3.71 \(\int -\frac{A^2+B^2}{B \left (1+w^2\right )^2 \left (1-\frac{\left (A^2+B^2\right ) w^2}{B^2 \left (1+w^2\right )}\right )} \, dw\)

Optimal. Leaf size=16 \[ -A \tanh ^{-1}\left (\frac{A w}{B}\right )-B \tan ^{-1}(w) \]

[Out]

-(B*ArcTan[w]) - A*ArcTanh[(A*w)/B]

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Rubi [A]  time = 0.199218, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 45, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ -A \tanh ^{-1}\left (\frac{A w}{B}\right )-B \tan ^{-1}(w) \]

Antiderivative was successfully verified.

[In]  Int[-((A^2 + B^2)/(B*(1 + w^2)^2*(1 - ((A^2 + B^2)*w^2)/(B^2*(1 + w^2))))),w]

[Out]

-(B*ArcTan[w]) - A*ArcTanh[(A*w)/B]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{\left (A^{2} + B^{2}\right ) \int \frac{1}{\left (1 - \frac{w^{2} \left (A^{2} + B^{2}\right )}{B^{2} \left (w^{2} + 1\right )}\right ) \left (w^{2} + 1\right )^{2}}\, dw}{B} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-A**2-B**2)/B/(w**2+1)**2/(1-(A**2+B**2)*w**2/B**2/(w**2+1)),w)

[Out]

-(A**2 + B**2)*Integral(1/((1 - w**2*(A**2 + B**2)/(B**2*(w**2 + 1)))*(w**2 + 1)
**2), w)/B

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Mathematica [B]  time = 0.0277822, 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[-((A^2 + B^2)/(B*(1 + w^2)^2*(1 - ((A^2 + B^2)*w^2)/(B^2*(1 + w^2))))),w]

[Out]

-((B*(A^2 + B^2)*(B*ArcTan[w] + A*ArcTanh[(A*w)/B]))/(A^2*B + B^3))

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Maple [B]  time = 0.012, 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((-A^2-B^2)/B/(w^2+1)^2/(1-(A^2+B^2)*w^2/B^2/(w^2+1)),w)

[Out]

-B/(A^2+B^2)*arctan(w)*A^2-1/(A^2+B^2)*arctan(w)*B^3+1/2*A^3/(A^2+B^2)*ln(A*w-B)
+1/2*A*B^2/(A^2+B^2)*ln(A*w-B)-1/2*A^3/(A^2+B^2)*ln(A*w+B)-1/2*A*B^2/(A^2+B^2)*l
n(A*w+B)

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Maxima [A]  time = 1.50595, size = 92, normalized size = 5.75 \[ -\frac{{\left (A^{2} + B^{2}\right )}{\left (\frac{2 \, B^{2} \arctan \left (w\right )}{A^{2} + B^{2}} + \frac{A B \log \left (A w + B\right )}{A^{2} + B^{2}} - \frac{A B \log \left (A w - B\right )}{A^{2} + B^{2}}\right )}}{2 \, B} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((A^2 + B^2)/((w^2 + 1)^2*B*((A^2 + B^2)*w^2/((w^2 + 1)*B^2) - 1)),w, algorithm="maxima")

[Out]

-1/2*(A^2 + B^2)*(2*B^2*arctan(w)/(A^2 + B^2) + A*B*log(A*w + B)/(A^2 + B^2) - A
*B*log(A*w - B)/(A^2 + B^2))/B

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Fricas [A]  time = 0.214495, 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)/((w^2 + 1)^2*B*((A^2 + B^2)*w^2/((w^2 + 1)*B^2) - 1)),w, algorithm="fricas")

[Out]

-B*arctan(w) - 1/2*A*log(A*w + B) + 1/2*A*log(A*w - B)

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Sympy [A]  time = 2.32441, 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((-A**2-B**2)/B/(w**2+1)**2/(1-(A**2+B**2)*w**2/B**2/(w**2+1)),w)

[Out]

(A**2*B + B**3)*(-A*log(w + (-A**9/(B*(A**2 + B**2)**3) - A**7*B/(A**2 + B**2)**
3 + A**5*B**3/(A**2 + B**2)**3 + A**5/(B*(A**2 + B**2)) + A**3*B**5/(A**2 + B**2
)**3 + A*B**3/(A**2 + B**2))/A**2)/(2*B*(A**2 + B**2)) + A*log(w + (A**9/(B*(A**
2 + B**2)**3) + A**7*B/(A**2 + B**2)**3 - A**5*B**3/(A**2 + B**2)**3 - A**5/(B*(
A**2 + B**2)) - A**3*B**5/(A**2 + B**2)**3 - A*B**3/(A**2 + B**2))/A**2)/(2*B*(A
**2 + B**2)) + I*log(w + (-I*A**6*B**2/(A**2 + B**2)**3 - I*A**4*B**4/(A**2 + B*
*2)**3 - I*A**4/(A**2 + B**2) + I*A**2*B**6/(A**2 + B**2)**3 + I*B**8/(A**2 + B*
*2)**3 - I*B**4/(A**2 + B**2))/A**2)/(2*(A**2 + B**2)) - I*log(w + (I*A**6*B**2/
(A**2 + B**2)**3 + I*A**4*B**4/(A**2 + B**2)**3 + I*A**4/(A**2 + B**2) - I*A**2*
B**6/(A**2 + B**2)**3 - I*B**8/(A**2 + B**2)**3 + I*B**4/(A**2 + B**2))/A**2)/(2
*(A**2 + B**2)))

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GIAC/XCAS [A]  time = 0.222453, size = 111, normalized size = 6.94 \[ -\frac{{\left (\frac{A^{3} B{\rm ln}\left ({\left | A w + B \right |}\right )}{A^{4} + A^{2} B^{2}} - \frac{A^{3} B{\rm ln}\left ({\left | A w - B \right |}\right )}{A^{4} + A^{2} B^{2}} + \frac{2 \, B^{2} \arctan \left (w\right )}{A^{2} + B^{2}}\right )}{\left (A^{2} + B^{2}\right )}}{2 \, B} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((A^2 + B^2)/((w^2 + 1)^2*B*((A^2 + B^2)*w^2/((w^2 + 1)*B^2) - 1)),w, algorithm="giac")

[Out]

-1/2*(A^3*B*ln(abs(A*w + B))/(A^4 + A^2*B^2) - A^3*B*ln(abs(A*w - B))/(A^4 + A^2
*B^2) + 2*B^2*arctan(w)/(A^2 + B^2))*(A^2 + B^2)/B