Integrand size = 25, antiderivative size = 248 \[ \int \frac {a+b \text {arccosh}(c x)}{x^4 \left (d-c^2 d x^2\right )^2} \, dx=-\frac {b c^3}{3 d^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c}{6 d^2 x^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {a+b \text {arccosh}(c x)}{3 d^2 x^3 \left (1-c^2 x^2\right )}-\frac {5 c^2 (a+b \text {arccosh}(c x))}{3 d^2 x \left (1-c^2 x^2\right )}+\frac {5 c^4 x (a+b \text {arccosh}(c x))}{2 d^2 \left (1-c^2 x^2\right )}+\frac {13 b c^3 \arctan \left (\sqrt {-1+c x} \sqrt {1+c x}\right )}{6 d^2}+\frac {5 c^3 (a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{\text {arccosh}(c x)}\right )}{d^2}+\frac {5 b c^3 \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )}{2 d^2}-\frac {5 b c^3 \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )}{2 d^2} \]
1/3*(-a-b*arccosh(c*x))/d^2/x^3/(-c^2*x^2+1)-5/3*c^2*(a+b*arccosh(c*x))/d^ 2/x/(-c^2*x^2+1)+5/2*c^4*x*(a+b*arccosh(c*x))/d^2/(-c^2*x^2+1)+13/6*b*c^3* arctan((c*x-1)^(1/2)*(c*x+1)^(1/2))/d^2+5*c^3*(a+b*arccosh(c*x))*arctanh(c *x+(c*x-1)^(1/2)*(c*x+1)^(1/2))/d^2+5/2*b*c^3*polylog(2,-c*x-(c*x-1)^(1/2) *(c*x+1)^(1/2))/d^2-5/2*b*c^3*polylog(2,c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))/d ^2-1/3*b*c^3/d^2/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/6*b*c/d^2/x^2/(c*x-1)^(1/2) /(c*x+1)^(1/2)
Time = 1.30 (sec) , antiderivative size = 377, normalized size of antiderivative = 1.52 \[ \int \frac {a+b \text {arccosh}(c x)}{x^4 \left (d-c^2 d x^2\right )^2} \, dx=-\frac {\frac {4 a}{x^3}+\frac {24 a c^2}{x}-3 b c^3 \sqrt {\frac {-1+c x}{1+c x}}+\frac {3 b c^3 \sqrt {\frac {-1+c x}{1+c x}}}{-1+c x}+\frac {3 b c^4 x \sqrt {\frac {-1+c x}{1+c x}}}{-1+c x}-\frac {2 b c^3}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b c}{x^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {6 a c^4 x}{-1+c^2 x^2}+\frac {4 b \text {arccosh}(c x)}{x^3}+\frac {24 b c^2 \text {arccosh}(c x)}{x}+\frac {3 b c^3 \text {arccosh}(c x)}{-1+c x}+\frac {3 b c^3 \text {arccosh}(c x)}{1+c x}-\frac {26 b c^3 \sqrt {-1+c^2 x^2} \arctan \left (\sqrt {-1+c^2 x^2}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+30 b c^3 \text {arccosh}(c x) \log \left (1-e^{\text {arccosh}(c x)}\right )-30 b c^3 \text {arccosh}(c x) \log \left (1+e^{\text {arccosh}(c x)}\right )+15 a c^3 \log (1-c x)-15 a c^3 \log (1+c x)-30 b c^3 \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )+30 b c^3 \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )}{12 d^2} \]
-1/12*((4*a)/x^3 + (24*a*c^2)/x - 3*b*c^3*Sqrt[(-1 + c*x)/(1 + c*x)] + (3* b*c^3*Sqrt[(-1 + c*x)/(1 + c*x)])/(-1 + c*x) + (3*b*c^4*x*Sqrt[(-1 + c*x)/ (1 + c*x)])/(-1 + c*x) - (2*b*c^3)/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (2*b*c )/(x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (6*a*c^4*x)/(-1 + c^2*x^2) + (4*b*A rcCosh[c*x])/x^3 + (24*b*c^2*ArcCosh[c*x])/x + (3*b*c^3*ArcCosh[c*x])/(-1 + c*x) + (3*b*c^3*ArcCosh[c*x])/(1 + c*x) - (26*b*c^3*Sqrt[-1 + c^2*x^2]*A rcTan[Sqrt[-1 + c^2*x^2]])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + 30*b*c^3*ArcCo sh[c*x]*Log[1 - E^ArcCosh[c*x]] - 30*b*c^3*ArcCosh[c*x]*Log[1 + E^ArcCosh[ c*x]] + 15*a*c^3*Log[1 - c*x] - 15*a*c^3*Log[1 + c*x] - 30*b*c^3*PolyLog[2 , -E^ArcCosh[c*x]] + 30*b*c^3*PolyLog[2, E^ArcCosh[c*x]])/d^2
Result contains complex when optimal does not.
Time = 1.29 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.26, number of steps used = 22, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.840, Rules used = {6347, 27, 114, 27, 115, 27, 103, 218, 6347, 115, 27, 103, 218, 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.
| \(\displaystyle \int \frac {a+b \text {arccosh}(c x)}{x^4 \left (d-c^2 d x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 6347 |
| \(\displaystyle \frac {5}{3} c^2 \int \frac {a+b \text {arccosh}(c x)}{d^2 x^2 \left (1-c^2 x^2\right )^2}dx-\frac {b c \int \frac {1}{x^3 (c x-1)^{3/2} (c x+1)^{3/2}}dx}{3 d^2}-\frac {a+b \text {arccosh}(c x)}{3 d^2 x^3 \left (1-c^2 x^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5 c^2 \int \frac {a+b \text {arccosh}(c x)}{x^2 \left (1-c^2 x^2\right )^2}dx}{3 d^2}-\frac {b c \int \frac {1}{x^3 (c x-1)^{3/2} (c x+1)^{3/2}}dx}{3 d^2}-\frac {a+b \text {arccosh}(c x)}{3 d^2 x^3 \left (1-c^2 x^2\right )}\) |
| \(\Big \downarrow \) 114 |
| \(\displaystyle \frac {5 c^2 \int \frac {a+b \text {arccosh}(c x)}{x^2 \left (1-c^2 x^2\right )^2}dx}{3 d^2}-\frac {b c \left (\frac {1}{2} \int \frac {3 c^2}{x (c x-1)^{3/2} (c x+1)^{3/2}}dx+\frac {1}{2 x^2 \sqrt {c x-1} \sqrt {c x+1}}\right )}{3 d^2}-\frac {a+b \text {arccosh}(c x)}{3 d^2 x^3 \left (1-c^2 x^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5 c^2 \int \frac {a+b \text {arccosh}(c x)}{x^2 \left (1-c^2 x^2\right )^2}dx}{3 d^2}-\frac {b c \left (\frac {3}{2} c^2 \int \frac {1}{x (c x-1)^{3/2} (c x+1)^{3/2}}dx+\frac {1}{2 x^2 \sqrt {c x-1} \sqrt {c x+1}}\right )}{3 d^2}-\frac {a+b \text {arccosh}(c x)}{3 d^2 x^3 \left (1-c^2 x^2\right )}\) |
| \(\Big \downarrow \) 115 |
| \(\displaystyle \frac {5 c^2 \int \frac {a+b \text {arccosh}(c x)}{x^2 \left (1-c^2 x^2\right )^2}dx}{3 d^2}-\frac {b c \left (\frac {3}{2} c^2 \left (-\frac {\int \frac {c}{x \sqrt {c x-1} \sqrt {c x+1}}dx}{c}-\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{2 x^2 \sqrt {c x-1} \sqrt {c x+1}}\right )}{3 d^2}-\frac {a+b \text {arccosh}(c x)}{3 d^2 x^3 \left (1-c^2 x^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5 c^2 \int \frac {a+b \text {arccosh}(c x)}{x^2 \left (1-c^2 x^2\right )^2}dx}{3 d^2}-\frac {b c \left (\frac {3}{2} c^2 \left (-\int \frac {1}{x \sqrt {c x-1} \sqrt {c x+1}}dx-\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{2 x^2 \sqrt {c x-1} \sqrt {c x+1}}\right )}{3 d^2}-\frac {a+b \text {arccosh}(c x)}{3 d^2 x^3 \left (1-c^2 x^2\right )}\) |
| \(\Big \downarrow \) 103 |
| \(\displaystyle \frac {5 c^2 \int \frac {a+b \text {arccosh}(c x)}{x^2 \left (1-c^2 x^2\right )^2}dx}{3 d^2}-\frac {b c \left (\frac {3}{2} c^2 \left (-c \int \frac {1}{(c x-1) (c x+1) c+c}d\left (\sqrt {c x-1} \sqrt {c x+1}\right )-\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{2 x^2 \sqrt {c x-1} \sqrt {c x+1}}\right )}{3 d^2}-\frac {a+b \text {arccosh}(c x)}{3 d^2 x^3 \left (1-c^2 x^2\right )}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {5 c^2 \int \frac {a+b \text {arccosh}(c x)}{x^2 \left (1-c^2 x^2\right )^2}dx}{3 d^2}-\frac {a+b \text {arccosh}(c x)}{3 d^2 x^3 \left (1-c^2 x^2\right )}-\frac {b c \left (\frac {3}{2} c^2 \left (-\arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )-\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{2 x^2 \sqrt {c x-1} \sqrt {c x+1}}\right )}{3 d^2}\) |
| \(\Big \downarrow \) 6347 |
| \(\displaystyle \frac {5 c^2 \left (3 c^2 \int \frac {a+b \text {arccosh}(c x)}{\left (1-c^2 x^2\right )^2}dx-b c \int \frac {1}{x (c x-1)^{3/2} (c x+1)^{3/2}}dx-\frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )}\right )}{3 d^2}-\frac {a+b \text {arccosh}(c x)}{3 d^2 x^3 \left (1-c^2 x^2\right )}-\frac {b c \left (\frac {3}{2} c^2 \left (-\arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )-\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{2 x^2 \sqrt {c x-1} \sqrt {c x+1}}\right )}{3 d^2}\) |
| \(\Big \downarrow \) 115 |
| \(\displaystyle \frac {5 c^2 \left (3 c^2 \int \frac {a+b \text {arccosh}(c x)}{\left (1-c^2 x^2\right )^2}dx-b c \left (-\frac {\int \frac {c}{x \sqrt {c x-1} \sqrt {c x+1}}dx}{c}-\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}\right )-\frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )}\right )}{3 d^2}-\frac {a+b \text {arccosh}(c x)}{3 d^2 x^3 \left (1-c^2 x^2\right )}-\frac {b c \left (\frac {3}{2} c^2 \left (-\arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )-\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{2 x^2 \sqrt {c x-1} \sqrt {c x+1}}\right )}{3 d^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5 c^2 \left (3 c^2 \int \frac {a+b \text {arccosh}(c x)}{\left (1-c^2 x^2\right )^2}dx-b c \left (-\int \frac {1}{x \sqrt {c x-1} \sqrt {c x+1}}dx-\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}\right )-\frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )}\right )}{3 d^2}-\frac {a+b \text {arccosh}(c x)}{3 d^2 x^3 \left (1-c^2 x^2\right )}-\frac {b c \left (\frac {3}{2} c^2 \left (-\arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )-\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{2 x^2 \sqrt {c x-1} \sqrt {c x+1}}\right )}{3 d^2}\) |
| \(\Big \downarrow \) 103 |
| \(\displaystyle \frac {5 c^2 \left (3 c^2 \int \frac {a+b \text {arccosh}(c x)}{\left (1-c^2 x^2\right )^2}dx-b c \left (-c \int \frac {1}{(c x-1) (c x+1) c+c}d\left (\sqrt {c x-1} \sqrt {c x+1}\right )-\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}\right )-\frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )}\right )}{3 d^2}-\frac {a+b \text {arccosh}(c x)}{3 d^2 x^3 \left (1-c^2 x^2\right )}-\frac {b c \left (\frac {3}{2} c^2 \left (-\arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )-\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{2 x^2 \sqrt {c x-1} \sqrt {c x+1}}\right )}{3 d^2}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {5 c^2 \left (3 c^2 \int \frac {a+b \text {arccosh}(c x)}{\left (1-c^2 x^2\right )^2}dx-\frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )}-b c \left (-\arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )-\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}\right )\right )}{3 d^2}-\frac {a+b \text {arccosh}(c x)}{3 d^2 x^3 \left (1-c^2 x^2\right )}-\frac {b c \left (\frac {3}{2} c^2 \left (-\arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )-\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{2 x^2 \sqrt {c x-1} \sqrt {c x+1}}\right )}{3 d^2}\) |
| \(\Big \downarrow \) 6316 |
| \(\displaystyle \frac {5 c^2 \left (3 c^2 \left (\frac {1}{2} \int \frac {a+b \text {arccosh}(c x)}{1-c^2 x^2}dx+\frac {1}{2} b c \int \frac {x}{(c x-1)^{3/2} (c x+1)^{3/2}}dx+\frac {x (a+b \text {arccosh}(c x))}{2 \left (1-c^2 x^2\right )}\right )-\frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )}-b c \left (-\arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )-\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}\right )\right )}{3 d^2}-\frac {a+b \text {arccosh}(c x)}{3 d^2 x^3 \left (1-c^2 x^2\right )}-\frac {b c \left (\frac {3}{2} c^2 \left (-\arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )-\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{2 x^2 \sqrt {c x-1} \sqrt {c x+1}}\right )}{3 d^2}\) |
| \(\Big \downarrow \) 83 |
| \(\displaystyle \frac {5 c^2 \left (3 c^2 \left (\frac {1}{2} \int \frac {a+b \text {arccosh}(c x)}{1-c^2 x^2}dx+\frac {x (a+b \text {arccosh}(c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {c x-1} \sqrt {c x+1}}\right )-\frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )}-b c \left (-\arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )-\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}\right )\right )}{3 d^2}-\frac {a+b \text {arccosh}(c x)}{3 d^2 x^3 \left (1-c^2 x^2\right )}-\frac {b c \left (\frac {3}{2} c^2 \left (-\arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )-\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{2 x^2 \sqrt {c x-1} \sqrt {c x+1}}\right )}{3 d^2}\) |
| \(\Big \downarrow \) 6318 |
| \(\displaystyle \frac {5 c^2 \left (3 c^2 \left (-\frac {\int \frac {a+b \text {arccosh}(c x)}{\sqrt {\frac {c x-1}{c x+1}} (c x+1)}d\text {arccosh}(c x)}{2 c}+\frac {x (a+b \text {arccosh}(c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {c x-1} \sqrt {c x+1}}\right )-\frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )}-b c \left (-\arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )-\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}\right )\right )}{3 d^2}-\frac {a+b \text {arccosh}(c x)}{3 d^2 x^3 \left (1-c^2 x^2\right )}-\frac {b c \left (\frac {3}{2} c^2 \left (-\arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )-\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{2 x^2 \sqrt {c x-1} \sqrt {c x+1}}\right )}{3 d^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {5 c^2 \left (3 c^2 \left (-\frac {\int i (a+b \text {arccosh}(c x)) \csc (i \text {arccosh}(c x))d\text {arccosh}(c x)}{2 c}+\frac {x (a+b \text {arccosh}(c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {c x-1} \sqrt {c x+1}}\right )-\frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )}-b c \left (-\arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )-\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}\right )\right )}{3 d^2}-\frac {a+b \text {arccosh}(c x)}{3 d^2 x^3 \left (1-c^2 x^2\right )}-\frac {b c \left (\frac {3}{2} c^2 \left (-\arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )-\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{2 x^2 \sqrt {c x-1} \sqrt {c x+1}}\right )}{3 d^2}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {5 c^2 \left (3 c^2 \left (-\frac {i \int (a+b \text {arccosh}(c x)) \csc (i \text {arccosh}(c x))d\text {arccosh}(c x)}{2 c}+\frac {x (a+b \text {arccosh}(c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {c x-1} \sqrt {c x+1}}\right )-\frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )}-b c \left (-\arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )-\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}\right )\right )}{3 d^2}-\frac {a+b \text {arccosh}(c x)}{3 d^2 x^3 \left (1-c^2 x^2\right )}-\frac {b c \left (\frac {3}{2} c^2 \left (-\arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )-\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{2 x^2 \sqrt {c x-1} \sqrt {c x+1}}\right )}{3 d^2}\) |
| \(\Big \downarrow \) 4670 |
| \(\displaystyle \frac {5 c^2 \left (3 c^2 \left (-\frac {i \left (i b \int \log \left (1-e^{\text {arccosh}(c x)}\right )d\text {arccosh}(c x)-i b \int \log \left (1+e^{\text {arccosh}(c x)}\right )d\text {arccosh}(c x)+2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )}{2 c}+\frac {x (a+b \text {arccosh}(c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {c x-1} \sqrt {c x+1}}\right )-\frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )}-b c \left (-\arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )-\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}\right )\right )}{3 d^2}-\frac {a+b \text {arccosh}(c x)}{3 d^2 x^3 \left (1-c^2 x^2\right )}-\frac {b c \left (\frac {3}{2} c^2 \left (-\arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )-\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{2 x^2 \sqrt {c x-1} \sqrt {c x+1}}\right )}{3 d^2}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle \frac {5 c^2 \left (3 c^2 \left (-\frac {i \left (i b \int e^{-\text {arccosh}(c x)} \log \left (1-e^{\text {arccosh}(c x)}\right )de^{\text {arccosh}(c x)}-i b \int e^{-\text {arccosh}(c x)} \log \left (1+e^{\text {arccosh}(c x)}\right )de^{\text {arccosh}(c x)}+2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )}{2 c}+\frac {x (a+b \text {arccosh}(c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {c x-1} \sqrt {c x+1}}\right )-\frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )}-b c \left (-\arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )-\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}\right )\right )}{3 d^2}-\frac {a+b \text {arccosh}(c x)}{3 d^2 x^3 \left (1-c^2 x^2\right )}-\frac {b c \left (\frac {3}{2} c^2 \left (-\arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )-\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{2 x^2 \sqrt {c x-1} \sqrt {c x+1}}\right )}{3 d^2}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {5 c^2 \left (3 c^2 \left (-\frac {i \left (2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )\right )}{2 c}+\frac {x (a+b \text {arccosh}(c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {c x-1} \sqrt {c x+1}}\right )-\frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )}-b c \left (-\arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )-\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}\right )\right )}{3 d^2}-\frac {a+b \text {arccosh}(c x)}{3 d^2 x^3 \left (1-c^2 x^2\right )}-\frac {b c \left (\frac {3}{2} c^2 \left (-\arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )-\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{2 x^2 \sqrt {c x-1} \sqrt {c x+1}}\right )}{3 d^2}\) |
-1/3*(a + b*ArcCosh[c*x])/(d^2*x^3*(1 - c^2*x^2)) - (b*c*(1/(2*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (3*c^2*(-(1/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])) - Arc Tan[Sqrt[-1 + c*x]*Sqrt[1 + c*x]]))/2))/(3*d^2) + (5*c^2*(-((a + b*ArcCosh [c*x])/(x*(1 - c^2*x^2))) - b*c*(-(1/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])) - Arc Tan[Sqrt[-1 + c*x]*Sqrt[1 + c*x]]) + 3*c^2*(-1/2*b/(c*Sqrt[-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)))/(3*d^2)
3.1.45.3.1 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] && ILtQ[m, -1] && (IntegerQ[n] || IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 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 = 0.84 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.01
| method | result | size |
| derivativedivides | \(c^{3} \left (\frac {a \left (-\frac {1}{3 c^{3} x^{3}}-\frac {2}{c x}-\frac {1}{4 \left (c x +1\right )}+\frac {5 \ln \left (c x +1\right )}{4}-\frac {1}{4 \left (c x -1\right )}-\frac {5 \ln \left (c x -1\right )}{4}\right )}{d^{2}}+\frac {b \left (-\frac {15 c^{4} x^{4} \operatorname {arccosh}\left (c x \right )+2 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{3}-10 c^{2} x^{2} \operatorname {arccosh}\left (c x \right )+\sqrt {c x -1}\, \sqrt {c x +1}\, c x -2 \,\operatorname {arccosh}\left (c x \right )}{6 c^{3} x^{3} \left (c^{2} x^{2}-1\right )}+\frac {13 \arctan \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{3}+\frac {5 \operatorname {dilog}\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2}+\frac {5 \operatorname {dilog}\left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2}+\frac {5 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2}\right )}{d^{2}}\right )\) | \(251\) |
| default | \(c^{3} \left (\frac {a \left (-\frac {1}{3 c^{3} x^{3}}-\frac {2}{c x}-\frac {1}{4 \left (c x +1\right )}+\frac {5 \ln \left (c x +1\right )}{4}-\frac {1}{4 \left (c x -1\right )}-\frac {5 \ln \left (c x -1\right )}{4}\right )}{d^{2}}+\frac {b \left (-\frac {15 c^{4} x^{4} \operatorname {arccosh}\left (c x \right )+2 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{3}-10 c^{2} x^{2} \operatorname {arccosh}\left (c x \right )+\sqrt {c x -1}\, \sqrt {c x +1}\, c x -2 \,\operatorname {arccosh}\left (c x \right )}{6 c^{3} x^{3} \left (c^{2} x^{2}-1\right )}+\frac {13 \arctan \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{3}+\frac {5 \operatorname {dilog}\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2}+\frac {5 \operatorname {dilog}\left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2}+\frac {5 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2}\right )}{d^{2}}\right )\) | \(251\) |
| parts | \(\frac {a \left (-\frac {c^{3}}{4 \left (c x +1\right )}+\frac {5 c^{3} \ln \left (c x +1\right )}{4}-\frac {1}{3 x^{3}}-\frac {2 c^{2}}{x}-\frac {c^{3}}{4 \left (c x -1\right )}-\frac {5 c^{3} \ln \left (c x -1\right )}{4}\right )}{d^{2}}+\frac {b \,c^{3} \left (-\frac {15 c^{4} x^{4} \operatorname {arccosh}\left (c x \right )+2 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{3}-10 c^{2} x^{2} \operatorname {arccosh}\left (c x \right )+\sqrt {c x -1}\, \sqrt {c x +1}\, c x -2 \,\operatorname {arccosh}\left (c x \right )}{6 c^{3} x^{3} \left (c^{2} x^{2}-1\right )}+\frac {13 \arctan \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{3}+\frac {5 \operatorname {dilog}\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2}+\frac {5 \operatorname {dilog}\left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2}+\frac {5 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2}\right )}{d^{2}}\) | \(259\) |
c^3*(a/d^2*(-1/3/c^3/x^3-2/c/x-1/4/(c*x+1)+5/4*ln(c*x+1)-1/4/(c*x-1)-5/4*l n(c*x-1))+b/d^2*(-1/6*(15*c^4*x^4*arccosh(c*x)+2*(c*x-1)^(1/2)*(c*x+1)^(1/ 2)*c^3*x^3-10*c^2*x^2*arccosh(c*x)+(c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x-2*arcco sh(c*x))/c^3/x^3/(c^2*x^2-1)+13/3*arctan(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))+ 5/2*dilog(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))+5/2*dilog(1+c*x+(c*x-1)^(1/2)*( c*x+1)^(1/2))+5/2*arccosh(c*x)*ln(1+c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))))
\[ \int \frac {a+b \text {arccosh}(c x)}{x^4 \left (d-c^2 d x^2\right )^2} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} - d\right )}^{2} x^{4}} \,d x } \]
\[ \int \frac {a+b \text {arccosh}(c x)}{x^4 \left (d-c^2 d x^2\right )^2} \, dx=\frac {\int \frac {a}{c^{4} x^{8} - 2 c^{2} x^{6} + x^{4}}\, dx + \int \frac {b \operatorname {acosh}{\left (c x \right )}}{c^{4} x^{8} - 2 c^{2} x^{6} + x^{4}}\, dx}{d^{2}} \]
(Integral(a/(c**4*x**8 - 2*c**2*x**6 + x**4), x) + Integral(b*acosh(c*x)/( c**4*x**8 - 2*c**2*x**6 + x**4), x))/d**2
\[ \int \frac {a+b \text {arccosh}(c x)}{x^4 \left (d-c^2 d x^2\right )^2} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} - d\right )}^{2} x^{4}} \,d x } \]
1/12*(15*c^3*log(c*x + 1)/d^2 - 15*c^3*log(c*x - 1)/d^2 - 2*(15*c^4*x^4 - 10*c^2*x^2 - 2)/(c^2*d^2*x^5 - d^2*x^3))*a + 1/192*(8640*c^7*integrate(1/2 4*x^5*log(c*x - 1)/(c^4*d^2*x^6 - 2*c^2*d^2*x^4 + d^2*x^2), x) - 120*c^6*( 2*x/(c^4*d^2*x^2 - c^2*d^2) + log(c*x + 1)/(c^3*d^2) - log(c*x - 1)/(c^3*d ^2)) - 2880*c^6*integrate(1/24*x^4*log(c*x - 1)/(c^4*d^2*x^6 - 2*c^2*d^2*x ^4 + d^2*x^2), x) + 45*(c*(2/(c^4*d^2*x - c^3*d^2) - log(c*x + 1)/(c^3*d^2 ) + log(c*x - 1)/(c^3*d^2)) + 4*log(c*x - 1)/(c^4*d^2*x^2 - c^2*d^2))*c^5 + 80*c^4*(2*x/(c^2*d^2*x^2 - d^2) - log(c*x + 1)/(c*d^2) + log(c*x - 1)/(c *d^2)) + 2880*c^4*integrate(1/24*x^2*log(c*x - 1)/(c^4*d^2*x^6 - 2*c^2*d^2 *x^4 + d^2*x^2), x) + 16*c^2*(2*(3*c^2*x^2 - 2)/(c^2*d^2*x^3 - d^2*x) - 3* c*log(c*x + 1)/d^2 + 3*c*log(c*x - 1)/d^2) - 4*(15*(c^5*x^5 - c^3*x^3)*log (c*x + 1)^2 + 30*(c^5*x^5 - c^3*x^3)*log(c*x + 1)*log(c*x - 1) + 4*(30*c^4 *x^4 - 20*c^2*x^2 - 15*(c^5*x^5 - c^3*x^3)*log(c*x + 1) + 15*(c^5*x^5 - c^ 3*x^3)*log(c*x - 1) - 4)*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1)))/(c^2*d^2* x^5 - d^2*x^3) + 192*integrate(-1/12*(30*c^5*x^4 - 20*c^3*x^2 - 15*(c^6*x^ 5 - c^4*x^3)*log(c*x + 1) + 15*(c^6*x^5 - c^4*x^3)*log(c*x - 1) - 4*c)/(c^ 5*d^2*x^8 - 2*c^3*d^2*x^6 + c*d^2*x^4 + (c^4*d^2*x^7 - 2*c^2*d^2*x^5 + d^2 *x^3)*sqrt(c*x + 1)*sqrt(c*x - 1)), x))*b
\[ \int \frac {a+b \text {arccosh}(c x)}{x^4 \left (d-c^2 d x^2\right )^2} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} - d\right )}^{2} x^{4}} \,d x } \]
Timed out. \[ \int \frac {a+b \text {arccosh}(c x)}{x^4 \left (d-c^2 d x^2\right )^2} \, dx=\int \frac {a+b\,\mathrm {acosh}\left (c\,x\right )}{x^4\,{\left (d-c^2\,d\,x^2\right )}^2} \,d x \]