Integrand size = 25, antiderivative size = 431 \[ \int \frac {(c e+d e x)^2}{(a+b \text {arccosh}(c+d x))^{7/2}} \, dx=-\frac {2 e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{5 b d (a+b \text {arccosh}(c+d x))^{5/2}}+\frac {8 e^2 (c+d x)}{15 b^2 d (a+b \text {arccosh}(c+d x))^{3/2}}-\frac {4 e^2 (c+d x)^3}{5 b^2 d (a+b \text {arccosh}(c+d x))^{3/2}}+\frac {16 e^2 \sqrt {-1+c+d x} \sqrt {1+c+d x}}{15 b^3 d \sqrt {a+b \text {arccosh}(c+d x)}}-\frac {24 e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{5 b^3 d \sqrt {a+b \text {arccosh}(c+d x)}}+\frac {e^2 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{15 b^{7/2} d}+\frac {3 e^2 e^{\frac {3 a}{b}} \sqrt {3 \pi } \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{5 b^{7/2} d}+\frac {e^2 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{15 b^{7/2} d}+\frac {3 e^2 e^{-\frac {3 a}{b}} \sqrt {3 \pi } \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{5 b^{7/2} d} \]
8/15*e^2*(d*x+c)/b^2/d/(a+b*arccosh(d*x+c))^(3/2)-4/5*e^2*(d*x+c)^3/b^2/d/ (a+b*arccosh(d*x+c))^(3/2)+1/15*e^2*exp(a/b)*erf((a+b*arccosh(d*x+c))^(1/2 )/b^(1/2))*Pi^(1/2)/b^(7/2)/d+1/15*e^2*erfi((a+b*arccosh(d*x+c))^(1/2)/b^( 1/2))*Pi^(1/2)/b^(7/2)/d/exp(a/b)+3/5*e^2*exp(3*a/b)*erf(3^(1/2)*(a+b*arcc osh(d*x+c))^(1/2)/b^(1/2))*3^(1/2)*Pi^(1/2)/b^(7/2)/d+3/5*e^2*erfi(3^(1/2) *(a+b*arccosh(d*x+c))^(1/2)/b^(1/2))*3^(1/2)*Pi^(1/2)/b^(7/2)/d/exp(3*a/b) -2/5*e^2*(d*x+c)^2*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)/b/d/(a+b*arccosh(d*x+c) )^(5/2)+16/15*e^2*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)/b^3/d/(a+b*arccosh(d*x+c ))^(1/2)-24/5*e^2*(d*x+c)^2*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)/b^3/d/(a+b*arc cosh(d*x+c))^(1/2)
Time = 1.94 (sec) , antiderivative size = 452, normalized size of antiderivative = 1.05 \[ \int \frac {(c e+d e x)^2}{(a+b \text {arccosh}(c+d x))^{7/2}} \, dx=\frac {e^2 \left (-6 b^2 \sqrt {\frac {-1+c+d x}{1+c+d x}} (1+c+d x)-2 e^{-\text {arccosh}(c+d x)} (a+b \text {arccosh}(c+d x)) \left (-2 a+b-2 b \text {arccosh}(c+d x)+2 e^{\frac {a}{b}+\text {arccosh}(c+d x)} \sqrt {\frac {a}{b}+\text {arccosh}(c+d x)} (a+b \text {arccosh}(c+d x)) \Gamma \left (\frac {1}{2},\frac {a}{b}+\text {arccosh}(c+d x)\right )\right )-2 e^{-\frac {a}{b}} (a+b \text {arccosh}(c+d x)) \left (e^{\frac {a}{b}+\text {arccosh}(c+d x)} (2 a+b+2 b \text {arccosh}(c+d x))+2 b \left (-\frac {a+b \text {arccosh}(c+d x)}{b}\right )^{3/2} \Gamma \left (\frac {1}{2},-\frac {a+b \text {arccosh}(c+d x)}{b}\right )\right )-3 (a+b \text {arccosh}(c+d x)) \left (12 \sqrt {3} b e^{-\frac {3 a}{b}} \left (-\frac {a+b \text {arccosh}(c+d x)}{b}\right )^{3/2} \Gamma \left (\frac {1}{2},-\frac {3 (a+b \text {arccosh}(c+d x))}{b}\right )+2 e^{-3 \text {arccosh}(c+d x)} \left (b+6 a \left (-1+e^{6 \text {arccosh}(c+d x)}\right )-6 b \text {arccosh}(c+d x)+b e^{6 \text {arccosh}(c+d x)} (1+6 \text {arccosh}(c+d x))+6 \sqrt {3} e^{3 \left (\frac {a}{b}+\text {arccosh}(c+d x)\right )} \sqrt {\frac {a}{b}+\text {arccosh}(c+d x)} (a+b \text {arccosh}(c+d x)) \Gamma \left (\frac {1}{2},\frac {3 (a+b \text {arccosh}(c+d x))}{b}\right )\right )\right )-6 b^2 \sinh (3 \text {arccosh}(c+d x))\right )}{60 b^3 d (a+b \text {arccosh}(c+d x))^{5/2}} \]
(e^2*(-6*b^2*Sqrt[(-1 + c + d*x)/(1 + c + d*x)]*(1 + c + d*x) - (2*(a + b* ArcCosh[c + d*x])*(-2*a + b - 2*b*ArcCosh[c + d*x] + 2*E^(a/b + ArcCosh[c + d*x])*Sqrt[a/b + ArcCosh[c + d*x]]*(a + b*ArcCosh[c + d*x])*Gamma[1/2, a /b + ArcCosh[c + d*x]]))/E^ArcCosh[c + d*x] - (2*(a + b*ArcCosh[c + d*x])* (E^(a/b + ArcCosh[c + d*x])*(2*a + b + 2*b*ArcCosh[c + d*x]) + 2*b*(-((a + b*ArcCosh[c + d*x])/b))^(3/2)*Gamma[1/2, -((a + b*ArcCosh[c + d*x])/b)])) /E^(a/b) - 3*(a + b*ArcCosh[c + d*x])*((12*Sqrt[3]*b*(-((a + b*ArcCosh[c + d*x])/b))^(3/2)*Gamma[1/2, (-3*(a + b*ArcCosh[c + d*x]))/b])/E^((3*a)/b) + (2*(b + 6*a*(-1 + E^(6*ArcCosh[c + d*x])) - 6*b*ArcCosh[c + d*x] + b*E^( 6*ArcCosh[c + d*x])*(1 + 6*ArcCosh[c + d*x]) + 6*Sqrt[3]*E^(3*(a/b + ArcCo sh[c + d*x]))*Sqrt[a/b + ArcCosh[c + d*x]]*(a + b*ArcCosh[c + d*x])*Gamma[ 1/2, (3*(a + b*ArcCosh[c + d*x]))/b]))/E^(3*ArcCosh[c + d*x])) - 6*b^2*Sin h[3*ArcCosh[c + d*x]]))/(60*b^3*d*(a + b*ArcCosh[c + d*x])^(5/2))
Time = 2.78 (sec) , antiderivative size = 507, normalized size of antiderivative = 1.18, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.560, Rules used = {6411, 27, 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.
\(\displaystyle \int \frac {(c e+d e x)^2}{(a+b \text {arccosh}(c+d x))^{7/2}} \, dx\) |
\(\Big \downarrow \) 6411 |
\(\displaystyle \frac {\int \frac {e^2 (c+d x)^2}{(a+b \text {arccosh}(c+d x))^{7/2}}d(c+d x)}{d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {e^2 \int \frac {(c+d x)^2}{(a+b \text {arccosh}(c+d x))^{7/2}}d(c+d x)}{d}\) |
\(\Big \downarrow \) 6301 |
\(\displaystyle \frac {e^2 \left (-\frac {4 \int \frac {c+d x}{\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^{5/2}}d(c+d x)}{5 b}+\frac {6 \int \frac {(c+d x)^3}{\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^{5/2}}d(c+d x)}{5 b}-\frac {2 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2}{5 b (a+b \text {arccosh}(c+d x))^{5/2}}\right )}{d}\) |
\(\Big \downarrow \) 6366 |
\(\displaystyle \frac {e^2 \left (-\frac {4 \left (\frac {2 \int \frac {1}{(a+b \text {arccosh}(c+d x))^{3/2}}d(c+d x)}{3 b}-\frac {2 (c+d x)}{3 b (a+b \text {arccosh}(c+d x))^{3/2}}\right )}{5 b}+\frac {6 \left (\frac {2 \int \frac {(c+d x)^2}{(a+b \text {arccosh}(c+d x))^{3/2}}d(c+d x)}{b}-\frac {2 (c+d x)^3}{3 b (a+b \text {arccosh}(c+d x))^{3/2}}\right )}{5 b}-\frac {2 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2}{5 b (a+b \text {arccosh}(c+d x))^{5/2}}\right )}{d}\) |
\(\Big \downarrow \) 6295 |
\(\displaystyle \frac {e^2 \left (\frac {6 \left (\frac {2 \int \frac {(c+d x)^2}{(a+b \text {arccosh}(c+d x))^{3/2}}d(c+d x)}{b}-\frac {2 (c+d x)^3}{3 b (a+b \text {arccosh}(c+d x))^{3/2}}\right )}{5 b}-\frac {4 \left (\frac {2 \left (\frac {2 \int \frac {c+d x}{\sqrt {c+d x-1} \sqrt {c+d x+1} \sqrt {a+b \text {arccosh}(c+d x)}}d(c+d x)}{b}-\frac {2 \sqrt {c+d x-1} \sqrt {c+d x+1}}{b \sqrt {a+b \text {arccosh}(c+d x)}}\right )}{3 b}-\frac {2 (c+d x)}{3 b (a+b \text {arccosh}(c+d x))^{3/2}}\right )}{5 b}-\frac {2 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2}{5 b (a+b \text {arccosh}(c+d x))^{5/2}}\right )}{d}\) |
\(\Big \downarrow \) 6300 |
\(\displaystyle \frac {e^2 \left (\frac {6 \left (\frac {2 \left (-\frac {2 \int \left (-\frac {3 \cosh \left (\frac {3 a}{b}-\frac {3 (a+b \text {arccosh}(c+d x))}{b}\right )}{4 \sqrt {a+b \text {arccosh}(c+d x)}}-\frac {\cosh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{4 \sqrt {a+b \text {arccosh}(c+d x)}}\right )d(a+b \text {arccosh}(c+d x))}{b^2}-\frac {2 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2}{b \sqrt {a+b \text {arccosh}(c+d x)}}\right )}{b}-\frac {2 (c+d x)^3}{3 b (a+b \text {arccosh}(c+d x))^{3/2}}\right )}{5 b}-\frac {4 \left (\frac {2 \left (\frac {2 \int \frac {c+d x}{\sqrt {c+d x-1} \sqrt {c+d x+1} \sqrt {a+b \text {arccosh}(c+d x)}}d(c+d x)}{b}-\frac {2 \sqrt {c+d x-1} \sqrt {c+d x+1}}{b \sqrt {a+b \text {arccosh}(c+d x)}}\right )}{3 b}-\frac {2 (c+d x)}{3 b (a+b \text {arccosh}(c+d x))^{3/2}}\right )}{5 b}-\frac {2 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2}{5 b (a+b \text {arccosh}(c+d x))^{5/2}}\right )}{d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {e^2 \left (-\frac {4 \left (\frac {2 \left (\frac {2 \int \frac {c+d x}{\sqrt {c+d x-1} \sqrt {c+d x+1} \sqrt {a+b \text {arccosh}(c+d x)}}d(c+d x)}{b}-\frac {2 \sqrt {c+d x-1} \sqrt {c+d x+1}}{b \sqrt {a+b \text {arccosh}(c+d x)}}\right )}{3 b}-\frac {2 (c+d x)}{3 b (a+b \text {arccosh}(c+d x))^{3/2}}\right )}{5 b}+\frac {6 \left (\frac {2 \left (-\frac {2 \left (-\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {3 \pi } \sqrt {b} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {3 \pi } \sqrt {b} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )\right )}{b^2}-\frac {2 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2}{b \sqrt {a+b \text {arccosh}(c+d x)}}\right )}{b}-\frac {2 (c+d x)^3}{3 b (a+b \text {arccosh}(c+d x))^{3/2}}\right )}{5 b}-\frac {2 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2}{5 b (a+b \text {arccosh}(c+d x))^{5/2}}\right )}{d}\) |
\(\Big \downarrow \) 6368 |
\(\displaystyle \frac {e^2 \left (-\frac {4 \left (\frac {2 \left (\frac {2 \int \frac {\cosh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{\sqrt {a+b \text {arccosh}(c+d x)}}d(a+b \text {arccosh}(c+d x))}{b^2}-\frac {2 \sqrt {c+d x-1} \sqrt {c+d x+1}}{b \sqrt {a+b \text {arccosh}(c+d x)}}\right )}{3 b}-\frac {2 (c+d x)}{3 b (a+b \text {arccosh}(c+d x))^{3/2}}\right )}{5 b}+\frac {6 \left (\frac {2 \left (-\frac {2 \left (-\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {3 \pi } \sqrt {b} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {3 \pi } \sqrt {b} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )\right )}{b^2}-\frac {2 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2}{b \sqrt {a+b \text {arccosh}(c+d x)}}\right )}{b}-\frac {2 (c+d x)^3}{3 b (a+b \text {arccosh}(c+d x))^{3/2}}\right )}{5 b}-\frac {2 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2}{5 b (a+b \text {arccosh}(c+d x))^{5/2}}\right )}{d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {e^2 \left (-\frac {4 \left (-\frac {2 (c+d x)}{3 b (a+b \text {arccosh}(c+d x))^{3/2}}+\frac {2 \left (-\frac {2 \sqrt {c+d x-1} \sqrt {c+d x+1}}{b \sqrt {a+b \text {arccosh}(c+d x)}}+\frac {2 \int \frac {\sin \left (\frac {i a}{b}-\frac {i (a+b \text {arccosh}(c+d x))}{b}+\frac {\pi }{2}\right )}{\sqrt {a+b \text {arccosh}(c+d x)}}d(a+b \text {arccosh}(c+d x))}{b^2}\right )}{3 b}\right )}{5 b}+\frac {6 \left (\frac {2 \left (-\frac {2 \left (-\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {3 \pi } \sqrt {b} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {3 \pi } \sqrt {b} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )\right )}{b^2}-\frac {2 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2}{b \sqrt {a+b \text {arccosh}(c+d x)}}\right )}{b}-\frac {2 (c+d x)^3}{3 b (a+b \text {arccosh}(c+d x))^{3/2}}\right )}{5 b}-\frac {2 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2}{5 b (a+b \text {arccosh}(c+d x))^{5/2}}\right )}{d}\) |
\(\Big \downarrow \) 3788 |
\(\displaystyle \frac {e^2 \left (-\frac {4 \left (-\frac {2 (c+d x)}{3 b (a+b \text {arccosh}(c+d x))^{3/2}}+\frac {2 \left (-\frac {2 \sqrt {c+d x-1} \sqrt {c+d x+1}}{b \sqrt {a+b \text {arccosh}(c+d x)}}+\frac {2 \left (\frac {1}{2} i \int -\frac {i e^{\frac {a-c-d x}{b}}}{\sqrt {a+b \text {arccosh}(c+d x)}}d(a+b \text {arccosh}(c+d x))-\frac {1}{2} i \int \frac {i e^{-\frac {a-c-d x}{b}}}{\sqrt {a+b \text {arccosh}(c+d x)}}d(a+b \text {arccosh}(c+d x))\right )}{b^2}\right )}{3 b}\right )}{5 b}+\frac {6 \left (\frac {2 \left (-\frac {2 \left (-\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {3 \pi } \sqrt {b} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {3 \pi } \sqrt {b} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )\right )}{b^2}-\frac {2 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2}{b \sqrt {a+b \text {arccosh}(c+d x)}}\right )}{b}-\frac {2 (c+d x)^3}{3 b (a+b \text {arccosh}(c+d x))^{3/2}}\right )}{5 b}-\frac {2 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2}{5 b (a+b \text {arccosh}(c+d x))^{5/2}}\right )}{d}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \frac {e^2 \left (-\frac {4 \left (\frac {2 \left (\frac {2 \left (\frac {1}{2} \int \frac {e^{-\frac {a-c-d x}{b}}}{\sqrt {a+b \text {arccosh}(c+d x)}}d(a+b \text {arccosh}(c+d x))+\frac {1}{2} \int \frac {e^{\frac {a-c-d x}{b}}}{\sqrt {a+b \text {arccosh}(c+d x)}}d(a+b \text {arccosh}(c+d x))\right )}{b^2}-\frac {2 \sqrt {c+d x-1} \sqrt {c+d x+1}}{b \sqrt {a+b \text {arccosh}(c+d x)}}\right )}{3 b}-\frac {2 (c+d x)}{3 b (a+b \text {arccosh}(c+d x))^{3/2}}\right )}{5 b}+\frac {6 \left (\frac {2 \left (-\frac {2 \left (-\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {3 \pi } \sqrt {b} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {3 \pi } \sqrt {b} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )\right )}{b^2}-\frac {2 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2}{b \sqrt {a+b \text {arccosh}(c+d x)}}\right )}{b}-\frac {2 (c+d x)^3}{3 b (a+b \text {arccosh}(c+d x))^{3/2}}\right )}{5 b}-\frac {2 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2}{5 b (a+b \text {arccosh}(c+d x))^{5/2}}\right )}{d}\) |
\(\Big \downarrow \) 2611 |
\(\displaystyle \frac {e^2 \left (-\frac {4 \left (\frac {2 \left (\frac {2 \left (\int e^{\frac {a}{b}-\frac {a+b \text {arccosh}(c+d x)}{b}}d\sqrt {a+b \text {arccosh}(c+d x)}+\int e^{\frac {a+b \text {arccosh}(c+d x)}{b}-\frac {a}{b}}d\sqrt {a+b \text {arccosh}(c+d x)}\right )}{b^2}-\frac {2 \sqrt {c+d x-1} \sqrt {c+d x+1}}{b \sqrt {a+b \text {arccosh}(c+d x)}}\right )}{3 b}-\frac {2 (c+d x)}{3 b (a+b \text {arccosh}(c+d x))^{3/2}}\right )}{5 b}+\frac {6 \left (\frac {2 \left (-\frac {2 \left (-\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {3 \pi } \sqrt {b} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {3 \pi } \sqrt {b} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )\right )}{b^2}-\frac {2 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2}{b \sqrt {a+b \text {arccosh}(c+d x)}}\right )}{b}-\frac {2 (c+d x)^3}{3 b (a+b \text {arccosh}(c+d x))^{3/2}}\right )}{5 b}-\frac {2 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2}{5 b (a+b \text {arccosh}(c+d x))^{5/2}}\right )}{d}\) |
\(\Big \downarrow \) 2633 |
\(\displaystyle \frac {e^2 \left (-\frac {4 \left (\frac {2 \left (\frac {2 \left (\int e^{\frac {a}{b}-\frac {a+b \text {arccosh}(c+d x)}{b}}d\sqrt {a+b \text {arccosh}(c+d x)}+\frac {1}{2} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )\right )}{b^2}-\frac {2 \sqrt {c+d x-1} \sqrt {c+d x+1}}{b \sqrt {a+b \text {arccosh}(c+d x)}}\right )}{3 b}-\frac {2 (c+d x)}{3 b (a+b \text {arccosh}(c+d x))^{3/2}}\right )}{5 b}+\frac {6 \left (\frac {2 \left (-\frac {2 \left (-\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {3 \pi } \sqrt {b} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {3 \pi } \sqrt {b} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )\right )}{b^2}-\frac {2 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2}{b \sqrt {a+b \text {arccosh}(c+d x)}}\right )}{b}-\frac {2 (c+d x)^3}{3 b (a+b \text {arccosh}(c+d x))^{3/2}}\right )}{5 b}-\frac {2 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2}{5 b (a+b \text {arccosh}(c+d x))^{5/2}}\right )}{d}\) |
\(\Big \downarrow \) 2634 |
\(\displaystyle \frac {e^2 \left (-\frac {4 \left (\frac {2 \left (\frac {2 \left (\frac {1}{2} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )+\frac {1}{2} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )\right )}{b^2}-\frac {2 \sqrt {c+d x-1} \sqrt {c+d x+1}}{b \sqrt {a+b \text {arccosh}(c+d x)}}\right )}{3 b}-\frac {2 (c+d x)}{3 b (a+b \text {arccosh}(c+d x))^{3/2}}\right )}{5 b}+\frac {6 \left (\frac {2 \left (-\frac {2 \left (-\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {3 \pi } \sqrt {b} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {3 \pi } \sqrt {b} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )\right )}{b^2}-\frac {2 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2}{b \sqrt {a+b \text {arccosh}(c+d x)}}\right )}{b}-\frac {2 (c+d x)^3}{3 b (a+b \text {arccosh}(c+d x))^{3/2}}\right )}{5 b}-\frac {2 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2}{5 b (a+b \text {arccosh}(c+d x))^{5/2}}\right )}{d}\) |
(e^2*((-2*Sqrt[-1 + c + d*x]*(c + d*x)^2*Sqrt[1 + c + d*x])/(5*b*(a + b*Ar cCosh[c + d*x])^(5/2)) - (4*((-2*(c + d*x))/(3*b*(a + b*ArcCosh[c + d*x])^ (3/2)) + (2*((-2*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x])/(b*Sqrt[a + b*ArcCo sh[c + d*x]]) + (2*((Sqrt[b]*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c + d *x]]/Sqrt[b]])/2 + (Sqrt[b]*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c + d*x]]/Sqr t[b]])/(2*E^(a/b))))/b^2))/(3*b)))/(5*b) + (6*((-2*(c + d*x)^3)/(3*b*(a + b*ArcCosh[c + d*x])^(3/2)) + (2*((-2*Sqrt[-1 + c + d*x]*(c + d*x)^2*Sqrt[1 + c + d*x])/(b*Sqrt[a + b*ArcCosh[c + d*x]]) - (2*(-1/8*(Sqrt[b]*E^(a/b)* Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]]) - (Sqrt[b]*E^((3*a)/b) *Sqrt[3*Pi]*Erf[(Sqrt[3]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/8 - (Sqrt [b]*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]])/(8*E^(a/b)) - (Sq rt[b]*Sqrt[3*Pi]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(8* E^((3*a)/b))))/b^2))/b))/(5*b)))/d
3.2.94.3.1 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[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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[((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]
\[\int \frac {\left (d e x +c e \right )^{2}}{\left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )^{\frac {7}{2}}}d x\]
Exception generated. \[ \int \frac {(c e+d e x)^2}{(a+b \text {arccosh}(c+d x))^{7/2}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
Timed out. \[ \int \frac {(c e+d e x)^2}{(a+b \text {arccosh}(c+d x))^{7/2}} \, dx=\text {Timed out} \]
\[ \int \frac {(c e+d e x)^2}{(a+b \text {arccosh}(c+d x))^{7/2}} \, dx=\int { \frac {{\left (d e x + c e\right )}^{2}}{{\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{\frac {7}{2}}} \,d x } \]
\[ \int \frac {(c e+d e x)^2}{(a+b \text {arccosh}(c+d x))^{7/2}} \, dx=\int { \frac {{\left (d e x + c e\right )}^{2}}{{\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{\frac {7}{2}}} \,d x } \]
Timed out. \[ \int \frac {(c e+d e x)^2}{(a+b \text {arccosh}(c+d x))^{7/2}} \, dx=\int \frac {{\left (c\,e+d\,e\,x\right )}^2}{{\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right )}^{7/2}} \,d x \]