Optimal. Leaf size=403 \[ -\frac{i b \text{PolyLog}\left (2,1-\frac{2 c \left (\sqrt{-d}-\sqrt{e} x\right )}{(1-i c x) \left (c \sqrt{-d}-i \sqrt{e}\right )}\right )}{4 e^2}-\frac{i b \text{PolyLog}\left (2,1-\frac{2 c \left (\sqrt{-d}+\sqrt{e} x\right )}{(1-i c x) \left (c \sqrt{-d}+i \sqrt{e}\right )}\right )}{4 e^2}+\frac{i b \text{PolyLog}\left (2,1-\frac{2}{1-i c x}\right )}{2 e^2}+\frac{d \left (a+b \tan ^{-1}(c x)\right )}{2 e^2 \left (d+e x^2\right )}+\frac{\left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2 c \left (\sqrt{-d}-\sqrt{e} x\right )}{(1-i c x) \left (c \sqrt{-d}-i \sqrt{e}\right )}\right )}{2 e^2}+\frac{\left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2 c \left (\sqrt{-d}+\sqrt{e} x\right )}{(1-i c x) \left (c \sqrt{-d}+i \sqrt{e}\right )}\right )}{2 e^2}-\frac{\log \left (\frac{2}{1-i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{e^2}-\frac{b c^2 d \tan ^{-1}(c x)}{2 e^2 \left (c^2 d-e\right )}+\frac{b c \sqrt{d} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{2 e^{3/2} \left (c^2 d-e\right )} \]
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Rubi [A] time = 0.448857, antiderivative size = 403, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 9, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {4980, 4974, 391, 203, 205, 4856, 2402, 2315, 2447} \[ -\frac{i b \text{PolyLog}\left (2,1-\frac{2 c \left (\sqrt{-d}-\sqrt{e} x\right )}{(1-i c x) \left (c \sqrt{-d}-i \sqrt{e}\right )}\right )}{4 e^2}-\frac{i b \text{PolyLog}\left (2,1-\frac{2 c \left (\sqrt{-d}+\sqrt{e} x\right )}{(1-i c x) \left (c \sqrt{-d}+i \sqrt{e}\right )}\right )}{4 e^2}+\frac{i b \text{PolyLog}\left (2,1-\frac{2}{1-i c x}\right )}{2 e^2}+\frac{d \left (a+b \tan ^{-1}(c x)\right )}{2 e^2 \left (d+e x^2\right )}+\frac{\left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2 c \left (\sqrt{-d}-\sqrt{e} x\right )}{(1-i c x) \left (c \sqrt{-d}-i \sqrt{e}\right )}\right )}{2 e^2}+\frac{\left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2 c \left (\sqrt{-d}+\sqrt{e} x\right )}{(1-i c x) \left (c \sqrt{-d}+i \sqrt{e}\right )}\right )}{2 e^2}-\frac{\log \left (\frac{2}{1-i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{e^2}-\frac{b c^2 d \tan ^{-1}(c x)}{2 e^2 \left (c^2 d-e\right )}+\frac{b c \sqrt{d} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{2 e^{3/2} \left (c^2 d-e\right )} \]
Antiderivative was successfully verified.
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Rule 4980
Rule 4974
Rule 391
Rule 203
Rule 205
Rule 4856
Rule 2402
Rule 2315
Rule 2447
Rubi steps
\begin{align*} \int \frac{x^3 \left (a+b \tan ^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx &=\int \left (-\frac{d x \left (a+b \tan ^{-1}(c x)\right )}{e \left (d+e x^2\right )^2}+\frac{x \left (a+b \tan ^{-1}(c x)\right )}{e \left (d+e x^2\right )}\right ) \, dx\\ &=\frac{\int \frac{x \left (a+b \tan ^{-1}(c x)\right )}{d+e x^2} \, dx}{e}-\frac{d \int \frac{x \left (a+b \tan ^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx}{e}\\ &=\frac{d \left (a+b \tan ^{-1}(c x)\right )}{2 e^2 \left (d+e x^2\right )}-\frac{(b c d) \int \frac{1}{\left (1+c^2 x^2\right ) \left (d+e x^2\right )} \, dx}{2 e^2}+\frac{\int \left (-\frac{a+b \tan ^{-1}(c x)}{2 \sqrt{e} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{a+b \tan ^{-1}(c x)}{2 \sqrt{e} \left (\sqrt{-d}+\sqrt{e} x\right )}\right ) \, dx}{e}\\ &=\frac{d \left (a+b \tan ^{-1}(c x)\right )}{2 e^2 \left (d+e x^2\right )}-\frac{\left (b c^3 d\right ) \int \frac{1}{1+c^2 x^2} \, dx}{2 \left (c^2 d-e\right ) e^2}-\frac{\int \frac{a+b \tan ^{-1}(c x)}{\sqrt{-d}-\sqrt{e} x} \, dx}{2 e^{3/2}}+\frac{\int \frac{a+b \tan ^{-1}(c x)}{\sqrt{-d}+\sqrt{e} x} \, dx}{2 e^{3/2}}+\frac{(b c d) \int \frac{1}{d+e x^2} \, dx}{2 \left (c^2 d-e\right ) e}\\ &=-\frac{b c^2 d \tan ^{-1}(c x)}{2 \left (c^2 d-e\right ) e^2}+\frac{d \left (a+b \tan ^{-1}(c x)\right )}{2 e^2 \left (d+e x^2\right )}+\frac{b c \sqrt{d} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{2 \left (c^2 d-e\right ) e^{3/2}}-\frac{\left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1-i c x}\right )}{e^2}+\frac{\left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2 c \left (\sqrt{-d}-\sqrt{e} x\right )}{\left (c \sqrt{-d}-i \sqrt{e}\right ) (1-i c x)}\right )}{2 e^2}+\frac{\left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2 c \left (\sqrt{-d}+\sqrt{e} x\right )}{\left (c \sqrt{-d}+i \sqrt{e}\right ) (1-i c x)}\right )}{2 e^2}+2 \frac{(b c) \int \frac{\log \left (\frac{2}{1-i c x}\right )}{1+c^2 x^2} \, dx}{2 e^2}-\frac{(b c) \int \frac{\log \left (\frac{2 c \left (\sqrt{-d}-\sqrt{e} x\right )}{\left (c \sqrt{-d}-i \sqrt{e}\right ) (1-i c x)}\right )}{1+c^2 x^2} \, dx}{2 e^2}-\frac{(b c) \int \frac{\log \left (\frac{2 c \left (\sqrt{-d}+\sqrt{e} x\right )}{\left (c \sqrt{-d}+i \sqrt{e}\right ) (1-i c x)}\right )}{1+c^2 x^2} \, dx}{2 e^2}\\ &=-\frac{b c^2 d \tan ^{-1}(c x)}{2 \left (c^2 d-e\right ) e^2}+\frac{d \left (a+b \tan ^{-1}(c x)\right )}{2 e^2 \left (d+e x^2\right )}+\frac{b c \sqrt{d} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{2 \left (c^2 d-e\right ) e^{3/2}}-\frac{\left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1-i c x}\right )}{e^2}+\frac{\left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2 c \left (\sqrt{-d}-\sqrt{e} x\right )}{\left (c \sqrt{-d}-i \sqrt{e}\right ) (1-i c x)}\right )}{2 e^2}+\frac{\left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2 c \left (\sqrt{-d}+\sqrt{e} x\right )}{\left (c \sqrt{-d}+i \sqrt{e}\right ) (1-i c x)}\right )}{2 e^2}-\frac{i b \text{Li}_2\left (1-\frac{2 c \left (\sqrt{-d}-\sqrt{e} x\right )}{\left (c \sqrt{-d}-i \sqrt{e}\right ) (1-i c x)}\right )}{4 e^2}-\frac{i b \text{Li}_2\left (1-\frac{2 c \left (\sqrt{-d}+\sqrt{e} x\right )}{\left (c \sqrt{-d}+i \sqrt{e}\right ) (1-i c x)}\right )}{4 e^2}+2 \frac{(i b) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1-i c x}\right )}{2 e^2}\\ &=-\frac{b c^2 d \tan ^{-1}(c x)}{2 \left (c^2 d-e\right ) e^2}+\frac{d \left (a+b \tan ^{-1}(c x)\right )}{2 e^2 \left (d+e x^2\right )}+\frac{b c \sqrt{d} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{2 \left (c^2 d-e\right ) e^{3/2}}-\frac{\left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1-i c x}\right )}{e^2}+\frac{\left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2 c \left (\sqrt{-d}-\sqrt{e} x\right )}{\left (c \sqrt{-d}-i \sqrt{e}\right ) (1-i c x)}\right )}{2 e^2}+\frac{\left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2 c \left (\sqrt{-d}+\sqrt{e} x\right )}{\left (c \sqrt{-d}+i \sqrt{e}\right ) (1-i c x)}\right )}{2 e^2}+\frac{i b \text{Li}_2\left (1-\frac{2}{1-i c x}\right )}{2 e^2}-\frac{i b \text{Li}_2\left (1-\frac{2 c \left (\sqrt{-d}-\sqrt{e} x\right )}{\left (c \sqrt{-d}-i \sqrt{e}\right ) (1-i c x)}\right )}{4 e^2}-\frac{i b \text{Li}_2\left (1-\frac{2 c \left (\sqrt{-d}+\sqrt{e} x\right )}{\left (c \sqrt{-d}+i \sqrt{e}\right ) (1-i c x)}\right )}{4 e^2}\\ \end{align*}
Mathematica [A] time = 7.91023, size = 522, normalized size = 1.3 \[ \frac{2 a \left (\frac{d}{d+e x^2}+\log \left (d+e x^2\right )\right )+b \left (-i \text{PolyLog}\left (2,\frac{c \left (\sqrt{d}-i \sqrt{e} x\right )}{c \sqrt{d}-\sqrt{e}}\right )+i \text{PolyLog}\left (2,\frac{c \left (\sqrt{d}-i \sqrt{e} x\right )}{c \sqrt{d}+\sqrt{e}}\right )+i \text{PolyLog}\left (2,\frac{c \left (\sqrt{d}+i \sqrt{e} x\right )}{c \sqrt{d}-\sqrt{e}}\right )-i \text{PolyLog}\left (2,\frac{c \left (\sqrt{d}+i \sqrt{e} x\right )}{c \sqrt{d}+\sqrt{e}}\right )-\frac{2 c^2 d \tan ^{-1}(c x)}{c^2 d-e}+\frac{2 c \sqrt{d} \sqrt{e} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{c^2 d-e}+\frac{2 d \tan ^{-1}(c x)}{d+e x^2}+i \log \left (x-\frac{i \sqrt{d}}{\sqrt{e}}\right ) \log \left (\frac{\sqrt{e} (-1-i c x)}{c \sqrt{d}-\sqrt{e}}\right )-i \log \left (x-\frac{i \sqrt{d}}{\sqrt{e}}\right ) \log \left (\frac{\sqrt{e} (1-i c x)}{c \sqrt{d}+\sqrt{e}}\right )-i \log \left (x+\frac{i \sqrt{d}}{\sqrt{e}}\right ) \log \left (\frac{\sqrt{e} (-1+i c x)}{c \sqrt{d}-\sqrt{e}}\right )+i \log \left (x+\frac{i \sqrt{d}}{\sqrt{e}}\right ) \log \left (\frac{\sqrt{e} (1+i c x)}{c \sqrt{d}+\sqrt{e}}\right )+2 \tan ^{-1}(c x) \log \left (x-\frac{i \sqrt{d}}{\sqrt{e}}\right )+2 \tan ^{-1}(c x) \log \left (x+\frac{i \sqrt{d}}{\sqrt{e}}\right )\right )}{4 e^2} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.203, size = 760, normalized size = 1.9 \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*} \frac{1}{2} \, a{\left (\frac{d}{e^{3} x^{2} + d e^{2}} + \frac{\log \left (e x^{2} + d\right )}{e^{2}}\right )} + 2 \, b \int \frac{x^{3} \arctan \left (c x\right )}{2 \,{\left (e^{2} x^{4} + 2 \, d e x^{2} + d^{2}\right )}}\,{d x} \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{b x^{3} \arctan \left (c x\right ) + a x^{3}}{e^{2} x^{4} + 2 \, d e x^{2} + d^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [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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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \arctan \left (c x\right ) + a\right )} x^{3}}{{\left (e x^{2} + d\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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