Integrand size = 20, antiderivative size = 101 \[ \int \frac {1}{(d x)^{3/2} \left (a+b \log \left (c x^n\right )\right )^2} \, dx=-\frac {e^{\frac {a}{2 b n}} \left (c x^n\right )^{\left .\frac {1}{2}\right /n} \operatorname {ExpIntegralEi}\left (\frac {-a-b \log \left (c x^n\right )}{2 b n}\right )}{2 b^2 d n^2 \sqrt {d x}}-\frac {1}{b d n \sqrt {d x} \left (a+b \log \left (c x^n\right )\right )} \]
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Time = 0.06 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.97, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {2343, 2347, 2209} \[ \int \frac {1}{(d x)^{3/2} \left (a+b \log \left (c x^n\right )\right )^2} \, dx=-\frac {e^{\frac {a}{2 b n}} \left (c x^n\right )^{\left .\frac {1}{2}\right /n} \operatorname {ExpIntegralEi}\left (-\frac {a+b \log \left (c x^n\right )}{2 b n}\right )}{2 b^2 d n^2 \sqrt {d x}}-\frac {1}{b d n \sqrt {d x} \left (a+b \log \left (c x^n\right )\right )} \]
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Rule 2209
Rule 2343
Rule 2347
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{b d n \sqrt {d x} \left (a+b \log \left (c x^n\right )\right )}-\frac {\int \frac {1}{(d x)^{3/2} \left (a+b \log \left (c x^n\right )\right )} \, dx}{2 b n} \\ & = -\frac {1}{b d n \sqrt {d x} \left (a+b \log \left (c x^n\right )\right )}-\frac {\left (c x^n\right )^{\left .\frac {1}{2}\right /n} \text {Subst}\left (\int \frac {e^{-\frac {x}{2 n}}}{a+b x} \, dx,x,\log \left (c x^n\right )\right )}{2 b d n^2 \sqrt {d x}} \\ & = -\frac {e^{\frac {a}{2 b n}} \left (c x^n\right )^{\left .\frac {1}{2}\right /n} \text {Ei}\left (-\frac {a+b \log \left (c x^n\right )}{2 b n}\right )}{2 b^2 d n^2 \sqrt {d x}}-\frac {1}{b d n \sqrt {d x} \left (a+b \log \left (c x^n\right )\right )} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.92 \[ \int \frac {1}{(d x)^{3/2} \left (a+b \log \left (c x^n\right )\right )^2} \, dx=-\frac {x \left (2 b n+e^{\frac {a}{2 b n}} \left (c x^n\right )^{\left .\frac {1}{2}\right /n} \operatorname {ExpIntegralEi}\left (-\frac {a+b \log \left (c x^n\right )}{2 b n}\right ) \left (a+b \log \left (c x^n\right )\right )\right )}{2 b^2 n^2 (d x)^{3/2} \left (a+b \log \left (c x^n\right )\right )} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.93 (sec) , antiderivative size = 429, normalized size of antiderivative = 4.25
method | result | size |
risch | \(-\frac {2}{b n \sqrt {d x}\, \left (2 a +2 b \ln \left (c \right )+2 b \ln \left ({\mathrm e}^{n \ln \left (x \right )}\right )-i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i {\mathrm e}^{n \ln \left (x \right )}\right ) \operatorname {csgn}\left (i c \,{\mathrm e}^{n \ln \left (x \right )}\right )+i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,{\mathrm e}^{n \ln \left (x \right )}\right )^{2}+i b \pi \,\operatorname {csgn}\left (i {\mathrm e}^{n \ln \left (x \right )}\right ) \operatorname {csgn}\left (i c \,{\mathrm e}^{n \ln \left (x \right )}\right )^{2}-i b \pi \operatorname {csgn}\left (i c \,{\mathrm e}^{n \ln \left (x \right )}\right )^{3}\right ) d}+\frac {{\mathrm e}^{-\frac {i \left (b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i {\mathrm e}^{n \ln \left (x \right )}\right ) \operatorname {csgn}\left (i c \,{\mathrm e}^{n \ln \left (x \right )}\right )-b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,{\mathrm e}^{n \ln \left (x \right )}\right )^{2}-b \pi \,\operatorname {csgn}\left (i {\mathrm e}^{n \ln \left (x \right )}\right ) \operatorname {csgn}\left (i c \,{\mathrm e}^{n \ln \left (x \right )}\right )^{2}+b \pi \operatorname {csgn}\left (i c \,{\mathrm e}^{n \ln \left (x \right )}\right )^{3}+2 i b n \left (\ln \left (x \right )-\ln \left (d x \right )\right )+2 i b \ln \left (c \right )+2 i b \left (\ln \left ({\mathrm e}^{n \ln \left (x \right )}\right )-n \ln \left (x \right )\right )+2 i a \right )}{4 b n}} \operatorname {Ei}_{1}\left (\frac {\ln \left (d x \right )}{2}-\frac {i \left (b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i {\mathrm e}^{n \ln \left (x \right )}\right ) \operatorname {csgn}\left (i c \,{\mathrm e}^{n \ln \left (x \right )}\right )-b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,{\mathrm e}^{n \ln \left (x \right )}\right )^{2}-b \pi \,\operatorname {csgn}\left (i {\mathrm e}^{n \ln \left (x \right )}\right ) \operatorname {csgn}\left (i c \,{\mathrm e}^{n \ln \left (x \right )}\right )^{2}+b \pi \operatorname {csgn}\left (i c \,{\mathrm e}^{n \ln \left (x \right )}\right )^{3}+2 i b n \left (\ln \left (x \right )-\ln \left (d x \right )\right )+2 i b \ln \left (c \right )+2 i b \left (\ln \left ({\mathrm e}^{n \ln \left (x \right )}\right )-n \ln \left (x \right )\right )+2 i a \right )}{4 b n}\right )}{2 b^{2} n^{2} d}\) | \(429\) |
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\[ \int \frac {1}{(d x)^{3/2} \left (a+b \log \left (c x^n\right )\right )^2} \, dx=\int { \frac {1}{\left (d x\right )^{\frac {3}{2}} {\left (b \log \left (c x^{n}\right ) + a\right )}^{2}} \,d x } \]
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\[ \int \frac {1}{(d x)^{3/2} \left (a+b \log \left (c x^n\right )\right )^2} \, dx=\int \frac {1}{\left (d x\right )^{\frac {3}{2}} \left (a + b \log {\left (c x^{n} \right )}\right )^{2}}\, dx \]
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\[ \int \frac {1}{(d x)^{3/2} \left (a+b \log \left (c x^n\right )\right )^2} \, dx=\int { \frac {1}{\left (d x\right )^{\frac {3}{2}} {\left (b \log \left (c x^{n}\right ) + a\right )}^{2}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 293 vs. \(2 (85) = 170\).
Time = 0.34 (sec) , antiderivative size = 293, normalized size of antiderivative = 2.90 \[ \int \frac {1}{(d x)^{3/2} \left (a+b \log \left (c x^n\right )\right )^2} \, dx=-\frac {\frac {b c^{\frac {1}{2 \, n}} n {\rm Ei}\left (-\frac {\log \left (c\right )}{2 \, n} - \frac {a}{2 \, b n} - \frac {1}{2} \, \log \left (x\right )\right ) e^{\left (\frac {a}{2 \, b n}\right )} \log \left (x\right )}{b^{3} \sqrt {d} n^{3} \log \left (x\right ) + b^{3} \sqrt {d} n^{2} \log \left (c\right ) + a b^{2} \sqrt {d} n^{2}} + \frac {b c^{\frac {1}{2 \, n}} {\rm Ei}\left (-\frac {\log \left (c\right )}{2 \, n} - \frac {a}{2 \, b n} - \frac {1}{2} \, \log \left (x\right )\right ) e^{\left (\frac {a}{2 \, b n}\right )} \log \left (c\right )}{b^{3} \sqrt {d} n^{3} \log \left (x\right ) + b^{3} \sqrt {d} n^{2} \log \left (c\right ) + a b^{2} \sqrt {d} n^{2}} + \frac {a c^{\frac {1}{2 \, n}} {\rm Ei}\left (-\frac {\log \left (c\right )}{2 \, n} - \frac {a}{2 \, b n} - \frac {1}{2} \, \log \left (x\right )\right ) e^{\left (\frac {a}{2 \, b n}\right )}}{b^{3} \sqrt {d} n^{3} \log \left (x\right ) + b^{3} \sqrt {d} n^{2} \log \left (c\right ) + a b^{2} \sqrt {d} n^{2}} + \frac {2 \, b n}{{\left (b^{3} \sqrt {d} n^{3} \log \left (x\right ) + b^{3} \sqrt {d} n^{2} \log \left (c\right ) + a b^{2} \sqrt {d} n^{2}\right )} \sqrt {x}}}{2 \, d} \]
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Timed out. \[ \int \frac {1}{(d x)^{3/2} \left (a+b \log \left (c x^n\right )\right )^2} \, dx=\int \frac {1}{{\left (d\,x\right )}^{3/2}\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2} \,d x \]
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