Optimal. Leaf size=574 \[ \frac{i b \text{PolyLog}\left (2,1-\frac{2 c \left (\sqrt{-d}-\sqrt{e} x\right )}{(1-i c x) \left (c \sqrt{-d}-i \sqrt{e}\right )}\right )}{4 d^3}+\frac{i b \text{PolyLog}\left (2,1-\frac{2 c \left (\sqrt{-d}+\sqrt{e} x\right )}{(1-i c x) \left (c \sqrt{-d}+i \sqrt{e}\right )}\right )}{4 d^3}+\frac{i b \text{PolyLog}(2,-i c x)}{2 d^3}-\frac{i b \text{PolyLog}(2,i c x)}{2 d^3}-\frac{i b \text{PolyLog}\left (2,1-\frac{2}{1-i c x}\right )}{2 d^3}+\frac{a+b \tan ^{-1}(c x)}{2 d^2 \left (d+e x^2\right )}-\frac{\left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2 c \left (\sqrt{-d}-\sqrt{e} x\right )}{(1-i c x) \left (c \sqrt{-d}-i \sqrt{e}\right )}\right )}{2 d^3}-\frac{\left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2 c \left (\sqrt{-d}+\sqrt{e} x\right )}{(1-i c x) \left (c \sqrt{-d}+i \sqrt{e}\right )}\right )}{2 d^3}+\frac{\log \left (\frac{2}{1-i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{d^3}+\frac{a+b \tan ^{-1}(c x)}{4 d \left (d+e x^2\right )^2}+\frac{a \log (x)}{d^3}+\frac{b c e x}{8 d^2 \left (c^2 d-e\right ) \left (d+e x^2\right )}-\frac{b c^2 \tan ^{-1}(c x)}{2 d^2 \left (c^2 d-e\right )}+\frac{b c \sqrt{e} \left (3 c^2 d-e\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{8 d^{5/2} \left (c^2 d-e\right )^2}+\frac{b c \sqrt{e} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{2 d^{5/2} \left (c^2 d-e\right )}-\frac{b c^4 \tan ^{-1}(c x)}{4 d \left (c^2 d-e\right )^2} \]
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Rubi [A] time = 0.631111, antiderivative size = 574, normalized size of antiderivative = 1., number of steps used = 24, number of rules used = 13, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.619, Rules used = {4980, 4848, 2391, 4974, 414, 522, 203, 205, 391, 4856, 2402, 2315, 2447} \[ \frac{i b \text{PolyLog}\left (2,1-\frac{2 c \left (\sqrt{-d}-\sqrt{e} x\right )}{(1-i c x) \left (c \sqrt{-d}-i \sqrt{e}\right )}\right )}{4 d^3}+\frac{i b \text{PolyLog}\left (2,1-\frac{2 c \left (\sqrt{-d}+\sqrt{e} x\right )}{(1-i c x) \left (c \sqrt{-d}+i \sqrt{e}\right )}\right )}{4 d^3}+\frac{i b \text{PolyLog}(2,-i c x)}{2 d^3}-\frac{i b \text{PolyLog}(2,i c x)}{2 d^3}-\frac{i b \text{PolyLog}\left (2,1-\frac{2}{1-i c x}\right )}{2 d^3}+\frac{a+b \tan ^{-1}(c x)}{2 d^2 \left (d+e x^2\right )}-\frac{\left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2 c \left (\sqrt{-d}-\sqrt{e} x\right )}{(1-i c x) \left (c \sqrt{-d}-i \sqrt{e}\right )}\right )}{2 d^3}-\frac{\left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2 c \left (\sqrt{-d}+\sqrt{e} x\right )}{(1-i c x) \left (c \sqrt{-d}+i \sqrt{e}\right )}\right )}{2 d^3}+\frac{\log \left (\frac{2}{1-i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{d^3}+\frac{a+b \tan ^{-1}(c x)}{4 d \left (d+e x^2\right )^2}+\frac{a \log (x)}{d^3}+\frac{b c e x}{8 d^2 \left (c^2 d-e\right ) \left (d+e x^2\right )}-\frac{b c^2 \tan ^{-1}(c x)}{2 d^2 \left (c^2 d-e\right )}+\frac{b c \sqrt{e} \left (3 c^2 d-e\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{8 d^{5/2} \left (c^2 d-e\right )^2}+\frac{b c \sqrt{e} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{2 d^{5/2} \left (c^2 d-e\right )}-\frac{b c^4 \tan ^{-1}(c x)}{4 d \left (c^2 d-e\right )^2} \]
Antiderivative was successfully verified.
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Rule 4980
Rule 4848
Rule 2391
Rule 4974
Rule 414
Rule 522
Rule 203
Rule 205
Rule 391
Rule 4856
Rule 2402
Rule 2315
Rule 2447
Rubi steps
\begin{align*} \int \frac{a+b \tan ^{-1}(c x)}{x \left (d+e x^2\right )^3} \, dx &=\int \left (\frac{a+b \tan ^{-1}(c x)}{d^3 x}-\frac{e x \left (a+b \tan ^{-1}(c x)\right )}{d \left (d+e x^2\right )^3}-\frac{e x \left (a+b \tan ^{-1}(c x)\right )}{d^2 \left (d+e x^2\right )^2}-\frac{e x \left (a+b \tan ^{-1}(c x)\right )}{d^3 \left (d+e x^2\right )}\right ) \, dx\\ &=\frac{\int \frac{a+b \tan ^{-1}(c x)}{x} \, dx}{d^3}-\frac{e \int \frac{x \left (a+b \tan ^{-1}(c x)\right )}{d+e x^2} \, dx}{d^3}-\frac{e \int \frac{x \left (a+b \tan ^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx}{d^2}-\frac{e \int \frac{x \left (a+b \tan ^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx}{d}\\ &=\frac{a+b \tan ^{-1}(c x)}{4 d \left (d+e x^2\right )^2}+\frac{a+b \tan ^{-1}(c x)}{2 d^2 \left (d+e x^2\right )}+\frac{a \log (x)}{d^3}+\frac{(i b) \int \frac{\log (1-i c x)}{x} \, dx}{2 d^3}-\frac{(i b) \int \frac{\log (1+i c x)}{x} \, dx}{2 d^3}-\frac{(b c) \int \frac{1}{\left (1+c^2 x^2\right ) \left (d+e x^2\right )} \, dx}{2 d^2}-\frac{(b c) \int \frac{1}{\left (1+c^2 x^2\right ) \left (d+e x^2\right )^2} \, dx}{4 d}-\frac{e \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}{d^3}\\ &=\frac{b c e x}{8 d^2 \left (c^2 d-e\right ) \left (d+e x^2\right )}+\frac{a+b \tan ^{-1}(c x)}{4 d \left (d+e x^2\right )^2}+\frac{a+b \tan ^{-1}(c x)}{2 d^2 \left (d+e x^2\right )}+\frac{a \log (x)}{d^3}+\frac{i b \text{Li}_2(-i c x)}{2 d^3}-\frac{i b \text{Li}_2(i c x)}{2 d^3}-\frac{(b c) \int \frac{2 c^2 d-e-c^2 e x^2}{\left (1+c^2 x^2\right ) \left (d+e x^2\right )} \, dx}{8 d^2 \left (c^2 d-e\right )}-\frac{\left (b c^3\right ) \int \frac{1}{1+c^2 x^2} \, dx}{2 d^2 \left (c^2 d-e\right )}+\frac{\sqrt{e} \int \frac{a+b \tan ^{-1}(c x)}{\sqrt{-d}-\sqrt{e} x} \, dx}{2 d^3}-\frac{\sqrt{e} \int \frac{a+b \tan ^{-1}(c x)}{\sqrt{-d}+\sqrt{e} x} \, dx}{2 d^3}+\frac{(b c e) \int \frac{1}{d+e x^2} \, dx}{2 d^2 \left (c^2 d-e\right )}\\ &=\frac{b c e x}{8 d^2 \left (c^2 d-e\right ) \left (d+e x^2\right )}-\frac{b c^2 \tan ^{-1}(c x)}{2 d^2 \left (c^2 d-e\right )}+\frac{a+b \tan ^{-1}(c x)}{4 d \left (d+e x^2\right )^2}+\frac{a+b \tan ^{-1}(c x)}{2 d^2 \left (d+e x^2\right )}+\frac{b c \sqrt{e} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{2 d^{5/2} \left (c^2 d-e\right )}+\frac{a \log (x)}{d^3}+\frac{\left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1-i c x}\right )}{d^3}-\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 d^3}-\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 d^3}+\frac{i b \text{Li}_2(-i c x)}{2 d^3}-\frac{i b \text{Li}_2(i c x)}{2 d^3}-2 \frac{(b c) \int \frac{\log \left (\frac{2}{1-i c x}\right )}{1+c^2 x^2} \, dx}{2 d^3}+\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 d^3}+\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 d^3}-\frac{\left (b c^5\right ) \int \frac{1}{1+c^2 x^2} \, dx}{4 d \left (c^2 d-e\right )^2}+\frac{\left (b c \left (3 c^2 d-e\right ) e\right ) \int \frac{1}{d+e x^2} \, dx}{8 d^2 \left (c^2 d-e\right )^2}\\ &=\frac{b c e x}{8 d^2 \left (c^2 d-e\right ) \left (d+e x^2\right )}-\frac{b c^4 \tan ^{-1}(c x)}{4 d \left (c^2 d-e\right )^2}-\frac{b c^2 \tan ^{-1}(c x)}{2 d^2 \left (c^2 d-e\right )}+\frac{a+b \tan ^{-1}(c x)}{4 d \left (d+e x^2\right )^2}+\frac{a+b \tan ^{-1}(c x)}{2 d^2 \left (d+e x^2\right )}+\frac{b c \sqrt{e} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{2 d^{5/2} \left (c^2 d-e\right )}+\frac{b c \left (3 c^2 d-e\right ) \sqrt{e} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{8 d^{5/2} \left (c^2 d-e\right )^2}+\frac{a \log (x)}{d^3}+\frac{\left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1-i c x}\right )}{d^3}-\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 d^3}-\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 d^3}+\frac{i b \text{Li}_2(-i c x)}{2 d^3}-\frac{i b \text{Li}_2(i c x)}{2 d^3}+\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 d^3}+\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 d^3}-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 d^3}\\ &=\frac{b c e x}{8 d^2 \left (c^2 d-e\right ) \left (d+e x^2\right )}-\frac{b c^4 \tan ^{-1}(c x)}{4 d \left (c^2 d-e\right )^2}-\frac{b c^2 \tan ^{-1}(c x)}{2 d^2 \left (c^2 d-e\right )}+\frac{a+b \tan ^{-1}(c x)}{4 d \left (d+e x^2\right )^2}+\frac{a+b \tan ^{-1}(c x)}{2 d^2 \left (d+e x^2\right )}+\frac{b c \sqrt{e} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{2 d^{5/2} \left (c^2 d-e\right )}+\frac{b c \left (3 c^2 d-e\right ) \sqrt{e} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{8 d^{5/2} \left (c^2 d-e\right )^2}+\frac{a \log (x)}{d^3}+\frac{\left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1-i c x}\right )}{d^3}-\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 d^3}-\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 d^3}+\frac{i b \text{Li}_2(-i c x)}{2 d^3}-\frac{i b \text{Li}_2(i c x)}{2 d^3}-\frac{i b \text{Li}_2\left (1-\frac{2}{1-i c x}\right )}{2 d^3}+\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 d^3}+\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 d^3}\\ \end{align*}
Mathematica [A] time = 12.8439, size = 645, normalized size = 1.12 \[ \frac{2 a \left (\frac{d \left (3 d+2 e x^2\right )}{\left (d+e x^2\right )^2}-2 \log \left (d+e x^2\right )+4 \log (x)\right )+b \left (2 i \text{PolyLog}\left (2,\frac{c \left (\sqrt{d}-i \sqrt{e} x\right )}{c \sqrt{d}-\sqrt{e}}\right )-2 i \text{PolyLog}\left (2,\frac{c \left (\sqrt{d}-i \sqrt{e} x\right )}{c \sqrt{d}+\sqrt{e}}\right )-2 i \text{PolyLog}\left (2,\frac{c \left (\sqrt{d}+i \sqrt{e} x\right )}{c \sqrt{d}-\sqrt{e}}\right )+2 i \text{PolyLog}\left (2,\frac{c \left (\sqrt{d}+i \sqrt{e} x\right )}{c \sqrt{d}+\sqrt{e}}\right )-4 i (-\text{PolyLog}(2,-i c x)+\text{PolyLog}(2,i c x)+\log (x) (\log (1-i c x)-\log (1+i c x)))+\frac{c d e x}{\left (c^2 d-e\right ) \left (d+e x^2\right )}+\frac{2 c^2 d \left (2 e-3 c^2 d\right ) \tan ^{-1}(c x)}{\left (e-c^2 d\right )^2}+\frac{c \sqrt{d} \sqrt{e} \left (7 c^2 d-5 e\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\left (e-c^2 d\right )^2}+\frac{2 d \tan ^{-1}(c x) \left (3 d+2 e x^2\right )}{\left (d+e x^2\right )^2}-2 i \log \left (x-\frac{i \sqrt{d}}{\sqrt{e}}\right ) \log \left (\frac{\sqrt{e} (-1-i c x)}{c \sqrt{d}-\sqrt{e}}\right )+2 i \log \left (x-\frac{i \sqrt{d}}{\sqrt{e}}\right ) \log \left (\frac{\sqrt{e} (1-i c x)}{c \sqrt{d}+\sqrt{e}}\right )+2 i \log \left (x+\frac{i \sqrt{d}}{\sqrt{e}}\right ) \log \left (\frac{\sqrt{e} (-1+i c x)}{c \sqrt{d}-\sqrt{e}}\right )-2 i \log \left (x+\frac{i \sqrt{d}}{\sqrt{e}}\right ) \log \left (\frac{\sqrt{e} (1+i c x)}{c \sqrt{d}+\sqrt{e}}\right )-4 \tan ^{-1}(c x) \log \left (x-\frac{i \sqrt{d}}{\sqrt{e}}\right )-4 \tan ^{-1}(c x) \log \left (x+\frac{i \sqrt{d}}{\sqrt{e}}\right )+8 \log (x) \tan ^{-1}(c x)\right )}{8 d^3} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.219, size = 1041, normalized size = 1.8 \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}{4} \, a{\left (\frac{2 \, e x^{2} + 3 \, d}{d^{2} e^{2} x^{4} + 2 \, d^{3} e x^{2} + d^{4}} - \frac{2 \, \log \left (e x^{2} + d\right )}{d^{3}} + \frac{4 \, \log \left (x\right )}{d^{3}}\right )} + 2 \, b \int \frac{\arctan \left (c x\right )}{2 \,{\left (e^{3} x^{7} + 3 \, d e^{2} x^{5} + 3 \, d^{2} e x^{3} + d^{3} x\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 \arctan \left (c x\right ) + a}{e^{3} x^{7} + 3 \, d e^{2} x^{5} + 3 \, d^{2} e x^{3} + d^{3} x}, 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{b \arctan \left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{3} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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