3.2.72 \(\int e^{2 i \text {ArcTan}(a+b x)} x^3 \, dx\) [172]

Optimal. Leaf size=72 \[ \frac {2 i (i+a)^2 x}{b^3}+\frac {(1-i a) x^2}{b^2}+\frac {2 i x^3}{3 b}-\frac {x^4}{4}-\frac {2 (1-i a)^3 \log (i+a+b x)}{b^4} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.04, antiderivative size = 72, normalized size of antiderivative = 1.00, 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 = {5203, 78} \begin {gather*} -\frac {2 (1-i a)^3 \log (a+b x+i)}{b^4}+\frac {2 i (a+i)^2 x}{b^3}+\frac {(1-i a) x^2}{b^2}+\frac {2 i x^3}{3 b}-\frac {x^4}{4} \end {gather*}

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 78

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]))))

Rule 5203

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]

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 = 0.05, size = 72, normalized size = 1.00 \begin {gather*} \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 {gather*}

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

________________________________________________________________________________________

Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 169 vs. \(2 (62 ) = 124\).
time = 0.13, size = 170, normalized size = 2.36

method	result	size
default	\(\frac {i \left (\frac {1}{4} i b^{3} x^{4}+\frac {2}{3} b^{2} x^{3}-i b \,x^{2}-a b \,x^{2}+4 i a x +2 a^{2} x -2 x \right )}{b^{3}}+\frac {\frac {\left (-2 i a^{3} b +6 a^{2} b +6 i a b -2 b \right ) \ln \left (b^{2} x^{2}+2 a b x +a^{2}+1\right )}{2 b^{2}}+\frac {\left (-2 i a^{4}+4 a^{3}+2 i+4 a -\frac {\left (-2 i a^{3} b +6 a^{2} b +6 i a b -2 b \right ) a}{b}\right ) \arctan \left (\frac {2 b^{2} x +2 a b}{2 b}\right )}{b}}{b^{3}}\)	\(170\)
risch	\(-\frac {x^{4}}{4}+\frac {2 i x^{3}}{3 b}+\frac {x^{2}}{b^{2}}-\frac {i a \,x^{2}}{b^{2}}-\frac {4 a x}{b^{3}}+\frac {2 i a^{2} x}{b^{3}}-\frac {2 i x}{b^{3}}+\frac {3 \ln \left (b^{2} x^{2}+2 a b x +a^{2}+1\right ) a^{2}}{b^{4}}-\frac {i \ln \left (b^{2} x^{2}+2 a b x +a^{2}+1\right ) a^{3}}{b^{4}}-\frac {\ln \left (b^{2} x^{2}+2 a b x +a^{2}+1\right )}{b^{4}}+\frac {3 i \ln \left (b^{2} x^{2}+2 a b x +a^{2}+1\right ) a}{b^{4}}-\frac {6 i \arctan \left (b x +a \right ) a^{2}}{b^{4}}-\frac {2 \arctan \left (b x +a \right ) a^{3}}{b^{4}}+\frac {2 i \arctan \left (b x +a \right )}{b^{4}}+\frac {6 \arctan \left (b x +a \right ) a}{b^{4}}\)	\(211\)

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,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

Maxima [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 116 vs. \(2 (56) = 112\).
time = 0.47, size = 116, normalized size = 1.61 \begin {gather*} -\frac {3 \, b^{3} x^{4} - 8 i \, b^{2} x^{3} + 12 \, {\left (i \, a - 1\right )} b x^{2} + 24 \, {\left (-i \, a^{2} + 2 \, a + i\right )} x}{12 \, b^{3}} - \frac {2 \, {\left (a^{3} + 3 i \, a^{2} - 3 \, a - 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 {gather*}

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 + 2*a + I)*x)/b^3 - 2*(a^3 + 3*I*a^2 - 3*a -
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

________________________________________________________________________________________

Fricas [A]
time = 2.33, size = 77, normalized size = 1.07 \begin {gather*} -\frac {3 \, b^{4} x^{4} - 8 i \, b^{3} x^{3} + 12 \, {\left (i \, a - 1\right )} b^{2} x^{2} + 24 \, {\left (-i \, a^{2} + 2 \, a + i\right )} b x + 24 \, {\left (i \, a^{3} - 3 \, a^{2} - 3 i \, a + 1\right )} \log \left (\frac {b x + a + i}{b}\right )}{12 \, b^{4}} \end {gather*}

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 + 2*a + I)*b*x + 24*(I*a^3 - 3*a^2 - 3*I*a
+ 1)*log((b*x + a + I)/b))/b^4

________________________________________________________________________________________

Sympy [A]
time = 0.19, size = 75, normalized size = 1.04 \begin {gather*} - \frac {x^{4}}{4} - x^{2} \left (\frac {i a}{b^{2}} - \frac {1}{b^{2}}\right ) - x \left (- \frac {2 i a^{2}}{b^{3}} + \frac {4 a}{b^{3}} + \frac {2 i}{b^{3}}\right ) + \frac {2 i x^{3}}{3 b} - \frac {2 i \left (a + i\right )^{3} \log {\left (a + b x + i \right )}}{b^{4}} \end {gather*}

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

________________________________________________________________________________________

Giac [A]
time = 0.45, size = 83, normalized size = 1.15 \begin {gather*} -\frac {2 \, {\left (i \, a^{3} - 3 \, a^{2} - 3 i \, a + 1\right )} \log \left (b x + a + i\right )}{b^{4}} - \frac {3 \, b^{4} x^{4} - 8 i \, b^{3} x^{3} + 12 i \, a b^{2} x^{2} - 24 i \, a^{2} b x - 12 \, b^{2} x^{2} + 48 \, a b x + 24 i \, b x}{12 \, b^{4}} \end {gather*}

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*(I*a^3 - 3*a^2 - 3*I*a + 1)*log(b*x + a + I)/b^4 - 1/12*(3*b^4*x^4 - 8*I*b^3*x^3 + 12*I*a*b^2*x^2 - 24*I*a^
2*b*x - 12*b^2*x^2 + 48*a*b*x + 24*I*b*x)/b^4

________________________________________________________________________________________

Mupad [B]
time = 0.53, size = 153, normalized size = 2.12 \begin {gather*} -x^3\,\left (\frac {\left (-1+a\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{3\,b}-\frac {\left (1+a\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{3\,b}\right )-\frac {x^4}{4}+\ln \left (x+\frac {a+1{}\mathrm {i}}{b}\right )\,\left (\frac {6\,a^2-2}{b^4}+\frac {\left (6\,a-2\,a^3\right )\,1{}\mathrm {i}}{b^4}\right )-\frac {x^2\,\left (-1+a\,1{}\mathrm {i}\right )\,\left (\frac {\left (-1+a\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{b}-\frac {\left (1+a\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{b}\right )\,1{}\mathrm {i}}{2\,b}+\frac {x\,{\left (-1+a\,1{}\mathrm {i}\right )}^2\,\left (\frac {\left (-1+a\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{b}-\frac {\left (1+a\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{b}\right )}{b^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3*(a*1i + b*x*1i + 1)^2)/((a + b*x)^2 + 1),x)

[Out]

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

________________________________________________________________________________________