Optimal. Leaf size=102 \[ \frac{i b^2 \text{PolyLog}\left (2,1-\frac{2}{1+i (c+d x)}\right )}{d}+\frac{(c+d x) \left (a+b \tan ^{-1}(c+d x)\right )^2}{d}+\frac{i \left (a+b \tan ^{-1}(c+d x)\right )^2}{d}+\frac{2 b \log \left (\frac{2}{1+i (c+d x)}\right ) \left (a+b \tan ^{-1}(c+d x)\right )}{d} \]
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Rubi [A] time = 0.106977, antiderivative size = 102, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {5039, 4846, 4920, 4854, 2402, 2315} \[ \frac{i b^2 \text{PolyLog}\left (2,1-\frac{2}{1+i (c+d x)}\right )}{d}+\frac{(c+d x) \left (a+b \tan ^{-1}(c+d x)\right )^2}{d}+\frac{i \left (a+b \tan ^{-1}(c+d x)\right )^2}{d}+\frac{2 b \log \left (\frac{2}{1+i (c+d x)}\right ) \left (a+b \tan ^{-1}(c+d x)\right )}{d} \]
Antiderivative was successfully verified.
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Rule 5039
Rule 4846
Rule 4920
Rule 4854
Rule 2402
Rule 2315
Rubi steps
\begin{align*} \int \left (a+b \tan ^{-1}(c+d x)\right )^2 \, dx &=\frac{\operatorname{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) \operatorname{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) \operatorname{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 ) \operatorname{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 ) \operatorname{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*}
Mathematica [A] time = 0.0730931, size = 109, normalized size = 1.07 \[ \frac{-i b^2 \text{PolyLog}\left (2,-e^{2 i \tan ^{-1}(c+d x)}\right )+a \left (a c+a d x+2 b \log \left (\frac{1}{\sqrt{(c+d x)^2+1}}\right )\right )+2 b \tan ^{-1}(c+d x) \left (a c+a d x+b \log \left (1+e^{2 i \tan ^{-1}(c+d x)}\right )\right )+b^2 (c+d x-i) \tan ^{-1}(c+d x)^2}{d} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.105, size = 180, normalized size = 1.8 \begin{align*} \left ( \arctan \left ( dx+c \right ) \right ) ^{2}x{b}^{2}-{\frac{i \left ( \arctan \left ( dx+c \right ) \right ) ^{2}{b}^{2}}{d}}+{\frac{ \left ( \arctan \left ( dx+c \right ) \right ) ^{2}{b}^{2}c}{d}}+2\,\arctan \left ( dx+c \right ) xab+2\,{\frac{\arctan \left ( dx+c \right ){b}^{2}}{d}\ln \left ({\frac{ \left ( 1+i \left ( dx+c \right ) \right ) ^{2}}{1+ \left ( dx+c \right ) ^{2}}}+1 \right ) }+2\,{\frac{\arctan \left ( dx+c \right ) abc}{d}}-{\frac{i{b}^{2}}{d}{\it polylog} \left ( 2,-{\frac{ \left ( 1+i \left ( dx+c \right ) \right ) ^{2}}{1+ \left ( dx+c \right ) ^{2}}} \right ) }+{a}^{2}x-{\frac{ab\ln \left ( 1+ \left ( dx+c \right ) ^{2} \right ) }{d}}+{\frac{{a}^{2}c}{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (b^{2} \arctan \left (d x + c\right )^{2} + 2 \, a b \arctan \left (d x + c\right ) + a^{2}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \operatorname{atan}{\left (c + d x \right )}\right )^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \arctan \left (d x + c\right ) + a\right )}^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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