Optimal. Leaf size=254 \[ -\frac {4 b^{3/2} p \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{3 a^{3/2}}-\frac {4 i b^{3/2} p^2 \text {Li}_2\left (1-\frac {2 \sqrt {a}}{i \sqrt {b} x+\sqrt {a}}\right )}{3 a^{3/2}}-\frac {4 i b^{3/2} p^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )^2}{3 a^{3/2}}+\frac {8 b^{3/2} p^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{3 a^{3/2}}-\frac {8 b^{3/2} p^2 \log \left (\frac {2 \sqrt {a}}{\sqrt {a}+i \sqrt {b} x}\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{3 a^{3/2}}-\frac {4 b p \log \left (c \left (a+b x^2\right )^p\right )}{3 a x}-\frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{3 x^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.29, antiderivative size = 254, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 10, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {2457, 2476, 2455, 205, 2470, 12, 4920, 4854, 2402, 2315} \[ -\frac {4 i b^{3/2} p^2 \text {PolyLog}\left (2,1-\frac {2 \sqrt {a}}{\sqrt {a}+i \sqrt {b} x}\right )}{3 a^{3/2}}-\frac {4 b^{3/2} p \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{3 a^{3/2}}-\frac {4 i b^{3/2} p^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )^2}{3 a^{3/2}}+\frac {8 b^{3/2} p^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{3 a^{3/2}}-\frac {8 b^{3/2} p^2 \log \left (\frac {2 \sqrt {a}}{\sqrt {a}+i \sqrt {b} x}\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{3 a^{3/2}}-\frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{3 x^3}-\frac {4 b p \log \left (c \left (a+b x^2\right )^p\right )}{3 a x} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 205
Rule 2315
Rule 2402
Rule 2455
Rule 2457
Rule 2470
Rule 2476
Rule 4854
Rule 4920
Rubi steps
\begin {align*} \int \frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{x^4} \, dx &=-\frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{3 x^3}+\frac {1}{3} (4 b p) \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{x^2 \left (a+b x^2\right )} \, dx\\ &=-\frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{3 x^3}+\frac {1}{3} (4 b p) \int \left (\frac {\log \left (c \left (a+b x^2\right )^p\right )}{a x^2}-\frac {b \log \left (c \left (a+b x^2\right )^p\right )}{a \left (a+b x^2\right )}\right ) \, dx\\ &=-\frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{3 x^3}+\frac {(4 b p) \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{x^2} \, dx}{3 a}-\frac {\left (4 b^2 p\right ) \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{a+b x^2} \, dx}{3 a}\\ &=-\frac {4 b p \log \left (c \left (a+b x^2\right )^p\right )}{3 a x}-\frac {4 b^{3/2} p \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{3 a^{3/2}}-\frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{3 x^3}+\frac {\left (8 b^2 p^2\right ) \int \frac {1}{a+b x^2} \, dx}{3 a}+\frac {\left (8 b^3 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}{3 a}\\ &=\frac {8 b^{3/2} p^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{3 a^{3/2}}-\frac {4 b p \log \left (c \left (a+b x^2\right )^p\right )}{3 a x}-\frac {4 b^{3/2} p \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{3 a^{3/2}}-\frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{3 x^3}+\frac {\left (8 b^{5/2} p^2\right ) \int \frac {x \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a+b x^2} \, dx}{3 a^{3/2}}\\ &=\frac {8 b^{3/2} p^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{3 a^{3/2}}-\frac {4 i b^{3/2} p^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )^2}{3 a^{3/2}}-\frac {4 b p \log \left (c \left (a+b x^2\right )^p\right )}{3 a x}-\frac {4 b^{3/2} p \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{3 a^{3/2}}-\frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{3 x^3}-\frac {\left (8 b^2 p^2\right ) \int \frac {\tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{i-\frac {\sqrt {b} x}{\sqrt {a}}} \, dx}{3 a^2}\\ &=\frac {8 b^{3/2} p^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{3 a^{3/2}}-\frac {4 i b^{3/2} p^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )^2}{3 a^{3/2}}-\frac {8 b^{3/2} 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 )}{3 a^{3/2}}-\frac {4 b p \log \left (c \left (a+b x^2\right )^p\right )}{3 a x}-\frac {4 b^{3/2} p \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{3 a^{3/2}}-\frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{3 x^3}+\frac {\left (8 b^2 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}{3 a^2}\\ &=\frac {8 b^{3/2} p^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{3 a^{3/2}}-\frac {4 i b^{3/2} p^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )^2}{3 a^{3/2}}-\frac {8 b^{3/2} 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 )}{3 a^{3/2}}-\frac {4 b p \log \left (c \left (a+b x^2\right )^p\right )}{3 a x}-\frac {4 b^{3/2} p \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{3 a^{3/2}}-\frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{3 x^3}-\frac {\left (8 i b^{3/2} 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 )}{3 a^{3/2}}\\ &=\frac {8 b^{3/2} p^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{3 a^{3/2}}-\frac {4 i b^{3/2} p^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )^2}{3 a^{3/2}}-\frac {8 b^{3/2} 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 )}{3 a^{3/2}}-\frac {4 b p \log \left (c \left (a+b x^2\right )^p\right )}{3 a x}-\frac {4 b^{3/2} p \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{3 a^{3/2}}-\frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{3 x^3}-\frac {4 i b^{3/2} p^2 \text {Li}_2\left (1-\frac {2 \sqrt {a}}{\sqrt {a}+i \sqrt {b} x}\right )}{3 a^{3/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.10, size = 207, normalized size = 0.81 \[ \frac {-4 b^{3/2} p x^3 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (\log \left (c \left (a+b x^2\right )^p\right )+2 p \log \left (\frac {2 \sqrt {a}}{\sqrt {a}+i \sqrt {b} x}\right )-2 p\right )-4 i b^{3/2} p^2 x^3 \text {Li}_2\left (\frac {\sqrt {b} x+i \sqrt {a}}{\sqrt {b} x-i \sqrt {a}}\right )-4 i b^{3/2} p^2 x^3 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )^2-\sqrt {a} \log \left (c \left (a+b x^2\right )^p\right ) \left (a \log \left (c \left (a+b x^2\right )^p\right )+4 b p x^2\right )}{3 a^{3/2} x^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{2}}{x^{4}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{2}}{x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.89, size = 0, normalized size = 0.00 \[ \int \frac {\ln \left (c \left (b \,x^{2}+a \right )^{p}\right )^{2}}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {p^{2} \log \left (b x^{2} + a\right )^{2}}{3 \, x^{3}} + \int \frac {3 \, b x^{2} \log \relax (c)^{2} + 3 \, a \log \relax (c)^{2} + 2 \, {\left ({\left (2 \, p^{2} + 3 \, p \log \relax (c)\right )} b x^{2} + 3 \, a p \log \relax (c)\right )} \log \left (b x^{2} + a\right )}{3 \, {\left (b x^{6} + a x^{4}\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\ln \left (c\,{\left (b\,x^2+a\right )}^p\right )}^2}{x^4} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log {\left (c \left (a + b x^{2}\right )^{p} \right )}^{2}}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________