Optimal. Leaf size=457 \[ \frac{i b^2 c \text{PolyLog}\left (2,1-\frac{2 c \left (\sqrt{-d}-\sqrt{e} x\right )}{(1-i c x) \left (c \sqrt{-d}-i \sqrt{e}\right )}\right )}{4 \sqrt{-d} \sqrt{e} \left (c^2 d-e\right )}-\frac{i b^2 c \text{PolyLog}\left (2,1-\frac{2 c \left (\sqrt{-d}+\sqrt{e} x\right )}{(1-i c x) \left (c \sqrt{-d}+i \sqrt{e}\right )}\right )}{4 \sqrt{-d} \sqrt{e} \left (c^2 d-e\right )}+\frac{c^2 \left (a+b \tan ^{-1}(c x)\right )^2}{2 e \left (c^2 d-e\right )}-\frac{b c \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2 c \left (\sqrt{-d}-\sqrt{e} x\right )}{(1-i c x) \left (c \sqrt{-d}-i \sqrt{e}\right )}\right )}{2 \sqrt{-d} \sqrt{e} \left (c^2 d-e\right )}+\frac{b c \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2 c \left (\sqrt{-d}+\sqrt{e} x\right )}{(1-i c x) \left (c \sqrt{-d}+i \sqrt{e}\right )}\right )}{2 \sqrt{-d} \sqrt{e} \left (c^2 d-e\right )}-\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{4 d e \left (1-\frac{\sqrt{e} x}{\sqrt{-d}}\right )}-\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{4 d e \left (\frac{\sqrt{e} x}{\sqrt{-d}}+1\right )} \]
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Rubi [A] time = 1.08768, antiderivative size = 457, normalized size of antiderivative = 1., number of steps used = 27, number of rules used = 10, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.476, Rules used = {4978, 4864, 4856, 2402, 2315, 2447, 4984, 4884, 4920, 4854} \[ \frac{i b^2 c \text{PolyLog}\left (2,1-\frac{2 c \left (\sqrt{-d}-\sqrt{e} x\right )}{(1-i c x) \left (c \sqrt{-d}-i \sqrt{e}\right )}\right )}{4 \sqrt{-d} \sqrt{e} \left (c^2 d-e\right )}-\frac{i b^2 c \text{PolyLog}\left (2,1-\frac{2 c \left (\sqrt{-d}+\sqrt{e} x\right )}{(1-i c x) \left (c \sqrt{-d}+i \sqrt{e}\right )}\right )}{4 \sqrt{-d} \sqrt{e} \left (c^2 d-e\right )}+\frac{c^2 \left (a+b \tan ^{-1}(c x)\right )^2}{2 e \left (c^2 d-e\right )}-\frac{b c \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2 c \left (\sqrt{-d}-\sqrt{e} x\right )}{(1-i c x) \left (c \sqrt{-d}-i \sqrt{e}\right )}\right )}{2 \sqrt{-d} \sqrt{e} \left (c^2 d-e\right )}+\frac{b c \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2 c \left (\sqrt{-d}+\sqrt{e} x\right )}{(1-i c x) \left (c \sqrt{-d}+i \sqrt{e}\right )}\right )}{2 \sqrt{-d} \sqrt{e} \left (c^2 d-e\right )}-\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{4 d e \left (1-\frac{\sqrt{e} x}{\sqrt{-d}}\right )}-\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{4 d e \left (\frac{\sqrt{e} x}{\sqrt{-d}}+1\right )} \]
Antiderivative was successfully verified.
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Rule 4978
Rule 4864
Rule 4856
Rule 2402
Rule 2315
Rule 2447
Rule 4984
Rule 4884
Rule 4920
Rule 4854
Rubi steps
\begin{align*} \int \frac{x \left (a+b \tan ^{-1}(c x)\right )^2}{\left (d+e x^2\right )^2} \, dx &=\frac{\int \frac{\left (a+b \tan ^{-1}(c x)\right )^2}{\left (1-\frac{\sqrt{e} x}{\sqrt{-d}}\right )^2} \, dx}{4 (-d)^{3/2} \sqrt{e}}-\frac{\int \frac{\left (a+b \tan ^{-1}(c x)\right )^2}{\left (1+\frac{\sqrt{e} x}{\sqrt{-d}}\right )^2} \, dx}{4 (-d)^{3/2} \sqrt{e}}\\ &=-\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{4 d e \left (1-\frac{\sqrt{e} x}{\sqrt{-d}}\right )}-\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{4 d e \left (1+\frac{\sqrt{e} x}{\sqrt{-d}}\right )}+\frac{(b c) \int \left (\frac{\sqrt{-d} e \left (a+b \tan ^{-1}(c x)\right )}{\left (c^2 d-e\right ) \left (-\sqrt{-d}+\sqrt{e} x\right )}+\frac{c^2 d \left (\sqrt{-d}+\sqrt{e} x\right ) \left (a+b \tan ^{-1}(c x)\right )}{\sqrt{-d} \left (c^2 d-e\right ) \left (1+c^2 x^2\right )}\right ) \, dx}{2 d e}+\frac{(b c) \int \left (\frac{\sqrt{-d} e \left (a+b \tan ^{-1}(c x)\right )}{\left (-c^2 d+e\right ) \left (\sqrt{-d}+\sqrt{e} x\right )}+\frac{c^2 \left (d+\sqrt{-d} \sqrt{e} x\right ) \left (a+b \tan ^{-1}(c x)\right )}{\left (c^2 d-e\right ) \left (1+c^2 x^2\right )}\right ) \, dx}{2 d e}\\ &=-\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{4 d e \left (1-\frac{\sqrt{e} x}{\sqrt{-d}}\right )}-\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{4 d e \left (1+\frac{\sqrt{e} x}{\sqrt{-d}}\right )}-\frac{(b c) \int \frac{a+b \tan ^{-1}(c x)}{-\sqrt{-d}+\sqrt{e} x} \, dx}{2 \sqrt{-d} \left (c^2 d-e\right )}+\frac{(b c) \int \frac{a+b \tan ^{-1}(c x)}{\sqrt{-d}+\sqrt{e} x} \, dx}{2 \sqrt{-d} \left (c^2 d-e\right )}+\frac{\left (b c^3\right ) \int \frac{\left (\sqrt{-d}+\sqrt{e} x\right ) \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{2 \sqrt{-d} \left (c^2 d-e\right ) e}+\frac{\left (b c^3\right ) \int \frac{\left (d+\sqrt{-d} \sqrt{e} x\right ) \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{2 d \left (c^2 d-e\right ) e}\\ &=-\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{4 d e \left (1-\frac{\sqrt{e} x}{\sqrt{-d}}\right )}-\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{4 d e \left (1+\frac{\sqrt{e} x}{\sqrt{-d}}\right )}-\frac{b c \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 \sqrt{-d} \left (c^2 d-e\right ) \sqrt{e}}+\frac{b c \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 \sqrt{-d} \left (c^2 d-e\right ) \sqrt{e}}+\frac{\left (b c^3\right ) \int \left (\frac{\sqrt{-d} \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2}+\frac{\sqrt{e} x \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2}\right ) \, dx}{2 \sqrt{-d} \left (c^2 d-e\right ) e}+\frac{\left (b c^3\right ) \int \left (\frac{d \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2}+\frac{\sqrt{-d} \sqrt{e} x \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2}\right ) \, dx}{2 d \left (c^2 d-e\right ) e}+\frac{\left (b^2 c^2\right ) \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 \sqrt{-d} \left (c^2 d-e\right ) \sqrt{e}}-\frac{\left (b^2 c^2\right ) \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 \sqrt{-d} \left (c^2 d-e\right ) \sqrt{e}}\\ &=-\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{4 d e \left (1-\frac{\sqrt{e} x}{\sqrt{-d}}\right )}-\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{4 d e \left (1+\frac{\sqrt{e} x}{\sqrt{-d}}\right )}-\frac{b c \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 \sqrt{-d} \left (c^2 d-e\right ) \sqrt{e}}+\frac{b c \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 \sqrt{-d} \left (c^2 d-e\right ) \sqrt{e}}+\frac{i b^2 c \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 \sqrt{-d} \left (c^2 d-e\right ) \sqrt{e}}-\frac{i b^2 c \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 \sqrt{-d} \left (c^2 d-e\right ) \sqrt{e}}+2 \frac{\left (b c^3\right ) \int \frac{a+b \tan ^{-1}(c x)}{1+c^2 x^2} \, dx}{2 \left (c^2 d-e\right ) e}\\ &=\frac{c^2 \left (a+b \tan ^{-1}(c x)\right )^2}{2 \left (c^2 d-e\right ) e}-\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{4 d e \left (1-\frac{\sqrt{e} x}{\sqrt{-d}}\right )}-\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{4 d e \left (1+\frac{\sqrt{e} x}{\sqrt{-d}}\right )}-\frac{b c \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 \sqrt{-d} \left (c^2 d-e\right ) \sqrt{e}}+\frac{b c \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 \sqrt{-d} \left (c^2 d-e\right ) \sqrt{e}}+\frac{i b^2 c \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 \sqrt{-d} \left (c^2 d-e\right ) \sqrt{e}}-\frac{i b^2 c \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 \sqrt{-d} \left (c^2 d-e\right ) \sqrt{e}}\\ \end{align*}
Mathematica [A] time = 8.77836, size = 885, normalized size = 1.94 \[ -\frac{a^2}{2 e \left (e x^2+d\right )}+2 b c^2 \left (\frac{c \tan ^{-1}(c x)-\frac{\sqrt{e} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{d}}}{2 e \left (c^3 d-c e\right )}-\frac{\tan ^{-1}(c x)}{2 e \left (e x^2 c^2+d c^2\right )}\right ) a+\frac{b^2 c^2 \left (\frac{4 \tan ^{-1}(c x)^2}{d c^2+e+\left (c^2 d-e\right ) \cos \left (2 \tan ^{-1}(c x)\right )}+\frac{4 \tan ^{-1}(c x) \tanh ^{-1}\left (\frac{\sqrt{-c^2 d e}}{c e x}\right )-2 \cos ^{-1}\left (-\frac{d c^2+e}{c^2 d-e}\right ) \tanh ^{-1}\left (\frac{c e x}{\sqrt{-c^2 d e}}\right )+\left (\cos ^{-1}\left (-\frac{d c^2+e}{c^2 d-e}\right )+2 i \tanh ^{-1}\left (\frac{c e x}{\sqrt{-c^2 d e}}\right )\right ) \log \left (\frac{2 c^2 d \left (\sqrt{-c^2 d e}-i e\right ) (c x-i)}{\left (c^2 d-e\right ) \left (d c^2+\sqrt{-c^2 d e} x c\right )}\right )+\left (\cos ^{-1}\left (-\frac{d c^2+e}{c^2 d-e}\right )-2 i \tanh ^{-1}\left (\frac{c e x}{\sqrt{-c^2 d e}}\right )\right ) \log \left (\frac{2 c^2 d \left (i e+\sqrt{-c^2 d e}\right ) (c x+i)}{\left (c^2 d-e\right ) \left (d c^2+\sqrt{-c^2 d e} x c\right )}\right )-\left (\cos ^{-1}\left (-\frac{d c^2+e}{c^2 d-e}\right )-2 i \left (\tanh ^{-1}\left (\frac{c d}{\sqrt{-c^2 d e} x}\right )+\tanh ^{-1}\left (\frac{c e x}{\sqrt{-c^2 d e}}\right )\right )\right ) \log \left (\frac{\sqrt{2} \sqrt{-c^2 d e} e^{-i \tan ^{-1}(c x)}}{\sqrt{c^2 d-e} \sqrt{d c^2+e+\left (c^2 d-e\right ) \cos \left (2 \tan ^{-1}(c x)\right )}}\right )-\left (\cos ^{-1}\left (-\frac{d c^2+e}{c^2 d-e}\right )+2 i \left (\tanh ^{-1}\left (\frac{c d}{\sqrt{-c^2 d e} x}\right )+\tanh ^{-1}\left (\frac{c e x}{\sqrt{-c^2 d e}}\right )\right )\right ) \log \left (\frac{\sqrt{2} \sqrt{-c^2 d e} e^{i \tan ^{-1}(c x)}}{\sqrt{c^2 d-e} \sqrt{d c^2+e+\left (c^2 d-e\right ) \cos \left (2 \tan ^{-1}(c x)\right )}}\right )-i \left (\text{PolyLog}\left (2,\frac{\left (d c^2+e-2 i \sqrt{-c^2 d e}\right ) \left (c^2 d-c \sqrt{-c^2 d e} x\right )}{\left (c^2 d-e\right ) \left (d c^2+\sqrt{-c^2 d e} x c\right )}\right )-\text{PolyLog}\left (2,\frac{\left (d c^2+e+2 i \sqrt{-c^2 d e}\right ) \left (c^2 d-c \sqrt{-c^2 d e} x\right )}{\left (c^2 d-e\right ) \left (d c^2+\sqrt{-c^2 d e} x c\right )}\right )\right )}{\sqrt{-c^2 d e}}\right )}{4 \left (c^2 d-e\right )} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.363, size = 1185, normalized size = 2.6 \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(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \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^{2} x \arctan \left (c x\right )^{2} + 2 \, a b x \arctan \left (c x\right ) + a^{2} x}{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 )}^{2} x}{{\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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