Optimal. Leaf size=1141 \[ \text{result too large to display} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 2.0471, antiderivative size = 1141, normalized size of antiderivative = 1., number of steps used = 42, number of rules used = 15, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.652, Rules used = {4980, 4852, 4924, 4868, 2447, 4914, 4864, 4856, 2402, 2315, 4984, 4884, 4920, 4854, 4858} \[ -\frac{i c e \text{PolyLog}\left (2,1-\frac{2}{1-i c x}\right ) b^2}{2 d^2 \left (c^2 d-e\right )}-\frac{i c \text{PolyLog}\left (2,\frac{2}{1-i c x}-1\right ) b^2}{d^2}-\frac{i c e \text{PolyLog}\left (2,1-\frac{2}{i c x+1}\right ) b^2}{2 d^2 \left (c^2 d-e\right )}+\frac{i c 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^2 \left (c^2 d-e\right )}+\frac{i c 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^2 \left (c^2 d-e\right )}-\frac{3 \sqrt{e} \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)^{5/2}}+\frac{3 \sqrt{e} \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)^{5/2}}+\frac{c e \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1-i c x}\right ) b}{d^2 \left (c^2 d-e\right )}-\frac{c e \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{i c x+1}\right ) b}{d^2 \left (c^2 d-e\right )}-\frac{c 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^2 \left (c^2 d-e\right )}-\frac{c 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^2 \left (c^2 d-e\right )}+\frac{2 c \left (a+b \tan ^{-1}(c x)\right ) \log \left (2-\frac{2}{1-i c x}\right ) b}{d^2}+\frac{3 i \sqrt{e} \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)^{5/2}}-\frac{3 i \sqrt{e} \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)^{5/2}}-\frac{i c e \left (a+b \tan ^{-1}(c x)\right )^2}{2 d^2 \left (c^2 d-e\right )}-\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{d^2 x}+\frac{\sqrt{e} \left (a+b \tan ^{-1}(c x)\right )^2}{4 d^2 \left (\sqrt{-d}-\sqrt{e} x\right )}-\frac{\sqrt{e} \left (a+b \tan ^{-1}(c x)\right )^2}{4 d^2 \left (\sqrt{e} x+\sqrt{-d}\right )}-\frac{i c \left (a+b \tan ^{-1}(c x)\right )^2}{d^2}-\frac{3 \sqrt{e} \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)^{5/2}}+\frac{3 \sqrt{e} \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)^{5/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4980
Rule 4852
Rule 4924
Rule 4868
Rule 2447
Rule 4914
Rule 4864
Rule 4856
Rule 2402
Rule 2315
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}{x^2 \left (d+e x^2\right )^2} \, dx &=\int \left (\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{d^2 x^2}-\frac{e \left (a+b \tan ^{-1}(c x)\right )^2}{d \left (d+e x^2\right )^2}-\frac{e \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^2} \, dx}{d^2}-\frac{e \int \frac{\left (a+b \tan ^{-1}(c x)\right )^2}{d+e x^2} \, dx}{d^2}-\frac{e \int \frac{\left (a+b \tan ^{-1}(c x)\right )^2}{\left (d+e x^2\right )^2} \, dx}{d}\\ &=-\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{d^2 x}+\frac{(2 b c) \int \frac{a+b \tan ^{-1}(c x)}{x \left (1+c^2 x^2\right )} \, dx}{d^2}-\frac{e \int \left (\frac{\sqrt{-d} \left (a+b \tan ^{-1}(c x)\right )^2}{2 d \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{\sqrt{-d} \left (a+b \tan ^{-1}(c x)\right )^2}{2 d \left (\sqrt{-d}+\sqrt{e} x\right )}\right ) \, dx}{d^2}-\frac{e \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}{d}\\ &=-\frac{i c \left (a+b \tan ^{-1}(c x)\right )^2}{d^2}-\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{d^2 x}+\frac{(2 i b c) \int \frac{a+b \tan ^{-1}(c x)}{x (i+c x)} \, dx}{d^2}+\frac{e \int \frac{\left (a+b \tan ^{-1}(c x)\right )^2}{\sqrt{-d}-\sqrt{e} x} \, dx}{2 (-d)^{5/2}}+\frac{e \int \frac{\left (a+b \tan ^{-1}(c x)\right )^2}{\sqrt{-d}+\sqrt{e} x} \, dx}{2 (-d)^{5/2}}+\frac{e^2 \int \frac{\left (a+b \tan ^{-1}(c x)\right )^2}{\left (\sqrt{-d} \sqrt{e}-e x\right )^2} \, dx}{4 d^2}+\frac{e^2 \int \frac{\left (a+b \tan ^{-1}(c x)\right )^2}{\left (\sqrt{-d} \sqrt{e}+e x\right )^2} \, dx}{4 d^2}+\frac{e^2 \int \frac{\left (a+b \tan ^{-1}(c x)\right )^2}{-d e-e^2 x^2} \, dx}{2 d^2}\\ &=-\frac{i c \left (a+b \tan ^{-1}(c x)\right )^2}{d^2}-\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{d^2 x}+\frac{\sqrt{e} \left (a+b \tan ^{-1}(c x)\right )^2}{4 d^2 \left (\sqrt{-d}-\sqrt{e} x\right )}-\frac{\sqrt{e} \left (a+b \tan ^{-1}(c x)\right )^2}{4 d^2 \left (\sqrt{-d}+\sqrt{e} x\right )}-\frac{\sqrt{e} \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)^{5/2}}+\frac{\sqrt{e} \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)^{5/2}}+\frac{2 b c \left (a+b \tan ^{-1}(c x)\right ) \log \left (2-\frac{2}{1-i c x}\right )}{d^2}+\frac{i b \sqrt{e} \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)^{5/2}}-\frac{i b \sqrt{e} \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)^{5/2}}-\frac{b^2 \sqrt{e} \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)^{5/2}}+\frac{b^2 \sqrt{e} \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)^{5/2}}-\frac{\left (2 b^2 c^2\right ) \int \frac{\log \left (2-\frac{2}{1-i c x}\right )}{1+c^2 x^2} \, dx}{d^2}+\frac{(b c e) \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^2}-\frac{(b c e) \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^2}+\frac{e^2 \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^2}\\ &=-\frac{i c \left (a+b \tan ^{-1}(c x)\right )^2}{d^2}-\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{d^2 x}+\frac{\sqrt{e} \left (a+b \tan ^{-1}(c x)\right )^2}{4 d^2 \left (\sqrt{-d}-\sqrt{e} x\right )}-\frac{\sqrt{e} \left (a+b \tan ^{-1}(c x)\right )^2}{4 d^2 \left (\sqrt{-d}+\sqrt{e} x\right )}-\frac{\sqrt{e} \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)^{5/2}}+\frac{\sqrt{e} \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)^{5/2}}+\frac{2 b c \left (a+b \tan ^{-1}(c x)\right ) \log \left (2-\frac{2}{1-i c x}\right )}{d^2}-\frac{i b^2 c \text{Li}_2\left (-1+\frac{2}{1-i c x}\right )}{d^2}+\frac{i b \sqrt{e} \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)^{5/2}}-\frac{i b \sqrt{e} \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)^{5/2}}-\frac{b^2 \sqrt{e} \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)^{5/2}}+\frac{b^2 \sqrt{e} \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)^{5/2}}+\frac{\left (b c^3 \sqrt{e}\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^2 \left (c^2 d-e\right )}+\frac{\left (b c^3 \sqrt{e}\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^2 \left (c^2 d-e\right )}+\frac{e \int \frac{\left (a+b \tan ^{-1}(c x)\right )^2}{\sqrt{-d}-\sqrt{e} x} \, dx}{4 (-d)^{5/2}}+\frac{e \int \frac{\left (a+b \tan ^{-1}(c x)\right )^2}{\sqrt{-d}+\sqrt{e} x} \, dx}{4 (-d)^{5/2}}-\frac{\left (b c e^{3/2}\right ) \int \frac{a+b \tan ^{-1}(c x)}{-\sqrt{-d}+\sqrt{e} x} \, dx}{2 d^2 \left (c^2 d-e\right )}-\frac{\left (b c e^{3/2}\right ) \int \frac{a+b \tan ^{-1}(c x)}{\sqrt{-d}+\sqrt{e} x} \, dx}{2 d^2 \left (c^2 d-e\right )}\\ &=-\frac{i c \left (a+b \tan ^{-1}(c x)\right )^2}{d^2}-\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{d^2 x}+\frac{\sqrt{e} \left (a+b \tan ^{-1}(c x)\right )^2}{4 d^2 \left (\sqrt{-d}-\sqrt{e} x\right )}-\frac{\sqrt{e} \left (a+b \tan ^{-1}(c x)\right )^2}{4 d^2 \left (\sqrt{-d}+\sqrt{e} x\right )}+\frac{b c e \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1-i c x}\right )}{d^2 \left (c^2 d-e\right )}-\frac{b c 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^2 \left (c^2 d-e\right )}-\frac{3 \sqrt{e} \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)^{5/2}}-\frac{b c 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^2 \left (c^2 d-e\right )}+\frac{3 \sqrt{e} \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)^{5/2}}+\frac{2 b c \left (a+b \tan ^{-1}(c x)\right ) \log \left (2-\frac{2}{1-i c x}\right )}{d^2}-\frac{i b^2 c \text{Li}_2\left (-1+\frac{2}{1-i c x}\right )}{d^2}+\frac{3 i b \sqrt{e} \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)^{5/2}}-\frac{3 i b \sqrt{e} \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)^{5/2}}-\frac{3 b^2 \sqrt{e} \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)^{5/2}}+\frac{3 b^2 \sqrt{e} \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)^{5/2}}+\frac{\left (b c^3 \sqrt{e}\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^2 \left (c^2 d-e\right )}+\frac{\left (b c^3 \sqrt{e}\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^2 \left (c^2 d-e\right )}-2 \frac{\left (b^2 c^2 e\right ) \int \frac{\log \left (\frac{2}{1-i c x}\right )}{1+c^2 x^2} \, dx}{2 d^2 \left (c^2 d-e\right )}+\frac{\left (b^2 c^2 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^2 \left (c^2 d-e\right )}+\frac{\left (b^2 c^2 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^2 \left (c^2 d-e\right )}\\ &=-\frac{i c \left (a+b \tan ^{-1}(c x)\right )^2}{d^2}-\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{d^2 x}+\frac{\sqrt{e} \left (a+b \tan ^{-1}(c x)\right )^2}{4 d^2 \left (\sqrt{-d}-\sqrt{e} x\right )}-\frac{\sqrt{e} \left (a+b \tan ^{-1}(c x)\right )^2}{4 d^2 \left (\sqrt{-d}+\sqrt{e} x\right )}+\frac{b c e \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1-i c x}\right )}{d^2 \left (c^2 d-e\right )}-\frac{b c 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^2 \left (c^2 d-e\right )}-\frac{3 \sqrt{e} \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)^{5/2}}-\frac{b c 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^2 \left (c^2 d-e\right )}+\frac{3 \sqrt{e} \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)^{5/2}}+\frac{2 b c \left (a+b \tan ^{-1}(c x)\right ) \log \left (2-\frac{2}{1-i c x}\right )}{d^2}-\frac{i b^2 c \text{Li}_2\left (-1+\frac{2}{1-i c x}\right )}{d^2}+\frac{i b^2 c 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^2 \left (c^2 d-e\right )}+\frac{3 i b \sqrt{e} \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)^{5/2}}+\frac{i b^2 c 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^2 \left (c^2 d-e\right )}-\frac{3 i b \sqrt{e} \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)^{5/2}}-\frac{3 b^2 \sqrt{e} \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)^{5/2}}+\frac{3 b^2 \sqrt{e} \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)^{5/2}}-2 \frac{\left (i b^2 c e\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1-i c x}\right )}{2 d^2 \left (c^2 d-e\right )}+2 \frac{\left (b c^3 e\right ) \int \frac{x \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{2 d^2 \left (c^2 d-e\right )}\\ &=-\frac{i c \left (a+b \tan ^{-1}(c x)\right )^2}{d^2}-\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{d^2 x}+\frac{\sqrt{e} \left (a+b \tan ^{-1}(c x)\right )^2}{4 d^2 \left (\sqrt{-d}-\sqrt{e} x\right )}-\frac{\sqrt{e} \left (a+b \tan ^{-1}(c x)\right )^2}{4 d^2 \left (\sqrt{-d}+\sqrt{e} x\right )}+\frac{b c e \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1-i c x}\right )}{d^2 \left (c^2 d-e\right )}-\frac{b c 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^2 \left (c^2 d-e\right )}-\frac{3 \sqrt{e} \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)^{5/2}}-\frac{b c 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^2 \left (c^2 d-e\right )}+\frac{3 \sqrt{e} \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)^{5/2}}+\frac{2 b c \left (a+b \tan ^{-1}(c x)\right ) \log \left (2-\frac{2}{1-i c x}\right )}{d^2}-\frac{i b^2 c e \text{Li}_2\left (1-\frac{2}{1-i c x}\right )}{2 d^2 \left (c^2 d-e\right )}-\frac{i b^2 c \text{Li}_2\left (-1+\frac{2}{1-i c x}\right )}{d^2}+\frac{i b^2 c 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^2 \left (c^2 d-e\right )}+\frac{3 i b \sqrt{e} \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)^{5/2}}+\frac{i b^2 c 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^2 \left (c^2 d-e\right )}-\frac{3 i b \sqrt{e} \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)^{5/2}}-\frac{3 b^2 \sqrt{e} \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)^{5/2}}+\frac{3 b^2 \sqrt{e} \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)^{5/2}}+2 \left (-\frac{i c e \left (a+b \tan ^{-1}(c x)\right )^2}{4 d^2 \left (c^2 d-e\right )}-\frac{\left (b c^2 e\right ) \int \frac{a+b \tan ^{-1}(c x)}{i-c x} \, dx}{2 d^2 \left (c^2 d-e\right )}\right )\\ &=-\frac{i c \left (a+b \tan ^{-1}(c x)\right )^2}{d^2}-\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{d^2 x}+\frac{\sqrt{e} \left (a+b \tan ^{-1}(c x)\right )^2}{4 d^2 \left (\sqrt{-d}-\sqrt{e} x\right )}-\frac{\sqrt{e} \left (a+b \tan ^{-1}(c x)\right )^2}{4 d^2 \left (\sqrt{-d}+\sqrt{e} x\right )}+\frac{b c e \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1-i c x}\right )}{d^2 \left (c^2 d-e\right )}-\frac{b c 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^2 \left (c^2 d-e\right )}-\frac{3 \sqrt{e} \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)^{5/2}}-\frac{b c 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^2 \left (c^2 d-e\right )}+\frac{3 \sqrt{e} \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)^{5/2}}+\frac{2 b c \left (a+b \tan ^{-1}(c x)\right ) \log \left (2-\frac{2}{1-i c x}\right )}{d^2}-\frac{i b^2 c e \text{Li}_2\left (1-\frac{2}{1-i c x}\right )}{2 d^2 \left (c^2 d-e\right )}-\frac{i b^2 c \text{Li}_2\left (-1+\frac{2}{1-i c x}\right )}{d^2}+\frac{i b^2 c 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^2 \left (c^2 d-e\right )}+\frac{3 i b \sqrt{e} \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)^{5/2}}+\frac{i b^2 c 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^2 \left (c^2 d-e\right )}-\frac{3 i b \sqrt{e} \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)^{5/2}}-\frac{3 b^2 \sqrt{e} \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)^{5/2}}+\frac{3 b^2 \sqrt{e} \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)^{5/2}}+2 \left (-\frac{i c e \left (a+b \tan ^{-1}(c x)\right )^2}{4 d^2 \left (c^2 d-e\right )}-\frac{b c e \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1+i c x}\right )}{2 d^2 \left (c^2 d-e\right )}+\frac{\left (b^2 c^2 e\right ) \int \frac{\log \left (\frac{2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{2 d^2 \left (c^2 d-e\right )}\right )\\ &=-\frac{i c \left (a+b \tan ^{-1}(c x)\right )^2}{d^2}-\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{d^2 x}+\frac{\sqrt{e} \left (a+b \tan ^{-1}(c x)\right )^2}{4 d^2 \left (\sqrt{-d}-\sqrt{e} x\right )}-\frac{\sqrt{e} \left (a+b \tan ^{-1}(c x)\right )^2}{4 d^2 \left (\sqrt{-d}+\sqrt{e} x\right )}+\frac{b c e \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1-i c x}\right )}{d^2 \left (c^2 d-e\right )}-\frac{b c 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^2 \left (c^2 d-e\right )}-\frac{3 \sqrt{e} \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)^{5/2}}-\frac{b c 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^2 \left (c^2 d-e\right )}+\frac{3 \sqrt{e} \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)^{5/2}}+\frac{2 b c \left (a+b \tan ^{-1}(c x)\right ) \log \left (2-\frac{2}{1-i c x}\right )}{d^2}-\frac{i b^2 c e \text{Li}_2\left (1-\frac{2}{1-i c x}\right )}{2 d^2 \left (c^2 d-e\right )}-\frac{i b^2 c \text{Li}_2\left (-1+\frac{2}{1-i c x}\right )}{d^2}+\frac{i b^2 c 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^2 \left (c^2 d-e\right )}+\frac{3 i b \sqrt{e} \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)^{5/2}}+\frac{i b^2 c 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^2 \left (c^2 d-e\right )}-\frac{3 i b \sqrt{e} \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)^{5/2}}-\frac{3 b^2 \sqrt{e} \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)^{5/2}}+\frac{3 b^2 \sqrt{e} \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)^{5/2}}+2 \left (-\frac{i c e \left (a+b \tan ^{-1}(c x)\right )^2}{4 d^2 \left (c^2 d-e\right )}-\frac{b c e \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1+i c x}\right )}{2 d^2 \left (c^2 d-e\right )}-\frac{\left (i b^2 c e\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+i c x}\right )}{2 d^2 \left (c^2 d-e\right )}\right )\\ &=-\frac{i c \left (a+b \tan ^{-1}(c x)\right )^2}{d^2}-\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{d^2 x}+\frac{\sqrt{e} \left (a+b \tan ^{-1}(c x)\right )^2}{4 d^2 \left (\sqrt{-d}-\sqrt{e} x\right )}-\frac{\sqrt{e} \left (a+b \tan ^{-1}(c x)\right )^2}{4 d^2 \left (\sqrt{-d}+\sqrt{e} x\right )}+\frac{b c e \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1-i c x}\right )}{d^2 \left (c^2 d-e\right )}-\frac{b c 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^2 \left (c^2 d-e\right )}-\frac{3 \sqrt{e} \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)^{5/2}}-\frac{b c 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^2 \left (c^2 d-e\right )}+\frac{3 \sqrt{e} \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)^{5/2}}+\frac{2 b c \left (a+b \tan ^{-1}(c x)\right ) \log \left (2-\frac{2}{1-i c x}\right )}{d^2}-\frac{i b^2 c e \text{Li}_2\left (1-\frac{2}{1-i c x}\right )}{2 d^2 \left (c^2 d-e\right )}-\frac{i b^2 c \text{Li}_2\left (-1+\frac{2}{1-i c x}\right )}{d^2}+2 \left (-\frac{i c e \left (a+b \tan ^{-1}(c x)\right )^2}{4 d^2 \left (c^2 d-e\right )}-\frac{b c e \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1+i c x}\right )}{2 d^2 \left (c^2 d-e\right )}-\frac{i b^2 c e \text{Li}_2\left (1-\frac{2}{1+i c x}\right )}{4 d^2 \left (c^2 d-e\right )}\right )+\frac{i b^2 c 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^2 \left (c^2 d-e\right )}+\frac{3 i b \sqrt{e} \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)^{5/2}}+\frac{i b^2 c 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^2 \left (c^2 d-e\right )}-\frac{3 i b \sqrt{e} \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)^{5/2}}-\frac{3 b^2 \sqrt{e} \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)^{5/2}}+\frac{3 b^2 \sqrt{e} \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)^{5/2}}\\ \end{align*}
Mathematica [F] time = 180.004, size = 0, normalized size = 0. \[ \text{\$Aborted} \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 3.653, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b\arctan \left ( cx \right ) \right ) ^{2}}{{x}^{2} \left ( e{x}^{2}+d \right ) ^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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^{6} + 2 \, d e x^{4} + d^{2} x^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]