Integrand size = 20, antiderivative size = 195 \[ \int \left (c-a^2 c x^2\right )^2 \text {arccosh}(a x)^2 \, dx=\frac {298 c^2 x}{225}-\frac {76}{675} a^2 c^2 x^3+\frac {2}{125} a^4 c^2 x^5-\frac {16 c^2 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)}{15 a}+\frac {8 c^2 (-1+a x)^{3/2} (1+a x)^{3/2} \text {arccosh}(a x)}{45 a}-\frac {2 c^2 (-1+a x)^{5/2} (1+a x)^{5/2} \text {arccosh}(a x)}{25 a}+\frac {8}{15} c^2 x \text {arccosh}(a x)^2+\frac {4}{15} c^2 x \left (1-a^2 x^2\right ) \text {arccosh}(a x)^2+\frac {1}{5} c^2 x \left (1-a^2 x^2\right )^2 \text {arccosh}(a x)^2 \]
[Out]
Time = 0.32 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {5897, 5879, 5915, 8, 41, 200} \[ \int \left (c-a^2 c x^2\right )^2 \text {arccosh}(a x)^2 \, dx=\frac {2}{125} a^4 c^2 x^5+\frac {1}{5} c^2 x \left (1-a^2 x^2\right )^2 \text {arccosh}(a x)^2+\frac {4}{15} c^2 x \left (1-a^2 x^2\right ) \text {arccosh}(a x)^2-\frac {76}{675} a^2 c^2 x^3+\frac {8}{15} c^2 x \text {arccosh}(a x)^2-\frac {2 c^2 (a x-1)^{5/2} (a x+1)^{5/2} \text {arccosh}(a x)}{25 a}+\frac {8 c^2 (a x-1)^{3/2} (a x+1)^{3/2} \text {arccosh}(a x)}{45 a}-\frac {16 c^2 \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)}{15 a}+\frac {298 c^2 x}{225} \]
[In]
[Out]
Rule 8
Rule 41
Rule 200
Rule 5879
Rule 5897
Rule 5915
Rubi steps \begin{align*} \text {integral}& = \frac {1}{5} c^2 x \left (1-a^2 x^2\right )^2 \text {arccosh}(a x)^2+\frac {1}{5} (4 c) \int \left (c-a^2 c x^2\right ) \text {arccosh}(a x)^2 \, dx-\frac {1}{5} \left (2 a c^2\right ) \int x (-1+a x)^{3/2} (1+a x)^{3/2} \text {arccosh}(a x) \, dx \\ & = -\frac {2 c^2 (-1+a x)^{5/2} (1+a x)^{5/2} \text {arccosh}(a x)}{25 a}+\frac {4}{15} c^2 x \left (1-a^2 x^2\right ) \text {arccosh}(a x)^2+\frac {1}{5} c^2 x \left (1-a^2 x^2\right )^2 \text {arccosh}(a x)^2+\frac {1}{25} \left (2 c^2\right ) \int (-1+a x)^2 (1+a x)^2 \, dx+\frac {1}{15} \left (8 c^2\right ) \int \text {arccosh}(a x)^2 \, dx+\frac {1}{15} \left (8 a c^2\right ) \int x \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x) \, dx \\ & = \frac {8 c^2 (-1+a x)^{3/2} (1+a x)^{3/2} \text {arccosh}(a x)}{45 a}-\frac {2 c^2 (-1+a x)^{5/2} (1+a x)^{5/2} \text {arccosh}(a x)}{25 a}+\frac {8}{15} c^2 x \text {arccosh}(a x)^2+\frac {4}{15} c^2 x \left (1-a^2 x^2\right ) \text {arccosh}(a x)^2+\frac {1}{5} c^2 x \left (1-a^2 x^2\right )^2 \text {arccosh}(a x)^2+\frac {1}{25} \left (2 c^2\right ) \int \left (-1+a^2 x^2\right )^2 \, dx-\frac {1}{45} \left (8 c^2\right ) \int (-1+a x) (1+a x) \, dx-\frac {1}{15} \left (16 a c^2\right ) \int \frac {x \text {arccosh}(a x)}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx \\ & = -\frac {16 c^2 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)}{15 a}+\frac {8 c^2 (-1+a x)^{3/2} (1+a x)^{3/2} \text {arccosh}(a x)}{45 a}-\frac {2 c^2 (-1+a x)^{5/2} (1+a x)^{5/2} \text {arccosh}(a x)}{25 a}+\frac {8}{15} c^2 x \text {arccosh}(a x)^2+\frac {4}{15} c^2 x \left (1-a^2 x^2\right ) \text {arccosh}(a x)^2+\frac {1}{5} c^2 x \left (1-a^2 x^2\right )^2 \text {arccosh}(a x)^2+\frac {1}{25} \left (2 c^2\right ) \int \left (1-2 a^2 x^2+a^4 x^4\right ) \, dx-\frac {1}{45} \left (8 c^2\right ) \int \left (-1+a^2 x^2\right ) \, dx+\frac {1}{15} \left (16 c^2\right ) \int 1 \, dx \\ & = \frac {298 c^2 x}{225}-\frac {76}{675} a^2 c^2 x^3+\frac {2}{125} a^4 c^2 x^5-\frac {16 c^2 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)}{15 a}+\frac {8 c^2 (-1+a x)^{3/2} (1+a x)^{3/2} \text {arccosh}(a x)}{45 a}-\frac {2 c^2 (-1+a x)^{5/2} (1+a x)^{5/2} \text {arccosh}(a x)}{25 a}+\frac {8}{15} c^2 x \text {arccosh}(a x)^2+\frac {4}{15} c^2 x \left (1-a^2 x^2\right ) \text {arccosh}(a x)^2+\frac {1}{5} c^2 x \left (1-a^2 x^2\right )^2 \text {arccosh}(a x)^2 \\ \end{align*}
Time = 0.25 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.52 \[ \int \left (c-a^2 c x^2\right )^2 \text {arccosh}(a x)^2 \, dx=\frac {c^2 \left (4470 a x-380 a^3 x^3+54 a^5 x^5-30 \sqrt {-1+a x} \sqrt {1+a x} \left (149-38 a^2 x^2+9 a^4 x^4\right ) \text {arccosh}(a x)+225 a x \left (15-10 a^2 x^2+3 a^4 x^4\right ) \text {arccosh}(a x)^2\right )}{3375 a} \]
[In]
[Out]
Time = 0.50 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.72
method | result | size |
derivativedivides | \(\frac {c^{2} \left (675 a^{5} x^{5} \operatorname {arccosh}\left (a x \right )^{2}-270 a^{4} x^{4} \operatorname {arccosh}\left (a x \right ) \sqrt {a x -1}\, \sqrt {a x +1}-2250 a^{3} x^{3} \operatorname {arccosh}\left (a x \right )^{2}+1140 a^{2} x^{2} \operatorname {arccosh}\left (a x \right ) \sqrt {a x -1}\, \sqrt {a x +1}+54 a^{5} x^{5}+3375 a x \operatorname {arccosh}\left (a x \right )^{2}-4470 \sqrt {a x -1}\, \sqrt {a x +1}\, \operatorname {arccosh}\left (a x \right )-380 a^{3} x^{3}+4470 a x \right )}{3375 a}\) | \(140\) |
default | \(\frac {c^{2} \left (675 a^{5} x^{5} \operatorname {arccosh}\left (a x \right )^{2}-270 a^{4} x^{4} \operatorname {arccosh}\left (a x \right ) \sqrt {a x -1}\, \sqrt {a x +1}-2250 a^{3} x^{3} \operatorname {arccosh}\left (a x \right )^{2}+1140 a^{2} x^{2} \operatorname {arccosh}\left (a x \right ) \sqrt {a x -1}\, \sqrt {a x +1}+54 a^{5} x^{5}+3375 a x \operatorname {arccosh}\left (a x \right )^{2}-4470 \sqrt {a x -1}\, \sqrt {a x +1}\, \operatorname {arccosh}\left (a x \right )-380 a^{3} x^{3}+4470 a x \right )}{3375 a}\) | \(140\) |
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.73 \[ \int \left (c-a^2 c x^2\right )^2 \text {arccosh}(a x)^2 \, dx=\frac {54 \, a^{5} c^{2} x^{5} - 380 \, a^{3} c^{2} x^{3} + 4470 \, a c^{2} x + 225 \, {\left (3 \, a^{5} c^{2} x^{5} - 10 \, a^{3} c^{2} x^{3} + 15 \, a c^{2} x\right )} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )^{2} - 30 \, {\left (9 \, a^{4} c^{2} x^{4} - 38 \, a^{2} c^{2} x^{2} + 149 \, c^{2}\right )} \sqrt {a^{2} x^{2} - 1} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )}{3375 \, a} \]
[In]
[Out]
\[ \int \left (c-a^2 c x^2\right )^2 \text {arccosh}(a x)^2 \, dx=c^{2} \left (\int \left (- 2 a^{2} x^{2} \operatorname {acosh}^{2}{\left (a x \right )}\right )\, dx + \int a^{4} x^{4} \operatorname {acosh}^{2}{\left (a x \right )}\, dx + \int \operatorname {acosh}^{2}{\left (a x \right )}\, dx\right ) \]
[In]
[Out]
none
Time = 0.25 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.69 \[ \int \left (c-a^2 c x^2\right )^2 \text {arccosh}(a x)^2 \, dx=\frac {2}{125} \, a^{4} c^{2} x^{5} - \frac {76}{675} \, a^{2} c^{2} x^{3} + \frac {298}{225} \, c^{2} x - \frac {2}{225} \, {\left (9 \, \sqrt {a^{2} x^{2} - 1} a^{2} c^{2} x^{4} - 38 \, \sqrt {a^{2} x^{2} - 1} c^{2} x^{2} + \frac {149 \, \sqrt {a^{2} x^{2} - 1} c^{2}}{a^{2}}\right )} a \operatorname {arcosh}\left (a x\right ) + \frac {1}{15} \, {\left (3 \, a^{4} c^{2} x^{5} - 10 \, a^{2} c^{2} x^{3} + 15 \, c^{2} x\right )} \operatorname {arcosh}\left (a x\right )^{2} \]
[In]
[Out]
Exception generated. \[ \int \left (c-a^2 c x^2\right )^2 \text {arccosh}(a x)^2 \, dx=\text {Exception raised: TypeError} \]
[In]
[Out]
Timed out. \[ \int \left (c-a^2 c x^2\right )^2 \text {arccosh}(a x)^2 \, dx=\int {\mathrm {acosh}\left (a\,x\right )}^2\,{\left (c-a^2\,c\,x^2\right )}^2 \,d x \]
[In]
[Out]