Optimal. Leaf size=189 \[ a^3 c \text {Li}_3\left (\frac {2}{1-i a x}-1\right )-2 i a^3 c \text {Li}_2\left (\frac {2}{1-i a x}-1\right ) \tan ^{-1}(a x)+a^3 c \log (x)-\frac {2}{3} i a^3 c \tan ^{-1}(a x)^3-\frac {1}{2} a^3 c \tan ^{-1}(a x)^2+2 a^3 c \log \left (2-\frac {2}{1-i a x}\right ) \tan ^{-1}(a x)^2-\frac {a^2 c \tan ^{-1}(a x)^3}{x}-\frac {a^2 c \tan ^{-1}(a x)}{x}-\frac {1}{2} a^3 c \log \left (a^2 x^2+1\right )-\frac {c \tan ^{-1}(a x)^3}{3 x^3}-\frac {a c \tan ^{-1}(a x)^2}{2 x^2} \]
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Rubi [A] time = 0.58, antiderivative size = 189, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 12, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {4950, 4852, 4918, 266, 36, 29, 31, 4884, 4924, 4868, 4992, 6610} \[ a^3 c \text {PolyLog}\left (3,-1+\frac {2}{1-i a x}\right )-2 i a^3 c \tan ^{-1}(a x) \text {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )-\frac {1}{2} a^3 c \log \left (a^2 x^2+1\right )+a^3 c \log (x)-\frac {2}{3} i a^3 c \tan ^{-1}(a x)^3-\frac {1}{2} a^3 c \tan ^{-1}(a x)^2-\frac {a^2 c \tan ^{-1}(a x)^3}{x}-\frac {a^2 c \tan ^{-1}(a x)}{x}+2 a^3 c \log \left (2-\frac {2}{1-i a x}\right ) \tan ^{-1}(a x)^2-\frac {a c \tan ^{-1}(a x)^2}{2 x^2}-\frac {c \tan ^{-1}(a x)^3}{3 x^3} \]
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 266
Rule 4852
Rule 4868
Rule 4884
Rule 4918
Rule 4924
Rule 4950
Rule 4992
Rule 6610
Rubi steps
\begin {align*} \int \frac {\left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^3}{x^4} \, dx &=c \int \frac {\tan ^{-1}(a x)^3}{x^4} \, dx+\left (a^2 c\right ) \int \frac {\tan ^{-1}(a x)^3}{x^2} \, dx\\ &=-\frac {c \tan ^{-1}(a x)^3}{3 x^3}-\frac {a^2 c \tan ^{-1}(a x)^3}{x}+(a c) \int \frac {\tan ^{-1}(a x)^2}{x^3 \left (1+a^2 x^2\right )} \, dx+\left (3 a^3 c\right ) \int \frac {\tan ^{-1}(a x)^2}{x \left (1+a^2 x^2\right )} \, dx\\ &=-i a^3 c \tan ^{-1}(a x)^3-\frac {c \tan ^{-1}(a x)^3}{3 x^3}-\frac {a^2 c \tan ^{-1}(a x)^3}{x}+(a c) \int \frac {\tan ^{-1}(a x)^2}{x^3} \, dx+\left (3 i a^3 c\right ) \int \frac {\tan ^{-1}(a x)^2}{x (i+a x)} \, dx-\left (a^3 c\right ) \int \frac {\tan ^{-1}(a x)^2}{x \left (1+a^2 x^2\right )} \, dx\\ &=-\frac {a c \tan ^{-1}(a x)^2}{2 x^2}-\frac {2}{3} i a^3 c \tan ^{-1}(a x)^3-\frac {c \tan ^{-1}(a x)^3}{3 x^3}-\frac {a^2 c \tan ^{-1}(a x)^3}{x}+3 a^3 c \tan ^{-1}(a x)^2 \log \left (2-\frac {2}{1-i a x}\right )+\left (a^2 c\right ) \int \frac {\tan ^{-1}(a x)}{x^2 \left (1+a^2 x^2\right )} \, dx-\left (i a^3 c\right ) \int \frac {\tan ^{-1}(a x)^2}{x (i+a x)} \, dx-\left (6 a^4 c\right ) \int \frac {\tan ^{-1}(a x) \log \left (2-\frac {2}{1-i a x}\right )}{1+a^2 x^2} \, dx\\ &=-\frac {a c \tan ^{-1}(a x)^2}{2 x^2}-\frac {2}{3} i a^3 c \tan ^{-1}(a x)^3-\frac {c \tan ^{-1}(a x)^3}{3 x^3}-\frac {a^2 c \tan ^{-1}(a x)^3}{x}+2 a^3 c \tan ^{-1}(a x)^2 \log \left (2-\frac {2}{1-i a x}\right )-3 i a^3 c \tan ^{-1}(a x) \text {Li}_2\left (-1+\frac {2}{1-i a x}\right )+\left (a^2 c\right ) \int \frac {\tan ^{-1}(a x)}{x^2} \, dx+\left (3 i a^4 c\right ) \int \frac {\text {Li}_2\left (-1+\frac {2}{1-i a x}\right )}{1+a^2 x^2} \, dx-\left (a^4 c\right ) \int \frac {\tan ^{-1}(a x)}{1+a^2 x^2} \, dx+\left (2 a^4 c\right ) \int \frac {\tan ^{-1}(a x) \log \left (2-\frac {2}{1-i a x}\right )}{1+a^2 x^2} \, dx\\ &=-\frac {a^2 c \tan ^{-1}(a x)}{x}-\frac {1}{2} a^3 c \tan ^{-1}(a x)^2-\frac {a c \tan ^{-1}(a x)^2}{2 x^2}-\frac {2}{3} i a^3 c \tan ^{-1}(a x)^3-\frac {c \tan ^{-1}(a x)^3}{3 x^3}-\frac {a^2 c \tan ^{-1}(a x)^3}{x}+2 a^3 c \tan ^{-1}(a x)^2 \log \left (2-\frac {2}{1-i a x}\right )-2 i a^3 c \tan ^{-1}(a x) \text {Li}_2\left (-1+\frac {2}{1-i a x}\right )+\frac {3}{2} a^3 c \text {Li}_3\left (-1+\frac {2}{1-i a x}\right )+\left (a^3 c\right ) \int \frac {1}{x \left (1+a^2 x^2\right )} \, dx-\left (i a^4 c\right ) \int \frac {\text {Li}_2\left (-1+\frac {2}{1-i a x}\right )}{1+a^2 x^2} \, dx\\ &=-\frac {a^2 c \tan ^{-1}(a x)}{x}-\frac {1}{2} a^3 c \tan ^{-1}(a x)^2-\frac {a c \tan ^{-1}(a x)^2}{2 x^2}-\frac {2}{3} i a^3 c \tan ^{-1}(a x)^3-\frac {c \tan ^{-1}(a x)^3}{3 x^3}-\frac {a^2 c \tan ^{-1}(a x)^3}{x}+2 a^3 c \tan ^{-1}(a x)^2 \log \left (2-\frac {2}{1-i a x}\right )-2 i a^3 c \tan ^{-1}(a x) \text {Li}_2\left (-1+\frac {2}{1-i a x}\right )+a^3 c \text {Li}_3\left (-1+\frac {2}{1-i a x}\right )+\frac {1}{2} \left (a^3 c\right ) \operatorname {Subst}\left (\int \frac {1}{x \left (1+a^2 x\right )} \, dx,x,x^2\right )\\ &=-\frac {a^2 c \tan ^{-1}(a x)}{x}-\frac {1}{2} a^3 c \tan ^{-1}(a x)^2-\frac {a c \tan ^{-1}(a x)^2}{2 x^2}-\frac {2}{3} i a^3 c \tan ^{-1}(a x)^3-\frac {c \tan ^{-1}(a x)^3}{3 x^3}-\frac {a^2 c \tan ^{-1}(a x)^3}{x}+2 a^3 c \tan ^{-1}(a x)^2 \log \left (2-\frac {2}{1-i a x}\right )-2 i a^3 c \tan ^{-1}(a x) \text {Li}_2\left (-1+\frac {2}{1-i a x}\right )+a^3 c \text {Li}_3\left (-1+\frac {2}{1-i a x}\right )+\frac {1}{2} \left (a^3 c\right ) \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )-\frac {1}{2} \left (a^5 c\right ) \operatorname {Subst}\left (\int \frac {1}{1+a^2 x} \, dx,x,x^2\right )\\ &=-\frac {a^2 c \tan ^{-1}(a x)}{x}-\frac {1}{2} a^3 c \tan ^{-1}(a x)^2-\frac {a c \tan ^{-1}(a x)^2}{2 x^2}-\frac {2}{3} i a^3 c \tan ^{-1}(a x)^3-\frac {c \tan ^{-1}(a x)^3}{3 x^3}-\frac {a^2 c \tan ^{-1}(a x)^3}{x}+a^3 c \log (x)-\frac {1}{2} a^3 c \log \left (1+a^2 x^2\right )+2 a^3 c \tan ^{-1}(a x)^2 \log \left (2-\frac {2}{1-i a x}\right )-2 i a^3 c \tan ^{-1}(a x) \text {Li}_2\left (-1+\frac {2}{1-i a x}\right )+a^3 c \text {Li}_3\left (-1+\frac {2}{1-i a x}\right )\\ \end {align*}
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Mathematica [A] time = 0.39, size = 177, normalized size = 0.94 \[ \frac {1}{12} c \left (24 i a^3 \tan ^{-1}(a x) \text {Li}_2\left (e^{-2 i \tan ^{-1}(a x)}\right )+12 a^3 \text {Li}_3\left (e^{-2 i \tan ^{-1}(a x)}\right )+8 i a^3 \tan ^{-1}(a x)^3-6 a^3 \tan ^{-1}(a x)^2+24 a^3 \tan ^{-1}(a x)^2 \log \left (1-e^{-2 i \tan ^{-1}(a x)}\right )-i \pi ^3 a^3-\frac {12 a^2 \tan ^{-1}(a x)^3}{x}-\frac {12 a^2 \tan ^{-1}(a x)}{x}+12 a^3 \log \left (\frac {a x}{\sqrt {a^2 x^2+1}}\right )-\frac {4 \tan ^{-1}(a x)^3}{x^3}-\frac {6 a \tan ^{-1}(a x)^2}{x^2}\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.77, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a^{2} c x^{2} + c\right )} \arctan \left (a x\right )^{3}}{x^{4}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 3.71, size = 5426, normalized size = 28.71 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {atan}\left (a\,x\right )}^3\,\left (c\,a^2\,x^2+c\right )}{x^4} \,d x \]
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 \[ c \left (\int \frac {\operatorname {atan}^{3}{\left (a x \right )}}{x^{4}}\, dx + \int \frac {a^{2} \operatorname {atan}^{3}{\left (a x \right )}}{x^{2}}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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