Integrand size = 16, antiderivative size = 642 \[ \int \frac {\cot ^{-1}(a+b x)}{c+d x^2} \, dx=-\frac {\log \left (\frac {i+a+b x}{a+b x}\right ) \log \left (-\frac {b \left (i \sqrt {c}-\sqrt {d} x\right )}{\left (b \sqrt {c}+(1-i a) \sqrt {d}\right ) (a+b x)}\right )}{4 \sqrt {c} \sqrt {d}}+\frac {\log \left (-\frac {i-a-b x}{a+b x}\right ) \log \left (\frac {i b \left (\sqrt {c}+i \sqrt {d} x\right )}{\left (b \sqrt {c}-(1+i a) \sqrt {d}\right ) (a+b x)}\right )}{4 \sqrt {c} \sqrt {d}}-\frac {\log \left (-\frac {i-a-b x}{a+b x}\right ) \log \left (\frac {b \left (i \sqrt {c}+\sqrt {d} x\right )}{\left (b \sqrt {c}+(1+i a) \sqrt {d}\right ) (a+b x)}\right )}{4 \sqrt {c} \sqrt {d}}+\frac {\log \left (\frac {i+a+b x}{a+b x}\right ) \log \left (-\frac {b \left (i \sqrt {c}+\sqrt {d} x\right )}{\left (b \sqrt {c}+i (i+a) \sqrt {d}\right ) (a+b x)}\right )}{4 \sqrt {c} \sqrt {d}}+\frac {\operatorname {PolyLog}\left (2,-\frac {\left (b \sqrt {c}-i a \sqrt {d}\right ) (i-a-b x)}{\left (b \sqrt {c}-(1+i a) \sqrt {d}\right ) (a+b x)}\right )}{4 \sqrt {c} \sqrt {d}}-\frac {\operatorname {PolyLog}\left (2,-\frac {\left (b \sqrt {c}+i a \sqrt {d}\right ) (i-a-b x)}{\left (b \sqrt {c}+(1+i a) \sqrt {d}\right ) (a+b x)}\right )}{4 \sqrt {c} \sqrt {d}}-\frac {\operatorname {PolyLog}\left (2,\frac {\left (b \sqrt {c}-i a \sqrt {d}\right ) (i+a+b x)}{\left (b \sqrt {c}+(1-i a) \sqrt {d}\right ) (a+b x)}\right )}{4 \sqrt {c} \sqrt {d}}+\frac {\operatorname {PolyLog}\left (2,\frac {\left (b \sqrt {c}+i a \sqrt {d}\right ) (i+a+b x)}{\left (b \sqrt {c}+i (i+a) \sqrt {d}\right ) (a+b x)}\right )}{4 \sqrt {c} \sqrt {d}} \]
-1/4*ln((I+a+b*x)/(b*x+a))*ln(-b*(I*c^(1/2)-x*d^(1/2))/(b*x+a)/(b*c^(1/2)+ (1-I*a)*d^(1/2)))/c^(1/2)/d^(1/2)+1/4*ln((-I+a+b*x)/(b*x+a))*ln(I*b*(c^(1/ 2)+I*x*d^(1/2))/(b*x+a)/(b*c^(1/2)-(1+I*a)*d^(1/2)))/c^(1/2)/d^(1/2)-1/4*l n((-I+a+b*x)/(b*x+a))*ln(b*(I*c^(1/2)+x*d^(1/2))/(b*x+a)/(b*c^(1/2)+(1+I*a )*d^(1/2)))/c^(1/2)/d^(1/2)+1/4*ln((I+a+b*x)/(b*x+a))*ln(-b*(I*c^(1/2)+x*d ^(1/2))/(b*x+a)/(b*c^(1/2)+I*(I+a)*d^(1/2)))/c^(1/2)/d^(1/2)-1/4*polylog(2 ,(I+a+b*x)*(b*c^(1/2)-I*a*d^(1/2))/(b*x+a)/(b*c^(1/2)+(1-I*a)*d^(1/2)))/c^ (1/2)/d^(1/2)+1/4*polylog(2,-(I-a-b*x)*(b*c^(1/2)-I*a*d^(1/2))/(b*x+a)/(b* c^(1/2)-(1+I*a)*d^(1/2)))/c^(1/2)/d^(1/2)-1/4*polylog(2,-(I-a-b*x)*(b*c^(1 /2)+I*a*d^(1/2))/(b*x+a)/(b*c^(1/2)+(1+I*a)*d^(1/2)))/c^(1/2)/d^(1/2)+1/4* polylog(2,(I+a+b*x)*(b*c^(1/2)+I*a*d^(1/2))/(b*x+a)/(b*c^(1/2)+I*(I+a)*d^( 1/2)))/c^(1/2)/d^(1/2)
Time = 0.41 (sec) , antiderivative size = 563, normalized size of antiderivative = 0.88 \[ \int \frac {\cot ^{-1}(a+b x)}{c+d x^2} \, dx=-\frac {i \left (\log \left (\frac {\sqrt {d} (-i+a+b x)}{b \sqrt {-c}+(-i+a) \sqrt {d}}\right ) \log \left (\sqrt {-c}-\sqrt {d} x\right )-\log \left (\frac {-i+a+b x}{a+b x}\right ) \log \left (\sqrt {-c}-\sqrt {d} x\right )-\log \left (\frac {\sqrt {d} (i+a+b x)}{b \sqrt {-c}+(i+a) \sqrt {d}}\right ) \log \left (\sqrt {-c}-\sqrt {d} x\right )+\log \left (\frac {i+a+b x}{a+b x}\right ) \log \left (\sqrt {-c}-\sqrt {d} x\right )-\log \left (-\frac {\sqrt {d} (-i+a+b x)}{b \sqrt {-c}-(-i+a) \sqrt {d}}\right ) \log \left (\sqrt {-c}+\sqrt {d} x\right )+\log \left (\frac {-i+a+b x}{a+b x}\right ) \log \left (\sqrt {-c}+\sqrt {d} x\right )+\log \left (-\frac {\sqrt {d} (i+a+b x)}{b \sqrt {-c}-(i+a) \sqrt {d}}\right ) \log \left (\sqrt {-c}+\sqrt {d} x\right )-\log \left (\frac {i+a+b x}{a+b x}\right ) \log \left (\sqrt {-c}+\sqrt {d} x\right )+\operatorname {PolyLog}\left (2,\frac {b \left (\sqrt {-c}-\sqrt {d} x\right )}{b \sqrt {-c}+(-i+a) \sqrt {d}}\right )-\operatorname {PolyLog}\left (2,\frac {b \left (\sqrt {-c}-\sqrt {d} x\right )}{b \sqrt {-c}+(i+a) \sqrt {d}}\right )-\operatorname {PolyLog}\left (2,\frac {b \left (\sqrt {-c}+\sqrt {d} x\right )}{b \sqrt {-c}-(-i+a) \sqrt {d}}\right )+\operatorname {PolyLog}\left (2,\frac {b \left (\sqrt {-c}+\sqrt {d} x\right )}{b \sqrt {-c}-(i+a) \sqrt {d}}\right )\right )}{4 \sqrt {-c} \sqrt {d}} \]
((-1/4*I)*(Log[(Sqrt[d]*(-I + a + b*x))/(b*Sqrt[-c] + (-I + a)*Sqrt[d])]*L og[Sqrt[-c] - Sqrt[d]*x] - Log[(-I + a + b*x)/(a + b*x)]*Log[Sqrt[-c] - Sq rt[d]*x] - Log[(Sqrt[d]*(I + a + b*x))/(b*Sqrt[-c] + (I + a)*Sqrt[d])]*Log [Sqrt[-c] - Sqrt[d]*x] + Log[(I + a + b*x)/(a + b*x)]*Log[Sqrt[-c] - Sqrt[ d]*x] - Log[-((Sqrt[d]*(-I + a + b*x))/(b*Sqrt[-c] - (-I + a)*Sqrt[d]))]*L og[Sqrt[-c] + Sqrt[d]*x] + Log[(-I + a + b*x)/(a + b*x)]*Log[Sqrt[-c] + Sq rt[d]*x] + Log[-((Sqrt[d]*(I + a + b*x))/(b*Sqrt[-c] - (I + a)*Sqrt[d]))]* Log[Sqrt[-c] + Sqrt[d]*x] - Log[(I + a + b*x)/(a + b*x)]*Log[Sqrt[-c] + Sq rt[d]*x] + PolyLog[2, (b*(Sqrt[-c] - Sqrt[d]*x))/(b*Sqrt[-c] + (-I + a)*Sq rt[d])] - PolyLog[2, (b*(Sqrt[-c] - Sqrt[d]*x))/(b*Sqrt[-c] + (I + a)*Sqrt [d])] - PolyLog[2, (b*(Sqrt[-c] + Sqrt[d]*x))/(b*Sqrt[-c] - (-I + a)*Sqrt[ d])] + PolyLog[2, (b*(Sqrt[-c] + Sqrt[d]*x))/(b*Sqrt[-c] - (I + a)*Sqrt[d] )]))/(Sqrt[-c]*Sqrt[d])
Time = 1.37 (sec) , antiderivative size = 721, normalized size of antiderivative = 1.12, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {5575, 2976, 2804, 2009, 2977, 2804, 2009}
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.
| \(\displaystyle \int \frac {\cot ^{-1}(a+b x)}{c+d x^2} \, dx\) |
| \(\Big \downarrow \) 5575 |
| \(\displaystyle \frac {1}{2} i \int \frac {\log \left (-\frac {-a-b x+i}{a+b x}\right )}{d x^2+c}dx-\frac {1}{2} i \int \frac {\log \left (\frac {a+b x+i}{a+b x}\right )}{d x^2+c}dx\) |
| \(\Big \downarrow \) 2976 |
| \(\displaystyle \frac {1}{2} i \int \frac {\log \left (-\frac {-a-b x+i}{a+b x}\right )}{d x^2+c}dx-\frac {1}{2} b \int \frac {\log \left (\frac {a+b x+i}{a+b x}\right )}{d (a+i)^2+\frac {\left (d a^2+b^2 c\right ) (a+b x+i)^2}{(a+b x)^2}+b^2 c-\frac {2 \left (c b^2+a (a+i) d\right ) (a+b x+i)}{a+b x}}d\frac {a+b x+i}{a+b x}\) |
| \(\Big \downarrow \) 2804 |
| \(\displaystyle \frac {1}{2} i \int \frac {\log \left (-\frac {-a-b x+i}{a+b x}\right )}{d x^2+c}dx-\frac {1}{2} b \int \left (\frac {\left (d a^2+b^2 c\right ) \log \left (\frac {a+b x+i}{a+b x}\right )}{b \sqrt {c} \sqrt {d} \left (2 d a^2+2 i d a+2 b^2 c-\frac {2 \left (d a^2+b^2 c\right ) (a+b x+i)}{a+b x}-2 b \sqrt {c} \sqrt {d}\right )}+\frac {\left (d a^2+b^2 c\right ) \log \left (\frac {a+b x+i}{a+b x}\right )}{b \sqrt {c} \sqrt {d} \left (-2 d a^2-2 i d a-2 b^2 c+\frac {2 \left (d a^2+b^2 c\right ) (a+b x+i)}{a+b x}-2 b \sqrt {c} \sqrt {d}\right )}\right )d\frac {a+b x+i}{a+b x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} i \int \frac {\log \left (-\frac {-a-b x+i}{a+b x}\right )}{d x^2+c}dx-\frac {1}{2} b \left (\frac {\operatorname {PolyLog}\left (2,\frac {\left (b \sqrt {c}-i a \sqrt {d}\right ) (a+b x+i)}{\left (\sqrt {d} (1-i a)+b \sqrt {c}\right ) (a+b x)}\right )}{2 b \sqrt {c} \sqrt {d}}-\frac {\operatorname {PolyLog}\left (2,\frac {\left (i \sqrt {d} a+b \sqrt {c}\right ) (a+b x+i)}{\left (i \sqrt {d} (a+i)+b \sqrt {c}\right ) (a+b x)}\right )}{2 b \sqrt {c} \sqrt {d}}+\frac {\log \left (\frac {a+b x+i}{a+b x}\right ) \log \left (1-\frac {(a+b x+i) \left (b \sqrt {c}-i a \sqrt {d}\right )}{(a+b x) \left (b \sqrt {c}+(1-i a) \sqrt {d}\right )}\right )}{2 b \sqrt {c} \sqrt {d}}-\frac {\log \left (\frac {a+b x+i}{a+b x}\right ) \log \left (1-\frac {(a+b x+i) \left (b \sqrt {c}+i a \sqrt {d}\right )}{(a+b x) \left (b \sqrt {c}+i (a+i) \sqrt {d}\right )}\right )}{2 b \sqrt {c} \sqrt {d}}\right )\) |
| \(\Big \downarrow \) 2977 |
| \(\displaystyle \frac {1}{2} b \int \frac {\log \left (-\frac {-a-b x+i}{a+b x}\right )}{d (i-a)^2+b^2 c+\frac {2 \left (b^2 c-(i-a) a d\right ) (-a-b x+i)}{a+b x}+\frac {\left (d a^2+b^2 c\right ) (-a-b x+i)^2}{(a+b x)^2}}d\frac {-a-b x+i}{a+b x}-\frac {1}{2} b \left (\frac {\operatorname {PolyLog}\left (2,\frac {\left (b \sqrt {c}-i a \sqrt {d}\right ) (a+b x+i)}{\left (\sqrt {d} (1-i a)+b \sqrt {c}\right ) (a+b x)}\right )}{2 b \sqrt {c} \sqrt {d}}-\frac {\operatorname {PolyLog}\left (2,\frac {\left (i \sqrt {d} a+b \sqrt {c}\right ) (a+b x+i)}{\left (i \sqrt {d} (a+i)+b \sqrt {c}\right ) (a+b x)}\right )}{2 b \sqrt {c} \sqrt {d}}+\frac {\log \left (\frac {a+b x+i}{a+b x}\right ) \log \left (1-\frac {(a+b x+i) \left (b \sqrt {c}-i a \sqrt {d}\right )}{(a+b x) \left (b \sqrt {c}+(1-i a) \sqrt {d}\right )}\right )}{2 b \sqrt {c} \sqrt {d}}-\frac {\log \left (\frac {a+b x+i}{a+b x}\right ) \log \left (1-\frac {(a+b x+i) \left (b \sqrt {c}+i a \sqrt {d}\right )}{(a+b x) \left (b \sqrt {c}+i (a+i) \sqrt {d}\right )}\right )}{2 b \sqrt {c} \sqrt {d}}\right )\) |
| \(\Big \downarrow \) 2804 |
| \(\displaystyle \frac {1}{2} b \int \left (\frac {\left (d a^2+b^2 c\right ) \log \left (-\frac {-a-b x+i}{a+b x}\right )}{b \sqrt {c} \sqrt {d} \left (-2 d a^2+2 i d a-2 b^2 c-2 b \sqrt {c} \sqrt {d}-\frac {2 \left (d a^2+b^2 c\right ) (-a-b x+i)}{a+b x}\right )}+\frac {\left (d a^2+b^2 c\right ) \log \left (-\frac {-a-b x+i}{a+b x}\right )}{b \sqrt {c} \sqrt {d} \left (2 d a^2-2 i d a+2 b^2 c-2 b \sqrt {c} \sqrt {d}+\frac {2 \left (d a^2+b^2 c\right ) (-a-b x+i)}{a+b x}\right )}\right )d\frac {-a-b x+i}{a+b x}-\frac {1}{2} b \left (\frac {\operatorname {PolyLog}\left (2,\frac {\left (b \sqrt {c}-i a \sqrt {d}\right ) (a+b x+i)}{\left (\sqrt {d} (1-i a)+b \sqrt {c}\right ) (a+b x)}\right )}{2 b \sqrt {c} \sqrt {d}}-\frac {\operatorname {PolyLog}\left (2,\frac {\left (i \sqrt {d} a+b \sqrt {c}\right ) (a+b x+i)}{\left (i \sqrt {d} (a+i)+b \sqrt {c}\right ) (a+b x)}\right )}{2 b \sqrt {c} \sqrt {d}}+\frac {\log \left (\frac {a+b x+i}{a+b x}\right ) \log \left (1-\frac {(a+b x+i) \left (b \sqrt {c}-i a \sqrt {d}\right )}{(a+b x) \left (b \sqrt {c}+(1-i a) \sqrt {d}\right )}\right )}{2 b \sqrt {c} \sqrt {d}}-\frac {\log \left (\frac {a+b x+i}{a+b x}\right ) \log \left (1-\frac {(a+b x+i) \left (b \sqrt {c}+i a \sqrt {d}\right )}{(a+b x) \left (b \sqrt {c}+i (a+i) \sqrt {d}\right )}\right )}{2 b \sqrt {c} \sqrt {d}}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} b \left (\frac {\operatorname {PolyLog}\left (2,-\frac {\left (b \sqrt {c}-i a \sqrt {d}\right ) (-a-b x+i)}{\left (b \sqrt {c}-(i a+1) \sqrt {d}\right ) (a+b x)}\right )}{2 b \sqrt {c} \sqrt {d}}-\frac {\operatorname {PolyLog}\left (2,-\frac {\left (i \sqrt {d} a+b \sqrt {c}\right ) (-a-b x+i)}{\left (\sqrt {d} (i a+1)+b \sqrt {c}\right ) (a+b x)}\right )}{2 b \sqrt {c} \sqrt {d}}+\frac {\log \left (-\frac {-a-b x+i}{a+b x}\right ) \log \left (1+\frac {(-a-b x+i) \left (b \sqrt {c}-i a \sqrt {d}\right )}{(a+b x) \left (b \sqrt {c}-(1+i a) \sqrt {d}\right )}\right )}{2 b \sqrt {c} \sqrt {d}}-\frac {\log \left (-\frac {-a-b x+i}{a+b x}\right ) \log \left (1+\frac {(-a-b x+i) \left (b \sqrt {c}+i a \sqrt {d}\right )}{(a+b x) \left (b \sqrt {c}+(1+i a) \sqrt {d}\right )}\right )}{2 b \sqrt {c} \sqrt {d}}\right )-\frac {1}{2} b \left (\frac {\operatorname {PolyLog}\left (2,\frac {\left (b \sqrt {c}-i a \sqrt {d}\right ) (a+b x+i)}{\left (\sqrt {d} (1-i a)+b \sqrt {c}\right ) (a+b x)}\right )}{2 b \sqrt {c} \sqrt {d}}-\frac {\operatorname {PolyLog}\left (2,\frac {\left (i \sqrt {d} a+b \sqrt {c}\right ) (a+b x+i)}{\left (i \sqrt {d} (a+i)+b \sqrt {c}\right ) (a+b x)}\right )}{2 b \sqrt {c} \sqrt {d}}+\frac {\log \left (\frac {a+b x+i}{a+b x}\right ) \log \left (1-\frac {(a+b x+i) \left (b \sqrt {c}-i a \sqrt {d}\right )}{(a+b x) \left (b \sqrt {c}+(1-i a) \sqrt {d}\right )}\right )}{2 b \sqrt {c} \sqrt {d}}-\frac {\log \left (\frac {a+b x+i}{a+b x}\right ) \log \left (1-\frac {(a+b x+i) \left (b \sqrt {c}+i a \sqrt {d}\right )}{(a+b x) \left (b \sqrt {c}+i (a+i) \sqrt {d}\right )}\right )}{2 b \sqrt {c} \sqrt {d}}\right )\) |
(b*((Log[-((I - a - b*x)/(a + b*x))]*Log[1 + ((b*Sqrt[c] - I*a*Sqrt[d])*(I - a - b*x))/((b*Sqrt[c] - (1 + I*a)*Sqrt[d])*(a + b*x))])/(2*b*Sqrt[c]*Sq rt[d]) - (Log[-((I - a - b*x)/(a + b*x))]*Log[1 + ((b*Sqrt[c] + I*a*Sqrt[d ])*(I - a - b*x))/((b*Sqrt[c] + (1 + I*a)*Sqrt[d])*(a + b*x))])/(2*b*Sqrt[ c]*Sqrt[d]) + PolyLog[2, -(((b*Sqrt[c] - I*a*Sqrt[d])*(I - a - b*x))/((b*S qrt[c] - (1 + I*a)*Sqrt[d])*(a + b*x)))]/(2*b*Sqrt[c]*Sqrt[d]) - PolyLog[2 , -(((b*Sqrt[c] + I*a*Sqrt[d])*(I - a - b*x))/((b*Sqrt[c] + (1 + I*a)*Sqrt [d])*(a + b*x)))]/(2*b*Sqrt[c]*Sqrt[d])))/2 - (b*((Log[(I + a + b*x)/(a + b*x)]*Log[1 - ((b*Sqrt[c] - I*a*Sqrt[d])*(I + a + b*x))/((b*Sqrt[c] + (1 - I*a)*Sqrt[d])*(a + b*x))])/(2*b*Sqrt[c]*Sqrt[d]) - (Log[(I + a + b*x)/(a + b*x)]*Log[1 - ((b*Sqrt[c] + I*a*Sqrt[d])*(I + a + b*x))/((b*Sqrt[c] + I* (I + a)*Sqrt[d])*(a + b*x))])/(2*b*Sqrt[c]*Sqrt[d]) + PolyLog[2, ((b*Sqrt[ c] - I*a*Sqrt[d])*(I + a + b*x))/((b*Sqrt[c] + (1 - I*a)*Sqrt[d])*(a + b*x ))]/(2*b*Sqrt[c]*Sqrt[d]) - PolyLog[2, ((b*Sqrt[c] + I*a*Sqrt[d])*(I + a + b*x))/((b*Sqrt[c] + I*(I + a)*Sqrt[d])*(a + b*x))]/(2*b*Sqrt[c]*Sqrt[d])) )/2
3.2.7.3.1 Defintions of rubi rules used
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{ u = ExpandIntegrand[(a + b*Log[c*x^n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] / ; FreeQ[{a, b, c, n}, x] && RationalFunctionQ[RFx, x] && IGtQ[p, 0]
Int[((A_.) + Log[(e_.)*(((a_.) + (b_.)*(x_))/((c_.) + (d_.)*(x_)))^(n_.)]*( B_.))^(p_.)*(P2x_)^(m_.), x_Symbol] :> With[{f = Coeff[P2x, x, 0], g = Coef f[P2x, x, 1], h = Coeff[P2x, x, 2]}, Simp[(b*c - a*d) Subst[Int[(b^2*f - a*b*g + a^2*h - (2*b*d*f - b*c*g - a*d*g + 2*a*c*h)*x + (d^2*f - c*d*g + c^ 2*h)*x^2)^m*((A + B*Log[e*x^n])^p/(b - d*x)^(2*(m + 1))), x], x, (a + b*x)/ (c + d*x)], x]] /; FreeQ[{a, b, c, d, e, A, B, n}, x] && PolyQ[P2x, x, 2] & & NeQ[b*c - a*d, 0] && IntegerQ[m] && IGtQ[p, 0]
Int[((A_.) + Log[(e_.)*((a_.) + (b_.)*(x_))^(n_.)*((c_.) + (d_.)*(x_))^(mn_ )]*(B_.))^(p_.)*(P2x_)^(m_.), x_Symbol] :> With[{f = Coeff[P2x, x, 0], g = Coeff[P2x, x, 1], h = Coeff[P2x, x, 2]}, Simp[(b*c - a*d) Subst[Int[(b^2* f - a*b*g + a^2*h - (2*b*d*f - b*c*g - a*d*g + 2*a*c*h)*x + (d^2*f - c*d*g + c^2*h)*x^2)^m*((A + B*Log[e*x^n])^p/(b - d*x)^(2*(m + 1))), x], x, (a + b *x)/(c + d*x)], x]] /; FreeQ[{a, b, c, d, e, A, B, n}, x] && PolyQ[P2x, x, 2] && EqQ[n + mn, 0] && IGtQ[n, 0] && NeQ[b*c - a*d, 0] && IntegerQ[m] && I GtQ[p, 0]
Int[ArcCot[(a_) + (b_.)*(x_)]/((c_) + (d_.)*(x_)^(n_.)), x_Symbol] :> Simp[ I/2 Int[Log[(-I + a + b*x)/(a + b*x)]/(c + d*x^n), x], x] - Simp[I/2 In t[Log[(I + a + b*x)/(a + b*x)]/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d}, x ] && RationalQ[n]
Time = 2.01 (sec) , antiderivative size = 591, normalized size of antiderivative = 0.92
| method | result | size |
| risch | \(-\frac {i b \pi \arctan \left (\frac {2 i a d +2 \left (-i b x -i a +1\right ) d -2 d}{2 \sqrt {-b^{2} c d}}\right )}{2 \sqrt {-b^{2} c d}}-\frac {\ln \left (-i b x -i a +1\right ) \ln \left (\frac {i a d -b \sqrt {c d}+\left (-i b x -i a +1\right ) d -d}{i a d -b \sqrt {c d}-d}\right ) \sqrt {c d}}{4 c d}+\frac {\ln \left (-i b x -i a +1\right ) \ln \left (\frac {i a d +b \sqrt {c d}+\left (-i b x -i a +1\right ) d -d}{i a d +b \sqrt {c d}-d}\right ) \sqrt {c d}}{4 c d}-\frac {\operatorname {dilog}\left (\frac {i a d -b \sqrt {c d}+\left (-i b x -i a +1\right ) d -d}{i a d -b \sqrt {c d}-d}\right ) \sqrt {c d}}{4 c d}+\frac {\operatorname {dilog}\left (\frac {i a d +b \sqrt {c d}+\left (-i b x -i a +1\right ) d -d}{i a d +b \sqrt {c d}-d}\right ) \sqrt {c d}}{4 c d}-\frac {\ln \left (i b x +i a +1\right ) \ln \left (\frac {i a d +b \sqrt {c d}-\left (i b x +i a +1\right ) d +d}{i a d +b \sqrt {c d}+d}\right ) \sqrt {c d}}{4 c d}+\frac {\ln \left (i b x +i a +1\right ) \ln \left (\frac {i a d -b \sqrt {c d}-\left (i b x +i a +1\right ) d +d}{i a d -b \sqrt {c d}+d}\right ) \sqrt {c d}}{4 c d}-\frac {\sqrt {c d}\, \operatorname {dilog}\left (\frac {i a d +b \sqrt {c d}-\left (i b x +i a +1\right ) d +d}{i a d +b \sqrt {c d}+d}\right )}{4 c d}+\frac {\sqrt {c d}\, \operatorname {dilog}\left (\frac {i a d -b \sqrt {c d}-\left (i b x +i a +1\right ) d +d}{i a d -b \sqrt {c d}+d}\right )}{4 c d}\) | \(591\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(2074\) |
| default | \(\text {Expression too large to display}\) | \(2074\) |
-1/2*I*b*Pi/(-b^2*c*d)^(1/2)*arctan(1/2*(2*I*a*d+2*(1-I*a-I*b*x)*d-2*d)/(- b^2*c*d)^(1/2))-1/4*ln(1-I*a-I*b*x)/c/d*ln((I*a*d-b*(c*d)^(1/2)+(1-I*a-I*b *x)*d-d)/(I*a*d-b*(c*d)^(1/2)-d))*(c*d)^(1/2)+1/4*ln(1-I*a-I*b*x)/c/d*ln(( I*a*d+b*(c*d)^(1/2)+(1-I*a-I*b*x)*d-d)/(I*a*d+b*(c*d)^(1/2)-d))*(c*d)^(1/2 )-1/4/c/d*dilog((I*a*d-b*(c*d)^(1/2)+(1-I*a-I*b*x)*d-d)/(I*a*d-b*(c*d)^(1/ 2)-d))*(c*d)^(1/2)+1/4/c/d*dilog((I*a*d+b*(c*d)^(1/2)+(1-I*a-I*b*x)*d-d)/( I*a*d+b*(c*d)^(1/2)-d))*(c*d)^(1/2)-1/4*ln(1+I*a+I*b*x)/c/d*ln((I*a*d+b*(c *d)^(1/2)-(1+I*a+I*b*x)*d+d)/(I*a*d+b*(c*d)^(1/2)+d))*(c*d)^(1/2)+1/4*ln(1 +I*a+I*b*x)/c/d*ln((I*a*d-b*(c*d)^(1/2)-(1+I*a+I*b*x)*d+d)/(I*a*d-b*(c*d)^ (1/2)+d))*(c*d)^(1/2)-1/4/c/d*(c*d)^(1/2)*dilog((I*a*d+b*(c*d)^(1/2)-(1+I* a+I*b*x)*d+d)/(I*a*d+b*(c*d)^(1/2)+d))+1/4/c/d*(c*d)^(1/2)*dilog((I*a*d-b* (c*d)^(1/2)-(1+I*a+I*b*x)*d+d)/(I*a*d-b*(c*d)^(1/2)+d))
\[ \int \frac {\cot ^{-1}(a+b x)}{c+d x^2} \, dx=\int { \frac {\operatorname {arccot}\left (b x + a\right )}{d x^{2} + c} \,d x } \]
Timed out. \[ \int \frac {\cot ^{-1}(a+b x)}{c+d x^2} \, dx=\text {Timed out} \]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 8519 vs. \(2 (456) = 912\).
Time = 4.03 (sec) , antiderivative size = 8519, normalized size of antiderivative = 13.27 \[ \int \frac {\cot ^{-1}(a+b x)}{c+d x^2} \, dx=\text {Too large to display} \]
-1/8*b*(8*arctan(d*x/sqrt(c*d))*arctan((b^2*x + a*b)/b)/b - (4*arctan(sqrt (d)*x/sqrt(c))*arctan2((2*a*b^2*c*d + (a*b^3*c + (a^3 + a)*b*d + (b^4*c + (a^2 + 3)*b^2*d)*x)*sqrt(c)*sqrt(d) + (3*b^3*c*d + (a^2 + 1)*b*d^2)*x)/(b^ 4*c^2 + 2*(a^2 + 3)*b^2*c*d + (a^4 + 2*a^2 + 1)*d^2 + 4*(b^3*c + (a^2 + 1) *b*d)*sqrt(c)*sqrt(d)), ((a^2 + 3)*b^2*c*d + (a^4 + 2*a^2 + 1)*d^2 + (2*a* b^2*d*x + b^3*c + 3*(a^2 + 1)*b*d)*sqrt(c)*sqrt(d) + (a*b^3*c*d + (a^3 + a )*b*d^2)*x)/(b^4*c^2 + 2*(a^2 + 3)*b^2*c*d + (a^4 + 2*a^2 + 1)*d^2 + 4*(b^ 3*c + (a^2 + 1)*b*d)*sqrt(c)*sqrt(d))) + 4*arctan(sqrt(d)*x/sqrt(c))*arcta n2((2*a*b^2*c*d - (a*b^3*c + (a^3 + a)*b*d + (b^4*c + (a^2 + 3)*b^2*d)*x)* sqrt(c)*sqrt(d) + (3*b^3*c*d + (a^2 + 1)*b*d^2)*x)/(b^4*c^2 + 2*(a^2 + 3)* b^2*c*d + (a^4 + 2*a^2 + 1)*d^2 - 4*(b^3*c + (a^2 + 1)*b*d)*sqrt(c)*sqrt(d )), ((a^2 + 3)*b^2*c*d + (a^4 + 2*a^2 + 1)*d^2 - (2*a*b^2*d*x + b^3*c + 3* (a^2 + 1)*b*d)*sqrt(c)*sqrt(d) + (a*b^3*c*d + (a^3 + a)*b*d^2)*x)/(b^4*c^2 + 2*(a^2 + 3)*b^2*c*d + (a^4 + 2*a^2 + 1)*d^2 - 4*(b^3*c + (a^2 + 1)*b*d) *sqrt(c)*sqrt(d))) + log(d*x^2 + c)*log(((a^2 + 1)*b^22*c^11*d + 11*(a^4 + 22*a^2 + 21)*b^20*c^10*d^2 + 55*(a^6 + 39*a^4 + 171*a^2 + 133)*b^18*c^9*d ^3 + 33*(5*a^8 + 260*a^6 + 1870*a^4 + 3876*a^2 + 2261)*b^16*c^8*d^4 + 330* (a^10 + 61*a^8 + 570*a^6 + 1802*a^4 + 2261*a^2 + 969)*b^14*c^7*d^5 + 22*(2 1*a^12 + 1386*a^10 + 15015*a^8 + 60060*a^6 + 109395*a^4 + 92378*a^2 + 2939 3)*b^12*c^6*d^6 + 22*(21*a^14 + 1407*a^12 + 16401*a^10 + 75075*a^8 + 16...
Timed out. \[ \int \frac {\cot ^{-1}(a+b x)}{c+d x^2} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {\cot ^{-1}(a+b x)}{c+d x^2} \, dx=\int \frac {\mathrm {acot}\left (a+b\,x\right )}{d\,x^2+c} \,d x \]