Optimal. Leaf size=220 \[ -i b e \text{PolyLog}\left (2,1-\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )+i b e \text{PolyLog}\left (2,-1+\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )-\frac{1}{2} b^2 e \text{PolyLog}\left (3,1-\frac{2}{1+i c x}\right )+\frac{1}{2} b^2 e \text{PolyLog}\left (3,-1+\frac{2}{1+i c x}\right )-\frac{1}{2} c^2 d \left (a+b \tan ^{-1}(c x)\right )^2-\frac{d \left (a+b \tan ^{-1}(c x)\right )^2}{2 x^2}-\frac{b c d \left (a+b \tan ^{-1}(c x)\right )}{x}+2 e \tanh ^{-1}\left (1-\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )^2-\frac{1}{2} b^2 c^2 d \log \left (c^2 x^2+1\right )+b^2 c^2 d \log (x) \]
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Rubi [A] time = 0.460768, antiderivative size = 220, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 12, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.571, Rules used = {4980, 4852, 4918, 266, 36, 29, 31, 4884, 4850, 4988, 4994, 6610} \[ -i b e \text{PolyLog}\left (2,1-\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )+i b e \text{PolyLog}\left (2,-1+\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )-\frac{1}{2} b^2 e \text{PolyLog}\left (3,1-\frac{2}{1+i c x}\right )+\frac{1}{2} b^2 e \text{PolyLog}\left (3,-1+\frac{2}{1+i c x}\right )-\frac{1}{2} c^2 d \left (a+b \tan ^{-1}(c x)\right )^2-\frac{d \left (a+b \tan ^{-1}(c x)\right )^2}{2 x^2}-\frac{b c d \left (a+b \tan ^{-1}(c x)\right )}{x}+2 e \tanh ^{-1}\left (1-\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )^2-\frac{1}{2} b^2 c^2 d \log \left (c^2 x^2+1\right )+b^2 c^2 d \log (x) \]
Antiderivative was successfully verified.
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Rule 4980
Rule 4852
Rule 4918
Rule 266
Rule 36
Rule 29
Rule 31
Rule 4884
Rule 4850
Rule 4988
Rule 4994
Rule 6610
Rubi steps
\begin{align*} \int \frac{\left (d+e x^2\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{x^3} \, dx &=\int \left (\frac{d \left (a+b \tan ^{-1}(c x)\right )^2}{x^3}+\frac{e \left (a+b \tan ^{-1}(c x)\right )^2}{x}\right ) \, dx\\ &=d \int \frac{\left (a+b \tan ^{-1}(c x)\right )^2}{x^3} \, dx+e \int \frac{\left (a+b \tan ^{-1}(c x)\right )^2}{x} \, dx\\ &=-\frac{d \left (a+b \tan ^{-1}(c x)\right )^2}{2 x^2}+2 e \left (a+b \tan ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac{2}{1+i c x}\right )+(b c d) \int \frac{a+b \tan ^{-1}(c x)}{x^2 \left (1+c^2 x^2\right )} \, dx-(4 b c e) \int \frac{\left (a+b \tan ^{-1}(c x)\right ) \tanh ^{-1}\left (1-\frac{2}{1+i c x}\right )}{1+c^2 x^2} \, dx\\ &=-\frac{d \left (a+b \tan ^{-1}(c x)\right )^2}{2 x^2}+2 e \left (a+b \tan ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac{2}{1+i c x}\right )+(b c d) \int \frac{a+b \tan ^{-1}(c x)}{x^2} \, dx-\left (b c^3 d\right ) \int \frac{a+b \tan ^{-1}(c x)}{1+c^2 x^2} \, dx+(2 b c e) \int \frac{\left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1+i c x}\right )}{1+c^2 x^2} \, dx-(2 b c e) \int \frac{\left (a+b \tan ^{-1}(c x)\right ) \log \left (2-\frac{2}{1+i c x}\right )}{1+c^2 x^2} \, dx\\ &=-\frac{b c d \left (a+b \tan ^{-1}(c x)\right )}{x}-\frac{1}{2} c^2 d \left (a+b \tan ^{-1}(c x)\right )^2-\frac{d \left (a+b \tan ^{-1}(c x)\right )^2}{2 x^2}+2 e \left (a+b \tan ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac{2}{1+i c x}\right )-i b e \left (a+b \tan ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1+i c x}\right )+i b e \left (a+b \tan ^{-1}(c x)\right ) \text{Li}_2\left (-1+\frac{2}{1+i c x}\right )+\left (b^2 c^2 d\right ) \int \frac{1}{x \left (1+c^2 x^2\right )} \, dx+\left (i b^2 c e\right ) \int \frac{\text{Li}_2\left (1-\frac{2}{1+i c x}\right )}{1+c^2 x^2} \, dx-\left (i b^2 c e\right ) \int \frac{\text{Li}_2\left (-1+\frac{2}{1+i c x}\right )}{1+c^2 x^2} \, dx\\ &=-\frac{b c d \left (a+b \tan ^{-1}(c x)\right )}{x}-\frac{1}{2} c^2 d \left (a+b \tan ^{-1}(c x)\right )^2-\frac{d \left (a+b \tan ^{-1}(c x)\right )^2}{2 x^2}+2 e \left (a+b \tan ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac{2}{1+i c x}\right )-i b e \left (a+b \tan ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1+i c x}\right )+i b e \left (a+b \tan ^{-1}(c x)\right ) \text{Li}_2\left (-1+\frac{2}{1+i c x}\right )-\frac{1}{2} b^2 e \text{Li}_3\left (1-\frac{2}{1+i c x}\right )+\frac{1}{2} b^2 e \text{Li}_3\left (-1+\frac{2}{1+i c x}\right )+\frac{1}{2} \left (b^2 c^2 d\right ) \operatorname{Subst}\left (\int \frac{1}{x \left (1+c^2 x\right )} \, dx,x,x^2\right )\\ &=-\frac{b c d \left (a+b \tan ^{-1}(c x)\right )}{x}-\frac{1}{2} c^2 d \left (a+b \tan ^{-1}(c x)\right )^2-\frac{d \left (a+b \tan ^{-1}(c x)\right )^2}{2 x^2}+2 e \left (a+b \tan ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac{2}{1+i c x}\right )-i b e \left (a+b \tan ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1+i c x}\right )+i b e \left (a+b \tan ^{-1}(c x)\right ) \text{Li}_2\left (-1+\frac{2}{1+i c x}\right )-\frac{1}{2} b^2 e \text{Li}_3\left (1-\frac{2}{1+i c x}\right )+\frac{1}{2} b^2 e \text{Li}_3\left (-1+\frac{2}{1+i c x}\right )+\frac{1}{2} \left (b^2 c^2 d\right ) \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,x^2\right )-\frac{1}{2} \left (b^2 c^4 d\right ) \operatorname{Subst}\left (\int \frac{1}{1+c^2 x} \, dx,x,x^2\right )\\ &=-\frac{b c d \left (a+b \tan ^{-1}(c x)\right )}{x}-\frac{1}{2} c^2 d \left (a+b \tan ^{-1}(c x)\right )^2-\frac{d \left (a+b \tan ^{-1}(c x)\right )^2}{2 x^2}+2 e \left (a+b \tan ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac{2}{1+i c x}\right )+b^2 c^2 d \log (x)-\frac{1}{2} b^2 c^2 d \log \left (1+c^2 x^2\right )-i b e \left (a+b \tan ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1+i c x}\right )+i b e \left (a+b \tan ^{-1}(c x)\right ) \text{Li}_2\left (-1+\frac{2}{1+i c x}\right )-\frac{1}{2} b^2 e \text{Li}_3\left (1-\frac{2}{1+i c x}\right )+\frac{1}{2} b^2 e \text{Li}_3\left (-1+\frac{2}{1+i c x}\right )\\ \end{align*}
Mathematica [A] time = 0.34584, size = 273, normalized size = 1.24 \[ i a b e (\text{PolyLog}(2,-i c x)-\text{PolyLog}(2,i c x))+\frac{1}{24} b^2 e \left (24 i \tan ^{-1}(c x) \text{PolyLog}\left (2,e^{-2 i \tan ^{-1}(c x)}\right )+24 i \tan ^{-1}(c x) \text{PolyLog}\left (2,-e^{2 i \tan ^{-1}(c x)}\right )+12 \text{PolyLog}\left (3,e^{-2 i \tan ^{-1}(c x)}\right )-12 \text{PolyLog}\left (3,-e^{2 i \tan ^{-1}(c x)}\right )+16 i \tan ^{-1}(c x)^3+24 \tan ^{-1}(c x)^2 \log \left (1-e^{-2 i \tan ^{-1}(c x)}\right )-24 \tan ^{-1}(c x)^2 \log \left (1+e^{2 i \tan ^{-1}(c x)}\right )-i \pi ^3\right )-\frac{a^2 d}{2 x^2}+a^2 e \log (x)-\frac{a b d \left (\tan ^{-1}(c x)+c x \left (c x \tan ^{-1}(c x)+1\right )\right )}{x^2}-\frac{b^2 d \left (-2 c^2 x^2 \log \left (\frac{c x}{\sqrt{c^2 x^2+1}}\right )+\left (c^2 x^2+1\right ) \tan ^{-1}(c x)^2+2 c x \tan ^{-1}(c x)\right )}{2 x^2} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 4.318, size = 1313, normalized size = 6. \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{a^{2} e x^{2} + a^{2} d +{\left (b^{2} e x^{2} + b^{2} d\right )} \arctan \left (c x\right )^{2} + 2 \,{\left (a b e x^{2} + a b d\right )} \arctan \left (c x\right )}{x^{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b \operatorname{atan}{\left (c x \right )}\right )^{2} \left (d + e x^{2}\right )}{x^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (e x^{2} + d\right )}{\left (b \arctan \left (c x\right ) + a\right )}^{2}}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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