Integrand size = 14, antiderivative size = 14 \[ \int \log ^3\left (c \left (a+b x^2\right )^p\right ) \, dx=-48 p^3 x+\frac {48 \sqrt {a} p^3 \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}-\frac {24 i \sqrt {a} p^3 \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )^2}{\sqrt {b}}-\frac {48 \sqrt {a} p^3 \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (\frac {2 \sqrt {a}}{\sqrt {a}+i \sqrt {b} x}\right )}{\sqrt {b}}+24 p^2 x \log \left (c \left (a+b x^2\right )^p\right )-\frac {24 \sqrt {a} p^2 \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{\sqrt {b}}-6 p x \log ^2\left (c \left (a+b x^2\right )^p\right )+x \log ^3\left (c \left (a+b x^2\right )^p\right )-\frac {24 i \sqrt {a} p^3 \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {a}}{\sqrt {a}+i \sqrt {b} x}\right )}{\sqrt {b}}+6 a p \text {Int}\left (\frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{a+b x^2},x\right ) \]
[Out]
Not integrable
Time = 0.28 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \log ^3\left (c \left (a+b x^2\right )^p\right ) \, dx=\int \log ^3\left (c \left (a+b x^2\right )^p\right ) \, dx \]
[In]
[Out]
Rubi steps \begin{align*} \text {integral}& = x \log ^3\left (c \left (a+b x^2\right )^p\right )-(6 b p) \int \frac {x^2 \log ^2\left (c \left (a+b x^2\right )^p\right )}{a+b x^2} \, dx \\ & = x \log ^3\left (c \left (a+b x^2\right )^p\right )-(6 b p) \int \left (\frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{b}-\frac {a \log ^2\left (c \left (a+b x^2\right )^p\right )}{b \left (a+b x^2\right )}\right ) \, dx \\ & = x \log ^3\left (c \left (a+b x^2\right )^p\right )-(6 p) \int \log ^2\left (c \left (a+b x^2\right )^p\right ) \, dx+(6 a p) \int \frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{a+b x^2} \, dx \\ & = -6 p x \log ^2\left (c \left (a+b x^2\right )^p\right )+x \log ^3\left (c \left (a+b x^2\right )^p\right )+(6 a p) \int \frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{a+b x^2} \, dx+\left (24 b p^2\right ) \int \frac {x^2 \log \left (c \left (a+b x^2\right )^p\right )}{a+b x^2} \, dx \\ & = -6 p x \log ^2\left (c \left (a+b x^2\right )^p\right )+x \log ^3\left (c \left (a+b x^2\right )^p\right )+(6 a p) \int \frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{a+b x^2} \, dx+\left (24 b p^2\right ) \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 \\ & = -6 p x \log ^2\left (c \left (a+b x^2\right )^p\right )+x \log ^3\left (c \left (a+b x^2\right )^p\right )+(6 a p) \int \frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{a+b x^2} \, dx+\left (24 p^2\right ) \int \log \left (c \left (a+b x^2\right )^p\right ) \, dx-\left (24 a p^2\right ) \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{a+b x^2} \, dx \\ & = 24 p^2 x \log \left (c \left (a+b x^2\right )^p\right )-\frac {24 \sqrt {a} p^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{\sqrt {b}}-6 p x \log ^2\left (c \left (a+b x^2\right )^p\right )+x \log ^3\left (c \left (a+b x^2\right )^p\right )+(6 a p) \int \frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{a+b x^2} \, dx-\left (48 b p^3\right ) \int \frac {x^2}{a+b x^2} \, dx+\left (48 a b p^3\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 \\ & = -48 p^3 x+24 p^2 x \log \left (c \left (a+b x^2\right )^p\right )-\frac {24 \sqrt {a} p^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{\sqrt {b}}-6 p x \log ^2\left (c \left (a+b x^2\right )^p\right )+x \log ^3\left (c \left (a+b x^2\right )^p\right )+(6 a p) \int \frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{a+b x^2} \, dx+\left (48 a p^3\right ) \int \frac {1}{a+b x^2} \, dx+\left (48 \sqrt {a} \sqrt {b} p^3\right ) \int \frac {x \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a+b x^2} \, dx \\ & = -48 p^3 x+\frac {48 \sqrt {a} p^3 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}-\frac {24 i \sqrt {a} p^3 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )^2}{\sqrt {b}}+24 p^2 x \log \left (c \left (a+b x^2\right )^p\right )-\frac {24 \sqrt {a} p^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{\sqrt {b}}-6 p x \log ^2\left (c \left (a+b x^2\right )^p\right )+x \log ^3\left (c \left (a+b x^2\right )^p\right )+(6 a p) \int \frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{a+b x^2} \, dx-\left (48 p^3\right ) \int \frac {\tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{i-\frac {\sqrt {b} x}{\sqrt {a}}} \, dx \\ & = -48 p^3 x+\frac {48 \sqrt {a} p^3 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}-\frac {24 i \sqrt {a} p^3 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )^2}{\sqrt {b}}-\frac {48 \sqrt {a} p^3 \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}}+24 p^2 x \log \left (c \left (a+b x^2\right )^p\right )-\frac {24 \sqrt {a} p^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{\sqrt {b}}-6 p x \log ^2\left (c \left (a+b x^2\right )^p\right )+x \log ^3\left (c \left (a+b x^2\right )^p\right )+(6 a p) \int \frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{a+b x^2} \, dx+\left (48 p^3\right ) \int \frac {\log \left (\frac {2}{1+\frac {i \sqrt {b} x}{\sqrt {a}}}\right )}{1+\frac {b x^2}{a}} \, dx \\ & = -48 p^3 x+\frac {48 \sqrt {a} p^3 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}-\frac {24 i \sqrt {a} p^3 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )^2}{\sqrt {b}}-\frac {48 \sqrt {a} p^3 \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}}+24 p^2 x \log \left (c \left (a+b x^2\right )^p\right )-\frac {24 \sqrt {a} p^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{\sqrt {b}}-6 p x \log ^2\left (c \left (a+b x^2\right )^p\right )+x \log ^3\left (c \left (a+b x^2\right )^p\right )+(6 a p) \int \frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{a+b x^2} \, dx-\frac {\left (48 i \sqrt {a} p^3\right ) \text {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}} \\ & = -48 p^3 x+\frac {48 \sqrt {a} p^3 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}-\frac {24 i \sqrt {a} p^3 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )^2}{\sqrt {b}}-\frac {48 \sqrt {a} p^3 \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}}+24 p^2 x \log \left (c \left (a+b x^2\right )^p\right )-\frac {24 \sqrt {a} p^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{\sqrt {b}}-6 p x \log ^2\left (c \left (a+b x^2\right )^p\right )+x \log ^3\left (c \left (a+b x^2\right )^p\right )-\frac {24 i \sqrt {a} p^3 \text {Li}_2\left (1-\frac {2 \sqrt {a}}{\sqrt {a}+i \sqrt {b} x}\right )}{\sqrt {b}}+(6 a p) \int \frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{a+b x^2} \, dx \\ \end{align*}
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(789\) vs. \(2(290)=580\).
Time = 2.67 (sec) , antiderivative size = 789, normalized size of antiderivative = 56.36 \[ \int \log ^3\left (c \left (a+b x^2\right )^p\right ) \, dx=\frac {6 \sqrt {a} p \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (-p \log \left (a+b x^2\right )+\log \left (c \left (a+b x^2\right )^p\right )\right )^2}{\sqrt {b}}+3 p x \log \left (a+b x^2\right ) \left (-p \log \left (a+b x^2\right )+\log \left (c \left (a+b x^2\right )^p\right )\right )^2+x \left (-p \log \left (a+b x^2\right )+\log \left (c \left (a+b x^2\right )^p\right )\right )^2 \left (-6 p-p \log \left (a+b x^2\right )+\log \left (c \left (a+b x^2\right )^p\right )\right )-\frac {3 p^2 \left (p \log \left (a+b x^2\right )-\log \left (c \left (a+b x^2\right )^p\right )\right ) \left (4 i \sqrt {a} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )^2+4 \sqrt {a} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (-2+2 \log \left (\frac {2 \sqrt {a}}{\sqrt {a}+i \sqrt {b} x}\right )+\log \left (a+b x^2\right )\right )+\sqrt {b} x \left (8-4 \log \left (a+b x^2\right )+\log ^2\left (a+b x^2\right )\right )+4 i \sqrt {a} \operatorname {PolyLog}\left (2,\frac {i \sqrt {a}+\sqrt {b} x}{-i \sqrt {a}+\sqrt {b} x}\right )\right )}{\sqrt {b}}+\frac {p^3 \left (-48 \sqrt {-a^2} \sqrt {\frac {b x^2}{a+b x^2}} \sqrt {a+b x^2} \arcsin \left (\frac {\sqrt {a}}{\sqrt {a+b x^2}}\right )+\sqrt {-a} b x^2 \left (-48+24 \log \left (a+b x^2\right )-6 \log ^2\left (a+b x^2\right )+\log ^3\left (a+b x^2\right )\right )-6 \sqrt {-a^2} \sqrt {\frac {b x^2}{a+b x^2}} \left (8 \sqrt {a} \, _4F_3\left (\frac {1}{2},\frac {1}{2},\frac {1}{2},\frac {1}{2};\frac {3}{2},\frac {3}{2},\frac {3}{2};\frac {a}{a+b x^2}\right )+\log \left (a+b x^2\right ) \left (4 \sqrt {a} \, _3F_2\left (\frac {1}{2},\frac {1}{2},\frac {1}{2};\frac {3}{2},\frac {3}{2};\frac {a}{a+b x^2}\right )+\sqrt {a+b x^2} \arcsin \left (\frac {\sqrt {a}}{\sqrt {a+b x^2}}\right ) \log \left (a+b x^2\right )\right )\right )+24 a \sqrt {b x^2} \text {arctanh}\left (\frac {\sqrt {b x^2}}{\sqrt {-a}}\right ) \left (\log \left (a+b x^2\right )-\log \left (1+\frac {b x^2}{a}\right )\right )+6 (-a)^{3/2} \sqrt {-\frac {b x^2}{a}} \left (\log ^2\left (1+\frac {b x^2}{a}\right )-4 \log \left (1+\frac {b x^2}{a}\right ) \log \left (\frac {1}{2} \left (1+\sqrt {-\frac {b x^2}{a}}\right )\right )+2 \log ^2\left (\frac {1}{2} \left (1+\sqrt {-\frac {b x^2}{a}}\right )\right )-4 \operatorname {PolyLog}\left (2,\frac {1}{2}-\frac {1}{2} \sqrt {-\frac {b x^2}{a}}\right )\right )\right )}{\sqrt {-a} b x} \]
[In]
[Out]
Not integrable
Time = 0.04 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00
\[\int {\ln \left (c \left (b \,x^{2}+a \right )^{p}\right )}^{3}d x\]
[In]
[Out]
Not integrable
Time = 0.32 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.14 \[ \int \log ^3\left (c \left (a+b x^2\right )^p\right ) \, dx=\int { \log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{3} \,d x } \]
[In]
[Out]
Not integrable
Time = 1.87 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \log ^3\left (c \left (a+b x^2\right )^p\right ) \, dx=\int \log {\left (c \left (a + b x^{2}\right )^{p} \right )}^{3}\, dx \]
[In]
[Out]
Not integrable
Time = 0.85 (sec) , antiderivative size = 111, normalized size of antiderivative = 7.93 \[ \int \log ^3\left (c \left (a+b x^2\right )^p\right ) \, dx=\int { \log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{3} \,d x } \]
[In]
[Out]
Not integrable
Time = 0.35 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.14 \[ \int \log ^3\left (c \left (a+b x^2\right )^p\right ) \, dx=\int { \log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{3} \,d x } \]
[In]
[Out]
Not integrable
Time = 1.31 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.14 \[ \int \log ^3\left (c \left (a+b x^2\right )^p\right ) \, dx=\int {\ln \left (c\,{\left (b\,x^2+a\right )}^p\right )}^3 \,d x \]
[In]
[Out]