Integrand size = 27, antiderivative size = 479 \[ \int \frac {a+b \text {arccosh}(c x)}{x^3 \left (d-c^2 d x^2\right )^{5/2}} \, dx=\frac {3 b c \sqrt {-1+c x} \sqrt {1+c x}}{4 d^2 x \sqrt {d-c^2 d x^2}}-\frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{4 d^2 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {5 b c^3 x \sqrt {-1+c x} \sqrt {1+c x}}{12 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {5 c^2 (a+b \text {arccosh}(c x))}{6 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {a+b \text {arccosh}(c x)}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}+\frac {5 c^2 (a+b \text {arccosh}(c x))}{2 d^2 \sqrt {d-c^2 d x^2}}+\frac {5 c^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \arctan \left (e^{\text {arccosh}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {13 b c^2 \sqrt {-1+c x} \sqrt {1+c x} \text {arctanh}(c x)}{6 d^2 \sqrt {d-c^2 d x^2}}-\frac {5 i b c^2 \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c x)}\right )}{2 d^2 \sqrt {d-c^2 d x^2}}+\frac {5 i b c^2 \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c x)}\right )}{2 d^2 \sqrt {d-c^2 d x^2}} \]
5/6*c^2*(a+b*arccosh(c*x))/d/(-c^2*d*x^2+d)^(3/2)+1/2*(-a-b*arccosh(c*x))/ d/x^2/(-c^2*d*x^2+d)^(3/2)+5/2*c^2*(a+b*arccosh(c*x))/d^2/(-c^2*d*x^2+d)^( 1/2)+3/4*b*c*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d^2/x/(-c^2*d*x^2+d)^(1/2)-1/4*b* c*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d^2/x/(-c^2*x^2+1)/(-c^2*d*x^2+d)^(1/2)+5/12 *b*c^3*x*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d^2/(-c^2*x^2+1)/(-c^2*d*x^2+d)^(1/2) +5*c^2*(a+b*arccosh(c*x))*arctan(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*(c*x-1)^ (1/2)*(c*x+1)^(1/2)/d^2/(-c^2*d*x^2+d)^(1/2)+13/6*b*c^2*arctanh(c*x)*(c*x- 1)^(1/2)*(c*x+1)^(1/2)/d^2/(-c^2*d*x^2+d)^(1/2)-5/2*I*b*c^2*polylog(2,-I*( c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d^2/(-c^2*d* x^2+d)^(1/2)+5/2*I*b*c^2*polylog(2,I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))*(c *x-1)^(1/2)*(c*x+1)^(1/2)/d^2/(-c^2*d*x^2+d)^(1/2)
Time = 7.21 (sec) , antiderivative size = 534, normalized size of antiderivative = 1.11 \[ \int \frac {a+b \text {arccosh}(c x)}{x^3 \left (d-c^2 d x^2\right )^{5/2}} \, dx=\sqrt {-d \left (-1+c^2 x^2\right )} \left (-\frac {a}{2 d^3 x^2}+\frac {a c^2}{3 d^3 \left (-1+c^2 x^2\right )^2}-\frac {2 a c^2}{d^3 \left (-1+c^2 x^2\right )}\right )+\frac {5 a c^2 \log (x)}{2 d^{5/2}}-\frac {5 a c^2 \log \left (d+\sqrt {d} \sqrt {-d \left (-1+c^2 x^2\right )}\right )}{2 d^{5/2}}+\frac {b c^2 \left (\frac {6 \sqrt {\frac {-1+c x}{1+c x}} (1+c x)}{c x}+\frac {6 (-1+c x) (1+c x) \text {arccosh}(c x)}{c^2 x^2}+26 \text {arccosh}(c x) \cosh ^2\left (\frac {1}{2} \text {arccosh}(c x)\right )-\coth \left (\frac {1}{2} \text {arccosh}(c x)\right )-\text {arccosh}(c x) \coth ^2\left (\frac {1}{2} \text {arccosh}(c x)\right )-30 i \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \text {arccosh}(c x) \log \left (1-i e^{-\text {arccosh}(c x)}\right )+30 i \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \text {arccosh}(c x) \log \left (1+i e^{-\text {arccosh}(c x)}\right )+26 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \log \left (\cosh \left (\frac {1}{2} \text {arccosh}(c x)\right )\right )-26 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \log \left (\sinh \left (\frac {1}{2} \text {arccosh}(c x)\right )\right )-30 i \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \operatorname {PolyLog}\left (2,-i e^{-\text {arccosh}(c x)}\right )+30 i \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \operatorname {PolyLog}\left (2,i e^{-\text {arccosh}(c x)}\right )-26 \text {arccosh}(c x) \sinh ^2\left (\frac {1}{2} \text {arccosh}(c x)\right )-\tanh \left (\frac {1}{2} \text {arccosh}(c x)\right )-\text {arccosh}(c x) \tanh ^2\left (\frac {1}{2} \text {arccosh}(c x)\right )\right )}{12 d^2 \sqrt {-d (-1+c x) (1+c x)}} \]
Sqrt[-(d*(-1 + c^2*x^2))]*(-1/2*a/(d^3*x^2) + (a*c^2)/(3*d^3*(-1 + c^2*x^2 )^2) - (2*a*c^2)/(d^3*(-1 + c^2*x^2))) + (5*a*c^2*Log[x])/(2*d^(5/2)) - (5 *a*c^2*Log[d + Sqrt[d]*Sqrt[-(d*(-1 + c^2*x^2))]])/(2*d^(5/2)) + (b*c^2*(( 6*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x))/(c*x) + (6*(-1 + c*x)*(1 + c*x)*Ar cCosh[c*x])/(c^2*x^2) + 26*ArcCosh[c*x]*Cosh[ArcCosh[c*x]/2]^2 - Coth[ArcC osh[c*x]/2] - ArcCosh[c*x]*Coth[ArcCosh[c*x]/2]^2 - (30*I)*Sqrt[(-1 + c*x) /(1 + c*x)]*(1 + c*x)*ArcCosh[c*x]*Log[1 - I/E^ArcCosh[c*x]] + (30*I)*Sqrt [(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x]*Log[1 + I/E^ArcCosh[c*x]] + 26*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*Log[Cosh[ArcCosh[c*x]/2]] - 26*Sqr t[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*Log[Sinh[ArcCosh[c*x]/2]] - (30*I)*Sqrt[ (-1 + c*x)/(1 + c*x)]*(1 + c*x)*PolyLog[2, (-I)/E^ArcCosh[c*x]] + (30*I)*S qrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*PolyLog[2, I/E^ArcCosh[c*x]] - 26*ArcC osh[c*x]*Sinh[ArcCosh[c*x]/2]^2 - Tanh[ArcCosh[c*x]/2] - ArcCosh[c*x]*Tanh [ArcCosh[c*x]/2]^2))/(12*d^2*Sqrt[-(d*(-1 + c*x)*(1 + c*x))])
Time = 1.67 (sec) , antiderivative size = 384, normalized size of antiderivative = 0.80, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {6347, 82, 253, 264, 219, 6351, 39, 215, 219, 6351, 25, 39, 219, 6361, 3042, 4668, 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.
| \(\displaystyle \int \frac {a+b \text {arccosh}(c x)}{x^3 \left (d-c^2 d x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 6347 |
| \(\displaystyle \frac {5}{2} c^2 \int \frac {a+b \text {arccosh}(c x)}{x \left (d-c^2 d x^2\right )^{5/2}}dx-\frac {b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {1}{x^2 (1-c x)^2 (c x+1)^2}dx}{2 d^2 \sqrt {d-c^2 d x^2}}-\frac {a+b \text {arccosh}(c x)}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 82 |
| \(\displaystyle \frac {5}{2} c^2 \int \frac {a+b \text {arccosh}(c x)}{x \left (d-c^2 d x^2\right )^{5/2}}dx-\frac {b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {1}{x^2 \left (1-c^2 x^2\right )^2}dx}{2 d^2 \sqrt {d-c^2 d x^2}}-\frac {a+b \text {arccosh}(c x)}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {5}{2} c^2 \int \frac {a+b \text {arccosh}(c x)}{x \left (d-c^2 d x^2\right )^{5/2}}dx-\frac {b c \sqrt {c x-1} \sqrt {c x+1} \left (\frac {3}{2} \int \frac {1}{x^2 \left (1-c^2 x^2\right )}dx+\frac {1}{2 x \left (1-c^2 x^2\right )}\right )}{2 d^2 \sqrt {d-c^2 d x^2}}-\frac {a+b \text {arccosh}(c x)}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 264 |
| \(\displaystyle \frac {5}{2} c^2 \int \frac {a+b \text {arccosh}(c x)}{x \left (d-c^2 d x^2\right )^{5/2}}dx-\frac {b c \sqrt {c x-1} \sqrt {c x+1} \left (\frac {3}{2} \left (c^2 \int \frac {1}{1-c^2 x^2}dx-\frac {1}{x}\right )+\frac {1}{2 x \left (1-c^2 x^2\right )}\right )}{2 d^2 \sqrt {d-c^2 d x^2}}-\frac {a+b \text {arccosh}(c x)}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {5}{2} c^2 \int \frac {a+b \text {arccosh}(c x)}{x \left (d-c^2 d x^2\right )^{5/2}}dx-\frac {a+b \text {arccosh}(c x)}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}-\frac {b c \sqrt {c x-1} \sqrt {c x+1} \left (\frac {3}{2} \left (c \text {arctanh}(c x)-\frac {1}{x}\right )+\frac {1}{2 x \left (1-c^2 x^2\right )}\right )}{2 d^2 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 6351 |
| \(\displaystyle \frac {5}{2} c^2 \left (\frac {\int \frac {a+b \text {arccosh}(c x)}{x \left (d-c^2 d x^2\right )^{3/2}}dx}{d}+\frac {b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {1}{(1-c x)^2 (c x+1)^2}dx}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {a+b \text {arccosh}(c x)}{3 d \left (d-c^2 d x^2\right )^{3/2}}\right )-\frac {a+b \text {arccosh}(c x)}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}-\frac {b c \sqrt {c x-1} \sqrt {c x+1} \left (\frac {3}{2} \left (c \text {arctanh}(c x)-\frac {1}{x}\right )+\frac {1}{2 x \left (1-c^2 x^2\right )}\right )}{2 d^2 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 39 |
| \(\displaystyle \frac {5}{2} c^2 \left (\frac {\int \frac {a+b \text {arccosh}(c x)}{x \left (d-c^2 d x^2\right )^{3/2}}dx}{d}+\frac {b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {1}{\left (1-c^2 x^2\right )^2}dx}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {a+b \text {arccosh}(c x)}{3 d \left (d-c^2 d x^2\right )^{3/2}}\right )-\frac {a+b \text {arccosh}(c x)}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}-\frac {b c \sqrt {c x-1} \sqrt {c x+1} \left (\frac {3}{2} \left (c \text {arctanh}(c x)-\frac {1}{x}\right )+\frac {1}{2 x \left (1-c^2 x^2\right )}\right )}{2 d^2 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 215 |
| \(\displaystyle \frac {5}{2} c^2 \left (\frac {\int \frac {a+b \text {arccosh}(c x)}{x \left (d-c^2 d x^2\right )^{3/2}}dx}{d}+\frac {b c \sqrt {c x-1} \sqrt {c x+1} \left (\frac {1}{2} \int \frac {1}{1-c^2 x^2}dx+\frac {x}{2 \left (1-c^2 x^2\right )}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {a+b \text {arccosh}(c x)}{3 d \left (d-c^2 d x^2\right )^{3/2}}\right )-\frac {a+b \text {arccosh}(c x)}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}-\frac {b c \sqrt {c x-1} \sqrt {c x+1} \left (\frac {3}{2} \left (c \text {arctanh}(c x)-\frac {1}{x}\right )+\frac {1}{2 x \left (1-c^2 x^2\right )}\right )}{2 d^2 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {5}{2} c^2 \left (\frac {\int \frac {a+b \text {arccosh}(c x)}{x \left (d-c^2 d x^2\right )^{3/2}}dx}{d}+\frac {a+b \text {arccosh}(c x)}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {b c \sqrt {c x-1} \sqrt {c x+1} \left (\frac {\text {arctanh}(c x)}{2 c}+\frac {x}{2 \left (1-c^2 x^2\right )}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}\right )-\frac {a+b \text {arccosh}(c x)}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}-\frac {b c \sqrt {c x-1} \sqrt {c x+1} \left (\frac {3}{2} \left (c \text {arctanh}(c x)-\frac {1}{x}\right )+\frac {1}{2 x \left (1-c^2 x^2\right )}\right )}{2 d^2 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 6351 |
| \(\displaystyle \frac {5}{2} c^2 \left (\frac {\frac {\int \frac {a+b \text {arccosh}(c x)}{x \sqrt {d-c^2 d x^2}}dx}{d}-\frac {b c \sqrt {c x-1} \sqrt {c x+1} \int -\frac {1}{(1-c x) (c x+1)}dx}{d \sqrt {d-c^2 d x^2}}+\frac {a+b \text {arccosh}(c x)}{d \sqrt {d-c^2 d x^2}}}{d}+\frac {a+b \text {arccosh}(c x)}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {b c \sqrt {c x-1} \sqrt {c x+1} \left (\frac {\text {arctanh}(c x)}{2 c}+\frac {x}{2 \left (1-c^2 x^2\right )}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}\right )-\frac {a+b \text {arccosh}(c x)}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}-\frac {b c \sqrt {c x-1} \sqrt {c x+1} \left (\frac {3}{2} \left (c \text {arctanh}(c x)-\frac {1}{x}\right )+\frac {1}{2 x \left (1-c^2 x^2\right )}\right )}{2 d^2 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {5}{2} c^2 \left (\frac {\frac {\int \frac {a+b \text {arccosh}(c x)}{x \sqrt {d-c^2 d x^2}}dx}{d}+\frac {b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {1}{(1-c x) (c x+1)}dx}{d \sqrt {d-c^2 d x^2}}+\frac {a+b \text {arccosh}(c x)}{d \sqrt {d-c^2 d x^2}}}{d}+\frac {a+b \text {arccosh}(c x)}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {b c \sqrt {c x-1} \sqrt {c x+1} \left (\frac {\text {arctanh}(c x)}{2 c}+\frac {x}{2 \left (1-c^2 x^2\right )}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}\right )-\frac {a+b \text {arccosh}(c x)}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}-\frac {b c \sqrt {c x-1} \sqrt {c x+1} \left (\frac {3}{2} \left (c \text {arctanh}(c x)-\frac {1}{x}\right )+\frac {1}{2 x \left (1-c^2 x^2\right )}\right )}{2 d^2 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 39 |
| \(\displaystyle \frac {5}{2} c^2 \left (\frac {\frac {\int \frac {a+b \text {arccosh}(c x)}{x \sqrt {d-c^2 d x^2}}dx}{d}+\frac {b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {1}{1-c^2 x^2}dx}{d \sqrt {d-c^2 d x^2}}+\frac {a+b \text {arccosh}(c x)}{d \sqrt {d-c^2 d x^2}}}{d}+\frac {a+b \text {arccosh}(c x)}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {b c \sqrt {c x-1} \sqrt {c x+1} \left (\frac {\text {arctanh}(c x)}{2 c}+\frac {x}{2 \left (1-c^2 x^2\right )}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}\right )-\frac {a+b \text {arccosh}(c x)}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}-\frac {b c \sqrt {c x-1} \sqrt {c x+1} \left (\frac {3}{2} \left (c \text {arctanh}(c x)-\frac {1}{x}\right )+\frac {1}{2 x \left (1-c^2 x^2\right )}\right )}{2 d^2 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {5}{2} c^2 \left (\frac {\frac {\int \frac {a+b \text {arccosh}(c x)}{x \sqrt {d-c^2 d x^2}}dx}{d}+\frac {a+b \text {arccosh}(c x)}{d \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {c x-1} \sqrt {c x+1} \text {arctanh}(c x)}{d \sqrt {d-c^2 d x^2}}}{d}+\frac {a+b \text {arccosh}(c x)}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {b c \sqrt {c x-1} \sqrt {c x+1} \left (\frac {\text {arctanh}(c x)}{2 c}+\frac {x}{2 \left (1-c^2 x^2\right )}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}\right )-\frac {a+b \text {arccosh}(c x)}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}-\frac {b c \sqrt {c x-1} \sqrt {c x+1} \left (\frac {3}{2} \left (c \text {arctanh}(c x)-\frac {1}{x}\right )+\frac {1}{2 x \left (1-c^2 x^2\right )}\right )}{2 d^2 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 6361 |
| \(\displaystyle \frac {5}{2} c^2 \left (\frac {\frac {\sqrt {c x-1} \sqrt {c x+1} \int \frac {a+b \text {arccosh}(c x)}{c x}d\text {arccosh}(c x)}{d \sqrt {d-c^2 d x^2}}+\frac {a+b \text {arccosh}(c x)}{d \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {c x-1} \sqrt {c x+1} \text {arctanh}(c x)}{d \sqrt {d-c^2 d x^2}}}{d}+\frac {a+b \text {arccosh}(c x)}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {b c \sqrt {c x-1} \sqrt {c x+1} \left (\frac {\text {arctanh}(c x)}{2 c}+\frac {x}{2 \left (1-c^2 x^2\right )}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}\right )-\frac {a+b \text {arccosh}(c x)}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}-\frac {b c \sqrt {c x-1} \sqrt {c x+1} \left (\frac {3}{2} \left (c \text {arctanh}(c x)-\frac {1}{x}\right )+\frac {1}{2 x \left (1-c^2 x^2\right )}\right )}{2 d^2 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {5}{2} c^2 \left (\frac {\frac {\sqrt {c x-1} \sqrt {c x+1} \int (a+b \text {arccosh}(c x)) \csc \left (i \text {arccosh}(c x)+\frac {\pi }{2}\right )d\text {arccosh}(c x)}{d \sqrt {d-c^2 d x^2}}+\frac {a+b \text {arccosh}(c x)}{d \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {c x-1} \sqrt {c x+1} \text {arctanh}(c x)}{d \sqrt {d-c^2 d x^2}}}{d}+\frac {a+b \text {arccosh}(c x)}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {b c \sqrt {c x-1} \sqrt {c x+1} \left (\frac {\text {arctanh}(c x)}{2 c}+\frac {x}{2 \left (1-c^2 x^2\right )}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}\right )-\frac {a+b \text {arccosh}(c x)}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}-\frac {b c \sqrt {c x-1} \sqrt {c x+1} \left (\frac {3}{2} \left (c \text {arctanh}(c x)-\frac {1}{x}\right )+\frac {1}{2 x \left (1-c^2 x^2\right )}\right )}{2 d^2 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 4668 |
| \(\displaystyle \frac {5}{2} c^2 \left (\frac {\frac {\sqrt {c x-1} \sqrt {c x+1} \left (-i b \int \log \left (1-i e^{\text {arccosh}(c x)}\right )d\text {arccosh}(c x)+i b \int \log \left (1+i e^{\text {arccosh}(c x)}\right )d\text {arccosh}(c x)+2 \arctan \left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )}{d \sqrt {d-c^2 d x^2}}+\frac {a+b \text {arccosh}(c x)}{d \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {c x-1} \sqrt {c x+1} \text {arctanh}(c x)}{d \sqrt {d-c^2 d x^2}}}{d}+\frac {a+b \text {arccosh}(c x)}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {b c \sqrt {c x-1} \sqrt {c x+1} \left (\frac {\text {arctanh}(c x)}{2 c}+\frac {x}{2 \left (1-c^2 x^2\right )}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}\right )-\frac {a+b \text {arccosh}(c x)}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}-\frac {b c \sqrt {c x-1} \sqrt {c x+1} \left (\frac {3}{2} \left (c \text {arctanh}(c x)-\frac {1}{x}\right )+\frac {1}{2 x \left (1-c^2 x^2\right )}\right )}{2 d^2 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle \frac {5}{2} c^2 \left (\frac {\frac {\sqrt {c x-1} \sqrt {c x+1} \left (-i b \int e^{-\text {arccosh}(c x)} \log \left (1-i e^{\text {arccosh}(c x)}\right )de^{\text {arccosh}(c x)}+i b \int e^{-\text {arccosh}(c x)} \log \left (1+i e^{\text {arccosh}(c x)}\right )de^{\text {arccosh}(c x)}+2 \arctan \left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )}{d \sqrt {d-c^2 d x^2}}+\frac {a+b \text {arccosh}(c x)}{d \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {c x-1} \sqrt {c x+1} \text {arctanh}(c x)}{d \sqrt {d-c^2 d x^2}}}{d}+\frac {a+b \text {arccosh}(c x)}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {b c \sqrt {c x-1} \sqrt {c x+1} \left (\frac {\text {arctanh}(c x)}{2 c}+\frac {x}{2 \left (1-c^2 x^2\right )}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}\right )-\frac {a+b \text {arccosh}(c x)}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}-\frac {b c \sqrt {c x-1} \sqrt {c x+1} \left (\frac {3}{2} \left (c \text {arctanh}(c x)-\frac {1}{x}\right )+\frac {1}{2 x \left (1-c^2 x^2\right )}\right )}{2 d^2 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {5}{2} c^2 \left (\frac {\frac {\sqrt {c x-1} \sqrt {c x+1} \left (2 \arctan \left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))-i b \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c x)}\right )+i b \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c x)}\right )\right )}{d \sqrt {d-c^2 d x^2}}+\frac {a+b \text {arccosh}(c x)}{d \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {c x-1} \sqrt {c x+1} \text {arctanh}(c x)}{d \sqrt {d-c^2 d x^2}}}{d}+\frac {a+b \text {arccosh}(c x)}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {b c \sqrt {c x-1} \sqrt {c x+1} \left (\frac {\text {arctanh}(c x)}{2 c}+\frac {x}{2 \left (1-c^2 x^2\right )}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}\right )-\frac {a+b \text {arccosh}(c x)}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}-\frac {b c \sqrt {c x-1} \sqrt {c x+1} \left (\frac {3}{2} \left (c \text {arctanh}(c x)-\frac {1}{x}\right )+\frac {1}{2 x \left (1-c^2 x^2\right )}\right )}{2 d^2 \sqrt {d-c^2 d x^2}}\) |
-1/2*(a + b*ArcCosh[c*x])/(d*x^2*(d - c^2*d*x^2)^(3/2)) - (b*c*Sqrt[-1 + c *x]*Sqrt[1 + c*x]*(1/(2*x*(1 - c^2*x^2)) + (3*(-x^(-1) + c*ArcTanh[c*x]))/ 2))/(2*d^2*Sqrt[d - c^2*d*x^2]) + (5*c^2*((a + b*ArcCosh[c*x])/(3*d*(d - c ^2*d*x^2)^(3/2)) + (b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(x/(2*(1 - c^2*x^2)) + ArcTanh[c*x]/(2*c)))/(3*d^2*Sqrt[d - c^2*d*x^2]) + ((a + b*ArcCosh[c*x]) /(d*Sqrt[d - c^2*d*x^2]) + (b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*ArcTanh[c*x])/( d*Sqrt[d - c^2*d*x^2]) + (Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(2*(a + b*ArcCosh[c *x])*ArcTan[E^ArcCosh[c*x]] - I*b*PolyLog[2, (-I)*E^ArcCosh[c*x]] + I*b*Po lyLog[2, I*E^ArcCosh[c*x]]))/(d*Sqrt[d - c^2*d*x^2]))/d))/2
3.2.32.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[( a*c + b*d*x^2)^m, x] /; FreeQ[{a, b, c, d, m}, x] && EqQ[b*c + a*d, 0] && ( IntegerQ[m] || (GtQ[a, 0] && GtQ[c, 0]))
Int[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_) )^(p_.), x_] :> Int[(a*c + b*d*x^2)^m*(e + f*x)^p, x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[n, m] && IntegerQ[m]
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) /(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1)) Int[(a + b*x^2)^(p + 1 ), x], x] /; FreeQ[{a, b}, x] && LtQ[p, -1] && (IntegerQ[4*p] || IntegerQ[6 *p])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(c*x )^(m + 1))*((a + b*x^2)^(p + 1)/(2*a*c*(p + 1))), x] + Simp[(m + 2*p + 3)/( 2*a*(p + 1)) Int[(c*x)^m*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, m }, x] && LtQ[p, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^( m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + 2*p + 3)/(a*c ^2*(m + 1))) Int[(c*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, p }, x] && LtQ[m, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
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_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_ ))^(m_.), x_Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)/E^( I*k*Pi)]/(f*fz*I)), x] + (-Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[ 1 - E^((-I)*e + f*fz*x)/E^(I*k*Pi)], x], x] + Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e + f*fz*x)/E^(I*k*Pi)], x], x]) /; FreeQ[{c , d, e, f, fz}, x] && IntegerQ[2*k] && IGtQ[m, 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])
Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_))/Sqrt[(d_) + (e_.) *(x_)^2], x_Symbol] :> Simp[(1/c^(m + 1))*Simp[Sqrt[1 + c*x]*(Sqrt[-1 + c*x ]/Sqrt[d + e*x^2])] Subst[Int[(a + b*x)^n*Cosh[x]^m, x], x, ArcCosh[c*x]] , x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0] && Int egerQ[m]
Time = 1.22 (sec) , antiderivative size = 655, normalized size of antiderivative = 1.37
| method | result | size |
| default | \(-\frac {a}{2 d \,x^{2} \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {5 a \,c^{2}}{6 d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {5 a \,c^{2}}{2 d^{2} \sqrt {-c^{2} d \,x^{2}+d}}-\frac {5 a \,c^{2} \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {-c^{2} d \,x^{2}+d}}{x}\right )}{2 d^{\frac {5}{2}}}+b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (15 c^{4} x^{4} \operatorname {arccosh}\left (c x \right )+2 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{3}-20 c^{2} x^{2} \operatorname {arccosh}\left (c x \right )-3 \sqrt {c x -1}\, \sqrt {c x +1}\, c x +3 \,\operatorname {arccosh}\left (c x \right )\right )}{6 d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right ) x^{2}}-\frac {13 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) c^{2}}{6 d^{3} \left (c^{2} x^{2}-1\right )}+\frac {13 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (\sqrt {c x -1}\, \sqrt {c x +1}+c x -1\right ) c^{2}}{6 d^{3} \left (c^{2} x^{2}-1\right )}-\frac {5 i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \ln \left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) c^{2}}{2 d^{3} \left (c^{2} x^{2}-1\right )}+\frac {5 i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {dilog}\left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) c^{2}}{2 d^{3} \left (c^{2} x^{2}-1\right )}-\frac {5 i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {dilog}\left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) c^{2}}{2 d^{3} \left (c^{2} x^{2}-1\right )}+\frac {5 i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \ln \left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) c^{2}}{2 d^{3} \left (c^{2} x^{2}-1\right )}\right )\) | \(655\) |
| parts | \(-\frac {a}{2 d \,x^{2} \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {5 a \,c^{2}}{6 d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {5 a \,c^{2}}{2 d^{2} \sqrt {-c^{2} d \,x^{2}+d}}-\frac {5 a \,c^{2} \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {-c^{2} d \,x^{2}+d}}{x}\right )}{2 d^{\frac {5}{2}}}+b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (15 c^{4} x^{4} \operatorname {arccosh}\left (c x \right )+2 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{3}-20 c^{2} x^{2} \operatorname {arccosh}\left (c x \right )-3 \sqrt {c x -1}\, \sqrt {c x +1}\, c x +3 \,\operatorname {arccosh}\left (c x \right )\right )}{6 d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right ) x^{2}}-\frac {13 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) c^{2}}{6 d^{3} \left (c^{2} x^{2}-1\right )}+\frac {13 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (\sqrt {c x -1}\, \sqrt {c x +1}+c x -1\right ) c^{2}}{6 d^{3} \left (c^{2} x^{2}-1\right )}-\frac {5 i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \ln \left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) c^{2}}{2 d^{3} \left (c^{2} x^{2}-1\right )}+\frac {5 i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {dilog}\left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) c^{2}}{2 d^{3} \left (c^{2} x^{2}-1\right )}-\frac {5 i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {dilog}\left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) c^{2}}{2 d^{3} \left (c^{2} x^{2}-1\right )}+\frac {5 i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \ln \left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) c^{2}}{2 d^{3} \left (c^{2} x^{2}-1\right )}\right )\) | \(655\) |
-1/2*a/d/x^2/(-c^2*d*x^2+d)^(3/2)+5/6*a*c^2/d/(-c^2*d*x^2+d)^(3/2)+5/2*a*c ^2/d^2/(-c^2*d*x^2+d)^(1/2)-5/2*a*c^2/d^(5/2)*ln((2*d+2*d^(1/2)*(-c^2*d*x^ 2+d)^(1/2))/x)+b*(-1/6*(-d*(c^2*x^2-1))^(1/2)*(15*c^4*x^4*arccosh(c*x)+2*( c*x-1)^(1/2)*(c*x+1)^(1/2)*c^3*x^3-20*c^2*x^2*arccosh(c*x)-3*(c*x-1)^(1/2) *(c*x+1)^(1/2)*c*x+3*arccosh(c*x))/d^3/(c^4*x^4-2*c^2*x^2+1)/x^2-13/6*(-d* (c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d^3/(c^2*x^2-1)*ln(1+c*x+(c *x-1)^(1/2)*(c*x+1)^(1/2))*c^2+13/6*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*( c*x+1)^(1/2)/d^3/(c^2*x^2-1)*ln((c*x-1)^(1/2)*(c*x+1)^(1/2)+c*x-1)*c^2-5/2 *I*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d^3/(c^2*x^2-1)*arcc osh(c*x)*ln(1-I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))*c^2+5/2*I*(-d*(c^2*x^2- 1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d^3/(c^2*x^2-1)*dilog(1+I*(c*x+(c*x- 1)^(1/2)*(c*x+1)^(1/2)))*c^2-5/2*I*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c *x+1)^(1/2)/d^3/(c^2*x^2-1)*dilog(1-I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))*c ^2+5/2*I*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d^3/(c^2*x^2-1 )*arccosh(c*x)*ln(1+I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))*c^2)
\[ \int \frac {a+b \text {arccosh}(c x)}{x^3 \left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{3}} \,d x } \]
integral(-sqrt(-c^2*d*x^2 + d)*(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. \[ \int \frac {a+b \text {arccosh}(c x)}{x^3 \left (d-c^2 d x^2\right )^{5/2}} \, dx=\text {Timed out} \]
\[ \int \frac {a+b \text {arccosh}(c x)}{x^3 \left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{3}} \,d x } \]
-1/6*a*(15*c^2*log(2*sqrt(-c^2*d*x^2 + d)*sqrt(d)/abs(x) + 2*d/abs(x))/d^( 5/2) - 15*c^2/(sqrt(-c^2*d*x^2 + d)*d^2) - 5*c^2/((-c^2*d*x^2 + d)^(3/2)*d ) + 3/((-c^2*d*x^2 + d)^(3/2)*d*x^2)) + b*integrate(log(c*x + sqrt(c*x + 1 )*sqrt(c*x - 1))/((-c^2*d*x^2 + d)^(5/2)*x^3), x)
\[ \int \frac {a+b \text {arccosh}(c x)}{x^3 \left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{3}} \,d x } \]
Timed out. \[ \int \frac {a+b \text {arccosh}(c x)}{x^3 \left (d-c^2 d x^2\right )^{5/2}} \, dx=\int \frac {a+b\,\mathrm {acosh}\left (c\,x\right )}{x^3\,{\left (d-c^2\,d\,x^2\right )}^{5/2}} \,d x \]