Integrand size = 16, antiderivative size = 361 \[ \int \frac {x^2}{(a+b \text {arccosh}(c x))^{7/2}} \, dx=-\frac {2 x^2 \sqrt {-1+c x} \sqrt {1+c x}}{5 b c (a+b \text {arccosh}(c x))^{5/2}}+\frac {8 x}{15 b^2 c^2 (a+b \text {arccosh}(c x))^{3/2}}-\frac {4 x^3}{5 b^2 (a+b \text {arccosh}(c x))^{3/2}}+\frac {16 \sqrt {-1+c x} \sqrt {1+c x}}{15 b^3 c^3 \sqrt {a+b \text {arccosh}(c x)}}-\frac {24 x^2 \sqrt {-1+c x} \sqrt {1+c x}}{5 b^3 c \sqrt {a+b \text {arccosh}(c x)}}+\frac {e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{15 b^{7/2} c^3}+\frac {3 e^{\frac {3 a}{b}} \sqrt {3 \pi } \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{5 b^{7/2} c^3}+\frac {e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{15 b^{7/2} c^3}+\frac {3 e^{-\frac {3 a}{b}} \sqrt {3 \pi } \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{5 b^{7/2} c^3} \]
8/15*x/b^2/c^2/(a+b*arccosh(c*x))^(3/2)-4/5*x^3/b^2/(a+b*arccosh(c*x))^(3/ 2)+1/15*exp(a/b)*erf((a+b*arccosh(c*x))^(1/2)/b^(1/2))*Pi^(1/2)/b^(7/2)/c^ 3+1/15*erfi((a+b*arccosh(c*x))^(1/2)/b^(1/2))*Pi^(1/2)/b^(7/2)/c^3/exp(a/b )+3/5*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^(7/2)/c^3+3/5*erfi(3^(1/2)*(a+b*arccosh(c*x))^(1/2)/b^(1/2))*3^(1/ 2)*Pi^(1/2)/b^(7/2)/c^3/exp(3*a/b)-2/5*x^2*(c*x-1)^(1/2)*(c*x+1)^(1/2)/b/c /(a+b*arccosh(c*x))^(5/2)+16/15*(c*x-1)^(1/2)*(c*x+1)^(1/2)/b^3/c^3/(a+b*a rccosh(c*x))^(1/2)-24/5*x^2*(c*x-1)^(1/2)*(c*x+1)^(1/2)/b^3/c/(a+b*arccosh (c*x))^(1/2)
Time = 1.69 (sec) , antiderivative size = 394, normalized size of antiderivative = 1.09 \[ \int \frac {x^2}{(a+b \text {arccosh}(c x))^{7/2}} \, dx=\frac {-6 b^2 \sqrt {\frac {-1+c x}{1+c x}} (1+c x)-2 e^{-\text {arccosh}(c x)} (a+b \text {arccosh}(c x)) \left (-2 a+b-2 b \text {arccosh}(c x)+2 e^{\frac {a}{b}+\text {arccosh}(c x)} \sqrt {\frac {a}{b}+\text {arccosh}(c x)} (a+b \text {arccosh}(c x)) \Gamma \left (\frac {1}{2},\frac {a}{b}+\text {arccosh}(c x)\right )\right )-2 e^{-\frac {a}{b}} (a+b \text {arccosh}(c x)) \left (e^{\frac {a}{b}+\text {arccosh}(c x)} (2 a+b+2 b \text {arccosh}(c x))+2 b \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^{3/2} \Gamma \left (\frac {1}{2},-\frac {a+b \text {arccosh}(c x)}{b}\right )\right )-3 (a+b \text {arccosh}(c x)) \left (12 \sqrt {3} b e^{-\frac {3 a}{b}} \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^{3/2} \Gamma \left (\frac {1}{2},-\frac {3 (a+b \text {arccosh}(c x))}{b}\right )+2 e^{-3 \text {arccosh}(c x)} \left (b+6 a \left (-1+e^{6 \text {arccosh}(c x)}\right )-6 b \text {arccosh}(c x)+b e^{6 \text {arccosh}(c x)} (1+6 \text {arccosh}(c x))+6 \sqrt {3} e^{3 \left (\frac {a}{b}+\text {arccosh}(c x)\right )} \sqrt {\frac {a}{b}+\text {arccosh}(c x)} (a+b \text {arccosh}(c x)) \Gamma \left (\frac {1}{2},\frac {3 (a+b \text {arccosh}(c x))}{b}\right )\right )\right )-6 b^2 \sinh (3 \text {arccosh}(c x))}{60 b^3 c^3 (a+b \text {arccosh}(c x))^{5/2}} \]
(-6*b^2*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x) - (2*(a + b*ArcCosh[c*x])*(-2 *a + b - 2*b*ArcCosh[c*x] + 2*E^(a/b + ArcCosh[c*x])*Sqrt[a/b + ArcCosh[c* x]]*(a + b*ArcCosh[c*x])*Gamma[1/2, a/b + ArcCosh[c*x]]))/E^ArcCosh[c*x] - (2*(a + b*ArcCosh[c*x])*(E^(a/b + ArcCosh[c*x])*(2*a + b + 2*b*ArcCosh[c* x]) + 2*b*(-((a + b*ArcCosh[c*x])/b))^(3/2)*Gamma[1/2, -((a + b*ArcCosh[c* x])/b)]))/E^(a/b) - 3*(a + b*ArcCosh[c*x])*((12*Sqrt[3]*b*(-((a + b*ArcCos h[c*x])/b))^(3/2)*Gamma[1/2, (-3*(a + b*ArcCosh[c*x]))/b])/E^((3*a)/b) + ( 2*(b + 6*a*(-1 + E^(6*ArcCosh[c*x])) - 6*b*ArcCosh[c*x] + b*E^(6*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 ]))/E^(3*ArcCosh[c*x])) - 6*b^2*Sinh[3*ArcCosh[c*x]])/(60*b^3*c^3*(a + b*A rcCosh[c*x])^(5/2))
Time = 3.18 (sec) , antiderivative size = 487, normalized size of antiderivative = 1.35, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {6301, 6366, 6295, 6300, 2009, 6368, 3042, 3788, 26, 2611, 2633, 2634}
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.
| \(\displaystyle \int \frac {x^2}{(a+b \text {arccosh}(c x))^{7/2}} \, dx\) |
| \(\Big \downarrow \) 6301 |
| \(\displaystyle \frac {6 c \int \frac {x^3}{\sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^{5/2}}dx}{5 b}-\frac {4 \int \frac {x}{\sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^{5/2}}dx}{5 b c}-\frac {2 x^2 \sqrt {c x-1} \sqrt {c x+1}}{5 b c (a+b \text {arccosh}(c x))^{5/2}}\) |
| \(\Big \downarrow \) 6366 |
| \(\displaystyle \frac {6 c \left (\frac {2 \int \frac {x^2}{(a+b \text {arccosh}(c x))^{3/2}}dx}{b c}-\frac {2 x^3}{3 b c (a+b \text {arccosh}(c x))^{3/2}}\right )}{5 b}-\frac {4 \left (\frac {2 \int \frac {1}{(a+b \text {arccosh}(c x))^{3/2}}dx}{3 b c}-\frac {2 x}{3 b c (a+b \text {arccosh}(c x))^{3/2}}\right )}{5 b c}-\frac {2 x^2 \sqrt {c x-1} \sqrt {c x+1}}{5 b c (a+b \text {arccosh}(c x))^{5/2}}\) |
| \(\Big \downarrow \) 6295 |
| \(\displaystyle \frac {6 c \left (\frac {2 \int \frac {x^2}{(a+b \text {arccosh}(c x))^{3/2}}dx}{b c}-\frac {2 x^3}{3 b c (a+b \text {arccosh}(c x))^{3/2}}\right )}{5 b}-\frac {4 \left (\frac {2 \left (\frac {2 c \int \frac {x}{\sqrt {c x-1} \sqrt {c x+1} \sqrt {a+b \text {arccosh}(c x)}}dx}{b}-\frac {2 \sqrt {c x-1} \sqrt {c x+1}}{b c \sqrt {a+b \text {arccosh}(c x)}}\right )}{3 b c}-\frac {2 x}{3 b c (a+b \text {arccosh}(c x))^{3/2}}\right )}{5 b c}-\frac {2 x^2 \sqrt {c x-1} \sqrt {c x+1}}{5 b c (a+b \text {arccosh}(c x))^{5/2}}\) |
| \(\Big \downarrow \) 6300 |
| \(\displaystyle \frac {6 c \left (\frac {2 \left (-\frac {2 \int \left (-\frac {3 \cosh \left (\frac {3 a}{b}-\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{4 \sqrt {a+b \text {arccosh}(c x)}}-\frac {\cosh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}{4 \sqrt {a+b \text {arccosh}(c x)}}\right )d(a+b \text {arccosh}(c x))}{b^2 c^3}-\frac {2 x^2 \sqrt {c x-1} \sqrt {c x+1}}{b c \sqrt {a+b \text {arccosh}(c x)}}\right )}{b c}-\frac {2 x^3}{3 b c (a+b \text {arccosh}(c x))^{3/2}}\right )}{5 b}-\frac {4 \left (\frac {2 \left (\frac {2 c \int \frac {x}{\sqrt {c x-1} \sqrt {c x+1} \sqrt {a+b \text {arccosh}(c x)}}dx}{b}-\frac {2 \sqrt {c x-1} \sqrt {c x+1}}{b c \sqrt {a+b \text {arccosh}(c x)}}\right )}{3 b c}-\frac {2 x}{3 b c (a+b \text {arccosh}(c x))^{3/2}}\right )}{5 b c}-\frac {2 x^2 \sqrt {c x-1} \sqrt {c x+1}}{5 b c (a+b \text {arccosh}(c x))^{5/2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {4 \left (\frac {2 \left (\frac {2 c \int \frac {x}{\sqrt {c x-1} \sqrt {c x+1} \sqrt {a+b \text {arccosh}(c x)}}dx}{b}-\frac {2 \sqrt {c x-1} \sqrt {c x+1}}{b c \sqrt {a+b \text {arccosh}(c x)}}\right )}{3 b c}-\frac {2 x}{3 b c (a+b \text {arccosh}(c x))^{3/2}}\right )}{5 b c}+\frac {6 c \left (\frac {2 \left (-\frac {2 \left (-\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {3 \pi } \sqrt {b} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {3 \pi } \sqrt {b} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )\right )}{b^2 c^3}-\frac {2 x^2 \sqrt {c x-1} \sqrt {c x+1}}{b c \sqrt {a+b \text {arccosh}(c x)}}\right )}{b c}-\frac {2 x^3}{3 b c (a+b \text {arccosh}(c x))^{3/2}}\right )}{5 b}-\frac {2 x^2 \sqrt {c x-1} \sqrt {c x+1}}{5 b c (a+b \text {arccosh}(c x))^{5/2}}\) |
| \(\Big \downarrow \) 6368 |
| \(\displaystyle -\frac {4 \left (\frac {2 \left (\frac {2 \int \frac {\cosh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}{\sqrt {a+b \text {arccosh}(c x)}}d(a+b \text {arccosh}(c x))}{b^2 c}-\frac {2 \sqrt {c x-1} \sqrt {c x+1}}{b c \sqrt {a+b \text {arccosh}(c x)}}\right )}{3 b c}-\frac {2 x}{3 b c (a+b \text {arccosh}(c x))^{3/2}}\right )}{5 b c}+\frac {6 c \left (\frac {2 \left (-\frac {2 \left (-\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {3 \pi } \sqrt {b} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {3 \pi } \sqrt {b} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )\right )}{b^2 c^3}-\frac {2 x^2 \sqrt {c x-1} \sqrt {c x+1}}{b c \sqrt {a+b \text {arccosh}(c x)}}\right )}{b c}-\frac {2 x^3}{3 b c (a+b \text {arccosh}(c x))^{3/2}}\right )}{5 b}-\frac {2 x^2 \sqrt {c x-1} \sqrt {c x+1}}{5 b c (a+b \text {arccosh}(c x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {4 \left (-\frac {2 x}{3 b c (a+b \text {arccosh}(c x))^{3/2}}+\frac {2 \left (-\frac {2 \sqrt {c x-1} \sqrt {c x+1}}{b c \sqrt {a+b \text {arccosh}(c x)}}+\frac {2 \int \frac {\sin \left (\frac {i a}{b}-\frac {i (a+b \text {arccosh}(c x))}{b}+\frac {\pi }{2}\right )}{\sqrt {a+b \text {arccosh}(c x)}}d(a+b \text {arccosh}(c x))}{b^2 c}\right )}{3 b c}\right )}{5 b c}+\frac {6 c \left (\frac {2 \left (-\frac {2 \left (-\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {3 \pi } \sqrt {b} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {3 \pi } \sqrt {b} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )\right )}{b^2 c^3}-\frac {2 x^2 \sqrt {c x-1} \sqrt {c x+1}}{b c \sqrt {a+b \text {arccosh}(c x)}}\right )}{b c}-\frac {2 x^3}{3 b c (a+b \text {arccosh}(c x))^{3/2}}\right )}{5 b}-\frac {2 x^2 \sqrt {c x-1} \sqrt {c x+1}}{5 b c (a+b \text {arccosh}(c x))^{5/2}}\) |
| \(\Big \downarrow \) 3788 |
| \(\displaystyle -\frac {4 \left (-\frac {2 x}{3 b c (a+b \text {arccosh}(c x))^{3/2}}+\frac {2 \left (-\frac {2 \sqrt {c x-1} \sqrt {c x+1}}{b c \sqrt {a+b \text {arccosh}(c x)}}+\frac {2 \left (\frac {1}{2} i \int -\frac {i e^{-\text {arccosh}(c x)}}{\sqrt {a+b \text {arccosh}(c x)}}d(a+b \text {arccosh}(c x))-\frac {1}{2} i \int \frac {i e^{\text {arccosh}(c x)}}{\sqrt {a+b \text {arccosh}(c x)}}d(a+b \text {arccosh}(c x))\right )}{b^2 c}\right )}{3 b c}\right )}{5 b c}+\frac {6 c \left (\frac {2 \left (-\frac {2 \left (-\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {3 \pi } \sqrt {b} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {3 \pi } \sqrt {b} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )\right )}{b^2 c^3}-\frac {2 x^2 \sqrt {c x-1} \sqrt {c x+1}}{b c \sqrt {a+b \text {arccosh}(c x)}}\right )}{b c}-\frac {2 x^3}{3 b c (a+b \text {arccosh}(c x))^{3/2}}\right )}{5 b}-\frac {2 x^2 \sqrt {c x-1} \sqrt {c x+1}}{5 b c (a+b \text {arccosh}(c x))^{5/2}}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {4 \left (\frac {2 \left (\frac {2 \left (\frac {1}{2} \int \frac {e^{-\text {arccosh}(c x)}}{\sqrt {a+b \text {arccosh}(c x)}}d(a+b \text {arccosh}(c x))+\frac {1}{2} \int \frac {e^{\text {arccosh}(c x)}}{\sqrt {a+b \text {arccosh}(c x)}}d(a+b \text {arccosh}(c x))\right )}{b^2 c}-\frac {2 \sqrt {c x-1} \sqrt {c x+1}}{b c \sqrt {a+b \text {arccosh}(c x)}}\right )}{3 b c}-\frac {2 x}{3 b c (a+b \text {arccosh}(c x))^{3/2}}\right )}{5 b c}+\frac {6 c \left (\frac {2 \left (-\frac {2 \left (-\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {3 \pi } \sqrt {b} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {3 \pi } \sqrt {b} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )\right )}{b^2 c^3}-\frac {2 x^2 \sqrt {c x-1} \sqrt {c x+1}}{b c \sqrt {a+b \text {arccosh}(c x)}}\right )}{b c}-\frac {2 x^3}{3 b c (a+b \text {arccosh}(c x))^{3/2}}\right )}{5 b}-\frac {2 x^2 \sqrt {c x-1} \sqrt {c x+1}}{5 b c (a+b \text {arccosh}(c x))^{5/2}}\) |
| \(\Big \downarrow \) 2611 |
| \(\displaystyle -\frac {4 \left (\frac {2 \left (\frac {2 \left (\int e^{\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}}d\sqrt {a+b \text {arccosh}(c x)}+\int e^{\frac {a+b \text {arccosh}(c x)}{b}-\frac {a}{b}}d\sqrt {a+b \text {arccosh}(c x)}\right )}{b^2 c}-\frac {2 \sqrt {c x-1} \sqrt {c x+1}}{b c \sqrt {a+b \text {arccosh}(c x)}}\right )}{3 b c}-\frac {2 x}{3 b c (a+b \text {arccosh}(c x))^{3/2}}\right )}{5 b c}+\frac {6 c \left (\frac {2 \left (-\frac {2 \left (-\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {3 \pi } \sqrt {b} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {3 \pi } \sqrt {b} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )\right )}{b^2 c^3}-\frac {2 x^2 \sqrt {c x-1} \sqrt {c x+1}}{b c \sqrt {a+b \text {arccosh}(c x)}}\right )}{b c}-\frac {2 x^3}{3 b c (a+b \text {arccosh}(c x))^{3/2}}\right )}{5 b}-\frac {2 x^2 \sqrt {c x-1} \sqrt {c x+1}}{5 b c (a+b \text {arccosh}(c x))^{5/2}}\) |
| \(\Big \downarrow \) 2633 |
| \(\displaystyle -\frac {4 \left (\frac {2 \left (\frac {2 \left (\int e^{\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}}d\sqrt {a+b \text {arccosh}(c x)}+\frac {1}{2} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )\right )}{b^2 c}-\frac {2 \sqrt {c x-1} \sqrt {c x+1}}{b c \sqrt {a+b \text {arccosh}(c x)}}\right )}{3 b c}-\frac {2 x}{3 b c (a+b \text {arccosh}(c x))^{3/2}}\right )}{5 b c}+\frac {6 c \left (\frac {2 \left (-\frac {2 \left (-\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {3 \pi } \sqrt {b} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {3 \pi } \sqrt {b} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )\right )}{b^2 c^3}-\frac {2 x^2 \sqrt {c x-1} \sqrt {c x+1}}{b c \sqrt {a+b \text {arccosh}(c x)}}\right )}{b c}-\frac {2 x^3}{3 b c (a+b \text {arccosh}(c x))^{3/2}}\right )}{5 b}-\frac {2 x^2 \sqrt {c x-1} \sqrt {c x+1}}{5 b c (a+b \text {arccosh}(c x))^{5/2}}\) |
| \(\Big \downarrow \) 2634 |
| \(\displaystyle \frac {6 c \left (\frac {2 \left (-\frac {2 \left (-\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {3 \pi } \sqrt {b} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {3 \pi } \sqrt {b} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )\right )}{b^2 c^3}-\frac {2 x^2 \sqrt {c x-1} \sqrt {c x+1}}{b c \sqrt {a+b \text {arccosh}(c x)}}\right )}{b c}-\frac {2 x^3}{3 b c (a+b \text {arccosh}(c x))^{3/2}}\right )}{5 b}-\frac {4 \left (\frac {2 \left (\frac {2 \left (\frac {1}{2} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )+\frac {1}{2} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )\right )}{b^2 c}-\frac {2 \sqrt {c x-1} \sqrt {c x+1}}{b c \sqrt {a+b \text {arccosh}(c x)}}\right )}{3 b c}-\frac {2 x}{3 b c (a+b \text {arccosh}(c x))^{3/2}}\right )}{5 b c}-\frac {2 x^2 \sqrt {c x-1} \sqrt {c x+1}}{5 b c (a+b \text {arccosh}(c x))^{5/2}}\) |
(-2*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(5*b*c*(a + b*ArcCosh[c*x])^(5/2)) - (4*((-2*x)/(3*b*c*(a + b*ArcCosh[c*x])^(3/2)) + (2*((-2*Sqrt[-1 + c*x]*Sq rt[1 + c*x])/(b*c*Sqrt[a + b*ArcCosh[c*x]]) + (2*((Sqrt[b]*E^(a/b)*Sqrt[Pi ]*Erf[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/2 + (Sqrt[b]*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(2*E^(a/b))))/(b^2*c)))/(3*b*c)))/(5*b*c) + ( 6*c*((-2*x^3)/(3*b*c*(a + b*ArcCosh[c*x])^(3/2)) + (2*((-2*x^2*Sqrt[-1 + c *x]*Sqrt[1 + c*x])/(b*c*Sqrt[a + b*ArcCosh[c*x]]) - (2*(-1/8*(Sqrt[b]*E^(a /b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]]) - (Sqrt[b]*E^((3*a)/b) *Sqrt[3*Pi]*Erf[(Sqrt[3]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/8 - (Sqrt[b]* Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(8*E^(a/b)) - (Sqrt[b]*Sq rt[3*Pi]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(8*E^((3*a)/b)) ))/(b^2*c^3)))/(b*c)))/(5*b)
3.2.60.3.1 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_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol ] :> Simp[I/2 Int[(c + d*x)^m/(E^(I*k*Pi)*E^(I*(e + f*x))), x], x] - Simp [I/2 Int[(c + d*x)^m*E^(I*k*Pi)*E^(I*(e + f*x)), x], x] /; FreeQ[{c, d, e , f, m}, x] && IntegerQ[2*k]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[Sqrt[1 + c* x]*Sqrt[-1 + c*x]*((a + b*ArcCosh[c*x])^(n + 1)/(b*c*(n + 1))), x] - Simp[c /(b*(n + 1)) Int[x*((a + b*ArcCosh[c*x])^(n + 1)/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])), x], x] /; FreeQ[{a, b, c}, x] && LtQ[n, -1]
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[1/(b^2*c^(m + 1)*(n + 1)) Subst[Int[ExpandTrigReduce[x^(n + 1), Cosh[-a/b + x/b]^(m - 1)*(m - (m + 1)*Cosh[-a/b + x/b]^2), x], x], x, a + b*ArcCosh[c*x]], x] /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && GeQ[n, -2] && LtQ[n, -1]
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_)*((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[((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]
\[\int \frac {x^{2}}{\left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{\frac {7}{2}}}d x\]
Exception generated. \[ \int \frac {x^2}{(a+b \text {arccosh}(c x))^{7/2}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
Timed out. \[ \int \frac {x^2}{(a+b \text {arccosh}(c x))^{7/2}} \, dx=\text {Timed out} \]
\[ \int \frac {x^2}{(a+b \text {arccosh}(c x))^{7/2}} \, dx=\int { \frac {x^{2}}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{\frac {7}{2}}} \,d x } \]
\[ \int \frac {x^2}{(a+b \text {arccosh}(c x))^{7/2}} \, dx=\int { \frac {x^{2}}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{\frac {7}{2}}} \,d x } \]
Timed out. \[ \int \frac {x^2}{(a+b \text {arccosh}(c x))^{7/2}} \, dx=\int \frac {x^2}{{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^{7/2}} \,d x \]