3.198 \(\int e^{-2 i \tan ^{-1}(a+b x)} x^4 \, dx\)
Optimal. Leaf size=99 \[ \frac{2 (1+i a) x^3}{3 b^2}-\frac{i (-a+i)^2 x^2}{b^3}-\frac{2 (1+i a)^3 x}{b^4}-\frac{2 i (-a+i)^4 \log (-a-b x+i)}{b^5}-\frac{i x^4}{2 b}-\frac{x^5}{5} \]
[Out]
(-2*(1 + I*a)^3*x)/b^4 - (I*(I - a)^2*x^2)/b^3 + (2*(1 + I*a)*x^3)/(3*b^2) - ((I/2)*x^4)/b - x^5/5 - ((2*I)*(I
- a)^4*Log[I - a - b*x])/b^5
________________________________________________________________________________________
Rubi [A] time = 0.0856157, antiderivative size = 99, 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, 77} \[ \frac{2 (1+i a) x^3}{3 b^2}-\frac{i (-a+i)^2 x^2}{b^3}-\frac{2 (1+i a)^3 x}{b^4}-\frac{2 i (-a+i)^4 \log (-a-b x+i)}{b^5}-\frac{i x^4}{2 b}-\frac{x^5}{5} \]
Antiderivative was successfully verified.
[In]
Int[x^4/E^((2*I)*ArcTan[a + b*x]),x]
[Out]
(-2*(1 + I*a)^3*x)/b^4 - (I*(I - a)^2*x^2)/b^3 + (2*(1 + I*a)*x^3)/(3*b^2) - ((I/2)*x^4)/b - x^5/5 - ((2*I)*(I
- a)^4*Log[I - a - b*x])/b^5
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 77
Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
Rubi steps
\begin{align*} \int e^{-2 i \tan ^{-1}(a+b x)} x^4 \, dx &=\int \frac{x^4 (1-i a-i b x)}{1+i a+i b x} \, dx\\ &=\int \left (\frac{2 (-1-i a)^3}{b^4}-\frac{2 i (-i+a)^2 x}{b^3}+\frac{2 (1+i a) x^2}{b^2}-\frac{2 i x^3}{b}-x^4-\frac{2 i (-i+a)^4}{b^4 (-i+a+b x)}\right ) \, dx\\ &=-\frac{2 (1+i a)^3 x}{b^4}-\frac{i (i-a)^2 x^2}{b^3}+\frac{2 (1+i a) x^3}{3 b^2}-\frac{i x^4}{2 b}-\frac{x^5}{5}-\frac{2 i (i-a)^4 \log (i-a-b x)}{b^5}\\ \end{align*}
Mathematica [A] time = 0.0700133, size = 95, normalized size = 0.96 \[ \frac{2 (1+i a) x^3}{3 b^2}-\frac{i (a-i)^2 x^2}{b^3}-\frac{2 (1+i a)^3 x}{b^4}-\frac{2 i (a-i)^4 \log (-a-b x+i)}{b^5}-\frac{i x^4}{2 b}-\frac{x^5}{5} \]
Antiderivative was successfully verified.
[In]
Integrate[x^4/E^((2*I)*ArcTan[a + b*x]),x]
[Out]
(-2*(1 + I*a)^3*x)/b^4 - (I*(-I + a)^2*x^2)/b^3 + (2*(1 + I*a)*x^3)/(3*b^2) - ((I/2)*x^4)/b - x^5/5 - ((2*I)*(
-I + a)^4*Log[I - a - b*x])/b^5
________________________________________________________________________________________
Maple [B] time = 0.043, size = 292, normalized size = 3. \begin{align*} -{\frac{{x}^{5}}{5}}-{\frac{i\ln \left ({b}^{2}{x}^{2}+2\,xab+{a}^{2}+1 \right ) }{{b}^{5}}}-{\frac{6\,iax}{{b}^{4}}}-{\frac{i{x}^{2}{a}^{2}}{{b}^{3}}}-{\frac{i\ln \left ({b}^{2}{x}^{2}+2\,xab+{a}^{2}+1 \right ){a}^{4}}{{b}^{5}}}+{\frac{2\,{x}^{3}}{3\,{b}^{2}}}+{\frac{{\frac{2\,i}{3}}{x}^{3}a}{{b}^{2}}}-2\,{\frac{a{x}^{2}}{{b}^{3}}}-{\frac{{\frac{i}{2}}{x}^{4}}{b}}+6\,{\frac{{a}^{2}x}{{b}^{4}}}-2\,{\frac{x}{{b}^{4}}}+2\,{\frac{\arctan \left ( bx+a \right ){a}^{4}}{{b}^{5}}}+{\frac{6\,i\ln \left ({b}^{2}{x}^{2}+2\,xab+{a}^{2}+1 \right ){a}^{2}}{{b}^{5}}}-12\,{\frac{\arctan \left ( bx+a \right ){a}^{2}}{{b}^{5}}}+{\frac{2\,i{a}^{3}x}{{b}^{4}}}+{\frac{8\,i\arctan \left ( bx+a \right ) a}{{b}^{5}}}-4\,{\frac{\ln \left ({b}^{2}{x}^{2}+2\,xab+{a}^{2}+1 \right ){a}^{3}}{{b}^{5}}}+2\,{\frac{\arctan \left ( bx+a \right ) }{{b}^{5}}}+{\frac{i{x}^{2}}{{b}^{3}}}-{\frac{8\,i\arctan \left ( bx+a \right ){a}^{3}}{{b}^{5}}}+4\,{\frac{\ln \left ({b}^{2}{x}^{2}+2\,xab+{a}^{2}+1 \right ) a}{{b}^{5}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^4/(1+I*(b*x+a))^2*(1+(b*x+a)^2),x)
[Out]
-1/5*x^5-I/b^5*ln(b^2*x^2+2*a*b*x+a^2+1)-6*I/b^4*a*x-I/b^3*x^2*a^2-I/b^5*ln(b^2*x^2+2*a*b*x+a^2+1)*a^4+2/3/b^2
*x^3+2/3*I/b^2*x^3*a-2/b^3*x^2*a-1/2*I*x^4/b+6/b^4*a^2*x-2/b^4*x+2/b^5*arctan(b*x+a)*a^4+6*I/b^5*ln(b^2*x^2+2*
a*b*x+a^2+1)*a^2-12/b^5*arctan(b*x+a)*a^2+2*I/b^4*a^3*x+8*I/b^5*arctan(b*x+a)*a-4/b^5*ln(b^2*x^2+2*a*b*x+a^2+1
)*a^3+2/b^5*arctan(b*x+a)+I/b^3*x^2-8*I/b^5*arctan(b*x+a)*a^3+4/b^5*ln(b^2*x^2+2*a*b*x+a^2+1)*a
________________________________________________________________________________________
Maxima [A] time = 1.0219, size = 138, normalized size = 1.39 \begin{align*} -\frac{6 \, b^{4} x^{5} + 15 i \, b^{3} x^{4} - 20 \,{\left (i \, a + 1\right )} b^{2} x^{3} +{\left (30 i \, a^{2} + 60 \, a - 30 i\right )} b x^{2} +{\left (-60 i \, a^{3} - 180 \, a^{2} + 180 i \, a + 60\right )} x}{30 \, b^{4}} + \frac{{\left (-2 i \, a^{4} - 8 \, a^{3} + 12 i \, a^{2} + 8 \, a - 2 i\right )} \log \left (i \, b x + i \, a + 1\right )}{b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^4/(1+I*(b*x+a))^2*(1+(b*x+a)^2),x, algorithm="maxima")
[Out]
-1/30*(6*b^4*x^5 + 15*I*b^3*x^4 - 20*(I*a + 1)*b^2*x^3 + (30*I*a^2 + 60*a - 30*I)*b*x^2 + (-60*I*a^3 - 180*a^2
+ 180*I*a + 60)*x)/b^4 + (-2*I*a^4 - 8*a^3 + 12*I*a^2 + 8*a - 2*I)*log(I*b*x + I*a + 1)/b^5
________________________________________________________________________________________
Fricas [A] time = 2.02636, size = 286, normalized size = 2.89 \begin{align*} -\frac{6 \, b^{5} x^{5} + 15 i \, b^{4} x^{4} + 20 \,{\left (-i \, a - 1\right )} b^{3} x^{3} -{\left (-30 i \, a^{2} - 60 \, a + 30 i\right )} b^{2} x^{2} -{\left (60 i \, a^{3} + 180 \, a^{2} - 180 i \, a - 60\right )} b x -{\left (-60 i \, a^{4} - 240 \, a^{3} + 360 i \, a^{2} + 240 \, a - 60 i\right )} \log \left (\frac{b x + a - i}{b}\right )}{30 \, b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^4/(1+I*(b*x+a))^2*(1+(b*x+a)^2),x, algorithm="fricas")
[Out]
-1/30*(6*b^5*x^5 + 15*I*b^4*x^4 + 20*(-I*a - 1)*b^3*x^3 - (-30*I*a^2 - 60*a + 30*I)*b^2*x^2 - (60*I*a^3 + 180*
a^2 - 180*I*a - 60)*b*x - (-60*I*a^4 - 240*a^3 + 360*I*a^2 + 240*a - 60*I)*log((b*x + a - I)/b))/b^5
________________________________________________________________________________________
Sympy [B] time = 13.4273, size = 2315, normalized size = 23.38 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x**4/(1+I*(b*x+a))**2*(1+(b*x+a)**2),x)
[Out]
-x**5/5 - x**4*(I*a**8 + 8*a**7 - 28*I*a**6 - 56*a**5 + 70*I*a**4 + 56*a**3 - 28*I*a**2 - 8*a + I)/(2*a**8*b -
16*I*a**7*b - 56*a**6*b + 112*I*a**5*b + 140*a**4*b - 112*I*a**3*b - 56*a**2*b + 16*I*a*b + 2*b) + x**3*(2*I*
a**17 + 34*a**16 - 272*I*a**15 - 1360*a**14 + 4760*I*a**13 + 12376*a**12 - 24752*I*a**11 - 38896*a**10 + 48620
*I*a**9 + 48620*a**8 - 38896*I*a**7 - 24752*a**6 + 12376*I*a**5 + 4760*a**4 - 1360*I*a**3 - 272*a**2 + 34*I*a
+ 2)/(3*a**16*b**2 - 48*I*a**15*b**2 - 360*a**14*b**2 + 1680*I*a**13*b**2 + 5460*a**12*b**2 - 13104*I*a**11*b*
*2 - 24024*a**10*b**2 + 34320*I*a**9*b**2 + 38610*a**8*b**2 - 34320*I*a**7*b**2 - 24024*a**6*b**2 + 13104*I*a*
*5*b**2 + 5460*a**4*b**2 - 1680*I*a**3*b**2 - 360*a**2*b**2 + 48*I*a*b**2 + 3*b**2) - x**2*(I*a**26 + 26*a**25
- 325*I*a**24 - 2600*a**23 + 14950*I*a**22 + 65780*a**21 - 230230*I*a**20 - 657800*a**19 + 1562275*I*a**18 +
3124550*a**17 - 5311735*I*a**16 - 7726160*a**15 + 9657700*I*a**14 + 10400600*a**13 - 9657700*I*a**12 - 7726160
*a**11 + 5311735*I*a**10 + 3124550*a**9 - 1562275*I*a**8 - 657800*a**7 + 230230*I*a**6 + 65780*a**5 - 14950*I*
a**4 - 2600*a**3 + 325*I*a**2 + 26*a - I)/(a**24*b**3 - 24*I*a**23*b**3 - 276*a**22*b**3 + 2024*I*a**21*b**3 +
10626*a**20*b**3 - 42504*I*a**19*b**3 - 134596*a**18*b**3 + 346104*I*a**17*b**3 + 735471*a**16*b**3 - 1307504
*I*a**15*b**3 - 1961256*a**14*b**3 + 2496144*I*a**13*b**3 + 2704156*a**12*b**3 - 2496144*I*a**11*b**3 - 196125
6*a**10*b**3 + 1307504*I*a**9*b**3 + 735471*a**8*b**3 - 346104*I*a**7*b**3 - 134596*a**6*b**3 + 42504*I*a**5*b
**3 + 10626*a**4*b**3 - 2024*I*a**3*b**3 - 276*a**2*b**3 + 24*I*a*b**3 + b**3) + x*(2*I*a**35 + 70*a**34 - 119
0*I*a**33 - 13090*a**32 + 104720*I*a**31 + 649264*a**30 - 3246320*I*a**29 - 13449040*a**28 + 47071640*I*a**27
+ 141214920*a**26 - 367158792*I*a**25 - 834451800*a**24 + 1668903600*I*a**23 + 2952675600*a**22 - 4639918800*I
*a**21 - 6495886320*a**20 + 8119857900*I*a**19 + 9075135300*a**18 - 9075135300*I*a**17 - 8119857900*a**16 + 64
95886320*I*a**15 + 4639918800*a**14 - 2952675600*I*a**13 - 1668903600*a**12 + 834451800*I*a**11 + 367158792*a*
*10 - 141214920*I*a**9 - 47071640*a**8 + 13449040*I*a**7 + 3246320*a**6 - 649264*I*a**5 - 104720*a**4 + 13090*
I*a**3 + 1190*a**2 - 70*I*a - 2)/(a**32*b**4 - 32*I*a**31*b**4 - 496*a**30*b**4 + 4960*I*a**29*b**4 + 35960*a*
*28*b**4 - 201376*I*a**27*b**4 - 906192*a**26*b**4 + 3365856*I*a**25*b**4 + 10518300*a**24*b**4 - 28048800*I*a
**23*b**4 - 64512240*a**22*b**4 + 129024480*I*a**21*b**4 + 225792840*a**20*b**4 - 347373600*I*a**19*b**4 - 471
435600*a**18*b**4 + 565722720*I*a**17*b**4 + 601080390*a**16*b**4 - 565722720*I*a**15*b**4 - 471435600*a**14*b
**4 + 347373600*I*a**13*b**4 + 225792840*a**12*b**4 - 129024480*I*a**11*b**4 - 64512240*a**10*b**4 + 28048800*
I*a**9*b**4 + 10518300*a**8*b**4 - 3365856*I*a**7*b**4 - 906192*a**6*b**4 + 201376*I*a**5*b**4 + 35960*a**4*b*
*4 - 4960*I*a**3*b**4 - 496*a**2*b**4 + 32*I*a*b**4 + b**4) + 2*(-I*a**44 - 44*a**43 + 946*I*a**42 + 13244*a**
41 - 135751*I*a**40 - 1086008*a**39 + 7059052*I*a**38 + 38320568*a**37 - 177232627*I*a**36 - 708930508*a**35 +
2481256778*I*a**34 + 7669339132*a**33 - 21090682613*I*a**32 - 51915526432*a**31 + 114955808528*I*a**30 + 2299
11617056*a**29 - 416714805914*I*a**28 - 686353797976*a**27 + 1029530696964*I*a**26 + 1408831480056*a**25 - 176
1039350070*I*a**24 - 2012616400080*a**23 + 2104098963720*I*a**22 + 2012616400080*a**21 - 1761039350070*I*a**20
- 1408831480056*a**19 + 1029530696964*I*a**18 + 686353797976*a**17 - 416714805914*I*a**16 - 229911617056*a**1
5 + 114955808528*I*a**14 + 51915526432*a**13 - 21090682613*I*a**12 - 7669339132*a**11 + 2481256778*I*a**10 + 7
08930508*a**9 - 177232627*I*a**8 - 38320568*a**7 + 7059052*I*a**6 + 1086008*a**5 - 135751*I*a**4 - 13244*a**3
+ 946*I*a**2 + 44*a - I)*log(-a**41 + 41*I*a**40 + 820*a**39 - 10660*I*a**38 - 101270*a**37 + 749398*I*a**36 +
4496388*a**35 - 22481940*I*a**34 - 95548245*a**33 + 350343565*I*a**32 + 1121099408*a**31 - 3159461968*I*a**30
- 7898654920*a**29 + 17620076360*I*a**28 + 35240152720*a**27 - 63432274896*I*a**26 - 103077446706*a**25 + 151
584480450*I*a**24 + 202112640600*a**23 - 244662670200*I*a**22 - 269128937220*a**21 + 269128937220*I*a**20 + 24
4662670200*a**19 - 202112640600*I*a**18 - 151584480450*a**17 + 103077446706*I*a**16 + 63432274896*a**15 - 3524
0152720*I*a**14 - 17620076360*a**13 + 7898654920*I*a**12 + 3159461968*a**11 - 1121099408*I*a**10 - 350343565*a
**9 + 95548245*I*a**8 + 22481940*a**7 - 4496388*I*a**6 - 749398*a**5 + 101270*I*a**4 + 10660*a**3 - 820*I*a**2
- 41*a + x*(-a**40*b + 40*I*a**39*b + 780*a**38*b - 9880*I*a**37*b - 91390*a**36*b + 658008*I*a**35*b + 38383
80*a**34*b - 18643560*I*a**33*b - 76904685*a**32*b + 273438880*I*a**31*b + 847660528*a**30*b - 2311801440*I*a*
*29*b - 5586853480*a**28*b + 12033222880*I*a**27*b + 23206929840*a**26*b - 40225345056*I*a**25*b - 62852101650
*a**24*b + 88732378800*I*a**23*b + 113380261800*a**22*b - 131282408400*I*a**21*b - 137846528820*a**20*b + 1312
82408400*I*a**19*b + 113380261800*a**18*b - 88732378800*I*a**17*b - 62852101650*a**16*b + 40225345056*I*a**15*
b + 23206929840*a**14*b - 12033222880*I*a**13*b - 5586853480*a**12*b + 2311801440*I*a**11*b + 847660528*a**10*
b - 273438880*I*a**9*b - 76904685*a**8*b + 18643560*I*a**7*b + 3838380*a**6*b - 658008*I*a**5*b - 91390*a**4*b
+ 9880*I*a**3*b + 780*a**2*b - 40*I*a*b - b) + I)/(b**5*(a**40 - 40*I*a**39 - 780*a**38 + 9880*I*a**37 + 9139
0*a**36 - 658008*I*a**35 - 3838380*a**34 + 18643560*I*a**33 + 76904685*a**32 - 273438880*I*a**31 - 847660528*a
**30 + 2311801440*I*a**29 + 5586853480*a**28 - 12033222880*I*a**27 - 23206929840*a**26 + 40225345056*I*a**25 +
62852101650*a**24 - 88732378800*I*a**23 - 113380261800*a**22 + 131282408400*I*a**21 + 137846528820*a**20 - 13
1282408400*I*a**19 - 113380261800*a**18 + 88732378800*I*a**17 + 62852101650*a**16 - 40225345056*I*a**15 - 2320
6929840*a**14 + 12033222880*I*a**13 + 5586853480*a**12 - 2311801440*I*a**11 - 847660528*a**10 + 273438880*I*a*
*9 + 76904685*a**8 - 18643560*I*a**7 - 3838380*a**6 + 658008*I*a**5 + 91390*a**4 - 9880*I*a**3 - 780*a**2 + 40
*I*a + 1))
________________________________________________________________________________________
Giac [B] time = 1.12598, size = 317, normalized size = 3.2 \begin{align*} \frac{2 \,{\left (a^{4} i + 4 \, a^{3} - 6 \, a^{2} i - 4 \, a + i\right )} \log \left (\frac{1}{\sqrt{{\left (b x + a\right )}^{2} + 1}{\left | b \right |}}\right )}{b^{5}} + \frac{{\left (b i x + a i + 1\right )}^{5}{\left (\frac{15 \,{\left (2 \, a b - 3 \, b i\right )} i}{{\left (b i x + a i + 1\right )} b} - \frac{20 \,{\left (3 \, a^{2} b^{2} - 10 \, a b^{2} i - 7 \, b^{2}\right )} i^{2}}{{\left (b i x + a i + 1\right )}^{2} b^{2}} + \frac{60 \,{\left (a^{3} b^{3} - 6 \, a^{2} b^{3} i - 9 \, a b^{3} + 4 \, b^{3} i\right )} i^{3}}{{\left (b i x + a i + 1\right )}^{3} b^{3}} - \frac{30 \,{\left (a^{4} b^{4} - 12 \, a^{3} b^{4} i - 30 \, a^{2} b^{4} + 28 \, a b^{4} i + 9 \, b^{4}\right )} i^{4}}{{\left (b i x + a i + 1\right )}^{4} b^{4}} - 6\right )}}{30 \, b^{5} i^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^4/(1+I*(b*x+a))^2*(1+(b*x+a)^2),x, algorithm="giac")
[Out]
2*(a^4*i + 4*a^3 - 6*a^2*i - 4*a + i)*log(1/(sqrt((b*x + a)^2 + 1)*abs(b)))/b^5 + 1/30*(b*i*x + a*i + 1)^5*(15
*(2*a*b - 3*b*i)*i/((b*i*x + a*i + 1)*b) - 20*(3*a^2*b^2 - 10*a*b^2*i - 7*b^2)*i^2/((b*i*x + a*i + 1)^2*b^2) +
60*(a^3*b^3 - 6*a^2*b^3*i - 9*a*b^3 + 4*b^3*i)*i^3/((b*i*x + a*i + 1)^3*b^3) - 30*(a^4*b^4 - 12*a^3*b^4*i - 3
0*a^2*b^4 + 28*a*b^4*i + 9*b^4)*i^4/((b*i*x + a*i + 1)^4*b^4) - 6)/(b^5*i^5)