Optimal. Leaf size=97 \[ -\frac {1}{2} i c \left (a+b \tan ^{-1}\left (c x^2\right )\right )^2-\frac {\left (a+b \tan ^{-1}\left (c x^2\right )\right )^2}{2 x^2}+b c \log \left (2-\frac {2}{1-i c x^2}\right ) \left (a+b \tan ^{-1}\left (c x^2\right )\right )-\frac {1}{2} i b^2 c \text {Li}_2\left (\frac {2}{1-i c x^2}-1\right ) \]
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Rubi [B] time = 0.65, antiderivative size = 290, normalized size of antiderivative = 2.99, number of steps used = 24, number of rules used = 13, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.812, Rules used = {5035, 2454, 2397, 2392, 2391, 2395, 36, 29, 31, 2439, 2416, 2394, 2393} \[ \frac {1}{2} i b^2 c \text {PolyLog}\left (2,-i c x^2\right )-\frac {1}{2} i b^2 c \text {PolyLog}\left (2,i c x^2\right )-\frac {1}{4} i b^2 c \text {PolyLog}\left (2,\frac {1}{2} \left (1-i c x^2\right )\right )+\frac {1}{4} i b^2 c \text {PolyLog}\left (2,\frac {1}{2} \left (1+i c x^2\right )\right )+\frac {1}{4} i b c \log \left (\frac {1}{2} \left (1+i c x^2\right )\right ) \left (2 i a-b \log \left (1-i c x^2\right )\right )+\frac {b \log \left (1+i c x^2\right ) \left (2 i a-b \log \left (1-i c x^2\right )\right )}{4 x^2}-\frac {\left (1-i c x^2\right ) \left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{8 x^2}+2 a b c \log (x)+\frac {b^2 \left (1+i c x^2\right ) \log ^2\left (1+i c x^2\right )}{8 x^2}+\frac {1}{4} i b^2 c \log \left (\frac {1}{2} \left (1-i c x^2\right )\right ) \log \left (1+i c x^2\right ) \]
Warning: Unable to verify antiderivative.
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Rule 29
Rule 31
Rule 36
Rule 2391
Rule 2392
Rule 2393
Rule 2394
Rule 2395
Rule 2397
Rule 2416
Rule 2439
Rule 2454
Rule 5035
Rubi steps
\begin {align*} \int \frac {\left (a+b \tan ^{-1}\left (c x^2\right )\right )^2}{x^3} \, dx &=\int \left (\frac {\left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{4 x^3}+\frac {b \left (-2 i a+b \log \left (1-i c x^2\right )\right ) \log \left (1+i c x^2\right )}{2 x^3}-\frac {b^2 \log ^2\left (1+i c x^2\right )}{4 x^3}\right ) \, dx\\ &=\frac {1}{4} \int \frac {\left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{x^3} \, dx+\frac {1}{2} b \int \frac {\left (-2 i a+b \log \left (1-i c x^2\right )\right ) \log \left (1+i c x^2\right )}{x^3} \, dx-\frac {1}{4} b^2 \int \frac {\log ^2\left (1+i c x^2\right )}{x^3} \, dx\\ &=\frac {1}{8} \operatorname {Subst}\left (\int \frac {(2 a+i b \log (1-i c x))^2}{x^2} \, dx,x,x^2\right )+\frac {1}{4} b \operatorname {Subst}\left (\int \frac {(-2 i a+b \log (1-i c x)) \log (1+i c x)}{x^2} \, dx,x,x^2\right )-\frac {1}{8} b^2 \operatorname {Subst}\left (\int \frac {\log ^2(1+i c x)}{x^2} \, dx,x,x^2\right )\\ &=-\frac {\left (1-i c x^2\right ) \left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{8 x^2}+\frac {b \left (2 i a-b \log \left (1-i c x^2\right )\right ) \log \left (1+i c x^2\right )}{4 x^2}+\frac {b^2 \left (1+i c x^2\right ) \log ^2\left (1+i c x^2\right )}{8 x^2}+\frac {1}{4} (i b c) \operatorname {Subst}\left (\int \frac {-2 i a+b \log (1-i c x)}{x (1+i c x)} \, dx,x,x^2\right )+\frac {1}{4} (b c) \operatorname {Subst}\left (\int \frac {2 a+i b \log (1-i c x)}{x} \, dx,x,x^2\right )-\frac {1}{4} \left (i b^2 c\right ) \operatorname {Subst}\left (\int \frac {\log (1+i c x)}{x} \, dx,x,x^2\right )-\frac {1}{4} \left (i b^2 c\right ) \operatorname {Subst}\left (\int \frac {\log (1+i c x)}{x (1-i c x)} \, dx,x,x^2\right )\\ &=a b c \log (x)-\frac {\left (1-i c x^2\right ) \left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{8 x^2}+\frac {b \left (2 i a-b \log \left (1-i c x^2\right )\right ) \log \left (1+i c x^2\right )}{4 x^2}+\frac {b^2 \left (1+i c x^2\right ) \log ^2\left (1+i c x^2\right )}{8 x^2}+\frac {1}{4} i b^2 c \text {Li}_2\left (-i c x^2\right )+\frac {1}{4} (i b c) \operatorname {Subst}\left (\int \left (\frac {-2 i a+b \log (1-i c x)}{x}-\frac {c (-2 i a+b \log (1-i c x))}{-i+c x}\right ) \, dx,x,x^2\right )+\frac {1}{4} \left (i b^2 c\right ) \operatorname {Subst}\left (\int \frac {\log (1-i c x)}{x} \, dx,x,x^2\right )-\frac {1}{4} \left (i b^2 c\right ) \operatorname {Subst}\left (\int \left (\frac {\log (1+i c x)}{x}-\frac {c \log (1+i c x)}{i+c x}\right ) \, dx,x,x^2\right )\\ &=a b c \log (x)-\frac {\left (1-i c x^2\right ) \left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{8 x^2}+\frac {b \left (2 i a-b \log \left (1-i c x^2\right )\right ) \log \left (1+i c x^2\right )}{4 x^2}+\frac {b^2 \left (1+i c x^2\right ) \log ^2\left (1+i c x^2\right )}{8 x^2}+\frac {1}{4} i b^2 c \text {Li}_2\left (-i c x^2\right )-\frac {1}{4} i b^2 c \text {Li}_2\left (i c x^2\right )+\frac {1}{4} (i b c) \operatorname {Subst}\left (\int \frac {-2 i a+b \log (1-i c x)}{x} \, dx,x,x^2\right )-\frac {1}{4} \left (i b^2 c\right ) \operatorname {Subst}\left (\int \frac {\log (1+i c x)}{x} \, dx,x,x^2\right )-\frac {1}{4} \left (i b c^2\right ) \operatorname {Subst}\left (\int \frac {-2 i a+b \log (1-i c x)}{-i+c x} \, dx,x,x^2\right )+\frac {1}{4} \left (i b^2 c^2\right ) \operatorname {Subst}\left (\int \frac {\log (1+i c x)}{i+c x} \, dx,x,x^2\right )\\ &=2 a b c \log (x)-\frac {\left (1-i c x^2\right ) \left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{8 x^2}+\frac {1}{4} i b c \left (2 i a-b \log \left (1-i c x^2\right )\right ) \log \left (\frac {1}{2} \left (1+i c x^2\right )\right )+\frac {1}{4} i b^2 c \log \left (\frac {1}{2} \left (1-i c x^2\right )\right ) \log \left (1+i c x^2\right )+\frac {b \left (2 i a-b \log \left (1-i c x^2\right )\right ) \log \left (1+i c x^2\right )}{4 x^2}+\frac {b^2 \left (1+i c x^2\right ) \log ^2\left (1+i c x^2\right )}{8 x^2}+\frac {1}{2} i b^2 c \text {Li}_2\left (-i c x^2\right )-\frac {1}{4} i b^2 c \text {Li}_2\left (i c x^2\right )+\frac {1}{4} \left (i b^2 c\right ) \operatorname {Subst}\left (\int \frac {\log (1-i c x)}{x} \, dx,x,x^2\right )+\frac {1}{4} \left (b^2 c^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (\frac {1}{2} i (-i+c x)\right )}{1-i c x} \, dx,x,x^2\right )+\frac {1}{4} \left (b^2 c^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (-\frac {1}{2} i (i+c x)\right )}{1+i c x} \, dx,x,x^2\right )\\ &=2 a b c \log (x)-\frac {\left (1-i c x^2\right ) \left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{8 x^2}+\frac {1}{4} i b c \left (2 i a-b \log \left (1-i c x^2\right )\right ) \log \left (\frac {1}{2} \left (1+i c x^2\right )\right )+\frac {1}{4} i b^2 c \log \left (\frac {1}{2} \left (1-i c x^2\right )\right ) \log \left (1+i c x^2\right )+\frac {b \left (2 i a-b \log \left (1-i c x^2\right )\right ) \log \left (1+i c x^2\right )}{4 x^2}+\frac {b^2 \left (1+i c x^2\right ) \log ^2\left (1+i c x^2\right )}{8 x^2}+\frac {1}{2} i b^2 c \text {Li}_2\left (-i c x^2\right )-\frac {1}{2} i b^2 c \text {Li}_2\left (i c x^2\right )+\frac {1}{4} \left (i b^2 c\right ) \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {x}{2}\right )}{x} \, dx,x,1-i c x^2\right )-\frac {1}{4} \left (i b^2 c\right ) \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {x}{2}\right )}{x} \, dx,x,1+i c x^2\right )\\ &=2 a b c \log (x)-\frac {\left (1-i c x^2\right ) \left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{8 x^2}+\frac {1}{4} i b c \left (2 i a-b \log \left (1-i c x^2\right )\right ) \log \left (\frac {1}{2} \left (1+i c x^2\right )\right )+\frac {1}{4} i b^2 c \log \left (\frac {1}{2} \left (1-i c x^2\right )\right ) \log \left (1+i c x^2\right )+\frac {b \left (2 i a-b \log \left (1-i c x^2\right )\right ) \log \left (1+i c x^2\right )}{4 x^2}+\frac {b^2 \left (1+i c x^2\right ) \log ^2\left (1+i c x^2\right )}{8 x^2}+\frac {1}{2} i b^2 c \text {Li}_2\left (-i c x^2\right )-\frac {1}{2} i b^2 c \text {Li}_2\left (i c x^2\right )-\frac {1}{4} i b^2 c \text {Li}_2\left (\frac {1}{2} \left (1-i c x^2\right )\right )+\frac {1}{4} i b^2 c \text {Li}_2\left (\frac {1}{2} \left (1+i c x^2\right )\right )\\ \end {align*}
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Mathematica [A] time = 0.16, size = 127, normalized size = 1.31 \[ -\frac {a^2}{2 x^2}+a b c \left (-\frac {1}{2} \log \left (c^2 x^4+1\right )+\log \left (c x^2\right )-\frac {\tan ^{-1}\left (c x^2\right )}{c x^2}\right )+\frac {1}{2} b^2 c \left (-i \left (\tan ^{-1}\left (c x^2\right )^2+\text {Li}_2\left (e^{2 i \tan ^{-1}\left (c x^2\right )}\right )\right )-\frac {\tan ^{-1}\left (c x^2\right )^2}{c x^2}+2 \tan ^{-1}\left (c x^2\right ) \log \left (1-e^{2 i \tan ^{-1}\left (c x^2\right )}\right )\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{2} \arctan \left (c x^{2}\right )^{2} + 2 \, a b \arctan \left (c x^{2}\right ) + a^{2}}{x^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \arctan \left (c x^{2}\right ) + a\right )}^{2}}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.60, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +b \arctan \left (c \,x^{2}\right )\right )^{2}}{x^{3}}\, dx \]
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 \[ -\frac {1}{2} \, {\left (c {\left (\log \left (c^{2} x^{4} + 1\right ) - \log \left (x^{4}\right )\right )} + \frac {2 \, \arctan \left (c x^{2}\right )}{x^{2}}\right )} a b + \frac {\frac {1}{4} \, {\left (8 \, x^{2} \int -\frac {12 \, c^{2} x^{4} \log \left (c^{2} x^{4} + 1\right ) - 56 \, c x^{2} \arctan \left (c x^{2}\right ) - 36 \, {\left (c^{2} x^{4} + 1\right )} \arctan \left (c x^{2}\right )^{2} - 3 \, {\left (c^{2} x^{4} + 1\right )} \log \left (c^{2} x^{4} + 1\right )^{2}}{4 \, {\left (c^{2} x^{7} + x^{3}\right )}}\,{d x} - 28 \, \arctan \left (c x^{2}\right )^{2} + 3 \, \log \left (c^{2} x^{4} + 1\right )^{2}\right )} b^{2}}{32 \, x^{2}} - \frac {a^{2}}{2 \, x^{2}} \]
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 \[ \int \frac {{\left (a+b\,\mathrm {atan}\left (c\,x^2\right )\right )}^2}{x^3} \,d x \]
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 \[ \int \frac {\left (a + b \operatorname {atan}{\left (c x^{2} \right )}\right )^{2}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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