3.173 \(\int e^{2 i \tan ^{-1}(a+b x)} x^2 \, dx\)

Optimal. Leaf size=54 \[ \frac{2 (1-i a) x}{b^2}+\frac{2 i (a+i)^2 \log (a+b x+i)}{b^3}+\frac{i x^2}{b}-\frac{x^3}{3} \]

[Out]

(2*(1 - I*a)*x)/b^2 + (I*x^2)/b - x^3/3 + ((2*I)*(I + a)^2*Log[I + a + b*x])/b^3

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Rubi [A]  time = 0.0478082, antiderivative size = 54, 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} \[ \frac{2 (1-i a) x}{b^2}+\frac{2 i (a+i)^2 \log (a+b x+i)}{b^3}+\frac{i x^2}{b}-\frac{x^3}{3} \]

Antiderivative was successfully verified.

[In]

Int[E^((2*I)*ArcTan[a + b*x])*x^2,x]

[Out]

(2*(1 - I*a)*x)/b^2 + (I*x^2)/b - x^3/3 + ((2*I)*(I + a)^2*Log[I + a + b*x])/b^3

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^2 \, dx &=\int \frac{x^2 (1+i a+i b x)}{1-i a-i b x} \, dx\\ &=\int \left (-\frac{2 i (i+a)}{b^2}+\frac{2 i x}{b}-x^2+\frac{2 i (i+a)^2}{b^2 (i+a+b x)}\right ) \, dx\\ &=\frac{2 (1-i a) x}{b^2}+\frac{i x^2}{b}-\frac{x^3}{3}+\frac{2 i (i+a)^2 \log (i+a+b x)}{b^3}\\ \end{align*}

Mathematica [A]  time = 0.0318999, size = 54, normalized size = 1. \[ \frac{2 (1-i a) x}{b^2}+\frac{2 i (a+i)^2 \log (a+b x+i)}{b^3}+\frac{i x^2}{b}-\frac{x^3}{3} \]

Antiderivative was successfully verified.

[In]

Integrate[E^((2*I)*ArcTan[a + b*x])*x^2,x]

[Out]

(2*(1 - I*a)*x)/b^2 + (I*x^2)/b - x^3/3 + ((2*I)*(I + a)^2*Log[I + a + b*x])/b^3

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Maple [B]  time = 0.037, size = 176, normalized size = 3.3 \begin{align*} -{\frac{{x}^{3}}{3}}+{\frac{i{x}^{2}}{b}}-{\frac{2\,iax}{{b}^{2}}}+2\,{\frac{x}{{b}^{2}}}+{\frac{i\ln \left ({b}^{2}{x}^{2}+2\,xab+{a}^{2}+1 \right ){a}^{2}}{{b}^{3}}}-{\frac{i\ln \left ({b}^{2}{x}^{2}+2\,xab+{a}^{2}+1 \right ) }{{b}^{3}}}-2\,{\frac{\ln \left ({b}^{2}{x}^{2}+2\,xab+{a}^{2}+1 \right ) a}{{b}^{3}}}+{\frac{4\,ia}{{b}^{3}}\arctan \left ({\frac{2\,{b}^{2}x+2\,ab}{2\,b}} \right ) }+2\,{\frac{{a}^{2}}{{b}^{3}}\arctan \left ( 1/2\,{\frac{2\,{b}^{2}x+2\,ab}{b}} \right ) }-2\,{\frac{1}{{b}^{3}}\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^2,x)

[Out]

-1/3*x^3+I*x^2/b-2*I/b^2*a*x+2/b^2*x+I/b^3*ln(b^2*x^2+2*a*b*x+a^2+1)*a^2-I/b^3*ln(b^2*x^2+2*a*b*x+a^2+1)-2/b^3
*ln(b^2*x^2+2*a*b*x+a^2+1)*a+4*I/b^3*arctan(1/2*(2*b^2*x+2*a*b)/b)*a+2/b^3*arctan(1/2*(2*b^2*x+2*a*b)/b)*a^2-2
/b^3*arctan(1/2*(2*b^2*x+2*a*b)/b)

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Maxima [B]  time = 1.50306, size = 117, normalized size = 2.17 \begin{align*} -\frac{b^{2} x^{3} - 3 i \, b x^{2} + 6 \,{\left (i \, a - 1\right )} x}{3 \, b^{2}} + \frac{2 \,{\left (a^{2} + 2 i \, a - 1\right )} \arctan \left (\frac{b^{2} x + a b}{b}\right )}{b^{3}} + \frac{{\left (i \, a^{2} - 2 \, a - i\right )} \log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}{b^{3}} \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^2,x, algorithm="maxima")

[Out]

-1/3*(b^2*x^3 - 3*I*b*x^2 + 6*(I*a - 1)*x)/b^2 + 2*(a^2 + 2*I*a - 1)*arctan((b^2*x + a*b)/b)/b^3 + (I*a^2 - 2*
a - I)*log(b^2*x^2 + 2*a*b*x + a^2 + 1)/b^3

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Fricas [A]  time = 1.61974, size = 132, normalized size = 2.44 \begin{align*} -\frac{b^{3} x^{3} - 3 i \, b^{2} x^{2} + 6 \,{\left (i \, a - 1\right )} b x -{\left (6 i \, a^{2} - 12 \, a - 6 i\right )} \log \left (\frac{b x + a + i}{b}\right )}{3 \, b^{3}} \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^2,x, algorithm="fricas")

[Out]

-1/3*(b^3*x^3 - 3*I*b^2*x^2 + 6*(I*a - 1)*b*x - (6*I*a^2 - 12*a - 6*I)*log((b*x + a + I)/b))/b^3

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Sympy [B]  time = 2.6935, size = 513, normalized size = 9.5 \begin{align*} - \frac{x^{3}}{3} + \frac{x^{2} \left (i a^{4} - 4 a^{3} - 6 i a^{2} + 4 a + i\right )}{a^{4} b + 4 i a^{3} b - 6 a^{2} b - 4 i a b + b} - \frac{x \left (2 i a^{9} - 18 a^{8} - 72 i a^{7} + 168 a^{6} + 252 i a^{5} - 252 a^{4} - 168 i a^{3} + 72 a^{2} + 18 i a - 2\right )}{a^{8} b^{2} + 8 i a^{7} b^{2} - 28 a^{6} b^{2} - 56 i a^{5} b^{2} + 70 a^{4} b^{2} + 56 i a^{3} b^{2} - 28 a^{2} b^{2} - 8 i a b^{2} + b^{2}} + \frac{2 \left (i a^{14} - 14 a^{13} - 91 i a^{12} + 364 a^{11} + 1001 i a^{10} - 2002 a^{9} - 3003 i a^{8} + 3432 a^{7} + 3003 i a^{6} - 2002 a^{5} - 1001 i a^{4} + 364 a^{3} + 91 i a^{2} - 14 a - i\right ) \log{\left (- a^{13} - 13 i a^{12} + 78 a^{11} + 286 i a^{10} - 715 a^{9} - 1287 i a^{8} + 1716 a^{7} + 1716 i a^{6} - 1287 a^{5} - 715 i a^{4} + 286 a^{3} + 78 i a^{2} - 13 a + x \left (- a^{12} b - 12 i a^{11} b + 66 a^{10} b + 220 i a^{9} b - 495 a^{8} b - 792 i a^{7} b + 924 a^{6} b + 792 i a^{5} b - 495 a^{4} b - 220 i a^{3} b + 66 a^{2} b + 12 i a b - b\right ) - i \right )}}{b^{3} \left (a^{12} + 12 i a^{11} - 66 a^{10} - 220 i a^{9} + 495 a^{8} + 792 i a^{7} - 924 a^{6} - 792 i a^{5} + 495 a^{4} + 220 i a^{3} - 66 a^{2} - 12 i a + 1\right )} \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**2,x)

[Out]

-x**3/3 + x**2*(I*a**4 - 4*a**3 - 6*I*a**2 + 4*a + I)/(a**4*b + 4*I*a**3*b - 6*a**2*b - 4*I*a*b + b) - x*(2*I*
a**9 - 18*a**8 - 72*I*a**7 + 168*a**6 + 252*I*a**5 - 252*a**4 - 168*I*a**3 + 72*a**2 + 18*I*a - 2)/(a**8*b**2
+ 8*I*a**7*b**2 - 28*a**6*b**2 - 56*I*a**5*b**2 + 70*a**4*b**2 + 56*I*a**3*b**2 - 28*a**2*b**2 - 8*I*a*b**2 +
b**2) + 2*(I*a**14 - 14*a**13 - 91*I*a**12 + 364*a**11 + 1001*I*a**10 - 2002*a**9 - 3003*I*a**8 + 3432*a**7 +
3003*I*a**6 - 2002*a**5 - 1001*I*a**4 + 364*a**3 + 91*I*a**2 - 14*a - I)*log(-a**13 - 13*I*a**12 + 78*a**11 +
286*I*a**10 - 715*a**9 - 1287*I*a**8 + 1716*a**7 + 1716*I*a**6 - 1287*a**5 - 715*I*a**4 + 286*a**3 + 78*I*a**2
- 13*a + x*(-a**12*b - 12*I*a**11*b + 66*a**10*b + 220*I*a**9*b - 495*a**8*b - 792*I*a**7*b + 924*a**6*b + 79
2*I*a**5*b - 495*a**4*b - 220*I*a**3*b + 66*a**2*b + 12*I*a*b - b) - I)/(b**3*(a**12 + 12*I*a**11 - 66*a**10 -
220*I*a**9 + 495*a**8 + 792*I*a**7 - 924*a**6 - 792*I*a**5 + 495*a**4 + 220*I*a**3 - 66*a**2 - 12*I*a + 1))

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Giac [A]  time = 1.09059, size = 77, normalized size = 1.43 \begin{align*} \frac{2 \,{\left (a^{2} i - 2 \, a - i\right )} \log \left (b x + a + i\right )}{b^{3}} - \frac{b^{3} x^{3} - 3 \, b^{2} i x^{2} + 6 \, a b i x - 6 \, b x}{3 \, b^{3}} \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^2,x, algorithm="giac")

[Out]

2*(a^2*i - 2*a - i)*log(b*x + a + i)/b^3 - 1/3*(b^3*x^3 - 3*b^2*i*x^2 + 6*a*b*i*x - 6*b*x)/b^3