Optimal. Leaf size=140 \[ \frac {1}{3} i c^3 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {2}{3} b c^3 \log \left (2-\frac {2}{1-i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )-\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{3 x^3}-\frac {b c \left (a+b \tan ^{-1}(c x)\right )}{3 x^2}+\frac {1}{3} i b^2 c^3 \text {Li}_2\left (\frac {2}{1-i c x}-1\right )-\frac {1}{3} b^2 c^3 \tan ^{-1}(c x)-\frac {b^2 c^2}{3 x} \]
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Rubi [A] time = 0.23, antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4852, 4918, 325, 203, 4924, 4868, 2447} \[ \frac {1}{3} i b^2 c^3 \text {PolyLog}\left (2,-1+\frac {2}{1-i c x}\right )+\frac {1}{3} i c^3 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {2}{3} b c^3 \log \left (2-\frac {2}{1-i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )-\frac {b c \left (a+b \tan ^{-1}(c x)\right )}{3 x^2}-\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{3 x^3}-\frac {b^2 c^2}{3 x}-\frac {1}{3} b^2 c^3 \tan ^{-1}(c x) \]
Antiderivative was successfully verified.
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Rule 203
Rule 325
Rule 2447
Rule 4852
Rule 4868
Rule 4918
Rule 4924
Rubi steps
\begin {align*} \int \frac {\left (a+b \tan ^{-1}(c x)\right )^2}{x^4} \, dx &=-\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{3 x^3}+\frac {1}{3} (2 b c) \int \frac {a+b \tan ^{-1}(c x)}{x^3 \left (1+c^2 x^2\right )} \, dx\\ &=-\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{3 x^3}+\frac {1}{3} (2 b c) \int \frac {a+b \tan ^{-1}(c x)}{x^3} \, dx-\frac {1}{3} \left (2 b c^3\right ) \int \frac {a+b \tan ^{-1}(c x)}{x \left (1+c^2 x^2\right )} \, dx\\ &=-\frac {b c \left (a+b \tan ^{-1}(c x)\right )}{3 x^2}+\frac {1}{3} i c^3 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{3 x^3}+\frac {1}{3} \left (b^2 c^2\right ) \int \frac {1}{x^2 \left (1+c^2 x^2\right )} \, dx-\frac {1}{3} \left (2 i b c^3\right ) \int \frac {a+b \tan ^{-1}(c x)}{x (i+c x)} \, dx\\ &=-\frac {b^2 c^2}{3 x}-\frac {b c \left (a+b \tan ^{-1}(c x)\right )}{3 x^2}+\frac {1}{3} i c^3 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{3 x^3}-\frac {2}{3} b c^3 \left (a+b \tan ^{-1}(c x)\right ) \log \left (2-\frac {2}{1-i c x}\right )-\frac {1}{3} \left (b^2 c^4\right ) \int \frac {1}{1+c^2 x^2} \, dx+\frac {1}{3} \left (2 b^2 c^4\right ) \int \frac {\log \left (2-\frac {2}{1-i c x}\right )}{1+c^2 x^2} \, dx\\ &=-\frac {b^2 c^2}{3 x}-\frac {1}{3} b^2 c^3 \tan ^{-1}(c x)-\frac {b c \left (a+b \tan ^{-1}(c x)\right )}{3 x^2}+\frac {1}{3} i c^3 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{3 x^3}-\frac {2}{3} b c^3 \left (a+b \tan ^{-1}(c x)\right ) \log \left (2-\frac {2}{1-i c x}\right )+\frac {1}{3} i b^2 c^3 \text {Li}_2\left (-1+\frac {2}{1-i c x}\right )\\ \end {align*}
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Mathematica [A] time = 0.39, size = 153, normalized size = 1.09 \[ -\frac {a^2+2 a b c^3 x^3 \log (c x)+b \tan ^{-1}(c x) \left (2 a+b c^3 x^3+2 b c^3 x^3 \log \left (1-e^{2 i \tan ^{-1}(c x)}\right )+b c x\right )-a b c^3 x^3 \log \left (c^2 x^2+1\right )+a b c x-i b^2 c^3 x^3 \text {Li}_2\left (e^{2 i \tan ^{-1}(c x)}\right )+b^2 \left (1-i c^3 x^3\right ) \tan ^{-1}(c x)^2+b^2 c^2 x^2}{3 x^3} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{2} \arctan \left (c x\right )^{2} + 2 \, a b \arctan \left (c x\right ) + a^{2}}{x^{4}}, x\right ) \]
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 [B] time = 0.02, size = 399, normalized size = 2.85 \[ -\frac {a^{2}}{3 x^{3}}-\frac {b^{2} \arctan \left (c x \right )^{2}}{3 x^{3}}-\frac {c \,b^{2} \arctan \left (c x \right )}{3 x^{2}}-\frac {2 c^{3} b^{2} \ln \left (c x \right ) \arctan \left (c x \right )}{3}+\frac {c^{3} b^{2} \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right )}{3}+\frac {i c^{3} b^{2} \ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )}{6}-\frac {i c^{3} b^{2} \dilog \left (i c x +1\right )}{3}-\frac {i c^{3} b^{2} \ln \left (c x -i\right )^{2}}{12}-\frac {i c^{3} b^{2} \ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )}{6}-\frac {i c^{3} b^{2} \ln \left (c x \right ) \ln \left (i c x +1\right )}{3}+\frac {i c^{3} b^{2} \ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )}{6}+\frac {i c^{3} b^{2} \dilog \left (-i c x +1\right )}{3}-\frac {i c^{3} b^{2} \dilog \left (-\frac {i \left (c x +i\right )}{2}\right )}{6}-\frac {b^{2} c^{2}}{3 x}-\frac {b^{2} c^{3} \arctan \left (c x \right )}{3}+\frac {i c^{3} b^{2} \ln \left (c x \right ) \ln \left (-i c x +1\right )}{3}+\frac {i c^{3} b^{2} \dilog \left (\frac {i \left (c x -i\right )}{2}\right )}{6}+\frac {i c^{3} b^{2} \ln \left (c x +i\right )^{2}}{12}-\frac {i c^{3} b^{2} \ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )}{6}-\frac {2 a b \arctan \left (c x \right )}{3 x^{3}}-\frac {c a b}{3 x^{2}}-\frac {2 c^{3} a b \ln \left (c x \right )}{3}+\frac {c^{3} a b \ln \left (c^{2} x^{2}+1\right )}{3} \]
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 \[ \frac {1}{3} \, {\left ({\left (c^{2} \log \left (c^{2} x^{2} + 1\right ) - c^{2} \log \left (x^{2}\right ) - \frac {1}{x^{2}}\right )} c - \frac {2 \, \arctan \left (c x\right )}{x^{3}}\right )} a b + \frac {\frac {1}{4} \, {\left (4 \, x^{3} \int -\frac {12 \, c^{2} x^{2} \log \left (c^{2} x^{2} + 1\right ) - 56 \, c x \arctan \left (c x\right ) - 108 \, {\left (c^{2} x^{2} + 1\right )} \arctan \left (c x\right )^{2} - 9 \, {\left (c^{2} x^{2} + 1\right )} \log \left (c^{2} x^{2} + 1\right )^{2}}{4 \, {\left (c^{2} x^{6} + x^{4}\right )}}\,{d x} - 28 \, \arctan \left (c x\right )^{2} + 3 \, \log \left (c^{2} x^{2} + 1\right )^{2}\right )} b^{2}}{48 \, x^{3}} - \frac {a^{2}}{3 \, x^{3}} \]
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 \[ \int \frac {{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^2}{x^4} \,d x \]
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 \[ \int \frac {\left (a + b \operatorname {atan}{\left (c x \right )}\right )^{2}}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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