Optimal. Leaf size=149 \[ -\frac {3}{4} i b c^2 \left (a+b \text {ArcTan}\left (c x^2\right )\right )^2-\frac {3 b c \left (a+b \text {ArcTan}\left (c x^2\right )\right )^2}{4 x^2}-\frac {1}{4} c^2 \left (a+b \text {ArcTan}\left (c x^2\right )\right )^3-\frac {\left (a+b \text {ArcTan}\left (c x^2\right )\right )^3}{4 x^4}+\frac {3}{2} b^2 c^2 \left (a+b \text {ArcTan}\left (c x^2\right )\right ) \log \left (2-\frac {2}{1-i c x^2}\right )-\frac {3}{4} i b^3 c^2 \text {PolyLog}\left (2,-1+\frac {2}{1-i c x^2}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.23, antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {4948, 4946,
5038, 5044, 4988, 2497, 5004} \begin {gather*} \frac {3}{2} b^2 c^2 \log \left (2-\frac {2}{1-i c x^2}\right ) \left (a+b \text {ArcTan}\left (c x^2\right )\right )-\frac {3}{4} i b c^2 \left (a+b \text {ArcTan}\left (c x^2\right )\right )^2-\frac {1}{4} c^2 \left (a+b \text {ArcTan}\left (c x^2\right )\right )^3-\frac {3 b c \left (a+b \text {ArcTan}\left (c x^2\right )\right )^2}{4 x^2}-\frac {\left (a+b \text {ArcTan}\left (c x^2\right )\right )^3}{4 x^4}-\frac {3}{4} i b^3 c^2 \text {Li}_2\left (\frac {2}{1-i c x^2}-1\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2497
Rule 4946
Rule 4948
Rule 4988
Rule 5004
Rule 5038
Rule 5044
Rubi steps
\begin {align*} \int \frac {\left (a+b \tan ^{-1}\left (c x^2\right )\right )^3}{x^5} \, dx &=\int \left (\frac {\left (2 a+i b \log \left (1-i c x^2\right )\right )^3}{8 x^5}+\frac {3 i b \left (-2 i a+b \log \left (1-i c x^2\right )\right )^2 \log \left (1+i c x^2\right )}{8 x^5}-\frac {3 i b^2 \left (-2 i a+b \log \left (1-i c x^2\right )\right ) \log ^2\left (1+i c x^2\right )}{8 x^5}+\frac {i b^3 \log ^3\left (1+i c x^2\right )}{8 x^5}\right ) \, dx\\ &=\frac {1}{8} \int \frac {\left (2 a+i b \log \left (1-i c x^2\right )\right )^3}{x^5} \, dx+\frac {1}{8} (3 i b) \int \frac {\left (-2 i a+b \log \left (1-i c x^2\right )\right )^2 \log \left (1+i c x^2\right )}{x^5} \, dx-\frac {1}{8} \left (3 i b^2\right ) \int \frac {\left (-2 i a+b \log \left (1-i c x^2\right )\right ) \log ^2\left (1+i c x^2\right )}{x^5} \, dx+\frac {1}{8} \left (i b^3\right ) \int \frac {\log ^3\left (1+i c x^2\right )}{x^5} \, dx\\ &=\frac {1}{16} \text {Subst}\left (\int \frac {(2 a+i b \log (1-i c x))^3}{x^3} \, dx,x,x^2\right )+\frac {1}{16} (3 i b) \text {Subst}\left (\int \frac {(-2 i a+b \log (1-i c x))^2 \log (1+i c x)}{x^3} \, dx,x,x^2\right )-\frac {1}{16} \left (3 i b^2\right ) \text {Subst}\left (\int \frac {(-2 i a+b \log (1-i c x)) \log ^2(1+i c x)}{x^3} \, dx,x,x^2\right )+\frac {1}{16} \left (i b^3\right ) \text {Subst}\left (\int \frac {\log ^3(1+i c x)}{x^3} \, dx,x,x^2\right )\\ &=-\frac {\left (2 a+i b \log \left (1-i c x^2\right )\right )^3}{32 x^4}-\frac {i b^3 \log ^3\left (1+i c x^2\right )}{32 x^4}+\frac {1}{16} (3 i b) \text {Subst}\left (\int \frac {(-2 i a+b \log (1-i c x))^2 \log (1+i c x)}{x^3} \, dx,x,x^2\right )-\frac {1}{16} \left (3 i b^2\right ) \text {Subst}\left (\int \frac {(-2 i a+b \log (1-i c x)) \log ^2(1+i c x)}{x^3} \, dx,x,x^2\right )+\frac {1}{32} (3 b c) \text {Subst}\left (\int \frac {(2 a+i b \log (1-i c x))^2}{x^2 (1-i c x)} \, dx,x,x^2\right )-\frac {1}{32} \left (3 b^3 c\right ) \text {Subst}\left (\int \frac {\log ^2(1+i c x)}{x^2 (1+i c x)} \, dx,x,x^2\right )\\ &=-\frac {\left (2 a+i b \log \left (1-i c x^2\right )\right )^3}{32 x^4}-\frac {i b^3 \log ^3\left (1+i c x^2\right )}{32 x^4}+\frac {1}{32} (3 i b) \text {Subst}\left (\int \frac {(2 a+i b \log (x))^2}{x \left (-\frac {i}{c}+\frac {i x}{c}\right )^2} \, dx,x,1-i c x^2\right )+\frac {1}{16} (3 i b) \text {Subst}\left (\int \frac {(-2 i a+b \log (1-i c x))^2 \log (1+i c x)}{x^3} \, dx,x,x^2\right )-\frac {1}{16} \left (3 i b^2\right ) \text {Subst}\left (\int \frac {(-2 i a+b \log (1-i c x)) \log ^2(1+i c x)}{x^3} \, dx,x,x^2\right )+\frac {1}{32} \left (3 i b^3\right ) \text {Subst}\left (\int \frac {\log ^2(x)}{x \left (\frac {i}{c}-\frac {i x}{c}\right )^2} \, dx,x,1+i c x^2\right )\\ &=-\frac {\left (2 a+i b \log \left (1-i c x^2\right )\right )^3}{32 x^4}-\frac {i b^3 \log ^3\left (1+i c x^2\right )}{32 x^4}+\frac {1}{32} (3 i b) \text {Subst}\left (\int \frac {(2 a+i b \log (x))^2}{\left (-\frac {i}{c}+\frac {i x}{c}\right )^2} \, dx,x,1-i c x^2\right )+\frac {1}{16} (3 i b) \text {Subst}\left (\int \frac {(-2 i a+b \log (1-i c x))^2 \log (1+i c x)}{x^3} \, dx,x,x^2\right )-\frac {1}{16} \left (3 i b^2\right ) \text {Subst}\left (\int \frac {(-2 i a+b \log (1-i c x)) \log ^2(1+i c x)}{x^3} \, dx,x,x^2\right )+\frac {1}{32} \left (3 i b^3\right ) \text {Subst}\left (\int \frac {\log ^2(x)}{\left (\frac {i}{c}-\frac {i x}{c}\right )^2} \, dx,x,1+i c x^2\right )-\frac {1}{32} (3 b c) \text {Subst}\left (\int \frac {(2 a+i b \log (x))^2}{x \left (-\frac {i}{c}+\frac {i x}{c}\right )} \, dx,x,1-i c x^2\right )+\frac {1}{32} \left (3 b^3 c\right ) \text {Subst}\left (\int \frac {\log ^2(x)}{x \left (\frac {i}{c}-\frac {i x}{c}\right )} \, dx,x,1+i c x^2\right )\\ &=-\frac {3 b c \left (1-i c x^2\right ) \left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{32 x^2}-\frac {\left (2 a+i b \log \left (1-i c x^2\right )\right )^3}{32 x^4}+\frac {3 b^3 c \left (1+i c x^2\right ) \log ^2\left (1+i c x^2\right )}{32 x^2}-\frac {i b^3 \log ^3\left (1+i c x^2\right )}{32 x^4}+\frac {1}{16} (3 i b) \text {Subst}\left (\int \frac {(-2 i a+b \log (1-i c x))^2 \log (1+i c x)}{x^3} \, dx,x,x^2\right )-\frac {1}{16} \left (3 i b^2\right ) \text {Subst}\left (\int \frac {(-2 i a+b \log (1-i c x)) \log ^2(1+i c x)}{x^3} \, dx,x,x^2\right )-\frac {1}{32} (3 b c) \text {Subst}\left (\int \frac {(2 a+i b \log (x))^2}{-\frac {i}{c}+\frac {i x}{c}} \, dx,x,1-i c x^2\right )+\frac {1}{16} \left (3 i b^2 c\right ) \text {Subst}\left (\int \frac {2 a+i b \log (x)}{-\frac {i}{c}+\frac {i x}{c}} \, dx,x,1-i c x^2\right )+\frac {1}{32} \left (3 b^3 c\right ) \text {Subst}\left (\int \frac {\log ^2(x)}{\frac {i}{c}-\frac {i x}{c}} \, dx,x,1+i c x^2\right )-\frac {1}{16} \left (3 b^3 c\right ) \text {Subst}\left (\int \frac {\log (x)}{\frac {i}{c}-\frac {i x}{c}} \, dx,x,1+i c x^2\right )-\frac {1}{32} \left (3 i b c^2\right ) \text {Subst}\left (\int \frac {(2 a+i b \log (x))^2}{x} \, dx,x,1-i c x^2\right )-\frac {1}{32} \left (3 i b^3 c^2\right ) \text {Subst}\left (\int \frac {\log ^2(x)}{x} \, dx,x,1+i c x^2\right )\\ &=\frac {3}{4} a b^2 c^2 \log (x)-\frac {3 b c \left (1-i c x^2\right ) \left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{32 x^2}+\frac {3}{32} i b c^2 \log \left (i c x^2\right ) \left (2 a+i b \log \left (1-i c x^2\right )\right )^2-\frac {\left (2 a+i b \log \left (1-i c x^2\right )\right )^3}{32 x^4}+\frac {3 b^3 c \left (1+i c x^2\right ) \log ^2\left (1+i c x^2\right )}{32 x^2}+\frac {3}{32} i b^3 c^2 \log \left (-i c x^2\right ) \log ^2\left (1+i c x^2\right )-\frac {i b^3 \log ^3\left (1+i c x^2\right )}{32 x^4}+\frac {3}{16} i b^3 c^2 \text {Li}_2\left (-i c x^2\right )+\frac {1}{16} (3 i b) \text {Subst}\left (\int \frac {(-2 i a+b \log (1-i c x))^2 \log (1+i c x)}{x^3} \, dx,x,x^2\right )-\frac {1}{16} \left (3 i b^2\right ) \text {Subst}\left (\int \frac {(-2 i a+b \log (1-i c x)) \log ^2(1+i c x)}{x^3} \, dx,x,x^2\right )-\frac {1}{16} \left (3 b^3 c\right ) \text {Subst}\left (\int \frac {\log (x)}{-\frac {i}{c}+\frac {i x}{c}} \, dx,x,1-i c x^2\right )-\frac {1}{32} \left (3 c^2\right ) \text {Subst}\left (\int x^2 \, dx,x,2 a+i b \log \left (1-i c x^2\right )\right )+\frac {1}{16} \left (3 b^2 c^2\right ) \text {Subst}\left (\int \frac {\log (1-x) (2 a+i b \log (x))}{x} \, dx,x,1-i c x^2\right )-\frac {1}{32} \left (3 i b^3 c^2\right ) \text {Subst}\left (\int x^2 \, dx,x,\log \left (1+i c x^2\right )\right )-\frac {1}{16} \left (3 i b^3 c^2\right ) \text {Subst}\left (\int \frac {\log (1-x) \log (x)}{x} \, dx,x,1+i c x^2\right )\\ &=\frac {3}{4} a b^2 c^2 \log (x)-\frac {3 b c \left (1-i c x^2\right ) \left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{32 x^2}+\frac {3}{32} i b c^2 \log \left (i c x^2\right ) \left (2 a+i b \log \left (1-i c x^2\right )\right )^2-\frac {1}{32} c^2 \left (2 a+i b \log \left (1-i c x^2\right )\right )^3-\frac {\left (2 a+i b \log \left (1-i c x^2\right )\right )^3}{32 x^4}+\frac {3 b^3 c \left (1+i c x^2\right ) \log ^2\left (1+i c x^2\right )}{32 x^2}+\frac {3}{32} i b^3 c^2 \log \left (-i c x^2\right ) \log ^2\left (1+i c x^2\right )-\frac {1}{32} i b^3 c^2 \log ^3\left (1+i c x^2\right )-\frac {i b^3 \log ^3\left (1+i c x^2\right )}{32 x^4}+\frac {3}{16} i b^3 c^2 \text {Li}_2\left (-i c x^2\right )-\frac {3}{16} i b^3 c^2 \text {Li}_2\left (i c x^2\right )-\frac {3}{16} b^2 c^2 \left (2 a+i b \log \left (1-i c x^2\right )\right ) \text {Li}_2\left (1-i c x^2\right )+\frac {3}{16} i b^3 c^2 \log \left (1+i c x^2\right ) \text {Li}_2\left (1+i c x^2\right )+\frac {1}{16} (3 i b) \text {Subst}\left (\int \frac {(-2 i a+b \log (1-i c x))^2 \log (1+i c x)}{x^3} \, dx,x,x^2\right )-\frac {1}{16} \left (3 i b^2\right ) \text {Subst}\left (\int \frac {(-2 i a+b \log (1-i c x)) \log ^2(1+i c x)}{x^3} \, dx,x,x^2\right )+\frac {1}{16} \left (3 i b^3 c^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,1-i c x^2\right )-\frac {1}{16} \left (3 i b^3 c^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,1+i c x^2\right )\\ &=\frac {3}{4} a b^2 c^2 \log (x)-\frac {3 b c \left (1-i c x^2\right ) \left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{32 x^2}+\frac {3}{32} i b c^2 \log \left (i c x^2\right ) \left (2 a+i b \log \left (1-i c x^2\right )\right )^2-\frac {1}{32} c^2 \left (2 a+i b \log \left (1-i c x^2\right )\right )^3-\frac {\left (2 a+i b \log \left (1-i c x^2\right )\right )^3}{32 x^4}+\frac {3 b^3 c \left (1+i c x^2\right ) \log ^2\left (1+i c x^2\right )}{32 x^2}+\frac {3}{32} i b^3 c^2 \log \left (-i c x^2\right ) \log ^2\left (1+i c x^2\right )-\frac {1}{32} i b^3 c^2 \log ^3\left (1+i c x^2\right )-\frac {i b^3 \log ^3\left (1+i c x^2\right )}{32 x^4}+\frac {3}{16} i b^3 c^2 \text {Li}_2\left (-i c x^2\right )-\frac {3}{16} i b^3 c^2 \text {Li}_2\left (i c x^2\right )-\frac {3}{16} b^2 c^2 \left (2 a+i b \log \left (1-i c x^2\right )\right ) \text {Li}_2\left (1-i c x^2\right )+\frac {3}{16} i b^3 c^2 \log \left (1+i c x^2\right ) \text {Li}_2\left (1+i c x^2\right )+\frac {3}{16} i b^3 c^2 \text {Li}_3\left (1-i c x^2\right )-\frac {3}{16} i b^3 c^2 \text {Li}_3\left (1+i c x^2\right )+\frac {1}{16} (3 i b) \text {Subst}\left (\int \frac {(-2 i a+b \log (1-i c x))^2 \log (1+i c x)}{x^3} \, dx,x,x^2\right )-\frac {1}{16} \left (3 i b^2\right ) \text {Subst}\left (\int \frac {(-2 i a+b \log (1-i c x)) \log ^2(1+i c x)}{x^3} \, dx,x,x^2\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.22, size = 196, normalized size = 1.32 \begin {gather*} -\frac {3 b^2 \left (a+a c^2 x^4+b c x^2 \left (1+i c x^2\right )\right ) \text {ArcTan}\left (c x^2\right )^2+b^3 \left (1+c^2 x^4\right ) \text {ArcTan}\left (c x^2\right )^3+3 b \text {ArcTan}\left (c x^2\right ) \left (a \left (a+2 b c x^2+a c^2 x^4\right )-2 b^2 c^2 x^4 \log \left (1-e^{2 i \text {ArcTan}\left (c x^2\right )}\right )\right )+a \left (a \left (a+3 b c x^2\right )-6 b^2 c^2 x^4 \log \left (\frac {c x^2}{\sqrt {1+c^2 x^4}}\right )\right )+3 i b^3 c^2 x^4 \text {PolyLog}\left (2,e^{2 i \text {ArcTan}\left (c x^2\right )}\right )}{4 x^4} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \arctan \left (c \,x^{2}\right )\right )^{3}}{x^{5}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b \operatorname {atan}{\left (c x^{2} \right )}\right )^{3}}{x^{5}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (a+b\,\mathrm {atan}\left (c\,x^2\right )\right )}^3}{x^5} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________