3.171 \(\int e^{2 i \tan ^{-1}(a+b x)} x^4 \, dx\)
Optimal. Leaf size=92 \[ \frac{2 (1-i a) x^3}{3 b^2}+\frac{i (a+i)^2 x^2}{b^3}-\frac{2 (1-i a)^3 x}{b^4}+\frac{2 i (a+i)^4 \log (a+b x+i)}{b^5}+\frac{i x^4}{2 b}-\frac{x^5}{5} \]
[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 = 0.0828601, antiderivative size = 92, 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[E^((2*I)*ArcTan[a + b*x])*x^4,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 = 0.066813, size = 92, normalized size = 1. \[ \frac{2 (1-i a) x^3}{3 b^2}+\frac{i (a+i)^2 x^2}{b^3}-\frac{2 (1-i a)^3 x}{b^4}+\frac{2 i (a+i)^4 \log (a+b x+i)}{b^5}+\frac{i x^4}{2 b}-\frac{x^5}{5} \]
Antiderivative was successfully verified.
[In]
Integrate[E^((2*I)*ArcTan[a + b*x])*x^4,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 = 0.039, size = 347, normalized size = 3.8 \begin{align*} -{\frac{{x}^{5}}{5}}-{\frac{8\,ia}{{b}^{5}}\arctan \left ({\frac{2\,{b}^{2}x+2\,ab}{2\,b}} \right ) }+{\frac{8\,i{a}^{3}}{{b}^{5}}\arctan \left ({\frac{2\,{b}^{2}x+2\,ab}{2\,b}} \right ) }-{\frac{{\frac{2\,i}{3}}{x}^{3}a}{{b}^{2}}}-{\frac{i{x}^{2}}{{b}^{3}}}+{\frac{2\,{x}^{3}}{3\,{b}^{2}}}+{\frac{6\,iax}{{b}^{4}}}-2\,{\frac{a{x}^{2}}{{b}^{3}}}+{\frac{i\ln \left ({b}^{2}{x}^{2}+2\,xab+{a}^{2}+1 \right ) }{{b}^{5}}}+6\,{\frac{{a}^{2}x}{{b}^{4}}}-2\,{\frac{x}{{b}^{4}}}-{\frac{2\,i{a}^{3}x}{{b}^{4}}}+{\frac{i\ln \left ({b}^{2}{x}^{2}+2\,xab+{a}^{2}+1 \right ){a}^{4}}{{b}^{5}}}-4\,{\frac{\ln \left ({b}^{2}{x}^{2}+2\,xab+{a}^{2}+1 \right ){a}^{3}}{{b}^{5}}}+{\frac{i{x}^{2}{a}^{2}}{{b}^{3}}}+4\,{\frac{\ln \left ({b}^{2}{x}^{2}+2\,xab+{a}^{2}+1 \right ) a}{{b}^{5}}}+{\frac{{\frac{i}{2}}{x}^{4}}{b}}+2\,{\frac{{a}^{4}}{{b}^{5}}\arctan \left ( 1/2\,{\frac{2\,{b}^{2}x+2\,ab}{b}} \right ) }-{\frac{6\,i\ln \left ({b}^{2}{x}^{2}+2\,xab+{a}^{2}+1 \right ){a}^{2}}{{b}^{5}}}-12\,{\frac{{a}^{2}}{{b}^{5}}\arctan \left ( 1/2\,{\frac{2\,{b}^{2}x+2\,ab}{b}} \right ) }+2\,{\frac{1}{{b}^{5}}\arctan \left ( 1/2\,{\frac{2\,{b}^{2}x+2\,ab}{b}} \right ) } \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^4,x)
[Out]
-1/5*x^5-8*I/b^5*arctan(1/2*(2*b^2*x+2*a*b)/b)*a+8*I/b^5*arctan(1/2*(2*b^2*x+2*a*b)/b)*a^3-2/3*I/b^2*x^3*a-I/b
^3*x^2+2/3/b^2*x^3+6*I/b^4*a*x-2/b^3*x^2*a+I/b^5*ln(b^2*x^2+2*a*b*x+a^2+1)+6/b^4*a^2*x-2/b^4*x-2*I/b^4*a^3*x+I
/b^5*ln(b^2*x^2+2*a*b*x+a^2+1)*a^4-4/b^5*ln(b^2*x^2+2*a*b*x+a^2+1)*a^3+I/b^3*x^2*a^2+4/b^5*ln(b^2*x^2+2*a*b*x+
a^2+1)*a+1/2*I*x^4/b+2/b^5*arctan(1/2*(2*b^2*x+2*a*b)/b)*a^4-6*I/b^5*ln(b^2*x^2+2*a*b*x+a^2+1)*a^2-12/b^5*arct
an(1/2*(2*b^2*x+2*a*b)/b)*a^2+2/b^5*arctan(1/2*(2*b^2*x+2*a*b)/b)
________________________________________________________________________________________
Maxima [B] time = 1.49865, size = 203, normalized size = 2.21 \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 \, a^{4} + 8 i \, a^{3} - 12 \, a^{2} - 8 i \, a + 2\right )} \arctan \left (\frac{b^{2} x + a b}{b}\right )}{b^{5}} + \frac{{\left (i \, a^{4} - 4 \, a^{3} - 6 i \, a^{2} + 4 \, a + i\right )} \log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}{b^{5}} \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^4,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*a^4 + 8*I*a^3 - 12*a^2 - 8*I*a + 2)*arctan((b^2*x + a*b)/b)/b^5 + (I*a^4 - 4*a^3
- 6*I*a^2 + 4*a + I)*log(b^2*x^2 + 2*a*b*x + a^2 + 1)/b^5
________________________________________________________________________________________
Fricas [A] time = 1.57646, size = 284, normalized size = 3.09 \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((1+I*(b*x+a))^2/(1+(b*x+a)^2)*x^4,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 = 13.4752, size = 2315, normalized size = 25.16 \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**4,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**4
1 + 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 + 22991
1617056*a**29 + 416714805914*I*a**28 - 686353797976*a**27 - 1029530696964*I*a**26 + 1408831480056*a**25 + 1761
039350070*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**15
- 114955808528*I*a**14 + 51915526432*a**13 + 21090682613*I*a**12 - 7669339132*a**11 - 2481256778*I*a**10 + 70
8930508*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 - 1515
84480450*I*a**24 + 202112640600*a**23 + 244662670200*I*a**22 - 269128937220*a**21 - 269128937220*I*a**20 + 244
662670200*a**19 + 202112640600*I*a**18 - 151584480450*a**17 - 103077446706*I*a**16 + 63432274896*a**15 + 35240
152720*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 + 383838
0*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 - 13128
2408400*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 + 91390
*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 + 131
282408400*I*a**19 - 113380261800*a**18 - 88732378800*I*a**17 + 62852101650*a**16 + 40225345056*I*a**15 - 23206
929840*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 [A] time = 1.11349, size = 176, normalized size = 1.91 \begin{align*} \frac{2 \,{\left (a^{4} i - 4 \, a^{3} - 6 \, a^{2} i + 4 \, a + i\right )} \log \left (b x + a + i\right )}{b^{5}} - \frac{6 \, b^{5} x^{5} - 15 \, b^{4} i x^{4} + 20 \, a b^{3} i x^{3} - 30 \, a^{2} b^{2} i x^{2} + 60 \, a^{3} b i x - 20 \, b^{3} x^{3} + 60 \, a b^{2} x^{2} + 30 \, b^{2} i x^{2} - 180 \, a^{2} b x - 180 \, a b i x + 60 \, b x}{30 \, b^{5}} \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^4,x, algorithm="giac")
[Out]
2*(a^4*i - 4*a^3 - 6*a^2*i + 4*a + i)*log(b*x + a + i)/b^5 - 1/30*(6*b^5*x^5 - 15*b^4*i*x^4 + 20*a*b^3*i*x^3 -
30*a^2*b^2*i*x^2 + 60*a^3*b*i*x - 20*b^3*x^3 + 60*a*b^2*x^2 + 30*b^2*i*x^2 - 180*a^2*b*x - 180*a*b*i*x + 60*b
*x)/b^5