Optimal. Leaf size=1087 \[ \frac {i c \sqrt {e} \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 ) b^2}{4 (-d)^{3/2} \left (c^2 d-e\right )}-\frac {i c \sqrt {e} \text {Li}_2\left (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)^{3/2} \left (c^2 d-e\right )}+\frac {\text {Li}_3\left (1-\frac {2}{1-i c x}\right ) b^2}{2 d^2}-\frac {\text {Li}_3\left (1-\frac {2}{i c x+1}\right ) b^2}{2 d^2}+\frac {\text {Li}_3\left (\frac {2}{i c x+1}-1\right ) b^2}{2 d^2}-\frac {\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 ) b^2}{4 d^2}-\frac {\text {Li}_3\left (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^2}-\frac {c \sqrt {e} \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)^{3/2} \left (c^2 d-e\right )}+\frac {c \sqrt {e} \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)^{3/2} \left (c^2 d-e\right )}-\frac {i \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-i c x}\right ) b}{d^2}-\frac {i \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{i c x+1}\right ) b}{d^2}+\frac {i \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (\frac {2}{i c x+1}-1\right ) b}{d^2}+\frac {i \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 ) b}{2 d^2}+\frac {i \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (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 d^2}-\frac {c^2 \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^2 \left (1-\frac {\sqrt {e} x}{\sqrt {-d}}\right )}+\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{4 d^2 \left (\frac {\sqrt {e} x}{\sqrt {-d}}+1\right )}+\frac {2 \left (a+b \tan ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac {2}{i c x+1}\right )}{d^2}+\frac {\left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2}{1-i c x}\right )}{d^2}-\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 )}{2 d^2}-\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 )}{2 d^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 1.86, antiderivative size = 1087, normalized size of antiderivative = 1.00, number of steps used = 39, number of rules used = 16, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.696, Rules used = {4980, 4850, 4988, 4884, 4994, 6610, 4978, 4864, 4856, 2402, 2315, 2447, 4984, 4920, 4854, 4858} \[ \frac {i c \sqrt {e} \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)^{3/2} \left (c^2 d-e\right )}-\frac {i c \sqrt {e} \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)^{3/2} \left (c^2 d-e\right )}+\frac {\text {PolyLog}\left (3,1-\frac {2}{1-i c x}\right ) b^2}{2 d^2}-\frac {\text {PolyLog}\left (3,1-\frac {2}{i c x+1}\right ) b^2}{2 d^2}+\frac {\text {PolyLog}\left (3,\frac {2}{i c x+1}-1\right ) b^2}{2 d^2}-\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}{4 d^2}-\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}{4 d^2}-\frac {c \sqrt {e} \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)^{3/2} \left (c^2 d-e\right )}+\frac {c \sqrt {e} \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)^{3/2} \left (c^2 d-e\right )}-\frac {i \left (a+b \tan ^{-1}(c x)\right ) \text {PolyLog}\left (2,1-\frac {2}{1-i c x}\right ) b}{d^2}-\frac {i \left (a+b \tan ^{-1}(c x)\right ) \text {PolyLog}\left (2,1-\frac {2}{i c x+1}\right ) b}{d^2}+\frac {i \left (a+b \tan ^{-1}(c x)\right ) \text {PolyLog}\left (2,\frac {2}{i c x+1}-1\right ) b}{d^2}+\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}{2 d^2}+\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}{2 d^2}-\frac {c^2 \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^2 \left (1-\frac {\sqrt {e} x}{\sqrt {-d}}\right )}+\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{4 d^2 \left (\frac {\sqrt {e} x}{\sqrt {-d}}+1\right )}+\frac {2 \left (a+b \tan ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac {2}{i c x+1}\right )}{d^2}+\frac {\left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2}{1-i c x}\right )}{d^2}-\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 )}{2 d^2}-\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 )}{2 d^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2315
Rule 2402
Rule 2447
Rule 4850
Rule 4854
Rule 4856
Rule 4858
Rule 4864
Rule 4884
Rule 4920
Rule 4978
Rule 4980
Rule 4984
Rule 4988
Rule 4994
Rule 6610
Rubi steps
\begin {align*} \int \frac {\left (a+b \tan ^{-1}(c x)\right )^2}{x \left (d+e x^2\right )^2} \, dx &=\int \left (\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{d^2 x}-\frac {e x \left (a+b \tan ^{-1}(c x)\right )^2}{d \left (d+e x^2\right )^2}-\frac {e x \left (a+b \tan ^{-1}(c x)\right )^2}{d^2 \left (d+e x^2\right )}\right ) \, dx\\ &=\frac {\int \frac {\left (a+b \tan ^{-1}(c x)\right )^2}{x} \, dx}{d^2}-\frac {e \int \frac {x \left (a+b \tan ^{-1}(c x)\right )^2}{d+e x^2} \, dx}{d^2}-\frac {e \int \frac {x \left (a+b \tan ^{-1}(c x)\right )^2}{\left (d+e x^2\right )^2} \, dx}{d}\\ &=\frac {2 \left (a+b \tan ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac {2}{1+i c x}\right )}{d^2}-\frac {(4 b c) \int \frac {\left (a+b \tan ^{-1}(c x)\right ) \tanh ^{-1}\left (1-\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{d^2}+\frac {\sqrt {e} \int \frac {\left (a+b \tan ^{-1}(c x)\right )^2}{\left (1-\frac {\sqrt {e} x}{\sqrt {-d}}\right )^2} \, dx}{4 (-d)^{5/2}}-\frac {\sqrt {e} \int \frac {\left (a+b \tan ^{-1}(c x)\right )^2}{\left (1+\frac {\sqrt {e} x}{\sqrt {-d}}\right )^2} \, dx}{4 (-d)^{5/2}}-\frac {e \int \left (-\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{2 \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{2 \sqrt {e} \left (\sqrt {-d}+\sqrt {e} x\right )}\right ) \, dx}{d^2}\\ &=\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{4 d^2 \left (1-\frac {\sqrt {e} x}{\sqrt {-d}}\right )}+\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{4 d^2 \left (1+\frac {\sqrt {e} x}{\sqrt {-d}}\right )}+\frac {2 \left (a+b \tan ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac {2}{1+i c x}\right )}{d^2}-\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^2}-\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^2}+\frac {(2 b c) \int \frac {\left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{d^2}-\frac {(2 b c) \int \frac {\left (a+b \tan ^{-1}(c x)\right ) \log \left (2-\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{d^2}+\frac {\sqrt {e} \int \frac {\left (a+b \tan ^{-1}(c x)\right )^2}{\sqrt {-d}-\sqrt {e} x} \, dx}{2 d^2}-\frac {\sqrt {e} \int \frac {\left (a+b \tan ^{-1}(c x)\right )^2}{\sqrt {-d}+\sqrt {e} x} \, dx}{2 d^2}\\ &=\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{4 d^2 \left (1-\frac {\sqrt {e} x}{\sqrt {-d}}\right )}+\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{4 d^2 \left (1+\frac {\sqrt {e} x}{\sqrt {-d}}\right )}+\frac {2 \left (a+b \tan ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac {2}{1+i c x}\right )}{d^2}+\frac {\left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2}{1-i c x}\right )}{d^2}-\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 )}{2 d^2}-\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 )}{2 d^2}-\frac {i b \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-i c x}\right )}{d^2}-\frac {i b \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{d^2}+\frac {i b \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (-1+\frac {2}{1+i c x}\right )}{d^2}+\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 )}{2 d^2}+\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 )}{2 d^2}+\frac {b^2 \text {Li}_3\left (1-\frac {2}{1-i c x}\right )}{2 d^2}-\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 )}{4 d^2}-\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 )}{4 d^2}+\frac {\left (i b^2 c\right ) \int \frac {\text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{d^2}-\frac {\left (i b^2 c\right ) \int \frac {\text {Li}_2\left (-1+\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{d^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)^{3/2} \left (c^2 d-e\right )}-\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^2 \left (c^2 d-e\right )}-\frac {(b c e) \int \frac {a+b \tan ^{-1}(c x)}{-\sqrt {-d}+\sqrt {e} x} \, dx}{2 (-d)^{3/2} \left (c^2 d-e\right )}+\frac {(b c e) \int \frac {a+b \tan ^{-1}(c x)}{\sqrt {-d}+\sqrt {e} x} \, dx}{2 (-d)^{3/2} \left (c^2 d-e\right )}\\ &=\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{4 d^2 \left (1-\frac {\sqrt {e} x}{\sqrt {-d}}\right )}+\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{4 d^2 \left (1+\frac {\sqrt {e} x}{\sqrt {-d}}\right )}+\frac {2 \left (a+b \tan ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac {2}{1+i c x}\right )}{d^2}+\frac {\left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2}{1-i c x}\right )}{d^2}-\frac {b c \sqrt {e} \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)^{3/2} \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 )}{2 d^2}+\frac {b c \sqrt {e} \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)^{3/2} \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 )}{2 d^2}-\frac {i b \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-i c x}\right )}{d^2}-\frac {i b \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{d^2}+\frac {i b \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (-1+\frac {2}{1+i c x}\right )}{d^2}+\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 )}{2 d^2}+\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 )}{2 d^2}+\frac {b^2 \text {Li}_3\left (1-\frac {2}{1-i c x}\right )}{2 d^2}-\frac {b^2 \text {Li}_3\left (1-\frac {2}{1+i c x}\right )}{2 d^2}+\frac {b^2 \text {Li}_3\left (-1+\frac {2}{1+i c x}\right )}{2 d^2}-\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 )}{4 d^2}-\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 )}{4 d^2}+\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)^{3/2} \left (c^2 d-e\right )}-\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^2 \left (c^2 d-e\right )}+\frac {\left (b^2 c^2 \sqrt {e}\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)^{3/2} \left (c^2 d-e\right )}-\frac {\left (b^2 c^2 \sqrt {e}\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)^{3/2} \left (c^2 d-e\right )}\\ &=\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{4 d^2 \left (1-\frac {\sqrt {e} x}{\sqrt {-d}}\right )}+\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{4 d^2 \left (1+\frac {\sqrt {e} x}{\sqrt {-d}}\right )}+\frac {2 \left (a+b \tan ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac {2}{1+i c x}\right )}{d^2}+\frac {\left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2}{1-i c x}\right )}{d^2}-\frac {b c \sqrt {e} \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)^{3/2} \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 )}{2 d^2}+\frac {b c \sqrt {e} \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)^{3/2} \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 )}{2 d^2}-\frac {i b \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-i c x}\right )}{d^2}-\frac {i b \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{d^2}+\frac {i b \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (-1+\frac {2}{1+i c x}\right )}{d^2}+\frac {i b^2 c \sqrt {e} \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} \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 )}{2 d^2}-\frac {i b^2 c \sqrt {e} \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} \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 )}{2 d^2}+\frac {b^2 \text {Li}_3\left (1-\frac {2}{1-i c x}\right )}{2 d^2}-\frac {b^2 \text {Li}_3\left (1-\frac {2}{1+i c x}\right )}{2 d^2}+\frac {b^2 \text {Li}_3\left (-1+\frac {2}{1+i c x}\right )}{2 d^2}-\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 )}{4 d^2}-\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 )}{4 d^2}-2 \frac {\left (b c^3\right ) \int \frac {a+b \tan ^{-1}(c x)}{1+c^2 x^2} \, dx}{2 d \left (c^2 d-e\right )}\\ &=-\frac {c^2 \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^2 \left (1-\frac {\sqrt {e} x}{\sqrt {-d}}\right )}+\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{4 d^2 \left (1+\frac {\sqrt {e} x}{\sqrt {-d}}\right )}+\frac {2 \left (a+b \tan ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac {2}{1+i c x}\right )}{d^2}+\frac {\left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2}{1-i c x}\right )}{d^2}-\frac {b c \sqrt {e} \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)^{3/2} \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 )}{2 d^2}+\frac {b c \sqrt {e} \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)^{3/2} \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 )}{2 d^2}-\frac {i b \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-i c x}\right )}{d^2}-\frac {i b \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{d^2}+\frac {i b \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (-1+\frac {2}{1+i c x}\right )}{d^2}+\frac {i b^2 c \sqrt {e} \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} \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 )}{2 d^2}-\frac {i b^2 c \sqrt {e} \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} \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 )}{2 d^2}+\frac {b^2 \text {Li}_3\left (1-\frac {2}{1-i c x}\right )}{2 d^2}-\frac {b^2 \text {Li}_3\left (1-\frac {2}{1+i c x}\right )}{2 d^2}+\frac {b^2 \text {Li}_3\left (-1+\frac {2}{1+i c x}\right )}{2 d^2}-\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 )}{4 d^2}-\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 )}{4 d^2}\\ \end {align*}
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Mathematica [F] time = 17.41, size = 0, normalized size = 0.00 \[ \int \frac {\left (a+b \tan ^{-1}(c x)\right )^2}{x \left (d+e x^2\right )^2} \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 0.40, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{2} \arctan \left (c x\right )^{2} + 2 \, a b \arctan \left (c x\right ) + a^{2}}{e^{2} x^{5} + 2 \, d e x^{3} + d^{2} x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 30.09, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +b \arctan \left (c x \right )\right )^{2}}{x \left (e \,x^{2}+d \right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{2} \, a^{2} {\left (\frac {1}{d e x^{2} + d^{2}} - \frac {\log \left (e x^{2} + d\right )}{d^{2}} + \frac {2 \, \log \relax (x)}{d^{2}}\right )} + \int \frac {b^{2} \arctan \left (c x\right )^{2} + 2 \, a b \arctan \left (c x\right )}{e^{2} x^{5} + 2 \, d e x^{3} + d^{2} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^2}{x\,{\left (e\,x^2+d\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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