Integrand size = 33, antiderivative size = 216 \[ \int \frac {\arctan (a+b x)}{\sqrt {\left (1+a^2\right ) c+2 a b c x+b^2 c x^2}} \, dx=-\frac {2 i \sqrt {1+(a+b x)^2} \arctan (a+b x) \arctan \left (\frac {\sqrt {1+i (a+b x)}}{\sqrt {1-i (a+b x)}}\right )}{b \sqrt {c+c (a+b x)^2}}+\frac {i \sqrt {1+(a+b x)^2} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1+i (a+b x)}}{\sqrt {1-i (a+b x)}}\right )}{b \sqrt {c+c (a+b x)^2}}-\frac {i \sqrt {1+(a+b x)^2} \operatorname {PolyLog}\left (2,\frac {i \sqrt {1+i (a+b x)}}{\sqrt {1-i (a+b x)}}\right )}{b \sqrt {c+c (a+b x)^2}} \] Output:
-2*I*(1+(b*x+a)^2)^(1/2)*arctan(b*x+a)*arctan((1+I*(b*x+a))^(1/2)/(1-I*(b* x+a))^(1/2))/b/(c+c*(b*x+a)^2)^(1/2)+I*(1+(b*x+a)^2)^(1/2)*polylog(2,-I*(1 +I*(b*x+a))^(1/2)/(1-I*(b*x+a))^(1/2))/b/(c+c*(b*x+a)^2)^(1/2)-I*(1+(b*x+a )^2)^(1/2)*polylog(2,I*(1+I*(b*x+a))^(1/2)/(1-I*(b*x+a))^(1/2))/b/(c+c*(b* x+a)^2)^(1/2)
Time = 0.08 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.44 \[ \int \frac {\arctan (a+b x)}{\sqrt {\left (1+a^2\right ) c+2 a b c x+b^2 c x^2}} \, dx=-\frac {i \sqrt {1+(a+b x)^2} \left (2 \arctan \left (e^{i \arctan (a+b x)}\right ) \arctan (a+b x)-\operatorname {PolyLog}\left (2,-i e^{i \arctan (a+b x)}\right )+\operatorname {PolyLog}\left (2,i e^{i \arctan (a+b x)}\right )\right )}{b \sqrt {c \left (1+(a+b x)^2\right )}} \] Input:
Integrate[ArcTan[a + b*x]/Sqrt[(1 + a^2)*c + 2*a*b*c*x + b^2*c*x^2],x]
Output:
((-I)*Sqrt[1 + (a + b*x)^2]*(2*ArcTan[E^(I*ArcTan[a + b*x])]*ArcTan[a + b* x] - PolyLog[2, (-I)*E^(I*ArcTan[a + b*x])] + PolyLog[2, I*E^(I*ArcTan[a + b*x])]))/(b*Sqrt[c*(1 + (a + b*x)^2)])
Time = 0.37 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.72, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {5578, 5425, 5421}
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.
\(\displaystyle \int \frac {\arctan (a+b x)}{\sqrt {\left (a^2+1\right ) c+2 a b c x+b^2 c x^2}} \, dx\) |
\(\Big \downarrow \) 5578 |
\(\displaystyle \frac {\int \frac {\arctan (a+b x)}{\sqrt {c (a+b x)^2+c}}d(a+b x)}{b}\) |
\(\Big \downarrow \) 5425 |
\(\displaystyle \frac {\sqrt {(a+b x)^2+1} \int \frac {\arctan (a+b x)}{\sqrt {(a+b x)^2+1}}d(a+b x)}{b \sqrt {c (a+b x)^2+c}}\) |
\(\Big \downarrow \) 5421 |
\(\displaystyle \frac {\sqrt {(a+b x)^2+1} \left (-2 i \arctan (a+b x) \arctan \left (\frac {\sqrt {1+i (a+b x)}}{\sqrt {1-i (a+b x)}}\right )+i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i (a+b x)+1}}{\sqrt {1-i (a+b x)}}\right )-i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i (a+b x)+1}}{\sqrt {1-i (a+b x)}}\right )\right )}{b \sqrt {c (a+b x)^2+c}}\) |
Input:
Int[ArcTan[a + b*x]/Sqrt[(1 + a^2)*c + 2*a*b*c*x + b^2*c*x^2],x]
Output:
(Sqrt[1 + (a + b*x)^2]*((-2*I)*ArcTan[a + b*x]*ArcTan[Sqrt[1 + I*(a + b*x) ]/Sqrt[1 - I*(a + b*x)]] + I*PolyLog[2, ((-I)*Sqrt[1 + I*(a + b*x)])/Sqrt[ 1 - I*(a + b*x)]] - I*PolyLog[2, (I*Sqrt[1 + I*(a + b*x)])/Sqrt[1 - I*(a + b*x)]]))/(b*Sqrt[c + c*(a + b*x)^2])
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[-2*I*(a + b*ArcTan[c*x])*(ArcTan[Sqrt[1 + I*c*x]/Sqrt[1 - I*c*x]]/ (c*Sqrt[d])), x] + (Simp[I*b*(PolyLog[2, (-I)*(Sqrt[1 + I*c*x]/Sqrt[1 - I*c *x])]/(c*Sqrt[d])), x] - Simp[I*b*(PolyLog[2, I*(Sqrt[1 + I*c*x]/Sqrt[1 - I *c*x])]/(c*Sqrt[d])), x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[d, 0]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_S ymbol] :> Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2] Int[(a + b*ArcTan[c*x])^ p/Sqrt[1 + c^2*x^2], x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] & & IGtQ[p, 0] && !GtQ[d, 0]
Int[((a_.) + ArcTan[(c_) + (d_.)*(x_)]*(b_.))^(p_.)*((A_.) + (B_.)*(x_) + ( C_.)*(x_)^2)^(q_.), x_Symbol] :> Simp[1/d Subst[Int[(C/d^2 + (C/d^2)*x^2) ^q*(a + b*ArcTan[x])^p, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, A, B, C, p, q}, x] && EqQ[B*(1 + c^2) - 2*A*c*d, 0] && EqQ[2*c*C - B*d, 0]
Time = 0.49 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.81
method | result | size |
default | \(-\frac {\left (\arctan \left (b x +a \right ) \ln \left (1+\frac {i \left (1+i \left (b x +a \right )\right )}{\sqrt {1+\left (b x +a \right )^{2}}}\right )-\arctan \left (b x +a \right ) \ln \left (1-\frac {i \left (1+i \left (b x +a \right )\right )}{\sqrt {1+\left (b x +a \right )^{2}}}\right )-i \operatorname {dilog}\left (1+\frac {i \left (1+i \left (b x +a \right )\right )}{\sqrt {1+\left (b x +a \right )^{2}}}\right )+i \operatorname {dilog}\left (1-\frac {i \left (1+i \left (b x +a \right )\right )}{\sqrt {1+\left (b x +a \right )^{2}}}\right )\right ) \sqrt {c \left (b x +a -i\right ) \left (b x +a +i\right )}}{\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, b c}\) | \(176\) |
Input:
int(arctan(b*x+a)/((a^2+1)*c+2*a*b*c*x+b^2*c*x^2)^(1/2),x,method=_RETURNVE RBOSE)
Output:
-(arctan(b*x+a)*ln(1+I*(1+I*(b*x+a))/(1+(b*x+a)^2)^(1/2))-arctan(b*x+a)*ln (1-I*(1+I*(b*x+a))/(1+(b*x+a)^2)^(1/2))-I*dilog(1+I*(1+I*(b*x+a))/(1+(b*x+ a)^2)^(1/2))+I*dilog(1-I*(1+I*(b*x+a))/(1+(b*x+a)^2)^(1/2)))*(c*(b*x+a-I)* (I+a+b*x))^(1/2)/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)/b/c
\[ \int \frac {\arctan (a+b x)}{\sqrt {\left (1+a^2\right ) c+2 a b c x+b^2 c x^2}} \, dx=\int { \frac {\arctan \left (b x + a\right )}{\sqrt {b^{2} c x^{2} + 2 \, a b c x + {\left (a^{2} + 1\right )} c}} \,d x } \] Input:
integrate(arctan(b*x+a)/((a^2+1)*c+2*a*b*c*x+b^2*c*x^2)^(1/2),x, algorithm ="fricas")
Output:
integral(arctan(b*x + a)/sqrt(b^2*c*x^2 + 2*a*b*c*x + (a^2 + 1)*c), x)
Timed out. \[ \int \frac {\arctan (a+b x)}{\sqrt {\left (1+a^2\right ) c+2 a b c x+b^2 c x^2}} \, dx=\text {Timed out} \] Input:
integrate(atan(b*x+a)/((a**2+1)*c+2*a*b*c*x+b**2*c*x**2)**(1/2),x)
Output:
Timed out
\[ \int \frac {\arctan (a+b x)}{\sqrt {\left (1+a^2\right ) c+2 a b c x+b^2 c x^2}} \, dx=\int { \frac {\arctan \left (b x + a\right )}{\sqrt {b^{2} c x^{2} + 2 \, a b c x + {\left (a^{2} + 1\right )} c}} \,d x } \] Input:
integrate(arctan(b*x+a)/((a^2+1)*c+2*a*b*c*x+b^2*c*x^2)^(1/2),x, algorithm ="maxima")
Output:
integrate(arctan(b*x + a)/sqrt(b^2*c*x^2 + 2*a*b*c*x + (a^2 + 1)*c), x)
\[ \int \frac {\arctan (a+b x)}{\sqrt {\left (1+a^2\right ) c+2 a b c x+b^2 c x^2}} \, dx=\int { \frac {\arctan \left (b x + a\right )}{\sqrt {b^{2} c x^{2} + 2 \, a b c x + {\left (a^{2} + 1\right )} c}} \,d x } \] Input:
integrate(arctan(b*x+a)/((a^2+1)*c+2*a*b*c*x+b^2*c*x^2)^(1/2),x, algorithm ="giac")
Output:
integrate(arctan(b*x + a)/sqrt(b^2*c*x^2 + 2*a*b*c*x + (a^2 + 1)*c), x)
Timed out. \[ \int \frac {\arctan (a+b x)}{\sqrt {\left (1+a^2\right ) c+2 a b c x+b^2 c x^2}} \, dx=\int \frac {\mathrm {atan}\left (a+b\,x\right )}{\sqrt {c\,b^2\,x^2+2\,a\,c\,b\,x+c\,\left (a^2+1\right )}} \,d x \] Input:
int(atan(a + b*x)/(c*(a^2 + 1) + b^2*c*x^2 + 2*a*b*c*x)^(1/2),x)
Output:
int(atan(a + b*x)/(c*(a^2 + 1) + b^2*c*x^2 + 2*a*b*c*x)^(1/2), x)
\[ \int \frac {\arctan (a+b x)}{\sqrt {\left (1+a^2\right ) c+2 a b c x+b^2 c x^2}} \, dx=\frac {\int \frac {\mathit {atan} \left (b x +a \right )}{\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}d x}{\sqrt {c}} \] Input:
int(atan(b*x+a)/((a^2+1)*c+2*a*b*c*x+b^2*c*x^2)^(1/2),x)
Output:
int(atan(a + b*x)/sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1),x)/sqrt(c)