Optimal. Leaf size=16 \[ -A \tanh ^{-1}\left (\frac{A w}{B}\right )-B \tan ^{-1}(w) \]
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Rubi [A] time = 0.0186529, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.129, Rules used = {12, 391, 203, 208} \[ -A \tanh ^{-1}\left (\frac{A w}{B}\right )-B \tan ^{-1}(w) \]
Antiderivative was successfully verified.
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Rule 12
Rule 391
Rule 203
Rule 208
Rubi steps
\begin{align*} \int -\frac{B \left (A^2+B^2\right )}{\left (1+w^2\right ) \left (B^2-A^2 w^2\right )} \, dw &=-\left (\left (B \left (A^2+B^2\right )\right ) \int \frac{1}{\left (1+w^2\right ) \left (B^2-A^2 w^2\right )} \, dw\right )\\ &=-\left (B \int \frac{1}{1+w^2} \, dw\right )-\left (A^2 B\right ) \int \frac{1}{B^2-A^2 w^2} \, dw\\ &=-B \tan ^{-1}(w)-A \tanh ^{-1}\left (\frac{A w}{B}\right )\\ \end{align*}
Mathematica [B] time = 0.0130182, size = 35, normalized size = 2.19 \[ -\frac{B \left (A^2+B^2\right ) \left (A \tanh ^{-1}\left (\frac{A w}{B}\right )+B \tan ^{-1}(w)\right )}{A^2 B+B^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.005, size = 121, normalized size = 7.6 \begin{align*}{\frac{{A}^{3}\ln \left ( Aw-B \right ) }{2\,{A}^{2}+2\,{B}^{2}}}+{\frac{A{B}^{2}\ln \left ( Aw-B \right ) }{2\,{A}^{2}+2\,{B}^{2}}}-{\frac{B\arctan \left ( w \right ){A}^{2}}{{A}^{2}+{B}^{2}}}-{\frac{\arctan \left ( w \right ){B}^{3}}{{A}^{2}+{B}^{2}}}-{\frac{{A}^{3}\ln \left ( Aw+B \right ) }{2\,{A}^{2}+2\,{B}^{2}}}-{\frac{A{B}^{2}\ln \left ( Aw+B \right ) }{2\,{A}^{2}+2\,{B}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.41032, size = 88, normalized size = 5.5 \begin{align*} -\frac{1}{2} \,{\left (A^{2} + B^{2}\right )} B{\left (\frac{A \log \left (A w + B\right )}{A^{2} B + B^{3}} - \frac{A \log \left (A w - B\right )}{A^{2} B + B^{3}} + \frac{2 \, \arctan \left (w\right )}{A^{2} + B^{2}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.43177, size = 76, normalized size = 4.75 \begin{align*} -B \arctan \left (w\right ) - \frac{1}{2} \, A \log \left (A w + B\right ) + \frac{1}{2} \, A \log \left (A w - B\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 1.27477, size = 422, normalized size = 26.38 \begin{align*} \left (A^{2} B + B^{3}\right ) \left (- \frac{A \log{\left (w + \frac{- \frac{A^{9}}{B \left (A^{2} + B^{2}\right )^{3}} - \frac{A^{7} B}{\left (A^{2} + B^{2}\right )^{3}} + \frac{A^{5} B^{3}}{\left (A^{2} + B^{2}\right )^{3}} + \frac{A^{5}}{B \left (A^{2} + B^{2}\right )} + \frac{A^{3} B^{5}}{\left (A^{2} + B^{2}\right )^{3}} + \frac{A B^{3}}{A^{2} + B^{2}}}{A^{2}} \right )}}{2 B \left (A^{2} + B^{2}\right )} + \frac{A \log{\left (w + \frac{\frac{A^{9}}{B \left (A^{2} + B^{2}\right )^{3}} + \frac{A^{7} B}{\left (A^{2} + B^{2}\right )^{3}} - \frac{A^{5} B^{3}}{\left (A^{2} + B^{2}\right )^{3}} - \frac{A^{5}}{B \left (A^{2} + B^{2}\right )} - \frac{A^{3} B^{5}}{\left (A^{2} + B^{2}\right )^{3}} - \frac{A B^{3}}{A^{2} + B^{2}}}{A^{2}} \right )}}{2 B \left (A^{2} + B^{2}\right )} + \frac{i \log{\left (w + \frac{- \frac{i A^{6} B^{2}}{\left (A^{2} + B^{2}\right )^{3}} - \frac{i A^{4} B^{4}}{\left (A^{2} + B^{2}\right )^{3}} - \frac{i A^{4}}{A^{2} + B^{2}} + \frac{i A^{2} B^{6}}{\left (A^{2} + B^{2}\right )^{3}} + \frac{i B^{8}}{\left (A^{2} + B^{2}\right )^{3}} - \frac{i B^{4}}{A^{2} + B^{2}}}{A^{2}} \right )}}{2 \left (A^{2} + B^{2}\right )} - \frac{i \log{\left (w + \frac{\frac{i A^{6} B^{2}}{\left (A^{2} + B^{2}\right )^{3}} + \frac{i A^{4} B^{4}}{\left (A^{2} + B^{2}\right )^{3}} + \frac{i A^{4}}{A^{2} + B^{2}} - \frac{i A^{2} B^{6}}{\left (A^{2} + B^{2}\right )^{3}} - \frac{i B^{8}}{\left (A^{2} + B^{2}\right )^{3}} + \frac{i B^{4}}{A^{2} + B^{2}}}{A^{2}} \right )}}{2 \left (A^{2} + B^{2}\right )}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.07565, size = 107, normalized size = 6.69 \begin{align*} -\frac{1}{2} \,{\left (\frac{A^{3} \log \left ({\left | A w + B \right |}\right )}{A^{4} B + A^{2} B^{3}} - \frac{A^{3} \log \left ({\left | A w - B \right |}\right )}{A^{4} B + A^{2} B^{3}} + \frac{2 \, \arctan \left (w\right )}{A^{2} + B^{2}}\right )}{\left (A^{2} + B^{2}\right )} B \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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