Optimal. Leaf size=1325 \[ d x a^2-\frac {2 (-1)^{3/4} b d \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) a}{\sqrt {c}}+\frac {2 (-1)^{3/4} b d \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) a}{\sqrt {c}}+i b d x \log \left (1-i c x^2\right ) a-i b d x \log \left (i c x^2+1\right ) a+\frac {(-1)^{3/4} b^2 d \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )^2}{\sqrt {c}}+\frac {1}{2} e x^2 \left (a+b \tan ^{-1}\left (c x^2\right )\right )^2+\frac {i e \left (a+b \tan ^{-1}\left (c x^2\right )\right )^2}{2 c}-\frac {\sqrt [4]{-1} b^2 d \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )^2}{\sqrt {c}}-\frac {1}{4} b^2 d x \log ^2\left (1-i c x^2\right )-\frac {1}{4} b^2 d x \log ^2\left (i c x^2+1\right )+\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}{\sqrt [4]{-1} \sqrt {c} x+1}\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 {c} x+\sqrt [4]{-1}\right )}{\sqrt [4]{-1} \sqrt {c} x+1}\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)^{3/4} \sqrt {c} x+1}\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 (\sqrt {c} x+(-1)^{3/4}\right )}{(-1)^{3/4} \sqrt {c} x+1}\right )}{\sqrt {c}}+\frac {\sqrt [4]{-1} b^2 d \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {(1+i) \left (\sqrt [4]{-1} \sqrt {c} x+1\right )}{(-1)^{3/4} \sqrt {c} x+1}\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)^{3/4} \sqrt {c} x+1\right )}{\sqrt [4]{-1} \sqrt {c} x+1}\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 {b e \left (a+b \tan ^{-1}\left (c x^2\right )\right ) \log \left (\frac {2}{i c x^2+1}\right )}{c}-\frac {\sqrt [4]{-1} b^2 d \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (i c x^2+1\right )}{\sqrt {c}}+\frac {\sqrt [4]{-1} b^2 d \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (i c x^2+1\right )}{\sqrt {c}}+\frac {1}{2} b^2 d x \log \left (1-i c x^2\right ) \log \left (i c x^2+1\right )+\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}{\sqrt [4]{-1} \sqrt {c} x+1}\right )}{\sqrt {c}}-\frac {(-1)^{3/4} b^2 d \text {Li}_2\left (1-\frac {\sqrt {2} \left (\sqrt {c} x+\sqrt [4]{-1}\right )}{\sqrt [4]{-1} \sqrt {c} x+1}\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)^{3/4} \sqrt {c} x+1}\right )}{\sqrt {c}}-\frac {\sqrt [4]{-1} b^2 d \text {Li}_2\left (\frac {\sqrt {2} \left (\sqrt {c} x+(-1)^{3/4}\right )}{(-1)^{3/4} \sqrt {c} x+1}+1\right )}{2 \sqrt {c}}-\frac {\sqrt [4]{-1} b^2 d \text {Li}_2\left (1-\frac {(1+i) \left (\sqrt [4]{-1} \sqrt {c} x+1\right )}{(-1)^{3/4} \sqrt {c} x+1}\right )}{2 \sqrt {c}}-\frac {(-1)^{3/4} b^2 d \text {Li}_2\left (1-\frac {(1-i) \left ((-1)^{3/4} \sqrt {c} x+1\right )}{\sqrt [4]{-1} \sqrt {c} x+1}\right )}{2 \sqrt {c}}+\frac {i b^2 e \text {Li}_2\left (1-\frac {2}{i c x^2+1}\right )}{2 c} \]
[Out]
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Rubi [A] time = 3.10, 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 12
Rule 30
Rule 43
Rule 203
Rule 204
Rule 205
Rule 206
Rule 208
Rule 260
Rule 297
Rule 321
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 2295
Rule 2296
Rule 2315
Rule 2389
Rule 2391
Rule 2393
Rule 2394
Rule 2402
Rule 2416
Rule 2447
Rule 2448
Rule 2450
Rule 2454
Rule 2470
Rule 2475
Rule 2476
Rule 2556
Rule 2557
Rule 4854
Rule 4856
Rule 4920
Rule 4928
Rule 5027
Rule 5029
Rule 5033
Rule 5035
Rule 5918
Rule 5920
Rule 5984
Rule 5992
Rule 6742
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*}
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Mathematica [C] time = 31.40, size = 5591, normalized size = 4.22 \[ \text {Result too large to show} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.75, size = 0, normalized size = 0.00 \[ {\rm integral}\left (a^{2} e x + a^{2} d + {\left (b^{2} e x + b^{2} d\right )} \arctan \left (c x^{2}\right )^{2} + 2 \, {\left (a b e x + a b d\right )} \arctan \left (c x^{2}\right ), x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (e x + d\right )} {\left (b \arctan \left (c x^{2}\right ) + a\right )}^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.42, size = 0, normalized size = 0.00 \[ \int \left (e x +d \right ) \left (a +b \arctan \left (c \,x^{2}\right )\right )^{2}\, dx \]
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 \[ 12 \, b^{2} c^{2} e \int \frac {x^{5} \arctan \left (c x^{2}\right )^{2}}{16 \, {\left (c^{2} x^{4} + 1\right )}}\,{d x} + b^{2} c^{2} e \int \frac {x^{5} \log \left (c^{2} x^{4} + 1\right )^{2}}{16 \, {\left (c^{2} x^{4} + 1\right )}}\,{d x} + 12 \, b^{2} c^{2} d \int \frac {x^{4} \arctan \left (c x^{2}\right )^{2}}{16 \, {\left (c^{2} x^{4} + 1\right )}}\,{d x} + 4 \, b^{2} c^{2} e \int \frac {x^{5} \log \left (c^{2} x^{4} + 1\right )}{16 \, {\left (c^{2} x^{4} + 1\right )}}\,{d x} + b^{2} c^{2} d \int \frac {x^{4} \log \left (c^{2} x^{4} + 1\right )^{2}}{16 \, {\left (c^{2} x^{4} + 1\right )}}\,{d x} + 8 \, b^{2} c^{2} d \int \frac {x^{4} \log \left (c^{2} x^{4} + 1\right )}{16 \, {\left (c^{2} x^{4} + 1\right )}}\,{d x} + \frac {1}{2} \, a^{2} e x^{2} + \frac {b^{2} e \arctan \left (c x^{2}\right )^{3}}{8 \, c} - 8 \, b^{2} c e \int \frac {x^{3} \arctan \left (c x^{2}\right )}{16 \, {\left (c^{2} x^{4} + 1\right )}}\,{d x} - 16 \, b^{2} c d \int \frac {x^{2} \arctan \left (c x^{2}\right )}{16 \, {\left (c^{2} x^{4} + 1\right )}}\,{d x} - \frac {1}{2} \, {\left (c {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (2 \, c x + \sqrt {2} \sqrt {c}\right )}}{2 \, \sqrt {c}}\right )}{c^{\frac {3}{2}}} + \frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (2 \, c x - \sqrt {2} \sqrt {c}\right )}}{2 \, \sqrt {c}}\right )}{c^{\frac {3}{2}}} - \frac {\sqrt {2} \log \left (c x^{2} + \sqrt {2} \sqrt {c} x + 1\right )}{c^{\frac {3}{2}}} + \frac {\sqrt {2} \log \left (c x^{2} - \sqrt {2} \sqrt {c} x + 1\right )}{c^{\frac {3}{2}}}\right )} - 4 \, x \arctan \left (c x^{2}\right )\right )} a b d + a^{2} d x + b^{2} e \int \frac {x \log \left (c^{2} x^{4} + 1\right )^{2}}{16 \, {\left (c^{2} x^{4} + 1\right )}}\,{d x} + 12 \, b^{2} d \int \frac {\arctan \left (c x^{2}\right )^{2}}{16 \, {\left (c^{2} x^{4} + 1\right )}}\,{d x} + b^{2} d \int \frac {\log \left (c^{2} x^{4} + 1\right )^{2}}{16 \, {\left (c^{2} x^{4} + 1\right )}}\,{d x} + \frac {{\left (2 \, c x^{2} \arctan \left (c x^{2}\right ) - \log \left (c^{2} x^{4} + 1\right )\right )} a b e}{2 \, c} + \frac {1}{8} \, {\left (b^{2} e x^{2} + 2 \, b^{2} d x\right )} \arctan \left (c x^{2}\right )^{2} - \frac {1}{32} \, {\left (b^{2} e x^{2} + 2 \, b^{2} d x\right )} \log \left (c^{2} x^{4} + 1\right )^{2} \]
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 \[ \int {\left (a+b\,\mathrm {atan}\left (c\,x^2\right )\right )}^2\,\left (d+e\,x\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \operatorname {atan}{\left (c x^{2} \right )}\right )^{2} \left (d + e x\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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