Optimal. Leaf size=231 \[ \frac {b^2 e x}{3 c^2}-\frac {b^2 e \text {ArcTan}(c x)}{3 c^3}-\frac {b e x^2 (a+b \text {ArcTan}(c x))}{3 c}+\frac {i d (a+b \text {ArcTan}(c x))^2}{c}-\frac {i e (a+b \text {ArcTan}(c x))^2}{3 c^3}+d x (a+b \text {ArcTan}(c x))^2+\frac {1}{3} e x^3 (a+b \text {ArcTan}(c x))^2+\frac {2 b d (a+b \text {ArcTan}(c x)) \log \left (\frac {2}{1+i c x}\right )}{c}-\frac {2 b e (a+b \text {ArcTan}(c x)) \log \left (\frac {2}{1+i c x}\right )}{3 c^3}+\frac {i b^2 d \text {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{c}-\frac {i b^2 e \text {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{3 c^3} \]
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Rubi [A]
time = 0.25, antiderivative size = 231, normalized size of antiderivative = 1.00, number of steps
used = 16, number of rules used = 10, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {5034, 4930,
5040, 4964, 2449, 2352, 4946, 5036, 327, 209} \begin {gather*} -\frac {i e (a+b \text {ArcTan}(c x))^2}{3 c^3}-\frac {2 b e \log \left (\frac {2}{1+i c x}\right ) (a+b \text {ArcTan}(c x))}{3 c^3}+d x (a+b \text {ArcTan}(c x))^2+\frac {i d (a+b \text {ArcTan}(c x))^2}{c}+\frac {2 b d \log \left (\frac {2}{1+i c x}\right ) (a+b \text {ArcTan}(c x))}{c}+\frac {1}{3} e x^3 (a+b \text {ArcTan}(c x))^2-\frac {b e x^2 (a+b \text {ArcTan}(c x))}{3 c}-\frac {b^2 e \text {ArcTan}(c x)}{3 c^3}-\frac {i b^2 e \text {Li}_2\left (1-\frac {2}{i c x+1}\right )}{3 c^3}+\frac {b^2 e x}{3 c^2}+\frac {i b^2 d \text {Li}_2\left (1-\frac {2}{i c x+1}\right )}{c} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 327
Rule 2352
Rule 2449
Rule 4930
Rule 4946
Rule 4964
Rule 5034
Rule 5036
Rule 5040
Rubi steps
\begin {align*} \int \left (d+e x^2\right ) \left (a+b \tan ^{-1}(c x)\right )^2 \, dx &=\int \left (d \left (a+b \tan ^{-1}(c x)\right )^2+e x^2 \left (a+b \tan ^{-1}(c x)\right )^2\right ) \, dx\\ &=d \int \left (a+b \tan ^{-1}(c x)\right )^2 \, dx+e \int x^2 \left (a+b \tan ^{-1}(c x)\right )^2 \, dx\\ &=d x \left (a+b \tan ^{-1}(c x)\right )^2+\frac {1}{3} e x^3 \left (a+b \tan ^{-1}(c x)\right )^2-(2 b c d) \int \frac {x \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx-\frac {1}{3} (2 b c e) \int \frac {x^3 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx\\ &=\frac {i d \left (a+b \tan ^{-1}(c x)\right )^2}{c}+d x \left (a+b \tan ^{-1}(c x)\right )^2+\frac {1}{3} e x^3 \left (a+b \tan ^{-1}(c x)\right )^2+(2 b d) \int \frac {a+b \tan ^{-1}(c x)}{i-c x} \, dx-\frac {(2 b e) \int x \left (a+b \tan ^{-1}(c x)\right ) \, dx}{3 c}+\frac {(2 b e) \int \frac {x \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{3 c}\\ &=-\frac {b e x^2 \left (a+b \tan ^{-1}(c x)\right )}{3 c}+\frac {i d \left (a+b \tan ^{-1}(c x)\right )^2}{c}-\frac {i e \left (a+b \tan ^{-1}(c x)\right )^2}{3 c^3}+d x \left (a+b \tan ^{-1}(c x)\right )^2+\frac {1}{3} e x^3 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {2 b d \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{c}-\left (2 b^2 d\right ) \int \frac {\log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx+\frac {1}{3} \left (b^2 e\right ) \int \frac {x^2}{1+c^2 x^2} \, dx-\frac {(2 b e) \int \frac {a+b \tan ^{-1}(c x)}{i-c x} \, dx}{3 c^2}\\ &=\frac {b^2 e x}{3 c^2}-\frac {b e x^2 \left (a+b \tan ^{-1}(c x)\right )}{3 c}+\frac {i d \left (a+b \tan ^{-1}(c x)\right )^2}{c}-\frac {i e \left (a+b \tan ^{-1}(c x)\right )^2}{3 c^3}+d x \left (a+b \tan ^{-1}(c x)\right )^2+\frac {1}{3} e x^3 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {2 b d \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{c}-\frac {2 b e \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{3 c^3}+\frac {\left (2 i b^2 d\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i c x}\right )}{c}-\frac {\left (b^2 e\right ) \int \frac {1}{1+c^2 x^2} \, dx}{3 c^2}+\frac {\left (2 b^2 e\right ) \int \frac {\log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{3 c^2}\\ &=\frac {b^2 e x}{3 c^2}-\frac {b^2 e \tan ^{-1}(c x)}{3 c^3}-\frac {b e x^2 \left (a+b \tan ^{-1}(c x)\right )}{3 c}+\frac {i d \left (a+b \tan ^{-1}(c x)\right )^2}{c}-\frac {i e \left (a+b \tan ^{-1}(c x)\right )^2}{3 c^3}+d x \left (a+b \tan ^{-1}(c x)\right )^2+\frac {1}{3} e x^3 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {2 b d \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{c}-\frac {2 b e \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{3 c^3}+\frac {i b^2 d \text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{c}-\frac {\left (2 i b^2 e\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i c x}\right )}{3 c^3}\\ &=\frac {b^2 e x}{3 c^2}-\frac {b^2 e \tan ^{-1}(c x)}{3 c^3}-\frac {b e x^2 \left (a+b \tan ^{-1}(c x)\right )}{3 c}+\frac {i d \left (a+b \tan ^{-1}(c x)\right )^2}{c}-\frac {i e \left (a+b \tan ^{-1}(c x)\right )^2}{3 c^3}+d x \left (a+b \tan ^{-1}(c x)\right )^2+\frac {1}{3} e x^3 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {2 b d \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{c}-\frac {2 b e \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{3 c^3}+\frac {i b^2 d \text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{c}-\frac {i b^2 e \text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{3 c^3}\\ \end {align*}
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Mathematica [A]
time = 0.29, size = 208, normalized size = 0.90 \begin {gather*} \frac {3 a^2 c^3 d x+b^2 c e x-a b c^2 e x^2+a^2 c^3 e x^3+b^2 \left (-3 i c^2 d+i e+c^3 \left (3 d x+e x^3\right )\right ) \text {ArcTan}(c x)^2-b \text {ArcTan}(c x) \left (-2 a c^3 x \left (3 d+e x^2\right )+b \left (e+c^2 e x^2\right )+2 b \left (-3 c^2 d+e\right ) \log \left (1+e^{2 i \text {ArcTan}(c x)}\right )\right )-3 a b c^2 d \log \left (1+c^2 x^2\right )+a b e \log \left (1+c^2 x^2\right )-i b^2 \left (3 c^2 d-e\right ) \text {PolyLog}\left (2,-e^{2 i \text {ArcTan}(c x)}\right )}{3 c^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 549 vs. \(2 (209 ) = 418\).
time = 0.31, size = 550, normalized size = 2.38 Too large to display
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 \right )}\right )^{2} \left (d + e x^{2}\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\right )\right )}^2\,\left (e\,x^2+d\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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