Optimal. Leaf size=164 \[ -\frac {3 i b e (a+b \text {ArcTan}(c+d x))^2}{2 d}-\frac {3 b e (c+d x) (a+b \text {ArcTan}(c+d x))^2}{2 d}+\frac {e (a+b \text {ArcTan}(c+d x))^3}{2 d}+\frac {e (c+d x)^2 (a+b \text {ArcTan}(c+d x))^3}{2 d}-\frac {3 b^2 e (a+b \text {ArcTan}(c+d x)) \log \left (\frac {2}{1+i (c+d x)}\right )}{d}-\frac {3 i b^3 e \text {PolyLog}\left (2,1-\frac {2}{1+i (c+d x)}\right )}{2 d} \]
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Rubi [A]
time = 0.18, antiderivative size = 164, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 10, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {5151, 12,
4946, 5036, 4930, 5040, 4964, 2449, 2352, 5004} \begin {gather*} -\frac {3 b^2 e \log \left (\frac {2}{1+i (c+d x)}\right ) (a+b \text {ArcTan}(c+d x))}{d}-\frac {3 i b e (a+b \text {ArcTan}(c+d x))^2}{2 d}-\frac {3 b e (c+d x) (a+b \text {ArcTan}(c+d x))^2}{2 d}+\frac {e (c+d x)^2 (a+b \text {ArcTan}(c+d x))^3}{2 d}+\frac {e (a+b \text {ArcTan}(c+d x))^3}{2 d}-\frac {3 i b^3 e \text {Li}_2\left (1-\frac {2}{i (c+d x)+1}\right )}{2 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2352
Rule 2449
Rule 4930
Rule 4946
Rule 4964
Rule 5004
Rule 5036
Rule 5040
Rule 5151
Rubi steps
\begin {align*} \int (c e+d e x) \left (a+b \tan ^{-1}(c+d x)\right )^3 \, dx &=\frac {\text {Subst}\left (\int e x \left (a+b \tan ^{-1}(x)\right )^3 \, dx,x,c+d x\right )}{d}\\ &=\frac {e \text {Subst}\left (\int x \left (a+b \tan ^{-1}(x)\right )^3 \, dx,x,c+d x\right )}{d}\\ &=\frac {e (c+d x)^2 \left (a+b \tan ^{-1}(c+d x)\right )^3}{2 d}-\frac {(3 b e) \text {Subst}\left (\int \frac {x^2 \left (a+b \tan ^{-1}(x)\right )^2}{1+x^2} \, dx,x,c+d x\right )}{2 d}\\ &=\frac {e (c+d x)^2 \left (a+b \tan ^{-1}(c+d x)\right )^3}{2 d}-\frac {(3 b e) \text {Subst}\left (\int \left (a+b \tan ^{-1}(x)\right )^2 \, dx,x,c+d x\right )}{2 d}+\frac {(3 b e) \text {Subst}\left (\int \frac {\left (a+b \tan ^{-1}(x)\right )^2}{1+x^2} \, dx,x,c+d x\right )}{2 d}\\ &=-\frac {3 b e (c+d x) \left (a+b \tan ^{-1}(c+d x)\right )^2}{2 d}+\frac {e \left (a+b \tan ^{-1}(c+d x)\right )^3}{2 d}+\frac {e (c+d x)^2 \left (a+b \tan ^{-1}(c+d x)\right )^3}{2 d}+\frac {\left (3 b^2 e\right ) \text {Subst}\left (\int \frac {x \left (a+b \tan ^{-1}(x)\right )}{1+x^2} \, dx,x,c+d x\right )}{d}\\ &=-\frac {3 i b e \left (a+b \tan ^{-1}(c+d x)\right )^2}{2 d}-\frac {3 b e (c+d x) \left (a+b \tan ^{-1}(c+d x)\right )^2}{2 d}+\frac {e \left (a+b \tan ^{-1}(c+d x)\right )^3}{2 d}+\frac {e (c+d x)^2 \left (a+b \tan ^{-1}(c+d x)\right )^3}{2 d}-\frac {\left (3 b^2 e\right ) \text {Subst}\left (\int \frac {a+b \tan ^{-1}(x)}{i-x} \, dx,x,c+d x\right )}{d}\\ &=-\frac {3 i b e \left (a+b \tan ^{-1}(c+d x)\right )^2}{2 d}-\frac {3 b e (c+d x) \left (a+b \tan ^{-1}(c+d x)\right )^2}{2 d}+\frac {e \left (a+b \tan ^{-1}(c+d x)\right )^3}{2 d}+\frac {e (c+d x)^2 \left (a+b \tan ^{-1}(c+d x)\right )^3}{2 d}-\frac {3 b^2 e \left (a+b \tan ^{-1}(c+d x)\right ) \log \left (\frac {2}{1+i (c+d x)}\right )}{d}+\frac {\left (3 b^3 e\right ) \text {Subst}\left (\int \frac {\log \left (\frac {2}{1+i x}\right )}{1+x^2} \, dx,x,c+d x\right )}{d}\\ &=-\frac {3 i b e \left (a+b \tan ^{-1}(c+d x)\right )^2}{2 d}-\frac {3 b e (c+d x) \left (a+b \tan ^{-1}(c+d x)\right )^2}{2 d}+\frac {e \left (a+b \tan ^{-1}(c+d x)\right )^3}{2 d}+\frac {e (c+d x)^2 \left (a+b \tan ^{-1}(c+d x)\right )^3}{2 d}-\frac {3 b^2 e \left (a+b \tan ^{-1}(c+d x)\right ) \log \left (\frac {2}{1+i (c+d x)}\right )}{d}-\frac {\left (3 i b^3 e\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i (c+d x)}\right )}{d}\\ &=-\frac {3 i b e \left (a+b \tan ^{-1}(c+d x)\right )^2}{2 d}-\frac {3 b e (c+d x) \left (a+b \tan ^{-1}(c+d x)\right )^2}{2 d}+\frac {e \left (a+b \tan ^{-1}(c+d x)\right )^3}{2 d}+\frac {e (c+d x)^2 \left (a+b \tan ^{-1}(c+d x)\right )^3}{2 d}-\frac {3 b^2 e \left (a+b \tan ^{-1}(c+d x)\right ) \log \left (\frac {2}{1+i (c+d x)}\right )}{d}-\frac {3 i b^3 e \text {Li}_2\left (1-\frac {2}{1+i (c+d x)}\right )}{2 d}\\ \end {align*}
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Mathematica [A]
time = 0.24, size = 196, normalized size = 1.20 \begin {gather*} \frac {e \left (3 b^2 (-i+c+d x) (-b+a (i+c+d x)) \text {ArcTan}(c+d x)^2+b^3 \left (1+c^2+2 c d x+d^2 x^2\right ) \text {ArcTan}(c+d x)^3+3 b \text {ArcTan}(c+d x) \left (a \left (-2 b (c+d x)+a \left (1+c^2+2 c d x+d^2 x^2\right )\right )-2 b^2 \log \left (1+e^{2 i \text {ArcTan}(c+d x)}\right )\right )+a \left (a (c+d x) (-3 b+a c+a d x)-6 b^2 \log \left (\frac {1}{\sqrt {1+(c+d x)^2}}\right )\right )+3 i b^3 \text {PolyLog}\left (2,-e^{2 i \text {ArcTan}(c+d x)}\right )\right )}{2 d} \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. 383 vs. \(2 (150 ) = 300\).
time = 0.27, size = 384, normalized size = 2.34
method | result | size |
derivativedivides | \(\frac {\frac {e \left (d x +c \right )^{2} a^{3}}{2}+\frac {e \,b^{3} \left (d x +c \right )^{2} \arctan \left (d x +c \right )^{3}}{2}+\frac {e \,b^{3} \arctan \left (d x +c \right )^{3}}{2}-\frac {3 e \,b^{3} \arctan \left (d x +c \right )^{2} \left (d x +c \right )}{2}+\frac {3 e \,b^{3} \arctan \left (d x +c \right ) \ln \left (1+\left (d x +c \right )^{2}\right )}{2}-\frac {3 i e \,b^{3} \ln \left (d x +c -i\right ) \ln \left (-\frac {i \left (d x +c +i\right )}{2}\right )}{4}+\frac {3 i e \,b^{3} \ln \left (d x +c +i\right ) \ln \left (\frac {i \left (d x +c -i\right )}{2}\right )}{4}+\frac {3 i e \,b^{3} \ln \left (d x +c +i\right )^{2}}{8}-\frac {3 i e \,b^{3} \dilog \left (-\frac {i \left (d x +c +i\right )}{2}\right )}{4}-\frac {3 i e \,b^{3} \ln \left (d x +c +i\right ) \ln \left (1+\left (d x +c \right )^{2}\right )}{4}+\frac {3 i e \,b^{3} \dilog \left (\frac {i \left (d x +c -i\right )}{2}\right )}{4}-\frac {3 i e \,b^{3} \ln \left (d x +c -i\right )^{2}}{8}+\frac {3 i e \,b^{3} \ln \left (d x +c -i\right ) \ln \left (1+\left (d x +c \right )^{2}\right )}{4}+\frac {3 e a \,b^{2} \left (d x +c \right )^{2} \arctan \left (d x +c \right )^{2}}{2}+\frac {3 e a \,b^{2} \arctan \left (d x +c \right )^{2}}{2}-3 e a \,b^{2} \left (d x +c \right ) \arctan \left (d x +c \right )+\frac {3 e a \,b^{2} \ln \left (1+\left (d x +c \right )^{2}\right )}{2}+\frac {3 e \,a^{2} b \left (d x +c \right )^{2} \arctan \left (d x +c \right )}{2}-\frac {3 e \left (d x +c \right ) a^{2} b}{2}+\frac {3 e \,a^{2} b \arctan \left (d x +c \right )}{2}}{d}\) | \(384\) |
default | \(\frac {\frac {e \left (d x +c \right )^{2} a^{3}}{2}+\frac {e \,b^{3} \left (d x +c \right )^{2} \arctan \left (d x +c \right )^{3}}{2}+\frac {e \,b^{3} \arctan \left (d x +c \right )^{3}}{2}-\frac {3 e \,b^{3} \arctan \left (d x +c \right )^{2} \left (d x +c \right )}{2}+\frac {3 e \,b^{3} \arctan \left (d x +c \right ) \ln \left (1+\left (d x +c \right )^{2}\right )}{2}-\frac {3 i e \,b^{3} \ln \left (d x +c -i\right ) \ln \left (-\frac {i \left (d x +c +i\right )}{2}\right )}{4}+\frac {3 i e \,b^{3} \ln \left (d x +c +i\right ) \ln \left (\frac {i \left (d x +c -i\right )}{2}\right )}{4}+\frac {3 i e \,b^{3} \ln \left (d x +c +i\right )^{2}}{8}-\frac {3 i e \,b^{3} \dilog \left (-\frac {i \left (d x +c +i\right )}{2}\right )}{4}-\frac {3 i e \,b^{3} \ln \left (d x +c +i\right ) \ln \left (1+\left (d x +c \right )^{2}\right )}{4}+\frac {3 i e \,b^{3} \dilog \left (\frac {i \left (d x +c -i\right )}{2}\right )}{4}-\frac {3 i e \,b^{3} \ln \left (d x +c -i\right )^{2}}{8}+\frac {3 i e \,b^{3} \ln \left (d x +c -i\right ) \ln \left (1+\left (d x +c \right )^{2}\right )}{4}+\frac {3 e a \,b^{2} \left (d x +c \right )^{2} \arctan \left (d x +c \right )^{2}}{2}+\frac {3 e a \,b^{2} \arctan \left (d x +c \right )^{2}}{2}-3 e a \,b^{2} \left (d x +c \right ) \arctan \left (d x +c \right )+\frac {3 e a \,b^{2} \ln \left (1+\left (d x +c \right )^{2}\right )}{2}+\frac {3 e \,a^{2} b \left (d x +c \right )^{2} \arctan \left (d x +c \right )}{2}-\frac {3 e \left (d x +c \right ) a^{2} b}{2}+\frac {3 e \,a^{2} b \arctan \left (d x +c \right )}{2}}{d}\) | \(384\) |
risch | \(\text {Expression too large to display}\) | \(1488\) |
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*} e \left (\int a^{3} c\, dx + \int a^{3} d x\, dx + \int b^{3} c \operatorname {atan}^{3}{\left (c + d x \right )}\, dx + \int 3 a b^{2} c \operatorname {atan}^{2}{\left (c + d x \right )}\, dx + \int 3 a^{2} b c \operatorname {atan}{\left (c + d x \right )}\, dx + \int b^{3} d x \operatorname {atan}^{3}{\left (c + d x \right )}\, dx + \int 3 a b^{2} d x \operatorname {atan}^{2}{\left (c + d x \right )}\, dx + \int 3 a^{2} b d x \operatorname {atan}{\left (c + d x \right )}\, dx\right ) \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 (c\,e+d\,e\,x\right )\,{\left (a+b\,\mathrm {atan}\left (c+d\,x\right )\right )}^3 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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