Integrand size = 29, antiderivative size = 440 \[ \int \frac {x^4 (a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {b^2 x (1-c x) (1+c x)}{4 c^4 d \sqrt {d-c^2 d x^2}}-\frac {b^2 \sqrt {-1+c x} \sqrt {1+c x} \text {arccosh}(c x)}{4 c^5 d \sqrt {d-c^2 d x^2}}+\frac {b x^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{2 c^3 d \sqrt {d-c^2 d x^2}}+\frac {x^3 (a+b \text {arccosh}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {\sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^2}{c^5 d \sqrt {d-c^2 d x^2}}+\frac {3 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 c^4 d^2}-\frac {\sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^3}{2 b c^5 d \sqrt {d-c^2 d x^2}}-\frac {2 b \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \log \left (1-e^{2 \text {arccosh}(c x)}\right )}{c^5 d \sqrt {d-c^2 d x^2}}-\frac {b^2 \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(c x)}\right )}{c^5 d \sqrt {d-c^2 d x^2}} \] Output:
1/4*b^2*x*(-c*x+1)*(c*x+1)/c^4/d/(-c^2*d*x^2+d)^(1/2)-1/4*b^2*(c*x-1)^(1/2 )*(c*x+1)^(1/2)*arccosh(c*x)/c^5/d/(-c^2*d*x^2+d)^(1/2)+1/2*b*x^2*(c*x-1)^ (1/2)*(c*x+1)^(1/2)*(a+b*arccosh(c*x))/c^3/d/(-c^2*d*x^2+d)^(1/2)+x^3*(a+b *arccosh(c*x))^2/c^2/d/(-c^2*d*x^2+d)^(1/2)+(c*x-1)^(1/2)*(c*x+1)^(1/2)*(a +b*arccosh(c*x))^2/c^5/d/(-c^2*d*x^2+d)^(1/2)+3/2*x*(-c^2*d*x^2+d)^(1/2)*( a+b*arccosh(c*x))^2/c^4/d^2-1/2*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(a+b*arccosh(c *x))^3/b/c^5/d/(-c^2*d*x^2+d)^(1/2)-2*b*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(a+b*a rccosh(c*x))*ln(1-(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2)/c^5/d/(-c^2*d*x^2+d )^(1/2)-b^2*(c*x-1)^(1/2)*(c*x+1)^(1/2)*polylog(2,(c*x+(c*x-1)^(1/2)*(c*x+ 1)^(1/2))^2)/c^5/d/(-c^2*d*x^2+d)^(1/2)
Time = 1.58 (sec) , antiderivative size = 343, normalized size of antiderivative = 0.78 \[ \int \frac {x^4 (a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {-4 a^2 c \sqrt {d} x \left (-3+c^2 x^2\right )+12 a^2 \sqrt {d-c^2 d x^2} \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )+2 a b \sqrt {d} \left (8 c x \text {arccosh}(c x)-\sqrt {\frac {-1+c x}{1+c x}} (1+c x) \left (6 \text {arccosh}(c x)^2-\cosh (2 \text {arccosh}(c x))+8 \log \left (\sqrt {\frac {-1+c x}{1+c x}} (1+c x)\right )+2 \text {arccosh}(c x) \sinh (2 \text {arccosh}(c x))\right )\right )+b^2 \sqrt {d} \left (8 c x \text {arccosh}(c x)^2+8 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \operatorname {PolyLog}\left (2,e^{-2 \text {arccosh}(c x)}\right )-\sqrt {\frac {-1+c x}{1+c x}} (1+c x) \left (4 \text {arccosh}(c x)^3-2 \text {arccosh}(c x) \left (\cosh (2 \text {arccosh}(c x))-8 \log \left (1-e^{-2 \text {arccosh}(c x)}\right )\right )+\sinh (2 \text {arccosh}(c x))+2 \text {arccosh}(c x)^2 (4+\sinh (2 \text {arccosh}(c x)))\right )\right )}{8 c^5 d^{3/2} \sqrt {d-c^2 d x^2}} \] Input:
Integrate[(x^4*(a + b*ArcCosh[c*x])^2)/(d - c^2*d*x^2)^(3/2),x]
Output:
(-4*a^2*c*Sqrt[d]*x*(-3 + c^2*x^2) + 12*a^2*Sqrt[d - c^2*d*x^2]*ArcTan[(c* x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] + 2*a*b*Sqrt[d]*(8*c*x*Ar cCosh[c*x] - Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*(6*ArcCosh[c*x]^2 - Cosh [2*ArcCosh[c*x]] + 8*Log[Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)] + 2*ArcCosh [c*x]*Sinh[2*ArcCosh[c*x]])) + b^2*Sqrt[d]*(8*c*x*ArcCosh[c*x]^2 + 8*Sqrt[ (-1 + c*x)/(1 + c*x)]*(1 + c*x)*PolyLog[2, E^(-2*ArcCosh[c*x])] - Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*(4*ArcCosh[c*x]^3 - 2*ArcCosh[c*x]*(Cosh[2*Ar cCosh[c*x]] - 8*Log[1 - E^(-2*ArcCosh[c*x])]) + Sinh[2*ArcCosh[c*x]] + 2*A rcCosh[c*x]^2*(4 + Sinh[2*ArcCosh[c*x]]))))/(8*c^5*d^(3/2)*Sqrt[d - c^2*d* x^2])
Result contains complex when optimal does not.
Time = 2.53 (sec) , antiderivative size = 412, normalized size of antiderivative = 0.94, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.621, Rules used = {6349, 25, 6327, 6353, 101, 43, 6298, 101, 43, 6307, 6328, 3042, 26, 4199, 25, 2620, 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.
| \(\displaystyle \int \frac {x^4 (a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 6349 |
| \(\displaystyle -\frac {3 \int \frac {x^2 (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}}dx}{c^2 d}-\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \int -\frac {x^3 (a+b \text {arccosh}(c x))}{(1-c x) (c x+1)}dx}{c d \sqrt {d-c^2 d x^2}}+\frac {x^3 (a+b \text {arccosh}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {3 \int \frac {x^2 (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}}dx}{c^2 d}+\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \int \frac {x^3 (a+b \text {arccosh}(c x))}{(1-c x) (c x+1)}dx}{c d \sqrt {d-c^2 d x^2}}+\frac {x^3 (a+b \text {arccosh}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 6327 |
| \(\displaystyle -\frac {3 \int \frac {x^2 (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}}dx}{c^2 d}+\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \int \frac {x^3 (a+b \text {arccosh}(c x))}{1-c^2 x^2}dx}{c d \sqrt {d-c^2 d x^2}}+\frac {x^3 (a+b \text {arccosh}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 6353 |
| \(\displaystyle \frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (\frac {\int \frac {x (a+b \text {arccosh}(c x))}{1-c^2 x^2}dx}{c^2}+\frac {b \int \frac {x^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{2 c}-\frac {x^2 (a+b \text {arccosh}(c x))}{2 c^2}\right )}{c d \sqrt {d-c^2 d x^2}}-\frac {3 \left (-\frac {b \sqrt {c x-1} \sqrt {c x+1} \int x (a+b \text {arccosh}(c x))dx}{c \sqrt {d-c^2 d x^2}}+\frac {\int \frac {(a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}}dx}{2 c^2}-\frac {x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 c^2 d}\right )}{c^2 d}+\frac {x^3 (a+b \text {arccosh}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 101 |
| \(\displaystyle \frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (\frac {\int \frac {x (a+b \text {arccosh}(c x))}{1-c^2 x^2}dx}{c^2}+\frac {b \left (\frac {\int \frac {1}{\sqrt {c x-1} \sqrt {c x+1}}dx}{2 c^2}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )}{2 c}-\frac {x^2 (a+b \text {arccosh}(c x))}{2 c^2}\right )}{c d \sqrt {d-c^2 d x^2}}-\frac {3 \left (-\frac {b \sqrt {c x-1} \sqrt {c x+1} \int x (a+b \text {arccosh}(c x))dx}{c \sqrt {d-c^2 d x^2}}+\frac {\int \frac {(a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}}dx}{2 c^2}-\frac {x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 c^2 d}\right )}{c^2 d}+\frac {x^3 (a+b \text {arccosh}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 43 |
| \(\displaystyle -\frac {3 \left (-\frac {b \sqrt {c x-1} \sqrt {c x+1} \int x (a+b \text {arccosh}(c x))dx}{c \sqrt {d-c^2 d x^2}}+\frac {\int \frac {(a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}}dx}{2 c^2}-\frac {x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 c^2 d}\right )}{c^2 d}+\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (\frac {\int \frac {x (a+b \text {arccosh}(c x))}{1-c^2 x^2}dx}{c^2}-\frac {x^2 (a+b \text {arccosh}(c x))}{2 c^2}+\frac {b \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )}{2 c}\right )}{c d \sqrt {d-c^2 d x^2}}+\frac {x^3 (a+b \text {arccosh}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 6298 |
| \(\displaystyle -\frac {3 \left (-\frac {b \sqrt {c x-1} \sqrt {c x+1} \left (\frac {1}{2} x^2 (a+b \text {arccosh}(c x))-\frac {1}{2} b c \int \frac {x^2}{\sqrt {c x-1} \sqrt {c x+1}}dx\right )}{c \sqrt {d-c^2 d x^2}}+\frac {\int \frac {(a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}}dx}{2 c^2}-\frac {x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 c^2 d}\right )}{c^2 d}+\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (\frac {\int \frac {x (a+b \text {arccosh}(c x))}{1-c^2 x^2}dx}{c^2}-\frac {x^2 (a+b \text {arccosh}(c x))}{2 c^2}+\frac {b \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )}{2 c}\right )}{c d \sqrt {d-c^2 d x^2}}+\frac {x^3 (a+b \text {arccosh}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 101 |
| \(\displaystyle -\frac {3 \left (-\frac {b \sqrt {c x-1} \sqrt {c x+1} \left (\frac {1}{2} x^2 (a+b \text {arccosh}(c x))-\frac {1}{2} b c \left (\frac {\int \frac {1}{\sqrt {c x-1} \sqrt {c x+1}}dx}{2 c^2}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )\right )}{c \sqrt {d-c^2 d x^2}}+\frac {\int \frac {(a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}}dx}{2 c^2}-\frac {x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 c^2 d}\right )}{c^2 d}+\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (\frac {\int \frac {x (a+b \text {arccosh}(c x))}{1-c^2 x^2}dx}{c^2}-\frac {x^2 (a+b \text {arccosh}(c x))}{2 c^2}+\frac {b \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )}{2 c}\right )}{c d \sqrt {d-c^2 d x^2}}+\frac {x^3 (a+b \text {arccosh}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 43 |
| \(\displaystyle \frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (\frac {\int \frac {x (a+b \text {arccosh}(c x))}{1-c^2 x^2}dx}{c^2}-\frac {x^2 (a+b \text {arccosh}(c x))}{2 c^2}+\frac {b \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )}{2 c}\right )}{c d \sqrt {d-c^2 d x^2}}-\frac {3 \left (\frac {\int \frac {(a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}}dx}{2 c^2}-\frac {x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 c^2 d}-\frac {b \sqrt {c x-1} \sqrt {c x+1} \left (\frac {1}{2} x^2 (a+b \text {arccosh}(c x))-\frac {1}{2} b c \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )\right )}{c \sqrt {d-c^2 d x^2}}\right )}{c^2 d}+\frac {x^3 (a+b \text {arccosh}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 6307 |
| \(\displaystyle \frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (\frac {\int \frac {x (a+b \text {arccosh}(c x))}{1-c^2 x^2}dx}{c^2}-\frac {x^2 (a+b \text {arccosh}(c x))}{2 c^2}+\frac {b \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )}{2 c}\right )}{c d \sqrt {d-c^2 d x^2}}+\frac {x^3 (a+b \text {arccosh}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {3 \left (-\frac {x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 c^2 d}+\frac {\sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^3}{6 b c^3 \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {c x-1} \sqrt {c x+1} \left (\frac {1}{2} x^2 (a+b \text {arccosh}(c x))-\frac {1}{2} b c \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )\right )}{c \sqrt {d-c^2 d x^2}}\right )}{c^2 d}\) |
| \(\Big \downarrow \) 6328 |
| \(\displaystyle \frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (-\frac {\int \frac {c x (a+b \text {arccosh}(c x))}{\sqrt {\frac {c x-1}{c x+1}} (c x+1)}d\text {arccosh}(c x)}{c^4}-\frac {x^2 (a+b \text {arccosh}(c x))}{2 c^2}+\frac {b \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )}{2 c}\right )}{c d \sqrt {d-c^2 d x^2}}+\frac {x^3 (a+b \text {arccosh}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {3 \left (-\frac {x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 c^2 d}+\frac {\sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^3}{6 b c^3 \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {c x-1} \sqrt {c x+1} \left (\frac {1}{2} x^2 (a+b \text {arccosh}(c x))-\frac {1}{2} b c \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )\right )}{c \sqrt {d-c^2 d x^2}}\right )}{c^2 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (-\frac {\int -i (a+b \text {arccosh}(c x)) \tan \left (i \text {arccosh}(c x)+\frac {\pi }{2}\right )d\text {arccosh}(c x)}{c^4}-\frac {x^2 (a+b \text {arccosh}(c x))}{2 c^2}+\frac {b \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )}{2 c}\right )}{c d \sqrt {d-c^2 d x^2}}+\frac {x^3 (a+b \text {arccosh}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {3 \left (-\frac {x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 c^2 d}+\frac {\sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^3}{6 b c^3 \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {c x-1} \sqrt {c x+1} \left (\frac {1}{2} x^2 (a+b \text {arccosh}(c x))-\frac {1}{2} b c \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )\right )}{c \sqrt {d-c^2 d x^2}}\right )}{c^2 d}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (\frac {i \int (a+b \text {arccosh}(c x)) \tan \left (i \text {arccosh}(c x)+\frac {\pi }{2}\right )d\text {arccosh}(c x)}{c^4}-\frac {x^2 (a+b \text {arccosh}(c x))}{2 c^2}+\frac {b \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )}{2 c}\right )}{c d \sqrt {d-c^2 d x^2}}+\frac {x^3 (a+b \text {arccosh}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {3 \left (-\frac {x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 c^2 d}+\frac {\sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^3}{6 b c^3 \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {c x-1} \sqrt {c x+1} \left (\frac {1}{2} x^2 (a+b \text {arccosh}(c x))-\frac {1}{2} b c \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )\right )}{c \sqrt {d-c^2 d x^2}}\right )}{c^2 d}\) |
| \(\Big \downarrow \) 4199 |
| \(\displaystyle \frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (\frac {i \left (2 i \int -\frac {e^{2 \text {arccosh}(c x)} (a+b \text {arccosh}(c x))}{1-e^{2 \text {arccosh}(c x)}}d\text {arccosh}(c x)-\frac {i (a+b \text {arccosh}(c x))^2}{2 b}\right )}{c^4}-\frac {x^2 (a+b \text {arccosh}(c x))}{2 c^2}+\frac {b \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )}{2 c}\right )}{c d \sqrt {d-c^2 d x^2}}+\frac {x^3 (a+b \text {arccosh}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {3 \left (-\frac {x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 c^2 d}+\frac {\sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^3}{6 b c^3 \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {c x-1} \sqrt {c x+1} \left (\frac {1}{2} x^2 (a+b \text {arccosh}(c x))-\frac {1}{2} b c \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )\right )}{c \sqrt {d-c^2 d x^2}}\right )}{c^2 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (\frac {i \left (-2 i \int \frac {e^{2 \text {arccosh}(c x)} (a+b \text {arccosh}(c x))}{1-e^{2 \text {arccosh}(c x)}}d\text {arccosh}(c x)-\frac {i (a+b \text {arccosh}(c x))^2}{2 b}\right )}{c^4}-\frac {x^2 (a+b \text {arccosh}(c x))}{2 c^2}+\frac {b \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )}{2 c}\right )}{c d \sqrt {d-c^2 d x^2}}+\frac {x^3 (a+b \text {arccosh}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {3 \left (-\frac {x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 c^2 d}+\frac {\sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^3}{6 b c^3 \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {c x-1} \sqrt {c x+1} \left (\frac {1}{2} x^2 (a+b \text {arccosh}(c x))-\frac {1}{2} b c \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )\right )}{c \sqrt {d-c^2 d x^2}}\right )}{c^2 d}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle \frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (\frac {i \left (-2 i \left (\frac {1}{2} b \int \log \left (1-e^{2 \text {arccosh}(c x)}\right )d\text {arccosh}(c x)-\frac {1}{2} \log \left (1-e^{2 \text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )-\frac {i (a+b \text {arccosh}(c x))^2}{2 b}\right )}{c^4}-\frac {x^2 (a+b \text {arccosh}(c x))}{2 c^2}+\frac {b \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )}{2 c}\right )}{c d \sqrt {d-c^2 d x^2}}+\frac {x^3 (a+b \text {arccosh}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {3 \left (-\frac {x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 c^2 d}+\frac {\sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^3}{6 b c^3 \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {c x-1} \sqrt {c x+1} \left (\frac {1}{2} x^2 (a+b \text {arccosh}(c x))-\frac {1}{2} b c \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )\right )}{c \sqrt {d-c^2 d x^2}}\right )}{c^2 d}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle \frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (\frac {i \left (-2 i \left (\frac {1}{4} b \int e^{-2 \text {arccosh}(c x)} \log \left (1-e^{2 \text {arccosh}(c x)}\right )de^{2 \text {arccosh}(c x)}-\frac {1}{2} \log \left (1-e^{2 \text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )-\frac {i (a+b \text {arccosh}(c x))^2}{2 b}\right )}{c^4}-\frac {x^2 (a+b \text {arccosh}(c x))}{2 c^2}+\frac {b \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )}{2 c}\right )}{c d \sqrt {d-c^2 d x^2}}+\frac {x^3 (a+b \text {arccosh}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {3 \left (-\frac {x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 c^2 d}+\frac {\sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^3}{6 b c^3 \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {c x-1} \sqrt {c x+1} \left (\frac {1}{2} x^2 (a+b \text {arccosh}(c x))-\frac {1}{2} b c \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )\right )}{c \sqrt {d-c^2 d x^2}}\right )}{c^2 d}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {x^3 (a+b \text {arccosh}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {3 \left (-\frac {x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 c^2 d}+\frac {\sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^3}{6 b c^3 \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {c x-1} \sqrt {c x+1} \left (\frac {1}{2} x^2 (a+b \text {arccosh}(c x))-\frac {1}{2} b c \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )\right )}{c \sqrt {d-c^2 d x^2}}\right )}{c^2 d}+\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (\frac {i \left (-2 i \left (-\frac {1}{2} \log \left (1-e^{2 \text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))-\frac {1}{4} b \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(c x)}\right )\right )-\frac {i (a+b \text {arccosh}(c x))^2}{2 b}\right )}{c^4}-\frac {x^2 (a+b \text {arccosh}(c x))}{2 c^2}+\frac {b \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )}{2 c}\right )}{c d \sqrt {d-c^2 d x^2}}\) |
Input:
Int[(x^4*(a + b*ArcCosh[c*x])^2)/(d - c^2*d*x^2)^(3/2),x]
Output:
(x^3*(a + b*ArcCosh[c*x])^2)/(c^2*d*Sqrt[d - c^2*d*x^2]) - (3*(-1/2*(x*Sqr t[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(c^2*d) + (Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^3)/(6*b*c^3*Sqrt[d - c^2*d*x^2]) - (b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*((x^2*(a + b*ArcCosh[c*x]))/2 - (b*c*((x*Sqrt[-1 + c* x]*Sqrt[1 + c*x])/(2*c^2) + ArcCosh[c*x]/(2*c^3)))/2))/(c*Sqrt[d - c^2*d*x ^2])))/(c^2*d) + (2*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-1/2*(x^2*(a + b*ArcCo sh[c*x]))/c^2 + (b*((x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(2*c^2) + ArcCosh[c*x ]/(2*c^3)))/(2*c) + (I*(((-1/2*I)*(a + b*ArcCosh[c*x])^2)/b - (2*I)*(-1/2* ((a + b*ArcCosh[c*x])*Log[1 - E^(2*ArcCosh[c*x])]) - (b*PolyLog[2, E^(2*Ar cCosh[c*x])])/4)))/c^4))/(c*d*Sqrt[d - c^2*d*x^2])
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[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_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( p_), x_] :> Simp[b*(a + b*x)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 3))), x] + Simp[1/(d*f*(n + p + 3)) Int[(c + d*x)^n*(e + f*x)^p*Simp [a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f *(n + p + 4) - b*(d*e*(n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 3, 0]
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_.) + Pi*(k_.) + (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 ))/E^(2*I*k*Pi))))/E^(2*I*k*Pi), x], x] /; FreeQ[{c, d, e, f, fz}, x] && In tegerQ[4*k] && IGtQ[m, 0]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcCosh[c*x])^n/(d*(m + 1))), x] - Simp[b*c* (n/(d*(m + 1))) Int[(d*x)^(m + 1)*((a + b*ArcCosh[c*x])^(n - 1)/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] & & NeQ[m, -1]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_ Symbol] :> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 + c*x]*(Sqrt[-1 + c*x]/Sqrt[d + e*x^2])]*(a + b*ArcCosh[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x ] && EqQ[c^2*d + e, 0] && 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_.))^(n_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[1/e Subst[Int[(a + b*x)^n*Coth[x], x], x, ArcCosh[c*x] ], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_ .)*(x_)^2)^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*((a + b*ArcCosh[c*x])^n/(2*e*(p + 1))), x] + (-Simp[f^2*((m - 1)/(2*e*(p + 1))) Int[(f*x)^(m - 2)*(d + e*x^2)^(p + 1)*(a + b*ArcCosh[c*x])^n, x], x] - S imp[b*f*(n/(2*c*(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}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[p, -1] && IGtQ[m, 1]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_ .)*(x_)^2)^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*((a + b*ArcCosh[c*x])^n/(e*(m + 2*p + 1))), x] + (Simp[f^2*((m - 1)/(c^2*(m + 2 *p + 1))) Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcCosh[c*x])^n, x], x] - Simp[b*f*(n/(c*(m + 2*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*Ar cCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[c^2* d + e, 0] && GtQ[n, 0] && IGtQ[m, 1] && NeQ[m + 2*p + 1, 0]
Time = 0.62 (sec) , antiderivative size = 756, normalized size of antiderivative = 1.72
| method | result | size |
| default | \(-\frac {a^{2} x^{3}}{2 c^{2} d \sqrt {-c^{2} d \,x^{2}+d}}+\frac {3 a^{2} x}{2 c^{4} d \sqrt {-c^{2} d \,x^{2}+d}}-\frac {3 a^{2} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{2 c^{4} d \sqrt {c^{2} d}}+\frac {b^{2} \sqrt {c x -1}\, \sqrt {c x +1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 \operatorname {arccosh}\left (c x \right )^{2} \sqrt {c x +1}\, \sqrt {c x -1}\, x^{3} c^{3}-2 \,\operatorname {arccosh}\left (c x \right ) c^{4} x^{4}+c^{3} x^{3} \sqrt {c x -1}\, \sqrt {c x +1}+2 \operatorname {arccosh}\left (c x \right )^{3} x^{2} c^{2}-6 \operatorname {arccosh}\left (c x \right )^{2} \sqrt {c x +1}\, \sqrt {c x -1}\, x c -4 \operatorname {arccosh}\left (c x \right )^{2} x^{2} c^{2}+8 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) x^{2} c^{2}+8 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right ) x^{2} c^{2}+3 c^{2} x^{2} \operatorname {arccosh}\left (c x \right )+8 \operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right ) x^{2} c^{2}+8 \operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) x^{2} c^{2}-\sqrt {c x -1}\, \sqrt {c x +1}\, c x -2 \operatorname {arccosh}\left (c x \right )^{3}+4 \operatorname {arccosh}\left (c x \right )^{2}-8 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )-8 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )-\operatorname {arccosh}\left (c x \right )-8 \operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )-8 \operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{4 \left (c^{2} x^{2}-1\right )^{2} d^{2} c^{5}}+\frac {a b \sqrt {c x -1}\, \sqrt {c x +1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (4 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x +1}\, \sqrt {c x -1}\, x^{3} c^{3}-2 c^{4} x^{4}+6 \operatorname {arccosh}\left (c x \right )^{2} x^{2} c^{2}-12 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x +1}\, \sqrt {c x -1}\, c x -8 c^{2} x^{2} \operatorname {arccosh}\left (c x \right )+8 \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}-1\right ) x^{2} c^{2}+3 c^{2} x^{2}-6 \operatorname {arccosh}\left (c x \right )^{2}+8 \,\operatorname {arccosh}\left (c x \right )-8 \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}-1\right )-1\right )}{4 \left (c^{2} x^{2}-1\right )^{2} d^{2} c^{5}}\) | \(756\) |
| parts | \(-\frac {a^{2} x^{3}}{2 c^{2} d \sqrt {-c^{2} d \,x^{2}+d}}+\frac {3 a^{2} x}{2 c^{4} d \sqrt {-c^{2} d \,x^{2}+d}}-\frac {3 a^{2} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{2 c^{4} d \sqrt {c^{2} d}}+\frac {b^{2} \sqrt {c x -1}\, \sqrt {c x +1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 \operatorname {arccosh}\left (c x \right )^{2} \sqrt {c x +1}\, \sqrt {c x -1}\, x^{3} c^{3}-2 \,\operatorname {arccosh}\left (c x \right ) c^{4} x^{4}+c^{3} x^{3} \sqrt {c x -1}\, \sqrt {c x +1}+2 \operatorname {arccosh}\left (c x \right )^{3} x^{2} c^{2}-6 \operatorname {arccosh}\left (c x \right )^{2} \sqrt {c x +1}\, \sqrt {c x -1}\, x c -4 \operatorname {arccosh}\left (c x \right )^{2} x^{2} c^{2}+8 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) x^{2} c^{2}+8 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right ) x^{2} c^{2}+3 c^{2} x^{2} \operatorname {arccosh}\left (c x \right )+8 \operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right ) x^{2} c^{2}+8 \operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) x^{2} c^{2}-\sqrt {c x -1}\, \sqrt {c x +1}\, c x -2 \operatorname {arccosh}\left (c x \right )^{3}+4 \operatorname {arccosh}\left (c x \right )^{2}-8 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )-8 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )-\operatorname {arccosh}\left (c x \right )-8 \operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )-8 \operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{4 \left (c^{2} x^{2}-1\right )^{2} d^{2} c^{5}}+\frac {a b \sqrt {c x -1}\, \sqrt {c x +1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (4 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x +1}\, \sqrt {c x -1}\, x^{3} c^{3}-2 c^{4} x^{4}+6 \operatorname {arccosh}\left (c x \right )^{2} x^{2} c^{2}-12 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x +1}\, \sqrt {c x -1}\, c x -8 c^{2} x^{2} \operatorname {arccosh}\left (c x \right )+8 \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}-1\right ) x^{2} c^{2}+3 c^{2} x^{2}-6 \operatorname {arccosh}\left (c x \right )^{2}+8 \,\operatorname {arccosh}\left (c x \right )-8 \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}-1\right )-1\right )}{4 \left (c^{2} x^{2}-1\right )^{2} d^{2} c^{5}}\) | \(756\) |
Input:
int(x^4*(a+b*arccosh(c*x))^2/(-c^2*d*x^2+d)^(3/2),x,method=_RETURNVERBOSE)
Output:
-1/2*a^2*x^3/c^2/d/(-c^2*d*x^2+d)^(1/2)+3/2*a^2/c^4*x/d/(-c^2*d*x^2+d)^(1/ 2)-3/2*a^2/c^4/d/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2) )+1/4*b^2*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)*(2*arccosh(c* x)^2*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^3*c^3-2*arccosh(c*x)*c^4*x^4+c^3*x^3*(c *x-1)^(1/2)*(c*x+1)^(1/2)+2*arccosh(c*x)^3*x^2*c^2-6*arccosh(c*x)^2*(c*x+1 )^(1/2)*(c*x-1)^(1/2)*x*c-4*arccosh(c*x)^2*x^2*c^2+8*arccosh(c*x)*ln(1+c*x +(c*x-1)^(1/2)*(c*x+1)^(1/2))*x^2*c^2+8*arccosh(c*x)*ln(1-c*x-(c*x-1)^(1/2 )*(c*x+1)^(1/2))*x^2*c^2+3*c^2*x^2*arccosh(c*x)+8*polylog(2,-c*x-(c*x-1)^( 1/2)*(c*x+1)^(1/2))*x^2*c^2+8*polylog(2,c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*x ^2*c^2-(c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x-2*arccosh(c*x)^3+4*arccosh(c*x)^2-8 *arccosh(c*x)*ln(1+c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))-8*arccosh(c*x)*ln(1-c* x-(c*x-1)^(1/2)*(c*x+1)^(1/2))-arccosh(c*x)-8*polylog(2,-c*x-(c*x-1)^(1/2) *(c*x+1)^(1/2))-8*polylog(2,c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))/(c^2*x^2-1)^ 2/d^2/c^5+1/4*a*b*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)*(4*ar ccosh(c*x)*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^3*c^3-2*c^4*x^4+6*arccosh(c*x)^2* x^2*c^2-12*arccosh(c*x)*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c*x-8*c^2*x^2*arccosh( c*x)+8*ln((c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2-1)*x^2*c^2+3*c^2*x^2-6*arcco sh(c*x)^2+8*arccosh(c*x)-8*ln((c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2-1)-1)/(c ^2*x^2-1)^2/d^2/c^5
\[ \int \frac {x^4 (a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2} x^{4}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(x^4*(a+b*arccosh(c*x))^2/(-c^2*d*x^2+d)^(3/2),x, algorithm="fric as")
Output:
integral((b^2*x^4*arccosh(c*x)^2 + 2*a*b*x^4*arccosh(c*x) + a^2*x^4)*sqrt( -c^2*d*x^2 + d)/(c^4*d^2*x^4 - 2*c^2*d^2*x^2 + d^2), x)
\[ \int \frac {x^4 (a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {x^{4} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}}{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(x**4*(a+b*acosh(c*x))**2/(-c**2*d*x**2+d)**(3/2),x)
Output:
Integral(x**4*(a + b*acosh(c*x))**2/(-d*(c*x - 1)*(c*x + 1))**(3/2), x)
\[ \int \frac {x^4 (a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2} x^{4}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(x^4*(a+b*arccosh(c*x))^2/(-c^2*d*x^2+d)^(3/2),x, algorithm="maxi ma")
Output:
-1/2*a^2*(x^3/(sqrt(-c^2*d*x^2 + d)*c^2*d) - 3*x/(sqrt(-c^2*d*x^2 + d)*c^4 *d) + 3*arcsin(c*x)/(c^5*d^(3/2))) + integrate(b^2*x^4*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))^2/(-c^2*d*x^2 + d)^(3/2) + 2*a*b*x^4*log(c*x + sqrt(c* x + 1)*sqrt(c*x - 1))/(-c^2*d*x^2 + d)^(3/2), x)
Exception generated. \[ \int \frac {x^4 (a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(x^4*(a+b*arccosh(c*x))^2/(-c^2*d*x^2+d)^(3/2),x, algorithm="giac ")
Output:
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. \[ \int \frac {x^4 (a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {x^4\,{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2}{{\left (d-c^2\,d\,x^2\right )}^{3/2}} \,d x \] Input:
int((x^4*(a + b*acosh(c*x))^2)/(d - c^2*d*x^2)^(3/2),x)
Output:
int((x^4*(a + b*acosh(c*x))^2)/(d - c^2*d*x^2)^(3/2), x)
\[ \int \frac {x^4 (a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {-3 \sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right ) a^{2}-4 \sqrt {-c^{2} x^{2}+1}\, \left (\int \frac {\mathit {acosh} \left (c x \right ) x^{4}}{\sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}-\sqrt {-c^{2} x^{2}+1}}d x \right ) a b \,c^{5}-2 \sqrt {-c^{2} x^{2}+1}\, \left (\int \frac {\mathit {acosh} \left (c x \right )^{2} x^{4}}{\sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}-\sqrt {-c^{2} x^{2}+1}}d x \right ) b^{2} c^{5}-a^{2} c^{3} x^{3}+3 a^{2} c x}{2 \sqrt {d}\, \sqrt {-c^{2} x^{2}+1}\, c^{5} d} \] Input:
int(x^4*(a+b*acosh(c*x))^2/(-c^2*d*x^2+d)^(3/2),x)
Output:
( - 3*sqrt( - c**2*x**2 + 1)*asin(c*x)*a**2 - 4*sqrt( - c**2*x**2 + 1)*int ((acosh(c*x)*x**4)/(sqrt( - c**2*x**2 + 1)*c**2*x**2 - sqrt( - c**2*x**2 + 1)),x)*a*b*c**5 - 2*sqrt( - c**2*x**2 + 1)*int((acosh(c*x)**2*x**4)/(sqrt ( - c**2*x**2 + 1)*c**2*x**2 - sqrt( - c**2*x**2 + 1)),x)*b**2*c**5 - a**2 *c**3*x**3 + 3*a**2*c*x)/(2*sqrt(d)*sqrt( - c**2*x**2 + 1)*c**5*d)