Optimal. Leaf size=98 \[ \frac {\sqrt {d x} e^{-\frac {a}{2 b n}} \left (c x^n\right )^{\left .-\frac {1}{2}\right /n} \text {Ei}\left (\frac {a+b \log \left (c x^n\right )}{2 b n}\right )}{2 b^2 d n^2}-\frac {\sqrt {d x}}{b d n \left (a+b \log \left (c x^n\right )\right )} \]
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Rubi [A] time = 0.09, antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {2306, 2310, 2178} \[ \frac {\sqrt {d x} e^{-\frac {a}{2 b n}} \left (c x^n\right )^{\left .-\frac {1}{2}\right /n} \text {Ei}\left (\frac {a+b \log \left (c x^n\right )}{2 b n}\right )}{2 b^2 d n^2}-\frac {\sqrt {d x}}{b d n \left (a+b \log \left (c x^n\right )\right )} \]
Antiderivative was successfully verified.
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Rule 2178
Rule 2306
Rule 2310
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {d x} \left (a+b \log \left (c x^n\right )\right )^2} \, dx &=-\frac {\sqrt {d x}}{b d n \left (a+b \log \left (c x^n\right )\right )}+\frac {\int \frac {1}{\sqrt {d x} \left (a+b \log \left (c x^n\right )\right )} \, dx}{2 b n}\\ &=-\frac {\sqrt {d x}}{b d n \left (a+b \log \left (c x^n\right )\right )}+\frac {\left (\sqrt {d x} \left (c x^n\right )^{\left .-\frac {1}{2}\right /n}\right ) \operatorname {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}\\ &=\frac {e^{-\frac {a}{2 b n}} \sqrt {d x} \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}-\frac {\sqrt {d x}}{b d n \left (a+b \log \left (c x^n\right )\right )}\\ \end {align*}
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Mathematica [A] time = 0.20, size = 83, normalized size = 0.85 \[ \frac {x \left (e^{-\frac {a}{2 b n}} \left (c x^n\right )^{\left .-\frac {1}{2}\right /n} \text {Ei}\left (\frac {a+b \log \left (c x^n\right )}{2 b n}\right )-\frac {2 b n}{a+b \log \left (c x^n\right )}\right )}{2 b^2 n^2 \sqrt {d x}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.41, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {d x}}{b^{2} d x \log \left (c x^{n}\right )^{2} + 2 \, a b d x \log \left (c x^{n}\right ) + a^{2} d x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {d x} {\left (b \log \left (c x^{n}\right ) + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 2.00, size = 427, normalized size = 4.36 \[ -\frac {2 x}{\sqrt {d x}\, \left (-i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i {\mathrm e}^{n \ln \relax (x )}\right ) \mathrm {csgn}\left (i c \,{\mathrm e}^{n \ln \relax (x )}\right )+i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,{\mathrm e}^{n \ln \relax (x )}\right )^{2}+i \pi b \,\mathrm {csgn}\left (i {\mathrm e}^{n \ln \relax (x )}\right ) \mathrm {csgn}\left (i c \,{\mathrm e}^{n \ln \relax (x )}\right )^{2}-i \pi b \mathrm {csgn}\left (i c \,{\mathrm e}^{n \ln \relax (x )}\right )^{3}+2 b \ln \relax (c )+2 b \ln \left ({\mathrm e}^{n \ln \relax (x )}\right )+2 a \right ) b n}-\frac {\Ei \left (1, -\frac {\ln \left (d x \right )}{2}+\frac {i \left (\pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i {\mathrm e}^{n \ln \relax (x )}\right ) \mathrm {csgn}\left (i c \,{\mathrm e}^{n \ln \relax (x )}\right )-\pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,{\mathrm e}^{n \ln \relax (x )}\right )^{2}-\pi b \,\mathrm {csgn}\left (i {\mathrm e}^{n \ln \relax (x )}\right ) \mathrm {csgn}\left (i c \,{\mathrm e}^{n \ln \relax (x )}\right )^{2}+\pi b \mathrm {csgn}\left (i c \,{\mathrm e}^{n \ln \relax (x )}\right )^{3}+2 i \left (\ln \relax (x )-\ln \left (d x \right )\right ) b n +2 i b \ln \relax (c )+2 i a +2 i \left (-n \ln \relax (x )+\ln \left ({\mathrm e}^{n \ln \relax (x )}\right )\right ) b \right )}{4 b n}\right ) {\mathrm e}^{\frac {i \left (\pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i {\mathrm e}^{n \ln \relax (x )}\right ) \mathrm {csgn}\left (i c \,{\mathrm e}^{n \ln \relax (x )}\right )-\pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,{\mathrm e}^{n \ln \relax (x )}\right )^{2}-\pi b \,\mathrm {csgn}\left (i {\mathrm e}^{n \ln \relax (x )}\right ) \mathrm {csgn}\left (i c \,{\mathrm e}^{n \ln \relax (x )}\right )^{2}+\pi b \mathrm {csgn}\left (i c \,{\mathrm e}^{n \ln \relax (x )}\right )^{3}+2 i \left (\ln \relax (x )-\ln \left (d x \right )\right ) b n +2 i b \ln \relax (c )+2 i a +2 i \left (-n \ln \relax (x )+\ln \left ({\mathrm e}^{n \ln \relax (x )}\right )\right ) b \right )}{4 b n}}}{2 b^{2} d \,n^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ 4 \, b n \int \frac {1}{{\left (b^{3} \sqrt {d} \log \relax (c)^{3} + b^{3} \sqrt {d} \log \left (x^{n}\right )^{3} + 3 \, a b^{2} \sqrt {d} \log \relax (c)^{2} + 3 \, a^{2} b \sqrt {d} \log \relax (c) + a^{3} \sqrt {d} + 3 \, {\left (b^{3} \sqrt {d} \log \relax (c) + a b^{2} \sqrt {d}\right )} \log \left (x^{n}\right )^{2} + 3 \, {\left (b^{3} \sqrt {d} \log \relax (c)^{2} + 2 \, a b^{2} \sqrt {d} \log \relax (c) + a^{2} b \sqrt {d}\right )} \log \left (x^{n}\right )\right )} \sqrt {x}}\,{d x} + \frac {2 \, \sqrt {x}}{b^{2} \sqrt {d} \log \relax (c)^{2} + b^{2} \sqrt {d} \log \left (x^{n}\right )^{2} + 2 \, a b \sqrt {d} \log \relax (c) + a^{2} \sqrt {d} + 2 \, {\left (b^{2} \sqrt {d} \log \relax (c) + a b \sqrt {d}\right )} \log \left (x^{n}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{\sqrt {d\,x}\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {d x} \left (a + b \log {\left (c x^{n} \right )}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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