Integrand size = 16, antiderivative size = 83 \[ \int \frac {x}{\log ^2\left (c \left (a+b x^2\right )^p\right )} \, dx=\frac {\left (a+b x^2\right ) \left (c \left (a+b x^2\right )^p\right )^{-1/p} \operatorname {ExpIntegralEi}\left (\frac {\log \left (c \left (a+b x^2\right )^p\right )}{p}\right )}{2 b p^2}-\frac {a+b x^2}{2 b p \log \left (c \left (a+b x^2\right )^p\right )} \]
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Time = 0.05 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {2504, 2436, 2334, 2337, 2209} \[ \int \frac {x}{\log ^2\left (c \left (a+b x^2\right )^p\right )} \, dx=\frac {\left (a+b x^2\right ) \left (c \left (a+b x^2\right )^p\right )^{-1/p} \operatorname {ExpIntegralEi}\left (\frac {\log \left (c \left (b x^2+a\right )^p\right )}{p}\right )}{2 b p^2}-\frac {a+b x^2}{2 b p \log \left (c \left (a+b x^2\right )^p\right )} \]
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Rule 2209
Rule 2334
Rule 2337
Rule 2436
Rule 2504
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {1}{\log ^2\left (c (a+b x)^p\right )} \, dx,x,x^2\right ) \\ & = \frac {\text {Subst}\left (\int \frac {1}{\log ^2\left (c x^p\right )} \, dx,x,a+b x^2\right )}{2 b} \\ & = -\frac {a+b x^2}{2 b p \log \left (c \left (a+b x^2\right )^p\right )}+\frac {\text {Subst}\left (\int \frac {1}{\log \left (c x^p\right )} \, dx,x,a+b x^2\right )}{2 b p} \\ & = -\frac {a+b x^2}{2 b p \log \left (c \left (a+b x^2\right )^p\right )}+\frac {\left (\left (a+b x^2\right ) \left (c \left (a+b x^2\right )^p\right )^{-1/p}\right ) \text {Subst}\left (\int \frac {e^{\frac {x}{p}}}{x} \, dx,x,\log \left (c \left (a+b x^2\right )^p\right )\right )}{2 b p^2} \\ & = \frac {\left (a+b x^2\right ) \left (c \left (a+b x^2\right )^p\right )^{-1/p} \text {Ei}\left (\frac {\log \left (c \left (a+b x^2\right )^p\right )}{p}\right )}{2 b p^2}-\frac {a+b x^2}{2 b p \log \left (c \left (a+b x^2\right )^p\right )} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.17 \[ \int \frac {x}{\log ^2\left (c \left (a+b x^2\right )^p\right )} \, dx=-\frac {\left (a+b x^2\right ) \left (c \left (a+b x^2\right )^p\right )^{-1/p} \left (p \left (c \left (a+b x^2\right )^p\right )^{\frac {1}{p}}-\operatorname {ExpIntegralEi}\left (\frac {\log \left (c \left (a+b x^2\right )^p\right )}{p}\right ) \log \left (c \left (a+b x^2\right )^p\right )\right )}{2 b p^2 \log \left (c \left (a+b x^2\right )^p\right )} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 1.77 (sec) , antiderivative size = 421, normalized size of antiderivative = 5.07
method | result | size |
risch | \(-\frac {b \,x^{2}+a}{\left (i \pi \,\operatorname {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}^{2}-i \pi \,\operatorname {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) \operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right ) \operatorname {csgn}\left (i c \right )-i \pi {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}^{3}+i \pi {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )+2 \ln \left (c \right )+2 \ln \left (\left (b \,x^{2}+a \right )^{p}\right )\right ) p b}-\frac {\left (b \,x^{2}+a \right ) c^{-\frac {1}{p}} {\left (\left (b \,x^{2}+a \right )^{p}\right )}^{-\frac {1}{p}} {\mathrm e}^{\frac {i \pi \,\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right ) \left (-\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )+\operatorname {csgn}\left (i c \right )\right ) \left (-\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )+\operatorname {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right )\right )}{2 p}} \operatorname {Ei}_{1}\left (-\ln \left (b \,x^{2}+a \right )-\frac {i \pi \,\operatorname {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}^{2}-i \pi \,\operatorname {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) \operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right ) \operatorname {csgn}\left (i c \right )-i \pi {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}^{3}+i \pi {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )+2 \ln \left (c \right )+2 \ln \left (\left (b \,x^{2}+a \right )^{p}\right )-2 p \ln \left (b \,x^{2}+a \right )}{2 p}\right )}{2 p^{2} b}\) | \(421\) |
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Time = 0.32 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.94 \[ \int \frac {x}{\log ^2\left (c \left (a+b x^2\right )^p\right )} \, dx=-\frac {{\left (b p x^{2} + a p\right )} c^{\left (\frac {1}{p}\right )} - {\left (p \log \left (b x^{2} + a\right ) + \log \left (c\right )\right )} \operatorname {log\_integral}\left ({\left (b x^{2} + a\right )} c^{\left (\frac {1}{p}\right )}\right )}{2 \, {\left (b p^{3} \log \left (b x^{2} + a\right ) + b p^{2} \log \left (c\right )\right )} c^{\left (\frac {1}{p}\right )}} \]
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\[ \int \frac {x}{\log ^2\left (c \left (a+b x^2\right )^p\right )} \, dx=\int \frac {x}{\log {\left (c \left (a + b x^{2}\right )^{p} \right )}^{2}}\, dx \]
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\[ \int \frac {x}{\log ^2\left (c \left (a+b x^2\right )^p\right )} \, dx=\int { \frac {x}{\log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{2}} \,d x } \]
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Time = 0.30 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.70 \[ \int \frac {x}{\log ^2\left (c \left (a+b x^2\right )^p\right )} \, dx=-\frac {{\left (b x^{2} + a\right )} p}{2 \, {\left (b p^{3} \log \left (b x^{2} + a\right ) + b p^{2} \log \left (c\right )\right )}} + \frac {p {\rm Ei}\left (\frac {\log \left (c\right )}{p} + \log \left (b x^{2} + a\right )\right ) \log \left (b x^{2} + a\right )}{2 \, {\left (b p^{3} \log \left (b x^{2} + a\right ) + b p^{2} \log \left (c\right )\right )} c^{\left (\frac {1}{p}\right )}} + \frac {{\rm Ei}\left (\frac {\log \left (c\right )}{p} + \log \left (b x^{2} + a\right )\right ) \log \left (c\right )}{2 \, {\left (b p^{3} \log \left (b x^{2} + a\right ) + b p^{2} \log \left (c\right )\right )} c^{\left (\frac {1}{p}\right )}} \]
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Timed out. \[ \int \frac {x}{\log ^2\left (c \left (a+b x^2\right )^p\right )} \, dx=\int \frac {x}{{\ln \left (c\,{\left (b\,x^2+a\right )}^p\right )}^2} \,d x \]
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