Integrand size = 29, antiderivative size = 426 \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{x^4} \, dx=\frac {b^2 c^2 d \sqrt {d-c^2 d x^2}}{3 x}-\frac {b^2 c^3 d \sqrt {d-c^2 d x^2} \text {arccosh}(c x)}{3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c d \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{3 x^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {c^2 d \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{x}-\frac {4 c^3 d \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{3 x^3}-\frac {c^3 d \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^3}{3 b \sqrt {-1+c x} \sqrt {1+c x}}-\frac {8 b c^3 d \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \log \left (1+e^{-2 \text {arccosh}(c x)}\right )}{3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {4 b^2 c^3 d \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c x)}\right )}{3 \sqrt {-1+c x} \sqrt {1+c x}} \]
-1/3*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))^2/x^3+1/3*b^2*c^2*d*(-c^2*d*x ^2+d)^(1/2)/x+c^2*d*(a+b*arccosh(c*x))^2*(-c^2*d*x^2+d)^(1/2)/x-1/3*b^2*c^ 3*d*arccosh(c*x)*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/3*b*c* d*(-c^2*x^2+1)*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/x^2/(c*x-1)^(1/2)/( c*x+1)^(1/2)-4/3*c^3*d*(a+b*arccosh(c*x))^2*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^( 1/2)/(c*x+1)^(1/2)-1/3*c^3*d*(a+b*arccosh(c*x))^3*(-c^2*d*x^2+d)^(1/2)/b/( c*x-1)^(1/2)/(c*x+1)^(1/2)-8/3*b*c^3*d*(a+b*arccosh(c*x))*ln(1+1/(c*x+(c*x -1)^(1/2)*(c*x+1)^(1/2))^2)*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/ 2)+4/3*b^2*c^3*d*polylog(2,-1/(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2)*(-c^2*d *x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)
Time = 1.87 (sec) , antiderivative size = 583, normalized size of antiderivative = 1.37 \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{x^4} \, dx=\frac {-a b c d^2 x+a b c^2 d^2 x^2-a^2 d^2 \sqrt {\frac {-1+c x}{1+c x}}+5 a^2 c^2 d^2 x^2 \sqrt {\frac {-1+c x}{1+c x}}+b^2 c^2 d^2 x^2 \sqrt {\frac {-1+c x}{1+c x}}-4 a^2 c^4 d^2 x^4 \sqrt {\frac {-1+c x}{1+c x}}-b^2 c^4 d^2 x^4 \sqrt {\frac {-1+c x}{1+c x}}-b d^2 (-1+c x) \left (-3 a c^3 x^3+b \left (-\sqrt {\frac {-1+c x}{1+c x}}-c x \sqrt {\frac {-1+c x}{1+c x}}+4 c^2 x^2 \sqrt {\frac {-1+c x}{1+c x}}+4 c^3 x^3 \left (-1+\sqrt {\frac {-1+c x}{1+c x}}\right )\right )\right ) \text {arccosh}(c x)^2+b^2 c^3 d^2 x^3 (-1+c x) \text {arccosh}(c x)^3-3 a^2 c^3 d^{3/2} x^3 \sqrt {\frac {-1+c x}{1+c x}} \sqrt {d-c^2 d x^2} \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )+b d^2 (-1+c x) \text {arccosh}(c x) \left (b c x+2 a \sqrt {\frac {-1+c x}{1+c x}} \left (1+c x-4 c^2 x^2-4 c^3 x^3\right )+8 b c^3 x^3 \log \left (1+e^{-2 \text {arccosh}(c x)}\right )\right )-8 a b c^3 d^2 x^3 \log (c x)+8 a b c^4 d^2 x^4 \log (c x)-4 b^2 c^3 d^2 x^3 (-1+c x) \operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c x)}\right )}{3 x^3 \sqrt {\frac {-1+c x}{1+c x}} \sqrt {d-c^2 d x^2}} \]
(-(a*b*c*d^2*x) + a*b*c^2*d^2*x^2 - a^2*d^2*Sqrt[(-1 + c*x)/(1 + c*x)] + 5 *a^2*c^2*d^2*x^2*Sqrt[(-1 + c*x)/(1 + c*x)] + b^2*c^2*d^2*x^2*Sqrt[(-1 + c *x)/(1 + c*x)] - 4*a^2*c^4*d^2*x^4*Sqrt[(-1 + c*x)/(1 + c*x)] - b^2*c^4*d^ 2*x^4*Sqrt[(-1 + c*x)/(1 + c*x)] - b*d^2*(-1 + c*x)*(-3*a*c^3*x^3 + b*(-Sq rt[(-1 + c*x)/(1 + c*x)] - c*x*Sqrt[(-1 + c*x)/(1 + c*x)] + 4*c^2*x^2*Sqrt [(-1 + c*x)/(1 + c*x)] + 4*c^3*x^3*(-1 + Sqrt[(-1 + c*x)/(1 + c*x)])))*Arc Cosh[c*x]^2 + b^2*c^3*d^2*x^3*(-1 + c*x)*ArcCosh[c*x]^3 - 3*a^2*c^3*d^(3/2 )*x^3*Sqrt[(-1 + c*x)/(1 + c*x)]*Sqrt[d - c^2*d*x^2]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] + b*d^2*(-1 + c*x)*ArcCosh[c*x]*(b*c *x + 2*a*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x - 4*c^2*x^2 - 4*c^3*x^3) + 8* b*c^3*x^3*Log[1 + E^(-2*ArcCosh[c*x])]) - 8*a*b*c^3*d^2*x^3*Log[c*x] + 8*a *b*c^4*d^2*x^4*Log[c*x] - 4*b^2*c^3*d^2*x^3*(-1 + c*x)*PolyLog[2, -E^(-2*A rcCosh[c*x])])/(3*x^3*Sqrt[(-1 + c*x)/(1 + c*x)]*Sqrt[d - c^2*d*x^2])
Result contains complex when optimal does not.
Time = 4.24 (sec) , antiderivative size = 403, normalized size of antiderivative = 0.95, number of steps used = 26, number of rules used = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.862, Rules used = {6343, 25, 6327, 6335, 108, 27, 43, 6297, 25, 3042, 26, 4201, 2620, 2715, 2838, 6339, 6297, 25, 3042, 26, 4201, 2620, 2715, 2838, 6308}
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.
| \(\displaystyle \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{x^4} \, dx\) |
| \(\Big \downarrow \) 6343 |
| \(\displaystyle c^2 (-d) \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{x^2}dx-\frac {2 b c d \sqrt {d-c^2 d x^2} \int -\frac {(1-c x) (c x+1) (a+b \text {arccosh}(c x))}{x^3}dx}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle c^2 (-d) \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{x^2}dx+\frac {2 b c d \sqrt {d-c^2 d x^2} \int \frac {(1-c x) (c x+1) (a+b \text {arccosh}(c x))}{x^3}dx}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 6327 |
| \(\displaystyle c^2 (-d) \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{x^2}dx+\frac {2 b c d \sqrt {d-c^2 d x^2} \int \frac {\left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))}{x^3}dx}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 6335 |
| \(\displaystyle c^2 (-d) \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{x^2}dx+\frac {2 b c d \sqrt {d-c^2 d x^2} \left (c^2 \left (-\int \frac {a+b \text {arccosh}(c x)}{x}dx\right )-\frac {1}{2} b c \int \frac {\sqrt {c x-1} \sqrt {c x+1}}{x^2}dx-\frac {\left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))}{2 x^2}\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle c^2 (-d) \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{x^2}dx+\frac {2 b c d \sqrt {d-c^2 d x^2} \left (c^2 \left (-\int \frac {a+b \text {arccosh}(c x)}{x}dx\right )-\frac {1}{2} b c \left (\int \frac {c^2}{\sqrt {c x-1} \sqrt {c x+1}}dx-\frac {\sqrt {c x-1} \sqrt {c x+1}}{x}\right )-\frac {\left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))}{2 x^2}\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle c^2 (-d) \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{x^2}dx+\frac {2 b c d \sqrt {d-c^2 d x^2} \left (c^2 \left (-\int \frac {a+b \text {arccosh}(c x)}{x}dx\right )-\frac {1}{2} b c \left (c^2 \int \frac {1}{\sqrt {c x-1} \sqrt {c x+1}}dx-\frac {\sqrt {c x-1} \sqrt {c x+1}}{x}\right )-\frac {\left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))}{2 x^2}\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 43 |
| \(\displaystyle c^2 (-d) \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{x^2}dx+\frac {2 b c d \sqrt {d-c^2 d x^2} \left (c^2 \left (-\int \frac {a+b \text {arccosh}(c x)}{x}dx\right )-\frac {\left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))}{2 x^2}-\frac {1}{2} b c \left (c \text {arccosh}(c x)-\frac {\sqrt {c x-1} \sqrt {c x+1}}{x}\right )\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 6297 |
| \(\displaystyle c^2 (-d) \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{x^2}dx+\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {c^2 \int -\left ((a+b \text {arccosh}(c x)) \tanh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )\right )d(a+b \text {arccosh}(c x))}{b}-\frac {\left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))}{2 x^2}-\frac {1}{2} b c \left (c \text {arccosh}(c x)-\frac {\sqrt {c x-1} \sqrt {c x+1}}{x}\right )\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle c^2 (-d) \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{x^2}dx+\frac {2 b c d \sqrt {d-c^2 d x^2} \left (\frac {c^2 \int (a+b \text {arccosh}(c x)) \tanh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )d(a+b \text {arccosh}(c x))}{b}-\frac {\left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))}{2 x^2}-\frac {1}{2} b c \left (c \text {arccosh}(c x)-\frac {\sqrt {c x-1} \sqrt {c x+1}}{x}\right )\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle c^2 (-d) \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{x^2}dx+\frac {2 b c d \sqrt {d-c^2 d x^2} \left (\frac {c^2 \int -i (a+b \text {arccosh}(c x)) \tan \left (\frac {i a}{b}-\frac {i (a+b \text {arccosh}(c x))}{b}\right )d(a+b \text {arccosh}(c x))}{b}-\frac {\left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))}{2 x^2}-\frac {1}{2} b c \left (c \text {arccosh}(c x)-\frac {\sqrt {c x-1} \sqrt {c x+1}}{x}\right )\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle c^2 (-d) \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{x^2}dx+\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {i c^2 \int (a+b \text {arccosh}(c x)) \tan \left (\frac {i a}{b}-\frac {i (a+b \text {arccosh}(c x))}{b}\right )d(a+b \text {arccosh}(c x))}{b}-\frac {\left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))}{2 x^2}-\frac {1}{2} b c \left (c \text {arccosh}(c x)-\frac {\sqrt {c x-1} \sqrt {c x+1}}{x}\right )\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 4201 |
| \(\displaystyle c^2 (-d) \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{x^2}dx+\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {i c^2 \left (2 i \int \frac {e^{-2 \text {arccosh}(c x)} (a+b \text {arccosh}(c x))}{1+e^{-2 \text {arccosh}(c x)}}d(a+b \text {arccosh}(c x))-\frac {1}{2} i (a+b \text {arccosh}(c x))^2\right )}{b}-\frac {\left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))}{2 x^2}-\frac {1}{2} b c \left (c \text {arccosh}(c x)-\frac {\sqrt {c x-1} \sqrt {c x+1}}{x}\right )\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle c^2 (-d) \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{x^2}dx+\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {i c^2 \left (2 i \left (\frac {1}{2} b \int \log \left (1+e^{-2 \text {arccosh}(c x)}\right )d(a+b \text {arccosh}(c x))-\frac {1}{2} b \log \left (e^{-2 \text {arccosh}(c x)}+1\right ) (a+b \text {arccosh}(c x))\right )-\frac {1}{2} i (a+b \text {arccosh}(c x))^2\right )}{b}-\frac {\left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))}{2 x^2}-\frac {1}{2} b c \left (c \text {arccosh}(c x)-\frac {\sqrt {c x-1} \sqrt {c x+1}}{x}\right )\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle \frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {i c^2 \left (2 i \left (-\frac {1}{4} b^2 \int e^{2 \text {arccosh}(c x)} \log \left (1+e^{-2 \text {arccosh}(c x)}\right )de^{-2 \text {arccosh}(c x)}-\frac {1}{2} b \log \left (e^{-2 \text {arccosh}(c x)}+1\right ) (a+b \text {arccosh}(c x))\right )-\frac {1}{2} i (a+b \text {arccosh}(c x))^2\right )}{b}-\frac {\left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))}{2 x^2}-\frac {1}{2} b c \left (c \text {arccosh}(c x)-\frac {\sqrt {c x-1} \sqrt {c x+1}}{x}\right )\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}+c^2 (-d) \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{x^2}dx-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle c^2 (-d) \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{x^2}dx+\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {i c^2 \left (2 i \left (\frac {1}{4} b^2 \operatorname {PolyLog}(2,-a-b \text {arccosh}(c x))-\frac {1}{2} b \log \left (e^{-2 \text {arccosh}(c x)}+1\right ) (a+b \text {arccosh}(c x))\right )-\frac {1}{2} i (a+b \text {arccosh}(c x))^2\right )}{b}-\frac {\left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))}{2 x^2}-\frac {1}{2} b c \left (c \text {arccosh}(c x)-\frac {\sqrt {c x-1} \sqrt {c x+1}}{x}\right )\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 6339 |
| \(\displaystyle c^2 (-d) \left (\frac {c^2 \sqrt {d-c^2 d x^2} \int \frac {(a+b \text {arccosh}(c x))^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{\sqrt {c x-1} \sqrt {c x+1}}+\frac {2 b c \sqrt {d-c^2 d x^2} \int \frac {a+b \text {arccosh}(c x)}{x}dx}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{x}\right )+\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {i c^2 \left (2 i \left (\frac {1}{4} b^2 \operatorname {PolyLog}(2,-a-b \text {arccosh}(c x))-\frac {1}{2} b \log \left (e^{-2 \text {arccosh}(c x)}+1\right ) (a+b \text {arccosh}(c x))\right )-\frac {1}{2} i (a+b \text {arccosh}(c x))^2\right )}{b}-\frac {\left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))}{2 x^2}-\frac {1}{2} b c \left (c \text {arccosh}(c x)-\frac {\sqrt {c x-1} \sqrt {c x+1}}{x}\right )\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 6297 |
| \(\displaystyle c^2 (-d) \left (\frac {c^2 \sqrt {d-c^2 d x^2} \int \frac {(a+b \text {arccosh}(c x))^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{\sqrt {c x-1} \sqrt {c x+1}}+\frac {2 c \sqrt {d-c^2 d x^2} \int -\left ((a+b \text {arccosh}(c x)) \tanh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )\right )d(a+b \text {arccosh}(c x))}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{x}\right )+\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {i c^2 \left (2 i \left (\frac {1}{4} b^2 \operatorname {PolyLog}(2,-a-b \text {arccosh}(c x))-\frac {1}{2} b \log \left (e^{-2 \text {arccosh}(c x)}+1\right ) (a+b \text {arccosh}(c x))\right )-\frac {1}{2} i (a+b \text {arccosh}(c x))^2\right )}{b}-\frac {\left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))}{2 x^2}-\frac {1}{2} b c \left (c \text {arccosh}(c x)-\frac {\sqrt {c x-1} \sqrt {c x+1}}{x}\right )\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle c^2 (-d) \left (\frac {c^2 \sqrt {d-c^2 d x^2} \int \frac {(a+b \text {arccosh}(c x))^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {2 c \sqrt {d-c^2 d x^2} \int (a+b \text {arccosh}(c x)) \tanh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )d(a+b \text {arccosh}(c x))}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{x}\right )+\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {i c^2 \left (2 i \left (\frac {1}{4} b^2 \operatorname {PolyLog}(2,-a-b \text {arccosh}(c x))-\frac {1}{2} b \log \left (e^{-2 \text {arccosh}(c x)}+1\right ) (a+b \text {arccosh}(c x))\right )-\frac {1}{2} i (a+b \text {arccosh}(c x))^2\right )}{b}-\frac {\left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))}{2 x^2}-\frac {1}{2} b c \left (c \text {arccosh}(c x)-\frac {\sqrt {c x-1} \sqrt {c x+1}}{x}\right )\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle c^2 (-d) \left (\frac {c^2 \sqrt {d-c^2 d x^2} \int \frac {(a+b \text {arccosh}(c x))^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {2 c \sqrt {d-c^2 d x^2} \int -i (a+b \text {arccosh}(c x)) \tan \left (\frac {i a}{b}-\frac {i (a+b \text {arccosh}(c x))}{b}\right )d(a+b \text {arccosh}(c x))}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{x}\right )+\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {i c^2 \left (2 i \left (\frac {1}{4} b^2 \operatorname {PolyLog}(2,-a-b \text {arccosh}(c x))-\frac {1}{2} b \log \left (e^{-2 \text {arccosh}(c x)}+1\right ) (a+b \text {arccosh}(c x))\right )-\frac {1}{2} i (a+b \text {arccosh}(c x))^2\right )}{b}-\frac {\left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))}{2 x^2}-\frac {1}{2} b c \left (c \text {arccosh}(c x)-\frac {\sqrt {c x-1} \sqrt {c x+1}}{x}\right )\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle c^2 (-d) \left (\frac {c^2 \sqrt {d-c^2 d x^2} \int \frac {(a+b \text {arccosh}(c x))^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{\sqrt {c x-1} \sqrt {c x+1}}+\frac {2 i c \sqrt {d-c^2 d x^2} \int (a+b \text {arccosh}(c x)) \tan \left (\frac {i a}{b}-\frac {i (a+b \text {arccosh}(c x))}{b}\right )d(a+b \text {arccosh}(c x))}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{x}\right )+\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {i c^2 \left (2 i \left (\frac {1}{4} b^2 \operatorname {PolyLog}(2,-a-b \text {arccosh}(c x))-\frac {1}{2} b \log \left (e^{-2 \text {arccosh}(c x)}+1\right ) (a+b \text {arccosh}(c x))\right )-\frac {1}{2} i (a+b \text {arccosh}(c x))^2\right )}{b}-\frac {\left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))}{2 x^2}-\frac {1}{2} b c \left (c \text {arccosh}(c x)-\frac {\sqrt {c x-1} \sqrt {c x+1}}{x}\right )\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 4201 |
| \(\displaystyle c^2 (-d) \left (\frac {c^2 \sqrt {d-c^2 d x^2} \int \frac {(a+b \text {arccosh}(c x))^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{\sqrt {c x-1} \sqrt {c x+1}}+\frac {2 i c \sqrt {d-c^2 d x^2} \left (2 i \int \frac {e^{-2 \text {arccosh}(c x)} (a+b \text {arccosh}(c x))}{1+e^{-2 \text {arccosh}(c x)}}d(a+b \text {arccosh}(c x))-\frac {1}{2} i (a+b \text {arccosh}(c x))^2\right )}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{x}\right )+\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {i c^2 \left (2 i \left (\frac {1}{4} b^2 \operatorname {PolyLog}(2,-a-b \text {arccosh}(c x))-\frac {1}{2} b \log \left (e^{-2 \text {arccosh}(c x)}+1\right ) (a+b \text {arccosh}(c x))\right )-\frac {1}{2} i (a+b \text {arccosh}(c x))^2\right )}{b}-\frac {\left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))}{2 x^2}-\frac {1}{2} b c \left (c \text {arccosh}(c x)-\frac {\sqrt {c x-1} \sqrt {c x+1}}{x}\right )\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle c^2 (-d) \left (\frac {c^2 \sqrt {d-c^2 d x^2} \int \frac {(a+b \text {arccosh}(c x))^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{\sqrt {c x-1} \sqrt {c x+1}}+\frac {2 i c \sqrt {d-c^2 d x^2} \left (2 i \left (\frac {1}{2} b \int \log \left (1+e^{-2 \text {arccosh}(c x)}\right )d(a+b \text {arccosh}(c x))-\frac {1}{2} b \log \left (e^{-2 \text {arccosh}(c x)}+1\right ) (a+b \text {arccosh}(c x))\right )-\frac {1}{2} i (a+b \text {arccosh}(c x))^2\right )}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{x}\right )+\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {i c^2 \left (2 i \left (\frac {1}{4} b^2 \operatorname {PolyLog}(2,-a-b \text {arccosh}(c x))-\frac {1}{2} b \log \left (e^{-2 \text {arccosh}(c x)}+1\right ) (a+b \text {arccosh}(c x))\right )-\frac {1}{2} i (a+b \text {arccosh}(c x))^2\right )}{b}-\frac {\left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))}{2 x^2}-\frac {1}{2} b c \left (c \text {arccosh}(c x)-\frac {\sqrt {c x-1} \sqrt {c x+1}}{x}\right )\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle c^2 (-d) \left (\frac {2 i c \sqrt {d-c^2 d x^2} \left (2 i \left (-\frac {1}{4} b^2 \int e^{2 \text {arccosh}(c x)} \log \left (1+e^{-2 \text {arccosh}(c x)}\right )de^{-2 \text {arccosh}(c x)}-\frac {1}{2} b \log \left (e^{-2 \text {arccosh}(c x)}+1\right ) (a+b \text {arccosh}(c x))\right )-\frac {1}{2} i (a+b \text {arccosh}(c x))^2\right )}{\sqrt {c x-1} \sqrt {c x+1}}+\frac {c^2 \sqrt {d-c^2 d x^2} \int \frac {(a+b \text {arccosh}(c x))^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{x}\right )+\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {i c^2 \left (2 i \left (\frac {1}{4} b^2 \operatorname {PolyLog}(2,-a-b \text {arccosh}(c x))-\frac {1}{2} b \log \left (e^{-2 \text {arccosh}(c x)}+1\right ) (a+b \text {arccosh}(c x))\right )-\frac {1}{2} i (a+b \text {arccosh}(c x))^2\right )}{b}-\frac {\left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))}{2 x^2}-\frac {1}{2} b c \left (c \text {arccosh}(c x)-\frac {\sqrt {c x-1} \sqrt {c x+1}}{x}\right )\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle c^2 (-d) \left (\frac {c^2 \sqrt {d-c^2 d x^2} \int \frac {(a+b \text {arccosh}(c x))^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{\sqrt {c x-1} \sqrt {c x+1}}+\frac {2 i c \sqrt {d-c^2 d x^2} \left (2 i \left (\frac {1}{4} b^2 \operatorname {PolyLog}(2,-a-b \text {arccosh}(c x))-\frac {1}{2} b \log \left (e^{-2 \text {arccosh}(c x)}+1\right ) (a+b \text {arccosh}(c x))\right )-\frac {1}{2} i (a+b \text {arccosh}(c x))^2\right )}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{x}\right )+\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {i c^2 \left (2 i \left (\frac {1}{4} b^2 \operatorname {PolyLog}(2,-a-b \text {arccosh}(c x))-\frac {1}{2} b \log \left (e^{-2 \text {arccosh}(c x)}+1\right ) (a+b \text {arccosh}(c x))\right )-\frac {1}{2} i (a+b \text {arccosh}(c x))^2\right )}{b}-\frac {\left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))}{2 x^2}-\frac {1}{2} b c \left (c \text {arccosh}(c x)-\frac {\sqrt {c x-1} \sqrt {c x+1}}{x}\right )\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 6308 |
| \(\displaystyle c^2 (-d) \left (\frac {2 i c \sqrt {d-c^2 d x^2} \left (2 i \left (\frac {1}{4} b^2 \operatorname {PolyLog}(2,-a-b \text {arccosh}(c x))-\frac {1}{2} b \log \left (e^{-2 \text {arccosh}(c x)}+1\right ) (a+b \text {arccosh}(c x))\right )-\frac {1}{2} i (a+b \text {arccosh}(c x))^2\right )}{\sqrt {c x-1} \sqrt {c x+1}}+\frac {c \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^3}{3 b \sqrt {c x-1} \sqrt {c x+1}}-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{x}\right )+\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {i c^2 \left (2 i \left (\frac {1}{4} b^2 \operatorname {PolyLog}(2,-a-b \text {arccosh}(c x))-\frac {1}{2} b \log \left (e^{-2 \text {arccosh}(c x)}+1\right ) (a+b \text {arccosh}(c x))\right )-\frac {1}{2} i (a+b \text {arccosh}(c x))^2\right )}{b}-\frac {\left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))}{2 x^2}-\frac {1}{2} b c \left (c \text {arccosh}(c x)-\frac {\sqrt {c x-1} \sqrt {c x+1}}{x}\right )\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{3 x^3}\) |
-1/3*((d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x])^2)/x^3 + (2*b*c*d*Sqrt[d - c^2*d*x^2]*(-1/2*((1 - c^2*x^2)*(a + b*ArcCosh[c*x]))/x^2 - (b*c*(-((Sqr t[-1 + c*x]*Sqrt[1 + c*x])/x) + c*ArcCosh[c*x]))/2 - (I*c^2*((-1/2*I)*(a + b*ArcCosh[c*x])^2 + (2*I)*(-1/2*(b*(a + b*ArcCosh[c*x])*Log[1 + E^(-2*Arc Cosh[c*x])]) + (b^2*PolyLog[2, -a - b*ArcCosh[c*x]])/4)))/b))/(3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - c^2*d*(-((Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^ 2)/x) + (c*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^3)/(3*b*Sqrt[-1 + c*x] *Sqrt[1 + c*x]) + ((2*I)*c*Sqrt[d - c^2*d*x^2]*((-1/2*I)*(a + b*ArcCosh[c* x])^2 + (2*I)*(-1/2*(b*(a + b*ArcCosh[c*x])*Log[1 + E^(-2*ArcCosh[c*x])]) + (b^2*PolyLog[2, -a - b*ArcCosh[c*x]])/4)))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x] ))
3.2.85.3.1 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[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ ArcCosh[b*(x/a)]/(b*Sqrt[d/b]), x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c + a *d, 0] && GtQ[a, 0] && GtQ[d/b, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^p/(b*(m + 1))) , x] - Simp[1/(b*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f* x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c , d, e, f}, x] && LtQ[m, -1] && GtQ[n, 0] && GtQ[p, 0] && (IntegersQ[2*m, 2 *n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])
Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ ((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp [((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si mp[d*(m/(b*f*g*n*Log[F])) Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x )))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 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[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (Complex[0, fz_])*(f_.)*(x_)], x _Symbol] :> Simp[(-I)*((c + d*x)^(m + 1)/(d*(m + 1))), x] + Simp[2*I Int[ (c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*fz*x)))), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> Simp[1/b Subst[Int[x^n*Tanh[-a/b + x/b], x], x, a + b*ArcCosh[c*x]], x] /; FreeQ[{a , b, c}, x] && IGtQ[n, 0]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(Sqrt[(d1_) + (e1_.)*(x_)]*Sq rt[(d2_) + (e2_.)*(x_)]), x_Symbol] :> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 + c*x]/Sqrt[d1 + e1*x]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]]*(a + b*ArcCosh[ c*x])^(n + 1), x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[e1, c*d1 ] && EqQ[e2, (-c)*d2] && NeQ[n, -1]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_.)*((d1_) + ( e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(x_))^(p_.), x_Symbol] :> Int[(f*x)^m*(d1 *d2 + e1*e2*x^2)^p*(a + b*ArcCosh[c*x])^n, x] /; FreeQ[{a, b, c, d1, e1, d2 , e2, f, m, n}, x] && EqQ[d2*e1 + d1*e2, 0] && IntegerQ[p]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_ )^2)^(p_.), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^p*((a + b*ArcCosh[c *x])/(f*(m + 1))), x] + (-Simp[b*c*((-d)^p/(f*(m + 1))) Int[(f*x)^(m + 1) *(1 + c*x)^(p - 1/2)*(-1 + c*x)^(p - 1/2), x], 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]), x], x]) / ; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0] && ILtQ[( m + 1)/2, 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 + 1))), x] + (-Simp[b*c*(n/(f*(m + 1)))*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] - Simp[(c^2/(f^2*(m + 1)))*Simp[Sqrt[d + e*x^2]/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])] Int[(f*x)^(m + 2)*((a + b*ArcCosh[c*x])^n/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^ 2*d + e, 0] && GtQ[n, 0] && LtQ[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*((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]
Time = 1.23 (sec) , antiderivative size = 500, normalized size of antiderivative = 1.17
| method | result | size |
| default | \(-\frac {a^{2} \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{3 d \,x^{3}}+\frac {2 a^{2} c^{2} \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{3 d x}+\frac {2 a^{2} c^{4} x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{3}+a^{2} c^{4} d x \sqrt {-c^{2} d \,x^{2}+d}+\frac {a^{2} c^{4} d^{2} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{\sqrt {c^{2} d}}-\frac {b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (\operatorname {arccosh}\left (c x \right )^{3} x^{3} c^{3}-4 \operatorname {arccosh}\left (c x \right )^{2} \sqrt {c x +1}\, \sqrt {c x -1}\, x^{2} c^{2}-4 \operatorname {arccosh}\left (c x \right )^{2} x^{3} c^{3}+8 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) x^{3} c^{3}+4 \operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) x^{3} c^{3}-\sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}+c^{3} x^{3}+\sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right )^{2}+c x \,\operatorname {arccosh}\left (c x \right )\right ) d}{3 \sqrt {c x -1}\, \sqrt {c x +1}\, x^{3}}-\frac {a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (3 \operatorname {arccosh}\left (c x \right )^{2} x^{3} c^{3}-8 \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, c^{2} x^{2}-8 c^{3} x^{3} \operatorname {arccosh}\left (c x \right )+8 \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) x^{3} c^{3}+2 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}+c x \right ) d}{3 \sqrt {c x -1}\, \sqrt {c x +1}\, x^{3}}\) | \(500\) |
| parts | \(-\frac {a^{2} \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{3 d \,x^{3}}+\frac {2 a^{2} c^{2} \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{3 d x}+\frac {2 a^{2} c^{4} x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{3}+a^{2} c^{4} d x \sqrt {-c^{2} d \,x^{2}+d}+\frac {a^{2} c^{4} d^{2} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{\sqrt {c^{2} d}}-\frac {b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (\operatorname {arccosh}\left (c x \right )^{3} x^{3} c^{3}-4 \operatorname {arccosh}\left (c x \right )^{2} \sqrt {c x +1}\, \sqrt {c x -1}\, x^{2} c^{2}-4 \operatorname {arccosh}\left (c x \right )^{2} x^{3} c^{3}+8 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) x^{3} c^{3}+4 \operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) x^{3} c^{3}-\sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}+c^{3} x^{3}+\sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right )^{2}+c x \,\operatorname {arccosh}\left (c x \right )\right ) d}{3 \sqrt {c x -1}\, \sqrt {c x +1}\, x^{3}}-\frac {a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (3 \operatorname {arccosh}\left (c x \right )^{2} x^{3} c^{3}-8 \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, c^{2} x^{2}-8 c^{3} x^{3} \operatorname {arccosh}\left (c x \right )+8 \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) x^{3} c^{3}+2 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}+c x \right ) d}{3 \sqrt {c x -1}\, \sqrt {c x +1}\, x^{3}}\) | \(500\) |
-1/3*a^2/d/x^3*(-c^2*d*x^2+d)^(5/2)+2/3*a^2*c^2/d/x*(-c^2*d*x^2+d)^(5/2)+2 /3*a^2*c^4*x*(-c^2*d*x^2+d)^(3/2)+a^2*c^4*d*x*(-c^2*d*x^2+d)^(1/2)+a^2*c^4 *d^2/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))-1/3*b^2*(- d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/x^3*(arccosh(c*x)^3*x^3*c ^3-4*arccosh(c*x)^2*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^2*c^2-4*arccosh(c*x)^2*x ^3*c^3+8*arccosh(c*x)*ln(1+(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2)*x^3*c^3+4* polylog(2,-(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2)*x^3*c^3-(c*x-1)^(1/2)*(c*x +1)^(1/2)*c^2*x^2+c^3*x^3+(c*x-1)^(1/2)*(c*x+1)^(1/2)*arccosh(c*x)^2+c*x*a rccosh(c*x))*d-1/3*a*b*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/ x^3*(3*arccosh(c*x)^2*x^3*c^3-8*(c*x+1)^(1/2)*arccosh(c*x)*(c*x-1)^(1/2)*c ^2*x^2-8*c^3*x^3*arccosh(c*x)+8*ln(1+(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2)* x^3*c^3+2*arccosh(c*x)*(c*x-1)^(1/2)*(c*x+1)^(1/2)+c*x)*d
\[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{x^4} \, dx=\int { \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}}{x^{4}} \,d x } \]
integral(-(a^2*c^2*d*x^2 - a^2*d + (b^2*c^2*d*x^2 - b^2*d)*arccosh(c*x)^2 + 2*(a*b*c^2*d*x^2 - a*b*d)*arccosh(c*x))*sqrt(-c^2*d*x^2 + d)/x^4, x)
\[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{x^4} \, dx=\int \frac {\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}}{x^{4}}\, dx \]
\[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{x^4} \, dx=\int { \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}}{x^{4}} \,d x } \]
1/3*(3*sqrt(-c^2*d*x^2 + d)*c^4*d*x + 3*c^3*d^(3/2)*arcsin(c*x) + 2*(-c^2* d*x^2 + d)^(3/2)*c^2/x - (-c^2*d*x^2 + d)^(5/2)/(d*x^3))*a^2 + integrate(( -c^2*d*x^2 + d)^(3/2)*b^2*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))^2/x^4 + 2 *(-c^2*d*x^2 + d)^(3/2)*a*b*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))/x^4, x)
Exception generated. \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{x^4} \, dx=\text {Exception raised: TypeError} \]
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. \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{x^4} \, dx=\int \frac {{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^{3/2}}{x^4} \,d x \]