3.176 \(\int \frac{e^{2 i \tan ^{-1}(a+b x)}}{x} \, dx\)
Optimal. Leaf size=38 \[ \frac{(-a+i) \log (x)}{a+i}-\frac{2 \log (a+b x+i)}{1-i a} \]
[Out]
((I - a)*Log[x])/(I + a) - (2*Log[I + a + b*x])/(1 - I*a)
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Rubi [A] time = 0.0339896, antiderivative size = 38, normalized size of antiderivative = 1.,
number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used =
{5095, 72} \[ \frac{(-a+i) \log (x)}{a+i}-\frac{2 \log (a+b x+i)}{1-i a} \]
Antiderivative was successfully verified.
[In]
Int[E^((2*I)*ArcTan[a + b*x])/x,x]
[Out]
((I - a)*Log[x])/(I + a) - (2*Log[I + a + b*x])/(1 - I*a)
Rule 5095
Int[E^(ArcTan[(c_.)*((a_) + (b_.)*(x_))]*(n_.))*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Int[((d + e*x)^m*(1 -
I*a*c - I*b*c*x)^((I*n)/2))/(1 + I*a*c + I*b*c*x)^((I*n)/2), x] /; FreeQ[{a, b, c, d, e, m, n}, x]
Rule 72
Int[((e_.) + (f_.)*(x_))^(p_.)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Int[ExpandIntegrand[(
e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[p]
Rubi steps
\begin{align*} \int \frac{e^{2 i \tan ^{-1}(a+b x)}}{x} \, dx &=\int \frac{1+i a+i b x}{x (1-i a-i b x)} \, dx\\ &=\int \left (\frac{i-a}{(i+a) x}-\frac{2 i b}{(i+a) (i+a+b x)}\right ) \, dx\\ &=\frac{(i-a) \log (x)}{i+a}-\frac{2 \log (i+a+b x)}{1-i a}\\ \end{align*}
Mathematica [A] time = 0.0180426, size = 31, normalized size = 0.82 \[ -\frac{2 i \log (a+b x+i)+(a-i) \log (x)}{a+i} \]
Antiderivative was successfully verified.
[In]
Integrate[E^((2*I)*ArcTan[a + b*x])/x,x]
[Out]
-(((-I + a)*Log[x] + (2*I)*Log[I + a + b*x])/(I + a))
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Maple [B] time = 0.043, size = 149, normalized size = 3.9 \begin{align*}{\frac{-i\ln \left ({b}^{2}{x}^{2}+2\,xab+{a}^{2}+1 \right ) a}{{a}^{2}+1}}-{\frac{\ln \left ({b}^{2}{x}^{2}+2\,xab+{a}^{2}+1 \right ) }{{a}^{2}+1}}+{\frac{2\,i}{{a}^{2}+1}\arctan \left ({\frac{2\,{b}^{2}x+2\,ab}{2\,b}} \right ) }-2\,{\frac{a}{{a}^{2}+1}\arctan \left ( 1/2\,{\frac{2\,{b}^{2}x+2\,ab}{b}} \right ) }+{\frac{2\,ia\ln \left ( x \right ) }{{a}^{2}+1}}-{\frac{{a}^{2}\ln \left ( x \right ) }{{a}^{2}+1}}+{\frac{\ln \left ( x \right ) }{{a}^{2}+1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((1+I*(b*x+a))^2/(1+(b*x+a)^2)/x,x)
[Out]
-I/(a^2+1)*ln(b^2*x^2+2*a*b*x+a^2+1)*a-1/(a^2+1)*ln(b^2*x^2+2*a*b*x+a^2+1)+2*I/(a^2+1)*arctan(1/2*(2*b^2*x+2*a
*b)/b)-2/(a^2+1)*arctan(1/2*(2*b^2*x+2*a*b)/b)*a+2*I/(a^2+1)*ln(x)*a-1/(a^2+1)*ln(x)*a^2+1/(a^2+1)*ln(x)
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Maxima [B] time = 1.47925, size = 108, normalized size = 2.84 \begin{align*} -\frac{{\left (2 \, a - 2 i\right )} \arctan \left (\frac{b^{2} x + a b}{b}\right )}{a^{2} + 1} - \frac{{\left (i \, a + 1\right )} \log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}{a^{2} + 1} - \frac{{\left (a^{2} - 2 i \, a - 1\right )} \log \left (x\right )}{a^{2} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((1+I*(b*x+a))^2/(1+(b*x+a)^2)/x,x, algorithm="maxima")
[Out]
-(2*a - 2*I)*arctan((b^2*x + a*b)/b)/(a^2 + 1) - (I*a + 1)*log(b^2*x^2 + 2*a*b*x + a^2 + 1)/(a^2 + 1) - (a^2 -
2*I*a - 1)*log(x)/(a^2 + 1)
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Fricas [A] time = 1.81157, size = 73, normalized size = 1.92 \begin{align*} -\frac{{\left (a - i\right )} \log \left (x\right ) + 2 i \, \log \left (\frac{b x + a + i}{b}\right )}{a + i} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((1+I*(b*x+a))^2/(1+(b*x+a)^2)/x,x, algorithm="fricas")
[Out]
-((a - I)*log(x) + 2*I*log((b*x + a + I)/b))/(a + I)
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Sympy [B] time = 2.54089, size = 1538, normalized size = 40.47 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((1+I*(b*x+a))**2/(1+(b*x+a)**2)/x,x)
[Out]
(-sqrt((-a**8 + 4*I*a**7 + 4*a**6 + 4*I*a**5 + 10*a**4 - 4*I*a**3 + 4*a**2 - 4*I*a - 1)*(-a**8 + 12*I*a**7 + 6
0*a**6 - 164*I*a**5 - 270*a**4 + 276*I*a**3 + 172*a**2 - 60*I*a - 9))/(2*(a**8 - 4*I*a**7 - 4*a**6 - 4*I*a**5
- 10*a**4 + 4*I*a**3 - 4*a**2 + 4*I*a + 1)) - 1/2)*log(x + (-sqrt((-a**8 + 4*I*a**7 + 4*a**6 + 4*I*a**5 + 10*a
**4 - 4*I*a**3 + 4*a**2 - 4*I*a - 1)*(-a**8 + 12*I*a**7 + 60*a**6 - 164*I*a**5 - 270*a**4 + 276*I*a**3 + 172*a
**2 - 60*I*a - 9))/(2*(a**8 - 4*I*a**7 - 4*a**6 - 4*I*a**5 - 10*a**4 + 4*I*a**3 - 4*a**2 + 4*I*a + 1)) - 1/2)*
(a**11 - 9*I*a**10 - 31*a**9 + 47*I*a**8 + 10*a**7 + 70*I*a**6 + 98*a**5 - 34*I*a**4 + 37*a**3 - 45*I*a**2 - 1
9*a + 3*I)/(a**10*b - 14*I*a**9*b - 85*a**8*b + 296*I*a**7*b + 658*a**6*b - 980*I*a**5*b - 994*a**4*b + 680*I*
a**3*b + 301*a**2*b - 78*I*a*b - 9*b) + (a**25 - 29*I*a**24 - 396*a**23 + 3388*I*a**22 + 20378*a**21 - 91602*I
*a**20 - 319116*a**19 + 880764*I*a**18 + 1948887*a**17 - 3465467*I*a**16 - 4901848*a**15 + 5325624*I*a**14 + 3
970316*a**13 - 967708*I*a**12 + 2488392*a**11 - 4876008*I*a**10 - 5388609*a**9 + 4348701*I*a**8 + 2724068*a**7
- 1346548*I*a**6 - 523782*a**5 + 157646*I*a**4 + 35524*a**3 - 5652*I*a**2 - 567*a + 27*I)/(a**24*b - 32*I*a**
23*b - 484*a**22*b + 4608*I*a**21*b + 31026*a**20*b - 157344*I*a**19*b - 624948*a**18*b + 1995456*I*a**17*b +
5216127*a**16*b - 11307584*I*a**15*b - 20514376*a**14*b + 31338752*I*a**13*b + 40461564*a**12*b - 44217408*I*a
**11*b - 40876296*a**10*b + 31876224*I*a**9*b + 20859663*a**8*b - 11361696*I*a**7*b - 5089492*a**6*b + 1842944
*I*a**5*b + 526066*a**4*b - 113952*I*a**3*b - 17604*a**2*b + 1728*I*a*b + 81*b)) + (sqrt((-a**8 + 4*I*a**7 + 4
*a**6 + 4*I*a**5 + 10*a**4 - 4*I*a**3 + 4*a**2 - 4*I*a - 1)*(-a**8 + 12*I*a**7 + 60*a**6 - 164*I*a**5 - 270*a*
*4 + 276*I*a**3 + 172*a**2 - 60*I*a - 9))/(2*(a**8 - 4*I*a**7 - 4*a**6 - 4*I*a**5 - 10*a**4 + 4*I*a**3 - 4*a**
2 + 4*I*a + 1)) - 1/2)*log(x + (sqrt((-a**8 + 4*I*a**7 + 4*a**6 + 4*I*a**5 + 10*a**4 - 4*I*a**3 + 4*a**2 - 4*I
*a - 1)*(-a**8 + 12*I*a**7 + 60*a**6 - 164*I*a**5 - 270*a**4 + 276*I*a**3 + 172*a**2 - 60*I*a - 9))/(2*(a**8 -
4*I*a**7 - 4*a**6 - 4*I*a**5 - 10*a**4 + 4*I*a**3 - 4*a**2 + 4*I*a + 1)) - 1/2)*(a**11 - 9*I*a**10 - 31*a**9
+ 47*I*a**8 + 10*a**7 + 70*I*a**6 + 98*a**5 - 34*I*a**4 + 37*a**3 - 45*I*a**2 - 19*a + 3*I)/(a**10*b - 14*I*a*
*9*b - 85*a**8*b + 296*I*a**7*b + 658*a**6*b - 980*I*a**5*b - 994*a**4*b + 680*I*a**3*b + 301*a**2*b - 78*I*a*
b - 9*b) + (a**25 - 29*I*a**24 - 396*a**23 + 3388*I*a**22 + 20378*a**21 - 91602*I*a**20 - 319116*a**19 + 88076
4*I*a**18 + 1948887*a**17 - 3465467*I*a**16 - 4901848*a**15 + 5325624*I*a**14 + 3970316*a**13 - 967708*I*a**12
+ 2488392*a**11 - 4876008*I*a**10 - 5388609*a**9 + 4348701*I*a**8 + 2724068*a**7 - 1346548*I*a**6 - 523782*a*
*5 + 157646*I*a**4 + 35524*a**3 - 5652*I*a**2 - 567*a + 27*I)/(a**24*b - 32*I*a**23*b - 484*a**22*b + 4608*I*a
**21*b + 31026*a**20*b - 157344*I*a**19*b - 624948*a**18*b + 1995456*I*a**17*b + 5216127*a**16*b - 11307584*I*
a**15*b - 20514376*a**14*b + 31338752*I*a**13*b + 40461564*a**12*b - 44217408*I*a**11*b - 40876296*a**10*b + 3
1876224*I*a**9*b + 20859663*a**8*b - 11361696*I*a**7*b - 5089492*a**6*b + 1842944*I*a**5*b + 526066*a**4*b - 1
13952*I*a**3*b - 17604*a**2*b + 1728*I*a*b + 81*b))
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Giac [A] time = 1.11587, size = 49, normalized size = 1.29 \begin{align*} -\frac{2 \, b i \log \left (b x + a + i\right )}{a b + b i} - \frac{{\left (a - i\right )} \log \left ({\left | x \right |}\right )}{a + i} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((1+I*(b*x+a))^2/(1+(b*x+a)^2)/x,x, algorithm="giac")
[Out]
-2*b*i*log(b*x + a + i)/(a*b + b*i) - (a - i)*log(abs(x))/(a + i)