Optimal. Leaf size=171 \[ -\frac {a b e x}{c}-\frac {b^2 e x \text {ArcTan}(c x)}{c}+\frac {i d (a+b \text {ArcTan}(c x))^2}{c}-\frac {\left (d^2-\frac {e^2}{c^2}\right ) (a+b \text {ArcTan}(c x))^2}{2 e}+\frac {(d+e x)^2 (a+b \text {ArcTan}(c x))^2}{2 e}+\frac {2 b d (a+b \text {ArcTan}(c x)) \log \left (\frac {2}{1+i c x}\right )}{c}+\frac {b^2 e \log \left (1+c^2 x^2\right )}{2 c^2}+\frac {i b^2 d \text {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{c} \]
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Rubi [A]
time = 0.20, antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 9, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.562, Rules used = {4974, 4930,
266, 5104, 5004, 5040, 4964, 2449, 2352} \begin {gather*} -\frac {\left (d^2-\frac {e^2}{c^2}\right ) (a+b \text {ArcTan}(c x))^2}{2 e}+\frac {(d+e x)^2 (a+b \text {ArcTan}(c x))^2}{2 e}+\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 {a b e x}{c}-\frac {b^2 e x \text {ArcTan}(c x)}{c}+\frac {b^2 e \log \left (c^2 x^2+1\right )}{2 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 266
Rule 2352
Rule 2449
Rule 4930
Rule 4964
Rule 4974
Rule 5004
Rule 5040
Rule 5104
Rubi steps
\begin {align*} \int (d+e x) \left (a+b \tan ^{-1}(c x)\right )^2 \, dx &=\frac {(d+e x)^2 \left (a+b \tan ^{-1}(c x)\right )^2}{2 e}-\frac {(b c) \int \left (\frac {e^2 \left (a+b \tan ^{-1}(c x)\right )}{c^2}+\frac {\left (c^2 d^2-e^2+2 c^2 d e x\right ) \left (a+b \tan ^{-1}(c x)\right )}{c^2 \left (1+c^2 x^2\right )}\right ) \, dx}{e}\\ &=\frac {(d+e x)^2 \left (a+b \tan ^{-1}(c x)\right )^2}{2 e}-\frac {b \int \frac {\left (c^2 d^2-e^2+2 c^2 d e x\right ) \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{c e}-\frac {(b e) \int \left (a+b \tan ^{-1}(c x)\right ) \, dx}{c}\\ &=-\frac {a b e x}{c}+\frac {(d+e x)^2 \left (a+b \tan ^{-1}(c x)\right )^2}{2 e}-\frac {b \int \left (\frac {c^2 d^2 \left (1-\frac {e^2}{c^2 d^2}\right ) \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2}+\frac {2 c^2 d e x \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2}\right ) \, dx}{c e}-\frac {\left (b^2 e\right ) \int \tan ^{-1}(c x) \, dx}{c}\\ &=-\frac {a b e x}{c}-\frac {b^2 e x \tan ^{-1}(c x)}{c}+\frac {(d+e x)^2 \left (a+b \tan ^{-1}(c x)\right )^2}{2 e}-(2 b c d) \int \frac {x \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx+\left (b^2 e\right ) \int \frac {x}{1+c^2 x^2} \, dx-\frac {(b (c d-e) (c d+e)) \int \frac {a+b \tan ^{-1}(c x)}{1+c^2 x^2} \, dx}{c e}\\ &=-\frac {a b e x}{c}-\frac {b^2 e x \tan ^{-1}(c x)}{c}+\frac {i d \left (a+b \tan ^{-1}(c x)\right )^2}{c}-\frac {\left (d^2-\frac {e^2}{c^2}\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{2 e}+\frac {(d+e x)^2 \left (a+b \tan ^{-1}(c x)\right )^2}{2 e}+\frac {b^2 e \log \left (1+c^2 x^2\right )}{2 c^2}+(2 b d) \int \frac {a+b \tan ^{-1}(c x)}{i-c x} \, dx\\ &=-\frac {a b e x}{c}-\frac {b^2 e x \tan ^{-1}(c x)}{c}+\frac {i d \left (a+b \tan ^{-1}(c x)\right )^2}{c}-\frac {\left (d^2-\frac {e^2}{c^2}\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{2 e}+\frac {(d+e x)^2 \left (a+b \tan ^{-1}(c x)\right )^2}{2 e}+\frac {2 b d \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{c}+\frac {b^2 e \log \left (1+c^2 x^2\right )}{2 c^2}-\left (2 b^2 d\right ) \int \frac {\log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx\\ &=-\frac {a b e x}{c}-\frac {b^2 e x \tan ^{-1}(c x)}{c}+\frac {i d \left (a+b \tan ^{-1}(c x)\right )^2}{c}-\frac {\left (d^2-\frac {e^2}{c^2}\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{2 e}+\frac {(d+e x)^2 \left (a+b \tan ^{-1}(c x)\right )^2}{2 e}+\frac {2 b d \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{c}+\frac {b^2 e \log \left (1+c^2 x^2\right )}{2 c^2}+\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 {a b e x}{c}-\frac {b^2 e x \tan ^{-1}(c x)}{c}+\frac {i d \left (a+b \tan ^{-1}(c x)\right )^2}{c}-\frac {\left (d^2-\frac {e^2}{c^2}\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{2 e}+\frac {(d+e x)^2 \left (a+b \tan ^{-1}(c x)\right )^2}{2 e}+\frac {2 b d \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{c}+\frac {b^2 e \log \left (1+c^2 x^2\right )}{2 c^2}+\frac {i b^2 d \text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{c}\\ \end {align*}
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Mathematica [A]
time = 0.14, size = 172, normalized size = 1.01 \begin {gather*} \frac {2 a^2 c^2 d x-2 a b c e x+a^2 c^2 e x^2+b^2 (-i+c x) (2 c d+i e+c e x) \text {ArcTan}(c x)^2+2 b \text {ArcTan}(c x) \left (-b c e x+a \left (e+2 c^2 d x+c^2 e x^2\right )+2 b c d \log \left (1+e^{2 i \text {ArcTan}(c x)}\right )\right )-2 a b c d \log \left (1+c^2 x^2\right )+b^2 e \log \left (1+c^2 x^2\right )-2 i b^2 c d \text {PolyLog}\left (2,-e^{2 i \text {ArcTan}(c x)}\right )}{2 c^2} \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. 339 vs. \(2 (161 ) = 322\).
time = 0.12, size = 340, normalized size = 1.99
method | result | size |
derivativedivides | \(\frac {\frac {a^{2} \left (d \,c^{2} x +\frac {1}{2} e \,c^{2} x^{2}\right )}{c}+b^{2} \arctan \left (c x \right )^{2} d c x +\frac {b^{2} c \arctan \left (c x \right )^{2} e \,x^{2}}{2}-b^{2} \ln \left (c^{2} x^{2}+1\right ) \arctan \left (c x \right ) d +\frac {b^{2} \arctan \left (c x \right )^{2} e}{2 c}-b^{2} \arctan \left (c x \right ) e x +\frac {b^{2} e \ln \left (c^{2} x^{2}+1\right )}{2 c}+\frac {i b^{2} d \ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )}{2}-\frac {i b^{2} d \dilog \left (\frac {i \left (c x -i\right )}{2}\right )}{2}-\frac {i b^{2} d \ln \left (c x +i\right )^{2}}{4}-\frac {i b^{2} d \ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )}{2}+\frac {i b^{2} d \ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )}{2}-\frac {i b^{2} d \ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )}{2}+\frac {i b^{2} d \dilog \left (-\frac {i \left (c x +i\right )}{2}\right )}{2}+\frac {i b^{2} d \ln \left (c x -i\right )^{2}}{4}+2 a b \arctan \left (c x \right ) d c x +a b c \arctan \left (c x \right ) e \,x^{2}-a b e x -a b d \ln \left (c^{2} x^{2}+1\right )+\frac {a b e \arctan \left (c x \right )}{c}}{c}\) | \(340\) |
default | \(\frac {\frac {a^{2} \left (d \,c^{2} x +\frac {1}{2} e \,c^{2} x^{2}\right )}{c}+b^{2} \arctan \left (c x \right )^{2} d c x +\frac {b^{2} c \arctan \left (c x \right )^{2} e \,x^{2}}{2}-b^{2} \ln \left (c^{2} x^{2}+1\right ) \arctan \left (c x \right ) d +\frac {b^{2} \arctan \left (c x \right )^{2} e}{2 c}-b^{2} \arctan \left (c x \right ) e x +\frac {b^{2} e \ln \left (c^{2} x^{2}+1\right )}{2 c}+\frac {i b^{2} d \ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )}{2}-\frac {i b^{2} d \dilog \left (\frac {i \left (c x -i\right )}{2}\right )}{2}-\frac {i b^{2} d \ln \left (c x +i\right )^{2}}{4}-\frac {i b^{2} d \ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )}{2}+\frac {i b^{2} d \ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )}{2}-\frac {i b^{2} d \ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )}{2}+\frac {i b^{2} d \dilog \left (-\frac {i \left (c x +i\right )}{2}\right )}{2}+\frac {i b^{2} d \ln \left (c x -i\right )^{2}}{4}+2 a b \arctan \left (c x \right ) d c x +a b c \arctan \left (c x \right ) e \,x^{2}-a b e x -a b d \ln \left (c^{2} x^{2}+1\right )+\frac {a b e \arctan \left (c x \right )}{c}}{c}\) | \(340\) |
risch | \(a^{2} d x +\frac {7 b^{2} e \ln \left (c^{2} x^{2}+1\right )}{16 c^{2}}-\frac {a b e x}{c}+\frac {e \,a^{2}}{2 c^{2}}+\frac {e b a \arctan \left (c x \right )}{c^{2}}-\frac {a b d \ln \left (c^{2} x^{2}+1\right )}{c}+\frac {a^{2} e \,x^{2}}{2}+\frac {i b^{2} \ln \left (\frac {1}{2}-\frac {i c x}{2}\right ) \ln \left (\frac {1}{2}+\frac {i c x}{2}\right ) d}{c}+i \ln \left (-i c x +1\right ) x a b d -\frac {i b^{2} \ln \left (-i c x +1\right ) \ln \left (\frac {1}{2}+\frac {i c x}{2}\right ) d}{c}+\frac {i e b a \ln \left (-i c x +1\right ) x^{2}}{2}-\frac {i e \,b^{2} \ln \left (-i c x +1\right ) x}{4 c}-\frac {i b^{2} \ln \left (-i c x +1\right ) \left (-i c x +1\right ) d}{2 c}+\frac {i d \,a^{2}}{c}-\frac {e \,b^{2} \ln \left (-i c x +1\right )^{2}}{8 c^{2}}-\frac {e \,b^{2} \ln \left (-i c x +1\right )^{2} x^{2}}{8}+\frac {e \,b^{2} \ln \left (-i c x +1\right ) x^{2}}{8}-\frac {\ln \left (-i c x +1\right )^{2} x \,b^{2} d}{4}-\frac {b^{2} \left (e \,c^{2} x^{2}+2 d \,c^{2} x -2 i c d +e \right ) \ln \left (i c x +1\right )^{2}}{8 c^{2}}+\frac {\ln \left (-i c x +1\right ) x \,b^{2} d}{2}+\frac {b^{2} d \arctan \left (c x \right )}{2 c}+\frac {i b^{2} \dilog \left (\frac {1}{2}-\frac {i c x}{2}\right ) d}{c}+\frac {i b^{2} d \ln \left (c^{2} x^{2}+1\right )}{4 c}+\frac {i e \,b^{2} \arctan \left (c x \right )}{8 c^{2}}-\frac {i e b a}{c^{2}}+\frac {b^{2} \ln \left (-i c x +1\right ) \left (-i c x +1\right )^{2} e}{8 c^{2}}-\frac {i \ln \left (-i c x +1\right )^{2} b^{2} d}{4 c}+\left (\frac {b^{2} x \left (e x +2 d \right ) \ln \left (-i c x +1\right )}{4}-\frac {i b \left (2 a \,c^{2} e \,x^{2}+4 a \,c^{2} d x -2 \ln \left (-i c x +1\right ) b c d -2 b c e x +i \ln \left (-i c x +1\right ) b e \right )}{4 c^{2}}\right ) \ln \left (i c x +1\right )\) | \(557\) |
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\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 = -0.01 \begin {gather*} \int {\left (a+b\,\mathrm {atan}\left (c\,x\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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