Integrand size = 25, antiderivative size = 230 \[ \int \frac {a+b \text {arccosh}(c x)}{x^2 \left (d-c^2 d x^2\right )^3} \, dx=\frac {b c}{12 d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}-\frac {7 b c}{8 d^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {a+b \text {arccosh}(c x)}{d^3 x \left (1-c^2 x^2\right )^2}+\frac {5 c^2 x (a+b \text {arccosh}(c x))}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac {15 c^2 x (a+b \text {arccosh}(c x))}{8 d^3 \left (1-c^2 x^2\right )}+\frac {b c \arctan \left (\sqrt {-1+c x} \sqrt {1+c x}\right )}{d^3}+\frac {15 c (a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{\text {arccosh}(c x)}\right )}{4 d^3}+\frac {15 b c \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )}{8 d^3}-\frac {15 b c \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )}{8 d^3} \]
1/12*b*c/d^3/(c*x-1)^(3/2)/(c*x+1)^(3/2)+(-a-b*arccosh(c*x))/d^3/x/(-c^2*x ^2+1)^2+5/4*c^2*x*(a+b*arccosh(c*x))/d^3/(-c^2*x^2+1)^2+15/8*c^2*x*(a+b*ar ccosh(c*x))/d^3/(-c^2*x^2+1)+b*c*arctan((c*x-1)^(1/2)*(c*x+1)^(1/2))/d^3+1 5/4*c*(a+b*arccosh(c*x))*arctanh(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))/d^3+15/8 *b*c*polylog(2,-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))/d^3-15/8*b*c*polylog(2,c* x+(c*x-1)^(1/2)*(c*x+1)^(1/2))/d^3-7/8*b*c/d^3/(c*x-1)^(1/2)/(c*x+1)^(1/2)
Time = 1.19 (sec) , antiderivative size = 362, normalized size of antiderivative = 1.57 \[ \int \frac {a+b \text {arccosh}(c x)}{x^2 \left (d-c^2 d x^2\right )^3} \, dx=\frac {-\frac {96 a}{x}+\frac {24 a c^2 x}{\left (-1+c^2 x^2\right )^2}-\frac {84 a c^2 x}{-1+c^2 x^2}-\frac {2 b c \left ((-2+c x) \sqrt {-1+c x} \sqrt {1+c x}-3 \text {arccosh}(c x)\right )}{(-1+c x)^2}+\frac {2 b c \left (\sqrt {-1+c x} \sqrt {1+c x} (2+c x)-3 \text {arccosh}(c x)\right )}{(1+c x)^2}-\frac {96 b \text {arccosh}(c x)}{x}+42 b c \left (-\frac {1}{\sqrt {\frac {-1+c x}{1+c x}}}+\frac {\text {arccosh}(c x)}{1-c x}\right )+42 b c \left (\sqrt {\frac {-1+c x}{1+c x}}-\frac {\text {arccosh}(c x)}{1+c x}\right )+\frac {96 b c \sqrt {-1+c^2 x^2} \arctan \left (\sqrt {-1+c^2 x^2}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-90 a c \log (1-c x)+90 a c \log (1+c x)-45 b c \left (\text {arccosh}(c x) \left (\text {arccosh}(c x)-4 \log \left (1+e^{\text {arccosh}(c x)}\right )\right )-4 \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )\right )+45 b c \left (\text {arccosh}(c x) \left (\text {arccosh}(c x)-4 \log \left (1-e^{\text {arccosh}(c x)}\right )\right )-4 \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )\right )}{96 d^3} \]
((-96*a)/x + (24*a*c^2*x)/(-1 + c^2*x^2)^2 - (84*a*c^2*x)/(-1 + c^2*x^2) - (2*b*c*((-2 + c*x)*Sqrt[-1 + c*x]*Sqrt[1 + c*x] - 3*ArcCosh[c*x]))/(-1 + c*x)^2 + (2*b*c*(Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(2 + c*x) - 3*ArcCosh[c*x])) /(1 + c*x)^2 - (96*b*ArcCosh[c*x])/x + 42*b*c*(-(1/Sqrt[(-1 + c*x)/(1 + c* x)]) + ArcCosh[c*x]/(1 - c*x)) + 42*b*c*(Sqrt[(-1 + c*x)/(1 + c*x)] - ArcC osh[c*x]/(1 + c*x)) + (96*b*c*Sqrt[-1 + c^2*x^2]*ArcTan[Sqrt[-1 + c^2*x^2] ])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - 90*a*c*Log[1 - c*x] + 90*a*c*Log[1 + c *x] - 45*b*c*(ArcCosh[c*x]*(ArcCosh[c*x] - 4*Log[1 + E^ArcCosh[c*x]]) - 4* PolyLog[2, -E^ArcCosh[c*x]]) + 45*b*c*(ArcCosh[c*x]*(ArcCosh[c*x] - 4*Log[ 1 - E^ArcCosh[c*x]]) - 4*PolyLog[2, E^ArcCosh[c*x]]))/(96*d^3)
Result contains complex when optimal does not.
Time = 1.13 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.17, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.720, Rules used = {6347, 27, 115, 27, 115, 27, 103, 218, 6316, 83, 6316, 83, 6318, 3042, 26, 4670, 2715, 2838}
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.
| \(\displaystyle \int \frac {a+b \text {arccosh}(c x)}{x^2 \left (d-c^2 d x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 6347 |
| \(\displaystyle 5 c^2 \int \frac {a+b \text {arccosh}(c x)}{d^3 \left (1-c^2 x^2\right )^3}dx+\frac {b c \int \frac {1}{x (c x-1)^{5/2} (c x+1)^{5/2}}dx}{d^3}-\frac {a+b \text {arccosh}(c x)}{d^3 x \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5 c^2 \int \frac {a+b \text {arccosh}(c x)}{\left (1-c^2 x^2\right )^3}dx}{d^3}+\frac {b c \int \frac {1}{x (c x-1)^{5/2} (c x+1)^{5/2}}dx}{d^3}-\frac {a+b \text {arccosh}(c x)}{d^3 x \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 115 |
| \(\displaystyle \frac {5 c^2 \int \frac {a+b \text {arccosh}(c x)}{\left (1-c^2 x^2\right )^3}dx}{d^3}+\frac {b c \left (-\frac {\int \frac {3 c}{x (c x-1)^{3/2} (c x+1)^{3/2}}dx}{3 c}-\frac {1}{3 (c x-1)^{3/2} (c x+1)^{3/2}}\right )}{d^3}-\frac {a+b \text {arccosh}(c x)}{d^3 x \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5 c^2 \int \frac {a+b \text {arccosh}(c x)}{\left (1-c^2 x^2\right )^3}dx}{d^3}+\frac {b c \left (-\int \frac {1}{x (c x-1)^{3/2} (c x+1)^{3/2}}dx-\frac {1}{3 (c x-1)^{3/2} (c x+1)^{3/2}}\right )}{d^3}-\frac {a+b \text {arccosh}(c x)}{d^3 x \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 115 |
| \(\displaystyle \frac {5 c^2 \int \frac {a+b \text {arccosh}(c x)}{\left (1-c^2 x^2\right )^3}dx}{d^3}+\frac {b c \left (\frac {\int \frac {c}{x \sqrt {c x-1} \sqrt {c x+1}}dx}{c}+\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {1}{3 (c x-1)^{3/2} (c x+1)^{3/2}}\right )}{d^3}-\frac {a+b \text {arccosh}(c x)}{d^3 x \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5 c^2 \int \frac {a+b \text {arccosh}(c x)}{\left (1-c^2 x^2\right )^3}dx}{d^3}+\frac {b c \left (\int \frac {1}{x \sqrt {c x-1} \sqrt {c x+1}}dx+\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {1}{3 (c x-1)^{3/2} (c x+1)^{3/2}}\right )}{d^3}-\frac {a+b \text {arccosh}(c x)}{d^3 x \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 103 |
| \(\displaystyle \frac {5 c^2 \int \frac {a+b \text {arccosh}(c x)}{\left (1-c^2 x^2\right )^3}dx}{d^3}+\frac {b c \left (c \int \frac {1}{(c x-1) (c x+1) c+c}d\left (\sqrt {c x-1} \sqrt {c x+1}\right )+\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {1}{3 (c x-1)^{3/2} (c x+1)^{3/2}}\right )}{d^3}-\frac {a+b \text {arccosh}(c x)}{d^3 x \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {5 c^2 \int \frac {a+b \text {arccosh}(c x)}{\left (1-c^2 x^2\right )^3}dx}{d^3}-\frac {a+b \text {arccosh}(c x)}{d^3 x \left (1-c^2 x^2\right )^2}+\frac {b c \left (\arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )+\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {1}{3 (c x-1)^{3/2} (c x+1)^{3/2}}\right )}{d^3}\) |
| \(\Big \downarrow \) 6316 |
| \(\displaystyle \frac {5 c^2 \left (\frac {3}{4} \int \frac {a+b \text {arccosh}(c x)}{\left (1-c^2 x^2\right )^2}dx-\frac {1}{4} b c \int \frac {x}{(c x-1)^{5/2} (c x+1)^{5/2}}dx+\frac {x (a+b \text {arccosh}(c x))}{4 \left (1-c^2 x^2\right )^2}\right )}{d^3}-\frac {a+b \text {arccosh}(c x)}{d^3 x \left (1-c^2 x^2\right )^2}+\frac {b c \left (\arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )+\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {1}{3 (c x-1)^{3/2} (c x+1)^{3/2}}\right )}{d^3}\) |
| \(\Big \downarrow \) 83 |
| \(\displaystyle \frac {5 c^2 \left (\frac {3}{4} \int \frac {a+b \text {arccosh}(c x)}{\left (1-c^2 x^2\right )^2}dx+\frac {x (a+b \text {arccosh}(c x))}{4 \left (1-c^2 x^2\right )^2}+\frac {b}{12 c (c x-1)^{3/2} (c x+1)^{3/2}}\right )}{d^3}-\frac {a+b \text {arccosh}(c x)}{d^3 x \left (1-c^2 x^2\right )^2}+\frac {b c \left (\arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )+\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {1}{3 (c x-1)^{3/2} (c x+1)^{3/2}}\right )}{d^3}\) |
| \(\Big \downarrow \) 6316 |
| \(\displaystyle \frac {5 c^2 \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {a+b \text {arccosh}(c x)}{1-c^2 x^2}dx+\frac {1}{2} b c \int \frac {x}{(c x-1)^{3/2} (c x+1)^{3/2}}dx+\frac {x (a+b \text {arccosh}(c x))}{2 \left (1-c^2 x^2\right )}\right )+\frac {x (a+b \text {arccosh}(c x))}{4 \left (1-c^2 x^2\right )^2}+\frac {b}{12 c (c x-1)^{3/2} (c x+1)^{3/2}}\right )}{d^3}-\frac {a+b \text {arccosh}(c x)}{d^3 x \left (1-c^2 x^2\right )^2}+\frac {b c \left (\arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )+\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {1}{3 (c x-1)^{3/2} (c x+1)^{3/2}}\right )}{d^3}\) |
| \(\Big \downarrow \) 83 |
| \(\displaystyle \frac {5 c^2 \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {a+b \text {arccosh}(c x)}{1-c^2 x^2}dx+\frac {x (a+b \text {arccosh}(c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {x (a+b \text {arccosh}(c x))}{4 \left (1-c^2 x^2\right )^2}+\frac {b}{12 c (c x-1)^{3/2} (c x+1)^{3/2}}\right )}{d^3}-\frac {a+b \text {arccosh}(c x)}{d^3 x \left (1-c^2 x^2\right )^2}+\frac {b c \left (\arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )+\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {1}{3 (c x-1)^{3/2} (c x+1)^{3/2}}\right )}{d^3}\) |
| \(\Big \downarrow \) 6318 |
| \(\displaystyle \frac {5 c^2 \left (\frac {3}{4} \left (-\frac {\int \frac {a+b \text {arccosh}(c x)}{\sqrt {\frac {c x-1}{c x+1}} (c x+1)}d\text {arccosh}(c x)}{2 c}+\frac {x (a+b \text {arccosh}(c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {x (a+b \text {arccosh}(c x))}{4 \left (1-c^2 x^2\right )^2}+\frac {b}{12 c (c x-1)^{3/2} (c x+1)^{3/2}}\right )}{d^3}-\frac {a+b \text {arccosh}(c x)}{d^3 x \left (1-c^2 x^2\right )^2}+\frac {b c \left (\arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )+\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {1}{3 (c x-1)^{3/2} (c x+1)^{3/2}}\right )}{d^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {5 c^2 \left (\frac {3}{4} \left (-\frac {\int i (a+b \text {arccosh}(c x)) \csc (i \text {arccosh}(c x))d\text {arccosh}(c x)}{2 c}+\frac {x (a+b \text {arccosh}(c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {x (a+b \text {arccosh}(c x))}{4 \left (1-c^2 x^2\right )^2}+\frac {b}{12 c (c x-1)^{3/2} (c x+1)^{3/2}}\right )}{d^3}-\frac {a+b \text {arccosh}(c x)}{d^3 x \left (1-c^2 x^2\right )^2}+\frac {b c \left (\arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )+\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {1}{3 (c x-1)^{3/2} (c x+1)^{3/2}}\right )}{d^3}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {5 c^2 \left (\frac {3}{4} \left (-\frac {i \int (a+b \text {arccosh}(c x)) \csc (i \text {arccosh}(c x))d\text {arccosh}(c x)}{2 c}+\frac {x (a+b \text {arccosh}(c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {x (a+b \text {arccosh}(c x))}{4 \left (1-c^2 x^2\right )^2}+\frac {b}{12 c (c x-1)^{3/2} (c x+1)^{3/2}}\right )}{d^3}-\frac {a+b \text {arccosh}(c x)}{d^3 x \left (1-c^2 x^2\right )^2}+\frac {b c \left (\arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )+\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {1}{3 (c x-1)^{3/2} (c x+1)^{3/2}}\right )}{d^3}\) |
| \(\Big \downarrow \) 4670 |
| \(\displaystyle \frac {5 c^2 \left (\frac {3}{4} \left (-\frac {i \left (i b \int \log \left (1-e^{\text {arccosh}(c x)}\right )d\text {arccosh}(c x)-i b \int \log \left (1+e^{\text {arccosh}(c x)}\right )d\text {arccosh}(c x)+2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )}{2 c}+\frac {x (a+b \text {arccosh}(c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {x (a+b \text {arccosh}(c x))}{4 \left (1-c^2 x^2\right )^2}+\frac {b}{12 c (c x-1)^{3/2} (c x+1)^{3/2}}\right )}{d^3}-\frac {a+b \text {arccosh}(c x)}{d^3 x \left (1-c^2 x^2\right )^2}+\frac {b c \left (\arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )+\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {1}{3 (c x-1)^{3/2} (c x+1)^{3/2}}\right )}{d^3}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle \frac {5 c^2 \left (\frac {3}{4} \left (-\frac {i \left (i b \int e^{-\text {arccosh}(c x)} \log \left (1-e^{\text {arccosh}(c x)}\right )de^{\text {arccosh}(c x)}-i b \int e^{-\text {arccosh}(c x)} \log \left (1+e^{\text {arccosh}(c x)}\right )de^{\text {arccosh}(c x)}+2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )}{2 c}+\frac {x (a+b \text {arccosh}(c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {x (a+b \text {arccosh}(c x))}{4 \left (1-c^2 x^2\right )^2}+\frac {b}{12 c (c x-1)^{3/2} (c x+1)^{3/2}}\right )}{d^3}-\frac {a+b \text {arccosh}(c x)}{d^3 x \left (1-c^2 x^2\right )^2}+\frac {b c \left (\arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )+\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {1}{3 (c x-1)^{3/2} (c x+1)^{3/2}}\right )}{d^3}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {5 c^2 \left (\frac {3}{4} \left (-\frac {i \left (2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )\right )}{2 c}+\frac {x (a+b \text {arccosh}(c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {x (a+b \text {arccosh}(c x))}{4 \left (1-c^2 x^2\right )^2}+\frac {b}{12 c (c x-1)^{3/2} (c x+1)^{3/2}}\right )}{d^3}-\frac {a+b \text {arccosh}(c x)}{d^3 x \left (1-c^2 x^2\right )^2}+\frac {b c \left (\arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )+\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {1}{3 (c x-1)^{3/2} (c x+1)^{3/2}}\right )}{d^3}\) |
-((a + b*ArcCosh[c*x])/(d^3*x*(1 - c^2*x^2)^2)) + (b*c*(-1/3*1/((-1 + c*x) ^(3/2)*(1 + c*x)^(3/2)) + 1/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + ArcTan[Sqrt[- 1 + c*x]*Sqrt[1 + c*x]]))/d^3 + (5*c^2*(b/(12*c*(-1 + c*x)^(3/2)*(1 + c*x) ^(3/2)) + (x*(a + b*ArcCosh[c*x]))/(4*(1 - c^2*x^2)^2) + (3*(-1/2*b/(c*Sqr t[-1 + c*x]*Sqrt[1 + c*x]) + (x*(a + b*ArcCosh[c*x]))/(2*(1 - c^2*x^2)) - ((I/2)*((2*I)*(a + b*ArcCosh[c*x])*ArcTanh[E^ArcCosh[c*x]] + I*b*PolyLog[2 , -E^ArcCosh[c*x]] - I*b*PolyLog[2, E^ArcCosh[c*x]]))/c))/4))/d^3
3.1.52.3.1 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[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0] && EqQ[a*d*f *(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)), 0]
Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_ ))), x_] :> Simp[b*f Subst[Int[1/(d*(b*e - a*f)^2 + b*f^2*x^2), x], x, Sq rt[a + b*x]*Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[2*b*d *e - f*(b*c + a*d), 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 )/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && LtQ[m, -1] && IntegersQ[2*m, 2 *n, 2*p]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Simp[1/(d*e*n*Log[F]) Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) ))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 , (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
Int[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x _Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)]/(f*fz*I)), x] + (-Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[1 - E^((-I)*e + f*fz*x )], x], x] + Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e + f*fz*x)], x], x]) /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_), x _Symbol] :> Simp[(-x)*(d + e*x^2)^(p + 1)*((a + b*ArcCosh[c*x])^n/(2*d*(p + 1))), x] + (Simp[(2*p + 3)/(2*d*(p + 1)) Int[(d + e*x^2)^(p + 1)*(a + b* ArcCosh[c*x])^n, x], x] - Simp[b*c*(n/(2*(p + 1)))*Simp[(d + e*x^2)^p/((1 + c*x)^p*(-1 + c*x)^p)] Int[x*(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, d, e}, x] && EqQ[c^2* d + e, 0] && GtQ[n, 0] && LtQ[p, -1] && NeQ[p, -3/2]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2), x_Symb ol] :> Simp[-(c*d)^(-1) Subst[Int[(a + b*x)^n*Csch[x], x], x, ArcCosh[c*x ]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_ .)*(x_)^2)^(p_), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^(p + 1)*((a + b*ArcCosh[c*x])^n/(d*f*(m + 1))), x] + (Simp[c^2*((m + 2*p + 3)/(f^2*(m + 1 ))) Int[(f*x)^(m + 2)*(d + e*x^2)^p*(a + b*ArcCosh[c*x])^n, x], x] + Simp [b*c*(n/(f*(m + 1)))*Simp[(d + e*x^2)^p/((1 + c*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, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && ILtQ[m, -1]
Time = 0.83 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.17
| method | result | size |
| derivativedivides | \(c \left (-\frac {a \left (\frac {1}{c x}+\frac {1}{16 \left (c x +1\right )^{2}}+\frac {7}{16 \left (c x +1\right )}-\frac {15 \ln \left (c x +1\right )}{16}-\frac {1}{16 \left (c x -1\right )^{2}}+\frac {7}{16 \left (c x -1\right )}+\frac {15 \ln \left (c x -1\right )}{16}\right )}{d^{3}}-\frac {b \left (\frac {45 c^{4} x^{4} \operatorname {arccosh}\left (c x \right )+21 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{3}-75 c^{2} x^{2} \operatorname {arccosh}\left (c x \right )-23 \sqrt {c x -1}\, \sqrt {c x +1}\, c x +24 \,\operatorname {arccosh}\left (c x \right )}{24 \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right ) c x}-2 \arctan \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )-\frac {15 \operatorname {dilog}\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8}-\frac {15 \operatorname {dilog}\left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8}-\frac {15 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8}\right )}{d^{3}}\right )\) | \(269\) |
| default | \(c \left (-\frac {a \left (\frac {1}{c x}+\frac {1}{16 \left (c x +1\right )^{2}}+\frac {7}{16 \left (c x +1\right )}-\frac {15 \ln \left (c x +1\right )}{16}-\frac {1}{16 \left (c x -1\right )^{2}}+\frac {7}{16 \left (c x -1\right )}+\frac {15 \ln \left (c x -1\right )}{16}\right )}{d^{3}}-\frac {b \left (\frac {45 c^{4} x^{4} \operatorname {arccosh}\left (c x \right )+21 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{3}-75 c^{2} x^{2} \operatorname {arccosh}\left (c x \right )-23 \sqrt {c x -1}\, \sqrt {c x +1}\, c x +24 \,\operatorname {arccosh}\left (c x \right )}{24 \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right ) c x}-2 \arctan \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )-\frac {15 \operatorname {dilog}\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8}-\frac {15 \operatorname {dilog}\left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8}-\frac {15 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8}\right )}{d^{3}}\right )\) | \(269\) |
| parts | \(-\frac {a \left (\frac {c}{16 \left (c x +1\right )^{2}}+\frac {7 c}{16 \left (c x +1\right )}-\frac {15 c \ln \left (c x +1\right )}{16}+\frac {1}{x}-\frac {c}{16 \left (c x -1\right )^{2}}+\frac {7 c}{16 \left (c x -1\right )}+\frac {15 c \ln \left (c x -1\right )}{16}\right )}{d^{3}}-\frac {b c \left (\frac {45 c^{4} x^{4} \operatorname {arccosh}\left (c x \right )+21 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{3}-75 c^{2} x^{2} \operatorname {arccosh}\left (c x \right )-23 \sqrt {c x -1}\, \sqrt {c x +1}\, c x +24 \,\operatorname {arccosh}\left (c x \right )}{24 \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right ) c x}-2 \arctan \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )-\frac {15 \operatorname {dilog}\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8}-\frac {15 \operatorname {dilog}\left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8}-\frac {15 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8}\right )}{d^{3}}\) | \(270\) |
c*(-a/d^3*(1/c/x+1/16/(c*x+1)^2+7/16/(c*x+1)-15/16*ln(c*x+1)-1/16/(c*x-1)^ 2+7/16/(c*x-1)+15/16*ln(c*x-1))-b/d^3*(1/24*(45*c^4*x^4*arccosh(c*x)+21*(c *x-1)^(1/2)*(c*x+1)^(1/2)*c^3*x^3-75*c^2*x^2*arccosh(c*x)-23*(c*x-1)^(1/2) *(c*x+1)^(1/2)*c*x+24*arccosh(c*x))/(c^4*x^4-2*c^2*x^2+1)/c/x-2*arctan(c*x +(c*x-1)^(1/2)*(c*x+1)^(1/2))-15/8*dilog(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))- 15/8*dilog(1+c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))-15/8*arccosh(c*x)*ln(1+c*x+( c*x-1)^(1/2)*(c*x+1)^(1/2))))
\[ \int \frac {a+b \text {arccosh}(c x)}{x^2 \left (d-c^2 d x^2\right )^3} \, dx=\int { -\frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} - d\right )}^{3} x^{2}} \,d x } \]
Timed out. \[ \int \frac {a+b \text {arccosh}(c x)}{x^2 \left (d-c^2 d x^2\right )^3} \, dx=\text {Timed out} \]
\[ \int \frac {a+b \text {arccosh}(c x)}{x^2 \left (d-c^2 d x^2\right )^3} \, dx=\int { -\frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} - d\right )}^{3} x^{2}} \,d x } \]
1/2048*(92160*c^7*integrate(1/32*x^5*log(c*x - 1)/(c^6*d^3*x^6 - 3*c^4*d^3 *x^4 + 3*c^2*d^3*x^2 - d^3), x) - 240*c^6*(2*(5*c^2*x^3 - 3*x)/(c^8*d^3*x^ 4 - 2*c^6*d^3*x^2 + c^4*d^3) + 3*log(c*x + 1)/(c^5*d^3) - 3*log(c*x - 1)/( c^5*d^3)) - 30720*c^6*integrate(1/32*x^4*log(c*x - 1)/(c^6*d^3*x^6 - 3*c^4 *d^3*x^4 + 3*c^2*d^3*x^2 - d^3), x) + 90*(c*(2*(5*c^2*x^2 + 3*c*x - 6)/(c^ 8*d^3*x^3 - c^7*d^3*x^2 - c^6*d^3*x + c^5*d^3) - 5*log(c*x + 1)/(c^5*d^3) + 5*log(c*x - 1)/(c^5*d^3)) + 16*(2*c^2*x^2 - 1)*log(c*x - 1)/(c^8*d^3*x^4 - 2*c^6*d^3*x^2 + c^4*d^3))*c^5 + 400*c^4*(2*(c^2*x^3 + x)/(c^6*d^3*x^4 - 2*c^4*d^3*x^2 + c^2*d^3) - log(c*x + 1)/(c^3*d^3) + log(c*x - 1)/(c^3*d^3 )) + 61440*c^4*integrate(1/32*x^2*log(c*x - 1)/(c^6*d^3*x^6 - 3*c^4*d^3*x^ 4 + 3*c^2*d^3*x^2 - d^3), x) + 45*(c*(2*(3*c^2*x^2 - 3*c*x - 2)/(c^6*d^3*x ^3 - c^5*d^3*x^2 - c^4*d^3*x + c^3*d^3) - 3*log(c*x + 1)/(c^3*d^3) + 3*log (c*x - 1)/(c^3*d^3)) - 16*log(c*x - 1)/(c^6*d^3*x^4 - 2*c^4*d^3*x^2 + c^2* d^3))*c^3 + 128*c^2*(2*(3*c^2*x^3 - 5*x)/(c^4*d^3*x^4 - 2*c^2*d^3*x^2 + d^ 3) - 3*log(c*x + 1)/(c*d^3) + 3*log(c*x - 1)/(c*d^3)) - 30720*c^2*integrat e(1/32*log(c*x - 1)/(c^6*d^3*x^6 - 3*c^4*d^3*x^4 + 3*c^2*d^3*x^2 - d^3), x ) - 32*(15*(c^5*x^5 - 2*c^3*x^3 + c*x)*log(c*x + 1)^2 + 30*(c^5*x^5 - 2*c^ 3*x^3 + c*x)*log(c*x + 1)*log(c*x - 1) + 4*(30*c^4*x^4 - 50*c^2*x^2 - 15*( c^5*x^5 - 2*c^3*x^3 + c*x)*log(c*x + 1) + 15*(c^5*x^5 - 2*c^3*x^3 + c*x)*l og(c*x - 1) + 16)*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1)))/(c^4*d^3*x^5 ...
\[ \int \frac {a+b \text {arccosh}(c x)}{x^2 \left (d-c^2 d x^2\right )^3} \, dx=\int { -\frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} - d\right )}^{3} x^{2}} \,d x } \]
Timed out. \[ \int \frac {a+b \text {arccosh}(c x)}{x^2 \left (d-c^2 d x^2\right )^3} \, dx=\int \frac {a+b\,\mathrm {acosh}\left (c\,x\right )}{x^2\,{\left (d-c^2\,d\,x^2\right )}^3} \,d x \]