Optimal. Leaf size=1325 \[ a^2 d x-\frac {2 (-1)^{3/4} a b d \text {ArcTan}\left ((-1)^{3/4} \sqrt {c} x\right )}{\sqrt {c}}+\frac {(-1)^{3/4} b^2 d \text {ArcTan}\left ((-1)^{3/4} \sqrt {c} x\right )^2}{\sqrt {c}}+\frac {i e \left (a+b \text {ArcTan}\left (c x^2\right )\right )^2}{2 c}+\frac {1}{2} e x^2 \left (a+b \text {ArcTan}\left (c x^2\right )\right )^2+\frac {2 (-1)^{3/4} a b 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 {2 \sqrt [4]{-1} b^2 d \text {ArcTan}\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 \text {ArcTan}\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 \text {ArcTan}\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 {(1+i) \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 \text {ArcTan}\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}}+i a b d x \log \left (1-i c x^2\right )+\frac {\sqrt [4]{-1} b^2 d \text {ArcTan}\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 {b e \left (a+b \text {ArcTan}\left (c x^2\right )\right ) \log \left (\frac {2}{1+i c x^2}\right )}{c}-i a b d x \log \left (1+i c x^2\right )-\frac {\sqrt [4]{-1} b^2 d \text {ArcTan}\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 d x \log ^2\left (1+i c x^2\right )+\frac {(-1)^{3/4} b^2 d \text {PolyLog}\left (2,1-\frac {2}{1-\sqrt [4]{-1} \sqrt {c} x}\right )}{\sqrt {c}}+\frac {(-1)^{3/4} b^2 d \text {PolyLog}\left (2,1-\frac {2}{1+\sqrt [4]{-1} \sqrt {c} x}\right )}{\sqrt {c}}-\frac {(-1)^{3/4} b^2 d \text {PolyLog}\left (2,1-\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 {PolyLog}\left (2,1-\frac {2}{1-(-1)^{3/4} \sqrt {c} x}\right )}{\sqrt {c}}+\frac {\sqrt [4]{-1} b^2 d \text {PolyLog}\left (2,1-\frac {2}{1+(-1)^{3/4} \sqrt {c} x}\right )}{\sqrt {c}}-\frac {\sqrt [4]{-1} b^2 d \text {PolyLog}\left (2,1+\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 {PolyLog}\left (2,1-\frac {(1+i) \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 {PolyLog}\left (2,1-\frac {(1-i) \left (1+(-1)^{3/4} \sqrt {c} x\right )}{1+\sqrt [4]{-1} \sqrt {c} x}\right )}{2 \sqrt {c}}+\frac {i b^2 e \text {PolyLog}\left (2,1-\frac {2}{1+i c x^2}\right )}{2 c} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 1.62, antiderivative size = 1325, normalized size of antiderivative = 1.00, number of steps
used = 77, number of rules used = 26, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.444, Rules used = {4982, 4932,
2498, 327, 209, 2500, 2526, 2520, 12, 5040, 4964, 2449, 2352, 212, 2636, 211, 5048, 4966, 2497,
214, 6139, 6057, 6131, 6055, 4948, 4930} \begin {gather*} d x a^2-\frac {2 (-1)^{3/4} b d \text {ArcTan}\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 \text {ArcTan}\left ((-1)^{3/4} \sqrt {c} x\right )^2}{\sqrt {c}}+\frac {1}{2} e x^2 \left (a+b \text {ArcTan}\left (c x^2\right )\right )^2+\frac {i e \left (a+b \text {ArcTan}\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 \text {ArcTan}\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 \text {ArcTan}\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 \text {ArcTan}\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 \text {ArcTan}\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 \text {ArcTan}\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 \text {ArcTan}\left (c x^2\right )\right ) \log \left (\frac {2}{i c x^2+1}\right )}{c}-\frac {\sqrt [4]{-1} b^2 d \text {ArcTan}\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} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 209
Rule 211
Rule 212
Rule 214
Rule 327
Rule 2352
Rule 2449
Rule 2497
Rule 2498
Rule 2500
Rule 2520
Rule 2526
Rule 2636
Rule 4930
Rule 4932
Rule 4948
Rule 4964
Rule 4966
Rule 4982
Rule 5040
Rule 5048
Rule 6055
Rule 6057
Rule 6131
Rule 6139
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 ) \text {Subst}\left (\int \log ^2(1-i c x) \, dx,x,x^2\right )-\frac {1}{8} \left (b^2 e\right ) \text {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 ) \text {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 ) \text {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 ) \text {Subst}\left (\int \log ^2(x) \, dx,x,1-i c x^2\right )}{8 c}+\frac {\left (i b^2 e\right ) \text {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 ) \text {Subst}\left (\int \log (x) \, dx,x,1-i c x^2\right )}{4 c}-\frac {\left (i b^2 e\right ) \text {Subst}\left (\int \log (x) \, dx,x,1+i c x^2\right )}{4 c}-\frac {1}{4} \left (b^2 c e\right ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {Subst}\left (\int \log (1-i c x) \, dx,x,x^2\right )-\frac {1}{4} \left (b^2 e\right ) \text {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 ) \text {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 ) \text {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 ) \text {Subst}\left (\int \log (x) \, dx,x,1-i c x^2\right )}{4 c}+\frac {\left (i b^2 e\right ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 [B] Both result and optimal contain complex but leaf count is larger than twice
the leaf count of optimal. \(5420\) vs. \(2(1325)=2650\).
time = 30.64, size = 5420, normalized size = 4.09 \begin {gather*} \text {Result too large to show} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.10, 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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \left (a + b \operatorname {atan}{\left (c x^{2} \right )}\right )^{2} \left (d + e x\right )\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int {\left (a+b\,\mathrm {atan}\left (c\,x^2\right )\right )}^2\,\left (d+e\,x\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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