Optimal. Leaf size=1039 \[ \text{result too large to display} \]
[Out]
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Rubi [A] time = 1.32887, antiderivative size = 1039, normalized size of antiderivative = 1., number of steps used = 32, number of rules used = 11, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.55, Rules used = {4914, 4864, 4856, 2402, 2315, 2447, 4984, 4884, 4920, 4854, 4858} \[ \frac{i c \text{PolyLog}\left (2,1-\frac{2}{1-i c x}\right ) b^2}{2 d \left (c^2 d-e\right )}+\frac{i c \text{PolyLog}\left (2,1-\frac{2}{i c x+1}\right ) b^2}{2 d \left (c^2 d-e\right )}-\frac{i c \text{PolyLog}\left (2,1-\frac{2 c \left (\sqrt{-d}-\sqrt{e} x\right )}{\left (c \sqrt{-d}-i \sqrt{e}\right ) (1-i c x)}\right ) b^2}{4 d \left (c^2 d-e\right )}-\frac{i c \text{PolyLog}\left (2,1-\frac{2 c \left (\sqrt{e} x+\sqrt{-d}\right )}{\left (\sqrt{-d} c+i \sqrt{e}\right ) (1-i c x)}\right ) b^2}{4 d \left (c^2 d-e\right )}-\frac{\text{PolyLog}\left (3,1-\frac{2 c \left (\sqrt{-d}-\sqrt{e} x\right )}{\left (c \sqrt{-d}-i \sqrt{e}\right ) (1-i c x)}\right ) b^2}{8 (-d)^{3/2} \sqrt{e}}+\frac{\text{PolyLog}\left (3,1-\frac{2 c \left (\sqrt{e} x+\sqrt{-d}\right )}{\left (\sqrt{-d} c+i \sqrt{e}\right ) (1-i c x)}\right ) b^2}{8 (-d)^{3/2} \sqrt{e}}-\frac{c \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1-i c x}\right ) b}{d \left (c^2 d-e\right )}+\frac{c \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{i c x+1}\right ) b}{d \left (c^2 d-e\right )}+\frac{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 ) b}{2 d \left (c^2 d-e\right )}+\frac{c \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2 c \left (\sqrt{e} x+\sqrt{-d}\right )}{\left (\sqrt{-d} c+i \sqrt{e}\right ) (1-i c x)}\right ) b}{2 d \left (c^2 d-e\right )}+\frac{i \left (a+b \tan ^{-1}(c x)\right ) \text{PolyLog}\left (2,1-\frac{2 c \left (\sqrt{-d}-\sqrt{e} x\right )}{\left (c \sqrt{-d}-i \sqrt{e}\right ) (1-i c x)}\right ) b}{4 (-d)^{3/2} \sqrt{e}}-\frac{i \left (a+b \tan ^{-1}(c x)\right ) \text{PolyLog}\left (2,1-\frac{2 c \left (\sqrt{e} x+\sqrt{-d}\right )}{\left (\sqrt{-d} c+i \sqrt{e}\right ) (1-i c x)}\right ) b}{4 (-d)^{3/2} \sqrt{e}}+\frac{i c \left (a+b \tan ^{-1}(c x)\right )^2}{2 d \left (c^2 d-e\right )}-\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{4 d \sqrt{e} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{4 d \sqrt{e} \left (\sqrt{e} x+\sqrt{-d}\right )}-\frac{\left (a+b \tan ^{-1}(c x)\right )^2 \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 )}{4 (-d)^{3/2} \sqrt{e}}+\frac{\left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac{2 c \left (\sqrt{e} x+\sqrt{-d}\right )}{\left (\sqrt{-d} c+i \sqrt{e}\right ) (1-i c x)}\right )}{4 (-d)^{3/2} \sqrt{e}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4914
Rule 4864
Rule 4856
Rule 2402
Rule 2315
Rule 2447
Rule 4984
Rule 4884
Rule 4920
Rule 4854
Rule 4858
Rubi steps
\begin{align*} \int \frac{\left (a+b \tan ^{-1}(c x)\right )^2}{\left (d+e x^2\right )^2} \, dx &=\int \left (-\frac{e \left (a+b \tan ^{-1}(c x)\right )^2}{4 d \left (\sqrt{-d} \sqrt{e}-e x\right )^2}-\frac{e \left (a+b \tan ^{-1}(c x)\right )^2}{4 d \left (\sqrt{-d} \sqrt{e}+e x\right )^2}-\frac{e \left (a+b \tan ^{-1}(c x)\right )^2}{2 d \left (-d e-e^2 x^2\right )}\right ) \, dx\\ &=-\frac{e \int \frac{\left (a+b \tan ^{-1}(c x)\right )^2}{\left (\sqrt{-d} \sqrt{e}-e x\right )^2} \, dx}{4 d}-\frac{e \int \frac{\left (a+b \tan ^{-1}(c x)\right )^2}{\left (\sqrt{-d} \sqrt{e}+e x\right )^2} \, dx}{4 d}-\frac{e \int \frac{\left (a+b \tan ^{-1}(c x)\right )^2}{-d e-e^2 x^2} \, dx}{2 d}\\ &=-\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{4 d \sqrt{e} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{4 d \sqrt{e} \left (\sqrt{-d}+\sqrt{e} x\right )}-\frac{(b c) \int \left (\frac{\sqrt{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 (-\sqrt{-d}+\sqrt{e} x\right ) \left (a+b \tan ^{-1}(c x)\right )}{\sqrt{e} \left (-c^2 d+e\right ) \left (1+c^2 x^2\right )}\right ) \, dx}{2 d}+\frac{(b c) \int \left (-\frac{\sqrt{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 (\sqrt{-d}+\sqrt{e} x\right ) \left (a+b \tan ^{-1}(c x)\right )}{\sqrt{e} \left (-c^2 d+e\right ) \left (1+c^2 x^2\right )}\right ) \, dx}{2 d}-\frac{e \int \left (-\frac{\sqrt{-d} \left (a+b \tan ^{-1}(c x)\right )^2}{2 d e \left (\sqrt{-d}-\sqrt{e} x\right )}-\frac{\sqrt{-d} \left (a+b \tan ^{-1}(c x)\right )^2}{2 d e \left (\sqrt{-d}+\sqrt{e} x\right )}\right ) \, dx}{2 d}\\ &=-\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{4 d \sqrt{e} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{4 d \sqrt{e} \left (\sqrt{-d}+\sqrt{e} x\right )}+\frac{\int \frac{\left (a+b \tan ^{-1}(c x)\right )^2}{\sqrt{-d}-\sqrt{e} x} \, dx}{4 (-d)^{3/2}}+\frac{\int \frac{\left (a+b \tan ^{-1}(c x)\right )^2}{\sqrt{-d}+\sqrt{e} x} \, dx}{4 (-d)^{3/2}}-\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 d \left (c^2 d-e\right ) \sqrt{e}}-\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 d \left (c^2 d-e\right ) \sqrt{e}}+\frac{\left (b c \sqrt{e}\right ) \int \frac{a+b \tan ^{-1}(c x)}{-\sqrt{-d}+\sqrt{e} x} \, dx}{2 d \left (c^2 d-e\right )}+\frac{\left (b c \sqrt{e}\right ) \int \frac{a+b \tan ^{-1}(c x)}{\sqrt{-d}+\sqrt{e} x} \, dx}{2 d \left (c^2 d-e\right )}\\ &=-\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{4 d \sqrt{e} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{4 d \sqrt{e} \left (\sqrt{-d}+\sqrt{e} x\right )}-\frac{b c \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1-i c x}\right )}{d \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 )}{\left (c \sqrt{-d}-i \sqrt{e}\right ) (1-i c x)}\right )}{2 d \left (c^2 d-e\right )}-\frac{\left (a+b \tan ^{-1}(c x)\right )^2 \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 )}{4 (-d)^{3/2} \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 d \left (c^2 d-e\right )}+\frac{\left (a+b \tan ^{-1}(c x)\right )^2 \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 )}{4 (-d)^{3/2} \sqrt{e}}+\frac{i b \left (a+b \tan ^{-1}(c x)\right ) \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} \sqrt{e}}-\frac{i b \left (a+b \tan ^{-1}(c x)\right ) \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} \sqrt{e}}-\frac{b^2 \text{Li}_3\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 )}{8 (-d)^{3/2} \sqrt{e}}+\frac{b^2 \text{Li}_3\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 )}{8 (-d)^{3/2} \sqrt{e}}+2 \frac{\left (b^2 c^2\right ) \int \frac{\log \left (\frac{2}{1-i c x}\right )}{1+c^2 x^2} \, dx}{2 d \left (c^2 d-e\right )}-\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 d \left (c^2 d-e\right )}-\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 d \left (c^2 d-e\right )}-\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 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 d \left (c^2 d-e\right ) \sqrt{e}}\\ &=-\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{4 d \sqrt{e} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{4 d \sqrt{e} \left (\sqrt{-d}+\sqrt{e} x\right )}-\frac{b c \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1-i c x}\right )}{d \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 )}{\left (c \sqrt{-d}-i \sqrt{e}\right ) (1-i c x)}\right )}{2 d \left (c^2 d-e\right )}-\frac{\left (a+b \tan ^{-1}(c x)\right )^2 \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 )}{4 (-d)^{3/2} \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 d \left (c^2 d-e\right )}+\frac{\left (a+b \tan ^{-1}(c x)\right )^2 \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 )}{4 (-d)^{3/2} \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 d \left (c^2 d-e\right )}+\frac{i b \left (a+b \tan ^{-1}(c x)\right ) \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} \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 d \left (c^2 d-e\right )}-\frac{i b \left (a+b \tan ^{-1}(c x)\right ) \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} \sqrt{e}}-\frac{b^2 \text{Li}_3\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 )}{8 (-d)^{3/2} \sqrt{e}}+\frac{b^2 \text{Li}_3\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 )}{8 (-d)^{3/2} \sqrt{e}}+2 \frac{\left (i b^2 c\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1-i c x}\right )}{2 d \left (c^2 d-e\right )}-2 \frac{\left (b c^3\right ) \int \frac{x \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{2 d \left (c^2 d-e\right )}\\ &=-\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{4 d \sqrt{e} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{4 d \sqrt{e} \left (\sqrt{-d}+\sqrt{e} x\right )}-\frac{b c \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1-i c x}\right )}{d \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 )}{\left (c \sqrt{-d}-i \sqrt{e}\right ) (1-i c x)}\right )}{2 d \left (c^2 d-e\right )}-\frac{\left (a+b \tan ^{-1}(c x)\right )^2 \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 )}{4 (-d)^{3/2} \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 d \left (c^2 d-e\right )}+\frac{\left (a+b \tan ^{-1}(c x)\right )^2 \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 )}{4 (-d)^{3/2} \sqrt{e}}+\frac{i b^2 c \text{Li}_2\left (1-\frac{2}{1-i c x}\right )}{2 d \left (c^2 d-e\right )}-\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 d \left (c^2 d-e\right )}+\frac{i b \left (a+b \tan ^{-1}(c x)\right ) \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} \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 d \left (c^2 d-e\right )}-\frac{i b \left (a+b \tan ^{-1}(c x)\right ) \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} \sqrt{e}}-\frac{b^2 \text{Li}_3\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 )}{8 (-d)^{3/2} \sqrt{e}}+\frac{b^2 \text{Li}_3\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 )}{8 (-d)^{3/2} \sqrt{e}}-2 \left (-\frac{i c \left (a+b \tan ^{-1}(c x)\right )^2}{4 d \left (c^2 d-e\right )}-\frac{\left (b c^2\right ) \int \frac{a+b \tan ^{-1}(c x)}{i-c x} \, dx}{2 d \left (c^2 d-e\right )}\right )\\ &=-\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{4 d \sqrt{e} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{4 d \sqrt{e} \left (\sqrt{-d}+\sqrt{e} x\right )}-\frac{b c \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1-i c x}\right )}{d \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 )}{\left (c \sqrt{-d}-i \sqrt{e}\right ) (1-i c x)}\right )}{2 d \left (c^2 d-e\right )}-\frac{\left (a+b \tan ^{-1}(c x)\right )^2 \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 )}{4 (-d)^{3/2} \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 d \left (c^2 d-e\right )}+\frac{\left (a+b \tan ^{-1}(c x)\right )^2 \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 )}{4 (-d)^{3/2} \sqrt{e}}+\frac{i b^2 c \text{Li}_2\left (1-\frac{2}{1-i c x}\right )}{2 d \left (c^2 d-e\right )}-\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 d \left (c^2 d-e\right )}+\frac{i b \left (a+b \tan ^{-1}(c x)\right ) \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} \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 d \left (c^2 d-e\right )}-\frac{i b \left (a+b \tan ^{-1}(c x)\right ) \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} \sqrt{e}}-\frac{b^2 \text{Li}_3\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 )}{8 (-d)^{3/2} \sqrt{e}}+\frac{b^2 \text{Li}_3\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 )}{8 (-d)^{3/2} \sqrt{e}}-2 \left (-\frac{i c \left (a+b \tan ^{-1}(c x)\right )^2}{4 d \left (c^2 d-e\right )}-\frac{b c \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1+i c x}\right )}{2 d \left (c^2 d-e\right )}+\frac{\left (b^2 c^2\right ) \int \frac{\log \left (\frac{2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{2 d \left (c^2 d-e\right )}\right )\\ &=-\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{4 d \sqrt{e} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{4 d \sqrt{e} \left (\sqrt{-d}+\sqrt{e} x\right )}-\frac{b c \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1-i c x}\right )}{d \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 )}{\left (c \sqrt{-d}-i \sqrt{e}\right ) (1-i c x)}\right )}{2 d \left (c^2 d-e\right )}-\frac{\left (a+b \tan ^{-1}(c x)\right )^2 \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 )}{4 (-d)^{3/2} \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 d \left (c^2 d-e\right )}+\frac{\left (a+b \tan ^{-1}(c x)\right )^2 \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 )}{4 (-d)^{3/2} \sqrt{e}}+\frac{i b^2 c \text{Li}_2\left (1-\frac{2}{1-i c x}\right )}{2 d \left (c^2 d-e\right )}-\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 d \left (c^2 d-e\right )}+\frac{i b \left (a+b \tan ^{-1}(c x)\right ) \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} \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 d \left (c^2 d-e\right )}-\frac{i b \left (a+b \tan ^{-1}(c x)\right ) \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} \sqrt{e}}-\frac{b^2 \text{Li}_3\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 )}{8 (-d)^{3/2} \sqrt{e}}+\frac{b^2 \text{Li}_3\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 )}{8 (-d)^{3/2} \sqrt{e}}-2 \left (-\frac{i c \left (a+b \tan ^{-1}(c x)\right )^2}{4 d \left (c^2 d-e\right )}-\frac{b c \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1+i c x}\right )}{2 d \left (c^2 d-e\right )}-\frac{\left (i b^2 c\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+i c x}\right )}{2 d \left (c^2 d-e\right )}\right )\\ &=-\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{4 d \sqrt{e} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{4 d \sqrt{e} \left (\sqrt{-d}+\sqrt{e} x\right )}-\frac{b c \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1-i c x}\right )}{d \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 )}{\left (c \sqrt{-d}-i \sqrt{e}\right ) (1-i c x)}\right )}{2 d \left (c^2 d-e\right )}-\frac{\left (a+b \tan ^{-1}(c x)\right )^2 \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 )}{4 (-d)^{3/2} \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 d \left (c^2 d-e\right )}+\frac{\left (a+b \tan ^{-1}(c x)\right )^2 \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 )}{4 (-d)^{3/2} \sqrt{e}}+\frac{i b^2 c \text{Li}_2\left (1-\frac{2}{1-i c x}\right )}{2 d \left (c^2 d-e\right )}-2 \left (-\frac{i c \left (a+b \tan ^{-1}(c x)\right )^2}{4 d \left (c^2 d-e\right )}-\frac{b c \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1+i c x}\right )}{2 d \left (c^2 d-e\right )}-\frac{i b^2 c \text{Li}_2\left (1-\frac{2}{1+i c x}\right )}{4 d \left (c^2 d-e\right )}\right )-\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 d \left (c^2 d-e\right )}+\frac{i b \left (a+b \tan ^{-1}(c x)\right ) \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} \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 d \left (c^2 d-e\right )}-\frac{i b \left (a+b \tan ^{-1}(c x)\right ) \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} \sqrt{e}}-\frac{b^2 \text{Li}_3\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 )}{8 (-d)^{3/2} \sqrt{e}}+\frac{b^2 \text{Li}_3\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 )}{8 (-d)^{3/2} \sqrt{e}}\\ \end{align*}
Mathematica [F] time = 23.0303, size = 0, normalized size = 0. \[ \int \frac{\left (a+b \tan ^{-1}(c x)\right )^2}{\left (d+e x^2\right )^2} \, dx \]
Verification is Not applicable to the result.
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Maple [C] time = 1.447, size = 6575, normalized size = 6.3 \begin{align*} \text{output 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} \arctan \left (c x\right )^{2} + 2 \, a b \arctan \left (c x\right ) + a^{2}}{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}}{{\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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