3.172 \(\int e^{2 i \tan ^{-1}(a+b x)} x^3 \, dx\)
Optimal. Leaf size=72 \[ \frac{(1-i a) x^2}{b^2}+\frac{2 i (a+i)^2 x}{b^3}-\frac{2 (1-i a)^3 \log (a+b x+i)}{b^4}+\frac{2 i x^3}{3 b}-\frac{x^4}{4} \]
[Out]
((2*I)*(I + a)^2*x)/b^3 + ((1 - I*a)*x^2)/b^2 + (((2*I)/3)*x^3)/b - x^4/4 - (2*(1 - I*a)^3*Log[I + a + b*x])/b
^4
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Rubi [A] time = 0.0586489, antiderivative size = 72, 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} \[ \frac{(1-i a) x^2}{b^2}+\frac{2 i (a+i)^2 x}{b^3}-\frac{2 (1-i a)^3 \log (a+b x+i)}{b^4}+\frac{2 i x^3}{3 b}-\frac{x^4}{4} \]
Antiderivative was successfully verified.
[In]
Int[E^((2*I)*ArcTan[a + b*x])*x^3,x]
[Out]
((2*I)*(I + a)^2*x)/b^3 + ((1 - I*a)*x^2)/b^2 + (((2*I)/3)*x^3)/b - x^4/4 - (2*(1 - I*a)^3*Log[I + a + b*x])/b
^4
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^3 \, dx &=\int \frac{x^3 (1+i a+i b x)}{1-i a-i b x} \, dx\\ &=\int \left (\frac{2 i (i+a)^2}{b^3}+\frac{2 (1-i a) x}{b^2}+\frac{2 i x^2}{b}-x^3+\frac{2 (-1+i a)^3}{b^3 (i+a+b x)}\right ) \, dx\\ &=\frac{2 i (i+a)^2 x}{b^3}+\frac{(1-i a) x^2}{b^2}+\frac{2 i x^3}{3 b}-\frac{x^4}{4}-\frac{2 (1-i a)^3 \log (i+a+b x)}{b^4}\\ \end{align*}
Mathematica [A] time = 0.0575975, size = 72, normalized size = 1. \[ \frac{(1-i a) x^2}{b^2}+\frac{2 i (a+i)^2 x}{b^3}-\frac{2 (1-i a)^3 \log (a+b x+i)}{b^4}+\frac{2 i x^3}{3 b}-\frac{x^4}{4} \]
Antiderivative was successfully verified.
[In]
Integrate[E^((2*I)*ArcTan[a + b*x])*x^3,x]
[Out]
((2*I)*(I + a)^2*x)/b^3 + ((1 - I*a)*x^2)/b^2 + (((2*I)/3)*x^3)/b - x^4/4 - (2*(1 - I*a)^3*Log[I + a + b*x])/b
^4
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Maple [B] time = 0.038, size = 255, normalized size = 3.5 \begin{align*} -{\frac{{x}^{4}}{4}}+{\frac{{\frac{2\,i}{3}}{x}^{3}}{b}}-{\frac{i{x}^{2}a}{{b}^{2}}}+{\frac{2\,i{a}^{2}x}{{b}^{3}}}+{\frac{{x}^{2}}{{b}^{2}}}-{\frac{2\,ix}{{b}^{3}}}-4\,{\frac{ax}{{b}^{3}}}-{\frac{i\ln \left ({b}^{2}{x}^{2}+2\,xab+{a}^{2}+1 \right ){a}^{3}}{{b}^{4}}}+{\frac{3\,i\ln \left ({b}^{2}{x}^{2}+2\,xab+{a}^{2}+1 \right ) a}{{b}^{4}}}+3\,{\frac{\ln \left ({b}^{2}{x}^{2}+2\,xab+{a}^{2}+1 \right ){a}^{2}}{{b}^{4}}}-{\frac{\ln \left ({b}^{2}{x}^{2}+2\,xab+{a}^{2}+1 \right ) }{{b}^{4}}}-{\frac{6\,i{a}^{2}}{{b}^{4}}\arctan \left ({\frac{2\,{b}^{2}x+2\,ab}{2\,b}} \right ) }-2\,{\frac{{a}^{3}}{{b}^{4}}\arctan \left ( 1/2\,{\frac{2\,{b}^{2}x+2\,ab}{b}} \right ) }+{\frac{2\,i}{{b}^{4}}\arctan \left ({\frac{2\,{b}^{2}x+2\,ab}{2\,b}} \right ) }+6\,{\frac{a}{{b}^{4}}\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^3,x)
[Out]
-1/4*x^4+2/3*I*x^3/b-I/b^2*x^2*a+2*I/b^3*a^2*x+1/b^2*x^2-2*I/b^3*x-4/b^3*a*x-I/b^4*ln(b^2*x^2+2*a*b*x+a^2+1)*a
^3+3*I/b^4*ln(b^2*x^2+2*a*b*x+a^2+1)*a+3/b^4*ln(b^2*x^2+2*a*b*x+a^2+1)*a^2-1/b^4*ln(b^2*x^2+2*a*b*x+a^2+1)-6*I
/b^4*arctan(1/2*(2*b^2*x+2*a*b)/b)*a^2-2/b^4*arctan(1/2*(2*b^2*x+2*a*b)/b)*a^3+2*I/b^4*arctan(1/2*(2*b^2*x+2*a
*b)/b)+6/b^4*arctan(1/2*(2*b^2*x+2*a*b)/b)*a
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Maxima [B] time = 1.45958, size = 159, normalized size = 2.21 \begin{align*} -\frac{3 \, b^{3} x^{4} - 8 i \, b^{2} x^{3} + 12 \,{\left (i \, a - 1\right )} b x^{2} -{\left (24 i \, a^{2} - 48 \, a - 24 i\right )} x}{12 \, b^{3}} - \frac{{\left (2 \, a^{3} + 6 i \, a^{2} - 6 \, a - 2 i\right )} \arctan \left (\frac{b^{2} x + a b}{b}\right )}{b^{4}} + \frac{{\left (-i \, a^{3} + 3 \, a^{2} + 3 i \, a - 1\right )} \log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}{b^{4}} \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^3,x, algorithm="maxima")
[Out]
-1/12*(3*b^3*x^4 - 8*I*b^2*x^3 + 12*(I*a - 1)*b*x^2 - (24*I*a^2 - 48*a - 24*I)*x)/b^3 - (2*a^3 + 6*I*a^2 - 6*a
- 2*I)*arctan((b^2*x + a*b)/b)/b^4 + (-I*a^3 + 3*a^2 + 3*I*a - 1)*log(b^2*x^2 + 2*a*b*x + a^2 + 1)/b^4
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Fricas [A] time = 1.55231, size = 201, normalized size = 2.79 \begin{align*} -\frac{3 \, b^{4} x^{4} - 8 i \, b^{3} x^{3} + 12 \,{\left (i \, a - 1\right )} b^{2} x^{2} -{\left (24 i \, a^{2} - 48 \, a - 24 i\right )} b x -{\left (-24 i \, a^{3} + 72 \, a^{2} + 72 i \, a - 24\right )} \log \left (\frac{b x + a + i}{b}\right )}{12 \, b^{4}} \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^3,x, algorithm="fricas")
[Out]
-1/12*(3*b^4*x^4 - 8*I*b^3*x^3 + 12*(I*a - 1)*b^2*x^2 - (24*I*a^2 - 48*a - 24*I)*b*x - (-24*I*a^3 + 72*a^2 + 7
2*I*a - 24)*log((b*x + a + I)/b))/b^4
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Sympy [B] time = 6.43326, size = 1212, normalized size = 16.83 \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**3,x)
[Out]
-x**4/4 + x**3*(2*I*a**6 - 12*a**5 - 30*I*a**4 + 40*a**3 + 30*I*a**2 - 12*a - 2*I)/(3*a**6*b + 18*I*a**5*b - 4
5*a**4*b - 60*I*a**3*b + 45*a**2*b + 18*I*a*b - 3*b) - x**2*(I*a**13 - 13*a**12 - 78*I*a**11 + 286*a**10 + 715
*I*a**9 - 1287*a**8 - 1716*I*a**7 + 1716*a**6 + 1287*I*a**5 - 715*a**4 - 286*I*a**3 + 78*a**2 + 13*I*a - 1)/(a
**12*b**2 + 12*I*a**11*b**2 - 66*a**10*b**2 - 220*I*a**9*b**2 + 495*a**8*b**2 + 792*I*a**7*b**2 - 924*a**6*b**
2 - 792*I*a**5*b**2 + 495*a**4*b**2 + 220*I*a**3*b**2 - 66*a**2*b**2 - 12*I*a*b**2 + b**2) + x*(2*I*a**20 - 40
*a**19 - 380*I*a**18 + 2280*a**17 + 9690*I*a**16 - 31008*a**15 - 77520*I*a**14 + 155040*a**13 + 251940*I*a**12
- 335920*a**11 - 369512*I*a**10 + 335920*a**9 + 251940*I*a**8 - 155040*a**7 - 77520*I*a**6 + 31008*a**5 + 969
0*I*a**4 - 2280*a**3 - 380*I*a**2 + 40*a + 2*I)/(a**18*b**3 + 18*I*a**17*b**3 - 153*a**16*b**3 - 816*I*a**15*b
**3 + 3060*a**14*b**3 + 8568*I*a**13*b**3 - 18564*a**12*b**3 - 31824*I*a**11*b**3 + 43758*a**10*b**3 + 48620*I
*a**9*b**3 - 43758*a**8*b**3 - 31824*I*a**7*b**3 + 18564*a**6*b**3 + 8568*I*a**5*b**3 - 3060*a**4*b**3 - 816*I
*a**3*b**3 + 153*a**2*b**3 + 18*I*a*b**3 - b**3) + 2*(-I*a**27 + 27*a**26 + 351*I*a**25 - 2925*a**24 - 17550*I
*a**23 + 80730*a**22 + 296010*I*a**21 - 888030*a**20 - 2220075*I*a**19 + 4686825*a**18 + 8436285*I*a**17 - 130
37895*a**16 - 17383860*I*a**15 + 20058300*a**14 + 20058300*I*a**13 - 17383860*a**12 - 13037895*I*a**11 + 84362
85*a**10 + 4686825*I*a**9 - 2220075*a**8 - 888030*I*a**7 + 296010*a**6 + 80730*I*a**5 - 17550*a**4 - 2925*I*a*
*3 + 351*a**2 + 27*I*a - 1)*log(-a**25 - 25*I*a**24 + 300*a**23 + 2300*I*a**22 - 12650*a**21 - 53130*I*a**20 +
177100*a**19 + 480700*I*a**18 - 1081575*a**17 - 2042975*I*a**16 + 3268760*a**15 + 4457400*I*a**14 - 5200300*a
**13 - 5200300*I*a**12 + 4457400*a**11 + 3268760*I*a**10 - 2042975*a**9 - 1081575*I*a**8 + 480700*a**7 + 17710
0*I*a**6 - 53130*a**5 - 12650*I*a**4 + 2300*a**3 + 300*I*a**2 - 25*a + x*(-a**24*b - 24*I*a**23*b + 276*a**22*
b + 2024*I*a**21*b - 10626*a**20*b - 42504*I*a**19*b + 134596*a**18*b + 346104*I*a**17*b - 735471*a**16*b - 13
07504*I*a**15*b + 1961256*a**14*b + 2496144*I*a**13*b - 2704156*a**12*b - 2496144*I*a**11*b + 1961256*a**10*b
+ 1307504*I*a**9*b - 735471*a**8*b - 346104*I*a**7*b + 134596*a**6*b + 42504*I*a**5*b - 10626*a**4*b - 2024*I*
a**3*b + 276*a**2*b + 24*I*a*b - b) - I)/(b**4*(a**24 + 24*I*a**23 - 276*a**22 - 2024*I*a**21 + 10626*a**20 +
42504*I*a**19 - 134596*a**18 - 346104*I*a**17 + 735471*a**16 + 1307504*I*a**15 - 1961256*a**14 - 2496144*I*a**
13 + 2704156*a**12 + 2496144*I*a**11 - 1961256*a**10 - 1307504*I*a**9 + 735471*a**8 + 346104*I*a**7 - 134596*a
**6 - 42504*I*a**5 + 10626*a**4 + 2024*I*a**3 - 276*a**2 - 24*I*a + 1))
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Giac [A] time = 1.09609, size = 119, normalized size = 1.65 \begin{align*} -\frac{2 \,{\left (a^{3} i - 3 \, a^{2} - 3 \, a i + 1\right )} \log \left (b x + a + i\right )}{b^{4}} - \frac{3 \, b^{4} x^{4} - 8 \, b^{3} i x^{3} + 12 \, a b^{2} i x^{2} - 24 \, a^{2} b i x - 12 \, b^{2} x^{2} + 48 \, a b x + 24 \, b i x}{12 \, b^{4}} \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^3,x, algorithm="giac")
[Out]
-2*(a^3*i - 3*a^2 - 3*a*i + 1)*log(b*x + a + i)/b^4 - 1/12*(3*b^4*x^4 - 8*b^3*i*x^3 + 12*a*b^2*i*x^2 - 24*a^2*
b*i*x - 12*b^2*x^2 + 48*a*b*x + 24*b*i*x)/b^4