Integrand size = 25, antiderivative size = 277 \[ \int \frac {a+b \text {arccosh}(c x)}{x^3 \left (d-c^2 d x^2\right )^3} \, dx=\frac {b c^3 x}{12 d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}-\frac {7 b c^3 x}{6 d^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{2 d^3 x}-\frac {a+b \text {arccosh}(c x)}{2 d^3 x^2}+\frac {c^2 \left (3-2 c^2 x^2\right )^2 (a+b \text {arccosh}(c x))}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac {6 c^2 (a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{2 \text {arccosh}(c x)}\right )}{d^3}-\frac {b c^2 \sqrt {-1+c^2 x^2} \text {arctanh}\left (\frac {c x}{\sqrt {-1+c^2 x^2}}\right )}{d^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 b c^2 \operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}(c x)}\right )}{2 d^3}-\frac {3 b c^2 \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(c x)}\right )}{2 d^3} \] Output:
1/12*b*c^3*x/d^3/(c*x-1)^(3/2)/(c*x+1)^(3/2)-7/6*b*c^3*x/d^3/(c*x-1)^(1/2) /(c*x+1)^(1/2)+1/2*b*c*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d^3/x-1/2*(a+b*arccosh( c*x))/d^3/x^2+1/4*c^2*(-2*c^2*x^2+3)^2*(a+b*arccosh(c*x))/d^3/(-c^2*x^2+1) ^2+6*c^2*(a+b*arccosh(c*x))*arctanh((c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2)/d ^3-b*c^2*(c^2*x^2-1)^(1/2)*arctanh(c*x/(c^2*x^2-1)^(1/2))/d^3/(c*x-1)^(1/2 )/(c*x+1)^(1/2)+3/2*b*c^2*polylog(2,-(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2)/ d^3-3/2*b*c^2*polylog(2,(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2)/d^3
Time = 1.01 (sec) , antiderivative size = 402, normalized size of antiderivative = 1.45 \[ \int \frac {a+b \text {arccosh}(c x)}{x^3 \left (d-c^2 d x^2\right )^3} \, dx=\frac {-\frac {2 a}{x^2}+\frac {a c^2}{\left (-1+c^2 x^2\right )^2}-\frac {4 a c^2}{-1+c^2 x^2}-\frac {b c^2 \left ((-2+c x) \sqrt {-1+c x} \sqrt {1+c x}-3 \text {arccosh}(c x)\right )}{12 (-1+c x)^2}-\frac {b c^2 \left (\sqrt {-1+c x} \sqrt {1+c x} (2+c x)-3 \text {arccosh}(c x)\right )}{12 (1+c x)^2}+\frac {2 b \left (c x \sqrt {-1+c x} \sqrt {1+c x}-\text {arccosh}(c x)\right )}{x^2}+\frac {9}{4} b c^2 \left (-\frac {1}{\sqrt {\frac {-1+c x}{1+c x}}}+\frac {\text {arccosh}(c x)}{1-c x}\right )-\frac {9}{4} b c^2 \left (\sqrt {\frac {-1+c x}{1+c x}}-\frac {\text {arccosh}(c x)}{1+c x}\right )+12 a c^2 \log (x)-6 a c^2 \log \left (1-c^2 x^2\right )+6 b c^2 \left (\text {arccosh}(c x) \left (\text {arccosh}(c x)+2 \log \left (1+e^{-2 \text {arccosh}(c x)}\right )\right )-\operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c x)}\right )\right )+3 b c^2 \left (\text {arccosh}(c x) \left (\text {arccosh}(c x)-4 \log \left (1+e^{\text {arccosh}(c x)}\right )\right )-4 \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )\right )+3 b c^2 \left (\text {arccosh}(c x) \left (\text {arccosh}(c x)-4 \log \left (1-e^{\text {arccosh}(c x)}\right )\right )-4 \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )\right )}{4 d^3} \] Input:
Integrate[(a + b*ArcCosh[c*x])/(x^3*(d - c^2*d*x^2)^3),x]
Output:
((-2*a)/x^2 + (a*c^2)/(-1 + c^2*x^2)^2 - (4*a*c^2)/(-1 + c^2*x^2) - (b*c^2 *((-2 + c*x)*Sqrt[-1 + c*x]*Sqrt[1 + c*x] - 3*ArcCosh[c*x]))/(12*(-1 + c*x )^2) - (b*c^2*(Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(2 + c*x) - 3*ArcCosh[c*x]))/( 12*(1 + c*x)^2) + (2*b*(c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x] - ArcCosh[c*x]))/ x^2 + (9*b*c^2*(-(1/Sqrt[(-1 + c*x)/(1 + c*x)]) + ArcCosh[c*x]/(1 - c*x))) /4 - (9*b*c^2*(Sqrt[(-1 + c*x)/(1 + c*x)] - ArcCosh[c*x]/(1 + c*x)))/4 + 1 2*a*c^2*Log[x] - 6*a*c^2*Log[1 - c^2*x^2] + 6*b*c^2*(ArcCosh[c*x]*(ArcCosh [c*x] + 2*Log[1 + E^(-2*ArcCosh[c*x])]) - PolyLog[2, -E^(-2*ArcCosh[c*x])] ) + 3*b*c^2*(ArcCosh[c*x]*(ArcCosh[c*x] - 4*Log[1 + E^ArcCosh[c*x]]) - 4*P olyLog[2, -E^ArcCosh[c*x]]) + 3*b*c^2*(ArcCosh[c*x]*(ArcCosh[c*x] - 4*Log[ 1 - E^ArcCosh[c*x]]) - 4*PolyLog[2, E^ArcCosh[c*x]]))/(4*d^3)
Result contains complex when optimal does not.
Time = 1.41 (sec) , antiderivative size = 312, normalized size of antiderivative = 1.13, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.720, Rules used = {6347, 27, 114, 27, 42, 41, 6351, 42, 41, 6351, 41, 6331, 5984, 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.
| \(\displaystyle \int \frac {a+b \text {arccosh}(c x)}{x^3 \left (d-c^2 d x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 6347 |
| \(\displaystyle 3 c^2 \int \frac {a+b \text {arccosh}(c x)}{d^3 x \left (1-c^2 x^2\right )^3}dx+\frac {b c \int \frac {1}{x^2 (c x-1)^{5/2} (c x+1)^{5/2}}dx}{2 d^3}-\frac {a+b \text {arccosh}(c x)}{2 d^3 x^2 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3 c^2 \int \frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )^3}dx}{d^3}+\frac {b c \int \frac {1}{x^2 (c x-1)^{5/2} (c x+1)^{5/2}}dx}{2 d^3}-\frac {a+b \text {arccosh}(c x)}{2 d^3 x^2 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 114 |
| \(\displaystyle \frac {3 c^2 \int \frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )^3}dx}{d^3}+\frac {b c \left (\int \frac {4 c^2}{(c x-1)^{5/2} (c x+1)^{5/2}}dx+\frac {1}{x (c x-1)^{3/2} (c x+1)^{3/2}}\right )}{2 d^3}-\frac {a+b \text {arccosh}(c x)}{2 d^3 x^2 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3 c^2 \int \frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )^3}dx}{d^3}+\frac {b c \left (4 c^2 \int \frac {1}{(c x-1)^{5/2} (c x+1)^{5/2}}dx+\frac {1}{x (c x-1)^{3/2} (c x+1)^{3/2}}\right )}{2 d^3}-\frac {a+b \text {arccosh}(c x)}{2 d^3 x^2 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 42 |
| \(\displaystyle \frac {3 c^2 \int \frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )^3}dx}{d^3}+\frac {b c \left (4 c^2 \left (-\frac {2}{3} \int \frac {1}{(c x-1)^{3/2} (c x+1)^{3/2}}dx-\frac {x}{3 (c x-1)^{3/2} (c x+1)^{3/2}}\right )+\frac {1}{x (c x-1)^{3/2} (c x+1)^{3/2}}\right )}{2 d^3}-\frac {a+b \text {arccosh}(c x)}{2 d^3 x^2 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 41 |
| \(\displaystyle \frac {3 c^2 \int \frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )^3}dx}{d^3}-\frac {a+b \text {arccosh}(c x)}{2 d^3 x^2 \left (1-c^2 x^2\right )^2}+\frac {b c \left (4 c^2 \left (\frac {2 x}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {x}{3 (c x-1)^{3/2} (c x+1)^{3/2}}\right )+\frac {1}{x (c x-1)^{3/2} (c x+1)^{3/2}}\right )}{2 d^3}\) |
| \(\Big \downarrow \) 6351 |
| \(\displaystyle \frac {3 c^2 \left (\int \frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )^2}dx-\frac {1}{4} b c \int \frac {1}{(c x-1)^{5/2} (c x+1)^{5/2}}dx+\frac {a+b \text {arccosh}(c x)}{4 \left (1-c^2 x^2\right )^2}\right )}{d^3}-\frac {a+b \text {arccosh}(c x)}{2 d^3 x^2 \left (1-c^2 x^2\right )^2}+\frac {b c \left (4 c^2 \left (\frac {2 x}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {x}{3 (c x-1)^{3/2} (c x+1)^{3/2}}\right )+\frac {1}{x (c x-1)^{3/2} (c x+1)^{3/2}}\right )}{2 d^3}\) |
| \(\Big \downarrow \) 42 |
| \(\displaystyle \frac {3 c^2 \left (\int \frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )^2}dx-\frac {1}{4} b c \left (-\frac {2}{3} \int \frac {1}{(c x-1)^{3/2} (c x+1)^{3/2}}dx-\frac {x}{3 (c x-1)^{3/2} (c x+1)^{3/2}}\right )+\frac {a+b \text {arccosh}(c x)}{4 \left (1-c^2 x^2\right )^2}\right )}{d^3}-\frac {a+b \text {arccosh}(c x)}{2 d^3 x^2 \left (1-c^2 x^2\right )^2}+\frac {b c \left (4 c^2 \left (\frac {2 x}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {x}{3 (c x-1)^{3/2} (c x+1)^{3/2}}\right )+\frac {1}{x (c x-1)^{3/2} (c x+1)^{3/2}}\right )}{2 d^3}\) |
| \(\Big \downarrow \) 41 |
| \(\displaystyle \frac {3 c^2 \left (\int \frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )^2}dx+\frac {a+b \text {arccosh}(c x)}{4 \left (1-c^2 x^2\right )^2}-\frac {1}{4} b c \left (\frac {2 x}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {x}{3 (c x-1)^{3/2} (c x+1)^{3/2}}\right )\right )}{d^3}-\frac {a+b \text {arccosh}(c x)}{2 d^3 x^2 \left (1-c^2 x^2\right )^2}+\frac {b c \left (4 c^2 \left (\frac {2 x}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {x}{3 (c x-1)^{3/2} (c x+1)^{3/2}}\right )+\frac {1}{x (c x-1)^{3/2} (c x+1)^{3/2}}\right )}{2 d^3}\) |
| \(\Big \downarrow \) 6351 |
| \(\displaystyle \frac {3 c^2 \left (\int \frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )}dx+\frac {1}{2} b c \int \frac {1}{(c x-1)^{3/2} (c x+1)^{3/2}}dx+\frac {a+b \text {arccosh}(c x)}{2 \left (1-c^2 x^2\right )}+\frac {a+b \text {arccosh}(c x)}{4 \left (1-c^2 x^2\right )^2}-\frac {1}{4} b c \left (\frac {2 x}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {x}{3 (c x-1)^{3/2} (c x+1)^{3/2}}\right )\right )}{d^3}-\frac {a+b \text {arccosh}(c x)}{2 d^3 x^2 \left (1-c^2 x^2\right )^2}+\frac {b c \left (4 c^2 \left (\frac {2 x}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {x}{3 (c x-1)^{3/2} (c x+1)^{3/2}}\right )+\frac {1}{x (c x-1)^{3/2} (c x+1)^{3/2}}\right )}{2 d^3}\) |
| \(\Big \downarrow \) 41 |
| \(\displaystyle \frac {3 c^2 \left (\int \frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )}dx+\frac {a+b \text {arccosh}(c x)}{2 \left (1-c^2 x^2\right )}+\frac {a+b \text {arccosh}(c x)}{4 \left (1-c^2 x^2\right )^2}-\frac {b c x}{2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {1}{4} b c \left (\frac {2 x}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {x}{3 (c x-1)^{3/2} (c x+1)^{3/2}}\right )\right )}{d^3}-\frac {a+b \text {arccosh}(c x)}{2 d^3 x^2 \left (1-c^2 x^2\right )^2}+\frac {b c \left (4 c^2 \left (\frac {2 x}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {x}{3 (c x-1)^{3/2} (c x+1)^{3/2}}\right )+\frac {1}{x (c x-1)^{3/2} (c x+1)^{3/2}}\right )}{2 d^3}\) |
| \(\Big \downarrow \) 6331 |
| \(\displaystyle \frac {3 c^2 \left (-\int \frac {a+b \text {arccosh}(c x)}{c x \sqrt {\frac {c x-1}{c x+1}} (c x+1)}d\text {arccosh}(c x)+\frac {a+b \text {arccosh}(c x)}{2 \left (1-c^2 x^2\right )}+\frac {a+b \text {arccosh}(c x)}{4 \left (1-c^2 x^2\right )^2}-\frac {b c x}{2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {1}{4} b c \left (\frac {2 x}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {x}{3 (c x-1)^{3/2} (c x+1)^{3/2}}\right )\right )}{d^3}-\frac {a+b \text {arccosh}(c x)}{2 d^3 x^2 \left (1-c^2 x^2\right )^2}+\frac {b c \left (4 c^2 \left (\frac {2 x}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {x}{3 (c x-1)^{3/2} (c x+1)^{3/2}}\right )+\frac {1}{x (c x-1)^{3/2} (c x+1)^{3/2}}\right )}{2 d^3}\) |
| \(\Big \downarrow \) 5984 |
| \(\displaystyle \frac {3 c^2 \left (-2 \int (a+b \text {arccosh}(c x)) \text {csch}(2 \text {arccosh}(c x))d\text {arccosh}(c x)+\frac {a+b \text {arccosh}(c x)}{2 \left (1-c^2 x^2\right )}+\frac {a+b \text {arccosh}(c x)}{4 \left (1-c^2 x^2\right )^2}-\frac {b c x}{2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {1}{4} b c \left (\frac {2 x}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {x}{3 (c x-1)^{3/2} (c x+1)^{3/2}}\right )\right )}{d^3}-\frac {a+b \text {arccosh}(c x)}{2 d^3 x^2 \left (1-c^2 x^2\right )^2}+\frac {b c \left (4 c^2 \left (\frac {2 x}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {x}{3 (c x-1)^{3/2} (c x+1)^{3/2}}\right )+\frac {1}{x (c x-1)^{3/2} (c x+1)^{3/2}}\right )}{2 d^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 c^2 \left (-2 \int i (a+b \text {arccosh}(c x)) \csc (2 i \text {arccosh}(c x))d\text {arccosh}(c x)+\frac {a+b \text {arccosh}(c x)}{2 \left (1-c^2 x^2\right )}+\frac {a+b \text {arccosh}(c x)}{4 \left (1-c^2 x^2\right )^2}-\frac {b c x}{2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {1}{4} b c \left (\frac {2 x}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {x}{3 (c x-1)^{3/2} (c x+1)^{3/2}}\right )\right )}{d^3}-\frac {a+b \text {arccosh}(c x)}{2 d^3 x^2 \left (1-c^2 x^2\right )^2}+\frac {b c \left (4 c^2 \left (\frac {2 x}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {x}{3 (c x-1)^{3/2} (c x+1)^{3/2}}\right )+\frac {1}{x (c x-1)^{3/2} (c x+1)^{3/2}}\right )}{2 d^3}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {3 c^2 \left (-2 i \int (a+b \text {arccosh}(c x)) \csc (2 i \text {arccosh}(c x))d\text {arccosh}(c x)+\frac {a+b \text {arccosh}(c x)}{2 \left (1-c^2 x^2\right )}+\frac {a+b \text {arccosh}(c x)}{4 \left (1-c^2 x^2\right )^2}-\frac {b c x}{2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {1}{4} b c \left (\frac {2 x}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {x}{3 (c x-1)^{3/2} (c x+1)^{3/2}}\right )\right )}{d^3}-\frac {a+b \text {arccosh}(c x)}{2 d^3 x^2 \left (1-c^2 x^2\right )^2}+\frac {b c \left (4 c^2 \left (\frac {2 x}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {x}{3 (c x-1)^{3/2} (c x+1)^{3/2}}\right )+\frac {1}{x (c x-1)^{3/2} (c x+1)^{3/2}}\right )}{2 d^3}\) |
| \(\Big \downarrow \) 4670 |
| \(\displaystyle \frac {3 c^2 \left (-2 i \left (\frac {1}{2} i b \int \log \left (1-e^{2 \text {arccosh}(c x)}\right )d\text {arccosh}(c x)-\frac {1}{2} i b \int \log \left (1+e^{2 \text {arccosh}(c x)}\right )d\text {arccosh}(c x)+i \text {arctanh}\left (e^{2 \text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )+\frac {a+b \text {arccosh}(c x)}{2 \left (1-c^2 x^2\right )}+\frac {a+b \text {arccosh}(c x)}{4 \left (1-c^2 x^2\right )^2}-\frac {b c x}{2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {1}{4} b c \left (\frac {2 x}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {x}{3 (c x-1)^{3/2} (c x+1)^{3/2}}\right )\right )}{d^3}-\frac {a+b \text {arccosh}(c x)}{2 d^3 x^2 \left (1-c^2 x^2\right )^2}+\frac {b c \left (4 c^2 \left (\frac {2 x}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {x}{3 (c x-1)^{3/2} (c x+1)^{3/2}}\right )+\frac {1}{x (c x-1)^{3/2} (c x+1)^{3/2}}\right )}{2 d^3}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle \frac {3 c^2 \left (-2 i \left (\frac {1}{4} i b \int e^{-2 \text {arccosh}(c x)} \log \left (1-e^{2 \text {arccosh}(c x)}\right )de^{2 \text {arccosh}(c x)}-\frac {1}{4} i b \int e^{-2 \text {arccosh}(c x)} \log \left (1+e^{2 \text {arccosh}(c x)}\right )de^{2 \text {arccosh}(c x)}+i \text {arctanh}\left (e^{2 \text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )+\frac {a+b \text {arccosh}(c x)}{2 \left (1-c^2 x^2\right )}+\frac {a+b \text {arccosh}(c x)}{4 \left (1-c^2 x^2\right )^2}-\frac {b c x}{2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {1}{4} b c \left (\frac {2 x}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {x}{3 (c x-1)^{3/2} (c x+1)^{3/2}}\right )\right )}{d^3}-\frac {a+b \text {arccosh}(c x)}{2 d^3 x^2 \left (1-c^2 x^2\right )^2}+\frac {b c \left (4 c^2 \left (\frac {2 x}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {x}{3 (c x-1)^{3/2} (c x+1)^{3/2}}\right )+\frac {1}{x (c x-1)^{3/2} (c x+1)^{3/2}}\right )}{2 d^3}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {3 c^2 \left (-2 i \left (i \text {arctanh}\left (e^{2 \text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))+\frac {1}{4} i b \operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}(c x)}\right )-\frac {1}{4} i b \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(c x)}\right )\right )+\frac {a+b \text {arccosh}(c x)}{2 \left (1-c^2 x^2\right )}+\frac {a+b \text {arccosh}(c x)}{4 \left (1-c^2 x^2\right )^2}-\frac {b c x}{2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {1}{4} b c \left (\frac {2 x}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {x}{3 (c x-1)^{3/2} (c x+1)^{3/2}}\right )\right )}{d^3}-\frac {a+b \text {arccosh}(c x)}{2 d^3 x^2 \left (1-c^2 x^2\right )^2}+\frac {b c \left (4 c^2 \left (\frac {2 x}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {x}{3 (c x-1)^{3/2} (c x+1)^{3/2}}\right )+\frac {1}{x (c x-1)^{3/2} (c x+1)^{3/2}}\right )}{2 d^3}\) |
Input:
Int[(a + b*ArcCosh[c*x])/(x^3*(d - c^2*d*x^2)^3),x]
Output:
(b*c*(1/(x*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2)) + 4*c^2*(-1/3*x/((-1 + c*x)^( 3/2)*(1 + c*x)^(3/2)) + (2*x)/(3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]))))/(2*d^3) - (a + b*ArcCosh[c*x])/(2*d^3*x^2*(1 - c^2*x^2)^2) + (3*c^2*(-1/2*(b*c*x)/ (Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c*(-1/3*x/((-1 + c*x)^(3/2)*(1 + c*x)^ (3/2)) + (2*x)/(3*Sqrt[-1 + c*x]*Sqrt[1 + c*x])))/4 + (a + b*ArcCosh[c*x]) /(4*(1 - c^2*x^2)^2) + (a + b*ArcCosh[c*x])/(2*(1 - c^2*x^2)) - (2*I)*(I*( a + b*ArcCosh[c*x])*ArcTanh[E^(2*ArcCosh[c*x])] + (I/4)*b*PolyLog[2, -E^(2 *ArcCosh[c*x])] - (I/4)*b*PolyLog[2, E^(2*ArcCosh[c*x])])))/d^3
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[1/(((a_) + (b_.)*(x_))^(3/2)*((c_) + (d_.)*(x_))^(3/2)), x_Symbol] :> S imp[x/(a*c*Sqrt[a + b*x]*Sqrt[c + d*x]), x] /; FreeQ[{a, b, c, d}, x] && Eq Q[b*c + a*d, 0]
Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[(- x)*(a + b*x)^(m + 1)*((c + d*x)^(m + 1)/(2*a*c*(m + 1))), x] + Simp[(2*m + 3)/(2*a*c*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(m + 1), x], x] /; Fre eQ[{a, b, c, d}, x] && EqQ[b*c + a*d, 0] && ILtQ[m + 3/2, 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[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[Csch[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sech[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Simp[2^n Int[(c + d*x)^m*Csch[2*a + 2*b*x ]^n, x], x] /; FreeQ[{a, b, c, d}, x] && RationalQ[m] && IntegerQ[n]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/((x_)*((d_) + (e_.)*(x_)^2)), x_Symbol] :> Simp[-d^(-1) Subst[Int[(a + b*x)^n/(Cosh[x]*Sinh[x]), x], x , ArcCosh[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IG tQ[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]
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/(2*d*f*(p + 1))), x] + (Simp[(m + 2*p + 3)/(2*d*(p + 1 )) Int[(f*x)^m*(d + e*x^2)^(p + 1)*(a + b*ArcCosh[c*x])^n, x], x] - Simp[ b*c*(n/(2*f*(p + 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, m}, x] && EqQ[c^2*d + e, 0] & & GtQ[n, 0] && LtQ[p, -1] && !GtQ[m, 1] && (IntegerQ[m] || IntegerQ[p] || EqQ[n, 1])
Time = 0.50 (sec) , antiderivative size = 391, normalized size of antiderivative = 1.41
| method | result | size |
| derivativedivides | \(c^{2} \left (-\frac {a \left (\frac {1}{2 c^{2} x^{2}}-3 \ln \left (c x \right )-\frac {1}{16 \left (c x -1\right )^{2}}+\frac {9}{16 \left (c x -1\right )}+\frac {3 \ln \left (c x -1\right )}{2}-\frac {1}{16 \left (c x +1\right )^{2}}-\frac {9}{16 \left (c x +1\right )}+\frac {3 \ln \left (c x +1\right )}{2}\right )}{d^{3}}-\frac {b \left (\frac {8 c^{5} x^{5} \sqrt {c x -1}\, \sqrt {c x +1}-8 c^{6} x^{6}+18 \,\operatorname {arccosh}\left (c x \right ) c^{4} x^{4}-3 c^{3} x^{3} \sqrt {c x -1}\, \sqrt {c x +1}+16 c^{4} x^{4}-27 c^{2} x^{2} \operatorname {arccosh}\left (c x \right )-6 \sqrt {c x -1}\, \sqrt {c x +1}\, c x -8 c^{2} x^{2}+6 \,\operatorname {arccosh}\left (c x \right )}{12 \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right ) c^{2} x^{2}}+3 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+3 \operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )-3 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )-\frac {3 \operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )}{2}+3 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )+3 \operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d^{3}}\right )\) | \(391\) |
| default | \(c^{2} \left (-\frac {a \left (\frac {1}{2 c^{2} x^{2}}-3 \ln \left (c x \right )-\frac {1}{16 \left (c x -1\right )^{2}}+\frac {9}{16 \left (c x -1\right )}+\frac {3 \ln \left (c x -1\right )}{2}-\frac {1}{16 \left (c x +1\right )^{2}}-\frac {9}{16 \left (c x +1\right )}+\frac {3 \ln \left (c x +1\right )}{2}\right )}{d^{3}}-\frac {b \left (\frac {8 c^{5} x^{5} \sqrt {c x -1}\, \sqrt {c x +1}-8 c^{6} x^{6}+18 \,\operatorname {arccosh}\left (c x \right ) c^{4} x^{4}-3 c^{3} x^{3} \sqrt {c x -1}\, \sqrt {c x +1}+16 c^{4} x^{4}-27 c^{2} x^{2} \operatorname {arccosh}\left (c x \right )-6 \sqrt {c x -1}\, \sqrt {c x +1}\, c x -8 c^{2} x^{2}+6 \,\operatorname {arccosh}\left (c x \right )}{12 \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right ) c^{2} x^{2}}+3 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+3 \operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )-3 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )-\frac {3 \operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )}{2}+3 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )+3 \operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d^{3}}\right )\) | \(391\) |
| parts | \(-\frac {a \left (\frac {1}{2 x^{2}}-3 c^{2} \ln \left (x \right )-\frac {c^{2}}{16 \left (c x +1\right )^{2}}-\frac {9 c^{2}}{16 \left (c x +1\right )}+\frac {3 c^{2} \ln \left (c x +1\right )}{2}-\frac {c^{2}}{16 \left (c x -1\right )^{2}}+\frac {9 c^{2}}{16 \left (c x -1\right )}+\frac {3 c^{2} \ln \left (c x -1\right )}{2}\right )}{d^{3}}-\frac {b \,c^{2} \left (\frac {8 c^{5} x^{5} \sqrt {c x -1}\, \sqrt {c x +1}-8 c^{6} x^{6}+18 \,\operatorname {arccosh}\left (c x \right ) c^{4} x^{4}-3 c^{3} x^{3} \sqrt {c x -1}\, \sqrt {c x +1}+16 c^{4} x^{4}-27 c^{2} x^{2} \operatorname {arccosh}\left (c x \right )-6 \sqrt {c x -1}\, \sqrt {c x +1}\, c x -8 c^{2} x^{2}+6 \,\operatorname {arccosh}\left (c x \right )}{12 \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right ) c^{2} x^{2}}+3 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+3 \operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )-3 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )-\frac {3 \operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )}{2}+3 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )+3 \operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d^{3}}\) | \(406\) |
Input:
int((a+b*arccosh(c*x))/x^3/(-c^2*d*x^2+d)^3,x,method=_RETURNVERBOSE)
Output:
c^2*(-a/d^3*(1/2/c^2/x^2-3*ln(c*x)-1/16/(c*x-1)^2+9/16/(c*x-1)+3/2*ln(c*x- 1)-1/16/(c*x+1)^2-9/16/(c*x+1)+3/2*ln(c*x+1))-b/d^3*(1/12/(c^4*x^4-2*c^2*x ^2+1)/c^2/x^2*(8*c^5*x^5*(c*x-1)^(1/2)*(c*x+1)^(1/2)-8*c^6*x^6+18*arccosh( c*x)*c^4*x^4-3*c^3*x^3*(c*x-1)^(1/2)*(c*x+1)^(1/2)+16*c^4*x^4-27*c^2*x^2*a rccosh(c*x)-6*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x-8*c^2*x^2+6*arccosh(c*x))+3* arccosh(c*x)*ln(1+c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))+3*polylog(2,-c*x-(c*x-1 )^(1/2)*(c*x+1)^(1/2))-3*arccosh(c*x)*ln(1+(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2 ))^2)-3/2*polylog(2,-(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2)+3*arccosh(c*x)*l n(1-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))+3*polylog(2,c*x+(c*x-1)^(1/2)*(c*x+1) ^(1/2))))
\[ \int \frac {a+b \text {arccosh}(c x)}{x^3 \left (d-c^2 d x^2\right )^3} \, dx=\int { -\frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} - d\right )}^{3} x^{3}} \,d x } \] Input:
integrate((a+b*arccosh(c*x))/x^3/(-c^2*d*x^2+d)^3,x, algorithm="fricas")
Output:
integral(-(b*arccosh(c*x) + a)/(c^6*d^3*x^9 - 3*c^4*d^3*x^7 + 3*c^2*d^3*x^ 5 - d^3*x^3), x)
Timed out. \[ \int \frac {a+b \text {arccosh}(c x)}{x^3 \left (d-c^2 d x^2\right )^3} \, dx=\text {Timed out} \] Input:
integrate((a+b*acosh(c*x))/x**3/(-c**2*d*x**2+d)**3,x)
Output:
Timed out
\[ \int \frac {a+b \text {arccosh}(c x)}{x^3 \left (d-c^2 d x^2\right )^3} \, dx=\int { -\frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} - d\right )}^{3} x^{3}} \,d x } \] Input:
integrate((a+b*arccosh(c*x))/x^3/(-c^2*d*x^2+d)^3,x, algorithm="maxima")
Output:
-1/4*a*((6*c^4*x^4 - 9*c^2*x^2 + 2)/(c^4*d^3*x^6 - 2*c^2*d^3*x^4 + d^3*x^2 ) + 6*c^2*log(c*x + 1)/d^3 + 6*c^2*log(c*x - 1)/d^3 - 12*c^2*log(x)/d^3) - b*integrate(log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))/(c^6*d^3*x^9 - 3*c^4*d ^3*x^7 + 3*c^2*d^3*x^5 - d^3*x^3), x)
\[ \int \frac {a+b \text {arccosh}(c x)}{x^3 \left (d-c^2 d x^2\right )^3} \, dx=\int { -\frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} - d\right )}^{3} x^{3}} \,d x } \] Input:
integrate((a+b*arccosh(c*x))/x^3/(-c^2*d*x^2+d)^3,x, algorithm="giac")
Output:
integrate(-(b*arccosh(c*x) + a)/((c^2*d*x^2 - d)^3*x^3), x)
Timed out. \[ \int \frac {a+b \text {arccosh}(c x)}{x^3 \left (d-c^2 d x^2\right )^3} \, dx=\int \frac {a+b\,\mathrm {acosh}\left (c\,x\right )}{x^3\,{\left (d-c^2\,d\,x^2\right )}^3} \,d x \] Input:
int((a + b*acosh(c*x))/(x^3*(d - c^2*d*x^2)^3),x)
Output:
int((a + b*acosh(c*x))/(x^3*(d - c^2*d*x^2)^3), x)
\[ \int \frac {a+b \text {arccosh}(c x)}{x^3 \left (d-c^2 d x^2\right )^3} \, dx=\frac {-4 \left (\int \frac {\mathit {acosh} \left (c x \right )}{c^{6} x^{9}-3 c^{4} x^{7}+3 c^{2} x^{5}-x^{3}}d x \right ) b \,c^{4} x^{6}+8 \left (\int \frac {\mathit {acosh} \left (c x \right )}{c^{6} x^{9}-3 c^{4} x^{7}+3 c^{2} x^{5}-x^{3}}d x \right ) b \,c^{2} x^{4}-4 \left (\int \frac {\mathit {acosh} \left (c x \right )}{c^{6} x^{9}-3 c^{4} x^{7}+3 c^{2} x^{5}-x^{3}}d x \right ) b \,x^{2}-6 \,\mathrm {log}\left (c^{2} x -c \right ) a \,c^{6} x^{6}+12 \,\mathrm {log}\left (c^{2} x -c \right ) a \,c^{4} x^{4}-6 \,\mathrm {log}\left (c^{2} x -c \right ) a \,c^{2} x^{2}-6 \,\mathrm {log}\left (c^{2} x +c \right ) a \,c^{6} x^{6}+12 \,\mathrm {log}\left (c^{2} x +c \right ) a \,c^{4} x^{4}-6 \,\mathrm {log}\left (c^{2} x +c \right ) a \,c^{2} x^{2}+12 \,\mathrm {log}\left (x \right ) a \,c^{6} x^{6}-24 \,\mathrm {log}\left (x \right ) a \,c^{4} x^{4}+12 \,\mathrm {log}\left (x \right ) a \,c^{2} x^{2}-3 a \,c^{6} x^{6}+6 a \,c^{2} x^{2}-2 a}{4 d^{3} x^{2} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )} \] Input:
int((a+b*acosh(c*x))/x^3/(-c^2*d*x^2+d)^3,x)
Output:
( - 4*int(acosh(c*x)/(c**6*x**9 - 3*c**4*x**7 + 3*c**2*x**5 - x**3),x)*b*c **4*x**6 + 8*int(acosh(c*x)/(c**6*x**9 - 3*c**4*x**7 + 3*c**2*x**5 - x**3) ,x)*b*c**2*x**4 - 4*int(acosh(c*x)/(c**6*x**9 - 3*c**4*x**7 + 3*c**2*x**5 - x**3),x)*b*x**2 - 6*log(c**2*x - c)*a*c**6*x**6 + 12*log(c**2*x - c)*a*c **4*x**4 - 6*log(c**2*x - c)*a*c**2*x**2 - 6*log(c**2*x + c)*a*c**6*x**6 + 12*log(c**2*x + c)*a*c**4*x**4 - 6*log(c**2*x + c)*a*c**2*x**2 + 12*log(x )*a*c**6*x**6 - 24*log(x)*a*c**4*x**4 + 12*log(x)*a*c**2*x**2 - 3*a*c**6*x **6 + 6*a*c**2*x**2 - 2*a)/(4*d**3*x**2*(c**4*x**4 - 2*c**2*x**2 + 1))