Integrand size = 27, antiderivative size = 407 \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{x^5} \, dx=-\frac {b c d^2 \sqrt {d-c^2 d x^2}}{12 x^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {9 b c^3 d^2 \sqrt {d-c^2 d x^2}}{8 x \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^5 d^2 x \sqrt {d-c^2 d x^2}}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {15}{8} c^4 d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))+\frac {5 c^2 d \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{8 x^2}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{4 x^4}-\frac {15 c^4 d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \arctan \left (e^{\text {arccosh}(c x)}\right )}{4 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {15 i b c^4 d^2 \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c x)}\right )}{8 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {15 i b c^4 d^2 \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c x)}\right )}{8 \sqrt {-1+c x} \sqrt {1+c x}} \]
5/8*c^2*d*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))/x^2-1/4*(-c^2*d*x^2+d)^( 5/2)*(a+b*arccosh(c*x))/x^4+15/8*c^4*d^2*(a+b*arccosh(c*x))*(-c^2*d*x^2+d) ^(1/2)-1/12*b*c*d^2*(-c^2*d*x^2+d)^(1/2)/x^3/(c*x-1)^(1/2)/(c*x+1)^(1/2)+9 /8*b*c^3*d^2*(-c^2*d*x^2+d)^(1/2)/x/(c*x-1)^(1/2)/(c*x+1)^(1/2)-b*c^5*d^2* x*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)-15/4*c^4*d^2*(a+b*arcco sh(c*x))*arctan(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*(-c^2*d*x^2+d)^(1/2)/(c*x -1)^(1/2)/(c*x+1)^(1/2)+15/8*I*b*c^4*d^2*polylog(2,-I*(c*x+(c*x-1)^(1/2)*( c*x+1)^(1/2)))*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)-15/8*I*b*c ^4*d^2*polylog(2,I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))*(-c^2*d*x^2+d)^(1/2) /(c*x-1)^(1/2)/(c*x+1)^(1/2)
Time = 1.47 (sec) , antiderivative size = 660, normalized size of antiderivative = 1.62 \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{x^5} \, dx=\frac {-2 b c d^3 x+2 b c^2 d^3 x^2+27 b c^3 d^3 x^3-27 b c^4 d^3 x^4-24 b c^5 d^3 x^5+24 b c^6 d^3 x^6-6 a d^3 \sqrt {\frac {-1+c x}{1+c x}}+33 a c^2 d^3 x^2 \sqrt {\frac {-1+c x}{1+c x}}-3 a c^4 d^3 x^4 \sqrt {\frac {-1+c x}{1+c x}}-24 a c^6 d^3 x^6 \sqrt {\frac {-1+c x}{1+c x}}-6 b d^3 \sqrt {\frac {-1+c x}{1+c x}} \text {arccosh}(c x)+33 b c^2 d^3 x^2 \sqrt {\frac {-1+c x}{1+c x}} \text {arccosh}(c x)-3 b c^4 d^3 x^4 \sqrt {\frac {-1+c x}{1+c x}} \text {arccosh}(c x)-24 b c^6 d^3 x^6 \sqrt {\frac {-1+c x}{1+c x}} \text {arccosh}(c x)+45 i b c^4 d^3 x^4 \text {arccosh}(c x) \log \left (1-i e^{-\text {arccosh}(c x)}\right )-45 i b c^5 d^3 x^5 \text {arccosh}(c x) \log \left (1-i e^{-\text {arccosh}(c x)}\right )-45 i b c^4 d^3 x^4 \text {arccosh}(c x) \log \left (1+i e^{-\text {arccosh}(c x)}\right )+45 i b c^5 d^3 x^5 \text {arccosh}(c x) \log \left (1+i e^{-\text {arccosh}(c x)}\right )+45 a c^4 d^{5/2} x^4 \sqrt {\frac {-1+c x}{1+c x}} \sqrt {d-c^2 d x^2} \log (x)-45 a c^4 d^{5/2} x^4 \sqrt {\frac {-1+c x}{1+c x}} \sqrt {d-c^2 d x^2} \log \left (d+\sqrt {d} \sqrt {d-c^2 d x^2}\right )-45 i b c^4 d^3 x^4 (-1+c x) \operatorname {PolyLog}\left (2,-i e^{-\text {arccosh}(c x)}\right )+45 i b c^4 d^3 x^4 (-1+c x) \operatorname {PolyLog}\left (2,i e^{-\text {arccosh}(c x)}\right )}{24 x^4 \sqrt {\frac {-1+c x}{1+c x}} \sqrt {d-c^2 d x^2}} \]
(-2*b*c*d^3*x + 2*b*c^2*d^3*x^2 + 27*b*c^3*d^3*x^3 - 27*b*c^4*d^3*x^4 - 24 *b*c^5*d^3*x^5 + 24*b*c^6*d^3*x^6 - 6*a*d^3*Sqrt[(-1 + c*x)/(1 + c*x)] + 3 3*a*c^2*d^3*x^2*Sqrt[(-1 + c*x)/(1 + c*x)] - 3*a*c^4*d^3*x^4*Sqrt[(-1 + c* x)/(1 + c*x)] - 24*a*c^6*d^3*x^6*Sqrt[(-1 + c*x)/(1 + c*x)] - 6*b*d^3*Sqrt [(-1 + c*x)/(1 + c*x)]*ArcCosh[c*x] + 33*b*c^2*d^3*x^2*Sqrt[(-1 + c*x)/(1 + c*x)]*ArcCosh[c*x] - 3*b*c^4*d^3*x^4*Sqrt[(-1 + c*x)/(1 + c*x)]*ArcCosh[ c*x] - 24*b*c^6*d^3*x^6*Sqrt[(-1 + c*x)/(1 + c*x)]*ArcCosh[c*x] + (45*I)*b *c^4*d^3*x^4*ArcCosh[c*x]*Log[1 - I/E^ArcCosh[c*x]] - (45*I)*b*c^5*d^3*x^5 *ArcCosh[c*x]*Log[1 - I/E^ArcCosh[c*x]] - (45*I)*b*c^4*d^3*x^4*ArcCosh[c*x ]*Log[1 + I/E^ArcCosh[c*x]] + (45*I)*b*c^5*d^3*x^5*ArcCosh[c*x]*Log[1 + I/ E^ArcCosh[c*x]] + 45*a*c^4*d^(5/2)*x^4*Sqrt[(-1 + c*x)/(1 + c*x)]*Sqrt[d - c^2*d*x^2]*Log[x] - 45*a*c^4*d^(5/2)*x^4*Sqrt[(-1 + c*x)/(1 + c*x)]*Sqrt[ d - c^2*d*x^2]*Log[d + Sqrt[d]*Sqrt[d - c^2*d*x^2]] - (45*I)*b*c^4*d^3*x^4 *(-1 + c*x)*PolyLog[2, (-I)/E^ArcCosh[c*x]] + (45*I)*b*c^4*d^3*x^4*(-1 + c *x)*PolyLog[2, I/E^ArcCosh[c*x]])/(24*x^4*Sqrt[(-1 + c*x)/(1 + c*x)]*Sqrt[ d - c^2*d*x^2])
Time = 1.72 (sec) , antiderivative size = 343, normalized size of antiderivative = 0.84, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.593, Rules used = {6343, 82, 244, 2009, 6343, 25, 82, 244, 2009, 6341, 24, 6362, 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.
| \(\displaystyle \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{x^5} \, dx\) |
| \(\Big \downarrow \) 6343 |
| \(\displaystyle -\frac {5}{4} c^2 d \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x^3}dx+\frac {b c d^2 \sqrt {d-c^2 d x^2} \int \frac {(1-c x)^2 (c x+1)^2}{x^4}dx}{4 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{4 x^4}\) |
| \(\Big \downarrow \) 82 |
| \(\displaystyle -\frac {5}{4} c^2 d \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x^3}dx+\frac {b c d^2 \sqrt {d-c^2 d x^2} \int \frac {\left (1-c^2 x^2\right )^2}{x^4}dx}{4 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{4 x^4}\) |
| \(\Big \downarrow \) 244 |
| \(\displaystyle -\frac {5}{4} c^2 d \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x^3}dx+\frac {b c d^2 \sqrt {d-c^2 d x^2} \int \left (c^4-\frac {2 c^2}{x^2}+\frac {1}{x^4}\right )dx}{4 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{4 x^4}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {5}{4} c^2 d \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x^3}dx-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{4 x^4}+\frac {b c d^2 \left (c^4 x+\frac {2 c^2}{x}-\frac {1}{3 x^3}\right ) \sqrt {d-c^2 d x^2}}{4 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 6343 |
| \(\displaystyle -\frac {5}{4} c^2 d \left (-\frac {3}{2} c^2 d \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{x}dx-\frac {b c d \sqrt {d-c^2 d x^2} \int -\frac {(1-c x) (c x+1)}{x^2}dx}{2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{2 x^2}\right )-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{4 x^4}+\frac {b c d^2 \left (c^4 x+\frac {2 c^2}{x}-\frac {1}{3 x^3}\right ) \sqrt {d-c^2 d x^2}}{4 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {5}{4} c^2 d \left (-\frac {3}{2} c^2 d \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{x}dx+\frac {b c d \sqrt {d-c^2 d x^2} \int \frac {(1-c x) (c x+1)}{x^2}dx}{2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{2 x^2}\right )-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{4 x^4}+\frac {b c d^2 \left (c^4 x+\frac {2 c^2}{x}-\frac {1}{3 x^3}\right ) \sqrt {d-c^2 d x^2}}{4 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 82 |
| \(\displaystyle -\frac {5}{4} c^2 d \left (-\frac {3}{2} c^2 d \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{x}dx+\frac {b c d \sqrt {d-c^2 d x^2} \int \frac {1-c^2 x^2}{x^2}dx}{2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{2 x^2}\right )-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{4 x^4}+\frac {b c d^2 \left (c^4 x+\frac {2 c^2}{x}-\frac {1}{3 x^3}\right ) \sqrt {d-c^2 d x^2}}{4 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 244 |
| \(\displaystyle -\frac {5}{4} c^2 d \left (-\frac {3}{2} c^2 d \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{x}dx+\frac {b c d \sqrt {d-c^2 d x^2} \int \left (\frac {1}{x^2}-c^2\right )dx}{2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{2 x^2}\right )-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{4 x^4}+\frac {b c d^2 \left (c^4 x+\frac {2 c^2}{x}-\frac {1}{3 x^3}\right ) \sqrt {d-c^2 d x^2}}{4 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {5}{4} c^2 d \left (-\frac {3}{2} c^2 d \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{x}dx-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{2 x^2}+\frac {b c d \left (c^2 (-x)-\frac {1}{x}\right ) \sqrt {d-c^2 d x^2}}{2 \sqrt {c x-1} \sqrt {c x+1}}\right )-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{4 x^4}+\frac {b c d^2 \left (c^4 x+\frac {2 c^2}{x}-\frac {1}{3 x^3}\right ) \sqrt {d-c^2 d x^2}}{4 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 6341 |
| \(\displaystyle -\frac {5}{4} c^2 d \left (-\frac {3}{2} c^2 d \left (-\frac {\sqrt {d-c^2 d x^2} \int \frac {a+b \text {arccosh}(c x)}{x \sqrt {c x-1} \sqrt {c x+1}}dx}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {b c \sqrt {d-c^2 d x^2} \int 1dx}{\sqrt {c x-1} \sqrt {c x+1}}+\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))\right )-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{2 x^2}+\frac {b c d \left (c^2 (-x)-\frac {1}{x}\right ) \sqrt {d-c^2 d x^2}}{2 \sqrt {c x-1} \sqrt {c x+1}}\right )-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{4 x^4}+\frac {b c d^2 \left (c^4 x+\frac {2 c^2}{x}-\frac {1}{3 x^3}\right ) \sqrt {d-c^2 d x^2}}{4 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle -\frac {5}{4} c^2 d \left (-\frac {3}{2} c^2 d \left (-\frac {\sqrt {d-c^2 d x^2} \int \frac {a+b \text {arccosh}(c x)}{x \sqrt {c x-1} \sqrt {c x+1}}dx}{\sqrt {c x-1} \sqrt {c x+1}}+\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {b c x \sqrt {d-c^2 d x^2}}{\sqrt {c x-1} \sqrt {c x+1}}\right )-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{2 x^2}+\frac {b c d \left (c^2 (-x)-\frac {1}{x}\right ) \sqrt {d-c^2 d x^2}}{2 \sqrt {c x-1} \sqrt {c x+1}}\right )-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{4 x^4}+\frac {b c d^2 \left (c^4 x+\frac {2 c^2}{x}-\frac {1}{3 x^3}\right ) \sqrt {d-c^2 d x^2}}{4 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 6362 |
| \(\displaystyle -\frac {5}{4} c^2 d \left (-\frac {3}{2} c^2 d \left (-\frac {\sqrt {d-c^2 d x^2} \int \frac {a+b \text {arccosh}(c x)}{c x}d\text {arccosh}(c x)}{\sqrt {c x-1} \sqrt {c x+1}}+\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {b c x \sqrt {d-c^2 d x^2}}{\sqrt {c x-1} \sqrt {c x+1}}\right )-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{2 x^2}+\frac {b c d \left (c^2 (-x)-\frac {1}{x}\right ) \sqrt {d-c^2 d x^2}}{2 \sqrt {c x-1} \sqrt {c x+1}}\right )-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{4 x^4}+\frac {b c d^2 \left (c^4 x+\frac {2 c^2}{x}-\frac {1}{3 x^3}\right ) \sqrt {d-c^2 d x^2}}{4 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {5}{4} c^2 d \left (-\frac {3}{2} c^2 d \left (-\frac {\sqrt {d-c^2 d x^2} \int (a+b \text {arccosh}(c x)) \csc \left (i \text {arccosh}(c x)+\frac {\pi }{2}\right )d\text {arccosh}(c x)}{\sqrt {c x-1} \sqrt {c x+1}}+\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {b c x \sqrt {d-c^2 d x^2}}{\sqrt {c x-1} \sqrt {c x+1}}\right )-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{2 x^2}+\frac {b c d \left (c^2 (-x)-\frac {1}{x}\right ) \sqrt {d-c^2 d x^2}}{2 \sqrt {c x-1} \sqrt {c x+1}}\right )-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{4 x^4}+\frac {b c d^2 \left (c^4 x+\frac {2 c^2}{x}-\frac {1}{3 x^3}\right ) \sqrt {d-c^2 d x^2}}{4 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 4668 |
| \(\displaystyle -\frac {5}{4} c^2 d \left (-\frac {3}{2} c^2 d \left (-\frac {\sqrt {d-c^2 d x^2} \left (-i b \int \log \left (1-i e^{\text {arccosh}(c x)}\right )d\text {arccosh}(c x)+i b \int \log \left (1+i e^{\text {arccosh}(c x)}\right )d\text {arccosh}(c x)+2 \arctan \left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )}{\sqrt {c x-1} \sqrt {c x+1}}+\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {b c x \sqrt {d-c^2 d x^2}}{\sqrt {c x-1} \sqrt {c x+1}}\right )-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{2 x^2}+\frac {b c d \left (c^2 (-x)-\frac {1}{x}\right ) \sqrt {d-c^2 d x^2}}{2 \sqrt {c x-1} \sqrt {c x+1}}\right )-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{4 x^4}+\frac {b c d^2 \left (c^4 x+\frac {2 c^2}{x}-\frac {1}{3 x^3}\right ) \sqrt {d-c^2 d x^2}}{4 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle -\frac {5}{4} c^2 d \left (-\frac {3}{2} c^2 d \left (-\frac {\sqrt {d-c^2 d x^2} \left (-i b \int e^{-\text {arccosh}(c x)} \log \left (1-i e^{\text {arccosh}(c x)}\right )de^{\text {arccosh}(c x)}+i b \int e^{-\text {arccosh}(c x)} \log \left (1+i e^{\text {arccosh}(c x)}\right )de^{\text {arccosh}(c x)}+2 \arctan \left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )}{\sqrt {c x-1} \sqrt {c x+1}}+\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {b c x \sqrt {d-c^2 d x^2}}{\sqrt {c x-1} \sqrt {c x+1}}\right )-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{2 x^2}+\frac {b c d \left (c^2 (-x)-\frac {1}{x}\right ) \sqrt {d-c^2 d x^2}}{2 \sqrt {c x-1} \sqrt {c x+1}}\right )-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{4 x^4}+\frac {b c d^2 \left (c^4 x+\frac {2 c^2}{x}-\frac {1}{3 x^3}\right ) \sqrt {d-c^2 d x^2}}{4 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle -\frac {5}{4} c^2 d \left (-\frac {3}{2} c^2 d \left (-\frac {\sqrt {d-c^2 d x^2} \left (2 \arctan \left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))-i b \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c x)}\right )+i b \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c x)}\right )\right )}{\sqrt {c x-1} \sqrt {c x+1}}+\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {b c x \sqrt {d-c^2 d x^2}}{\sqrt {c x-1} \sqrt {c x+1}}\right )-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{2 x^2}+\frac {b c d \left (c^2 (-x)-\frac {1}{x}\right ) \sqrt {d-c^2 d x^2}}{2 \sqrt {c x-1} \sqrt {c x+1}}\right )-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{4 x^4}+\frac {b c d^2 \left (c^4 x+\frac {2 c^2}{x}-\frac {1}{3 x^3}\right ) \sqrt {d-c^2 d x^2}}{4 \sqrt {c x-1} \sqrt {c x+1}}\) |
(b*c*d^2*(-1/3*1/x^3 + (2*c^2)/x + c^4*x)*Sqrt[d - c^2*d*x^2])/(4*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - ((d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]))/(4*x^ 4) - (5*c^2*d*((b*c*d*(-x^(-1) - c^2*x)*Sqrt[d - c^2*d*x^2])/(2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - ((d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]))/(2*x^2) - (3*c^2*d*(-((b*c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])) + Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]) - (Sqrt[d - c^2*d*x^2]*(2*(a + b*ArcCosh[c*x])*ArcTan[E^ArcCosh[c*x]] - I*b*PolyLog[2, (-I)*E^ArcCosh[c* x]] + I*b*PolyLog[2, I*E^ArcCosh[c*x]]))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])))/ 2))/4
3.2.2.3.1 Defintions of rubi rules used
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[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand Integrand[(c*x)^m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && IGtQ[p , 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_.) + 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_)*Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(f*x)^(m + 1)*Sqrt[d + e*x^2]*((a + b*Arc Cosh[c*x])^n/(f*(m + 2))), x] + (-Simp[(1/(m + 2))*Simp[Sqrt[d + e*x^2]/(Sq rt[1 + c*x]*Sqrt[-1 + c*x])] Int[(f*x)^m*((a + b*ArcCosh[c*x])^n/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])), x], x] - Simp[b*c*(n/(f*(m + 2)))*Simp[Sqrt[d + e* x^2]/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])] Int[(f*x)^(m + 1)*(a + b*ArcCosh[c*x ])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0] && (IGtQ[m, -2] || EqQ[n, 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*((a + b*Arc Cosh[c*x])^n/(f*(m + 1))), x] + (-Simp[2*e*(p/(f^2*(m + 1))) Int[(f*x)^(m + 2)*(d + e*x^2)^(p - 1)*(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}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && G tQ[p, 0] && LtQ[m, -1]
Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_))/(Sqrt[(d1_) + (e1 _.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)]), x_Symbol] :> Simp[(1/c^(m + 1))*Simp[ Sqrt[1 + c*x]/Sqrt[d1 + e1*x]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]] Subst [Int[(a + b*x)^n*Cosh[x]^m, x], x, ArcCosh[c*x]], x] /; FreeQ[{a, b, c, d1, e1, d2, e2}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && IGtQ[n, 0] && Inte gerQ[m]
Time = 1.08 (sec) , antiderivative size = 691, normalized size of antiderivative = 1.70
| method | result | size |
| default | \(-\frac {a \left (-c^{2} d \,x^{2}+d \right )^{\frac {7}{2}}}{4 d \,x^{4}}+\frac {3 a \,c^{2} \left (-c^{2} d \,x^{2}+d \right )^{\frac {7}{2}}}{8 d \,x^{2}}+\frac {3 a \,c^{4} \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{8}+\frac {5 a \,c^{4} d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{8}-\frac {15 a \,c^{4} d^{\frac {5}{2}} \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {-c^{2} d \,x^{2}+d}}{x}\right )}{8}+\frac {15 a \,c^{4} d^{2} \sqrt {-c^{2} d \,x^{2}+d}}{8}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, c^{6} d^{2} \operatorname {arccosh}\left (c x \right ) x^{2}}{\left (c x +1\right ) \left (c x -1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, c^{5} d^{2} x}{\sqrt {c x +1}\, \sqrt {c x -1}}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, c^{4} d^{2} \operatorname {arccosh}\left (c x \right )}{8 \left (c x +1\right ) \left (c x -1\right )}+\frac {9 b \,d^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, c^{3}}{8 \sqrt {c x +1}\, x \sqrt {c x -1}}-\frac {11 b \,d^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right ) c^{2}}{8 \left (c x +1\right ) x^{2} \left (c x -1\right )}-\frac {b \,d^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, c}{12 \sqrt {c x +1}\, x^{3} \sqrt {c x -1}}+\frac {b \,d^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right )}{4 \left (c x +1\right ) x^{4} \left (c x -1\right )}+\frac {15 i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {dilog}\left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) c^{4} d^{2}}{8 \sqrt {c x -1}\, \sqrt {c x +1}}-\frac {15 i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right ) \ln \left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) c^{4} d^{2}}{8 \sqrt {c x -1}\, \sqrt {c x +1}}-\frac {15 i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {dilog}\left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) c^{4} d^{2}}{8 \sqrt {c x -1}\, \sqrt {c x +1}}+\frac {15 i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right ) \ln \left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) c^{4} d^{2}}{8 \sqrt {c x -1}\, \sqrt {c x +1}}\) | \(691\) |
| parts | \(-\frac {a \left (-c^{2} d \,x^{2}+d \right )^{\frac {7}{2}}}{4 d \,x^{4}}+\frac {3 a \,c^{2} \left (-c^{2} d \,x^{2}+d \right )^{\frac {7}{2}}}{8 d \,x^{2}}+\frac {3 a \,c^{4} \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{8}+\frac {5 a \,c^{4} d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{8}-\frac {15 a \,c^{4} d^{\frac {5}{2}} \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {-c^{2} d \,x^{2}+d}}{x}\right )}{8}+\frac {15 a \,c^{4} d^{2} \sqrt {-c^{2} d \,x^{2}+d}}{8}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, c^{6} d^{2} \operatorname {arccosh}\left (c x \right ) x^{2}}{\left (c x +1\right ) \left (c x -1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, c^{5} d^{2} x}{\sqrt {c x +1}\, \sqrt {c x -1}}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, c^{4} d^{2} \operatorname {arccosh}\left (c x \right )}{8 \left (c x +1\right ) \left (c x -1\right )}+\frac {9 b \,d^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, c^{3}}{8 \sqrt {c x +1}\, x \sqrt {c x -1}}-\frac {11 b \,d^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right ) c^{2}}{8 \left (c x +1\right ) x^{2} \left (c x -1\right )}-\frac {b \,d^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, c}{12 \sqrt {c x +1}\, x^{3} \sqrt {c x -1}}+\frac {b \,d^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right )}{4 \left (c x +1\right ) x^{4} \left (c x -1\right )}+\frac {15 i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {dilog}\left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) c^{4} d^{2}}{8 \sqrt {c x -1}\, \sqrt {c x +1}}-\frac {15 i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right ) \ln \left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) c^{4} d^{2}}{8 \sqrt {c x -1}\, \sqrt {c x +1}}-\frac {15 i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {dilog}\left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) c^{4} d^{2}}{8 \sqrt {c x -1}\, \sqrt {c x +1}}+\frac {15 i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right ) \ln \left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) c^{4} d^{2}}{8 \sqrt {c x -1}\, \sqrt {c x +1}}\) | \(691\) |
-1/4*a/d/x^4*(-c^2*d*x^2+d)^(7/2)+3/8*a*c^2/d/x^2*(-c^2*d*x^2+d)^(7/2)+3/8 *a*c^4*(-c^2*d*x^2+d)^(5/2)+5/8*a*c^4*d*(-c^2*d*x^2+d)^(3/2)-15/8*a*c^4*d^ (5/2)*ln((2*d+2*d^(1/2)*(-c^2*d*x^2+d)^(1/2))/x)+15/8*a*c^4*d^2*(-c^2*d*x^ 2+d)^(1/2)+b*(-d*(c^2*x^2-1))^(1/2)*c^6*d^2/(c*x+1)/(c*x-1)*arccosh(c*x)*x ^2-b*(-d*(c^2*x^2-1))^(1/2)*c^5*d^2/(c*x+1)^(1/2)/(c*x-1)^(1/2)*x+1/8*b*(- d*(c^2*x^2-1))^(1/2)*c^4*d^2/(c*x+1)/(c*x-1)*arccosh(c*x)+9/8*b*d^2*(-d*(c ^2*x^2-1))^(1/2)/(c*x+1)^(1/2)/x/(c*x-1)^(1/2)*c^3-11/8*b*d^2*(-d*(c^2*x^2 -1))^(1/2)/(c*x+1)/x^2/(c*x-1)*arccosh(c*x)*c^2-1/12*b*d^2*(-d*(c^2*x^2-1) )^(1/2)/(c*x+1)^(1/2)/x^3/(c*x-1)^(1/2)*c+1/4*b*d^2*(-d*(c^2*x^2-1))^(1/2) /(c*x+1)/x^4/(c*x-1)*arccosh(c*x)+15/8*I*b*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^ (1/2)/(c*x+1)^(1/2)*dilog(1+I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))*c^4*d^2-1 5/8*I*b*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)*arccosh(c*x)*ln (1-I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))*c^4*d^2-15/8*I*b*(-d*(c^2*x^2-1))^ (1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)*dilog(1-I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/ 2)))*c^4*d^2+15/8*I*b*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)*a rccosh(c*x)*ln(1+I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))*c^4*d^2
\[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{x^5} \, dx=\int { \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{x^{5}} \,d x } \]
integral((a*c^4*d^2*x^4 - 2*a*c^2*d^2*x^2 + a*d^2 + (b*c^4*d^2*x^4 - 2*b*c ^2*d^2*x^2 + b*d^2)*arccosh(c*x))*sqrt(-c^2*d*x^2 + d)/x^5, x)
Timed out. \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{x^5} \, dx=\text {Timed out} \]
\[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{x^5} \, dx=\int { \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{x^{5}} \,d x } \]
-1/8*(15*c^4*d^(5/2)*log(2*sqrt(-c^2*d*x^2 + d)*sqrt(d)/abs(x) + 2*d/abs(x )) - 3*(-c^2*d*x^2 + d)^(5/2)*c^4 - 5*(-c^2*d*x^2 + d)^(3/2)*c^4*d - 15*sq rt(-c^2*d*x^2 + d)*c^4*d^2 - 3*(-c^2*d*x^2 + d)^(7/2)*c^2/(d*x^2) + 2*(-c^ 2*d*x^2 + d)^(7/2)/(d*x^4))*a + b*integrate((-c^2*d*x^2 + d)^(5/2)*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))/x^5, x)
Exception generated. \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{x^5} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{x^5} \, dx=\int \frac {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{5/2}}{x^5} \,d x \]