Optimal. Leaf size=102 \[ \frac {i (a+b \text {ArcTan}(c+d x))^2}{d}+\frac {(c+d x) (a+b \text {ArcTan}(c+d x))^2}{d}+\frac {2 b (a+b \text {ArcTan}(c+d x)) \log \left (\frac {2}{1+i (c+d x)}\right )}{d}+\frac {i b^2 \text {PolyLog}\left (2,1-\frac {2}{1+i (c+d x)}\right )}{d} \]
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Rubi [A]
time = 0.08, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5147, 4930,
5040, 4964, 2449, 2352} \begin {gather*} \frac {(c+d x) (a+b \text {ArcTan}(c+d x))^2}{d}+\frac {i (a+b \text {ArcTan}(c+d x))^2}{d}+\frac {2 b \log \left (\frac {2}{1+i (c+d x)}\right ) (a+b \text {ArcTan}(c+d x))}{d}+\frac {i b^2 \text {Li}_2\left (1-\frac {2}{i (c+d x)+1}\right )}{d} \end {gather*}
Antiderivative was successfully verified.
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Rule 2352
Rule 2449
Rule 4930
Rule 4964
Rule 5040
Rule 5147
Rubi steps
\begin {align*} \int \left (a+b \tan ^{-1}(c+d x)\right )^2 \, dx &=\frac {\text {Subst}\left (\int \left (a+b \tan ^{-1}(x)\right )^2 \, dx,x,c+d x\right )}{d}\\ &=\frac {(c+d x) \left (a+b \tan ^{-1}(c+d x)\right )^2}{d}-\frac {(2 b) \text {Subst}\left (\int \frac {x \left (a+b \tan ^{-1}(x)\right )}{1+x^2} \, dx,x,c+d x\right )}{d}\\ &=\frac {i \left (a+b \tan ^{-1}(c+d x)\right )^2}{d}+\frac {(c+d x) \left (a+b \tan ^{-1}(c+d x)\right )^2}{d}+\frac {(2 b) \text {Subst}\left (\int \frac {a+b \tan ^{-1}(x)}{i-x} \, dx,x,c+d x\right )}{d}\\ &=\frac {i \left (a+b \tan ^{-1}(c+d x)\right )^2}{d}+\frac {(c+d x) \left (a+b \tan ^{-1}(c+d x)\right )^2}{d}+\frac {2 b \left (a+b \tan ^{-1}(c+d x)\right ) \log \left (\frac {2}{1+i (c+d x)}\right )}{d}-\frac {\left (2 b^2\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 {i \left (a+b \tan ^{-1}(c+d x)\right )^2}{d}+\frac {(c+d x) \left (a+b \tan ^{-1}(c+d x)\right )^2}{d}+\frac {2 b \left (a+b \tan ^{-1}(c+d x)\right ) \log \left (\frac {2}{1+i (c+d x)}\right )}{d}+\frac {\left (2 i b^2\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i (c+d x)}\right )}{d}\\ &=\frac {i \left (a+b \tan ^{-1}(c+d x)\right )^2}{d}+\frac {(c+d x) \left (a+b \tan ^{-1}(c+d x)\right )^2}{d}+\frac {2 b \left (a+b \tan ^{-1}(c+d x)\right ) \log \left (\frac {2}{1+i (c+d x)}\right )}{d}+\frac {i b^2 \text {Li}_2\left (1-\frac {2}{1+i (c+d x)}\right )}{d}\\ \end {align*}
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Mathematica [A]
time = 0.13, size = 104, normalized size = 1.02 \begin {gather*} \frac {a^2 (c+d x)}{d}+\frac {b \left (b (-i+c+d x) \text {ArcTan}(c+d x)^2+2 \text {ArcTan}(c+d x) \left (a (c+d x)+b \log \left (1+e^{2 i \text {ArcTan}(c+d x)}\right )\right )-a \log \left (1+(c+d x)^2\right )-i b \text {PolyLog}\left (2,-e^{2 i \text {ArcTan}(c+d x)}\right )\right )}{d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 146, normalized size = 1.43
method | result | size |
derivativedivides | \(\frac {\left (d x +c \right ) a^{2}-i \arctan \left (d x +c \right )^{2} b^{2}+\arctan \left (d x +c \right )^{2} b^{2} \left (d x +c \right )-i \polylog \left (2, -\frac {\left (1+i \left (d x +c \right )\right )^{2}}{1+\left (d x +c \right )^{2}}\right ) b^{2}+2 \arctan \left (d x +c \right ) \ln \left (1+\frac {\left (1+i \left (d x +c \right )\right )^{2}}{1+\left (d x +c \right )^{2}}\right ) b^{2}+2 a b \left (d x +c \right ) \arctan \left (d x +c \right )-a b \ln \left (1+\left (d x +c \right )^{2}\right )}{d}\) | \(146\) |
default | \(\frac {\left (d x +c \right ) a^{2}-i \arctan \left (d x +c \right )^{2} b^{2}+\arctan \left (d x +c \right )^{2} b^{2} \left (d x +c \right )-i \polylog \left (2, -\frac {\left (1+i \left (d x +c \right )\right )^{2}}{1+\left (d x +c \right )^{2}}\right ) b^{2}+2 \arctan \left (d x +c \right ) \ln \left (1+\frac {\left (1+i \left (d x +c \right )\right )^{2}}{1+\left (d x +c \right )^{2}}\right ) b^{2}+2 a b \left (d x +c \right ) \arctan \left (d x +c \right )-a b \ln \left (1+\left (d x +c \right )^{2}\right )}{d}\) | \(146\) |
risch | \(-\frac {\ln \left (-i d x -i c +1\right )^{2} x \,b^{2}}{4}+\frac {\ln \left (-i d x -i c +1\right ) x \,b^{2}}{2}-\frac {i b^{2} \ln \left (-i d x -i c +1\right ) \left (-i d x -i c +1\right )}{2 d}+\frac {i b^{2} \ln \left (d^{2} x^{2}+2 c d x +c^{2}+1\right )}{4 d}+\frac {i b^{2} \dilog \left (\frac {1}{2}-\frac {1}{2} i d x -\frac {1}{2} i c \right )}{d}-\frac {b^{2} \left (d x +c -i\right ) \ln \left (1+i \left (d x +c \right )\right )^{2}}{4 d}-\frac {\ln \left (-i d x -i c +1\right )^{2} b^{2} c}{4 d}+\left (\frac {b^{2} x \ln \left (1-i \left (d x +c \right )\right )}{2}-\frac {i b \left (2 a d x -b \ln \left (1-i \left (d x +c \right )\right )+i \ln \left (1-i \left (d x +c \right )\right ) b c \right )}{2 d}\right ) \ln \left (1+i \left (d x +c \right )\right )+\frac {a^{2} c}{d}-\frac {i b^{2} \ln \left (\frac {1}{2} i d x +\frac {1}{2} i c +\frac {1}{2}\right ) \ln \left (-i d x -i c +1\right )}{d}+\frac {b^{2} \arctan \left (d x +c \right )}{2 d}+\frac {2 a b c \arctan \left (d x +c \right )}{d}+a^{2} x -\frac {i b^{2} c \arctan \left (d x +c \right )}{2 d}+\frac {i a^{2}}{d}+i \ln \left (-i d x -i c +1\right ) x a b -\frac {i \ln \left (-i d x -i c +1\right )^{2} b^{2}}{4 d}+\frac {i b^{2} \ln \left (\frac {1}{2} i d x +\frac {1}{2} i c +\frac {1}{2}\right ) \ln \left (\frac {1}{2}-\frac {1}{2} i d x -\frac {1}{2} i c \right )}{d}-\frac {a b \ln \left (d^{2} x^{2}+2 c d x +c^{2}+1\right )}{d}+\frac {b^{2} c \ln \left (d^{2} x^{2}+2 c d x +c^{2}+1\right )}{4 d}\) | \(469\) |
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 + d x \right )}\right )^{2}\, 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+d\,x\right )\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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