Optimal. Leaf size=433 \[ \frac{4 b \text{PolyLog}\left (2,1-\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{c^5 d^2}+\frac{10 i b^2 \text{PolyLog}\left (2,1-\frac{2}{1+i c x}\right )}{3 c^5 d^2}-\frac{2 i b^2 \text{PolyLog}\left (3,1-\frac{2}{1+i c x}\right )}{c^5 d^2}-\frac{x^3 \left (a+b \tan ^{-1}(c x)\right )^2}{3 c^2 d^2}-\frac{i x^2 \left (a+b \tan ^{-1}(c x)\right )^2}{c^3 d^2}+\frac{b x^2 \left (a+b \tan ^{-1}(c x)\right )}{3 c^3 d^2}+\frac{2 i a b x}{c^4 d^2}+\frac{3 x \left (a+b \tan ^{-1}(c x)\right )^2}{c^4 d^2}-\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{c^5 d^2 (-c x+i)}+\frac{11 i \left (a+b \tan ^{-1}(c x)\right )^2}{6 c^5 d^2}+\frac{i b \left (a+b \tan ^{-1}(c x)\right )}{c^5 d^2 (-c x+i)}-\frac{4 i \log \left (\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{c^5 d^2}+\frac{20 b \log \left (\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{3 c^5 d^2}-\frac{i b^2 \log \left (c^2 x^2+1\right )}{c^5 d^2}-\frac{b^2 x}{3 c^4 d^2}+\frac{b^2}{2 c^5 d^2 (-c x+i)}+\frac{2 i b^2 x \tan ^{-1}(c x)}{c^4 d^2}-\frac{b^2 \tan ^{-1}(c x)}{6 c^5 d^2} \]
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Rubi [A] time = 0.82717, antiderivative size = 433, normalized size of antiderivative = 1., number of steps used = 33, number of rules used = 18, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.72, Rules used = {4876, 4846, 4920, 4854, 2402, 2315, 4852, 4916, 260, 4884, 321, 203, 4864, 4862, 627, 44, 4994, 6610} \[ \frac{4 b \text{PolyLog}\left (2,1-\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{c^5 d^2}+\frac{10 i b^2 \text{PolyLog}\left (2,1-\frac{2}{1+i c x}\right )}{3 c^5 d^2}-\frac{2 i b^2 \text{PolyLog}\left (3,1-\frac{2}{1+i c x}\right )}{c^5 d^2}-\frac{x^3 \left (a+b \tan ^{-1}(c x)\right )^2}{3 c^2 d^2}-\frac{i x^2 \left (a+b \tan ^{-1}(c x)\right )^2}{c^3 d^2}+\frac{b x^2 \left (a+b \tan ^{-1}(c x)\right )}{3 c^3 d^2}+\frac{2 i a b x}{c^4 d^2}+\frac{3 x \left (a+b \tan ^{-1}(c x)\right )^2}{c^4 d^2}-\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{c^5 d^2 (-c x+i)}+\frac{11 i \left (a+b \tan ^{-1}(c x)\right )^2}{6 c^5 d^2}+\frac{i b \left (a+b \tan ^{-1}(c x)\right )}{c^5 d^2 (-c x+i)}-\frac{4 i \log \left (\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{c^5 d^2}+\frac{20 b \log \left (\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{3 c^5 d^2}-\frac{i b^2 \log \left (c^2 x^2+1\right )}{c^5 d^2}-\frac{b^2 x}{3 c^4 d^2}+\frac{b^2}{2 c^5 d^2 (-c x+i)}+\frac{2 i b^2 x \tan ^{-1}(c x)}{c^4 d^2}-\frac{b^2 \tan ^{-1}(c x)}{6 c^5 d^2} \]
Antiderivative was successfully verified.
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Rule 4876
Rule 4846
Rule 4920
Rule 4854
Rule 2402
Rule 2315
Rule 4852
Rule 4916
Rule 260
Rule 4884
Rule 321
Rule 203
Rule 4864
Rule 4862
Rule 627
Rule 44
Rule 4994
Rule 6610
Rubi steps
\begin{align*} \int \frac{x^4 \left (a+b \tan ^{-1}(c x)\right )^2}{(d+i c d x)^2} \, dx &=\int \left (\frac{3 \left (a+b \tan ^{-1}(c x)\right )^2}{c^4 d^2}-\frac{2 i x \left (a+b \tan ^{-1}(c x)\right )^2}{c^3 d^2}-\frac{x^2 \left (a+b \tan ^{-1}(c x)\right )^2}{c^2 d^2}-\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{c^4 d^2 (-i+c x)^2}+\frac{4 i \left (a+b \tan ^{-1}(c x)\right )^2}{c^4 d^2 (-i+c x)}\right ) \, dx\\ &=\frac{(4 i) \int \frac{\left (a+b \tan ^{-1}(c x)\right )^2}{-i+c x} \, dx}{c^4 d^2}-\frac{\int \frac{\left (a+b \tan ^{-1}(c x)\right )^2}{(-i+c x)^2} \, dx}{c^4 d^2}+\frac{3 \int \left (a+b \tan ^{-1}(c x)\right )^2 \, dx}{c^4 d^2}-\frac{(2 i) \int x \left (a+b \tan ^{-1}(c x)\right )^2 \, dx}{c^3 d^2}-\frac{\int x^2 \left (a+b \tan ^{-1}(c x)\right )^2 \, dx}{c^2 d^2}\\ &=\frac{3 x \left (a+b \tan ^{-1}(c x)\right )^2}{c^4 d^2}-\frac{i x^2 \left (a+b \tan ^{-1}(c x)\right )^2}{c^3 d^2}-\frac{x^3 \left (a+b \tan ^{-1}(c x)\right )^2}{3 c^2 d^2}-\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{c^5 d^2 (i-c x)}-\frac{4 i \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac{2}{1+i c x}\right )}{c^5 d^2}+\frac{(8 i b) \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}{c^4 d^2}-\frac{(2 b) \int \left (-\frac{i \left (a+b \tan ^{-1}(c x)\right )}{2 (-i+c x)^2}+\frac{i \left (a+b \tan ^{-1}(c x)\right )}{2 \left (1+c^2 x^2\right )}\right ) \, dx}{c^4 d^2}-\frac{(6 b) \int \frac{x \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{c^3 d^2}+\frac{(2 i b) \int \frac{x^2 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{c^2 d^2}+\frac{(2 b) \int \frac{x^3 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{3 c d^2}\\ &=\frac{3 i \left (a+b \tan ^{-1}(c x)\right )^2}{c^5 d^2}+\frac{3 x \left (a+b \tan ^{-1}(c x)\right )^2}{c^4 d^2}-\frac{i x^2 \left (a+b \tan ^{-1}(c x)\right )^2}{c^3 d^2}-\frac{x^3 \left (a+b \tan ^{-1}(c x)\right )^2}{3 c^2 d^2}-\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{c^5 d^2 (i-c x)}-\frac{4 i \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac{2}{1+i c x}\right )}{c^5 d^2}+\frac{4 b \left (a+b \tan ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1+i c x}\right )}{c^5 d^2}+\frac{(i b) \int \frac{a+b \tan ^{-1}(c x)}{(-i+c x)^2} \, dx}{c^4 d^2}-\frac{(i b) \int \frac{a+b \tan ^{-1}(c x)}{1+c^2 x^2} \, dx}{c^4 d^2}+\frac{(2 i b) \int \left (a+b \tan ^{-1}(c x)\right ) \, dx}{c^4 d^2}-\frac{(2 i b) \int \frac{a+b \tan ^{-1}(c x)}{1+c^2 x^2} \, dx}{c^4 d^2}+\frac{(6 b) \int \frac{a+b \tan ^{-1}(c x)}{i-c x} \, dx}{c^4 d^2}-\frac{\left (4 b^2\right ) \int \frac{\text{Li}_2\left (1-\frac{2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{c^4 d^2}+\frac{(2 b) \int x \left (a+b \tan ^{-1}(c x)\right ) \, dx}{3 c^3 d^2}-\frac{(2 b) \int \frac{x \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{3 c^3 d^2}\\ &=\frac{2 i a b x}{c^4 d^2}+\frac{b x^2 \left (a+b \tan ^{-1}(c x)\right )}{3 c^3 d^2}+\frac{i b \left (a+b \tan ^{-1}(c x)\right )}{c^5 d^2 (i-c x)}+\frac{11 i \left (a+b \tan ^{-1}(c x)\right )^2}{6 c^5 d^2}+\frac{3 x \left (a+b \tan ^{-1}(c x)\right )^2}{c^4 d^2}-\frac{i x^2 \left (a+b \tan ^{-1}(c x)\right )^2}{c^3 d^2}-\frac{x^3 \left (a+b \tan ^{-1}(c x)\right )^2}{3 c^2 d^2}-\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{c^5 d^2 (i-c x)}+\frac{6 b \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1+i c x}\right )}{c^5 d^2}-\frac{4 i \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac{2}{1+i c x}\right )}{c^5 d^2}+\frac{4 b \left (a+b \tan ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1+i c x}\right )}{c^5 d^2}-\frac{2 i b^2 \text{Li}_3\left (1-\frac{2}{1+i c x}\right )}{c^5 d^2}+\frac{(2 b) \int \frac{a+b \tan ^{-1}(c x)}{i-c x} \, dx}{3 c^4 d^2}+\frac{\left (i b^2\right ) \int \frac{1}{(-i+c x) \left (1+c^2 x^2\right )} \, dx}{c^4 d^2}+\frac{\left (2 i b^2\right ) \int \tan ^{-1}(c x) \, dx}{c^4 d^2}-\frac{\left (6 b^2\right ) \int \frac{\log \left (\frac{2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{c^4 d^2}-\frac{b^2 \int \frac{x^2}{1+c^2 x^2} \, dx}{3 c^2 d^2}\\ &=\frac{2 i a b x}{c^4 d^2}-\frac{b^2 x}{3 c^4 d^2}+\frac{2 i b^2 x \tan ^{-1}(c x)}{c^4 d^2}+\frac{b x^2 \left (a+b \tan ^{-1}(c x)\right )}{3 c^3 d^2}+\frac{i b \left (a+b \tan ^{-1}(c x)\right )}{c^5 d^2 (i-c x)}+\frac{11 i \left (a+b \tan ^{-1}(c x)\right )^2}{6 c^5 d^2}+\frac{3 x \left (a+b \tan ^{-1}(c x)\right )^2}{c^4 d^2}-\frac{i x^2 \left (a+b \tan ^{-1}(c x)\right )^2}{c^3 d^2}-\frac{x^3 \left (a+b \tan ^{-1}(c x)\right )^2}{3 c^2 d^2}-\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{c^5 d^2 (i-c x)}+\frac{20 b \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1+i c x}\right )}{3 c^5 d^2}-\frac{4 i \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac{2}{1+i c x}\right )}{c^5 d^2}+\frac{4 b \left (a+b \tan ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1+i c x}\right )}{c^5 d^2}-\frac{2 i b^2 \text{Li}_3\left (1-\frac{2}{1+i c x}\right )}{c^5 d^2}+\frac{\left (6 i b^2\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+i c x}\right )}{c^5 d^2}+\frac{\left (i b^2\right ) \int \frac{1}{(-i+c x)^2 (i+c x)} \, dx}{c^4 d^2}+\frac{b^2 \int \frac{1}{1+c^2 x^2} \, dx}{3 c^4 d^2}-\frac{\left (2 b^2\right ) \int \frac{\log \left (\frac{2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{3 c^4 d^2}-\frac{\left (2 i b^2\right ) \int \frac{x}{1+c^2 x^2} \, dx}{c^3 d^2}\\ &=\frac{2 i a b x}{c^4 d^2}-\frac{b^2 x}{3 c^4 d^2}+\frac{b^2 \tan ^{-1}(c x)}{3 c^5 d^2}+\frac{2 i b^2 x \tan ^{-1}(c x)}{c^4 d^2}+\frac{b x^2 \left (a+b \tan ^{-1}(c x)\right )}{3 c^3 d^2}+\frac{i b \left (a+b \tan ^{-1}(c x)\right )}{c^5 d^2 (i-c x)}+\frac{11 i \left (a+b \tan ^{-1}(c x)\right )^2}{6 c^5 d^2}+\frac{3 x \left (a+b \tan ^{-1}(c x)\right )^2}{c^4 d^2}-\frac{i x^2 \left (a+b \tan ^{-1}(c x)\right )^2}{c^3 d^2}-\frac{x^3 \left (a+b \tan ^{-1}(c x)\right )^2}{3 c^2 d^2}-\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{c^5 d^2 (i-c x)}+\frac{20 b \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1+i c x}\right )}{3 c^5 d^2}-\frac{4 i \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac{2}{1+i c x}\right )}{c^5 d^2}-\frac{i b^2 \log \left (1+c^2 x^2\right )}{c^5 d^2}+\frac{3 i b^2 \text{Li}_2\left (1-\frac{2}{1+i c x}\right )}{c^5 d^2}+\frac{4 b \left (a+b \tan ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1+i c x}\right )}{c^5 d^2}-\frac{2 i b^2 \text{Li}_3\left (1-\frac{2}{1+i c x}\right )}{c^5 d^2}+\frac{\left (2 i b^2\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+i c x}\right )}{3 c^5 d^2}+\frac{\left (i b^2\right ) \int \left (-\frac{i}{2 (-i+c x)^2}+\frac{i}{2 \left (1+c^2 x^2\right )}\right ) \, dx}{c^4 d^2}\\ &=\frac{2 i a b x}{c^4 d^2}-\frac{b^2 x}{3 c^4 d^2}+\frac{b^2}{2 c^5 d^2 (i-c x)}+\frac{b^2 \tan ^{-1}(c x)}{3 c^5 d^2}+\frac{2 i b^2 x \tan ^{-1}(c x)}{c^4 d^2}+\frac{b x^2 \left (a+b \tan ^{-1}(c x)\right )}{3 c^3 d^2}+\frac{i b \left (a+b \tan ^{-1}(c x)\right )}{c^5 d^2 (i-c x)}+\frac{11 i \left (a+b \tan ^{-1}(c x)\right )^2}{6 c^5 d^2}+\frac{3 x \left (a+b \tan ^{-1}(c x)\right )^2}{c^4 d^2}-\frac{i x^2 \left (a+b \tan ^{-1}(c x)\right )^2}{c^3 d^2}-\frac{x^3 \left (a+b \tan ^{-1}(c x)\right )^2}{3 c^2 d^2}-\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{c^5 d^2 (i-c x)}+\frac{20 b \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1+i c x}\right )}{3 c^5 d^2}-\frac{4 i \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac{2}{1+i c x}\right )}{c^5 d^2}-\frac{i b^2 \log \left (1+c^2 x^2\right )}{c^5 d^2}+\frac{10 i b^2 \text{Li}_2\left (1-\frac{2}{1+i c x}\right )}{3 c^5 d^2}+\frac{4 b \left (a+b \tan ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1+i c x}\right )}{c^5 d^2}-\frac{2 i b^2 \text{Li}_3\left (1-\frac{2}{1+i c x}\right )}{c^5 d^2}-\frac{b^2 \int \frac{1}{1+c^2 x^2} \, dx}{2 c^4 d^2}\\ &=\frac{2 i a b x}{c^4 d^2}-\frac{b^2 x}{3 c^4 d^2}+\frac{b^2}{2 c^5 d^2 (i-c x)}-\frac{b^2 \tan ^{-1}(c x)}{6 c^5 d^2}+\frac{2 i b^2 x \tan ^{-1}(c x)}{c^4 d^2}+\frac{b x^2 \left (a+b \tan ^{-1}(c x)\right )}{3 c^3 d^2}+\frac{i b \left (a+b \tan ^{-1}(c x)\right )}{c^5 d^2 (i-c x)}+\frac{11 i \left (a+b \tan ^{-1}(c x)\right )^2}{6 c^5 d^2}+\frac{3 x \left (a+b \tan ^{-1}(c x)\right )^2}{c^4 d^2}-\frac{i x^2 \left (a+b \tan ^{-1}(c x)\right )^2}{c^3 d^2}-\frac{x^3 \left (a+b \tan ^{-1}(c x)\right )^2}{3 c^2 d^2}-\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{c^5 d^2 (i-c x)}+\frac{20 b \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1+i c x}\right )}{3 c^5 d^2}-\frac{4 i \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac{2}{1+i c x}\right )}{c^5 d^2}-\frac{i b^2 \log \left (1+c^2 x^2\right )}{c^5 d^2}+\frac{10 i b^2 \text{Li}_2\left (1-\frac{2}{1+i c x}\right )}{3 c^5 d^2}+\frac{4 b \left (a+b \tan ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1+i c x}\right )}{c^5 d^2}-\frac{2 i b^2 \text{Li}_3\left (1-\frac{2}{1+i c x}\right )}{c^5 d^2}\\ \end{align*}
Mathematica [A] time = 2.37743, size = 502, normalized size = 1.16 \[ -\frac{2 a b \left (24 \text{PolyLog}\left (2,-e^{2 i \tan ^{-1}(c x)}\right )-2 c^2 x^2+20 \log \left (c^2 x^2+1\right )+2 \tan ^{-1}(c x) \left (2 c^3 x^3+6 i c^2 x^2-18 c x+24 i \log \left (1+e^{2 i \tan ^{-1}(c x)}\right )-3 \sin \left (2 \tan ^{-1}(c x)\right )-3 i \cos \left (2 \tan ^{-1}(c x)\right )+6 i\right )-12 i c x+48 \tan ^{-1}(c x)^2+3 i \sin \left (2 \tan ^{-1}(c x)\right )-3 \cos \left (2 \tan ^{-1}(c x)\right )-2\right )+b^2 \left (8 \left (6 \tan ^{-1}(c x)+5 i\right ) \text{PolyLog}\left (2,-e^{2 i \tan ^{-1}(c x)}\right )+24 i \text{PolyLog}\left (3,-e^{2 i \tan ^{-1}(c x)}\right )+12 i \log \left (c^2 x^2+1\right )+4 c^3 x^3 \tan ^{-1}(c x)^2+12 i c^2 x^2 \tan ^{-1}(c x)^2-4 c^2 x^2 \tan ^{-1}(c x)+4 c x-36 c x \tan ^{-1}(c x)^2-24 i c x \tan ^{-1}(c x)+32 \tan ^{-1}(c x)^3+52 i \tan ^{-1}(c x)^2-4 \tan ^{-1}(c x)+48 i \tan ^{-1}(c x)^2 \log \left (1+e^{2 i \tan ^{-1}(c x)}\right )-80 \tan ^{-1}(c x) \log \left (1+e^{2 i \tan ^{-1}(c x)}\right )-6 \tan ^{-1}(c x)^2 \sin \left (2 \tan ^{-1}(c x)\right )+6 i \tan ^{-1}(c x) \sin \left (2 \tan ^{-1}(c x)\right )+3 \sin \left (2 \tan ^{-1}(c x)\right )-6 i \tan ^{-1}(c x)^2 \cos \left (2 \tan ^{-1}(c x)\right )-6 \tan ^{-1}(c x) \cos \left (2 \tan ^{-1}(c x)\right )+3 i \cos \left (2 \tan ^{-1}(c x)\right )\right )+4 a^2 c^3 x^3+12 i a^2 c^2 x^2-24 i a^2 \log \left (c^2 x^2+1\right )-36 a^2 c x-\frac{12 a^2}{c x-i}+48 a^2 \tan ^{-1}(c x)}{12 c^5 d^2} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 2.018, size = 1498, normalized size = 3.5 \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(-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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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b^{2} x^{4} \log \left (-\frac{c x + i}{c x - i}\right )^{2} - 4 i \, a b x^{4} \log \left (-\frac{c x + i}{c x - i}\right ) - 4 \, a^{2} x^{4}}{4 \, c^{2} d^{2} x^{2} - 8 i \, c d^{2} x - 4 \, d^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \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^{4}}{{\left (i \, c d x + d\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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