Integrand size = 23, antiderivative size = 352 \[ \int \frac {(c e+d e x)^2}{(a+b \text {arccosh}(c+d x))^4} \, dx=-\frac {e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{3 b d (a+b \text {arccosh}(c+d x))^3}+\frac {e^2 (c+d x)}{3 b^2 d (a+b \text {arccosh}(c+d x))^2}-\frac {e^2 (c+d x)^3}{2 b^2 d (a+b \text {arccosh}(c+d x))^2}+\frac {e^2 \sqrt {-1+c+d x} \sqrt {1+c+d x}}{3 b^3 d (a+b \text {arccosh}(c+d x))}-\frac {3 e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{2 b^3 d (a+b \text {arccosh}(c+d x))}+\frac {e^2 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{24 b^4 d}+\frac {9 e^2 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c+d x))}{b}\right )}{8 b^4 d}-\frac {e^2 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{24 b^4 d}-\frac {9 e^2 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c+d x))}{b}\right )}{8 b^4 d} \] Output:
-1/3*e^2*(d*x+c-1)^(1/2)*(d*x+c)^2*(d*x+c+1)^(1/2)/b/d/(a+b*arccosh(d*x+c) )^3+1/3*e^2*(d*x+c)/b^2/d/(a+b*arccosh(d*x+c))^2-1/2*e^2*(d*x+c)^3/b^2/d/( a+b*arccosh(d*x+c))^2+1/3*e^2*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)/b^3/d/(a+b*a rccosh(d*x+c))-3/2*e^2*(d*x+c-1)^(1/2)*(d*x+c)^2*(d*x+c+1)^(1/2)/b^3/d/(a+ b*arccosh(d*x+c))+1/24*e^2*cosh(a/b)*Chi((a+b*arccosh(d*x+c))/b)/b^4/d+9/8 *e^2*cosh(3*a/b)*Chi(3*(a+b*arccosh(d*x+c))/b)/b^4/d-1/24*e^2*sinh(a/b)*Sh i((a+b*arccosh(d*x+c))/b)/b^4/d-9/8*e^2*sinh(3*a/b)*Shi(3*(a+b*arccosh(d*x +c))/b)/b^4/d
Time = 1.11 (sec) , antiderivative size = 272, normalized size of antiderivative = 0.77 \[ \int \frac {(c e+d e x)^2}{(a+b \text {arccosh}(c+d x))^4} \, dx=\frac {e^2 \left (-\frac {8 b^3 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{(a+b \text {arccosh}(c+d x))^3}+\frac {4 b^2 \left (2 (c+d x)-3 (c+d x)^3\right )}{(a+b \text {arccosh}(c+d x))^2}-\frac {4 b \sqrt {-1+c+d x} \sqrt {1+c+d x} \left (-2+9 (c+d x)^2\right )}{a+b \text {arccosh}(c+d x)}-80 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {arccosh}(c+d x)\right )+80 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arccosh}(c+d x)\right )+27 \left (3 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {arccosh}(c+d x)\right )+\cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (3 \left (\frac {a}{b}+\text {arccosh}(c+d x)\right )\right )-3 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arccosh}(c+d x)\right )-\sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {arccosh}(c+d x)\right )\right )\right )\right )}{24 b^4 d} \] Input:
Integrate[(c*e + d*e*x)^2/(a + b*ArcCosh[c + d*x])^4,x]
Output:
(e^2*((-8*b^3*Sqrt[-1 + c + d*x]*(c + d*x)^2*Sqrt[1 + c + d*x])/(a + b*Arc Cosh[c + d*x])^3 + (4*b^2*(2*(c + d*x) - 3*(c + d*x)^3))/(a + b*ArcCosh[c + d*x])^2 - (4*b*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x]*(-2 + 9*(c + d*x)^2) )/(a + b*ArcCosh[c + d*x]) - 80*Cosh[a/b]*CoshIntegral[a/b + ArcCosh[c + d *x]] + 80*Sinh[a/b]*SinhIntegral[a/b + ArcCosh[c + d*x]] + 27*(3*Cosh[a/b] *CoshIntegral[a/b + ArcCosh[c + d*x]] + Cosh[(3*a)/b]*CoshIntegral[3*(a/b + ArcCosh[c + d*x])] - 3*Sinh[a/b]*SinhIntegral[a/b + ArcCosh[c + d*x]] - Sinh[(3*a)/b]*SinhIntegral[3*(a/b + ArcCosh[c + d*x])])))/(24*b^4*d)
Time = 2.28 (sec) , antiderivative size = 374, normalized size of antiderivative = 1.06, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.652, Rules used = {6411, 27, 6301, 6366, 6295, 6300, 2009, 6368, 3042, 3784, 26, 3042, 26, 3779, 3782}
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.
| \(\displaystyle \int \frac {(c e+d e x)^2}{(a+b \text {arccosh}(c+d x))^4} \, dx\) |
| \(\Big \downarrow \) 6411 |
| \(\displaystyle \frac {\int \frac {e^2 (c+d x)^2}{(a+b \text {arccosh}(c+d x))^4}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e^2 \int \frac {(c+d x)^2}{(a+b \text {arccosh}(c+d x))^4}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 6301 |
| \(\displaystyle \frac {e^2 \left (-\frac {2 \int \frac {c+d x}{\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^3}d(c+d x)}{3 b}+\frac {\int \frac {(c+d x)^3}{\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^3}d(c+d x)}{b}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2}{3 b (a+b \text {arccosh}(c+d x))^3}\right )}{d}\) |
| \(\Big \downarrow \) 6366 |
| \(\displaystyle \frac {e^2 \left (-\frac {2 \left (\frac {\int \frac {1}{(a+b \text {arccosh}(c+d x))^2}d(c+d x)}{2 b}-\frac {c+d x}{2 b (a+b \text {arccosh}(c+d x))^2}\right )}{3 b}+\frac {\frac {3 \int \frac {(c+d x)^2}{(a+b \text {arccosh}(c+d x))^2}d(c+d x)}{2 b}-\frac {(c+d x)^3}{2 b (a+b \text {arccosh}(c+d x))^2}}{b}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2}{3 b (a+b \text {arccosh}(c+d x))^3}\right )}{d}\) |
| \(\Big \downarrow \) 6295 |
| \(\displaystyle \frac {e^2 \left (\frac {\frac {3 \int \frac {(c+d x)^2}{(a+b \text {arccosh}(c+d x))^2}d(c+d x)}{2 b}-\frac {(c+d x)^3}{2 b (a+b \text {arccosh}(c+d x))^2}}{b}-\frac {2 \left (\frac {\frac {\int \frac {c+d x}{\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))}d(c+d x)}{b}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1}}{b (a+b \text {arccosh}(c+d x))}}{2 b}-\frac {c+d x}{2 b (a+b \text {arccosh}(c+d x))^2}\right )}{3 b}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2}{3 b (a+b \text {arccosh}(c+d x))^3}\right )}{d}\) |
| \(\Big \downarrow \) 6300 |
| \(\displaystyle \frac {e^2 \left (\frac {\frac {3 \left (-\frac {\int \left (-\frac {3 \cosh \left (\frac {3 a}{b}-\frac {3 (a+b \text {arccosh}(c+d x))}{b}\right )}{4 (a+b \text {arccosh}(c+d x))}-\frac {\cosh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{4 (a+b \text {arccosh}(c+d x))}\right )d(a+b \text {arccosh}(c+d x))}{b^2}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2}{b (a+b \text {arccosh}(c+d x))}\right )}{2 b}-\frac {(c+d x)^3}{2 b (a+b \text {arccosh}(c+d x))^2}}{b}-\frac {2 \left (\frac {\frac {\int \frac {c+d x}{\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))}d(c+d x)}{b}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1}}{b (a+b \text {arccosh}(c+d x))}}{2 b}-\frac {c+d x}{2 b (a+b \text {arccosh}(c+d x))^2}\right )}{3 b}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2}{3 b (a+b \text {arccosh}(c+d x))^3}\right )}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {e^2 \left (-\frac {2 \left (\frac {\frac {\int \frac {c+d x}{\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))}d(c+d x)}{b}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1}}{b (a+b \text {arccosh}(c+d x))}}{2 b}-\frac {c+d x}{2 b (a+b \text {arccosh}(c+d x))^2}\right )}{3 b}+\frac {\frac {3 \left (-\frac {-\frac {1}{4} \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )-\frac {3}{4} \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c+d x))}{b}\right )+\frac {1}{4} \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )+\frac {3}{4} \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c+d x))}{b}\right )}{b^2}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2}{b (a+b \text {arccosh}(c+d x))}\right )}{2 b}-\frac {(c+d x)^3}{2 b (a+b \text {arccosh}(c+d x))^2}}{b}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2}{3 b (a+b \text {arccosh}(c+d x))^3}\right )}{d}\) |
| \(\Big \downarrow \) 6368 |
| \(\displaystyle \frac {e^2 \left (-\frac {2 \left (\frac {\frac {\int \frac {\cosh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))}{b^2}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1}}{b (a+b \text {arccosh}(c+d x))}}{2 b}-\frac {c+d x}{2 b (a+b \text {arccosh}(c+d x))^2}\right )}{3 b}+\frac {\frac {3 \left (-\frac {-\frac {1}{4} \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )-\frac {3}{4} \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c+d x))}{b}\right )+\frac {1}{4} \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )+\frac {3}{4} \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c+d x))}{b}\right )}{b^2}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2}{b (a+b \text {arccosh}(c+d x))}\right )}{2 b}-\frac {(c+d x)^3}{2 b (a+b \text {arccosh}(c+d x))^2}}{b}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2}{3 b (a+b \text {arccosh}(c+d x))^3}\right )}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {e^2 \left (-\frac {2 \left (-\frac {c+d x}{2 b (a+b \text {arccosh}(c+d x))^2}+\frac {-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1}}{b (a+b \text {arccosh}(c+d x))}+\frac {\int \frac {\sin \left (\frac {i a}{b}-\frac {i (a+b \text {arccosh}(c+d x))}{b}+\frac {\pi }{2}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))}{b^2}}{2 b}\right )}{3 b}+\frac {\frac {3 \left (-\frac {-\frac {1}{4} \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )-\frac {3}{4} \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c+d x))}{b}\right )+\frac {1}{4} \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )+\frac {3}{4} \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c+d x))}{b}\right )}{b^2}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2}{b (a+b \text {arccosh}(c+d x))}\right )}{2 b}-\frac {(c+d x)^3}{2 b (a+b \text {arccosh}(c+d x))^2}}{b}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2}{3 b (a+b \text {arccosh}(c+d x))^3}\right )}{d}\) |
| \(\Big \downarrow \) 3784 |
| \(\displaystyle \frac {e^2 \left (-\frac {2 \left (-\frac {c+d x}{2 b (a+b \text {arccosh}(c+d x))^2}+\frac {-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1}}{b (a+b \text {arccosh}(c+d x))}+\frac {\cosh \left (\frac {a}{b}\right ) \int \frac {\cosh \left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))-i \sinh \left (\frac {a}{b}\right ) \int -\frac {i \sinh \left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))}{b^2}}{2 b}\right )}{3 b}+\frac {\frac {3 \left (-\frac {-\frac {1}{4} \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )-\frac {3}{4} \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c+d x))}{b}\right )+\frac {1}{4} \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )+\frac {3}{4} \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c+d x))}{b}\right )}{b^2}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2}{b (a+b \text {arccosh}(c+d x))}\right )}{2 b}-\frac {(c+d x)^3}{2 b (a+b \text {arccosh}(c+d x))^2}}{b}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2}{3 b (a+b \text {arccosh}(c+d x))^3}\right )}{d}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {e^2 \left (-\frac {2 \left (\frac {\frac {\cosh \left (\frac {a}{b}\right ) \int \frac {\cosh \left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))-\sinh \left (\frac {a}{b}\right ) \int \frac {\sinh \left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))}{b^2}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1}}{b (a+b \text {arccosh}(c+d x))}}{2 b}-\frac {c+d x}{2 b (a+b \text {arccosh}(c+d x))^2}\right )}{3 b}+\frac {\frac {3 \left (-\frac {-\frac {1}{4} \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )-\frac {3}{4} \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c+d x))}{b}\right )+\frac {1}{4} \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )+\frac {3}{4} \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c+d x))}{b}\right )}{b^2}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2}{b (a+b \text {arccosh}(c+d x))}\right )}{2 b}-\frac {(c+d x)^3}{2 b (a+b \text {arccosh}(c+d x))^2}}{b}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2}{3 b (a+b \text {arccosh}(c+d x))^3}\right )}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {e^2 \left (-\frac {2 \left (-\frac {c+d x}{2 b (a+b \text {arccosh}(c+d x))^2}+\frac {-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1}}{b (a+b \text {arccosh}(c+d x))}+\frac {\cosh \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {i (a+b \text {arccosh}(c+d x))}{b}+\frac {\pi }{2}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))-\sinh \left (\frac {a}{b}\right ) \int -\frac {i \sin \left (\frac {i (a+b \text {arccosh}(c+d x))}{b}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))}{b^2}}{2 b}\right )}{3 b}+\frac {\frac {3 \left (-\frac {-\frac {1}{4} \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )-\frac {3}{4} \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c+d x))}{b}\right )+\frac {1}{4} \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )+\frac {3}{4} \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c+d x))}{b}\right )}{b^2}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2}{b (a+b \text {arccosh}(c+d x))}\right )}{2 b}-\frac {(c+d x)^3}{2 b (a+b \text {arccosh}(c+d x))^2}}{b}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2}{3 b (a+b \text {arccosh}(c+d x))^3}\right )}{d}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {e^2 \left (-\frac {2 \left (-\frac {c+d x}{2 b (a+b \text {arccosh}(c+d x))^2}+\frac {-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1}}{b (a+b \text {arccosh}(c+d x))}+\frac {i \sinh \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {i (a+b \text {arccosh}(c+d x))}{b}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))+\cosh \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {i (a+b \text {arccosh}(c+d x))}{b}+\frac {\pi }{2}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))}{b^2}}{2 b}\right )}{3 b}+\frac {\frac {3 \left (-\frac {-\frac {1}{4} \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )-\frac {3}{4} \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c+d x))}{b}\right )+\frac {1}{4} \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )+\frac {3}{4} \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c+d x))}{b}\right )}{b^2}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2}{b (a+b \text {arccosh}(c+d x))}\right )}{2 b}-\frac {(c+d x)^3}{2 b (a+b \text {arccosh}(c+d x))^2}}{b}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2}{3 b (a+b \text {arccosh}(c+d x))^3}\right )}{d}\) |
| \(\Big \downarrow \) 3779 |
| \(\displaystyle \frac {e^2 \left (-\frac {2 \left (-\frac {c+d x}{2 b (a+b \text {arccosh}(c+d x))^2}+\frac {-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1}}{b (a+b \text {arccosh}(c+d x))}+\frac {-\sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )+\cosh \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {i (a+b \text {arccosh}(c+d x))}{b}+\frac {\pi }{2}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))}{b^2}}{2 b}\right )}{3 b}+\frac {\frac {3 \left (-\frac {-\frac {1}{4} \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )-\frac {3}{4} \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c+d x))}{b}\right )+\frac {1}{4} \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )+\frac {3}{4} \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c+d x))}{b}\right )}{b^2}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2}{b (a+b \text {arccosh}(c+d x))}\right )}{2 b}-\frac {(c+d x)^3}{2 b (a+b \text {arccosh}(c+d x))^2}}{b}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2}{3 b (a+b \text {arccosh}(c+d x))^3}\right )}{d}\) |
| \(\Big \downarrow \) 3782 |
| \(\displaystyle \frac {e^2 \left (-\frac {2 \left (\frac {\frac {\cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )-\sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{b^2}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1}}{b (a+b \text {arccosh}(c+d x))}}{2 b}-\frac {c+d x}{2 b (a+b \text {arccosh}(c+d x))^2}\right )}{3 b}+\frac {\frac {3 \left (-\frac {-\frac {1}{4} \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )-\frac {3}{4} \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c+d x))}{b}\right )+\frac {1}{4} \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )+\frac {3}{4} \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c+d x))}{b}\right )}{b^2}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2}{b (a+b \text {arccosh}(c+d x))}\right )}{2 b}-\frac {(c+d x)^3}{2 b (a+b \text {arccosh}(c+d x))^2}}{b}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2}{3 b (a+b \text {arccosh}(c+d x))^3}\right )}{d}\) |
Input:
Int[(c*e + d*e*x)^2/(a + b*ArcCosh[c + d*x])^4,x]
Output:
(e^2*(-1/3*(Sqrt[-1 + c + d*x]*(c + d*x)^2*Sqrt[1 + c + d*x])/(b*(a + b*Ar cCosh[c + d*x])^3) - (2*(-1/2*(c + d*x)/(b*(a + b*ArcCosh[c + d*x])^2) + ( -((Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x])/(b*(a + b*ArcCosh[c + d*x]))) + ( Cosh[a/b]*CoshIntegral[(a + b*ArcCosh[c + d*x])/b] - Sinh[a/b]*SinhIntegra l[(a + b*ArcCosh[c + d*x])/b])/b^2)/(2*b)))/(3*b) + (-1/2*(c + d*x)^3/(b*( a + b*ArcCosh[c + d*x])^2) + (3*(-((Sqrt[-1 + c + d*x]*(c + d*x)^2*Sqrt[1 + c + d*x])/(b*(a + b*ArcCosh[c + d*x]))) - (-1/4*(Cosh[a/b]*CoshIntegral[ (a + b*ArcCosh[c + d*x])/b]) - (3*Cosh[(3*a)/b]*CoshIntegral[(3*(a + b*Arc Cosh[c + d*x]))/b])/4 + (Sinh[a/b]*SinhIntegral[(a + b*ArcCosh[c + d*x])/b ])/4 + (3*Sinh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcCosh[c + d*x]))/b])/4)/b ^2))/(2*b))/b))/d
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[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbo l] :> Simp[I*(SinhIntegral[c*f*(fz/d) + f*fz*x]/d), x] /; FreeQ[{c, d, e, f , fz}, x] && EqQ[d*e - c*f*fz*I, 0]
Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbo l] :> Simp[CoshIntegral[c*f*(fz/d) + f*fz*x]/d, x] /; FreeQ[{c, d, e, f, fz }, x] && EqQ[d*(e - Pi/2) - c*f*fz*I, 0]
Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[Cos[(d* e - c*f)/d] Int[Sin[c*(f/d) + f*x]/(c + d*x), x], x] + Simp[Sin[(d*e - c* f)/d] Int[Cos[c*(f/d) + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f}, x] && NeQ[d*e - c*f, 0]
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]
Leaf count of result is larger than twice the leaf count of optimal. \(776\) vs. \(2(322)=644\).
Time = 0.16 (sec) , antiderivative size = 777, normalized size of antiderivative = 2.21
| method | result | size |
| derivativedivides | \(\frac {\frac {\left (-4 \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (d x +c \right )^{2}+\sqrt {d x +c -1}\, \sqrt {d x +c +1}+4 \left (d x +c \right )^{3}-3 d x -3 c \right ) e^{2} \left (9 b^{2} \operatorname {arccosh}\left (d x +c \right )^{2}+18 a b \,\operatorname {arccosh}\left (d x +c \right )-3 b^{2} \operatorname {arccosh}\left (d x +c \right )+9 a^{2}-3 a b +2 b^{2}\right )}{48 b^{3} \left (b^{3} \operatorname {arccosh}\left (d x +c \right )^{3}+3 a \,b^{2} \operatorname {arccosh}\left (d x +c \right )^{2}+3 a^{2} b \,\operatorname {arccosh}\left (d x +c \right )+a^{3}\right )}-\frac {9 e^{2} {\mathrm e}^{\frac {3 a}{b}} \operatorname {expIntegral}_{1}\left (3 \,\operatorname {arccosh}\left (d x +c \right )+\frac {3 a}{b}\right )}{16 b^{4}}+\frac {\left (-\sqrt {d x +c -1}\, \sqrt {d x +c +1}+d x +c \right ) e^{2} \left (b^{2} \operatorname {arccosh}\left (d x +c \right )^{2}+2 a b \,\operatorname {arccosh}\left (d x +c \right )-b^{2} \operatorname {arccosh}\left (d x +c \right )+a^{2}-a b +2 b^{2}\right )}{48 b^{3} \left (b^{3} \operatorname {arccosh}\left (d x +c \right )^{3}+3 a \,b^{2} \operatorname {arccosh}\left (d x +c \right )^{2}+3 a^{2} b \,\operatorname {arccosh}\left (d x +c \right )+a^{3}\right )}-\frac {e^{2} {\mathrm e}^{\frac {a}{b}} \operatorname {expIntegral}_{1}\left (\operatorname {arccosh}\left (d x +c \right )+\frac {a}{b}\right )}{48 b^{4}}-\frac {e^{2} \left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )}{24 b \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )^{3}}-\frac {e^{2} \left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )}{48 b^{2} \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )^{2}}-\frac {e^{2} \left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )}{48 b^{3} \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )}-\frac {e^{2} {\mathrm e}^{-\frac {a}{b}} \operatorname {expIntegral}_{1}\left (-\operatorname {arccosh}\left (d x +c \right )-\frac {a}{b}\right )}{48 b^{4}}-\frac {e^{2} \left (4 \left (d x +c \right )^{3}-3 d x -3 c +4 \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (d x +c \right )^{2}-\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )}{24 b \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )^{3}}-\frac {e^{2} \left (4 \left (d x +c \right )^{3}-3 d x -3 c +4 \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (d x +c \right )^{2}-\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )}{16 b^{2} \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )^{2}}-\frac {3 e^{2} \left (4 \left (d x +c \right )^{3}-3 d x -3 c +4 \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (d x +c \right )^{2}-\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )}{16 b^{3} \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )}-\frac {9 e^{2} {\mathrm e}^{-\frac {3 a}{b}} \operatorname {expIntegral}_{1}\left (-3 \,\operatorname {arccosh}\left (d x +c \right )-\frac {3 a}{b}\right )}{16 b^{4}}}{d}\) | \(777\) |
| default | \(\frac {\frac {\left (-4 \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (d x +c \right )^{2}+\sqrt {d x +c -1}\, \sqrt {d x +c +1}+4 \left (d x +c \right )^{3}-3 d x -3 c \right ) e^{2} \left (9 b^{2} \operatorname {arccosh}\left (d x +c \right )^{2}+18 a b \,\operatorname {arccosh}\left (d x +c \right )-3 b^{2} \operatorname {arccosh}\left (d x +c \right )+9 a^{2}-3 a b +2 b^{2}\right )}{48 b^{3} \left (b^{3} \operatorname {arccosh}\left (d x +c \right )^{3}+3 a \,b^{2} \operatorname {arccosh}\left (d x +c \right )^{2}+3 a^{2} b \,\operatorname {arccosh}\left (d x +c \right )+a^{3}\right )}-\frac {9 e^{2} {\mathrm e}^{\frac {3 a}{b}} \operatorname {expIntegral}_{1}\left (3 \,\operatorname {arccosh}\left (d x +c \right )+\frac {3 a}{b}\right )}{16 b^{4}}+\frac {\left (-\sqrt {d x +c -1}\, \sqrt {d x +c +1}+d x +c \right ) e^{2} \left (b^{2} \operatorname {arccosh}\left (d x +c \right )^{2}+2 a b \,\operatorname {arccosh}\left (d x +c \right )-b^{2} \operatorname {arccosh}\left (d x +c \right )+a^{2}-a b +2 b^{2}\right )}{48 b^{3} \left (b^{3} \operatorname {arccosh}\left (d x +c \right )^{3}+3 a \,b^{2} \operatorname {arccosh}\left (d x +c \right )^{2}+3 a^{2} b \,\operatorname {arccosh}\left (d x +c \right )+a^{3}\right )}-\frac {e^{2} {\mathrm e}^{\frac {a}{b}} \operatorname {expIntegral}_{1}\left (\operatorname {arccosh}\left (d x +c \right )+\frac {a}{b}\right )}{48 b^{4}}-\frac {e^{2} \left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )}{24 b \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )^{3}}-\frac {e^{2} \left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )}{48 b^{2} \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )^{2}}-\frac {e^{2} \left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )}{48 b^{3} \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )}-\frac {e^{2} {\mathrm e}^{-\frac {a}{b}} \operatorname {expIntegral}_{1}\left (-\operatorname {arccosh}\left (d x +c \right )-\frac {a}{b}\right )}{48 b^{4}}-\frac {e^{2} \left (4 \left (d x +c \right )^{3}-3 d x -3 c +4 \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (d x +c \right )^{2}-\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )}{24 b \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )^{3}}-\frac {e^{2} \left (4 \left (d x +c \right )^{3}-3 d x -3 c +4 \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (d x +c \right )^{2}-\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )}{16 b^{2} \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )^{2}}-\frac {3 e^{2} \left (4 \left (d x +c \right )^{3}-3 d x -3 c +4 \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (d x +c \right )^{2}-\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )}{16 b^{3} \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )}-\frac {9 e^{2} {\mathrm e}^{-\frac {3 a}{b}} \operatorname {expIntegral}_{1}\left (-3 \,\operatorname {arccosh}\left (d x +c \right )-\frac {3 a}{b}\right )}{16 b^{4}}}{d}\) | \(777\) |
Input:
int((d*e*x+c*e)^2/(a+b*arccosh(d*x+c))^4,x,method=_RETURNVERBOSE)
Output:
1/d*(1/48*(-4*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)*(d*x+c)^2+(d*x+c-1)^(1/2)*(d *x+c+1)^(1/2)+4*(d*x+c)^3-3*d*x-3*c)*e^2*(9*b^2*arccosh(d*x+c)^2+18*a*b*ar ccosh(d*x+c)-3*b^2*arccosh(d*x+c)+9*a^2-3*a*b+2*b^2)/b^3/(b^3*arccosh(d*x+ c)^3+3*a*b^2*arccosh(d*x+c)^2+3*a^2*b*arccosh(d*x+c)+a^3)-9/16*e^2/b^4*exp (3*a/b)*Ei(1,3*arccosh(d*x+c)+3*a/b)+1/48*(-(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2 )+d*x+c)*e^2*(b^2*arccosh(d*x+c)^2+2*a*b*arccosh(d*x+c)-b^2*arccosh(d*x+c) +a^2-a*b+2*b^2)/b^3/(b^3*arccosh(d*x+c)^3+3*a*b^2*arccosh(d*x+c)^2+3*a^2*b *arccosh(d*x+c)+a^3)-1/48*e^2/b^4*exp(a/b)*Ei(1,arccosh(d*x+c)+a/b)-1/24/b *e^2*(d*x+c+(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2))/(a+b*arccosh(d*x+c))^3-1/48/b ^2*e^2*(d*x+c+(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2))/(a+b*arccosh(d*x+c))^2-1/48 /b^3*e^2*(d*x+c+(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2))/(a+b*arccosh(d*x+c))-1/48 /b^4*e^2*exp(-a/b)*Ei(1,-arccosh(d*x+c)-a/b)-1/24/b*e^2*(4*(d*x+c)^3-3*d*x -3*c+4*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)*(d*x+c)^2-(d*x+c-1)^(1/2)*(d*x+c+1) ^(1/2))/(a+b*arccosh(d*x+c))^3-1/16/b^2*e^2*(4*(d*x+c)^3-3*d*x-3*c+4*(d*x+ c-1)^(1/2)*(d*x+c+1)^(1/2)*(d*x+c)^2-(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2))/(a+b *arccosh(d*x+c))^2-3/16/b^3*e^2*(4*(d*x+c)^3-3*d*x-3*c+4*(d*x+c-1)^(1/2)*( d*x+c+1)^(1/2)*(d*x+c)^2-(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2))/(a+b*arccosh(d*x +c))-9/16/b^4*e^2*exp(-3*a/b)*Ei(1,-3*arccosh(d*x+c)-3*a/b))
\[ \int \frac {(c e+d e x)^2}{(a+b \text {arccosh}(c+d x))^4} \, dx=\int { \frac {{\left (d e x + c e\right )}^{2}}{{\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{4}} \,d x } \] Input:
integrate((d*e*x+c*e)^2/(a+b*arccosh(d*x+c))^4,x, algorithm="fricas")
Output:
integral((d^2*e^2*x^2 + 2*c*d*e^2*x + c^2*e^2)/(b^4*arccosh(d*x + c)^4 + 4 *a*b^3*arccosh(d*x + c)^3 + 6*a^2*b^2*arccosh(d*x + c)^2 + 4*a^3*b*arccosh (d*x + c) + a^4), x)
\[ \int \frac {(c e+d e x)^2}{(a+b \text {arccosh}(c+d x))^4} \, dx=e^{2} \left (\int \frac {c^{2}}{a^{4} + 4 a^{3} b \operatorname {acosh}{\left (c + d x \right )} + 6 a^{2} b^{2} \operatorname {acosh}^{2}{\left (c + d x \right )} + 4 a b^{3} \operatorname {acosh}^{3}{\left (c + d x \right )} + b^{4} \operatorname {acosh}^{4}{\left (c + d x \right )}}\, dx + \int \frac {d^{2} x^{2}}{a^{4} + 4 a^{3} b \operatorname {acosh}{\left (c + d x \right )} + 6 a^{2} b^{2} \operatorname {acosh}^{2}{\left (c + d x \right )} + 4 a b^{3} \operatorname {acosh}^{3}{\left (c + d x \right )} + b^{4} \operatorname {acosh}^{4}{\left (c + d x \right )}}\, dx + \int \frac {2 c d x}{a^{4} + 4 a^{3} b \operatorname {acosh}{\left (c + d x \right )} + 6 a^{2} b^{2} \operatorname {acosh}^{2}{\left (c + d x \right )} + 4 a b^{3} \operatorname {acosh}^{3}{\left (c + d x \right )} + b^{4} \operatorname {acosh}^{4}{\left (c + d x \right )}}\, dx\right ) \] Input:
integrate((d*e*x+c*e)**2/(a+b*acosh(d*x+c))**4,x)
Output:
e**2*(Integral(c**2/(a**4 + 4*a**3*b*acosh(c + d*x) + 6*a**2*b**2*acosh(c + d*x)**2 + 4*a*b**3*acosh(c + d*x)**3 + b**4*acosh(c + d*x)**4), x) + Int egral(d**2*x**2/(a**4 + 4*a**3*b*acosh(c + d*x) + 6*a**2*b**2*acosh(c + d* x)**2 + 4*a*b**3*acosh(c + d*x)**3 + b**4*acosh(c + d*x)**4), x) + Integra l(2*c*d*x/(a**4 + 4*a**3*b*acosh(c + d*x) + 6*a**2*b**2*acosh(c + d*x)**2 + 4*a*b**3*acosh(c + d*x)**3 + b**4*acosh(c + d*x)**4), x))
Timed out. \[ \int \frac {(c e+d e x)^2}{(a+b \text {arccosh}(c+d x))^4} \, dx=\text {Timed out} \] Input:
integrate((d*e*x+c*e)^2/(a+b*arccosh(d*x+c))^4,x, algorithm="maxima")
Output:
Timed out
\[ \int \frac {(c e+d e x)^2}{(a+b \text {arccosh}(c+d x))^4} \, dx=\int { \frac {{\left (d e x + c e\right )}^{2}}{{\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{4}} \,d x } \] Input:
integrate((d*e*x+c*e)^2/(a+b*arccosh(d*x+c))^4,x, algorithm="giac")
Output:
integrate((d*e*x + c*e)^2/(b*arccosh(d*x + c) + a)^4, x)
Timed out. \[ \int \frac {(c e+d e x)^2}{(a+b \text {arccosh}(c+d x))^4} \, dx=\int \frac {{\left (c\,e+d\,e\,x\right )}^2}{{\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right )}^4} \,d x \] Input:
int((c*e + d*e*x)^2/(a + b*acosh(c + d*x))^4,x)
Output:
int((c*e + d*e*x)^2/(a + b*acosh(c + d*x))^4, x)
\[ \int \frac {(c e+d e x)^2}{(a+b \text {arccosh}(c+d x))^4} \, dx=e^{2} \left (\left (\int \frac {x^{2}}{\mathit {acosh} \left (d x +c \right )^{4} b^{4}+4 \mathit {acosh} \left (d x +c \right )^{3} a \,b^{3}+6 \mathit {acosh} \left (d x +c \right )^{2} a^{2} b^{2}+4 \mathit {acosh} \left (d x +c \right ) a^{3} b +a^{4}}d x \right ) d^{2}+2 \left (\int \frac {x}{\mathit {acosh} \left (d x +c \right )^{4} b^{4}+4 \mathit {acosh} \left (d x +c \right )^{3} a \,b^{3}+6 \mathit {acosh} \left (d x +c \right )^{2} a^{2} b^{2}+4 \mathit {acosh} \left (d x +c \right ) a^{3} b +a^{4}}d x \right ) c d +\left (\int \frac {1}{\mathit {acosh} \left (d x +c \right )^{4} b^{4}+4 \mathit {acosh} \left (d x +c \right )^{3} a \,b^{3}+6 \mathit {acosh} \left (d x +c \right )^{2} a^{2} b^{2}+4 \mathit {acosh} \left (d x +c \right ) a^{3} b +a^{4}}d x \right ) c^{2}\right ) \] Input:
int((d*e*x+c*e)^2/(a+b*acosh(d*x+c))^4,x)
Output:
e**2*(int(x**2/(acosh(c + d*x)**4*b**4 + 4*acosh(c + d*x)**3*a*b**3 + 6*ac osh(c + d*x)**2*a**2*b**2 + 4*acosh(c + d*x)*a**3*b + a**4),x)*d**2 + 2*in t(x/(acosh(c + d*x)**4*b**4 + 4*acosh(c + d*x)**3*a*b**3 + 6*acosh(c + d*x )**2*a**2*b**2 + 4*acosh(c + d*x)*a**3*b + a**4),x)*c*d + int(1/(acosh(c + d*x)**4*b**4 + 4*acosh(c + d*x)**3*a*b**3 + 6*acosh(c + d*x)**2*a**2*b**2 + 4*acosh(c + d*x)*a**3*b + a**4),x)*c**2)