Optimal. Leaf size=819 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a+b \tan ^{-1}(c x)\right )}{2 d^{3/2} \sqrt{e}}+\frac{x \left (a+b \tan ^{-1}(c x)\right )}{2 d \left (e x^2+d\right )}+\frac{i b c \log \left (\frac{\sqrt{e} \left (1-\sqrt{-c^2} x\right )}{i \sqrt{-c^2} \sqrt{d}+\sqrt{e}}\right ) \log \left (1-\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{8 \sqrt{-c^2} d^{3/2} \sqrt{e}}-\frac{i b c \log \left (-\frac{\sqrt{e} \left (\sqrt{-c^2} x+1\right )}{i \sqrt{-c^2} \sqrt{d}-\sqrt{e}}\right ) \log \left (1-\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{8 \sqrt{-c^2} d^{3/2} \sqrt{e}}-\frac{i b c \log \left (-\frac{\sqrt{e} \left (1-\sqrt{-c^2} x\right )}{i \sqrt{-c^2} \sqrt{d}-\sqrt{e}}\right ) \log \left (\frac{i \sqrt{e} x}{\sqrt{d}}+1\right )}{8 \sqrt{-c^2} d^{3/2} \sqrt{e}}+\frac{i b c \log \left (\frac{\sqrt{e} \left (\sqrt{-c^2} x+1\right )}{i \sqrt{-c^2} \sqrt{d}+\sqrt{e}}\right ) \log \left (\frac{i \sqrt{e} x}{\sqrt{d}}+1\right )}{8 \sqrt{-c^2} d^{3/2} \sqrt{e}}-\frac{b c \log \left (c^2 x^2+1\right )}{4 d \left (c^2 d-e\right )}+\frac{b c \log \left (e x^2+d\right )}{4 d \left (c^2 d-e\right )}+\frac{i b c \text{PolyLog}\left (2,\frac{\sqrt{-c^2} \left (\sqrt{d}-i \sqrt{e} x\right )}{\sqrt{-c^2} \sqrt{d}-i \sqrt{e}}\right )}{8 \sqrt{-c^2} d^{3/2} \sqrt{e}}-\frac{i b c \text{PolyLog}\left (2,\frac{\sqrt{-c^2} \left (\sqrt{d}-i \sqrt{e} x\right )}{\sqrt{-c^2} \sqrt{d}+i \sqrt{e}}\right )}{8 \sqrt{-c^2} d^{3/2} \sqrt{e}}+\frac{i b c \text{PolyLog}\left (2,\frac{\sqrt{-c^2} \left (i \sqrt{e} x+\sqrt{d}\right )}{\sqrt{-c^2} \sqrt{d}-i \sqrt{e}}\right )}{8 \sqrt{-c^2} d^{3/2} \sqrt{e}}-\frac{i b c \text{PolyLog}\left (2,\frac{\sqrt{-c^2} \left (i \sqrt{e} x+\sqrt{d}\right )}{\sqrt{-c^2} \sqrt{d}+i \sqrt{e}}\right )}{8 \sqrt{-c^2} d^{3/2} \sqrt{e}} \]
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Rubi [A] time = 0.892897, antiderivative size = 819, normalized size of antiderivative = 1., number of steps used = 24, number of rules used = 12, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667, Rules used = {199, 205, 4912, 6725, 444, 36, 31, 4908, 2409, 2394, 2393, 2391} \[ \frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a+b \tan ^{-1}(c x)\right )}{2 d^{3/2} \sqrt{e}}+\frac{x \left (a+b \tan ^{-1}(c x)\right )}{2 d \left (e x^2+d\right )}+\frac{i b c \log \left (\frac{\sqrt{e} \left (1-\sqrt{-c^2} x\right )}{i \sqrt{-c^2} \sqrt{d}+\sqrt{e}}\right ) \log \left (1-\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{8 \sqrt{-c^2} d^{3/2} \sqrt{e}}-\frac{i b c \log \left (-\frac{\sqrt{e} \left (\sqrt{-c^2} x+1\right )}{i \sqrt{-c^2} \sqrt{d}-\sqrt{e}}\right ) \log \left (1-\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{8 \sqrt{-c^2} d^{3/2} \sqrt{e}}-\frac{i b c \log \left (-\frac{\sqrt{e} \left (1-\sqrt{-c^2} x\right )}{i \sqrt{-c^2} \sqrt{d}-\sqrt{e}}\right ) \log \left (\frac{i \sqrt{e} x}{\sqrt{d}}+1\right )}{8 \sqrt{-c^2} d^{3/2} \sqrt{e}}+\frac{i b c \log \left (\frac{\sqrt{e} \left (\sqrt{-c^2} x+1\right )}{i \sqrt{-c^2} \sqrt{d}+\sqrt{e}}\right ) \log \left (\frac{i \sqrt{e} x}{\sqrt{d}}+1\right )}{8 \sqrt{-c^2} d^{3/2} \sqrt{e}}-\frac{b c \log \left (c^2 x^2+1\right )}{4 d \left (c^2 d-e\right )}+\frac{b c \log \left (e x^2+d\right )}{4 d \left (c^2 d-e\right )}+\frac{i b c \text{PolyLog}\left (2,\frac{\sqrt{-c^2} \left (\sqrt{d}-i \sqrt{e} x\right )}{\sqrt{-c^2} \sqrt{d}-i \sqrt{e}}\right )}{8 \sqrt{-c^2} d^{3/2} \sqrt{e}}-\frac{i b c \text{PolyLog}\left (2,\frac{\sqrt{-c^2} \left (\sqrt{d}-i \sqrt{e} x\right )}{\sqrt{-c^2} \sqrt{d}+i \sqrt{e}}\right )}{8 \sqrt{-c^2} d^{3/2} \sqrt{e}}+\frac{i b c \text{PolyLog}\left (2,\frac{\sqrt{-c^2} \left (i \sqrt{e} x+\sqrt{d}\right )}{\sqrt{-c^2} \sqrt{d}-i \sqrt{e}}\right )}{8 \sqrt{-c^2} d^{3/2} \sqrt{e}}-\frac{i b c \text{PolyLog}\left (2,\frac{\sqrt{-c^2} \left (i \sqrt{e} x+\sqrt{d}\right )}{\sqrt{-c^2} \sqrt{d}+i \sqrt{e}}\right )}{8 \sqrt{-c^2} d^{3/2} \sqrt{e}} \]
Antiderivative was successfully verified.
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Rule 199
Rule 205
Rule 4912
Rule 6725
Rule 444
Rule 36
Rule 31
Rule 4908
Rule 2409
Rule 2394
Rule 2393
Rule 2391
Rubi steps
\begin{align*} \int \frac{a+b \tan ^{-1}(c x)}{\left (d+e x^2\right )^2} \, dx &=\frac{x \left (a+b \tan ^{-1}(c x)\right )}{2 d \left (d+e x^2\right )}+\frac{\left (a+b \tan ^{-1}(c x)\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{2 d^{3/2} \sqrt{e}}-(b c) \int \frac{\frac{x}{2 d \left (d+e x^2\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{2 d^{3/2} \sqrt{e}}}{1+c^2 x^2} \, dx\\ &=\frac{x \left (a+b \tan ^{-1}(c x)\right )}{2 d \left (d+e x^2\right )}+\frac{\left (a+b \tan ^{-1}(c x)\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{2 d^{3/2} \sqrt{e}}-(b c) \int \left (\frac{x}{2 d \left (1+c^2 x^2\right ) \left (d+e x^2\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{2 d^{3/2} \sqrt{e} \left (1+c^2 x^2\right )}\right ) \, dx\\ &=\frac{x \left (a+b \tan ^{-1}(c x)\right )}{2 d \left (d+e x^2\right )}+\frac{\left (a+b \tan ^{-1}(c x)\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{2 d^{3/2} \sqrt{e}}-\frac{(b c) \int \frac{x}{\left (1+c^2 x^2\right ) \left (d+e x^2\right )} \, dx}{2 d}-\frac{(b c) \int \frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{1+c^2 x^2} \, dx}{2 d^{3/2} \sqrt{e}}\\ &=\frac{x \left (a+b \tan ^{-1}(c x)\right )}{2 d \left (d+e x^2\right )}+\frac{\left (a+b \tan ^{-1}(c x)\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{2 d^{3/2} \sqrt{e}}-\frac{(b c) \operatorname{Subst}\left (\int \frac{1}{\left (1+c^2 x\right ) (d+e x)} \, dx,x,x^2\right )}{4 d}-\frac{(i b c) \int \frac{\log \left (1-\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{1+c^2 x^2} \, dx}{4 d^{3/2} \sqrt{e}}+\frac{(i b c) \int \frac{\log \left (1+\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{1+c^2 x^2} \, dx}{4 d^{3/2} \sqrt{e}}\\ &=\frac{x \left (a+b \tan ^{-1}(c x)\right )}{2 d \left (d+e x^2\right )}+\frac{\left (a+b \tan ^{-1}(c x)\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{2 d^{3/2} \sqrt{e}}-\frac{\left (b c^3\right ) \operatorname{Subst}\left (\int \frac{1}{1+c^2 x} \, dx,x,x^2\right )}{4 d \left (c^2 d-e\right )}-\frac{(i b c) \int \left (\frac{\log \left (1-\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{2 \left (1-\sqrt{-c^2} x\right )}+\frac{\log \left (1-\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{2 \left (1+\sqrt{-c^2} x\right )}\right ) \, dx}{4 d^{3/2} \sqrt{e}}+\frac{(i b c) \int \left (\frac{\log \left (1+\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{2 \left (1-\sqrt{-c^2} x\right )}+\frac{\log \left (1+\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{2 \left (1+\sqrt{-c^2} x\right )}\right ) \, dx}{4 d^{3/2} \sqrt{e}}+\frac{(b c e) \operatorname{Subst}\left (\int \frac{1}{d+e x} \, dx,x,x^2\right )}{4 d \left (c^2 d-e\right )}\\ &=\frac{x \left (a+b \tan ^{-1}(c x)\right )}{2 d \left (d+e x^2\right )}+\frac{\left (a+b \tan ^{-1}(c x)\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{2 d^{3/2} \sqrt{e}}-\frac{b c \log \left (1+c^2 x^2\right )}{4 d \left (c^2 d-e\right )}+\frac{b c \log \left (d+e x^2\right )}{4 d \left (c^2 d-e\right )}-\frac{(i b c) \int \frac{\log \left (1-\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{1-\sqrt{-c^2} x} \, dx}{8 d^{3/2} \sqrt{e}}-\frac{(i b c) \int \frac{\log \left (1-\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{1+\sqrt{-c^2} x} \, dx}{8 d^{3/2} \sqrt{e}}+\frac{(i b c) \int \frac{\log \left (1+\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{1-\sqrt{-c^2} x} \, dx}{8 d^{3/2} \sqrt{e}}+\frac{(i b c) \int \frac{\log \left (1+\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{1+\sqrt{-c^2} x} \, dx}{8 d^{3/2} \sqrt{e}}\\ &=\frac{x \left (a+b \tan ^{-1}(c x)\right )}{2 d \left (d+e x^2\right )}+\frac{\left (a+b \tan ^{-1}(c x)\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{2 d^{3/2} \sqrt{e}}+\frac{i b c \log \left (\frac{\sqrt{e} \left (1-\sqrt{-c^2} x\right )}{i \sqrt{-c^2} \sqrt{d}+\sqrt{e}}\right ) \log \left (1-\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{8 \sqrt{-c^2} d^{3/2} \sqrt{e}}-\frac{i b c \log \left (-\frac{\sqrt{e} \left (1+\sqrt{-c^2} x\right )}{i \sqrt{-c^2} \sqrt{d}-\sqrt{e}}\right ) \log \left (1-\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{8 \sqrt{-c^2} d^{3/2} \sqrt{e}}-\frac{i b c \log \left (-\frac{\sqrt{e} \left (1-\sqrt{-c^2} x\right )}{i \sqrt{-c^2} \sqrt{d}-\sqrt{e}}\right ) \log \left (1+\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{8 \sqrt{-c^2} d^{3/2} \sqrt{e}}+\frac{i b c \log \left (\frac{\sqrt{e} \left (1+\sqrt{-c^2} x\right )}{i \sqrt{-c^2} \sqrt{d}+\sqrt{e}}\right ) \log \left (1+\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{8 \sqrt{-c^2} d^{3/2} \sqrt{e}}-\frac{b c \log \left (1+c^2 x^2\right )}{4 d \left (c^2 d-e\right )}+\frac{b c \log \left (d+e x^2\right )}{4 d \left (c^2 d-e\right )}-\frac{(b c) \int \frac{\log \left (-\frac{i \sqrt{e} \left (1-\sqrt{-c^2} x\right )}{\sqrt{d} \left (\sqrt{-c^2}-\frac{i \sqrt{e}}{\sqrt{d}}\right )}\right )}{1-\frac{i \sqrt{e} x}{\sqrt{d}}} \, dx}{8 \sqrt{-c^2} d^2}-\frac{(b c) \int \frac{\log \left (\frac{i \sqrt{e} \left (1-\sqrt{-c^2} x\right )}{\sqrt{d} \left (\sqrt{-c^2}+\frac{i \sqrt{e}}{\sqrt{d}}\right )}\right )}{1+\frac{i \sqrt{e} x}{\sqrt{d}}} \, dx}{8 \sqrt{-c^2} d^2}+\frac{(b c) \int \frac{\log \left (-\frac{i \sqrt{e} \left (1+\sqrt{-c^2} x\right )}{\sqrt{d} \left (-\sqrt{-c^2}-\frac{i \sqrt{e}}{\sqrt{d}}\right )}\right )}{1-\frac{i \sqrt{e} x}{\sqrt{d}}} \, dx}{8 \sqrt{-c^2} d^2}+\frac{(b c) \int \frac{\log \left (\frac{i \sqrt{e} \left (1+\sqrt{-c^2} x\right )}{\sqrt{d} \left (-\sqrt{-c^2}+\frac{i \sqrt{e}}{\sqrt{d}}\right )}\right )}{1+\frac{i \sqrt{e} x}{\sqrt{d}}} \, dx}{8 \sqrt{-c^2} d^2}\\ &=\frac{x \left (a+b \tan ^{-1}(c x)\right )}{2 d \left (d+e x^2\right )}+\frac{\left (a+b \tan ^{-1}(c x)\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{2 d^{3/2} \sqrt{e}}+\frac{i b c \log \left (\frac{\sqrt{e} \left (1-\sqrt{-c^2} x\right )}{i \sqrt{-c^2} \sqrt{d}+\sqrt{e}}\right ) \log \left (1-\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{8 \sqrt{-c^2} d^{3/2} \sqrt{e}}-\frac{i b c \log \left (-\frac{\sqrt{e} \left (1+\sqrt{-c^2} x\right )}{i \sqrt{-c^2} \sqrt{d}-\sqrt{e}}\right ) \log \left (1-\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{8 \sqrt{-c^2} d^{3/2} \sqrt{e}}-\frac{i b c \log \left (-\frac{\sqrt{e} \left (1-\sqrt{-c^2} x\right )}{i \sqrt{-c^2} \sqrt{d}-\sqrt{e}}\right ) \log \left (1+\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{8 \sqrt{-c^2} d^{3/2} \sqrt{e}}+\frac{i b c \log \left (\frac{\sqrt{e} \left (1+\sqrt{-c^2} x\right )}{i \sqrt{-c^2} \sqrt{d}+\sqrt{e}}\right ) \log \left (1+\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{8 \sqrt{-c^2} d^{3/2} \sqrt{e}}-\frac{b c \log \left (1+c^2 x^2\right )}{4 d \left (c^2 d-e\right )}+\frac{b c \log \left (d+e x^2\right )}{4 d \left (c^2 d-e\right )}+\frac{(i b c) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{\sqrt{-c^2} x}{-\sqrt{-c^2}-\frac{i \sqrt{e}}{\sqrt{d}}}\right )}{x} \, dx,x,1-\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{8 \sqrt{-c^2} d^{3/2} \sqrt{e}}-\frac{(i b c) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{\sqrt{-c^2} x}{\sqrt{-c^2}-\frac{i \sqrt{e}}{\sqrt{d}}}\right )}{x} \, dx,x,1-\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{8 \sqrt{-c^2} d^{3/2} \sqrt{e}}-\frac{(i b c) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{\sqrt{-c^2} x}{-\sqrt{-c^2}+\frac{i \sqrt{e}}{\sqrt{d}}}\right )}{x} \, dx,x,1+\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{8 \sqrt{-c^2} d^{3/2} \sqrt{e}}+\frac{(i b c) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{\sqrt{-c^2} x}{\sqrt{-c^2}+\frac{i \sqrt{e}}{\sqrt{d}}}\right )}{x} \, dx,x,1+\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{8 \sqrt{-c^2} d^{3/2} \sqrt{e}}\\ &=\frac{x \left (a+b \tan ^{-1}(c x)\right )}{2 d \left (d+e x^2\right )}+\frac{\left (a+b \tan ^{-1}(c x)\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{2 d^{3/2} \sqrt{e}}+\frac{i b c \log \left (\frac{\sqrt{e} \left (1-\sqrt{-c^2} x\right )}{i \sqrt{-c^2} \sqrt{d}+\sqrt{e}}\right ) \log \left (1-\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{8 \sqrt{-c^2} d^{3/2} \sqrt{e}}-\frac{i b c \log \left (-\frac{\sqrt{e} \left (1+\sqrt{-c^2} x\right )}{i \sqrt{-c^2} \sqrt{d}-\sqrt{e}}\right ) \log \left (1-\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{8 \sqrt{-c^2} d^{3/2} \sqrt{e}}-\frac{i b c \log \left (-\frac{\sqrt{e} \left (1-\sqrt{-c^2} x\right )}{i \sqrt{-c^2} \sqrt{d}-\sqrt{e}}\right ) \log \left (1+\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{8 \sqrt{-c^2} d^{3/2} \sqrt{e}}+\frac{i b c \log \left (\frac{\sqrt{e} \left (1+\sqrt{-c^2} x\right )}{i \sqrt{-c^2} \sqrt{d}+\sqrt{e}}\right ) \log \left (1+\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{8 \sqrt{-c^2} d^{3/2} \sqrt{e}}-\frac{b c \log \left (1+c^2 x^2\right )}{4 d \left (c^2 d-e\right )}+\frac{b c \log \left (d+e x^2\right )}{4 d \left (c^2 d-e\right )}+\frac{i b c \text{Li}_2\left (\frac{\sqrt{-c^2} \left (\sqrt{d}-i \sqrt{e} x\right )}{\sqrt{-c^2} \sqrt{d}-i \sqrt{e}}\right )}{8 \sqrt{-c^2} d^{3/2} \sqrt{e}}-\frac{i b c \text{Li}_2\left (\frac{\sqrt{-c^2} \left (\sqrt{d}-i \sqrt{e} x\right )}{\sqrt{-c^2} \sqrt{d}+i \sqrt{e}}\right )}{8 \sqrt{-c^2} d^{3/2} \sqrt{e}}+\frac{i b c \text{Li}_2\left (\frac{\sqrt{-c^2} \left (\sqrt{d}+i \sqrt{e} x\right )}{\sqrt{-c^2} \sqrt{d}-i \sqrt{e}}\right )}{8 \sqrt{-c^2} d^{3/2} \sqrt{e}}-\frac{i b c \text{Li}_2\left (\frac{\sqrt{-c^2} \left (\sqrt{d}+i \sqrt{e} x\right )}{\sqrt{-c^2} \sqrt{d}+i \sqrt{e}}\right )}{8 \sqrt{-c^2} d^{3/2} \sqrt{e}}\\ \end{align*}
Mathematica [A] time = 10.0501, size = 861, normalized size = 1.05 \[ \frac{a x}{2 d \left (e x^2+d\right )}+\frac{a \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{2 d^{3/2} \sqrt{e}}+\frac{b c \left (\frac{2 \log \left (\frac{\left (c^2 d-e\right ) \cos \left (2 \tan ^{-1}(c x)\right )}{d c^2+e}+1\right )}{c^2 d-e}+\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}}+\frac{4 \tan ^{-1}(c x) \sin \left (2 \tan ^{-1}(c x)\right )}{d c^2+e+\left (c^2 d-e\right ) \cos \left (2 \tan ^{-1}(c x)\right )}\right )}{8 d} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.767, size = 2315, normalized size = 2.8 \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(-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 \arctan \left (c x\right ) + a}{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{b \arctan \left (c x\right ) + a}{{\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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