Integrand size = 13, antiderivative size = 61 \[ \int \frac {f^{a+\frac {b}{x}}}{x^4} \, dx=-\frac {2 f^{a+\frac {b}{x}}}{b^3 \log ^3(f)}+\frac {2 f^{a+\frac {b}{x}}}{b^2 x \log ^2(f)}-\frac {f^{a+\frac {b}{x}}}{b x^2 \log (f)} \]
[Out]
Time = 0.04 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2243, 2240} \[ \int \frac {f^{a+\frac {b}{x}}}{x^4} \, dx=-\frac {2 f^{a+\frac {b}{x}}}{b^3 \log ^3(f)}+\frac {2 f^{a+\frac {b}{x}}}{b^2 x \log ^2(f)}-\frac {f^{a+\frac {b}{x}}}{b x^2 \log (f)} \]
[In]
[Out]
Rule 2240
Rule 2243
Rubi steps \begin{align*} \text {integral}& = -\frac {f^{a+\frac {b}{x}}}{b x^2 \log (f)}-\frac {2 \int \frac {f^{a+\frac {b}{x}}}{x^3} \, dx}{b \log (f)} \\ & = \frac {2 f^{a+\frac {b}{x}}}{b^2 x \log ^2(f)}-\frac {f^{a+\frac {b}{x}}}{b x^2 \log (f)}+\frac {2 \int \frac {f^{a+\frac {b}{x}}}{x^2} \, dx}{b^2 \log ^2(f)} \\ & = -\frac {2 f^{a+\frac {b}{x}}}{b^3 \log ^3(f)}+\frac {2 f^{a+\frac {b}{x}}}{b^2 x \log ^2(f)}-\frac {f^{a+\frac {b}{x}}}{b x^2 \log (f)} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.67 \[ \int \frac {f^{a+\frac {b}{x}}}{x^4} \, dx=-\frac {f^{a+\frac {b}{x}} \left (2 x^2-2 b x \log (f)+b^2 \log ^2(f)\right )}{b^3 x^2 \log ^3(f)} \]
[In]
[Out]
Time = 0.04 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.72
method | result | size |
risch | \(-\frac {\left (\ln \left (f \right )^{2} b^{2}-2 \ln \left (f \right ) b x +2 x^{2}\right ) f^{\frac {a x +b}{x}}}{\ln \left (f \right )^{3} b^{3} x^{2}}\) | \(44\) |
meijerg | \(\frac {f^{a} \left (2-\frac {\left (\frac {3 b^{2} \ln \left (f \right )^{2}}{x^{2}}-\frac {6 b \ln \left (f \right )}{x}+6\right ) {\mathrm e}^{\frac {b \ln \left (f \right )}{x}}}{3}\right )}{\ln \left (f \right )^{3} b^{3}}\) | \(46\) |
parallelrisch | \(\frac {-\ln \left (f \right )^{2} f^{a +\frac {b}{x}} b^{2}+2 b \,f^{a +\frac {b}{x}} x \ln \left (f \right )-2 f^{a +\frac {b}{x}} x^{2}}{x^{2} \ln \left (f \right )^{3} b^{3}}\) | \(60\) |
norman | \(\frac {-\frac {x \,{\mathrm e}^{\left (a +\frac {b}{x}\right ) \ln \left (f \right )}}{b \ln \left (f \right )}+\frac {2 x^{2} {\mathrm e}^{\left (a +\frac {b}{x}\right ) \ln \left (f \right )}}{b^{2} \ln \left (f \right )^{2}}-\frac {2 x^{3} {\mathrm e}^{\left (a +\frac {b}{x}\right ) \ln \left (f \right )}}{b^{3} \ln \left (f \right )^{3}}}{x^{3}}\) | \(73\) |
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.70 \[ \int \frac {f^{a+\frac {b}{x}}}{x^4} \, dx=-\frac {{\left (b^{2} \log \left (f\right )^{2} - 2 \, b x \log \left (f\right ) + 2 \, x^{2}\right )} f^{\frac {a x + b}{x}}}{b^{3} x^{2} \log \left (f\right )^{3}} \]
[In]
[Out]
Time = 0.07 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.64 \[ \int \frac {f^{a+\frac {b}{x}}}{x^4} \, dx=\frac {f^{a + \frac {b}{x}} \left (- b^{2} \log {\left (f \right )}^{2} + 2 b x \log {\left (f \right )} - 2 x^{2}\right )}{b^{3} x^{2} \log {\left (f \right )}^{3}} \]
[In]
[Out]
Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.23 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.36 \[ \int \frac {f^{a+\frac {b}{x}}}{x^4} \, dx=-\frac {f^{a} \Gamma \left (3, -\frac {b \log \left (f\right )}{x}\right )}{b^{3} \log \left (f\right )^{3}} \]
[In]
[Out]
\[ \int \frac {f^{a+\frac {b}{x}}}{x^4} \, dx=\int { \frac {f^{a + \frac {b}{x}}}{x^{4}} \,d x } \]
[In]
[Out]
Time = 0.17 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.74 \[ \int \frac {f^{a+\frac {b}{x}}}{x^4} \, dx=-\frac {f^{a+\frac {b}{x}}\,\left (\frac {1}{b\,\ln \left (f\right )}+\frac {2\,x^2}{b^3\,{\ln \left (f\right )}^3}-\frac {2\,x}{b^2\,{\ln \left (f\right )}^2}\right )}{x^2} \]
[In]
[Out]