Optimal. Leaf size=237 \[ x \log ^2\left (c \left (a+b x^2\right )^p\right )-4 p x \log \left (c \left (a+b x^2\right )^p\right )+\frac {4 \sqrt {a} p \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{\sqrt {b}}+\frac {4 i \sqrt {a} p^2 \text {Li}_2\left (1-\frac {2 \sqrt {a}}{i \sqrt {b} x+\sqrt {a}}\right )}{\sqrt {b}}+\frac {4 i \sqrt {a} p^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )^2}{\sqrt {b}}-\frac {8 \sqrt {a} p^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}+\frac {8 \sqrt {a} p^2 \log \left (\frac {2 \sqrt {a}}{\sqrt {a}+i \sqrt {b} x}\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}+8 p^2 x \]
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Rubi [A] time = 0.27, antiderivative size = 237, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 11, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.786, Rules used = {2450, 2476, 2448, 321, 205, 2470, 12, 4920, 4854, 2402, 2315} \[ \frac {4 i \sqrt {a} p^2 \text {PolyLog}\left (2,1-\frac {2 \sqrt {a}}{\sqrt {a}+i \sqrt {b} x}\right )}{\sqrt {b}}+x \log ^2\left (c \left (a+b x^2\right )^p\right )-4 p x \log \left (c \left (a+b x^2\right )^p\right )+\frac {4 \sqrt {a} p \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{\sqrt {b}}+\frac {4 i \sqrt {a} p^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )^2}{\sqrt {b}}-\frac {8 \sqrt {a} p^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}+\frac {8 \sqrt {a} p^2 \log \left (\frac {2 \sqrt {a}}{\sqrt {a}+i \sqrt {b} x}\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}+8 p^2 x \]
Antiderivative was successfully verified.
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Rule 12
Rule 205
Rule 321
Rule 2315
Rule 2402
Rule 2448
Rule 2450
Rule 2470
Rule 2476
Rule 4854
Rule 4920
Rubi steps
\begin {align*} \int \log ^2\left (c \left (a+b x^2\right )^p\right ) \, dx &=x \log ^2\left (c \left (a+b x^2\right )^p\right )-(4 b p) \int \frac {x^2 \log \left (c \left (a+b x^2\right )^p\right )}{a+b x^2} \, dx\\ &=x \log ^2\left (c \left (a+b x^2\right )^p\right )-(4 b p) \int \left (\frac {\log \left (c \left (a+b x^2\right )^p\right )}{b}-\frac {a \log \left (c \left (a+b x^2\right )^p\right )}{b \left (a+b x^2\right )}\right ) \, dx\\ &=x \log ^2\left (c \left (a+b x^2\right )^p\right )-(4 p) \int \log \left (c \left (a+b x^2\right )^p\right ) \, dx+(4 a p) \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{a+b x^2} \, dx\\ &=-4 p x \log \left (c \left (a+b x^2\right )^p\right )+\frac {4 \sqrt {a} p \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{\sqrt {b}}+x \log ^2\left (c \left (a+b x^2\right )^p\right )+\left (8 b p^2\right ) \int \frac {x^2}{a+b x^2} \, dx-\left (8 a b p^2\right ) \int \frac {x \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b} \left (a+b x^2\right )} \, dx\\ &=8 p^2 x-4 p x \log \left (c \left (a+b x^2\right )^p\right )+\frac {4 \sqrt {a} p \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{\sqrt {b}}+x \log ^2\left (c \left (a+b x^2\right )^p\right )-\left (8 a p^2\right ) \int \frac {1}{a+b x^2} \, dx-\left (8 \sqrt {a} \sqrt {b} p^2\right ) \int \frac {x \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a+b x^2} \, dx\\ &=8 p^2 x-\frac {8 \sqrt {a} p^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}+\frac {4 i \sqrt {a} p^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )^2}{\sqrt {b}}-4 p x \log \left (c \left (a+b x^2\right )^p\right )+\frac {4 \sqrt {a} p \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{\sqrt {b}}+x \log ^2\left (c \left (a+b x^2\right )^p\right )+\left (8 p^2\right ) \int \frac {\tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{i-\frac {\sqrt {b} x}{\sqrt {a}}} \, dx\\ &=8 p^2 x-\frac {8 \sqrt {a} p^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}+\frac {4 i \sqrt {a} p^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )^2}{\sqrt {b}}+\frac {8 \sqrt {a} p^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (\frac {2 \sqrt {a}}{\sqrt {a}+i \sqrt {b} x}\right )}{\sqrt {b}}-4 p x \log \left (c \left (a+b x^2\right )^p\right )+\frac {4 \sqrt {a} p \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{\sqrt {b}}+x \log ^2\left (c \left (a+b x^2\right )^p\right )-\left (8 p^2\right ) \int \frac {\log \left (\frac {2}{1+\frac {i \sqrt {b} x}{\sqrt {a}}}\right )}{1+\frac {b x^2}{a}} \, dx\\ &=8 p^2 x-\frac {8 \sqrt {a} p^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}+\frac {4 i \sqrt {a} p^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )^2}{\sqrt {b}}+\frac {8 \sqrt {a} p^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (\frac {2 \sqrt {a}}{\sqrt {a}+i \sqrt {b} x}\right )}{\sqrt {b}}-4 p x \log \left (c \left (a+b x^2\right )^p\right )+\frac {4 \sqrt {a} p \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{\sqrt {b}}+x \log ^2\left (c \left (a+b x^2\right )^p\right )+\frac {\left (8 i \sqrt {a} p^2\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+\frac {i \sqrt {b} x}{\sqrt {a}}}\right )}{\sqrt {b}}\\ &=8 p^2 x-\frac {8 \sqrt {a} p^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}+\frac {4 i \sqrt {a} p^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )^2}{\sqrt {b}}+\frac {8 \sqrt {a} p^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (\frac {2 \sqrt {a}}{\sqrt {a}+i \sqrt {b} x}\right )}{\sqrt {b}}-4 p x \log \left (c \left (a+b x^2\right )^p\right )+\frac {4 \sqrt {a} p \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{\sqrt {b}}+x \log ^2\left (c \left (a+b x^2\right )^p\right )+\frac {4 i \sqrt {a} p^2 \text {Li}_2\left (1-\frac {2 \sqrt {a}}{\sqrt {a}+i \sqrt {b} x}\right )}{\sqrt {b}}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 193, normalized size = 0.81 \[ \frac {\sqrt {b} x \left (\log ^2\left (c \left (a+b x^2\right )^p\right )-4 p \log \left (c \left (a+b x^2\right )^p\right )+8 p^2\right )+4 \sqrt {a} p \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (\log \left (c \left (a+b x^2\right )^p\right )+2 p \log \left (\frac {2 \sqrt {a}}{\sqrt {a}+i \sqrt {b} x}\right )-2 p\right )+4 i \sqrt {a} p^2 \text {Li}_2\left (\frac {\sqrt {b} x+i \sqrt {a}}{\sqrt {b} x-i \sqrt {a}}\right )+4 i \sqrt {a} p^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )^2}{\sqrt {b}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{2}, x\right ) \]
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 \[ \int \log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.72, size = 0, normalized size = 0.00 \[ \int \ln \left (c \left (b \,x^{2}+a \right )^{p}\right )^{2}\, dx \]
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 \[ p^{2} x \log \left (b x^{2} + a\right )^{2} + \int \frac {b x^{2} \log \relax (c)^{2} + a \log \relax (c)^{2} - 2 \, {\left ({\left (2 \, p^{2} - p \log \relax (c)\right )} b x^{2} - a p \log \relax (c)\right )} \log \left (b x^{2} + a\right )}{b x^{2} + a}\,{d x} \]
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 \[ \int {\ln \left (c\,{\left (b\,x^2+a\right )}^p\right )}^2 \,d x \]
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 \[ \int \log {\left (c \left (a + b x^{2}\right )^{p} \right )}^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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