Integrand size = 48, antiderivative size = 16 \[ \int \frac {-A^2-B^2}{B \left (1+w^2\right )^2 \left (1-\frac {\left (A^2+B^2\right ) w^2}{B^2 \left (1+w^2\right )}\right )} \, dw=-B \arctan (w)-A \text {arctanh}\left (\frac {A w}{B}\right ) \] Output:
-B*arctan(w)-A*arctanh(A*w/B)
Leaf count is larger than twice the leaf count of optimal. \(35\) vs. \(2(16)=32\).
Time = 0.01 (sec) , antiderivative size = 35, normalized size of antiderivative = 2.19 \[ \int \frac {-A^2-B^2}{B \left (1+w^2\right )^2 \left (1-\frac {\left (A^2+B^2\right ) w^2}{B^2 \left (1+w^2\right )}\right )} \, dw=-\frac {B \left (A^2+B^2\right ) \left (B \arctan (w)+A \text {arctanh}\left (\frac {A w}{B}\right )\right )}{A^2 B+B^3} \] Input:
Integrate[(-A^2 - B^2)/(B*(1 + w^2)^2*(1 - ((A^2 + B^2)*w^2)/(B^2*(1 + w^2 )))),w]
Output:
-((B*(A^2 + B^2)*(B*ArcTan[w] + A*ArcTanh[(A*w)/B]))/(A^2*B + B^3))
Leaf count is larger than twice the leaf count of optimal. \(44\) vs. \(2(16)=32\).
Time = 0.31 (sec) , antiderivative size = 44, normalized size of antiderivative = 2.75, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {27, 7239, 27, 303, 216, 221}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {-A^2-B^2}{B \left (w^2+1\right )^2 \left (1-\frac {w^2 \left (A^2+B^2\right )}{B^2 \left (w^2+1\right )}\right )} \, dw\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\left (A^2+B^2\right ) \int \frac {1}{\left (w^2+1\right )^2 \left (1-\frac {\left (A^2+B^2\right ) w^2}{B^2 \left (w^2+1\right )}\right )}dw}{B}\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle -\frac {\left (A^2+B^2\right ) \int \frac {B^2}{\left (w^2+1\right ) \left (B^2-A^2 w^2\right )}dw}{B}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -B \left (A^2+B^2\right ) \int \frac {1}{\left (w^2+1\right ) \left (B^2-A^2 w^2\right )}dw\) |
\(\Big \downarrow \) 303 |
\(\displaystyle -B \left (A^2+B^2\right ) \left (\frac {A^2 \int \frac {1}{B^2-A^2 w^2}dw}{A^2+B^2}+\frac {\int \frac {1}{w^2+1}dw}{A^2+B^2}\right )\) |
\(\Big \downarrow \) 216 |
\(\displaystyle -B \left (A^2+B^2\right ) \left (\frac {A^2 \int \frac {1}{B^2-A^2 w^2}dw}{A^2+B^2}+\frac {\arctan (w)}{A^2+B^2}\right )\) |
\(\Big \downarrow \) 221 |
\(\displaystyle -B \left (A^2+B^2\right ) \left (\frac {\arctan (w)}{A^2+B^2}+\frac {A \text {arctanh}\left (\frac {A w}{B}\right )}{B \left (A^2+B^2\right )}\right )\) |
Input:
Int[(-A^2 - B^2)/(B*(1 + w^2)^2*(1 - ((A^2 + B^2)*w^2)/(B^2*(1 + w^2)))),w ]
Output:
-(B*(A^2 + B^2)*(ArcTan[w]/(A^2 + B^2) + (A*ArcTanh[(A*w)/B])/(B*(A^2 + B^ 2))))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
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]
Int[1/(((a_) + (b_.)*(x_)^2)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Simp[b/(b *c - a*d) Int[1/(a + b*x^2), x], x] - Simp[d/(b*c - a*d) Int[1/(c + d*x ^2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Result contains complex when optimal does not.
Time = 0.12 (sec) , antiderivative size = 70, normalized size of antiderivative = 4.38
method | result | size |
parallelrisch | \(\frac {\left (-A^{2}-B^{2}\right ) \left (-i B^{2} \ln \left (-i+w \right )+i B^{2} \ln \left (w +i\right )-A B \ln \left (A w -B \right )+A B \ln \left (A w +B \right )\right )}{2 B \left (A^{2}+B^{2}\right )}\) | \(70\) |
default | \(\left (-A^{2}-B^{2}\right ) B \left (-\frac {A \ln \left (A w -B \right )}{2 B \left (A^{2}+B^{2}\right )}+\frac {A \ln \left (A w +B \right )}{2 B \left (A^{2}+B^{2}\right )}+\frac {\arctan \left (w \right )}{A^{2}+B^{2}}\right )\) | \(71\) |
risch | \(-\frac {A^{3} \ln \left (-A w -B \right )}{2 \left (A^{2}+B^{2}\right )}-\frac {A \ln \left (-A w -B \right ) B^{2}}{2 \left (A^{2}+B^{2}\right )}+\frac {A^{3} \ln \left (-A w +B \right )}{2 A^{2}+2 B^{2}}+\frac {A \ln \left (-A w +B \right ) B^{2}}{2 A^{2}+2 B^{2}}-\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (A^{4}+2 A^{2} B^{2}+B^{4}\right ) \textit {\_Z}^{2}+B^{4}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (-A^{6}-B^{2} A^{4}+A^{2} B^{4}+B^{6}\right ) \textit {\_R}^{2}-2 A^{2} B^{4}\right ) w +\left (-B^{2} A^{4}-2 A^{2} B^{4}-B^{6}\right ) \textit {\_R} \right )\right ) A^{2}}{2 B}-\frac {B \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (A^{4}+2 A^{2} B^{2}+B^{4}\right ) \textit {\_Z}^{2}+B^{4}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (-A^{6}-B^{2} A^{4}+A^{2} B^{4}+B^{6}\right ) \textit {\_R}^{2}-2 A^{2} B^{4}\right ) w +\left (-B^{2} A^{4}-2 A^{2} B^{4}-B^{6}\right ) \textit {\_R} \right )\right )}{2}\) | \(291\) |
Input:
int((-A^2-B^2)/B/(w^2+1)^2/(1-(A^2+B^2)*w^2/B^2/(w^2+1)),w,method=_RETURNV ERBOSE)
Output:
1/2*(-A^2-B^2)/B*(-I*B^2*ln(-I+w)+I*B^2*ln(w+I)-A*B*ln(A*w-B)+A*B*ln(A*w+B ))/(A^2+B^2)
Time = 0.08 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.62 \[ \int \frac {-A^2-B^2}{B \left (1+w^2\right )^2 \left (1-\frac {\left (A^2+B^2\right ) w^2}{B^2 \left (1+w^2\right )}\right )} \, dw=-B \arctan \left (w\right ) - \frac {1}{2} \, A \log \left (A w + B\right ) + \frac {1}{2} \, A \log \left (A w - B\right ) \] Input:
integrate((-A^2-B^2)/B/(w^2+1)^2/(1-(A^2+B^2)*w^2/B^2/(w^2+1)),w, algorith m="fricas")
Output:
-B*arctan(w) - 1/2*A*log(A*w + B) + 1/2*A*log(A*w - B)
Result contains complex when optimal does not.
Time = 0.89 (sec) , antiderivative size = 422, normalized size of antiderivative = 26.38 \[ \int \frac {-A^2-B^2}{B \left (1+w^2\right )^2 \left (1-\frac {\left (A^2+B^2\right ) w^2}{B^2 \left (1+w^2\right )}\right )} \, dw=\left (A^{2} B + B^{3}\right ) \left (- \frac {A \log {\left (w + \frac {- \frac {A^{9}}{B \left (A^{2} + B^{2}\right )^{3}} - \frac {A^{7} B}{\left (A^{2} + B^{2}\right )^{3}} + \frac {A^{5} B^{3}}{\left (A^{2} + B^{2}\right )^{3}} + \frac {A^{5}}{B \left (A^{2} + B^{2}\right )} + \frac {A^{3} B^{5}}{\left (A^{2} + B^{2}\right )^{3}} + \frac {A B^{3}}{A^{2} + B^{2}}}{A^{2}} \right )}}{2 B \left (A^{2} + B^{2}\right )} + \frac {A \log {\left (w + \frac {\frac {A^{9}}{B \left (A^{2} + B^{2}\right )^{3}} + \frac {A^{7} B}{\left (A^{2} + B^{2}\right )^{3}} - \frac {A^{5} B^{3}}{\left (A^{2} + B^{2}\right )^{3}} - \frac {A^{5}}{B \left (A^{2} + B^{2}\right )} - \frac {A^{3} B^{5}}{\left (A^{2} + B^{2}\right )^{3}} - \frac {A B^{3}}{A^{2} + B^{2}}}{A^{2}} \right )}}{2 B \left (A^{2} + B^{2}\right )} + \frac {i \log {\left (w + \frac {- \frac {i A^{6} B^{2}}{\left (A^{2} + B^{2}\right )^{3}} - \frac {i A^{4} B^{4}}{\left (A^{2} + B^{2}\right )^{3}} - \frac {i A^{4}}{A^{2} + B^{2}} + \frac {i A^{2} B^{6}}{\left (A^{2} + B^{2}\right )^{3}} + \frac {i B^{8}}{\left (A^{2} + B^{2}\right )^{3}} - \frac {i B^{4}}{A^{2} + B^{2}}}{A^{2}} \right )}}{2 \left (A^{2} + B^{2}\right )} - \frac {i \log {\left (w + \frac {\frac {i A^{6} B^{2}}{\left (A^{2} + B^{2}\right )^{3}} + \frac {i A^{4} B^{4}}{\left (A^{2} + B^{2}\right )^{3}} + \frac {i A^{4}}{A^{2} + B^{2}} - \frac {i A^{2} B^{6}}{\left (A^{2} + B^{2}\right )^{3}} - \frac {i B^{8}}{\left (A^{2} + B^{2}\right )^{3}} + \frac {i B^{4}}{A^{2} + B^{2}}}{A^{2}} \right )}}{2 \left (A^{2} + B^{2}\right )}\right ) \] Input:
integrate((-A**2-B**2)/B/(w**2+1)**2/(1-(A**2+B**2)*w**2/B**2/(w**2+1)),w)
Output:
(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) ))
Leaf count of result is larger than twice the leaf count of optimal. 68 vs. \(2 (16) = 32\).
Time = 0.11 (sec) , antiderivative size = 68, normalized size of antiderivative = 4.25 \[ \int \frac {-A^2-B^2}{B \left (1+w^2\right )^2 \left (1-\frac {\left (A^2+B^2\right ) w^2}{B^2 \left (1+w^2\right )}\right )} \, dw=-\frac {{\left (A^{2} + B^{2}\right )} {\left (\frac {2 \, B^{2} \arctan \left (w\right )}{A^{2} + B^{2}} + \frac {A B \log \left (A w + B\right )}{A^{2} + B^{2}} - \frac {A B \log \left (A w - B\right )}{A^{2} + B^{2}}\right )}}{2 \, B} \] Input:
integrate((-A^2-B^2)/B/(w^2+1)^2/(1-(A^2+B^2)*w^2/B^2/(w^2+1)),w, algorith m="maxima")
Output:
-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
Leaf count of result is larger than twice the leaf count of optimal. 82 vs. \(2 (16) = 32\).
Time = 0.12 (sec) , antiderivative size = 82, normalized size of antiderivative = 5.12 \[ \int \frac {-A^2-B^2}{B \left (1+w^2\right )^2 \left (1-\frac {\left (A^2+B^2\right ) w^2}{B^2 \left (1+w^2\right )}\right )} \, dw=-\frac {{\left (\frac {A^{3} B \log \left ({\left | A w + B \right |}\right )}{A^{4} + A^{2} B^{2}} - \frac {A^{3} B \log \left ({\left | A w - B \right |}\right )}{A^{4} + A^{2} B^{2}} + \frac {2 \, B^{2} \arctan \left (w\right )}{A^{2} + B^{2}}\right )} {\left (A^{2} + B^{2}\right )}}{2 \, B} \] Input:
integrate((-A^2-B^2)/B/(w^2+1)^2/(1-(A^2+B^2)*w^2/B^2/(w^2+1)),w, algorith m="giac")
Output:
-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
Time = 0.13 (sec) , antiderivative size = 352, normalized size of antiderivative = 22.00 \[ \int \frac {-A^2-B^2}{B \left (1+w^2\right )^2 \left (1-\frac {\left (A^2+B^2\right ) w^2}{B^2 \left (1+w^2\right )}\right )} \, dw=-A\,\mathrm {atanh}\left (\frac {2\,A^{13}\,w}{2\,A^{12}\,B+6\,A^{10}\,B^3+6\,A^8\,B^5+2\,A^6\,B^7}+\frac {2\,A^7\,B^6\,w}{2\,A^{12}\,B+6\,A^{10}\,B^3+6\,A^8\,B^5+2\,A^6\,B^7}+\frac {6\,A^9\,B^4\,w}{2\,A^{12}\,B+6\,A^{10}\,B^3+6\,A^8\,B^5+2\,A^6\,B^7}+\frac {6\,A^{11}\,B^2\,w}{2\,A^{12}\,B+6\,A^{10}\,B^3+6\,A^8\,B^5+2\,A^6\,B^7}\right )-B\,\mathrm {atan}\left (\frac {2\,A^4\,B^9\,w}{2\,A^{10}\,B^3+6\,A^8\,B^5+6\,A^6\,B^7+2\,A^4\,B^9}+\frac {6\,A^6\,B^7\,w}{2\,A^{10}\,B^3+6\,A^8\,B^5+6\,A^6\,B^7+2\,A^4\,B^9}+\frac {6\,A^8\,B^5\,w}{2\,A^{10}\,B^3+6\,A^8\,B^5+6\,A^6\,B^7+2\,A^4\,B^9}+\frac {2\,A^{10}\,B^3\,w}{2\,A^{10}\,B^3+6\,A^8\,B^5+6\,A^6\,B^7+2\,A^4\,B^9}\right ) \] Input:
int((A^2 + B^2)/(B*(w^2 + 1)^2*((w^2*(A^2 + B^2))/(B^2*(w^2 + 1)) - 1)),w)
Output:
- 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^12* 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))
Time = 0.16 (sec) , antiderivative size = 33, normalized size of antiderivative = 2.06 \[ \int \frac {-A^2-B^2}{B \left (1+w^2\right )^2 \left (1-\frac {\left (A^2+B^2\right ) w^2}{B^2 \left (1+w^2\right )}\right )} \, dw=-\mathit {atan} \left (w \right ) b +\frac {\mathrm {log}\left (a^{2} w -a b \right ) a}{2}-\frac {\mathrm {log}\left (a^{2} w +a b \right ) a}{2} \] Input:
int((-A^2-B^2)/B/(w^2+1)^2/(1-(A^2+B^2)*w^2/B^2/(w^2+1)),w)
Output:
( - 2*atan(w)*b + log(a**2*w - a*b)*a - log(a**2*w + a*b)*a)/2