Optimal. Leaf size=16 \[ -A \tanh ^{-1}\left (\frac{A w}{B}\right )-B \tan ^{-1}(w) \]
[Out]
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Rubi [A] time = 0.0405668, 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 \[ -A \tanh ^{-1}\left (\frac{A w}{B}\right )-B \tan ^{-1}(w) \]
Antiderivative was successfully verified.
[In] Int[-((B*(A^2 + B^2))/((1 + w^2)*(B^2 - A^2*w^2))),w]
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Rubi in Sympy [A] time = 9.15516, size = 14, normalized size = 0.88 \[ - A \operatorname{atanh}{\left (\frac{A w}{B} \right )} - B \operatorname{atan}{\left (w \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(-B*(A**2+B**2)/(w**2+1)/(-A**2*w**2+B**2),w)
[Out]
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Mathematica [B] time = 0.0199897, 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.
[In] Integrate[-((B*(A^2 + B^2))/((1 + w^2)*(B^2 - A^2*w^2))),w]
[Out]
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Maple [B] time = 0.01, size = 121, normalized size = 7.6 \[ -{\frac{B\arctan \left ( w \right ){A}^{2}}{{A}^{2}+{B}^{2}}}-{\frac{\arctan \left ( w \right ){B}^{3}}{{A}^{2}+{B}^{2}}}+{\frac{{A}^{3}\ln \left ( Aw-B \right ) }{2\,{A}^{2}+2\,{B}^{2}}}+{\frac{A{B}^{2}\ln \left ( Aw-B \right ) }{2\,{A}^{2}+2\,{B}^{2}}}-{\frac{{A}^{3}\ln \left ( Aw+B \right ) }{2\,{A}^{2}+2\,{B}^{2}}}-{\frac{A{B}^{2}\ln \left ( Aw+B \right ) }{2\,{A}^{2}+2\,{B}^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(-B*(A^2+B^2)/(w^2+1)/(-A^2*w^2+B^2),w)
[Out]
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Maxima [A] time = 1.50452, size = 88, normalized size = 5.5 \[ -\frac{1}{2} \,{\left (A^{2} + B^{2}\right )} B{\left (\frac{A \log \left (A w + B\right )}{A^{2} B + B^{3}} - \frac{A \log \left (A w - B\right )}{A^{2} B + B^{3}} + \frac{2 \, \arctan \left (w\right )}{A^{2} + B^{2}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((A^2 + B^2)*B/((A^2*w^2 - B^2)*(w^2 + 1)),w, algorithm="maxima")
[Out]
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Fricas [A] time = 0.215967, size = 35, normalized size = 2.19 \[ -B \arctan \left (w\right ) - \frac{1}{2} \, A \log \left (A w + B\right ) + \frac{1}{2} \, A \log \left (A w - B\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((A^2 + B^2)*B/((A^2*w^2 - B^2)*(w^2 + 1)),w, algorithm="fricas")
[Out]
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Sympy [A] time = 2.27451, size = 422, normalized size = 26.38 \[ \left (A^{2} B + B^{3}\right ) \left (- \frac{A \log{\left (w + \frac{- \frac{A^{9}}{B \left (A^{2} + B^{2}\right )^{3}} - \frac{A^{7} B}{\left (A^{2} + B^{2}\right )^{3}} + \frac{A^{5} B^{3}}{\left (A^{2} + B^{2}\right )^{3}} + \frac{A^{5}}{B \left (A^{2} + B^{2}\right )} + \frac{A^{3} B^{5}}{\left (A^{2} + B^{2}\right )^{3}} + \frac{A B^{3}}{A^{2} + B^{2}}}{A^{2}} \right )}}{2 B \left (A^{2} + B^{2}\right )} + \frac{A \log{\left (w + \frac{\frac{A^{9}}{B \left (A^{2} + B^{2}\right )^{3}} + \frac{A^{7} B}{\left (A^{2} + B^{2}\right )^{3}} - \frac{A^{5} B^{3}}{\left (A^{2} + B^{2}\right )^{3}} - \frac{A^{5}}{B \left (A^{2} + B^{2}\right )} - \frac{A^{3} B^{5}}{\left (A^{2} + B^{2}\right )^{3}} - \frac{A B^{3}}{A^{2} + B^{2}}}{A^{2}} \right )}}{2 B \left (A^{2} + B^{2}\right )} + \frac{i \log{\left (w + \frac{- \frac{i A^{6} B^{2}}{\left (A^{2} + B^{2}\right )^{3}} - \frac{i A^{4} B^{4}}{\left (A^{2} + B^{2}\right )^{3}} - \frac{i A^{4}}{A^{2} + B^{2}} + \frac{i A^{2} B^{6}}{\left (A^{2} + B^{2}\right )^{3}} + \frac{i B^{8}}{\left (A^{2} + B^{2}\right )^{3}} - \frac{i B^{4}}{A^{2} + B^{2}}}{A^{2}} \right )}}{2 \left (A^{2} + B^{2}\right )} - \frac{i \log{\left (w + \frac{\frac{i A^{6} B^{2}}{\left (A^{2} + B^{2}\right )^{3}} + \frac{i A^{4} B^{4}}{\left (A^{2} + B^{2}\right )^{3}} + \frac{i A^{4}}{A^{2} + B^{2}} - \frac{i A^{2} B^{6}}{\left (A^{2} + B^{2}\right )^{3}} - \frac{i B^{8}}{\left (A^{2} + B^{2}\right )^{3}} + \frac{i B^{4}}{A^{2} + B^{2}}}{A^{2}} \right )}}{2 \left (A^{2} + B^{2}\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-B*(A**2+B**2)/(w**2+1)/(-A**2*w**2+B**2),w)
[Out]
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GIAC/XCAS [A] time = 0.208557, size = 107, normalized size = 6.69 \[ -\frac{1}{2} \,{\left (\frac{A^{3}{\rm ln}\left ({\left | A w + B \right |}\right )}{A^{4} B + A^{2} B^{3}} - \frac{A^{3}{\rm ln}\left ({\left | A w - B \right |}\right )}{A^{4} B + A^{2} B^{3}} + \frac{2 \, \arctan \left (w\right )}{A^{2} + B^{2}}\right )}{\left (A^{2} + B^{2}\right )} B \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((A^2 + B^2)*B/((A^2*w^2 - B^2)*(w^2 + 1)),w, algorithm="giac")
[Out]