Optimal. Leaf size=301 \[ \frac {b n \sqrt {1-\frac {e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt {1-\frac {e^2 x^2}{d^2}}\right )^2}{2 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\sqrt {1-\frac {e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt {1-\frac {e^2 x^2}{d^2}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d-e x} \sqrt {d+e x}}-\frac {b n \sqrt {1-\frac {e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt {1-\frac {e^2 x^2}{d^2}}\right ) \log \left (\frac {2}{1-\sqrt {1-\frac {e^2 x^2}{d^2}}}\right )}{\sqrt {d-e x} \sqrt {d+e x}}-\frac {b n \sqrt {1-\frac {e^2 x^2}{d^2}} \text {Li}_2\left (-\frac {1+\sqrt {1-\frac {e^2 x^2}{d^2}}}{1-\sqrt {1-\frac {e^2 x^2}{d^2}}}\right )}{2 \sqrt {d-e x} \sqrt {d+e x}} \]
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Rubi [A]
time = 0.40, antiderivative size = 301, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 9, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {2387, 272, 65,
214, 2390, 6131, 6055, 2449, 2352} \begin {gather*} -\frac {b n \sqrt {1-\frac {e^2 x^2}{d^2}} \text {PolyLog}\left (2,-\frac {\sqrt {1-\frac {e^2 x^2}{d^2}}+1}{1-\sqrt {1-\frac {e^2 x^2}{d^2}}}\right )}{2 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\sqrt {1-\frac {e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt {1-\frac {e^2 x^2}{d^2}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d-e x} \sqrt {d+e x}}+\frac {b n \sqrt {1-\frac {e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt {1-\frac {e^2 x^2}{d^2}}\right )^2}{2 \sqrt {d-e x} \sqrt {d+e x}}-\frac {b n \sqrt {1-\frac {e^2 x^2}{d^2}} \log \left (\frac {2}{1-\sqrt {1-\frac {e^2 x^2}{d^2}}}\right ) \tanh ^{-1}\left (\sqrt {1-\frac {e^2 x^2}{d^2}}\right )}{\sqrt {d-e x} \sqrt {d+e x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 214
Rule 272
Rule 2352
Rule 2387
Rule 2390
Rule 2449
Rule 6055
Rule 6131
Rubi steps
\begin {align*} \int \frac {a+b \log \left (c x^n\right )}{x \sqrt {d-e x} \sqrt {d+e x}} \, dx &=\frac {\sqrt {1-\frac {e^2 x^2}{d^2}} \int \frac {a+b \log \left (c x^n\right )}{x \sqrt {1-\frac {e^2 x^2}{d^2}}} \, dx}{\sqrt {d-e x} \sqrt {d+e x}}\\ &=-\frac {\sqrt {1-\frac {e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt {1-\frac {e^2 x^2}{d^2}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d-e x} \sqrt {d+e x}}+\frac {\left (b n \sqrt {1-\frac {e^2 x^2}{d^2}}\right ) \int \frac {\tanh ^{-1}\left (\sqrt {1-\frac {e^2 x^2}{d^2}}\right )}{x} \, dx}{\sqrt {d-e x} \sqrt {d+e x}}\\ &=-\frac {\sqrt {1-\frac {e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt {1-\frac {e^2 x^2}{d^2}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d-e x} \sqrt {d+e x}}+\frac {\left (b n \sqrt {1-\frac {e^2 x^2}{d^2}}\right ) \text {Subst}\left (\int \frac {\tanh ^{-1}\left (\sqrt {1-\frac {e^2 x}{d^2}}\right )}{x} \, dx,x,x^2\right )}{2 \sqrt {d-e x} \sqrt {d+e x}}\\ &=-\frac {\sqrt {1-\frac {e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt {1-\frac {e^2 x^2}{d^2}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d-e x} \sqrt {d+e x}}+\frac {\left (b n \sqrt {1-\frac {e^2 x^2}{d^2}}\right ) \text {Subst}\left (\int \frac {x \tanh ^{-1}(x)}{-1+x^2} \, dx,x,\sqrt {1-\frac {e^2 x^2}{d^2}}\right )}{\sqrt {d-e x} \sqrt {d+e x}}\\ &=\frac {b n \sqrt {1-\frac {e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt {1-\frac {e^2 x^2}{d^2}}\right )^2}{2 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\sqrt {1-\frac {e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt {1-\frac {e^2 x^2}{d^2}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d-e x} \sqrt {d+e x}}-\frac {\left (b n \sqrt {1-\frac {e^2 x^2}{d^2}}\right ) \text {Subst}\left (\int \frac {\tanh ^{-1}(x)}{1-x} \, dx,x,\sqrt {1-\frac {e^2 x^2}{d^2}}\right )}{\sqrt {d-e x} \sqrt {d+e x}}\\ &=\frac {b n \sqrt {1-\frac {e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt {1-\frac {e^2 x^2}{d^2}}\right )^2}{2 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\sqrt {1-\frac {e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt {1-\frac {e^2 x^2}{d^2}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d-e x} \sqrt {d+e x}}-\frac {b n \sqrt {1-\frac {e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt {1-\frac {e^2 x^2}{d^2}}\right ) \log \left (\frac {2}{1-\sqrt {1-\frac {e^2 x^2}{d^2}}}\right )}{\sqrt {d-e x} \sqrt {d+e x}}+\frac {\left (b n \sqrt {1-\frac {e^2 x^2}{d^2}}\right ) \text {Subst}\left (\int \frac {\log \left (\frac {2}{1-x}\right )}{1-x^2} \, dx,x,\sqrt {1-\frac {e^2 x^2}{d^2}}\right )}{\sqrt {d-e x} \sqrt {d+e x}}\\ &=\frac {b n \sqrt {1-\frac {e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt {1-\frac {e^2 x^2}{d^2}}\right )^2}{2 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\sqrt {1-\frac {e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt {1-\frac {e^2 x^2}{d^2}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d-e x} \sqrt {d+e x}}-\frac {b n \sqrt {1-\frac {e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt {1-\frac {e^2 x^2}{d^2}}\right ) \log \left (\frac {2}{1-\sqrt {1-\frac {e^2 x^2}{d^2}}}\right )}{\sqrt {d-e x} \sqrt {d+e x}}-\frac {\left (b n \sqrt {1-\frac {e^2 x^2}{d^2}}\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-\sqrt {1-\frac {e^2 x^2}{d^2}}}\right )}{\sqrt {d-e x} \sqrt {d+e x}}\\ &=\frac {b n \sqrt {1-\frac {e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt {1-\frac {e^2 x^2}{d^2}}\right )^2}{2 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\sqrt {1-\frac {e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt {1-\frac {e^2 x^2}{d^2}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d-e x} \sqrt {d+e x}}-\frac {b n \sqrt {1-\frac {e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt {1-\frac {e^2 x^2}{d^2}}\right ) \log \left (\frac {2}{1-\sqrt {1-\frac {e^2 x^2}{d^2}}}\right )}{\sqrt {d-e x} \sqrt {d+e x}}-\frac {b n \sqrt {1-\frac {e^2 x^2}{d^2}} \text {Li}_2\left (1-\frac {2}{1-\sqrt {1-\frac {e^2 x^2}{d^2}}}\right )}{2 \sqrt {d-e x} \sqrt {d+e x}}\\ \end {align*}
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Mathematica [A]
time = 1.18, size = 310, normalized size = 1.03 \begin {gather*} \frac {\log (x) \left (a-b n \log (x)+b \log \left (c x^n\right )\right )}{d}-\frac {\left (a-b n \log (x)+b \log \left (c x^n\right )\right ) \log \left (d+\sqrt {d-e x} \sqrt {d+e x}\right )}{d}+\frac {b n \sqrt {-d^2+e^2 x^2} \left (-\frac {4 \tanh ^{-1}\left (\frac {\sqrt {-d^2+e^2 x^2}}{\sqrt {-d^2}}\right ) \left (2 \log (x)-\log \left (\frac {e^2 x^2}{d^2}\right )\right )}{\sqrt {-d^2}}+\frac {\sqrt {1-\frac {e^2 x^2}{d^2}} \left (\log ^2\left (\frac {e^2 x^2}{d^2}\right )-4 \log \left (\frac {e^2 x^2}{d^2}\right ) \log \left (\frac {1}{2} \left (1+\sqrt {1-\frac {e^2 x^2}{d^2}}\right )\right )+2 \log ^2\left (\frac {1}{2} \left (1+\sqrt {1-\frac {e^2 x^2}{d^2}}\right )\right )-4 \text {Li}_2\left (\frac {1}{2}-\frac {1}{2} \sqrt {1-\frac {e^2 x^2}{d^2}}\right )\right )}{\sqrt {-d^2+e^2 x^2}}\right )}{8 \sqrt {d-e x} \sqrt {d+e x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {a +b \ln \left (c \,x^{n}\right )}{x \sqrt {-e x +d}\, \sqrt {e x +d}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {a + b \log {\left (c x^{n} \right )}}{x \sqrt {d - e x} \sqrt {d + e x}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {a+b\,\ln \left (c\,x^n\right )}{x\,\sqrt {d+e\,x}\,\sqrt {d-e\,x}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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