Optimal. Leaf size=172 \[ -i b^2 c d \text{PolyLog}\left (2,-1+\frac{2}{1-i c x}\right )+\frac{i b^2 e \text{PolyLog}\left (2,1-\frac{2}{1+i c x}\right )}{c}-i c d \left (a+b \tan ^{-1}(c x)\right )^2-\frac{d \left (a+b \tan ^{-1}(c x)\right )^2}{x}+2 b c d \log \left (2-\frac{2}{1-i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )+\frac{i e \left (a+b \tan ^{-1}(c x)\right )^2}{c}+e x \left (a+b \tan ^{-1}(c x)\right )^2+\frac{2 b e \log \left (\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{c} \]
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Rubi [A] time = 0.326766, antiderivative size = 172, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 10, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.476, Rules used = {4980, 4846, 4920, 4854, 2402, 2315, 4852, 4924, 4868, 2447} \[ -i b^2 c d \text{PolyLog}\left (2,-1+\frac{2}{1-i c x}\right )+\frac{i b^2 e \text{PolyLog}\left (2,1-\frac{2}{1+i c x}\right )}{c}-i c d \left (a+b \tan ^{-1}(c x)\right )^2-\frac{d \left (a+b \tan ^{-1}(c x)\right )^2}{x}+2 b c d \log \left (2-\frac{2}{1-i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )+\frac{i e \left (a+b \tan ^{-1}(c x)\right )^2}{c}+e x \left (a+b \tan ^{-1}(c x)\right )^2+\frac{2 b e \log \left (\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{c} \]
Antiderivative was successfully verified.
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Rule 4980
Rule 4846
Rule 4920
Rule 4854
Rule 2402
Rule 2315
Rule 4852
Rule 4924
Rule 4868
Rule 2447
Rubi steps
\begin{align*} \int \frac{\left (d+e x^2\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{x^2} \, dx &=\int \left (e \left (a+b \tan ^{-1}(c x)\right )^2+\frac{d \left (a+b \tan ^{-1}(c x)\right )^2}{x^2}\right ) \, dx\\ &=d \int \frac{\left (a+b \tan ^{-1}(c x)\right )^2}{x^2} \, dx+e \int \left (a+b \tan ^{-1}(c x)\right )^2 \, dx\\ &=-\frac{d \left (a+b \tan ^{-1}(c x)\right )^2}{x}+e x \left (a+b \tan ^{-1}(c x)\right )^2+(2 b c d) \int \frac{a+b \tan ^{-1}(c x)}{x \left (1+c^2 x^2\right )} \, dx-(2 b c e) \int \frac{x \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx\\ &=-i c d \left (a+b \tan ^{-1}(c x)\right )^2+\frac{i e \left (a+b \tan ^{-1}(c x)\right )^2}{c}-\frac{d \left (a+b \tan ^{-1}(c x)\right )^2}{x}+e x \left (a+b \tan ^{-1}(c x)\right )^2+(2 i b c d) \int \frac{a+b \tan ^{-1}(c x)}{x (i+c x)} \, dx+(2 b e) \int \frac{a+b \tan ^{-1}(c x)}{i-c x} \, dx\\ &=-i c d \left (a+b \tan ^{-1}(c x)\right )^2+\frac{i e \left (a+b \tan ^{-1}(c x)\right )^2}{c}-\frac{d \left (a+b \tan ^{-1}(c x)\right )^2}{x}+e x \left (a+b \tan ^{-1}(c x)\right )^2+\frac{2 b e \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1+i c x}\right )}{c}+2 b c d \left (a+b \tan ^{-1}(c x)\right ) \log \left (2-\frac{2}{1-i c x}\right )-\left (2 b^2 c^2 d\right ) \int \frac{\log \left (2-\frac{2}{1-i c x}\right )}{1+c^2 x^2} \, dx-\left (2 b^2 e\right ) \int \frac{\log \left (\frac{2}{1+i c x}\right )}{1+c^2 x^2} \, dx\\ &=-i c d \left (a+b \tan ^{-1}(c x)\right )^2+\frac{i e \left (a+b \tan ^{-1}(c x)\right )^2}{c}-\frac{d \left (a+b \tan ^{-1}(c x)\right )^2}{x}+e x \left (a+b \tan ^{-1}(c x)\right )^2+\frac{2 b e \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1+i c x}\right )}{c}+2 b c d \left (a+b \tan ^{-1}(c x)\right ) \log \left (2-\frac{2}{1-i c x}\right )-i b^2 c d \text{Li}_2\left (-1+\frac{2}{1-i c x}\right )+\frac{\left (2 i b^2 e\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+i c x}\right )}{c}\\ &=-i c d \left (a+b \tan ^{-1}(c x)\right )^2+\frac{i e \left (a+b \tan ^{-1}(c x)\right )^2}{c}-\frac{d \left (a+b \tan ^{-1}(c x)\right )^2}{x}+e x \left (a+b \tan ^{-1}(c x)\right )^2+\frac{2 b e \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1+i c x}\right )}{c}+2 b c d \left (a+b \tan ^{-1}(c x)\right ) \log \left (2-\frac{2}{1-i c x}\right )-i b^2 c d \text{Li}_2\left (-1+\frac{2}{1-i c x}\right )+\frac{i b^2 e \text{Li}_2\left (1-\frac{2}{1+i c x}\right )}{c}\\ \end{align*}
Mathematica [A] time = 0.272527, size = 204, normalized size = 1.19 \[ \frac{-b^2 c d \left (i c x \left (\tan ^{-1}(c x)^2+\text{PolyLog}\left (2,e^{2 i \tan ^{-1}(c x)}\right )\right )+\tan ^{-1}(c x)^2-2 c x \tan ^{-1}(c x) \log \left (1-e^{2 i \tan ^{-1}(c x)}\right )\right )+b^2 e x \left (\tan ^{-1}(c x) \left ((c x-i) \tan ^{-1}(c x)+2 \log \left (1+e^{2 i \tan ^{-1}(c x)}\right )\right )-i \text{PolyLog}\left (2,-e^{2 i \tan ^{-1}(c x)}\right )\right )-a^2 c d+a^2 c e x^2+a b c d \left (c x \left (2 \log (c x)-\log \left (c^2 x^2+1\right )\right )-2 \tan ^{-1}(c x)\right )+a b e x \left (2 c x \tan ^{-1}(c x)-\log \left (c^2 x^2+1\right )\right )}{c x} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.132, size = 597, normalized size = 3.5 \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(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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^{2}}, 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^{2}}\, 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^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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