Optimal. Leaf size=1325 \[ \text{result too large to display} \]
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Rubi [A] time = 3.09881, antiderivative size = 1554, normalized size of antiderivative = 1.17, number of steps used = 110, number of rules used = 46, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 2.556, Rules used = {6742, 5027, 297, 1162, 617, 204, 1165, 628, 5033, 260, 5029, 2450, 2476, 2448, 321, 203, 2470, 12, 4920, 4854, 2402, 2315, 206, 2556, 205, 4928, 4856, 2447, 208, 5992, 5920, 5984, 5918, 5035, 2454, 2389, 2296, 2295, 30, 2557, 2475, 43, 2416, 2394, 2393, 2391} \[ \text{result too large to display} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
Rule 6742
Rule 5027
Rule 297
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rule 5033
Rule 260
Rule 5029
Rule 2450
Rule 2476
Rule 2448
Rule 321
Rule 203
Rule 2470
Rule 12
Rule 4920
Rule 4854
Rule 2402
Rule 2315
Rule 206
Rule 2556
Rule 205
Rule 4928
Rule 4856
Rule 2447
Rule 208
Rule 5992
Rule 5920
Rule 5984
Rule 5918
Rule 5035
Rule 2454
Rule 2389
Rule 2296
Rule 2295
Rule 30
Rule 2557
Rule 2475
Rule 43
Rule 2416
Rule 2394
Rule 2393
Rule 2391
Rubi steps
\begin{align*} \int (d+e x) \left (a+b \tan ^{-1}\left (c x^2\right )\right )^2 \, dx &=\int \left (a^2 (d+e x)+2 a b (d+e x) \tan ^{-1}\left (c x^2\right )+b^2 (d+e x) \tan ^{-1}\left (c x^2\right )^2\right ) \, dx\\ &=\frac{a^2 (d+e x)^2}{2 e}+(2 a b) \int (d+e x) \tan ^{-1}\left (c x^2\right ) \, dx+b^2 \int (d+e x) \tan ^{-1}\left (c x^2\right )^2 \, dx\\ &=\frac{a^2 (d+e x)^2}{2 e}+(2 a b) \int \left (d \tan ^{-1}\left (c x^2\right )+e x \tan ^{-1}\left (c x^2\right )\right ) \, dx+b^2 \int \left (d \tan ^{-1}\left (c x^2\right )^2+e x \tan ^{-1}\left (c x^2\right )^2\right ) \, dx\\ &=\frac{a^2 (d+e x)^2}{2 e}+(2 a b d) \int \tan ^{-1}\left (c x^2\right ) \, dx+\left (b^2 d\right ) \int \tan ^{-1}\left (c x^2\right )^2 \, dx+(2 a b e) \int x \tan ^{-1}\left (c x^2\right ) \, dx+\left (b^2 e\right ) \int x \tan ^{-1}\left (c x^2\right )^2 \, dx\\ &=\frac{a^2 (d+e x)^2}{2 e}+2 a b d x \tan ^{-1}\left (c x^2\right )+a b e x^2 \tan ^{-1}\left (c x^2\right )+\left (b^2 d\right ) \int \left (-\frac{1}{4} \log ^2\left (1-i c x^2\right )+\frac{1}{2} \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right )-\frac{1}{4} \log ^2\left (1+i c x^2\right )\right ) \, dx-(4 a b c d) \int \frac{x^2}{1+c^2 x^4} \, dx+\left (b^2 e\right ) \int \left (-\frac{1}{4} x \log ^2\left (1-i c x^2\right )+\frac{1}{2} x \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right )-\frac{1}{4} x \log ^2\left (1+i c x^2\right )\right ) \, dx-(2 a b c e) \int \frac{x^3}{1+c^2 x^4} \, dx\\ &=\frac{a^2 (d+e x)^2}{2 e}+2 a b d x \tan ^{-1}\left (c x^2\right )+a b e x^2 \tan ^{-1}\left (c x^2\right )-\frac{a b e \log \left (1+c^2 x^4\right )}{2 c}+(2 a b d) \int \frac{1-c x^2}{1+c^2 x^4} \, dx-(2 a b d) \int \frac{1+c x^2}{1+c^2 x^4} \, dx-\frac{1}{4} \left (b^2 d\right ) \int \log ^2\left (1-i c x^2\right ) \, dx-\frac{1}{4} \left (b^2 d\right ) \int \log ^2\left (1+i c x^2\right ) \, dx+\frac{1}{2} \left (b^2 d\right ) \int \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right ) \, dx-\frac{1}{4} \left (b^2 e\right ) \int x \log ^2\left (1-i c x^2\right ) \, dx-\frac{1}{4} \left (b^2 e\right ) \int x \log ^2\left (1+i c x^2\right ) \, dx+\frac{1}{2} \left (b^2 e\right ) \int x \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right ) \, dx\\ &=\frac{a^2 (d+e x)^2}{2 e}+2 a b d x \tan ^{-1}\left (c x^2\right )+a b e x^2 \tan ^{-1}\left (c x^2\right )-\frac{1}{4} b^2 d x \log ^2\left (1-i c x^2\right )+\frac{1}{2} b^2 d x \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right )+\frac{1}{4} b^2 e x^2 \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right )-\frac{1}{4} b^2 d x \log ^2\left (1+i c x^2\right )-\frac{a b e \log \left (1+c^2 x^4\right )}{2 c}-\frac{1}{2} \left (b^2 d\right ) \int \frac{2 c x^2 \log \left (1-i c x^2\right )}{-i+c x^2} \, dx-\frac{1}{2} \left (b^2 d\right ) \int \frac{2 c x^2 \log \left (1+i c x^2\right )}{i+c x^2} \, dx-\frac{(a b d) \int \frac{1}{\frac{1}{c}-\frac{\sqrt{2} x}{\sqrt{c}}+x^2} \, dx}{c}-\frac{(a b d) \int \frac{1}{\frac{1}{c}+\frac{\sqrt{2} x}{\sqrt{c}}+x^2} \, dx}{c}-\frac{(a b d) \int \frac{\frac{\sqrt{2}}{\sqrt{c}}+2 x}{-\frac{1}{c}-\frac{\sqrt{2} x}{\sqrt{c}}-x^2} \, dx}{\sqrt{2} \sqrt{c}}-\frac{(a b d) \int \frac{\frac{\sqrt{2}}{\sqrt{c}}-2 x}{-\frac{1}{c}+\frac{\sqrt{2} x}{\sqrt{c}}-x^2} \, dx}{\sqrt{2} \sqrt{c}}-\left (i b^2 c d\right ) \int \frac{x^2 \log \left (1-i c x^2\right )}{1-i c x^2} \, dx+\left (i b^2 c d\right ) \int \frac{x^2 \log \left (1+i c x^2\right )}{1+i c x^2} \, dx-\frac{1}{8} \left (b^2 e\right ) \operatorname{Subst}\left (\int \log ^2(1-i c x) \, dx,x,x^2\right )-\frac{1}{8} \left (b^2 e\right ) \operatorname{Subst}\left (\int \log ^2(1+i c x) \, dx,x,x^2\right )-\frac{1}{2} \left (b^2 e\right ) \int \frac{c x^3 \log \left (1-i c x^2\right )}{-i+c x^2} \, dx-\frac{1}{2} \left (b^2 e\right ) \int \frac{c x^3 \log \left (1+i c x^2\right )}{i+c x^2} \, dx\\ &=\frac{a^2 (d+e x)^2}{2 e}+2 a b d x \tan ^{-1}\left (c x^2\right )+a b e x^2 \tan ^{-1}\left (c x^2\right )-\frac{1}{4} b^2 d x \log ^2\left (1-i c x^2\right )+\frac{1}{2} b^2 d x \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right )+\frac{1}{4} b^2 e x^2 \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right )-\frac{1}{4} b^2 d x \log ^2\left (1+i c x^2\right )-\frac{a b d \log \left (1-\sqrt{2} \sqrt{c} x+c x^2\right )}{\sqrt{2} \sqrt{c}}+\frac{a b d \log \left (1+\sqrt{2} \sqrt{c} x+c x^2\right )}{\sqrt{2} \sqrt{c}}-\frac{a b e \log \left (1+c^2 x^4\right )}{2 c}-\frac{\left (\sqrt{2} a b d\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\sqrt{2} \sqrt{c} x\right )}{\sqrt{c}}+\frac{\left (\sqrt{2} a b d\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\sqrt{2} \sqrt{c} x\right )}{\sqrt{c}}-\left (i b^2 c d\right ) \int \left (\frac{i \log \left (1-i c x^2\right )}{c}-\frac{i \log \left (1-i c x^2\right )}{c \left (1-i c x^2\right )}\right ) \, dx+\left (i b^2 c d\right ) \int \left (-\frac{i \log \left (1+i c x^2\right )}{c}+\frac{i \log \left (1+i c x^2\right )}{c \left (1+i c x^2\right )}\right ) \, dx-\left (b^2 c d\right ) \int \frac{x^2 \log \left (1-i c x^2\right )}{-i+c x^2} \, dx-\left (b^2 c d\right ) \int \frac{x^2 \log \left (1+i c x^2\right )}{i+c x^2} \, dx-\frac{\left (i b^2 e\right ) \operatorname{Subst}\left (\int \log ^2(x) \, dx,x,1-i c x^2\right )}{8 c}+\frac{\left (i b^2 e\right ) \operatorname{Subst}\left (\int \log ^2(x) \, dx,x,1+i c x^2\right )}{8 c}-\frac{1}{2} \left (b^2 c e\right ) \int \frac{x^3 \log \left (1-i c x^2\right )}{-i+c x^2} \, dx-\frac{1}{2} \left (b^2 c e\right ) \int \frac{x^3 \log \left (1+i c x^2\right )}{i+c x^2} \, dx\\ &=\frac{a^2 (d+e x)^2}{2 e}+2 a b d x \tan ^{-1}\left (c x^2\right )+a b e x^2 \tan ^{-1}\left (c x^2\right )+\frac{\sqrt{2} a b d \tan ^{-1}\left (1-\sqrt{2} \sqrt{c} x\right )}{\sqrt{c}}-\frac{\sqrt{2} a b d \tan ^{-1}\left (1+\sqrt{2} \sqrt{c} x\right )}{\sqrt{c}}-\frac{1}{4} b^2 d x \log ^2\left (1-i c x^2\right )-\frac{i b^2 e \left (1-i c x^2\right ) \log ^2\left (1-i c x^2\right )}{8 c}+\frac{1}{2} b^2 d x \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right )+\frac{1}{4} b^2 e x^2 \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right )-\frac{1}{4} b^2 d x \log ^2\left (1+i c x^2\right )+\frac{i b^2 e \left (1+i c x^2\right ) \log ^2\left (1+i c x^2\right )}{8 c}-\frac{a b d \log \left (1-\sqrt{2} \sqrt{c} x+c x^2\right )}{\sqrt{2} \sqrt{c}}+\frac{a b d \log \left (1+\sqrt{2} \sqrt{c} x+c x^2\right )}{\sqrt{2} \sqrt{c}}-\frac{a b e \log \left (1+c^2 x^4\right )}{2 c}+\left (b^2 d\right ) \int \log \left (1-i c x^2\right ) \, dx-\left (b^2 d\right ) \int \frac{\log \left (1-i c x^2\right )}{1-i c x^2} \, dx+\left (b^2 d\right ) \int \log \left (1+i c x^2\right ) \, dx-\left (b^2 d\right ) \int \frac{\log \left (1+i c x^2\right )}{1+i c x^2} \, dx-\left (b^2 c d\right ) \int \left (\frac{\log \left (1-i c x^2\right )}{c}+\frac{i \log \left (1-i c x^2\right )}{c \left (-i+c x^2\right )}\right ) \, dx-\left (b^2 c d\right ) \int \left (\frac{\log \left (1+i c x^2\right )}{c}-\frac{i \log \left (1+i c x^2\right )}{c \left (i+c x^2\right )}\right ) \, dx+\frac{\left (i b^2 e\right ) \operatorname{Subst}\left (\int \log (x) \, dx,x,1-i c x^2\right )}{4 c}-\frac{\left (i b^2 e\right ) \operatorname{Subst}\left (\int \log (x) \, dx,x,1+i c x^2\right )}{4 c}-\frac{1}{4} \left (b^2 c e\right ) \operatorname{Subst}\left (\int \frac{x \log (1-i c x)}{-i+c x} \, dx,x,x^2\right )-\frac{1}{4} \left (b^2 c e\right ) \operatorname{Subst}\left (\int \frac{x \log (1+i c x)}{i+c x} \, dx,x,x^2\right )\\ &=-\frac{1}{2} b^2 e x^2+\frac{a^2 (d+e x)^2}{2 e}+2 a b d x \tan ^{-1}\left (c x^2\right )+a b e x^2 \tan ^{-1}\left (c x^2\right )+\frac{\sqrt{2} a b d \tan ^{-1}\left (1-\sqrt{2} \sqrt{c} x\right )}{\sqrt{c}}-\frac{\sqrt{2} a b d \tan ^{-1}\left (1+\sqrt{2} \sqrt{c} x\right )}{\sqrt{c}}+b^2 d x \log \left (1-i c x^2\right )+\frac{i b^2 e \left (1-i c x^2\right ) \log \left (1-i c x^2\right )}{4 c}+\frac{\sqrt [4]{-1} b^2 d \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (1-i c x^2\right )}{\sqrt{c}}-\frac{1}{4} b^2 d x \log ^2\left (1-i c x^2\right )-\frac{i b^2 e \left (1-i c x^2\right ) \log ^2\left (1-i c x^2\right )}{8 c}+b^2 d x \log \left (1+i c x^2\right )-\frac{i b^2 e \left (1+i c x^2\right ) \log \left (1+i c x^2\right )}{4 c}+\frac{\sqrt [4]{-1} b^2 d \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (1+i c x^2\right )}{\sqrt{c}}+\frac{1}{2} b^2 d x \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right )+\frac{1}{4} b^2 e x^2 \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right )-\frac{1}{4} b^2 d x \log ^2\left (1+i c x^2\right )+\frac{i b^2 e \left (1+i c x^2\right ) \log ^2\left (1+i c x^2\right )}{8 c}-\frac{a b d \log \left (1-\sqrt{2} \sqrt{c} x+c x^2\right )}{\sqrt{2} \sqrt{c}}+\frac{a b d \log \left (1+\sqrt{2} \sqrt{c} x+c x^2\right )}{\sqrt{2} \sqrt{c}}-\frac{a b e \log \left (1+c^2 x^4\right )}{2 c}-\left (i b^2 d\right ) \int \frac{\log \left (1-i c x^2\right )}{-i+c x^2} \, dx+\left (i b^2 d\right ) \int \frac{\log \left (1+i c x^2\right )}{i+c x^2} \, dx-\left (b^2 d\right ) \int \log \left (1-i c x^2\right ) \, dx-\left (b^2 d\right ) \int \log \left (1+i c x^2\right ) \, dx+\left (2 i b^2 c d\right ) \int \frac{x^2}{1-i c x^2} \, dx-\left (2 i b^2 c d\right ) \int \frac{x^2}{1+i c x^2} \, dx+\left (2 i b^2 c d\right ) \int \frac{\sqrt [4]{-1} x \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )}{\sqrt{c} \left (1-i c x^2\right )} \, dx-\left (2 i b^2 c d\right ) \int \frac{\sqrt [4]{-1} x \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )}{\sqrt{c} \left (1+i c x^2\right )} \, dx-\frac{1}{4} \left (b^2 c e\right ) \operatorname{Subst}\left (\int \left (\frac{\log (1-i c x)}{c}+\frac{i \log (1-i c x)}{c (-i+c x)}\right ) \, dx,x,x^2\right )-\frac{1}{4} \left (b^2 c e\right ) \operatorname{Subst}\left (\int \left (\frac{\log (1+i c x)}{c}-\frac{i \log (1+i c x)}{c (i+c x)}\right ) \, dx,x,x^2\right )\\ &=-4 b^2 d x-\frac{1}{2} b^2 e x^2+\frac{a^2 (d+e x)^2}{2 e}+2 a b d x \tan ^{-1}\left (c x^2\right )+a b e x^2 \tan ^{-1}\left (c x^2\right )+\frac{\sqrt{2} a b d \tan ^{-1}\left (1-\sqrt{2} \sqrt{c} x\right )}{\sqrt{c}}-\frac{\sqrt{2} a b d \tan ^{-1}\left (1+\sqrt{2} \sqrt{c} x\right )}{\sqrt{c}}+\frac{i b^2 e \left (1-i c x^2\right ) \log \left (1-i c x^2\right )}{4 c}+\frac{\sqrt [4]{-1} b^2 d \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (1-i c x^2\right )}{\sqrt{c}}-\frac{\sqrt [4]{-1} b^2 d \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (1-i c x^2\right )}{\sqrt{c}}-\frac{1}{4} b^2 d x \log ^2\left (1-i c x^2\right )-\frac{i b^2 e \left (1-i c x^2\right ) \log ^2\left (1-i c x^2\right )}{8 c}-\frac{i b^2 e \left (1+i c x^2\right ) \log \left (1+i c x^2\right )}{4 c}-\frac{\sqrt [4]{-1} b^2 d \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (1+i c x^2\right )}{\sqrt{c}}+\frac{\sqrt [4]{-1} b^2 d \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (1+i c x^2\right )}{\sqrt{c}}+\frac{1}{2} b^2 d x \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right )+\frac{1}{4} b^2 e x^2 \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right )-\frac{1}{4} b^2 d x \log ^2\left (1+i c x^2\right )+\frac{i b^2 e \left (1+i c x^2\right ) \log ^2\left (1+i c x^2\right )}{8 c}-\frac{a b d \log \left (1-\sqrt{2} \sqrt{c} x+c x^2\right )}{\sqrt{2} \sqrt{c}}+\frac{a b d \log \left (1+\sqrt{2} \sqrt{c} x+c x^2\right )}{\sqrt{2} \sqrt{c}}-\frac{a b e \log \left (1+c^2 x^4\right )}{2 c}+\left (2 b^2 d\right ) \int \frac{1}{1-i c x^2} \, dx+\left (2 b^2 d\right ) \int \frac{1}{1+i c x^2} \, dx+\left (2 (-1)^{3/4} b^2 \sqrt{c} d\right ) \int \frac{x \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )}{1-i c x^2} \, dx-\left (2 (-1)^{3/4} b^2 \sqrt{c} d\right ) \int \frac{x \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )}{1+i c x^2} \, dx-\left (2 i b^2 c d\right ) \int \frac{x^2}{1-i c x^2} \, dx+\left (2 i b^2 c d\right ) \int \frac{x^2}{1+i c x^2} \, dx+\left (2 b^2 c d\right ) \int \frac{(-1)^{3/4} x \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )}{\sqrt{c} \left (1+i c x^2\right )} \, dx-\left (2 b^2 c d\right ) \int \frac{(-1)^{3/4} x \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )}{\sqrt{c} \left (1-i c x^2\right )} \, dx-\frac{1}{4} \left (i b^2 e\right ) \operatorname{Subst}\left (\int \frac{\log (1-i c x)}{-i+c x} \, dx,x,x^2\right )+\frac{1}{4} \left (i b^2 e\right ) \operatorname{Subst}\left (\int \frac{\log (1+i c x)}{i+c x} \, dx,x,x^2\right )-\frac{1}{4} \left (b^2 e\right ) \operatorname{Subst}\left (\int \log (1-i c x) \, dx,x,x^2\right )-\frac{1}{4} \left (b^2 e\right ) \operatorname{Subst}\left (\int \log (1+i c x) \, dx,x,x^2\right )\\ &=-\frac{1}{2} b^2 e x^2+\frac{a^2 (d+e x)^2}{2 e}-\frac{2 \sqrt [4]{-1} b^2 d \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )}{\sqrt{c}}+\frac{(-1)^{3/4} b^2 d \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )^2}{\sqrt{c}}+2 a b d x \tan ^{-1}\left (c x^2\right )+a b e x^2 \tan ^{-1}\left (c x^2\right )+\frac{\sqrt{2} a b d \tan ^{-1}\left (1-\sqrt{2} \sqrt{c} x\right )}{\sqrt{c}}-\frac{\sqrt{2} a b d \tan ^{-1}\left (1+\sqrt{2} \sqrt{c} x\right )}{\sqrt{c}}-\frac{2 \sqrt [4]{-1} b^2 d \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )}{\sqrt{c}}-\frac{\sqrt [4]{-1} b^2 d \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )^2}{\sqrt{c}}+\frac{i b^2 e \left (1-i c x^2\right ) \log \left (1-i c x^2\right )}{4 c}+\frac{\sqrt [4]{-1} b^2 d \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (1-i c x^2\right )}{\sqrt{c}}-\frac{\sqrt [4]{-1} b^2 d \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (1-i c x^2\right )}{\sqrt{c}}-\frac{1}{4} b^2 d x \log ^2\left (1-i c x^2\right )-\frac{i b^2 e \left (1-i c x^2\right ) \log ^2\left (1-i c x^2\right )}{8 c}-\frac{i b^2 e \log \left (1-i c x^2\right ) \log \left (\frac{1}{2} \left (1+i c x^2\right )\right )}{4 c}-\frac{i b^2 e \left (1+i c x^2\right ) \log \left (1+i c x^2\right )}{4 c}-\frac{\sqrt [4]{-1} b^2 d \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (1+i c x^2\right )}{\sqrt{c}}+\frac{\sqrt [4]{-1} b^2 d \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (1+i c x^2\right )}{\sqrt{c}}+\frac{i b^2 e \log \left (\frac{1}{2} \left (1-i c x^2\right )\right ) \log \left (1+i c x^2\right )}{4 c}+\frac{1}{2} b^2 d x \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right )+\frac{1}{4} b^2 e x^2 \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right )-\frac{1}{4} b^2 d x \log ^2\left (1+i c x^2\right )+\frac{i b^2 e \left (1+i c x^2\right ) \log ^2\left (1+i c x^2\right )}{8 c}-\frac{a b d \log \left (1-\sqrt{2} \sqrt{c} x+c x^2\right )}{\sqrt{2} \sqrt{c}}+\frac{a b d \log \left (1+\sqrt{2} \sqrt{c} x+c x^2\right )}{\sqrt{2} \sqrt{c}}-\frac{a b e \log \left (1+c^2 x^4\right )}{2 c}-\left (2 b^2 d\right ) \int \frac{1}{1-i c x^2} \, dx-\left (2 b^2 d\right ) \int \frac{1}{1+i c x^2} \, dx-\left (2 b^2 d\right ) \int \frac{\tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )}{i-(-1)^{3/4} \sqrt{c} x} \, dx-\left (2 b^2 d\right ) \int \frac{\tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )}{1-(-1)^{3/4} \sqrt{c} x} \, dx+\left (2 (-1)^{3/4} b^2 \sqrt{c} d\right ) \int \frac{x \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )}{1+i c x^2} \, dx-\left (2 (-1)^{3/4} b^2 \sqrt{c} d\right ) \int \frac{x \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )}{1-i c x^2} \, dx+\frac{1}{4} \left (b^2 e\right ) \operatorname{Subst}\left (\int \frac{\log \left (\frac{1}{2} i (-i+c x)\right )}{1-i c x} \, dx,x,x^2\right )+\frac{1}{4} \left (b^2 e\right ) \operatorname{Subst}\left (\int \frac{\log \left (-\frac{1}{2} i (i+c x)\right )}{1+i c x} \, dx,x,x^2\right )-\frac{\left (i b^2 e\right ) \operatorname{Subst}\left (\int \log (x) \, dx,x,1-i c x^2\right )}{4 c}+\frac{\left (i b^2 e\right ) \operatorname{Subst}\left (\int \log (x) \, dx,x,1+i c x^2\right )}{4 c}\\ &=\frac{a^2 (d+e x)^2}{2 e}+\frac{(-1)^{3/4} b^2 d \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )^2}{\sqrt{c}}+2 a b d x \tan ^{-1}\left (c x^2\right )+a b e x^2 \tan ^{-1}\left (c x^2\right )+\frac{\sqrt{2} a b d \tan ^{-1}\left (1-\sqrt{2} \sqrt{c} x\right )}{\sqrt{c}}-\frac{\sqrt{2} a b d \tan ^{-1}\left (1+\sqrt{2} \sqrt{c} x\right )}{\sqrt{c}}-\frac{\sqrt [4]{-1} b^2 d \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )^2}{\sqrt{c}}+\frac{2 \sqrt [4]{-1} b^2 d \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (\frac{2}{1-\sqrt [4]{-1} \sqrt{c} x}\right )}{\sqrt{c}}+\frac{2 \sqrt [4]{-1} b^2 d \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (\frac{2}{1-(-1)^{3/4} \sqrt{c} x}\right )}{\sqrt{c}}+\frac{\sqrt [4]{-1} b^2 d \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (1-i c x^2\right )}{\sqrt{c}}-\frac{\sqrt [4]{-1} b^2 d \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (1-i c x^2\right )}{\sqrt{c}}-\frac{1}{4} b^2 d x \log ^2\left (1-i c x^2\right )-\frac{i b^2 e \left (1-i c x^2\right ) \log ^2\left (1-i c x^2\right )}{8 c}-\frac{i b^2 e \log \left (1-i c x^2\right ) \log \left (\frac{1}{2} \left (1+i c x^2\right )\right )}{4 c}-\frac{\sqrt [4]{-1} b^2 d \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (1+i c x^2\right )}{\sqrt{c}}+\frac{\sqrt [4]{-1} b^2 d \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (1+i c x^2\right )}{\sqrt{c}}+\frac{i b^2 e \log \left (\frac{1}{2} \left (1-i c x^2\right )\right ) \log \left (1+i c x^2\right )}{4 c}+\frac{1}{2} b^2 d x \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right )+\frac{1}{4} b^2 e x^2 \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right )-\frac{1}{4} b^2 d x \log ^2\left (1+i c x^2\right )+\frac{i b^2 e \left (1+i c x^2\right ) \log ^2\left (1+i c x^2\right )}{8 c}-\frac{a b d \log \left (1-\sqrt{2} \sqrt{c} x+c x^2\right )}{\sqrt{2} \sqrt{c}}+\frac{a b d \log \left (1+\sqrt{2} \sqrt{c} x+c x^2\right )}{\sqrt{2} \sqrt{c}}-\frac{a b e \log \left (1+c^2 x^4\right )}{2 c}+\left (2 b^2 d\right ) \int \frac{\log \left (\frac{2}{1-\sqrt [4]{-1} \sqrt{c} x}\right )}{1-i c x^2} \, dx+\left (2 b^2 d\right ) \int \frac{\log \left (\frac{2}{1-(-1)^{3/4} \sqrt{c} x}\right )}{1+i c x^2} \, dx+\left (2 (-1)^{3/4} b^2 \sqrt{c} d\right ) \int \left (\frac{i \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )}{2 \sqrt{c} \left (\sqrt [4]{-1}-\sqrt{c} x\right )}-\frac{i \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )}{2 \sqrt{c} \left (\sqrt [4]{-1}+\sqrt{c} x\right )}\right ) \, dx-\left (2 (-1)^{3/4} b^2 \sqrt{c} d\right ) \int \left (-\frac{i \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )}{2 \sqrt{c} \left (-(-1)^{3/4}-\sqrt{c} x\right )}+\frac{i \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )}{2 \sqrt{c} \left (-(-1)^{3/4}+\sqrt{c} x\right )}\right ) \, dx+\frac{\left (i b^2 e\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{x}{2}\right )}{x} \, dx,x,1-i c x^2\right )}{4 c}-\frac{\left (i b^2 e\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{x}{2}\right )}{x} \, dx,x,1+i c x^2\right )}{4 c}\\ &=\frac{a^2 (d+e x)^2}{2 e}+\frac{(-1)^{3/4} b^2 d \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )^2}{\sqrt{c}}+2 a b d x \tan ^{-1}\left (c x^2\right )+a b e x^2 \tan ^{-1}\left (c x^2\right )+\frac{\sqrt{2} a b d \tan ^{-1}\left (1-\sqrt{2} \sqrt{c} x\right )}{\sqrt{c}}-\frac{\sqrt{2} a b d \tan ^{-1}\left (1+\sqrt{2} \sqrt{c} x\right )}{\sqrt{c}}-\frac{\sqrt [4]{-1} b^2 d \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )^2}{\sqrt{c}}+\frac{2 \sqrt [4]{-1} b^2 d \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (\frac{2}{1-\sqrt [4]{-1} \sqrt{c} x}\right )}{\sqrt{c}}+\frac{2 \sqrt [4]{-1} b^2 d \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (\frac{2}{1-(-1)^{3/4} \sqrt{c} x}\right )}{\sqrt{c}}+\frac{\sqrt [4]{-1} b^2 d \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (1-i c x^2\right )}{\sqrt{c}}-\frac{\sqrt [4]{-1} b^2 d \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (1-i c x^2\right )}{\sqrt{c}}-\frac{1}{4} b^2 d x \log ^2\left (1-i c x^2\right )-\frac{i b^2 e \left (1-i c x^2\right ) \log ^2\left (1-i c x^2\right )}{8 c}-\frac{i b^2 e \log \left (1-i c x^2\right ) \log \left (\frac{1}{2} \left (1+i c x^2\right )\right )}{4 c}-\frac{\sqrt [4]{-1} b^2 d \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (1+i c x^2\right )}{\sqrt{c}}+\frac{\sqrt [4]{-1} b^2 d \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (1+i c x^2\right )}{\sqrt{c}}+\frac{i b^2 e \log \left (\frac{1}{2} \left (1-i c x^2\right )\right ) \log \left (1+i c x^2\right )}{4 c}+\frac{1}{2} b^2 d x \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right )+\frac{1}{4} b^2 e x^2 \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right )-\frac{1}{4} b^2 d x \log ^2\left (1+i c x^2\right )+\frac{i b^2 e \left (1+i c x^2\right ) \log ^2\left (1+i c x^2\right )}{8 c}-\frac{a b d \log \left (1-\sqrt{2} \sqrt{c} x+c x^2\right )}{\sqrt{2} \sqrt{c}}+\frac{a b d \log \left (1+\sqrt{2} \sqrt{c} x+c x^2\right )}{\sqrt{2} \sqrt{c}}-\frac{a b e \log \left (1+c^2 x^4\right )}{2 c}-\frac{i b^2 e \text{Li}_2\left (\frac{1}{2} \left (1-i c x^2\right )\right )}{4 c}+\frac{i b^2 e \text{Li}_2\left (\frac{1}{2} \left (1+i c x^2\right )\right )}{4 c}-\left (\sqrt [4]{-1} b^2 d\right ) \int \frac{\tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )}{\sqrt [4]{-1}-\sqrt{c} x} \, dx+\left (\sqrt [4]{-1} b^2 d\right ) \int \frac{\tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )}{\sqrt [4]{-1}+\sqrt{c} x} \, dx-\left (\sqrt [4]{-1} b^2 d\right ) \int \frac{\tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )}{-(-1)^{3/4}-\sqrt{c} x} \, dx+\left (\sqrt [4]{-1} b^2 d\right ) \int \frac{\tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )}{-(-1)^{3/4}+\sqrt{c} x} \, dx+\frac{\left (2 \sqrt [4]{-1} b^2 d\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1-(-1)^{3/4} \sqrt{c} x}\right )}{\sqrt{c}}+\frac{\left (2 (-1)^{3/4} b^2 d\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1-\sqrt [4]{-1} \sqrt{c} x}\right )}{\sqrt{c}}\\ &=\frac{a^2 (d+e x)^2}{2 e}+\frac{(-1)^{3/4} b^2 d \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )^2}{\sqrt{c}}+2 a b d x \tan ^{-1}\left (c x^2\right )+a b e x^2 \tan ^{-1}\left (c x^2\right )+\frac{\sqrt{2} a b d \tan ^{-1}\left (1-\sqrt{2} \sqrt{c} x\right )}{\sqrt{c}}-\frac{\sqrt{2} a b d \tan ^{-1}\left (1+\sqrt{2} \sqrt{c} x\right )}{\sqrt{c}}-\frac{\sqrt [4]{-1} b^2 d \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )^2}{\sqrt{c}}+\frac{2 \sqrt [4]{-1} b^2 d \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (\frac{2}{1-\sqrt [4]{-1} \sqrt{c} x}\right )}{\sqrt{c}}-\frac{2 \sqrt [4]{-1} b^2 d \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (\frac{2}{1+\sqrt [4]{-1} \sqrt{c} x}\right )}{\sqrt{c}}+\frac{\sqrt [4]{-1} b^2 d \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (\frac{\sqrt{2} \left (\sqrt [4]{-1}+\sqrt{c} x\right )}{1+\sqrt [4]{-1} \sqrt{c} x}\right )}{\sqrt{c}}+\frac{2 \sqrt [4]{-1} b^2 d \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (\frac{2}{1-(-1)^{3/4} \sqrt{c} x}\right )}{\sqrt{c}}-\frac{2 \sqrt [4]{-1} b^2 d \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (\frac{2}{1+(-1)^{3/4} \sqrt{c} x}\right )}{\sqrt{c}}+\frac{\sqrt [4]{-1} b^2 d \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (-\frac{\sqrt{2} \left ((-1)^{3/4}+\sqrt{c} x\right )}{1+(-1)^{3/4} \sqrt{c} x}\right )}{\sqrt{c}}+\frac{\sqrt [4]{-1} b^2 d \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (\frac{\sqrt [4]{-1} \sqrt{2} \left (1+\sqrt [4]{-1} \sqrt{c} x\right )}{1+(-1)^{3/4} \sqrt{c} x}\right )}{\sqrt{c}}+\frac{\sqrt [4]{-1} b^2 d \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (\frac{(1-i) \left (1+(-1)^{3/4} \sqrt{c} x\right )}{1+\sqrt [4]{-1} \sqrt{c} x}\right )}{\sqrt{c}}+\frac{\sqrt [4]{-1} b^2 d \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (1-i c x^2\right )}{\sqrt{c}}-\frac{\sqrt [4]{-1} b^2 d \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (1-i c x^2\right )}{\sqrt{c}}-\frac{1}{4} b^2 d x \log ^2\left (1-i c x^2\right )-\frac{i b^2 e \left (1-i c x^2\right ) \log ^2\left (1-i c x^2\right )}{8 c}-\frac{i b^2 e \log \left (1-i c x^2\right ) \log \left (\frac{1}{2} \left (1+i c x^2\right )\right )}{4 c}-\frac{\sqrt [4]{-1} b^2 d \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (1+i c x^2\right )}{\sqrt{c}}+\frac{\sqrt [4]{-1} b^2 d \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (1+i c x^2\right )}{\sqrt{c}}+\frac{i b^2 e \log \left (\frac{1}{2} \left (1-i c x^2\right )\right ) \log \left (1+i c x^2\right )}{4 c}+\frac{1}{2} b^2 d x \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right )+\frac{1}{4} b^2 e x^2 \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right )-\frac{1}{4} b^2 d x \log ^2\left (1+i c x^2\right )+\frac{i b^2 e \left (1+i c x^2\right ) \log ^2\left (1+i c x^2\right )}{8 c}-\frac{a b d \log \left (1-\sqrt{2} \sqrt{c} x+c x^2\right )}{\sqrt{2} \sqrt{c}}+\frac{a b d \log \left (1+\sqrt{2} \sqrt{c} x+c x^2\right )}{\sqrt{2} \sqrt{c}}-\frac{a b e \log \left (1+c^2 x^4\right )}{2 c}-\frac{i b^2 e \text{Li}_2\left (\frac{1}{2} \left (1-i c x^2\right )\right )}{4 c}+\frac{i b^2 e \text{Li}_2\left (\frac{1}{2} \left (1+i c x^2\right )\right )}{4 c}+\frac{(-1)^{3/4} b^2 d \text{Li}_2\left (1-\frac{2}{1-\sqrt [4]{-1} \sqrt{c} x}\right )}{\sqrt{c}}+\frac{\sqrt [4]{-1} b^2 d \text{Li}_2\left (1-\frac{2}{1-(-1)^{3/4} \sqrt{c} x}\right )}{\sqrt{c}}-2 \left (\left (b^2 d\right ) \int \frac{\log \left (\frac{2}{1+\sqrt [4]{-1} \sqrt{c} x}\right )}{1-i c x^2} \, dx\right )+\left (b^2 d\right ) \int \frac{\log \left (-\frac{(1-i) (-1)^{3/4} \left (\sqrt [4]{-1}-\sqrt{c} x\right )}{1+\sqrt [4]{-1} \sqrt{c} x}\right )}{1-i c x^2} \, dx+\left (b^2 d\right ) \int \frac{\log \left (-\frac{(1+i) (-1)^{3/4} \left (\sqrt [4]{-1}+\sqrt{c} x\right )}{1+\sqrt [4]{-1} \sqrt{c} x}\right )}{1-i c x^2} \, dx-2 \left (\left (b^2 d\right ) \int \frac{\log \left (\frac{2}{1+(-1)^{3/4} \sqrt{c} x}\right )}{1+i c x^2} \, dx\right )+\left (b^2 d\right ) \int \frac{\log \left (-\frac{(1+i) (-1)^{3/4} \left (-(-1)^{3/4}-\sqrt{c} x\right )}{1+(-1)^{3/4} \sqrt{c} x}\right )}{1+i c x^2} \, dx+\left (b^2 d\right ) \int \frac{\log \left (\frac{(1-i) (-1)^{3/4} \left (-(-1)^{3/4}+\sqrt{c} x\right )}{1+(-1)^{3/4} \sqrt{c} x}\right )}{1+i c x^2} \, dx\\ &=\frac{a^2 (d+e x)^2}{2 e}+\frac{(-1)^{3/4} b^2 d \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )^2}{\sqrt{c}}+2 a b d x \tan ^{-1}\left (c x^2\right )+a b e x^2 \tan ^{-1}\left (c x^2\right )+\frac{\sqrt{2} a b d \tan ^{-1}\left (1-\sqrt{2} \sqrt{c} x\right )}{\sqrt{c}}-\frac{\sqrt{2} a b d \tan ^{-1}\left (1+\sqrt{2} \sqrt{c} x\right )}{\sqrt{c}}-\frac{\sqrt [4]{-1} b^2 d \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )^2}{\sqrt{c}}+\frac{2 \sqrt [4]{-1} b^2 d \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (\frac{2}{1-\sqrt [4]{-1} \sqrt{c} x}\right )}{\sqrt{c}}-\frac{2 \sqrt [4]{-1} b^2 d \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (\frac{2}{1+\sqrt [4]{-1} \sqrt{c} x}\right )}{\sqrt{c}}+\frac{\sqrt [4]{-1} b^2 d \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (\frac{\sqrt{2} \left (\sqrt [4]{-1}+\sqrt{c} x\right )}{1+\sqrt [4]{-1} \sqrt{c} x}\right )}{\sqrt{c}}+\frac{2 \sqrt [4]{-1} b^2 d \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (\frac{2}{1-(-1)^{3/4} \sqrt{c} x}\right )}{\sqrt{c}}-\frac{2 \sqrt [4]{-1} b^2 d \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (\frac{2}{1+(-1)^{3/4} \sqrt{c} x}\right )}{\sqrt{c}}+\frac{\sqrt [4]{-1} b^2 d \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (-\frac{\sqrt{2} \left ((-1)^{3/4}+\sqrt{c} x\right )}{1+(-1)^{3/4} \sqrt{c} x}\right )}{\sqrt{c}}+\frac{\sqrt [4]{-1} b^2 d \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (\frac{\sqrt [4]{-1} \sqrt{2} \left (1+\sqrt [4]{-1} \sqrt{c} x\right )}{1+(-1)^{3/4} \sqrt{c} x}\right )}{\sqrt{c}}+\frac{\sqrt [4]{-1} b^2 d \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (\frac{(1-i) \left (1+(-1)^{3/4} \sqrt{c} x\right )}{1+\sqrt [4]{-1} \sqrt{c} x}\right )}{\sqrt{c}}+\frac{\sqrt [4]{-1} b^2 d \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (1-i c x^2\right )}{\sqrt{c}}-\frac{\sqrt [4]{-1} b^2 d \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (1-i c x^2\right )}{\sqrt{c}}-\frac{1}{4} b^2 d x \log ^2\left (1-i c x^2\right )-\frac{i b^2 e \left (1-i c x^2\right ) \log ^2\left (1-i c x^2\right )}{8 c}-\frac{i b^2 e \log \left (1-i c x^2\right ) \log \left (\frac{1}{2} \left (1+i c x^2\right )\right )}{4 c}-\frac{\sqrt [4]{-1} b^2 d \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (1+i c x^2\right )}{\sqrt{c}}+\frac{\sqrt [4]{-1} b^2 d \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (1+i c x^2\right )}{\sqrt{c}}+\frac{i b^2 e \log \left (\frac{1}{2} \left (1-i c x^2\right )\right ) \log \left (1+i c x^2\right )}{4 c}+\frac{1}{2} b^2 d x \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right )+\frac{1}{4} b^2 e x^2 \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right )-\frac{1}{4} b^2 d x \log ^2\left (1+i c x^2\right )+\frac{i b^2 e \left (1+i c x^2\right ) \log ^2\left (1+i c x^2\right )}{8 c}-\frac{a b d \log \left (1-\sqrt{2} \sqrt{c} x+c x^2\right )}{\sqrt{2} \sqrt{c}}+\frac{a b d \log \left (1+\sqrt{2} \sqrt{c} x+c x^2\right )}{\sqrt{2} \sqrt{c}}-\frac{a b e \log \left (1+c^2 x^4\right )}{2 c}-\frac{i b^2 e \text{Li}_2\left (\frac{1}{2} \left (1-i c x^2\right )\right )}{4 c}+\frac{i b^2 e \text{Li}_2\left (\frac{1}{2} \left (1+i c x^2\right )\right )}{4 c}+\frac{(-1)^{3/4} b^2 d \text{Li}_2\left (1-\frac{2}{1-\sqrt [4]{-1} \sqrt{c} x}\right )}{\sqrt{c}}-\frac{(-1)^{3/4} b^2 d \text{Li}_2\left (1-\frac{\sqrt{2} \left (\sqrt [4]{-1}+\sqrt{c} x\right )}{1+\sqrt [4]{-1} \sqrt{c} x}\right )}{2 \sqrt{c}}+\frac{\sqrt [4]{-1} b^2 d \text{Li}_2\left (1-\frac{2}{1-(-1)^{3/4} \sqrt{c} x}\right )}{\sqrt{c}}-\frac{\sqrt [4]{-1} b^2 d \text{Li}_2\left (1+\frac{\sqrt{2} \left ((-1)^{3/4}+\sqrt{c} x\right )}{1+(-1)^{3/4} \sqrt{c} x}\right )}{2 \sqrt{c}}-\frac{\sqrt [4]{-1} b^2 d \text{Li}_2\left (1-\frac{\sqrt [4]{-1} \sqrt{2} \left (1+\sqrt [4]{-1} \sqrt{c} x\right )}{1+(-1)^{3/4} \sqrt{c} x}\right )}{2 \sqrt{c}}-\frac{(-1)^{3/4} b^2 d \text{Li}_2\left (1-\frac{(1-i) \left (1+(-1)^{3/4} \sqrt{c} x\right )}{1+\sqrt [4]{-1} \sqrt{c} x}\right )}{2 \sqrt{c}}+2 \frac{\left (\sqrt [4]{-1} b^2 d\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+(-1)^{3/4} \sqrt{c} x}\right )}{\sqrt{c}}+2 \frac{\left ((-1)^{3/4} b^2 d\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+\sqrt [4]{-1} \sqrt{c} x}\right )}{\sqrt{c}}\\ &=\frac{a^2 (d+e x)^2}{2 e}+\frac{(-1)^{3/4} b^2 d \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )^2}{\sqrt{c}}+2 a b d x \tan ^{-1}\left (c x^2\right )+a b e x^2 \tan ^{-1}\left (c x^2\right )+\frac{\sqrt{2} a b d \tan ^{-1}\left (1-\sqrt{2} \sqrt{c} x\right )}{\sqrt{c}}-\frac{\sqrt{2} a b d \tan ^{-1}\left (1+\sqrt{2} \sqrt{c} x\right )}{\sqrt{c}}-\frac{\sqrt [4]{-1} b^2 d \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )^2}{\sqrt{c}}+\frac{2 \sqrt [4]{-1} b^2 d \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (\frac{2}{1-\sqrt [4]{-1} \sqrt{c} x}\right )}{\sqrt{c}}-\frac{2 \sqrt [4]{-1} b^2 d \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (\frac{2}{1+\sqrt [4]{-1} \sqrt{c} x}\right )}{\sqrt{c}}+\frac{\sqrt [4]{-1} b^2 d \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (\frac{\sqrt{2} \left (\sqrt [4]{-1}+\sqrt{c} x\right )}{1+\sqrt [4]{-1} \sqrt{c} x}\right )}{\sqrt{c}}+\frac{2 \sqrt [4]{-1} b^2 d \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (\frac{2}{1-(-1)^{3/4} \sqrt{c} x}\right )}{\sqrt{c}}-\frac{2 \sqrt [4]{-1} b^2 d \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (\frac{2}{1+(-1)^{3/4} \sqrt{c} x}\right )}{\sqrt{c}}+\frac{\sqrt [4]{-1} b^2 d \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (-\frac{\sqrt{2} \left ((-1)^{3/4}+\sqrt{c} x\right )}{1+(-1)^{3/4} \sqrt{c} x}\right )}{\sqrt{c}}+\frac{\sqrt [4]{-1} b^2 d \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (\frac{\sqrt [4]{-1} \sqrt{2} \left (1+\sqrt [4]{-1} \sqrt{c} x\right )}{1+(-1)^{3/4} \sqrt{c} x}\right )}{\sqrt{c}}+\frac{\sqrt [4]{-1} b^2 d \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (\frac{(1-i) \left (1+(-1)^{3/4} \sqrt{c} x\right )}{1+\sqrt [4]{-1} \sqrt{c} x}\right )}{\sqrt{c}}+\frac{\sqrt [4]{-1} b^2 d \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (1-i c x^2\right )}{\sqrt{c}}-\frac{\sqrt [4]{-1} b^2 d \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (1-i c x^2\right )}{\sqrt{c}}-\frac{1}{4} b^2 d x \log ^2\left (1-i c x^2\right )-\frac{i b^2 e \left (1-i c x^2\right ) \log ^2\left (1-i c x^2\right )}{8 c}-\frac{i b^2 e \log \left (1-i c x^2\right ) \log \left (\frac{1}{2} \left (1+i c x^2\right )\right )}{4 c}-\frac{\sqrt [4]{-1} b^2 d \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (1+i c x^2\right )}{\sqrt{c}}+\frac{\sqrt [4]{-1} b^2 d \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (1+i c x^2\right )}{\sqrt{c}}+\frac{i b^2 e \log \left (\frac{1}{2} \left (1-i c x^2\right )\right ) \log \left (1+i c x^2\right )}{4 c}+\frac{1}{2} b^2 d x \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right )+\frac{1}{4} b^2 e x^2 \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right )-\frac{1}{4} b^2 d x \log ^2\left (1+i c x^2\right )+\frac{i b^2 e \left (1+i c x^2\right ) \log ^2\left (1+i c x^2\right )}{8 c}-\frac{a b d \log \left (1-\sqrt{2} \sqrt{c} x+c x^2\right )}{\sqrt{2} \sqrt{c}}+\frac{a b d \log \left (1+\sqrt{2} \sqrt{c} x+c x^2\right )}{\sqrt{2} \sqrt{c}}-\frac{a b e \log \left (1+c^2 x^4\right )}{2 c}-\frac{i b^2 e \text{Li}_2\left (\frac{1}{2} \left (1-i c x^2\right )\right )}{4 c}+\frac{i b^2 e \text{Li}_2\left (\frac{1}{2} \left (1+i c x^2\right )\right )}{4 c}+\frac{(-1)^{3/4} b^2 d \text{Li}_2\left (1-\frac{2}{1-\sqrt [4]{-1} \sqrt{c} x}\right )}{\sqrt{c}}+\frac{(-1)^{3/4} b^2 d \text{Li}_2\left (1-\frac{2}{1+\sqrt [4]{-1} \sqrt{c} x}\right )}{\sqrt{c}}-\frac{(-1)^{3/4} b^2 d \text{Li}_2\left (1-\frac{\sqrt{2} \left (\sqrt [4]{-1}+\sqrt{c} x\right )}{1+\sqrt [4]{-1} \sqrt{c} x}\right )}{2 \sqrt{c}}+\frac{\sqrt [4]{-1} b^2 d \text{Li}_2\left (1-\frac{2}{1-(-1)^{3/4} \sqrt{c} x}\right )}{\sqrt{c}}+\frac{\sqrt [4]{-1} b^2 d \text{Li}_2\left (1-\frac{2}{1+(-1)^{3/4} \sqrt{c} x}\right )}{\sqrt{c}}-\frac{\sqrt [4]{-1} b^2 d \text{Li}_2\left (1+\frac{\sqrt{2} \left ((-1)^{3/4}+\sqrt{c} x\right )}{1+(-1)^{3/4} \sqrt{c} x}\right )}{2 \sqrt{c}}-\frac{\sqrt [4]{-1} b^2 d \text{Li}_2\left (1-\frac{\sqrt [4]{-1} \sqrt{2} \left (1+\sqrt [4]{-1} \sqrt{c} x\right )}{1+(-1)^{3/4} \sqrt{c} x}\right )}{2 \sqrt{c}}-\frac{(-1)^{3/4} b^2 d \text{Li}_2\left (1-\frac{(1-i) \left (1+(-1)^{3/4} \sqrt{c} x\right )}{1+\sqrt [4]{-1} \sqrt{c} x}\right )}{2 \sqrt{c}}\\ \end{align*}
Mathematica [C] time = 32.1986, size = 5593, normalized size = 4.22 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.277, size = 0, normalized size = 0. \begin{align*} \int \left ( ex+d \right ) \left ( a+b\arctan \left ( c{x}^{2} \right ) \right ) ^{2}\, dx \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 (a^{2} e x + a^{2} d +{\left (b^{2} e x + b^{2} d\right )} \arctan \left (c x^{2}\right )^{2} + 2 \,{\left (a b e x + a b d\right )} \arctan \left (c x^{2}\right ), x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \operatorname{atan}{\left (c x^{2} \right )}\right )^{2} \left (d + e x\right )\, dx \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{\left (e x + d\right )}{\left (b \arctan \left (c x^{2}\right ) + a\right )}^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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