1.23 \[ \int \frac {x^2}{(a+b \text {arccosh}(c x))^{5/2}} \, dx=\frac {e^{-3 \left (\frac {a}{b}+\text {arccosh}(c x)\right )} \left (2 e^{\frac {4 a}{b}+3 \text {arccosh}(c x)} \sqrt {\frac {a}{b}+\text {arccosh}(c x)} (a+b \text {arccosh}(c x)) \Gamma \left (\frac {1}{2},\frac {a}{b}+\text {arccosh}(c x)\right )-6 \sqrt {3} b e^{3 \text {arccosh}(c x)} \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^{3/2} \Gamma \left (\frac {1}{2},-\frac {3 (a+b \text {arccosh}(c x))}{b}\right )-2 b e^{\frac {2 a}{b}+3 \text {arccosh}(c x)} \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^{3/2} \Gamma \left (\frac {1}{2},-\frac {a+b \text {arccosh}(c x)}{b}\right )+e^{\frac {3 a}{b}} \left (-\left (\left (1+e^{2 \text {arccosh}(c x)}\right ) \left (a \left (6-4 e^{2 \text {arccosh}(c x)}+6 e^{4 \text {arccosh}(c x)}\right )+b \left (-1+6 \text {arccosh}(c x)-4 e^{2 \text {arccosh}(c x)} \text {arccosh}(c x)+e^{4 \text {arccosh}(c x)} (1+6 \text {arccosh}(c x))\right )\right )\right )+6 \sqrt {3} e^{3 \left (\frac {a}{b}+\text {arccosh}(c x)\right )} \sqrt {\frac {a}{b}+\text {arccosh}(c x)} (a+b \text {arccosh}(c x)) \Gamma \left (\frac {1}{2},\frac {3 (a+b \text {arccosh}(c x))}{b}\right )\right )\right )}{12 b^2 c^3 (a+b \text {arccosh}(c x))^{3/2}} \]
(2*E^((4*a)/b + 3*ArcCosh[c*x])*Sqrt[a/b + ArcCosh[c*x]]*(a + b*ArcCosh[c* x])*Gamma[1/2, a/b + ArcCosh[c*x]] - 6*Sqrt[3]*b*E^(3*ArcCosh[c*x])*(-((a + b*ArcCosh[c*x])/b))^(3/2)*Gamma[1/2, (-3*(a + b*ArcCosh[c*x]))/b] - 2*b* E^((2*a)/b + 3*ArcCosh[c*x])*(-((a + b*ArcCosh[c*x])/b))^(3/2)*Gamma[1/2, -((a + b*ArcCosh[c*x])/b)] + E^((3*a)/b)*(-((1 + E^(2*ArcCosh[c*x]))*(a*(6 - 4*E^(2*ArcCosh[c*x]) + 6*E^(4*ArcCosh[c*x])) + b*(-1 + 6*ArcCosh[c*x] - 4*E^(2*ArcCosh[c*x])*ArcCosh[c*x] + E^(4*ArcCosh[c*x])*(1 + 6*ArcCosh[c*x ])))) + 6*Sqrt[3]*E^(3*(a/b + ArcCosh[c*x]))*Sqrt[a/b + ArcCosh[c*x]]*(a + b*ArcCosh[c*x])*Gamma[1/2, (3*(a + b*ArcCosh[c*x]))/b]))/(12*b^2*c^3*E^(3 *(a/b + ArcCosh[c*x]))*(a + b*ArcCosh[c*x])^(3/2))
Result contains complex when optimal does not.
Time = 2.32 (sec) , antiderivative size = 392, normalized size of antiderivative = 1.42, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.875, Rules used = {6301, 6366, 6296, 25, 3042, 26, 3789, 2611, 2633, 2634, 6302, 25, 5971, 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.
| \(\displaystyle \int \frac {x^2}{(a+b \text {arccosh}(c x))^{5/2}} \, dx\) |
| \(\Big \downarrow \) 6301 |
| \(\displaystyle \frac {2 c \int \frac {x^3}{\sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^{3/2}}dx}{b}-\frac {4 \int \frac {x}{\sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^{3/2}}dx}{3 b c}-\frac {2 x^2 \sqrt {c x-1} \sqrt {c x+1}}{3 b c (a+b \text {arccosh}(c x))^{3/2}}\) |
| \(\Big \downarrow \) 6366 |
| \(\displaystyle \frac {2 c \left (\frac {6 \int \frac {x^2}{\sqrt {a+b \text {arccosh}(c x)}}dx}{b c}-\frac {2 x^3}{b c \sqrt {a+b \text {arccosh}(c x)}}\right )}{b}-\frac {4 \left (\frac {2 \int \frac {1}{\sqrt {a+b \text {arccosh}(c x)}}dx}{b c}-\frac {2 x}{b c \sqrt {a+b \text {arccosh}(c x)}}\right )}{3 b c}-\frac {2 x^2 \sqrt {c x-1} \sqrt {c x+1}}{3 b c (a+b \text {arccosh}(c x))^{3/2}}\) |
| \(\Big \downarrow \) 6296 |
| \(\displaystyle -\frac {4 \left (\frac {2 \int -\frac {\sinh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}{\sqrt {a+b \text {arccosh}(c x)}}d(a+b \text {arccosh}(c x))}{b^2 c^2}-\frac {2 x}{b c \sqrt {a+b \text {arccosh}(c x)}}\right )}{3 b c}+\frac {2 c \left (\frac {6 \int \frac {x^2}{\sqrt {a+b \text {arccosh}(c x)}}dx}{b c}-\frac {2 x^3}{b c \sqrt {a+b \text {arccosh}(c x)}}\right )}{b}-\frac {2 x^2 \sqrt {c x-1} \sqrt {c x+1}}{3 b c (a+b \text {arccosh}(c x))^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {4 \left (-\frac {2 \int \frac {\sinh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}{\sqrt {a+b \text {arccosh}(c x)}}d(a+b \text {arccosh}(c x))}{b^2 c^2}-\frac {2 x}{b c \sqrt {a+b \text {arccosh}(c x)}}\right )}{3 b c}+\frac {2 c \left (\frac {6 \int \frac {x^2}{\sqrt {a+b \text {arccosh}(c x)}}dx}{b c}-\frac {2 x^3}{b c \sqrt {a+b \text {arccosh}(c x)}}\right )}{b}-\frac {2 x^2 \sqrt {c x-1} \sqrt {c x+1}}{3 b c (a+b \text {arccosh}(c x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {4 \left (-\frac {2 x}{b c \sqrt {a+b \text {arccosh}(c x)}}-\frac {2 \int -\frac {i \sin \left (\frac {i a}{b}-\frac {i (a+b \text {arccosh}(c x))}{b}\right )}{\sqrt {a+b \text {arccosh}(c x)}}d(a+b \text {arccosh}(c x))}{b^2 c^2}\right )}{3 b c}+\frac {2 c \left (\frac {6 \int \frac {x^2}{\sqrt {a+b \text {arccosh}(c x)}}dx}{b c}-\frac {2 x^3}{b c \sqrt {a+b \text {arccosh}(c x)}}\right )}{b}-\frac {2 x^2 \sqrt {c x-1} \sqrt {c x+1}}{3 b c (a+b \text {arccosh}(c x))^{3/2}}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {4 \left (-\frac {2 x}{b c \sqrt {a+b \text {arccosh}(c x)}}+\frac {2 i \int \frac {\sin \left (\frac {i a}{b}-\frac {i (a+b \text {arccosh}(c x))}{b}\right )}{\sqrt {a+b \text {arccosh}(c x)}}d(a+b \text {arccosh}(c x))}{b^2 c^2}\right )}{3 b c}+\frac {2 c \left (\frac {6 \int \frac {x^2}{\sqrt {a+b \text {arccosh}(c x)}}dx}{b c}-\frac {2 x^3}{b c \sqrt {a+b \text {arccosh}(c x)}}\right )}{b}-\frac {2 x^2 \sqrt {c x-1} \sqrt {c x+1}}{3 b c (a+b \text {arccosh}(c x))^{3/2}}\) |
| \(\Big \downarrow \) 3789 |
| \(\displaystyle -\frac {4 \left (-\frac {2 x}{b c \sqrt {a+b \text {arccosh}(c x)}}+\frac {2 i \left (\frac {1}{2} i \int \frac {e^{-\text {arccosh}(c x)}}{\sqrt {a+b \text {arccosh}(c x)}}d(a+b \text {arccosh}(c x))-\frac {1}{2} i \int \frac {e^{\text {arccosh}(c x)}}{\sqrt {a+b \text {arccosh}(c x)}}d(a+b \text {arccosh}(c x))\right )}{b^2 c^2}\right )}{3 b c}+\frac {2 c \left (\frac {6 \int \frac {x^2}{\sqrt {a+b \text {arccosh}(c x)}}dx}{b c}-\frac {2 x^3}{b c \sqrt {a+b \text {arccosh}(c x)}}\right )}{b}-\frac {2 x^2 \sqrt {c x-1} \sqrt {c x+1}}{3 b c (a+b \text {arccosh}(c x))^{3/2}}\) |
| \(\Big \downarrow \) 2611 |
| \(\displaystyle -\frac {4 \left (-\frac {2 x}{b c \sqrt {a+b \text {arccosh}(c x)}}+\frac {2 i \left (i \int e^{\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}}d\sqrt {a+b \text {arccosh}(c x)}-i \int e^{\frac {a+b \text {arccosh}(c x)}{b}-\frac {a}{b}}d\sqrt {a+b \text {arccosh}(c x)}\right )}{b^2 c^2}\right )}{3 b c}+\frac {2 c \left (\frac {6 \int \frac {x^2}{\sqrt {a+b \text {arccosh}(c x)}}dx}{b c}-\frac {2 x^3}{b c \sqrt {a+b \text {arccosh}(c x)}}\right )}{b}-\frac {2 x^2 \sqrt {c x-1} \sqrt {c x+1}}{3 b c (a+b \text {arccosh}(c x))^{3/2}}\) |
| \(\Big \downarrow \) 2633 |
| \(\displaystyle -\frac {4 \left (-\frac {2 x}{b c \sqrt {a+b \text {arccosh}(c x)}}+\frac {2 i \left (i \int e^{\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}}d\sqrt {a+b \text {arccosh}(c x)}-\frac {1}{2} i \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )\right )}{b^2 c^2}\right )}{3 b c}+\frac {2 c \left (\frac {6 \int \frac {x^2}{\sqrt {a+b \text {arccosh}(c x)}}dx}{b c}-\frac {2 x^3}{b c \sqrt {a+b \text {arccosh}(c x)}}\right )}{b}-\frac {2 x^2 \sqrt {c x-1} \sqrt {c x+1}}{3 b c (a+b \text {arccosh}(c x))^{3/2}}\) |
| \(\Big \downarrow \) 2634 |
| \(\displaystyle \frac {2 c \left (\frac {6 \int \frac {x^2}{\sqrt {a+b \text {arccosh}(c x)}}dx}{b c}-\frac {2 x^3}{b c \sqrt {a+b \text {arccosh}(c x)}}\right )}{b}-\frac {4 \left (-\frac {2 x}{b c \sqrt {a+b \text {arccosh}(c x)}}+\frac {2 i \left (\frac {1}{2} i \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )-\frac {1}{2} i \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )\right )}{b^2 c^2}\right )}{3 b c}-\frac {2 x^2 \sqrt {c x-1} \sqrt {c x+1}}{3 b c (a+b \text {arccosh}(c x))^{3/2}}\) |
| \(\Big \downarrow \) 6302 |
| \(\displaystyle \frac {2 c \left (\frac {6 \int -\frac {\cosh ^2\left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right ) \sinh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}{\sqrt {a+b \text {arccosh}(c x)}}d(a+b \text {arccosh}(c x))}{b^2 c^4}-\frac {2 x^3}{b c \sqrt {a+b \text {arccosh}(c x)}}\right )}{b}-\frac {4 \left (-\frac {2 x}{b c \sqrt {a+b \text {arccosh}(c x)}}+\frac {2 i \left (\frac {1}{2} i \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )-\frac {1}{2} i \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )\right )}{b^2 c^2}\right )}{3 b c}-\frac {2 x^2 \sqrt {c x-1} \sqrt {c x+1}}{3 b c (a+b \text {arccosh}(c x))^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 c \left (-\frac {6 \int \frac {\cosh ^2\left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right ) \sinh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}{\sqrt {a+b \text {arccosh}(c x)}}d(a+b \text {arccosh}(c x))}{b^2 c^4}-\frac {2 x^3}{b c \sqrt {a+b \text {arccosh}(c x)}}\right )}{b}-\frac {4 \left (-\frac {2 x}{b c \sqrt {a+b \text {arccosh}(c x)}}+\frac {2 i \left (\frac {1}{2} i \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )-\frac {1}{2} i \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )\right )}{b^2 c^2}\right )}{3 b c}-\frac {2 x^2 \sqrt {c x-1} \sqrt {c x+1}}{3 b c (a+b \text {arccosh}(c x))^{3/2}}\) |
| \(\Big \downarrow \) 5971 |
| \(\displaystyle \frac {2 c \left (-\frac {6 \int \left (\frac {\sinh \left (\frac {3 a}{b}-\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{4 \sqrt {a+b \text {arccosh}(c x)}}+\frac {\sinh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}{4 \sqrt {a+b \text {arccosh}(c x)}}\right )d(a+b \text {arccosh}(c x))}{b^2 c^4}-\frac {2 x^3}{b c \sqrt {a+b \text {arccosh}(c x)}}\right )}{b}-\frac {4 \left (-\frac {2 x}{b c \sqrt {a+b \text {arccosh}(c x)}}+\frac {2 i \left (\frac {1}{2} i \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )-\frac {1}{2} i \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )\right )}{b^2 c^2}\right )}{3 b c}-\frac {2 x^2 \sqrt {c x-1} \sqrt {c x+1}}{3 b c (a+b \text {arccosh}(c x))^{3/2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 c \left (\frac {6 \left (-\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\frac {\pi }{3}} \sqrt {b} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \sqrt {b} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )\right )}{b^2 c^4}-\frac {2 x^3}{b c \sqrt {a+b \text {arccosh}(c x)}}\right )}{b}-\frac {4 \left (-\frac {2 x}{b c \sqrt {a+b \text {arccosh}(c x)}}+\frac {2 i \left (\frac {1}{2} i \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )-\frac {1}{2} i \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )\right )}{b^2 c^2}\right )}{3 b c}-\frac {2 x^2 \sqrt {c x-1} \sqrt {c x+1}}{3 b c (a+b \text {arccosh}(c x))^{3/2}}\) |
(-2*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(3*b*c*(a + b*ArcCosh[c*x])^(3/2)) - (4*((-2*x)/(b*c*Sqrt[a + b*ArcCosh[c*x]]) + ((2*I)*((I/2)*Sqrt[b]*E^(a/b) *Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]] - ((I/2)*Sqrt[b]*Sqrt[Pi]* Erfi[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/E^(a/b)))/(b^2*c^2)))/(3*b*c) + (2 *c*((-2*x^3)/(b*c*Sqrt[a + b*ArcCosh[c*x]]) + (6*(-1/8*(Sqrt[b]*E^(a/b)*Sq rt[Pi]*Erf[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]]) - (Sqrt[b]*E^((3*a)/b)*Sqrt[ Pi/3]*Erf[(Sqrt[3]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/8 + (Sqrt[b]*Sqrt[P i]*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))))/(b^ 2*c^4)))/b
3.2.57.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] : > Simp[2/d Subst[Int[F^(g*(e - c*(f/d)) + f*g*(x^2/d)), x], x, Sqrt[c + d *x]], x] /; FreeQ[{F, c, d, e, f, g}, x] && !TrueQ[$UseGamma]
Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt [Pi]*(Erfi[(c + d*x)*Rt[b*Log[F], 2]]/(2*d*Rt[b*Log[F], 2])), x] /; FreeQ[{ F, a, b, c, d}, x] && PosQ[b]
Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt [Pi]*(Erf[(c + d*x)*Rt[(-b)*Log[F], 2]]/(2*d*Rt[(-b)*Log[F], 2])), x] /; Fr eeQ[{F, a, b, c, d}, x] && NegQ[b]
Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[I /2 Int[(c + d*x)^m/E^(I*(e + f*x)), x], x] - Simp[I/2 Int[(c + d*x)^m*E ^(I*(e + f*x)), x], x] /; FreeQ[{c, d, e, f, m}, x]
Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sinh[a + b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] & & IGtQ[p, 0]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[1/(b*c) S ubst[Int[x^n*Sinh[-a/b + x/b], x], x, a + b*ArcCosh[c*x]], x] /; FreeQ[{a, b, c, n}, x]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[ x^m*Sqrt[1 + c*x]*Sqrt[-1 + c*x]*((a + b*ArcCosh[c*x])^(n + 1)/(b*c*(n + 1) )), x] + (-Simp[c*((m + 1)/(b*(n + 1))) Int[x^(m + 1)*((a + b*ArcCosh[c*x ])^(n + 1)/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])), x], x] + Simp[m/(b*c*(n + 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] && LtQ[n, -2]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[ 1/(b*c^(m + 1)) Subst[Int[x^n*Cosh[-a/b + x/b]^m*Sinh[-a/b + x/b], x], x, a + b*ArcCosh[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[m, 0]
Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*((f_.)*(x_))^(m_.))/(Sqrt[(d1 _) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)]), x_Symbol] :> Simp[(f*x)^m*((a + b*ArcCosh[c*x])^(n + 1)/(b*c*(n + 1)))*Simp[Sqrt[1 + c*x]/Sqrt[d1 + e1*x ]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]], x] - Simp[f*(m/(b*c*(n + 1)))*Simp [Sqrt[1 + c*x]/Sqrt[d1 + e1*x]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]] Int[ (f*x)^(m - 1)*(a + b*ArcCosh[c*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d1, e 1, d2, e2, f, m}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && LtQ[n, -1]
\[\int \frac {x^{2}}{\left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{\frac {5}{2}}}d x\]
Exception generated. \[ \int \frac {x^2}{(a+b \text {arccosh}(c x))^{5/2}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int \frac {x^2}{(a+b \text {arccosh}(c x))^{5/2}} \, dx=\int \frac {x^{2}}{\left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{\frac {5}{2}}}\, dx \]
\[ \int \frac {x^2}{(a+b \text {arccosh}(c x))^{5/2}} \, dx=\int { \frac {x^{2}}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {x^2}{(a+b \text {arccosh}(c x))^{5/2}} \, dx=\int { \frac {x^{2}}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {x^2}{(a+b \text {arccosh}(c x))^{5/2}} \, dx=\int \frac {x^2}{{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^{5/2}} \,d x \]