Integrand size = 29, antiderivative size = 330 \[ \int x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2 \, dx=-\frac {856 b^2 \sqrt {d-c^2 d x^2}}{3375 c^4}+\frac {22 b^2 x^2 \sqrt {d-c^2 d x^2}}{3375 c^2}+\frac {2}{125} b^2 x^4 \sqrt {d-c^2 d x^2}+\frac {4 b x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{15 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{45 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b c x^5 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{25 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{15 c^4}-\frac {x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{15 c^2}+\frac {1}{5} x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2 \] Output:
-856/3375*b^2*(-c^2*d*x^2+d)^(1/2)/c^4+22/3375*b^2*x^2*(-c^2*d*x^2+d)^(1/2 )/c^2+2/125*b^2*x^4*(-c^2*d*x^2+d)^(1/2)+4/15*b*x*(-c^2*d*x^2+d)^(1/2)*(a+ b*arccosh(c*x))/c^3/(c*x-1)^(1/2)/(c*x+1)^(1/2)+2/45*b*x^3*(-c^2*d*x^2+d)^ (1/2)*(a+b*arccosh(c*x))/c/(c*x-1)^(1/2)/(c*x+1)^(1/2)-2/25*b*c*x^5*(-c^2* d*x^2+d)^(1/2)*(a+b*arccosh(c*x))/(c*x-1)^(1/2)/(c*x+1)^(1/2)-2/15*(-c^2*d *x^2+d)^(1/2)*(a+b*arccosh(c*x))^2/c^4-1/15*x^2*(-c^2*d*x^2+d)^(1/2)*(a+b* arccosh(c*x))^2/c^2+1/5*x^4*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))^2
Time = 0.39 (sec) , antiderivative size = 237, normalized size of antiderivative = 0.72 \[ \int x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2 \, dx=\frac {\sqrt {d-c^2 d x^2} \left (225 a^2 \left (-1+c^2 x^2\right )^2 \left (2+3 c^2 x^2\right )-30 a b c x \sqrt {-1+c x} \sqrt {1+c x} \left (-30-5 c^2 x^2+9 c^4 x^4\right )+2 b^2 \left (428-439 c^2 x^2-16 c^4 x^4+27 c^6 x^6\right )+30 b \left (15 a \left (-1+c^2 x^2\right )^2 \left (2+3 c^2 x^2\right )+b c x \sqrt {-1+c x} \sqrt {1+c x} \left (30+5 c^2 x^2-9 c^4 x^4\right )\right ) \text {arccosh}(c x)+225 b^2 \left (-1+c^2 x^2\right )^2 \left (2+3 c^2 x^2\right ) \text {arccosh}(c x)^2\right )}{3375 c^4 \left (-1+c^2 x^2\right )} \] Input:
Integrate[x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2,x]
Output:
(Sqrt[d - c^2*d*x^2]*(225*a^2*(-1 + c^2*x^2)^2*(2 + 3*c^2*x^2) - 30*a*b*c* x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-30 - 5*c^2*x^2 + 9*c^4*x^4) + 2*b^2*(428 - 439*c^2*x^2 - 16*c^4*x^4 + 27*c^6*x^6) + 30*b*(15*a*(-1 + c^2*x^2)^2*(2 + 3*c^2*x^2) + b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(30 + 5*c^2*x^2 - 9*c^4* x^4))*ArcCosh[c*x] + 225*b^2*(-1 + c^2*x^2)^2*(2 + 3*c^2*x^2)*ArcCosh[c*x] ^2))/(3375*c^4*(-1 + c^2*x^2))
Time = 2.49 (sec) , antiderivative size = 425, normalized size of antiderivative = 1.29, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.483, Rules used = {6341, 6298, 111, 27, 111, 27, 83, 6354, 6298, 111, 27, 83, 6330, 2009}
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.
| \(\displaystyle \int x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2 \, dx\) |
| \(\Big \downarrow \) 6341 |
| \(\displaystyle -\frac {2 b c \sqrt {d-c^2 d x^2} \int x^4 (a+b \text {arccosh}(c x))dx}{5 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\sqrt {d-c^2 d x^2} \int \frac {x^3 (a+b \text {arccosh}(c x))^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{5 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{5} x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 6298 |
| \(\displaystyle -\frac {2 b c \sqrt {d-c^2 d x^2} \left (\frac {1}{5} x^5 (a+b \text {arccosh}(c x))-\frac {1}{5} b c \int \frac {x^5}{\sqrt {c x-1} \sqrt {c x+1}}dx\right )}{5 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\sqrt {d-c^2 d x^2} \int \frac {x^3 (a+b \text {arccosh}(c x))^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{5 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{5} x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 111 |
| \(\displaystyle -\frac {\sqrt {d-c^2 d x^2} \int \frac {x^3 (a+b \text {arccosh}(c x))^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{5 \sqrt {c x-1} \sqrt {c x+1}}-\frac {2 b c \sqrt {d-c^2 d x^2} \left (\frac {1}{5} x^5 (a+b \text {arccosh}(c x))-\frac {1}{5} b c \left (\frac {\int \frac {4 x^3}{\sqrt {c x-1} \sqrt {c x+1}}dx}{5 c^2}+\frac {x^4 \sqrt {c x-1} \sqrt {c x+1}}{5 c^2}\right )\right )}{5 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{5} x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\sqrt {d-c^2 d x^2} \int \frac {x^3 (a+b \text {arccosh}(c x))^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{5 \sqrt {c x-1} \sqrt {c x+1}}-\frac {2 b c \sqrt {d-c^2 d x^2} \left (\frac {1}{5} x^5 (a+b \text {arccosh}(c x))-\frac {1}{5} b c \left (\frac {4 \int \frac {x^3}{\sqrt {c x-1} \sqrt {c x+1}}dx}{5 c^2}+\frac {x^4 \sqrt {c x-1} \sqrt {c x+1}}{5 c^2}\right )\right )}{5 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{5} x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 111 |
| \(\displaystyle -\frac {\sqrt {d-c^2 d x^2} \int \frac {x^3 (a+b \text {arccosh}(c x))^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{5 \sqrt {c x-1} \sqrt {c x+1}}-\frac {2 b c \sqrt {d-c^2 d x^2} \left (\frac {1}{5} x^5 (a+b \text {arccosh}(c x))-\frac {1}{5} b c \left (\frac {4 \left (\frac {\int \frac {2 x}{\sqrt {c x-1} \sqrt {c x+1}}dx}{3 c^2}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}\right )}{5 c^2}+\frac {x^4 \sqrt {c x-1} \sqrt {c x+1}}{5 c^2}\right )\right )}{5 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{5} x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\sqrt {d-c^2 d x^2} \int \frac {x^3 (a+b \text {arccosh}(c x))^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{5 \sqrt {c x-1} \sqrt {c x+1}}-\frac {2 b c \sqrt {d-c^2 d x^2} \left (\frac {1}{5} x^5 (a+b \text {arccosh}(c x))-\frac {1}{5} b c \left (\frac {4 \left (\frac {2 \int \frac {x}{\sqrt {c x-1} \sqrt {c x+1}}dx}{3 c^2}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}\right )}{5 c^2}+\frac {x^4 \sqrt {c x-1} \sqrt {c x+1}}{5 c^2}\right )\right )}{5 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{5} x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 83 |
| \(\displaystyle -\frac {\sqrt {d-c^2 d x^2} \int \frac {x^3 (a+b \text {arccosh}(c x))^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{5 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{5} x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {2 b c \sqrt {d-c^2 d x^2} \left (\frac {1}{5} x^5 (a+b \text {arccosh}(c x))-\frac {1}{5} b c \left (\frac {x^4 \sqrt {c x-1} \sqrt {c x+1}}{5 c^2}+\frac {4 \left (\frac {2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^4}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}\right )}{5 c^2}\right )\right )}{5 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 6354 |
| \(\displaystyle -\frac {\sqrt {d-c^2 d x^2} \left (\frac {2 \int \frac {x (a+b \text {arccosh}(c x))^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{3 c^2}-\frac {2 b \int x^2 (a+b \text {arccosh}(c x))dx}{3 c}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^2}{3 c^2}\right )}{5 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{5} x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {2 b c \sqrt {d-c^2 d x^2} \left (\frac {1}{5} x^5 (a+b \text {arccosh}(c x))-\frac {1}{5} b c \left (\frac {x^4 \sqrt {c x-1} \sqrt {c x+1}}{5 c^2}+\frac {4 \left (\frac {2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^4}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}\right )}{5 c^2}\right )\right )}{5 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 6298 |
| \(\displaystyle -\frac {\sqrt {d-c^2 d x^2} \left (\frac {2 \int \frac {x (a+b \text {arccosh}(c x))^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{3 c^2}-\frac {2 b \left (\frac {1}{3} x^3 (a+b \text {arccosh}(c x))-\frac {1}{3} b c \int \frac {x^3}{\sqrt {c x-1} \sqrt {c x+1}}dx\right )}{3 c}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^2}{3 c^2}\right )}{5 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{5} x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {2 b c \sqrt {d-c^2 d x^2} \left (\frac {1}{5} x^5 (a+b \text {arccosh}(c x))-\frac {1}{5} b c \left (\frac {x^4 \sqrt {c x-1} \sqrt {c x+1}}{5 c^2}+\frac {4 \left (\frac {2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^4}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}\right )}{5 c^2}\right )\right )}{5 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 111 |
| \(\displaystyle -\frac {\sqrt {d-c^2 d x^2} \left (-\frac {2 b \left (\frac {1}{3} x^3 (a+b \text {arccosh}(c x))-\frac {1}{3} b c \left (\frac {\int \frac {2 x}{\sqrt {c x-1} \sqrt {c x+1}}dx}{3 c^2}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}\right )\right )}{3 c}+\frac {2 \int \frac {x (a+b \text {arccosh}(c x))^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{3 c^2}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^2}{3 c^2}\right )}{5 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{5} x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {2 b c \sqrt {d-c^2 d x^2} \left (\frac {1}{5} x^5 (a+b \text {arccosh}(c x))-\frac {1}{5} b c \left (\frac {x^4 \sqrt {c x-1} \sqrt {c x+1}}{5 c^2}+\frac {4 \left (\frac {2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^4}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}\right )}{5 c^2}\right )\right )}{5 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\sqrt {d-c^2 d x^2} \left (-\frac {2 b \left (\frac {1}{3} x^3 (a+b \text {arccosh}(c x))-\frac {1}{3} b c \left (\frac {2 \int \frac {x}{\sqrt {c x-1} \sqrt {c x+1}}dx}{3 c^2}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}\right )\right )}{3 c}+\frac {2 \int \frac {x (a+b \text {arccosh}(c x))^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{3 c^2}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^2}{3 c^2}\right )}{5 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{5} x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {2 b c \sqrt {d-c^2 d x^2} \left (\frac {1}{5} x^5 (a+b \text {arccosh}(c x))-\frac {1}{5} b c \left (\frac {x^4 \sqrt {c x-1} \sqrt {c x+1}}{5 c^2}+\frac {4 \left (\frac {2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^4}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}\right )}{5 c^2}\right )\right )}{5 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 83 |
| \(\displaystyle -\frac {\sqrt {d-c^2 d x^2} \left (\frac {2 \int \frac {x (a+b \text {arccosh}(c x))^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{3 c^2}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^2}{3 c^2}-\frac {2 b \left (\frac {1}{3} x^3 (a+b \text {arccosh}(c x))-\frac {1}{3} b c \left (\frac {2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^4}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}\right )\right )}{3 c}\right )}{5 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{5} x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {2 b c \sqrt {d-c^2 d x^2} \left (\frac {1}{5} x^5 (a+b \text {arccosh}(c x))-\frac {1}{5} b c \left (\frac {x^4 \sqrt {c x-1} \sqrt {c x+1}}{5 c^2}+\frac {4 \left (\frac {2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^4}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}\right )}{5 c^2}\right )\right )}{5 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 6330 |
| \(\displaystyle -\frac {\sqrt {d-c^2 d x^2} \left (\frac {2 \left (\frac {\sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^2}{c^2}-\frac {2 b \int (a+b \text {arccosh}(c x))dx}{c}\right )}{3 c^2}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^2}{3 c^2}-\frac {2 b \left (\frac {1}{3} x^3 (a+b \text {arccosh}(c x))-\frac {1}{3} b c \left (\frac {2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^4}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}\right )\right )}{3 c}\right )}{5 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{5} x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {2 b c \sqrt {d-c^2 d x^2} \left (\frac {1}{5} x^5 (a+b \text {arccosh}(c x))-\frac {1}{5} b c \left (\frac {x^4 \sqrt {c x-1} \sqrt {c x+1}}{5 c^2}+\frac {4 \left (\frac {2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^4}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}\right )}{5 c^2}\right )\right )}{5 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{5} x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {\sqrt {d-c^2 d x^2} \left (\frac {x^2 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^2}{3 c^2}+\frac {2 \left (\frac {\sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^2}{c^2}-\frac {2 b \left (a x+b x \text {arccosh}(c x)-\frac {b \sqrt {c x-1} \sqrt {c x+1}}{c}\right )}{c}\right )}{3 c^2}-\frac {2 b \left (\frac {1}{3} x^3 (a+b \text {arccosh}(c x))-\frac {1}{3} b c \left (\frac {2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^4}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}\right )\right )}{3 c}\right )}{5 \sqrt {c x-1} \sqrt {c x+1}}-\frac {2 b c \sqrt {d-c^2 d x^2} \left (\frac {1}{5} x^5 (a+b \text {arccosh}(c x))-\frac {1}{5} b c \left (\frac {x^4 \sqrt {c x-1} \sqrt {c x+1}}{5 c^2}+\frac {4 \left (\frac {2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^4}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}\right )}{5 c^2}\right )\right )}{5 \sqrt {c x-1} \sqrt {c x+1}}\) |
Input:
Int[x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2,x]
Output:
(x^4*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/5 - (2*b*c*Sqrt[d - c^2*d *x^2]*(-1/5*(b*c*((x^4*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(5*c^2) + (4*((2*Sqrt [-1 + c*x]*Sqrt[1 + c*x])/(3*c^4) + (x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(3* c^2)))/(5*c^2))) + (x^5*(a + b*ArcCosh[c*x]))/5))/(5*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (Sqrt[d - c^2*d*x^2]*((x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b* ArcCosh[c*x])^2)/(3*c^2) - (2*b*(-1/3*(b*c*((2*Sqrt[-1 + c*x]*Sqrt[1 + c*x ])/(3*c^4) + (x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(3*c^2))) + (x^3*(a + b*Ar cCosh[c*x]))/3))/(3*c) + (2*((Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[ c*x])^2)/c^2 - (2*b*(a*x - (b*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/c + b*x*ArcCos h[c*x]))/c))/(3*c^2)))/(5*Sqrt[-1 + c*x]*Sqrt[1 + c*x])
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0] && EqQ[a*d*f *(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)), 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[b*(a + b*x)^(m - 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 )/(d*f*(m + n + p + 1))), x] + Simp[1/(d*f*(m + n + p + 1)) Int[(a + b*x) ^(m - 2)*(c + d*x)^n*(e + f*x)^p*Simp[a^2*d*f*(m + n + p + 1) - b*(b*c*e*(m - 1) + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(2*m + n + p) - b*(d*e*(m + n) + c*f*(m + p)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] & & GtQ[m, 1] && NeQ[m + n + p + 1, 0] && IntegerQ[m]
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_.)*(x_)*((d1_) + (e1_.)*(x_))^(p _)*((d2_) + (e2_.)*(x_))^(p_), x_Symbol] :> Simp[(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*((a + b*ArcCosh[c*x])^n/(2*e1*e2*(p + 1))), x] - Simp[b*(n/(2 *c*(p + 1)))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*x)^p/(-1 + c*x)^ p] Int[(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, d1, e1, d2, e2, p}, x] && EqQ[e1, c*d1] && E qQ[e2, (-c)*d2] && GtQ[n, 0] && NeQ[p, -1]
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_)*((d1_) + (e 1_.)*(x_))^(p_)*((d2_) + (e2_.)*(x_))^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 1)*(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*((a + b*ArcCosh[c*x])^n/(e1*e2*( m + 2*p + 1))), x] + (Simp[f^2*((m - 1)/(c^2*(m + 2*p + 1))) Int[(f*x)^(m - 2)*(d1 + e1*x)^p*(d2 + e2*x)^p*(a + b*ArcCosh[c*x])^n, x], x] - Simp[b*f *(n/(c*(m + 2*p + 1)))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*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, d1, e1, d2, e2, f, p}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && GtQ[n, 0] && IGtQ[m, 1] && N eQ[m + 2*p + 1, 0]
Time = 0.86 (sec) , antiderivative size = 557, normalized size of antiderivative = 1.69
| method | result | size |
| orering | \(\frac {\left (1647 c^{8} x^{8}-2131 c^{6} x^{6}-8610 c^{4} x^{4}+13060 c^{2} x^{2}-5136\right ) \sqrt {-c^{2} d \,x^{2}+d}\, \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{2}}{3375 \left (c^{2} x^{2}-1\right ) c^{6} x^{2}}-\frac {4 \left (81 c^{6} x^{6}-40 c^{4} x^{4}-878 c^{2} x^{2}+642\right ) \left (3 x^{2} \sqrt {-c^{2} d \,x^{2}+d}\, \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{2}-\frac {x^{4} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{2} c^{2} d}{\sqrt {-c^{2} d \,x^{2}+d}}+\frac {2 b c \,x^{3} \sqrt {-c^{2} d \,x^{2}+d}\, \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}{\sqrt {c x -1}\, \sqrt {c x +1}}\right )}{3375 c^{6} x^{4}}+\frac {\left (27 c^{4} x^{4}+11 c^{2} x^{2}-428\right ) \left (c x -1\right ) \left (c x +1\right ) \left (6 x \sqrt {-c^{2} d \,x^{2}+d}\, \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{2}-\frac {7 x^{3} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{2} c^{2} d}{\sqrt {-c^{2} d \,x^{2}+d}}+\frac {12 b c \,x^{2} \sqrt {-c^{2} d \,x^{2}+d}\, \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}{\sqrt {c x -1}\, \sqrt {c x +1}}-\frac {x^{5} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{2} c^{4} d^{2}}{\left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}-\frac {4 x^{4} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right ) c^{3} d b}{\sqrt {-c^{2} d \,x^{2}+d}\, \sqrt {c x -1}\, \sqrt {c x +1}}+\frac {2 b^{2} c^{2} x^{3} \sqrt {-c^{2} d \,x^{2}+d}}{\left (c x -1\right ) \left (c x +1\right )}-\frac {b \,c^{2} x^{3} \sqrt {-c^{2} d \,x^{2}+d}\, \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}{\left (c x -1\right )^{\frac {3}{2}} \sqrt {c x +1}}-\frac {b \,c^{2} x^{3} \sqrt {-c^{2} d \,x^{2}+d}\, \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}{\sqrt {c x -1}\, \left (c x +1\right )^{\frac {3}{2}}}\right )}{3375 c^{6} x^{3}}\) | \(557\) |
| default | \(\text {Expression too large to display}\) | \(1284\) |
| parts | \(\text {Expression too large to display}\) | \(1284\) |
Input:
int(x^3*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))^2,x,method=_RETURNVERBOSE)
Output:
1/3375*(1647*c^8*x^8-2131*c^6*x^6-8610*c^4*x^4+13060*c^2*x^2-5136)/(c^2*x^ 2-1)/c^6/x^2*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))^2-4/3375*(81*c^6*x^6- 40*c^4*x^4-878*c^2*x^2+642)/c^6/x^4*(3*x^2*(-c^2*d*x^2+d)^(1/2)*(a+b*arcco sh(c*x))^2-x^4/(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))^2*c^2*d+2*b*c*x^3*( -c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))/(c*x-1)^(1/2)/(c*x+1)^(1/2))+1/3375 *(27*c^4*x^4+11*c^2*x^2-428)/c^6*(c*x-1)*(c*x+1)/x^3*(6*x*(-c^2*d*x^2+d)^( 1/2)*(a+b*arccosh(c*x))^2-7*x^3/(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))^2* c^2*d+12*b*c*x^2*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))/(c*x-1)^(1/2)/(c* x+1)^(1/2)-x^5/(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))^2*c^4*d^2-4*x^4/(-c ^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))*c^3*d*b/(c*x-1)^(1/2)/(c*x+1)^(1/2)+2 *b^2*c^2*x^3*(-c^2*d*x^2+d)^(1/2)/(c*x-1)/(c*x+1)-b*c^2*x^3*(-c^2*d*x^2+d) ^(1/2)*(a+b*arccosh(c*x))/(c*x-1)^(3/2)/(c*x+1)^(1/2)-b*c^2*x^3*(-c^2*d*x^ 2+d)^(1/2)*(a+b*arccosh(c*x))/(c*x-1)^(1/2)/(c*x+1)^(3/2))
Time = 0.10 (sec) , antiderivative size = 349, normalized size of antiderivative = 1.06 \[ \int x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2 \, dx=\frac {225 \, {\left (3 \, b^{2} c^{6} x^{6} - 4 \, b^{2} c^{4} x^{4} - b^{2} c^{2} x^{2} + 2 \, b^{2}\right )} \sqrt {-c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right )^{2} - 30 \, {\left (9 \, a b c^{5} x^{5} - 5 \, a b c^{3} x^{3} - 30 \, a b c x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} - 30 \, {\left ({\left (9 \, b^{2} c^{5} x^{5} - 5 \, b^{2} c^{3} x^{3} - 30 \, b^{2} c x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} - 15 \, {\left (3 \, a b c^{6} x^{6} - 4 \, a b c^{4} x^{4} - a b c^{2} x^{2} + 2 \, a b\right )} \sqrt {-c^{2} d x^{2} + d}\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) + {\left (27 \, {\left (25 \, a^{2} + 2 \, b^{2}\right )} c^{6} x^{6} - 4 \, {\left (225 \, a^{2} + 8 \, b^{2}\right )} c^{4} x^{4} - {\left (225 \, a^{2} + 878 \, b^{2}\right )} c^{2} x^{2} + 450 \, a^{2} + 856 \, b^{2}\right )} \sqrt {-c^{2} d x^{2} + d}}{3375 \, {\left (c^{6} x^{2} - c^{4}\right )}} \] Input:
integrate(x^3*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))^2,x, algorithm="fric as")
Output:
1/3375*(225*(3*b^2*c^6*x^6 - 4*b^2*c^4*x^4 - b^2*c^2*x^2 + 2*b^2)*sqrt(-c^ 2*d*x^2 + d)*log(c*x + sqrt(c^2*x^2 - 1))^2 - 30*(9*a*b*c^5*x^5 - 5*a*b*c^ 3*x^3 - 30*a*b*c*x)*sqrt(-c^2*d*x^2 + d)*sqrt(c^2*x^2 - 1) - 30*((9*b^2*c^ 5*x^5 - 5*b^2*c^3*x^3 - 30*b^2*c*x)*sqrt(-c^2*d*x^2 + d)*sqrt(c^2*x^2 - 1) - 15*(3*a*b*c^6*x^6 - 4*a*b*c^4*x^4 - a*b*c^2*x^2 + 2*a*b)*sqrt(-c^2*d*x^ 2 + d))*log(c*x + sqrt(c^2*x^2 - 1)) + (27*(25*a^2 + 2*b^2)*c^6*x^6 - 4*(2 25*a^2 + 8*b^2)*c^4*x^4 - (225*a^2 + 878*b^2)*c^2*x^2 + 450*a^2 + 856*b^2) *sqrt(-c^2*d*x^2 + d))/(c^6*x^2 - c^4)
\[ \int x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2 \, dx=\int x^{3} \sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}\, dx \] Input:
integrate(x**3*(-c**2*d*x**2+d)**(1/2)*(a+b*acosh(c*x))**2,x)
Output:
Integral(x**3*sqrt(-d*(c*x - 1)*(c*x + 1))*(a + b*acosh(c*x))**2, x)
Time = 0.14 (sec) , antiderivative size = 326, normalized size of antiderivative = 0.99 \[ \int x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2 \, dx=-\frac {1}{15} \, b^{2} {\left (\frac {3 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x^{2}}{c^{2} d} + \frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}}{c^{4} d}\right )} \operatorname {arcosh}\left (c x\right )^{2} - \frac {2}{15} \, a b {\left (\frac {3 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x^{2}}{c^{2} d} + \frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}}{c^{4} d}\right )} \operatorname {arcosh}\left (c x\right ) - \frac {1}{15} \, a^{2} {\left (\frac {3 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x^{2}}{c^{2} d} + \frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}}{c^{4} d}\right )} + \frac {2}{3375} \, b^{2} {\left (\frac {27 \, \sqrt {c^{2} x^{2} - 1} c^{2} \sqrt {-d} x^{4} + 11 \, \sqrt {c^{2} x^{2} - 1} \sqrt {-d} x^{2} - \frac {428 \, \sqrt {c^{2} x^{2} - 1} \sqrt {-d}}{c^{2}}}{c^{2}} - \frac {15 \, {\left (9 \, c^{4} \sqrt {-d} x^{5} - 5 \, c^{2} \sqrt {-d} x^{3} - 30 \, \sqrt {-d} x\right )} \operatorname {arcosh}\left (c x\right )}{c^{3}}\right )} - \frac {2 \, {\left (9 \, c^{4} \sqrt {-d} x^{5} - 5 \, c^{2} \sqrt {-d} x^{3} - 30 \, \sqrt {-d} x\right )} a b}{225 \, c^{3}} \] Input:
integrate(x^3*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))^2,x, algorithm="maxi ma")
Output:
-1/15*b^2*(3*(-c^2*d*x^2 + d)^(3/2)*x^2/(c^2*d) + 2*(-c^2*d*x^2 + d)^(3/2) /(c^4*d))*arccosh(c*x)^2 - 2/15*a*b*(3*(-c^2*d*x^2 + d)^(3/2)*x^2/(c^2*d) + 2*(-c^2*d*x^2 + d)^(3/2)/(c^4*d))*arccosh(c*x) - 1/15*a^2*(3*(-c^2*d*x^2 + d)^(3/2)*x^2/(c^2*d) + 2*(-c^2*d*x^2 + d)^(3/2)/(c^4*d)) + 2/3375*b^2*( (27*sqrt(c^2*x^2 - 1)*c^2*sqrt(-d)*x^4 + 11*sqrt(c^2*x^2 - 1)*sqrt(-d)*x^2 - 428*sqrt(c^2*x^2 - 1)*sqrt(-d)/c^2)/c^2 - 15*(9*c^4*sqrt(-d)*x^5 - 5*c^ 2*sqrt(-d)*x^3 - 30*sqrt(-d)*x)*arccosh(c*x)/c^3) - 2/225*(9*c^4*sqrt(-d)* x^5 - 5*c^2*sqrt(-d)*x^3 - 30*sqrt(-d)*x)*a*b/c^3
Exception generated. \[ \int x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2 \, dx=\text {Exception raised: TypeError} \] Input:
integrate(x^3*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))^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. \[ \int x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2 \, dx=\int x^3\,{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2\,\sqrt {d-c^2\,d\,x^2} \,d x \] Input:
int(x^3*(a + b*acosh(c*x))^2*(d - c^2*d*x^2)^(1/2),x)
Output:
int(x^3*(a + b*acosh(c*x))^2*(d - c^2*d*x^2)^(1/2), x)
\[ \int x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2 \, dx=\frac {\sqrt {d}\, \left (3 \sqrt {-c^{2} x^{2}+1}\, a^{2} c^{4} x^{4}-\sqrt {-c^{2} x^{2}+1}\, a^{2} c^{2} x^{2}-2 \sqrt {-c^{2} x^{2}+1}\, a^{2}+30 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right ) x^{3}d x \right ) a b \,c^{4}+15 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right )^{2} x^{3}d x \right ) b^{2} c^{4}\right )}{15 c^{4}} \] Input:
int(x^3*(-c^2*d*x^2+d)^(1/2)*(a+b*acosh(c*x))^2,x)
Output:
(sqrt(d)*(3*sqrt( - c**2*x**2 + 1)*a**2*c**4*x**4 - sqrt( - c**2*x**2 + 1) *a**2*c**2*x**2 - 2*sqrt( - c**2*x**2 + 1)*a**2 + 30*int(sqrt( - c**2*x**2 + 1)*acosh(c*x)*x**3,x)*a*b*c**4 + 15*int(sqrt( - c**2*x**2 + 1)*acosh(c* x)**2*x**3,x)*b**2*c**4))/(15*c**4)