Integrand size = 25, antiderivative size = 408 \[ \int (c e+d e x)^2 (a+b \text {arccosh}(c+d x))^{5/2} \, dx=\frac {5 b^2 e^2 (c+d x) \sqrt {a+b \text {arccosh}(c+d x)}}{6 d}+\frac {5 b^2 e^2 (c+d x)^3 \sqrt {a+b \text {arccosh}(c+d x)}}{36 d}-\frac {5 b e^2 \sqrt {-1+c+d x} \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))^{3/2}}{9 d}-\frac {5 b e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))^{3/2}}{18 d}+\frac {e^2 (c+d x)^3 (a+b \text {arccosh}(c+d x))^{5/2}}{3 d}-\frac {15 b^{5/2} e^2 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{64 d}-\frac {5 b^{5/2} e^2 e^{\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{576 d}-\frac {15 b^{5/2} e^2 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{64 d}-\frac {5 b^{5/2} e^2 e^{-\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{576 d} \]
1/3*e^2*(d*x+c)^3*(a+b*arccosh(d*x+c))^(5/2)/d-5/1728*b^(5/2)*e^2*exp(3*a/ b)*erf(3^(1/2)*(a+b*arccosh(d*x+c))^(1/2)/b^(1/2))*3^(1/2)*Pi^(1/2)/d-5/17 28*b^(5/2)*e^2*erfi(3^(1/2)*(a+b*arccosh(d*x+c))^(1/2)/b^(1/2))*3^(1/2)*Pi ^(1/2)/d/exp(3*a/b)-15/64*b^(5/2)*e^2*exp(a/b)*erf((a+b*arccosh(d*x+c))^(1 /2)/b^(1/2))*Pi^(1/2)/d-15/64*b^(5/2)*e^2*erfi((a+b*arccosh(d*x+c))^(1/2)/ b^(1/2))*Pi^(1/2)/d/exp(a/b)-5/9*b*e^2*(a+b*arccosh(d*x+c))^(3/2)*(d*x+c-1 )^(1/2)*(d*x+c+1)^(1/2)/d-5/18*b*e^2*(d*x+c)^2*(a+b*arccosh(d*x+c))^(3/2)* (d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)/d+5/6*b^2*e^2*(d*x+c)*(a+b*arccosh(d*x+c)) ^(1/2)/d+5/36*b^2*e^2*(d*x+c)^3*(a+b*arccosh(d*x+c))^(1/2)/d
Leaf count is larger than twice the leaf count of optimal. \(1008\) vs. \(2(408)=816\).
Time = 7.72 (sec) , antiderivative size = 1008, normalized size of antiderivative = 2.47 \[ \int (c e+d e x)^2 (a+b \text {arccosh}(c+d x))^{5/2} \, dx=e^2 \left (\frac {a^2 e^{-\frac {3 a}{b}} \sqrt {a+b \text {arccosh}(c+d x)} \left (9 e^{\frac {4 a}{b}} \sqrt {-\frac {a+b \text {arccosh}(c+d x)}{b}} \Gamma \left (\frac {3}{2},\frac {a}{b}+\text {arccosh}(c+d x)\right )+\sqrt {3} \sqrt {\frac {a}{b}+\text {arccosh}(c+d x)} \Gamma \left (\frac {3}{2},-\frac {3 (a+b \text {arccosh}(c+d x))}{b}\right )+9 e^{\frac {2 a}{b}} \sqrt {\frac {a}{b}+\text {arccosh}(c+d x)} \Gamma \left (\frac {3}{2},-\frac {a+b \text {arccosh}(c+d x)}{b}\right )+\sqrt {3} e^{\frac {6 a}{b}} \sqrt {-\frac {a+b \text {arccosh}(c+d x)}{b}} \Gamma \left (\frac {3}{2},\frac {3 (a+b \text {arccosh}(c+d x))}{b}\right )\right )}{72 d \sqrt {-\frac {(a+b \text {arccosh}(c+d x))^2}{b^2}}}+\frac {a \sqrt {b} \left (9 \left (-12 \sqrt {b} \sqrt {\frac {-1+c+d x}{1+c+d x}} (1+c+d x) \sqrt {a+b \text {arccosh}(c+d x)}+8 \sqrt {b} (c+d x) \text {arccosh}(c+d x) \sqrt {a+b \text {arccosh}(c+d x)}+(2 a+3 b) \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {a}{b}\right )-\sinh \left (\frac {a}{b}\right )\right )+(2 a-3 b) \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {a}{b}\right )+\sinh \left (\frac {a}{b}\right )\right )\right )+(2 a+b) \sqrt {3 \pi } \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {3 a}{b}\right )-\sinh \left (\frac {3 a}{b}\right )\right )+(2 a-b) \sqrt {3 \pi } \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {3 a}{b}\right )+\sinh \left (\frac {3 a}{b}\right )\right )+12 \sqrt {b} \sqrt {a+b \text {arccosh}(c+d x)} (2 \text {arccosh}(c+d x) \cosh (3 \text {arccosh}(c+d x))-\sinh (3 \text {arccosh}(c+d x)))\right )}{144 d}+\frac {-27 \left (-4 b \sqrt {a+b \text {arccosh}(c+d x)} \left (2 \sqrt {\frac {-1+c+d x}{1+c+d x}} (1+c+d x) (a-5 b \text {arccosh}(c+d x))+b (c+d x) \left (15+4 \text {arccosh}(c+d x)^2\right )\right )+\sqrt {b} \left (4 a^2+12 a b+15 b^2\right ) \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {a}{b}\right )-\sinh \left (\frac {a}{b}\right )\right )+\sqrt {b} \left (4 a^2-12 a b+15 b^2\right ) \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {a}{b}\right )+\sinh \left (\frac {a}{b}\right )\right )\right )-\sqrt {b} \left (12 a^2+12 a b+5 b^2\right ) \sqrt {3 \pi } \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {3 a}{b}\right )-\sinh \left (\frac {3 a}{b}\right )\right )-\sqrt {b} \left (12 a^2-12 a b+5 b^2\right ) \sqrt {3 \pi } \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {3 a}{b}\right )+\sinh \left (\frac {3 a}{b}\right )\right )+12 b \sqrt {a+b \text {arccosh}(c+d x)} \left (b \left (5+12 \text {arccosh}(c+d x)^2\right ) \cosh (3 \text {arccosh}(c+d x))+2 (a-5 b \text {arccosh}(c+d x)) \sinh (3 \text {arccosh}(c+d x))\right )}{1728 d}\right ) \]
e^2*((a^2*Sqrt[a + b*ArcCosh[c + d*x]]*(9*E^((4*a)/b)*Sqrt[-((a + b*ArcCos h[c + d*x])/b)]*Gamma[3/2, a/b + ArcCosh[c + d*x]] + Sqrt[3]*Sqrt[a/b + Ar cCosh[c + d*x]]*Gamma[3/2, (-3*(a + b*ArcCosh[c + d*x]))/b] + 9*E^((2*a)/b )*Sqrt[a/b + ArcCosh[c + d*x]]*Gamma[3/2, -((a + b*ArcCosh[c + d*x])/b)] + Sqrt[3]*E^((6*a)/b)*Sqrt[-((a + b*ArcCosh[c + d*x])/b)]*Gamma[3/2, (3*(a + b*ArcCosh[c + d*x]))/b]))/(72*d*E^((3*a)/b)*Sqrt[-((a + b*ArcCosh[c + d* x])^2/b^2)]) + (a*Sqrt[b]*(9*(-12*Sqrt[b]*Sqrt[(-1 + c + d*x)/(1 + c + d*x )]*(1 + c + d*x)*Sqrt[a + b*ArcCosh[c + d*x]] + 8*Sqrt[b]*(c + d*x)*ArcCos h[c + d*x]*Sqrt[a + b*ArcCosh[c + d*x]] + (2*a + 3*b)*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]]*(Cosh[a/b] - Sinh[a/b]) + (2*a - 3*b)*Sqrt [Pi]*Erf[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]]*(Cosh[a/b] + Sinh[a/b])) + (2*a + b)*Sqrt[3*Pi]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]]* (Cosh[(3*a)/b] - Sinh[(3*a)/b]) + (2*a - b)*Sqrt[3*Pi]*Erf[(Sqrt[3]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]]*(Cosh[(3*a)/b] + Sinh[(3*a)/b]) + 12*Sqrt [b]*Sqrt[a + b*ArcCosh[c + d*x]]*(2*ArcCosh[c + d*x]*Cosh[3*ArcCosh[c + d* x]] - Sinh[3*ArcCosh[c + d*x]])))/(144*d) + (-27*(-4*b*Sqrt[a + b*ArcCosh[ c + d*x]]*(2*Sqrt[(-1 + c + d*x)/(1 + c + d*x)]*(1 + c + d*x)*(a - 5*b*Arc Cosh[c + d*x]) + b*(c + d*x)*(15 + 4*ArcCosh[c + d*x]^2)) + Sqrt[b]*(4*a^2 + 12*a*b + 15*b^2)*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]]*(C osh[a/b] - Sinh[a/b]) + Sqrt[b]*(4*a^2 - 12*a*b + 15*b^2)*Sqrt[Pi]*Erf[...
Time = 3.47 (sec) , antiderivative size = 465, normalized size of antiderivative = 1.14, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.640, Rules used = {6411, 27, 6299, 6354, 6299, 6330, 6294, 6368, 3042, 3788, 26, 2611, 2633, 2634, 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.
| \(\displaystyle \int (c e+d e x)^2 (a+b \text {arccosh}(c+d x))^{5/2} \, dx\) |
| \(\Big \downarrow \) 6411 |
| \(\displaystyle \frac {\int e^2 (c+d x)^2 (a+b \text {arccosh}(c+d x))^{5/2}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e^2 \int (c+d x)^2 (a+b \text {arccosh}(c+d x))^{5/2}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 6299 |
| \(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \text {arccosh}(c+d x))^{5/2}-\frac {5}{6} b \int \frac {(c+d x)^3 (a+b \text {arccosh}(c+d x))^{3/2}}{\sqrt {c+d x-1} \sqrt {c+d x+1}}d(c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 6354 |
| \(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \text {arccosh}(c+d x))^{5/2}-\frac {5}{6} b \left (-\frac {1}{2} b \int (c+d x)^2 \sqrt {a+b \text {arccosh}(c+d x)}d(c+d x)+\frac {2}{3} \int \frac {(c+d x) (a+b \text {arccosh}(c+d x))^{3/2}}{\sqrt {c+d x-1} \sqrt {c+d x+1}}d(c+d x)+\frac {1}{3} \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2 (a+b \text {arccosh}(c+d x))^{3/2}\right )\right )}{d}\) |
| \(\Big \downarrow \) 6299 |
| \(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \text {arccosh}(c+d x))^{5/2}-\frac {5}{6} b \left (-\frac {1}{2} b \left (\frac {1}{3} (c+d x)^3 \sqrt {a+b \text {arccosh}(c+d x)}-\frac {1}{6} b \int \frac {(c+d x)^3}{\sqrt {c+d x-1} \sqrt {c+d x+1} \sqrt {a+b \text {arccosh}(c+d x)}}d(c+d x)\right )+\frac {2}{3} \int \frac {(c+d x) (a+b \text {arccosh}(c+d x))^{3/2}}{\sqrt {c+d x-1} \sqrt {c+d x+1}}d(c+d x)+\frac {1}{3} \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2 (a+b \text {arccosh}(c+d x))^{3/2}\right )\right )}{d}\) |
| \(\Big \downarrow \) 6330 |
| \(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \text {arccosh}(c+d x))^{5/2}-\frac {5}{6} b \left (-\frac {1}{2} b \left (\frac {1}{3} (c+d x)^3 \sqrt {a+b \text {arccosh}(c+d x)}-\frac {1}{6} b \int \frac {(c+d x)^3}{\sqrt {c+d x-1} \sqrt {c+d x+1} \sqrt {a+b \text {arccosh}(c+d x)}}d(c+d x)\right )+\frac {2}{3} \left (\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^{3/2}-\frac {3}{2} b \int \sqrt {a+b \text {arccosh}(c+d x)}d(c+d x)\right )+\frac {1}{3} \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2 (a+b \text {arccosh}(c+d x))^{3/2}\right )\right )}{d}\) |
| \(\Big \downarrow \) 6294 |
| \(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \text {arccosh}(c+d x))^{5/2}-\frac {5}{6} b \left (\frac {2}{3} \left (\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^{3/2}-\frac {3}{2} b \left ((c+d x) \sqrt {a+b \text {arccosh}(c+d x)}-\frac {1}{2} b \int \frac {c+d x}{\sqrt {c+d x-1} \sqrt {c+d x+1} \sqrt {a+b \text {arccosh}(c+d x)}}d(c+d x)\right )\right )-\frac {1}{2} b \left (\frac {1}{3} (c+d x)^3 \sqrt {a+b \text {arccosh}(c+d x)}-\frac {1}{6} b \int \frac {(c+d x)^3}{\sqrt {c+d x-1} \sqrt {c+d x+1} \sqrt {a+b \text {arccosh}(c+d x)}}d(c+d x)\right )+\frac {1}{3} \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2 (a+b \text {arccosh}(c+d x))^{3/2}\right )\right )}{d}\) |
| \(\Big \downarrow \) 6368 |
| \(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \text {arccosh}(c+d x))^{5/2}-\frac {5}{6} b \left (-\frac {1}{2} b \left (\frac {1}{3} (c+d x)^3 \sqrt {a+b \text {arccosh}(c+d x)}-\frac {1}{6} \int \frac {\cosh ^3\left (\frac {a}{b}-\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{\sqrt {a+b \text {arccosh}(c+d x)}}d(a+b \text {arccosh}(c+d x))\right )+\frac {2}{3} \left (\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^{3/2}-\frac {3}{2} b \left ((c+d x) \sqrt {a+b \text {arccosh}(c+d x)}-\frac {1}{2} \int \frac {\cosh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{\sqrt {a+b \text {arccosh}(c+d x)}}d(a+b \text {arccosh}(c+d x))\right )\right )+\frac {1}{3} \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2 (a+b \text {arccosh}(c+d x))^{3/2}\right )\right )}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \text {arccosh}(c+d x))^{5/2}-\frac {5}{6} b \left (\frac {2}{3} \left (\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^{3/2}-\frac {3}{2} b \left ((c+d x) \sqrt {a+b \text {arccosh}(c+d x)}-\frac {1}{2} \int \frac {\sin \left (\frac {i a}{b}-\frac {i (a+b \text {arccosh}(c+d x))}{b}+\frac {\pi }{2}\right )}{\sqrt {a+b \text {arccosh}(c+d x)}}d(a+b \text {arccosh}(c+d x))\right )\right )-\frac {1}{2} b \left (\frac {1}{3} (c+d x)^3 \sqrt {a+b \text {arccosh}(c+d x)}-\frac {1}{6} \int \frac {\sin \left (\frac {i a}{b}-\frac {i (a+b \text {arccosh}(c+d x))}{b}+\frac {\pi }{2}\right )^3}{\sqrt {a+b \text {arccosh}(c+d x)}}d(a+b \text {arccosh}(c+d x))\right )+\frac {1}{3} \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2 (a+b \text {arccosh}(c+d x))^{3/2}\right )\right )}{d}\) |
| \(\Big \downarrow \) 3788 |
| \(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \text {arccosh}(c+d x))^{5/2}-\frac {5}{6} b \left (\frac {2}{3} \left (\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^{3/2}-\frac {3}{2} b \left ((c+d x) \sqrt {a+b \text {arccosh}(c+d x)}+\frac {1}{2} \left (\frac {1}{2} i \int \frac {i e^{-\frac {a-c-d x}{b}}}{\sqrt {a+b \text {arccosh}(c+d x)}}d(a+b \text {arccosh}(c+d x))-\frac {1}{2} i \int -\frac {i e^{\frac {a-c-d x}{b}}}{\sqrt {a+b \text {arccosh}(c+d x)}}d(a+b \text {arccosh}(c+d x))\right )\right )\right )-\frac {1}{2} b \left (\frac {1}{3} (c+d x)^3 \sqrt {a+b \text {arccosh}(c+d x)}-\frac {1}{6} \int \frac {\sin \left (\frac {i a}{b}-\frac {i (a+b \text {arccosh}(c+d x))}{b}+\frac {\pi }{2}\right )^3}{\sqrt {a+b \text {arccosh}(c+d x)}}d(a+b \text {arccosh}(c+d x))\right )+\frac {1}{3} \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2 (a+b \text {arccosh}(c+d x))^{3/2}\right )\right )}{d}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \text {arccosh}(c+d x))^{5/2}-\frac {5}{6} b \left (\frac {2}{3} \left (\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^{3/2}-\frac {3}{2} b \left (\frac {1}{2} \left (-\frac {1}{2} \int \frac {e^{-\frac {a-c-d x}{b}}}{\sqrt {a+b \text {arccosh}(c+d x)}}d(a+b \text {arccosh}(c+d x))-\frac {1}{2} \int \frac {e^{\frac {a-c-d x}{b}}}{\sqrt {a+b \text {arccosh}(c+d x)}}d(a+b \text {arccosh}(c+d x))\right )+(c+d x) \sqrt {a+b \text {arccosh}(c+d x)}\right )\right )-\frac {1}{2} b \left (\frac {1}{3} (c+d x)^3 \sqrt {a+b \text {arccosh}(c+d x)}-\frac {1}{6} \int \frac {\sin \left (\frac {i a}{b}-\frac {i (a+b \text {arccosh}(c+d x))}{b}+\frac {\pi }{2}\right )^3}{\sqrt {a+b \text {arccosh}(c+d x)}}d(a+b \text {arccosh}(c+d x))\right )+\frac {1}{3} \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2 (a+b \text {arccosh}(c+d x))^{3/2}\right )\right )}{d}\) |
| \(\Big \downarrow \) 2611 |
| \(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \text {arccosh}(c+d x))^{5/2}-\frac {5}{6} b \left (\frac {2}{3} \left (\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^{3/2}-\frac {3}{2} b \left (\frac {1}{2} \left (-\int e^{\frac {a}{b}-\frac {a+b \text {arccosh}(c+d x)}{b}}d\sqrt {a+b \text {arccosh}(c+d x)}-\int e^{\frac {a+b \text {arccosh}(c+d x)}{b}-\frac {a}{b}}d\sqrt {a+b \text {arccosh}(c+d x)}\right )+(c+d x) \sqrt {a+b \text {arccosh}(c+d x)}\right )\right )-\frac {1}{2} b \left (\frac {1}{3} (c+d x)^3 \sqrt {a+b \text {arccosh}(c+d x)}-\frac {1}{6} \int \frac {\sin \left (\frac {i a}{b}-\frac {i (a+b \text {arccosh}(c+d x))}{b}+\frac {\pi }{2}\right )^3}{\sqrt {a+b \text {arccosh}(c+d x)}}d(a+b \text {arccosh}(c+d x))\right )+\frac {1}{3} \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2 (a+b \text {arccosh}(c+d x))^{3/2}\right )\right )}{d}\) |
| \(\Big \downarrow \) 2633 |
| \(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \text {arccosh}(c+d x))^{5/2}-\frac {5}{6} b \left (\frac {2}{3} \left (\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^{3/2}-\frac {3}{2} b \left (\frac {1}{2} \left (-\int e^{\frac {a}{b}-\frac {a+b \text {arccosh}(c+d x)}{b}}d\sqrt {a+b \text {arccosh}(c+d x)}-\frac {1}{2} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )\right )+(c+d x) \sqrt {a+b \text {arccosh}(c+d x)}\right )\right )-\frac {1}{2} b \left (\frac {1}{3} (c+d x)^3 \sqrt {a+b \text {arccosh}(c+d x)}-\frac {1}{6} \int \frac {\sin \left (\frac {i a}{b}-\frac {i (a+b \text {arccosh}(c+d x))}{b}+\frac {\pi }{2}\right )^3}{\sqrt {a+b \text {arccosh}(c+d x)}}d(a+b \text {arccosh}(c+d x))\right )+\frac {1}{3} \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2 (a+b \text {arccosh}(c+d x))^{3/2}\right )\right )}{d}\) |
| \(\Big \downarrow \) 2634 |
| \(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \text {arccosh}(c+d x))^{5/2}-\frac {5}{6} b \left (-\frac {1}{2} b \left (\frac {1}{3} (c+d x)^3 \sqrt {a+b \text {arccosh}(c+d x)}-\frac {1}{6} \int \frac {\sin \left (\frac {i a}{b}-\frac {i (a+b \text {arccosh}(c+d x))}{b}+\frac {\pi }{2}\right )^3}{\sqrt {a+b \text {arccosh}(c+d x)}}d(a+b \text {arccosh}(c+d x))\right )+\frac {2}{3} \left (\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^{3/2}-\frac {3}{2} b \left (\frac {1}{2} \left (-\frac {1}{2} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{2} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )\right )+(c+d x) \sqrt {a+b \text {arccosh}(c+d x)}\right )\right )+\frac {1}{3} \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2 (a+b \text {arccosh}(c+d x))^{3/2}\right )\right )}{d}\) |
| \(\Big \downarrow \) 3793 |
| \(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \text {arccosh}(c+d x))^{5/2}-\frac {5}{6} b \left (-\frac {1}{2} b \left (\frac {1}{3} (c+d x)^3 \sqrt {a+b \text {arccosh}(c+d x)}-\frac {1}{6} \int \left (\frac {\cosh \left (\frac {3 a}{b}-\frac {3 (a+b \text {arccosh}(c+d x))}{b}\right )}{4 \sqrt {a+b \text {arccosh}(c+d x)}}+\frac {3 \cosh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{4 \sqrt {a+b \text {arccosh}(c+d x)}}\right )d(a+b \text {arccosh}(c+d x))\right )+\frac {2}{3} \left (\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^{3/2}-\frac {3}{2} b \left (\frac {1}{2} \left (-\frac {1}{2} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{2} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )\right )+(c+d x) \sqrt {a+b \text {arccosh}(c+d x)}\right )\right )+\frac {1}{3} \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2 (a+b \text {arccosh}(c+d x))^{3/2}\right )\right )}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \text {arccosh}(c+d x))^{5/2}-\frac {5}{6} b \left (\frac {2}{3} \left (\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^{3/2}-\frac {3}{2} b \left (\frac {1}{2} \left (-\frac {1}{2} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{2} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )\right )+(c+d x) \sqrt {a+b \text {arccosh}(c+d x)}\right )\right )-\frac {1}{2} b \left (\frac {1}{6} \left (-\frac {3}{8} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\frac {\pi }{3}} \sqrt {b} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )-\frac {3}{8} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\frac {\pi }{3}} \sqrt {b} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )\right )+\frac {1}{3} (c+d x)^3 \sqrt {a+b \text {arccosh}(c+d x)}\right )+\frac {1}{3} \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2 (a+b \text {arccosh}(c+d x))^{3/2}\right )\right )}{d}\) |
(e^2*(((c + d*x)^3*(a + b*ArcCosh[c + d*x])^(5/2))/3 - (5*b*((Sqrt[-1 + c + d*x]*(c + d*x)^2*Sqrt[1 + c + d*x]*(a + b*ArcCosh[c + d*x])^(3/2))/3 + ( 2*(Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x]*(a + b*ArcCosh[c + d*x])^(3/2) - ( 3*b*((c + d*x)*Sqrt[a + b*ArcCosh[c + d*x]] + (-1/2*(Sqrt[b]*E^(a/b)*Sqrt[ Pi]*Erf[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]]) - (Sqrt[b]*Sqrt[Pi]*Erfi[Sq rt[a + b*ArcCosh[c + d*x]]/Sqrt[b]])/(2*E^(a/b)))/2))/2))/3 - (b*(((c + d* x)^3*Sqrt[a + b*ArcCosh[c + d*x]])/3 + ((-3*Sqrt[b]*E^(a/b)*Sqrt[Pi]*Erf[S qrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]])/8 - (Sqrt[b]*E^((3*a)/b)*Sqrt[Pi/3]* Erf[(Sqrt[3]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/8 - (3*Sqrt[b]*Sqrt[P i]*Erfi[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]])/(8*E^(a/b)) - (Sqrt[b]*Sqrt [Pi/3]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(8*E^((3*a)/b )))/6))/2))/6))/d
3.2.66.3.1 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[((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_Symbol] :> Simp[x*(a + b*A rcCosh[c*x])^n, x] - Simp[b*c*n Int[x*((a + b*ArcCosh[c*x])^(n - 1)/(Sqrt [1 + c*x]*Sqrt[-1 + c*x])), x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]
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_)*((d1_) + (e1_.)*(x_))^(p _)*((d2_) + (e2_.)*(x_))^(p_), x_Symbol] :> Simp[(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*((a + b*ArcCosh[c*x])^n/(2*e1*e2*(p + 1))), x] - Simp[b*(n/(2 *c*(p + 1)))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*x)^p/(-1 + c*x)^ p] Int[(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, p}, x] && EqQ[e1, c*d1] && E qQ[e2, (-c)*d2] && GtQ[n, 0] && NeQ[p, -1]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d1_) + (e 1_.)*(x_))^(p_)*((d2_) + (e2_.)*(x_))^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 1)*(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*((a + b*ArcCosh[c*x])^n/(e1*e2*( m + 2*p + 1))), x] + (Simp[f^2*((m - 1)/(c^2*(m + 2*p + 1))) Int[(f*x)^(m - 2)*(d1 + e1*x)^p*(d2 + e2*x)^p*(a + b*ArcCosh[c*x])^n, x], x] - Simp[b*f *(n/(c*(m + 2*p + 1)))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*x)^p/( -1 + c*x)^p] Int[(f*x)^(m - 1)*(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*( a + b*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d1, e1, d2, e2, f, p}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && GtQ[n, 0] && IGtQ[m, 1] && N eQ[m + 2*p + 1, 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[((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]
\[\int \left (d e x +c e \right )^{2} \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )^{\frac {5}{2}}d x\]
Exception generated. \[ \int (c e+d e x)^2 (a+b \text {arccosh}(c+d x))^{5/2} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
Timed out. \[ \int (c e+d e x)^2 (a+b \text {arccosh}(c+d x))^{5/2} \, dx=\text {Timed out} \]
\[ \int (c e+d e x)^2 (a+b \text {arccosh}(c+d x))^{5/2} \, dx=\int { {\left (d e x + c e\right )}^{2} {\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{\frac {5}{2}} \,d x } \]
\[ \int (c e+d e x)^2 (a+b \text {arccosh}(c+d x))^{5/2} \, dx=\int { {\left (d e x + c e\right )}^{2} {\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{\frac {5}{2}} \,d x } \]
Timed out. \[ \int (c e+d e x)^2 (a+b \text {arccosh}(c+d x))^{5/2} \, dx=\int {\left (c\,e+d\,e\,x\right )}^2\,{\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right )}^{5/2} \,d x \]