Integrand size = 27, antiderivative size = 221 \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^4} \, dx=-\frac {\sqrt {c-\frac {c}{a^2 x^2}}}{4 a \sqrt {1-\frac {1}{a^2 x^2}} x^4}+\frac {\sqrt {c-\frac {c}{a^2 x^2}}}{\sqrt {1-\frac {1}{a^2 x^2}} x^3}-\frac {2 a \sqrt {c-\frac {c}{a^2 x^2}}}{\sqrt {1-\frac {1}{a^2 x^2}} x^2}+\frac {4 a^2 \sqrt {c-\frac {c}{a^2 x^2}}}{\sqrt {1-\frac {1}{a^2 x^2}} x}+\frac {4 a^3 \sqrt {c-\frac {c}{a^2 x^2}} \log (x)}{\sqrt {1-\frac {1}{a^2 x^2}}}-\frac {4 a^3 \sqrt {c-\frac {c}{a^2 x^2}} \log (1+a x)}{\sqrt {1-\frac {1}{a^2 x^2}}} \]
[Out]
Time = 0.21 (sec) , antiderivative size = 221, 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 = {6332, 6328, 90} \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^4} \, dx=\frac {4 a^2 \sqrt {c-\frac {c}{a^2 x^2}}}{x \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {2 a \sqrt {c-\frac {c}{a^2 x^2}}}{x^2 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {\sqrt {c-\frac {c}{a^2 x^2}}}{4 a x^4 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {c-\frac {c}{a^2 x^2}}}{x^3 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {4 a^3 \log (x) \sqrt {c-\frac {c}{a^2 x^2}}}{\sqrt {1-\frac {1}{a^2 x^2}}}-\frac {4 a^3 \sqrt {c-\frac {c}{a^2 x^2}} \log (a x+1)}{\sqrt {1-\frac {1}{a^2 x^2}}} \]
[In]
[Out]
Rule 90
Rule 6328
Rule 6332
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {c-\frac {c}{a^2 x^2}} \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {1-\frac {1}{a^2 x^2}}}{x^4} \, dx}{\sqrt {1-\frac {1}{a^2 x^2}}} \\ & = \frac {\sqrt {c-\frac {c}{a^2 x^2}} \int \frac {(-1+a x)^2}{x^5 (1+a x)} \, dx}{a \sqrt {1-\frac {1}{a^2 x^2}}} \\ & = \frac {\sqrt {c-\frac {c}{a^2 x^2}} \int \left (\frac {1}{x^5}-\frac {3 a}{x^4}+\frac {4 a^2}{x^3}-\frac {4 a^3}{x^2}+\frac {4 a^4}{x}-\frac {4 a^5}{1+a x}\right ) \, dx}{a \sqrt {1-\frac {1}{a^2 x^2}}} \\ & = -\frac {\sqrt {c-\frac {c}{a^2 x^2}}}{4 a \sqrt {1-\frac {1}{a^2 x^2}} x^4}+\frac {\sqrt {c-\frac {c}{a^2 x^2}}}{\sqrt {1-\frac {1}{a^2 x^2}} x^3}-\frac {2 a \sqrt {c-\frac {c}{a^2 x^2}}}{\sqrt {1-\frac {1}{a^2 x^2}} x^2}+\frac {4 a^2 \sqrt {c-\frac {c}{a^2 x^2}}}{\sqrt {1-\frac {1}{a^2 x^2}} x}+\frac {4 a^3 \sqrt {c-\frac {c}{a^2 x^2}} \log (x)}{\sqrt {1-\frac {1}{a^2 x^2}}}-\frac {4 a^3 \sqrt {c-\frac {c}{a^2 x^2}} \log (1+a x)}{\sqrt {1-\frac {1}{a^2 x^2}}} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.34 \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^4} \, dx=\frac {\sqrt {c-\frac {c}{a^2 x^2}} \left (-\frac {1}{4 a x^4}+\frac {1}{x^3}-\frac {2 a}{x^2}+\frac {4 a^2}{x}+4 a^3 \log (x)-4 a^3 \log (1+a x)\right )}{\sqrt {1-\frac {1}{a^2 x^2}}} \]
[In]
[Out]
Time = 0.06 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.44
method | result | size |
default | \(-\frac {\left (16 \ln \left (a x +1\right ) x^{4} a^{4}-16 \ln \left (x \right ) x^{4} a^{4}-16 a^{3} x^{3}+8 a^{2} x^{2}-4 a x +1\right ) \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}\, \left (a x +1\right ) \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{4 \left (a x -1\right )^{2} x^{3}}\) | \(98\) |
[In]
[Out]
none
Time = 0.24 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.45 \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^4} \, dx=\frac {16 \, a^{5} \sqrt {c} x^{4} \log \left (\frac {2 \, a^{3} c x^{2} + 2 \, a^{2} c x - \sqrt {a^{2} c} {\left (2 \, a x + 1\right )} \sqrt {c} + a c}{a x^{2} + x}\right ) + {\left (16 \, a^{3} x^{3} - 8 \, a^{2} x^{2} + 4 \, a x - 1\right )} \sqrt {a^{2} c}}{4 \, a^{2} x^{4}} \]
[In]
[Out]
Timed out. \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^4} \, dx=\text {Timed out} \]
[In]
[Out]
\[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^4} \, dx=\int { \frac {\sqrt {c - \frac {c}{a^{2} x^{2}}} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}{x^{4}} \,d x } \]
[In]
[Out]
\[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^4} \, dx=\int { \frac {\sqrt {c - \frac {c}{a^{2} x^{2}}} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}{x^{4}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^4} \, dx=\int \frac {\sqrt {c-\frac {c}{a^2\,x^2}}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{x^4} \,d x \]
[In]
[Out]