3.1.71 \(\int \frac {-A^2-B^2}{B (1+w^2)^2 (1-\frac {(A^2+B^2) w^2}{B^2 (1+w^2)})} \, dw\) [71]

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

[Out]

-B*arctan(w)-A*arctanh(A*w/B)

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Rubi [A]
time = 0.08, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 48, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.104, Rules used = {12, 6820, 400, 209, 214} \begin {gather*} -A \tanh ^{-1}\left (\frac {A w}{B}\right )-B \tan ^{-1}(w) \end {gather*}

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]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rule 400

Int[1/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^(n_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x^n),
x], x] - Dist[d/(b*c - a*d), Int[1/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0]

Rule 6820

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rubi steps

\begin {align*} \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 &=-\frac {\left (A^2+B^2\right ) \int \frac {1}{\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}{B}\\ &=-\frac {\left (A^2+B^2\right ) \int \frac {B^2}{\left (1+w^2\right ) \left (B^2-A^2 w^2\right )} \, dw}{B}\\ &=-\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*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(35\) vs. \(2(16)=32\).
time = 0.01, size = 35, normalized size = 2.19 \begin {gather*} -\frac {B \left (A^2+B^2\right ) \left (B \tan ^{-1}(w)+A \tanh ^{-1}\left (\frac {A w}{B}\right )\right )}{A^2 B+B^3} \end {gather*}

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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Mathics [C] Result contains complex when optimal does not.
time = 9.03, size = 43, normalized size = 2.69 \begin {gather*} \frac {A \left (\text {Log}\left [\frac {A w-B}{A}\right ]-\text {Log}\left [\frac {A w+B}{A}\right ]\right )}{2}+\frac {I}{2} B \left (\text {Log}\left [-I+w\right ]-\text {Log}\left [I+w\right ]\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[-(A^2 + B^2)/(B*(1 + w^2)^2*(1 - ((A^2 + B^2)/B^2)*(w^2/(1 + w^2)))),w]')

[Out]

A (Log[(A w - B) / A] - Log[(A w + B) / A]) / 2 + I / 2 B (Log[-I + w] - Log[I + w])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(70\) vs. \(2(16)=32\).
time = 0.06, size = 71, normalized size = 4.44

method	result	size
default	\(\left (-A^{2}-B^{2}\right ) B \left (\frac {\arctan \left (w \right )}{A^{2}+B^{2}}-\frac {A \ln \left (A w -B \right )}{2 B \left (A^{2}+B^{2}\right )}+\frac {A \ln \left (A w +B \right )}{2 B \left (A^{2}+B^{2}\right )}\right )\)	\(71\)
risch	\(-\frac {A^{3} \ln \left (-A w -B \right )}{2 \left (A^{2}+B^{2}\right )}-\frac {A \ln \left (-A w -B \right ) B^{2}}{2 \left (A^{2}+B^{2}\right )}+\frac {A^{3} \ln \left (-A w +B \right )}{2 A^{2}+2 B^{2}}+\frac {A \ln \left (-A w +B \right ) B^{2}}{2 A^{2}+2 B^{2}}-\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\left (A^{4}+2 A^{2} B^{2}+B^{4}\right ) \textit {\_Z}^{2}+B^{4}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (-A^{6}-B^{2} A^{4}+A^{2} B^{4}+B^{6}\right ) \textit {\_R}^{2}-2 A^{2} B^{4}\right ) w +\left (-B^{2} A^{4}-2 A^{2} B^{4}-B^{6}\right ) \textit {\_R} \right )\right ) A^{2}}{2 B}-\frac {B \left (\munderset {\textit {\_R} =\RootOf \left (\left (A^{4}+2 A^{2} B^{2}+B^{4}\right ) \textit {\_Z}^{2}+B^{4}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (-A^{6}-B^{2} A^{4}+A^{2} B^{4}+B^{6}\right ) \textit {\_R}^{2}-2 A^{2} B^{4}\right ) w +\left (-B^{2} A^{4}-2 A^{2} B^{4}-B^{6}\right ) \textit {\_R} \right )\right )}{2}\)	\(291\)

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,method=_RETURNVERBOSE)

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 68 vs. \(2 (16) = 32\).
time = 0.36, size = 68, normalized size = 4.25 \begin {gather*} -\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} \end {gather*}

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, 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 = 0.32, size = 26, normalized size = 1.62 \begin {gather*} -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 {gather*}

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, 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 [C] Result contains complex when optimal does not.
time = 1.09, size = 422, normalized size = 26.38 \begin {gather*} \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 {gather*}

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*l
og(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 [B] Leaf count of result is larger than twice the leaf count of optimal. 82 vs. \(2 (16) = 32\).
time = 0.00, size = 89, normalized size = 5.56 \begin {gather*} \frac {\left (-A^{2}-B^{2}\right ) \left (-\frac {A^{3} B \ln \left |w A-B\right |}{2 A^{4}+2 A^{2} B^{2}}-\frac {A^{3} B \ln \left |w A+B\right |}{-2 A^{4}-2 A^{2} B^{2}}+\frac {2 B^{2} \arctan w}{2 \left (A^{2}+B^{2}\right )}\right )}{B} \end {gather*}

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]

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

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Mupad [B]
time = 0.24, size = 352, normalized size = 22.00 \begin {gather*} -A\,\mathrm {atanh}\left (\frac {2\,A^{13}\,w}{2\,A^{12}\,B+6\,A^{10}\,B^3+6\,A^8\,B^5+2\,A^6\,B^7}+\frac {2\,A^7\,B^6\,w}{2\,A^{12}\,B+6\,A^{10}\,B^3+6\,A^8\,B^5+2\,A^6\,B^7}+\frac {6\,A^9\,B^4\,w}{2\,A^{12}\,B+6\,A^{10}\,B^3+6\,A^8\,B^5+2\,A^6\,B^7}+\frac {6\,A^{11}\,B^2\,w}{2\,A^{12}\,B+6\,A^{10}\,B^3+6\,A^8\,B^5+2\,A^6\,B^7}\right )-B\,\mathrm {atan}\left (\frac {2\,A^4\,B^9\,w}{2\,A^{10}\,B^3+6\,A^8\,B^5+6\,A^6\,B^7+2\,A^4\,B^9}+\frac {6\,A^6\,B^7\,w}{2\,A^{10}\,B^3+6\,A^8\,B^5+6\,A^6\,B^7+2\,A^4\,B^9}+\frac {6\,A^8\,B^5\,w}{2\,A^{10}\,B^3+6\,A^8\,B^5+6\,A^6\,B^7+2\,A^4\,B^9}+\frac {2\,A^{10}\,B^3\,w}{2\,A^{10}\,B^3+6\,A^8\,B^5+6\,A^6\,B^7+2\,A^4\,B^9}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- A*atanh((2*A^13*w)/(2*A^12*B + 2*A^6*B^7 + 6*A^8*B^5 + 6*A^10*B^3) + (2*A^7*B^6*w)/(2*A^12*B + 2*A^6*B^7 + 6
*A^8*B^5 + 6*A^10*B^3) + (6*A^9*B^4*w)/(2*A^12*B + 2*A^6*B^7 + 6*A^8*B^5 + 6*A^10*B^3) + (6*A^11*B^2*w)/(2*A^1
2*B + 2*A^6*B^7 + 6*A^8*B^5 + 6*A^10*B^3)) - B*atan((2*A^4*B^9*w)/(2*A^4*B^9 + 6*A^6*B^7 + 6*A^8*B^5 + 2*A^10*
B^3) + (6*A^6*B^7*w)/(2*A^4*B^9 + 6*A^6*B^7 + 6*A^8*B^5 + 2*A^10*B^3) + (6*A^8*B^5*w)/(2*A^4*B^9 + 6*A^6*B^7 +
6*A^8*B^5 + 2*A^10*B^3) + (2*A^10*B^3*w)/(2*A^4*B^9 + 6*A^6*B^7 + 6*A^8*B^5 + 2*A^10*B^3))

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