Integrand size = 24, antiderivative size = 315 \[ \int \frac {x^4 \text {arccosh}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx=-\frac {45 x^2 \sqrt {-1+a x}}{128 a^3 \sqrt {1-a x}}-\frac {3 x^4 \sqrt {-1+a x}}{128 a \sqrt {1-a x}}-\frac {45 x \sqrt {1-a x} \sqrt {1+a x} \text {arccosh}(a x)}{64 a^4}-\frac {3 x^3 \sqrt {1-a x} \sqrt {1+a x} \text {arccosh}(a x)}{32 a^2}+\frac {45 \sqrt {-1+a x} \text {arccosh}(a x)^2}{128 a^5 \sqrt {1-a x}}-\frac {9 x^2 \sqrt {-1+a x} \text {arccosh}(a x)^2}{16 a^3 \sqrt {1-a x}}-\frac {3 x^4 \sqrt {-1+a x} \text {arccosh}(a x)^2}{16 a \sqrt {1-a x}}-\frac {3 x \sqrt {1-a^2 x^2} \text {arccosh}(a x)^3}{8 a^4}-\frac {x^3 \sqrt {1-a^2 x^2} \text {arccosh}(a x)^3}{4 a^2}+\frac {3 \sqrt {-1+a x} \text {arccosh}(a x)^4}{32 a^5 \sqrt {1-a x}} \]
[Out]
Time = 0.65 (sec) , antiderivative size = 315, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5938, 5892, 5883, 5939, 5893, 30} \[ \int \frac {x^4 \text {arccosh}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx=\frac {3 \sqrt {a x-1} \text {arccosh}(a x)^4}{32 a^5 \sqrt {1-a x}}+\frac {45 \sqrt {a x-1} \text {arccosh}(a x)^2}{128 a^5 \sqrt {1-a x}}-\frac {45 x \sqrt {1-a x} \sqrt {a x+1} \text {arccosh}(a x)}{64 a^4}-\frac {9 x^2 \sqrt {a x-1} \text {arccosh}(a x)^2}{16 a^3 \sqrt {1-a x}}-\frac {45 x^2 \sqrt {a x-1}}{128 a^3 \sqrt {1-a x}}-\frac {3 x^3 \sqrt {1-a x} \sqrt {a x+1} \text {arccosh}(a x)}{32 a^2}-\frac {x^3 \sqrt {1-a^2 x^2} \text {arccosh}(a x)^3}{4 a^2}-\frac {3 x \sqrt {1-a^2 x^2} \text {arccosh}(a x)^3}{8 a^4}-\frac {3 x^4 \sqrt {a x-1} \text {arccosh}(a x)^2}{16 a \sqrt {1-a x}}-\frac {3 x^4 \sqrt {a x-1}}{128 a \sqrt {1-a x}} \]
[In]
[Out]
Rule 30
Rule 5883
Rule 5892
Rule 5893
Rule 5938
Rule 5939
Rubi steps \begin{align*} \text {integral}& = -\frac {x^3 \sqrt {1-a^2 x^2} \text {arccosh}(a x)^3}{4 a^2}+\frac {3 \int \frac {x^2 \text {arccosh}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx}{4 a^2}-\frac {\left (3 \sqrt {-1+a x}\right ) \int x^3 \text {arccosh}(a x)^2 \, dx}{4 a \sqrt {1-a x}} \\ & = -\frac {3 x^4 \sqrt {-1+a x} \text {arccosh}(a x)^2}{16 a \sqrt {1-a x}}-\frac {3 x \sqrt {1-a^2 x^2} \text {arccosh}(a x)^3}{8 a^4}-\frac {x^3 \sqrt {1-a^2 x^2} \text {arccosh}(a x)^3}{4 a^2}+\frac {3 \int \frac {\text {arccosh}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx}{8 a^4}+\frac {\left (3 \sqrt {-1+a x}\right ) \int \frac {x^4 \text {arccosh}(a x)}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{8 \sqrt {1-a x}}-\frac {\left (9 \sqrt {-1+a x}\right ) \int x \text {arccosh}(a x)^2 \, dx}{8 a^3 \sqrt {1-a x}} \\ & = -\frac {3 x^3 \sqrt {1-a x} \sqrt {1+a x} \text {arccosh}(a x)}{32 a^2}-\frac {9 x^2 \sqrt {-1+a x} \text {arccosh}(a x)^2}{16 a^3 \sqrt {1-a x}}-\frac {3 x^4 \sqrt {-1+a x} \text {arccosh}(a x)^2}{16 a \sqrt {1-a x}}-\frac {3 x \sqrt {1-a^2 x^2} \text {arccosh}(a x)^3}{8 a^4}-\frac {x^3 \sqrt {1-a^2 x^2} \text {arccosh}(a x)^3}{4 a^2}+\frac {3 \sqrt {-1+a x} \text {arccosh}(a x)^4}{32 a^5 \sqrt {1-a x}}+\frac {\left (9 \sqrt {-1+a x}\right ) \int \frac {x^2 \text {arccosh}(a x)}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{32 a^2 \sqrt {1-a x}}+\frac {\left (9 \sqrt {-1+a x}\right ) \int \frac {x^2 \text {arccosh}(a x)}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{8 a^2 \sqrt {1-a x}}-\frac {\left (3 \sqrt {-1+a x}\right ) \int x^3 \, dx}{32 a \sqrt {1-a x}} \\ & = -\frac {3 x^4 \sqrt {-1+a x}}{128 a \sqrt {1-a x}}-\frac {45 x \sqrt {1-a x} \sqrt {1+a x} \text {arccosh}(a x)}{64 a^4}-\frac {3 x^3 \sqrt {1-a x} \sqrt {1+a x} \text {arccosh}(a x)}{32 a^2}-\frac {9 x^2 \sqrt {-1+a x} \text {arccosh}(a x)^2}{16 a^3 \sqrt {1-a x}}-\frac {3 x^4 \sqrt {-1+a x} \text {arccosh}(a x)^2}{16 a \sqrt {1-a x}}-\frac {3 x \sqrt {1-a^2 x^2} \text {arccosh}(a x)^3}{8 a^4}-\frac {x^3 \sqrt {1-a^2 x^2} \text {arccosh}(a x)^3}{4 a^2}+\frac {3 \sqrt {-1+a x} \text {arccosh}(a x)^4}{32 a^5 \sqrt {1-a x}}+\frac {\left (9 \sqrt {-1+a x}\right ) \int \frac {\text {arccosh}(a x)}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{64 a^4 \sqrt {1-a x}}+\frac {\left (9 \sqrt {-1+a x}\right ) \int \frac {\text {arccosh}(a x)}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{16 a^4 \sqrt {1-a x}}-\frac {\left (9 \sqrt {-1+a x}\right ) \int x \, dx}{64 a^3 \sqrt {1-a x}}-\frac {\left (9 \sqrt {-1+a x}\right ) \int x \, dx}{16 a^3 \sqrt {1-a x}} \\ & = -\frac {45 x^2 \sqrt {-1+a x}}{128 a^3 \sqrt {1-a x}}-\frac {3 x^4 \sqrt {-1+a x}}{128 a \sqrt {1-a x}}-\frac {45 x \sqrt {1-a x} \sqrt {1+a x} \text {arccosh}(a x)}{64 a^4}-\frac {3 x^3 \sqrt {1-a x} \sqrt {1+a x} \text {arccosh}(a x)}{32 a^2}+\frac {45 \sqrt {-1+a x} \text {arccosh}(a x)^2}{128 a^5 \sqrt {1-a x}}-\frac {9 x^2 \sqrt {-1+a x} \text {arccosh}(a x)^2}{16 a^3 \sqrt {1-a x}}-\frac {3 x^4 \sqrt {-1+a x} \text {arccosh}(a x)^2}{16 a \sqrt {1-a x}}-\frac {3 x \sqrt {1-a^2 x^2} \text {arccosh}(a x)^3}{8 a^4}-\frac {x^3 \sqrt {1-a^2 x^2} \text {arccosh}(a x)^3}{4 a^2}+\frac {3 \sqrt {-1+a x} \text {arccosh}(a x)^4}{32 a^5 \sqrt {1-a x}} \\ \end{align*}
Time = 0.57 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.43 \[ \int \frac {x^4 \text {arccosh}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx=\frac {\sqrt {\frac {-1+a x}{1+a x}} (1+a x) \left (-192 \left (1+2 \text {arccosh}(a x)^2\right ) \cosh (2 \text {arccosh}(a x))-3 \left (1+8 \text {arccosh}(a x)^2\right ) \cosh (4 \text {arccosh}(a x))+4 \text {arccosh}(a x) \left (24 \text {arccosh}(a x)^3+32 \left (3+2 \text {arccosh}(a x)^2\right ) \sinh (2 \text {arccosh}(a x))+\left (3+8 \text {arccosh}(a x)^2\right ) \sinh (4 \text {arccosh}(a x))\right )\right )}{1024 a^5 \sqrt {-((-1+a x) (1+a x))}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(519\) vs. \(2(259)=518\).
Time = 0.89 (sec) , antiderivative size = 520, normalized size of antiderivative = 1.65
method | result | size |
default | \(-\frac {3 \sqrt {-a^{2} x^{2}+1}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \operatorname {arccosh}\left (a x \right )^{4}}{32 a^{5} \left (a^{2} x^{2}-1\right )}-\frac {\sqrt {-a^{2} x^{2}+1}\, \left (8 a^{5} x^{5}-12 a^{3} x^{3}+8 \sqrt {a x -1}\, \sqrt {a x +1}\, a^{4} x^{4}+4 a x -8 a^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}+\sqrt {a x -1}\, \sqrt {a x +1}\right ) \left (32 \operatorname {arccosh}\left (a x \right )^{3}-24 \operatorname {arccosh}\left (a x \right )^{2}+12 \,\operatorname {arccosh}\left (a x \right )-3\right )}{2048 a^{5} \left (a^{2} x^{2}-1\right )}-\frac {\sqrt {-a^{2} x^{2}+1}\, \left (2 a^{3} x^{3}-2 a x +2 a^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}-\sqrt {a x -1}\, \sqrt {a x +1}\right ) \left (4 \operatorname {arccosh}\left (a x \right )^{3}-6 \operatorname {arccosh}\left (a x \right )^{2}+6 \,\operatorname {arccosh}\left (a x \right )-3\right )}{32 a^{5} \left (a^{2} x^{2}-1\right )}-\frac {\sqrt {-a^{2} x^{2}+1}\, \left (2 a^{3} x^{3}-2 a x -2 a^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}+\sqrt {a x -1}\, \sqrt {a x +1}\right ) \left (4 \operatorname {arccosh}\left (a x \right )^{3}+6 \operatorname {arccosh}\left (a x \right )^{2}+6 \,\operatorname {arccosh}\left (a x \right )+3\right )}{32 a^{5} \left (a^{2} x^{2}-1\right )}-\frac {\sqrt {-a^{2} x^{2}+1}\, \left (8 a^{5} x^{5}-12 a^{3} x^{3}-8 \sqrt {a x -1}\, \sqrt {a x +1}\, a^{4} x^{4}+4 a x +8 a^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}-\sqrt {a x -1}\, \sqrt {a x +1}\right ) \left (32 \operatorname {arccosh}\left (a x \right )^{3}+24 \operatorname {arccosh}\left (a x \right )^{2}+12 \,\operatorname {arccosh}\left (a x \right )+3\right )}{2048 a^{5} \left (a^{2} x^{2}-1\right )}\) | \(520\) |
[In]
[Out]
\[ \int \frac {x^4 \text {arccosh}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx=\int { \frac {x^{4} \operatorname {arcosh}\left (a x\right )^{3}}{\sqrt {-a^{2} x^{2} + 1}} \,d x } \]
[In]
[Out]
\[ \int \frac {x^4 \text {arccosh}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx=\int \frac {x^{4} \operatorname {acosh}^{3}{\left (a x \right )}}{\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \]
[In]
[Out]
Exception generated. \[ \int \frac {x^4 \text {arccosh}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx=\text {Exception raised: RuntimeError} \]
[In]
[Out]
\[ \int \frac {x^4 \text {arccosh}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx=\int { \frac {x^{4} \operatorname {arcosh}\left (a x\right )^{3}}{\sqrt {-a^{2} x^{2} + 1}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {x^4 \text {arccosh}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx=\int \frac {x^4\,{\mathrm {acosh}\left (a\,x\right )}^3}{\sqrt {1-a^2\,x^2}} \,d x \]
[In]
[Out]