Optimal. Leaf size=200 \[ \frac{3}{2} i b d^2 e \text{PolyLog}(2,-i c x)-\frac{3}{2} i b d^2 e \text{PolyLog}(2,i c x)-\frac{d^3 \left (a+b \tan ^{-1}(c x)\right )}{2 x^2}+\frac{3}{2} d e^2 x^2 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{4} e^3 x^4 \left (a+b \tan ^{-1}(c x)\right )+3 a d^2 e \log (x)-\frac{1}{2} b c^2 d^3 \tan ^{-1}(c x)+\frac{3 b d e^2 \tan ^{-1}(c x)}{2 c^2}+\frac{b e^3 x}{4 c^3}-\frac{b e^3 \tan ^{-1}(c x)}{4 c^4}-\frac{b c d^3}{2 x}-\frac{3 b d e^2 x}{2 c}-\frac{b e^3 x^3}{12 c} \]
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Rubi [A] time = 0.210537, antiderivative size = 200, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 8, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.381, Rules used = {4980, 4852, 325, 203, 4848, 2391, 321, 302} \[ \frac{3}{2} i b d^2 e \text{PolyLog}(2,-i c x)-\frac{3}{2} i b d^2 e \text{PolyLog}(2,i c x)-\frac{d^3 \left (a+b \tan ^{-1}(c x)\right )}{2 x^2}+\frac{3}{2} d e^2 x^2 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{4} e^3 x^4 \left (a+b \tan ^{-1}(c x)\right )+3 a d^2 e \log (x)-\frac{1}{2} b c^2 d^3 \tan ^{-1}(c x)+\frac{3 b d e^2 \tan ^{-1}(c x)}{2 c^2}+\frac{b e^3 x}{4 c^3}-\frac{b e^3 \tan ^{-1}(c x)}{4 c^4}-\frac{b c d^3}{2 x}-\frac{3 b d e^2 x}{2 c}-\frac{b e^3 x^3}{12 c} \]
Antiderivative was successfully verified.
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Rule 4980
Rule 4852
Rule 325
Rule 203
Rule 4848
Rule 2391
Rule 321
Rule 302
Rubi steps
\begin{align*} \int \frac{\left (d+e x^2\right )^3 \left (a+b \tan ^{-1}(c x)\right )}{x^3} \, dx &=\int \left (\frac{d^3 \left (a+b \tan ^{-1}(c x)\right )}{x^3}+\frac{3 d^2 e \left (a+b \tan ^{-1}(c x)\right )}{x}+3 d e^2 x \left (a+b \tan ^{-1}(c x)\right )+e^3 x^3 \left (a+b \tan ^{-1}(c x)\right )\right ) \, dx\\ &=d^3 \int \frac{a+b \tan ^{-1}(c x)}{x^3} \, dx+\left (3 d^2 e\right ) \int \frac{a+b \tan ^{-1}(c x)}{x} \, dx+\left (3 d e^2\right ) \int x \left (a+b \tan ^{-1}(c x)\right ) \, dx+e^3 \int x^3 \left (a+b \tan ^{-1}(c x)\right ) \, dx\\ &=-\frac{d^3 \left (a+b \tan ^{-1}(c x)\right )}{2 x^2}+\frac{3}{2} d e^2 x^2 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{4} e^3 x^4 \left (a+b \tan ^{-1}(c x)\right )+3 a d^2 e \log (x)+\frac{1}{2} \left (b c d^3\right ) \int \frac{1}{x^2 \left (1+c^2 x^2\right )} \, dx+\frac{1}{2} \left (3 i b d^2 e\right ) \int \frac{\log (1-i c x)}{x} \, dx-\frac{1}{2} \left (3 i b d^2 e\right ) \int \frac{\log (1+i c x)}{x} \, dx-\frac{1}{2} \left (3 b c d e^2\right ) \int \frac{x^2}{1+c^2 x^2} \, dx-\frac{1}{4} \left (b c e^3\right ) \int \frac{x^4}{1+c^2 x^2} \, dx\\ &=-\frac{b c d^3}{2 x}-\frac{3 b d e^2 x}{2 c}-\frac{d^3 \left (a+b \tan ^{-1}(c x)\right )}{2 x^2}+\frac{3}{2} d e^2 x^2 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{4} e^3 x^4 \left (a+b \tan ^{-1}(c x)\right )+3 a d^2 e \log (x)+\frac{3}{2} i b d^2 e \text{Li}_2(-i c x)-\frac{3}{2} i b d^2 e \text{Li}_2(i c x)-\frac{1}{2} \left (b c^3 d^3\right ) \int \frac{1}{1+c^2 x^2} \, dx+\frac{\left (3 b d e^2\right ) \int \frac{1}{1+c^2 x^2} \, dx}{2 c}-\frac{1}{4} \left (b c e^3\right ) \int \left (-\frac{1}{c^4}+\frac{x^2}{c^2}+\frac{1}{c^4 \left (1+c^2 x^2\right )}\right ) \, dx\\ &=-\frac{b c d^3}{2 x}-\frac{3 b d e^2 x}{2 c}+\frac{b e^3 x}{4 c^3}-\frac{b e^3 x^3}{12 c}-\frac{1}{2} b c^2 d^3 \tan ^{-1}(c x)+\frac{3 b d e^2 \tan ^{-1}(c x)}{2 c^2}-\frac{d^3 \left (a+b \tan ^{-1}(c x)\right )}{2 x^2}+\frac{3}{2} d e^2 x^2 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{4} e^3 x^4 \left (a+b \tan ^{-1}(c x)\right )+3 a d^2 e \log (x)+\frac{3}{2} i b d^2 e \text{Li}_2(-i c x)-\frac{3}{2} i b d^2 e \text{Li}_2(i c x)-\frac{\left (b e^3\right ) \int \frac{1}{1+c^2 x^2} \, dx}{4 c^3}\\ &=-\frac{b c d^3}{2 x}-\frac{3 b d e^2 x}{2 c}+\frac{b e^3 x}{4 c^3}-\frac{b e^3 x^3}{12 c}-\frac{1}{2} b c^2 d^3 \tan ^{-1}(c x)+\frac{3 b d e^2 \tan ^{-1}(c x)}{2 c^2}-\frac{b e^3 \tan ^{-1}(c x)}{4 c^4}-\frac{d^3 \left (a+b \tan ^{-1}(c x)\right )}{2 x^2}+\frac{3}{2} d e^2 x^2 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{4} e^3 x^4 \left (a+b \tan ^{-1}(c x)\right )+3 a d^2 e \log (x)+\frac{3}{2} i b d^2 e \text{Li}_2(-i c x)-\frac{3}{2} i b d^2 e \text{Li}_2(i c x)\\ \end{align*}
Mathematica [C] time = 0.160364, size = 170, normalized size = 0.85 \[ \frac{1}{12} \left (-\frac{6 b c d^3 \text{Hypergeometric2F1}\left (-\frac{1}{2},1,\frac{1}{2},-c^2 x^2\right )}{x}+18 i b d^2 e \text{PolyLog}(2,-i c x)-18 i b d^2 e \text{PolyLog}(2,i c x)-\frac{6 d^3 \left (a+b \tan ^{-1}(c x)\right )}{x^2}+18 d e^2 x^2 \left (a+b \tan ^{-1}(c x)\right )+3 e^3 x^4 \left (a+b \tan ^{-1}(c x)\right )+36 a d^2 e \log (x)-\frac{18 b d e^2 \left (c x-\tan ^{-1}(c x)\right )}{c^2}-\frac{b e^3 \left (c^3 x^3-3 c x+3 \tan ^{-1}(c x)\right )}{c^4}\right ) \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.056, size = 251, normalized size = 1.3 \begin{align*}{\frac{a{e}^{3}{x}^{4}}{4}}+{\frac{3\,a{x}^{2}d{e}^{2}}{2}}-{\frac{{d}^{3}a}{2\,{x}^{2}}}+3\,a{d}^{2}e\ln \left ( cx \right ) +{\frac{b\arctan \left ( cx \right ){e}^{3}{x}^{4}}{4}}+{\frac{3\,b\arctan \left ( cx \right ){x}^{2}d{e}^{2}}{2}}-{\frac{b{d}^{3}\arctan \left ( cx \right ) }{2\,{x}^{2}}}+3\,b\arctan \left ( cx \right ){d}^{2}e\ln \left ( cx \right ) -{\frac{b{e}^{3}{x}^{3}}{12\,c}}-{\frac{3\,bd{e}^{2}x}{2\,c}}+{\frac{b{e}^{3}x}{4\,{c}^{3}}}-{\frac{b{c}^{2}{d}^{3}\arctan \left ( cx \right ) }{2}}+{\frac{3\,bd{e}^{2}\arctan \left ( cx \right ) }{2\,{c}^{2}}}-{\frac{b\arctan \left ( cx \right ){e}^{3}}{4\,{c}^{4}}}-{\frac{bc{d}^{3}}{2\,x}}-{\frac{3\,i}{2}}b{d}^{2}e{\it dilog} \left ( 1-icx \right ) +{\frac{3\,i}{2}}b{d}^{2}e{\it dilog} \left ( 1+icx \right ) -{\frac{3\,i}{2}}b{d}^{2}e\ln \left ( cx \right ) \ln \left ( 1-icx \right ) +{\frac{3\,i}{2}}b{d}^{2}e\ln \left ( cx \right ) \ln \left ( 1+icx \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.20896, size = 321, normalized size = 1.6 \begin{align*} \frac{1}{4} \, a e^{3} x^{4} + \frac{3}{2} \, a d e^{2} x^{2} - \frac{1}{2} \,{\left ({\left (c \arctan \left (c x\right ) + \frac{1}{x}\right )} c + \frac{\arctan \left (c x\right )}{x^{2}}\right )} b d^{3} + 3 \, a d^{2} e \log \left (x\right ) - \frac{a d^{3}}{2 \, x^{2}} - \frac{b c^{3} e^{3} x^{3} + 9 \, \pi b c^{4} d^{2} e \log \left (c^{2} x^{2} + 1\right ) - 36 \, b c^{4} d^{2} e \arctan \left (c x\right ) \log \left (x{\left | c \right |}\right ) + 18 i \, b c^{4} d^{2} e{\rm Li}_2\left (i \, c x + 1\right ) - 18 i \, b c^{4} d^{2} e{\rm Li}_2\left (-i \, c x + 1\right ) + 3 \,{\left (6 \, b c^{3} d e^{2} - b c e^{3}\right )} x -{\left (3 \, b c^{4} e^{3} x^{4} + 18 \, b c^{4} d e^{2} x^{2} + 36 i \, b c^{4} d^{2} e \arctan \left (0, c\right ) + 18 \, b c^{2} d e^{2} - 3 \, b e^{3}\right )} \arctan \left (c x\right )}{12 \, c^{4}} \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 e^{3} x^{6} + 3 \, a d e^{2} x^{4} + 3 \, a d^{2} e x^{2} + a d^{3} +{\left (b e^{3} x^{6} + 3 \, b d e^{2} x^{4} + 3 \, b d^{2} e x^{2} + b d^{3}\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 ) \left (d + e x^{2}\right )^{3}}{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 )}^{3}{\left (b \arctan \left (c x\right ) + a\right )}}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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