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

Optimal. Leaf size=59 \[ \frac{2 (1+i a) x}{b^2}-\frac{2 i (-a+i)^2 \log (-a-b x+i)}{b^3}-\frac{i x^2}{b}-\frac{x^3}{3} \]

[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 = 0.047512, antiderivative size = 59, 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[x^2/E^((2*I)*ArcTan[a + b*x]),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 = 0.0340041, size = 55, normalized size = 0.93 \[ \frac{b x \left (6 i a-b^2 x^2-3 i b x+6\right )-6 i (a-i)^2 \log (-a-b x+i)}{3 b^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*x*(6 + (6*I)*a - (3*I)*b*x - b^2*x^2) - (6*I)*(-I + a)^2*Log[I - a - b*x])/(3*b^3)

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Maple [B]  time = 0.043, size = 143, normalized size = 2.4 \begin{align*} -{\frac{{x}^{3}}{3}}-{\frac{i{x}^{2}}{b}}+{\frac{2\,iax}{{b}^{2}}}+2\,{\frac{x}{{b}^{2}}}+2\,{\frac{\arctan \left ( bx+a \right ){a}^{2}}{{b}^{3}}}-{\frac{i\ln \left ({b}^{2}{x}^{2}+2\,xab+{a}^{2}+1 \right ){a}^{2}}{{b}^{3}}}-2\,{\frac{\arctan \left ( bx+a \right ) }{{b}^{3}}}-{\frac{4\,i\arctan \left ( bx+a \right ) a}{{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}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(1+I*(b*x+a))^2*(1+(b*x+a)^2),x)

[Out]

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

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Maxima [A]  time = 1.02958, size = 70, normalized size = 1.19 \begin{align*} -\frac{b^{2} x^{3} + 3 i \, b x^{2} + 6 \,{\left (-i \, a - 1\right )} x}{3 \, b^{2}} + \frac{{\left (-2 i \, a^{2} - 4 \, a + 2 i\right )} \log \left (i \, b x + i \, a + 1\right )}{b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(1+I*(b*x+a))^2*(1+(b*x+a)^2),x, algorithm="maxima")

[Out]

-1/3*(b^2*x^3 + 3*I*b*x^2 + 6*(-I*a - 1)*x)/b^2 + (-2*I*a^2 - 4*a + 2*I)*log(I*b*x + I*a + 1)/b^3

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Fricas [A]  time = 2.12139, size = 135, normalized size = 2.29 \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(x^2/(1+I*(b*x+a))^2*(1+(b*x+a)^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 = 2.69192, size = 513, normalized size = 8.69 \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(x**2/(1+I*(b*x+a))**2*(1+(b*x+a)**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 - 7
92*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 [B]  time = 1.12451, size = 162, normalized size = 2.75 \begin{align*} \frac{2 \,{\left (a^{2} i + 2 \, a - i\right )} \log \left (\frac{1}{\sqrt{{\left (b x + a\right )}^{2} + 1}{\left | b \right |}}\right )}{b^{3}} + \frac{{\left (b i x + a i + 1\right )}^{3}{\left (\frac{3 \,{\left (a b - 2 \, b i\right )} i}{{\left (b i x + a i + 1\right )} b} - \frac{3 \,{\left (a^{2} b^{2} - 6 \, a b^{2} i - 5 \, b^{2}\right )} i^{2}}{{\left (b i x + a i + 1\right )}^{2} b^{2}} - 1\right )}}{3 \, b^{3} i^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(1+I*(b*x+a))^2*(1+(b*x+a)^2),x, algorithm="giac")

[Out]

2*(a^2*i + 2*a - i)*log(1/(sqrt((b*x + a)^2 + 1)*abs(b)))/b^3 + 1/3*(b*i*x + a*i + 1)^3*(3*(a*b - 2*b*i)*i/((b
*i*x + a*i + 1)*b) - 3*(a^2*b^2 - 6*a*b^2*i - 5*b^2)*i^2/((b*i*x + a*i + 1)^2*b^2) - 1)/(b^3*i^3)