Integrand size = 27, antiderivative size = 186 \[ \int e^{-3 \coth ^{-1}(a x)} x^2 \sqrt {c-a^2 c x^2} \, dx=-\frac {4 \sqrt {c-a^2 c x^2}}{a^3 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {2 x \sqrt {c-a^2 c x^2}}{a^2 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {x^2 \sqrt {c-a^2 c x^2}}{a \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {x^3 \sqrt {c-a^2 c x^2}}{4 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {4 \sqrt {c-a^2 c x^2} \log (1+a x)}{a^4 \sqrt {1-\frac {1}{a^2 x^2}} x} \]
[Out]
Time = 0.19 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {6327, 6328, 90} \[ \int e^{-3 \coth ^{-1}(a x)} x^2 \sqrt {c-a^2 c x^2} \, dx=-\frac {x^2 \sqrt {c-a^2 c x^2}}{a \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {2 x \sqrt {c-a^2 c x^2}}{a^2 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {x^3 \sqrt {c-a^2 c x^2}}{4 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {4 \sqrt {c-a^2 c x^2} \log (a x+1)}{a^4 x \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {4 \sqrt {c-a^2 c x^2}}{a^3 \sqrt {1-\frac {1}{a^2 x^2}}} \]
[In]
[Out]
Rule 90
Rule 6327
Rule 6328
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {c-a^2 c x^2} \int e^{-3 \coth ^{-1}(a x)} \sqrt {1-\frac {1}{a^2 x^2}} x^3 \, dx}{\sqrt {1-\frac {1}{a^2 x^2}} x} \\ & = \frac {\sqrt {c-a^2 c x^2} \int \frac {x^2 (-1+a x)^2}{1+a x} \, dx}{a \sqrt {1-\frac {1}{a^2 x^2}} x} \\ & = \frac {\sqrt {c-a^2 c x^2} \int \left (-\frac {4}{a^2}+\frac {4 x}{a}-3 x^2+a x^3+\frac {4}{a^2 (1+a x)}\right ) \, dx}{a \sqrt {1-\frac {1}{a^2 x^2}} x} \\ & = -\frac {4 \sqrt {c-a^2 c x^2}}{a^3 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {2 x \sqrt {c-a^2 c x^2}}{a^2 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {x^2 \sqrt {c-a^2 c x^2}}{a \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {x^3 \sqrt {c-a^2 c x^2}}{4 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {4 \sqrt {c-a^2 c x^2} \log (1+a x)}{a^4 \sqrt {1-\frac {1}{a^2 x^2}} x} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.40 \[ \int e^{-3 \coth ^{-1}(a x)} x^2 \sqrt {c-a^2 c x^2} \, dx=\frac {\sqrt {c-a^2 c x^2} \left (-\frac {4 x}{a^3}+\frac {2 x^2}{a^2}-\frac {x^3}{a}+\frac {x^4}{4}+\frac {4 \log (1+a x)}{a^4}\right )}{\sqrt {1-\frac {1}{a^2 x^2}} x} \]
[In]
[Out]
Time = 0.55 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.45
method | result | size |
default | \(\frac {\left (a^{4} x^{4}-4 a^{3} x^{3}+8 a^{2} x^{2}-16 a x +16 \ln \left (a x +1\right )\right ) \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (a x +1\right ) \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{4 a^{3} \left (a x -1\right )^{2}}\) | \(83\) |
[In]
[Out]
none
Time = 0.25 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.26 \[ \int e^{-3 \coth ^{-1}(a x)} x^2 \sqrt {c-a^2 c x^2} \, dx=\frac {{\left (a^{4} x^{4} - 4 \, a^{3} x^{3} + 8 \, a^{2} x^{2} - 16 \, a x + 16 \, \log \left (a x + 1\right )\right )} \sqrt {-a^{2} c}}{4 \, a^{4}} \]
[In]
[Out]
Timed out. \[ \int e^{-3 \coth ^{-1}(a x)} x^2 \sqrt {c-a^2 c x^2} \, dx=\text {Timed out} \]
[In]
[Out]
\[ \int e^{-3 \coth ^{-1}(a x)} x^2 \sqrt {c-a^2 c x^2} \, dx=\int { \sqrt {-a^{2} c x^{2} + c} x^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} \,d x } \]
[In]
[Out]
\[ \int e^{-3 \coth ^{-1}(a x)} x^2 \sqrt {c-a^2 c x^2} \, dx=\int { \sqrt {-a^{2} c x^{2} + c} x^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} \,d x } \]
[In]
[Out]
Timed out. \[ \int e^{-3 \coth ^{-1}(a x)} x^2 \sqrt {c-a^2 c x^2} \, dx=\int x^2\,\sqrt {c-a^2\,c\,x^2}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2} \,d x \]
[In]
[Out]