Integrand size = 14, antiderivative size = 102 \[ \int e^{-3 i \arctan (a x)} x^2 \, dx=-\frac {i (1-i a x)^3}{a^3 \sqrt {1+a^2 x^2}}-\frac {i (3-i a x)^2 \sqrt {1+a^2 x^2}}{3 a^3}-\frac {(28 i+3 a x) \sqrt {1+a^2 x^2}}{6 a^3}+\frac {11 \text {arcsinh}(a x)}{2 a^3} \]
11/2*arcsinh(a*x)/a^3-I*(1-I*a*x)^3/a^3/(a^2*x^2+1)^(1/2)-1/3*I*(3-I*a*x)^ 2*(a^2*x^2+1)^(1/2)/a^3-1/6*(28*I+3*a*x)*(a^2*x^2+1)^(1/2)/a^3
Time = 0.04 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.62 \[ \int e^{-3 i \arctan (a x)} x^2 \, dx=\frac {\frac {\sqrt {1+a^2 x^2} \left (-52-19 i a x-7 a^2 x^2+2 i a^3 x^3\right )}{-i+a x}+33 \text {arcsinh}(a x)}{6 a^3} \]
((Sqrt[1 + a^2*x^2]*(-52 - (19*I)*a*x - 7*a^2*x^2 + (2*I)*a^3*x^3))/(-I + a*x) + 33*ArcSinh[a*x])/(6*a^3)
Time = 0.70 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.23, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.857, Rules used = {5583, 2164, 25, 2027, 2164, 27, 563, 2346, 2346, 27, 455, 222}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int x^2 e^{-3 i \arctan (a x)} \, dx\) |
| \(\Big \downarrow \) 5583 |
| \(\displaystyle \int \frac {x^2 (1-i a x)^2}{(1+i a x) \sqrt {a^2 x^2+1}}dx\) |
| \(\Big \downarrow \) 2164 |
| \(\displaystyle i a \int -\frac {\sqrt {a^2 x^2+1} \left (x^3+\frac {i x^2}{a}\right )}{(i a x+1)^2}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -i a \int \frac {\sqrt {a^2 x^2+1} \left (x^3+\frac {i x^2}{a}\right )}{(i a x+1)^2}dx\) |
| \(\Big \downarrow \) 2027 |
| \(\displaystyle -i a \int \frac {x^2 \left (x+\frac {i}{a}\right ) \sqrt {a^2 x^2+1}}{(i a x+1)^2}dx\) |
| \(\Big \downarrow \) 2164 |
| \(\displaystyle a^2 \int \frac {x^2 \left (a^2 x^2+1\right )^{3/2}}{a^2 (i a x+1)^3}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {x^2 \left (a^2 x^2+1\right )^{3/2}}{(1+i a x)^3}dx\) |
| \(\Big \downarrow \) 563 |
| \(\displaystyle \frac {\int \frac {i a^3 x^3-3 a^2 x^2-4 i a x+4}{\sqrt {a^2 x^2+1}}dx}{a^2}-\frac {4 i \sqrt {a^2 x^2+1}}{a^3 (1+i a x)}\) |
| \(\Big \downarrow \) 2346 |
| \(\displaystyle \frac {\frac {\int \frac {-9 x^2 a^4-14 i x a^3+12 a^2}{\sqrt {a^2 x^2+1}}dx}{3 a^2}+\frac {1}{3} i a x^2 \sqrt {a^2 x^2+1}}{a^2}-\frac {4 i \sqrt {a^2 x^2+1}}{a^3 (1+i a x)}\) |
| \(\Big \downarrow \) 2346 |
| \(\displaystyle \frac {\frac {-\frac {9}{2} a^2 x \sqrt {a^2 x^2+1}+\frac {\int \frac {a^4 (33-28 i a x)}{\sqrt {a^2 x^2+1}}dx}{2 a^2}}{3 a^2}+\frac {1}{3} i a x^2 \sqrt {a^2 x^2+1}}{a^2}-\frac {4 i \sqrt {a^2 x^2+1}}{a^3 (1+i a x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {-\frac {9}{2} a^2 x \sqrt {a^2 x^2+1}+\frac {1}{2} a^2 \int \frac {33-28 i a x}{\sqrt {a^2 x^2+1}}dx}{3 a^2}+\frac {1}{3} i a x^2 \sqrt {a^2 x^2+1}}{a^2}-\frac {4 i \sqrt {a^2 x^2+1}}{a^3 (1+i a x)}\) |
| \(\Big \downarrow \) 455 |
| \(\displaystyle \frac {\frac {-\frac {9}{2} a^2 x \sqrt {a^2 x^2+1}+\frac {1}{2} a^2 \left (33 \int \frac {1}{\sqrt {a^2 x^2+1}}dx-\frac {28 i \sqrt {a^2 x^2+1}}{a}\right )}{3 a^2}+\frac {1}{3} i a x^2 \sqrt {a^2 x^2+1}}{a^2}-\frac {4 i \sqrt {a^2 x^2+1}}{a^3 (1+i a x)}\) |
| \(\Big \downarrow \) 222 |
| \(\displaystyle \frac {\frac {-\frac {9}{2} a^2 x \sqrt {a^2 x^2+1}+\frac {1}{2} a^2 \left (\frac {33 \text {arcsinh}(a x)}{a}-\frac {28 i \sqrt {a^2 x^2+1}}{a}\right )}{3 a^2}+\frac {1}{3} i a x^2 \sqrt {a^2 x^2+1}}{a^2}-\frac {4 i \sqrt {a^2 x^2+1}}{a^3 (1+i a x)}\) |
((-4*I)*Sqrt[1 + a^2*x^2])/(a^3*(1 + I*a*x)) + ((I/3)*a*x^2*Sqrt[1 + a^2*x ^2] + ((-9*a^2*x*Sqrt[1 + a^2*x^2])/2 + (a^2*(((-28*I)*Sqrt[1 + a^2*x^2])/ a + (33*ArcSinh[a*x])/a))/2)/(3*a^2))/a^2
3.1.53.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt [a])]/Rt[b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b]
Int[((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[d*(( a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] + Simp[c Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && !LeQ[p, -1]
Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo l] :> Simp[(-(-c)^(m - n - 2))*d^(2*n - m + 3)*(Sqrt[a + b*x^2]/(2^(n + 1)* b^(n + 2)*(c + d*x))), x] - Simp[d^(2*n - m + 2)/b^(n + 1) Int[(1/Sqrt[a + b*x^2])*ExpandToSum[(2^(-n - 1)*(-c)^(m - n - 1) - d^m*x^m*(-c + d*x)^(-n - 1))/(c + d*x), x], x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c^2 + a*d^2 , 0] && IGtQ[m, 0] && ILtQ[n, 0] && EqQ[n + p, -3/2]
Int[(Fx_.)*((a_.)*(x_)^(r_.) + (b_.)*(x_)^(s_.))^(p_.), x_Symbol] :> Int[x^ (p*r)*(a + b*x^(s - r))^p*Fx, x] /; FreeQ[{a, b, r, s}, x] && IntegerQ[p] & & PosQ[s - r] && !(EqQ[p, 1] && EqQ[u, 1])
Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[d*e Int[(d + e*x)^(m - 1)*PolynomialQuotient[Pq, a*e + b*d*x, x]* (a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, d, e, m, p}, x] && PolyQ[Pq, x] && EqQ[b*d^2 + a*e^2, 0] && EqQ[PolynomialRemainder[Pq, a*e + b*d*x, x], 0 ]
Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], e = Coeff[Pq, x, Expon[Pq, x]]}, Simp[e*x^(q - 1)*((a + b*x^2)^(p + 1)/(b*( q + 2*p + 1))), x] + Simp[1/(b*(q + 2*p + 1)) Int[(a + b*x^2)^p*ExpandToS um[b*(q + 2*p + 1)*Pq - a*e*(q - 1)*x^(q - 2) - b*e*(q + 2*p + 1)*x^q, x], x], x]] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x] && !LeQ[p, -1]
Int[E^(ArcTan[(a_.)*(x_)]*(n_))*(x_)^(m_.), x_Symbol] :> Int[x^m*((1 - I*a* x)^((I*n + 1)/2)/((1 + I*a*x)^((I*n - 1)/2)*Sqrt[1 + a^2*x^2])), x] /; Free Q[{a, m}, x] && IntegerQ[(I*n - 1)/2]
Time = 0.36 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.09
| method | result | size |
| risch | \(\frac {i \left (2 a^{2} x^{2}+9 i a x -28\right ) \sqrt {a^{2} x^{2}+1}}{6 a^{3}}+\frac {11 \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}+1}\right )}{2 a^{2} \sqrt {a^{2}}}-\frac {4 \sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )}}{a^{4} \left (x -\frac {i}{a}\right )}\) | \(111\) |
| default | \(-\frac {2 \left (-\frac {i \left (\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )\right )^{\frac {5}{2}}}{a \left (x -\frac {i}{a}\right )^{2}}+3 i a \left (\frac {\left (\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )\right )^{\frac {3}{2}}}{3}+i a \left (\frac {\left (2 \left (x -\frac {i}{a}\right ) a^{2}+2 i a \right ) \sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )}}{4 a^{2}}+\frac {\ln \left (\frac {i a +\left (x -\frac {i}{a}\right ) a^{2}}{\sqrt {a^{2}}}+\sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )}\right )}{2 \sqrt {a^{2}}}\right )\right )\right )}{a^{4}}+\frac {i \left (\frac {\left (\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )\right )^{\frac {3}{2}}}{3}+i a \left (\frac {\left (2 \left (x -\frac {i}{a}\right ) a^{2}+2 i a \right ) \sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )}}{4 a^{2}}+\frac {\ln \left (\frac {i a +\left (x -\frac {i}{a}\right ) a^{2}}{\sqrt {a^{2}}}+\sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )}\right )}{2 \sqrt {a^{2}}}\right )\right )}{a^{3}}-\frac {i \left (\frac {i \left (\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )\right )^{\frac {5}{2}}}{a \left (x -\frac {i}{a}\right )^{3}}-2 i a \left (-\frac {i \left (\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )\right )^{\frac {5}{2}}}{a \left (x -\frac {i}{a}\right )^{2}}+3 i a \left (\frac {\left (\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )\right )^{\frac {3}{2}}}{3}+i a \left (\frac {\left (2 \left (x -\frac {i}{a}\right ) a^{2}+2 i a \right ) \sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )}}{4 a^{2}}+\frac {\ln \left (\frac {i a +\left (x -\frac {i}{a}\right ) a^{2}}{\sqrt {a^{2}}}+\sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )}\right )}{2 \sqrt {a^{2}}}\right )\right )\right )\right )}{a^{5}}\) | \(619\) |
1/6*I*(2*a^2*x^2+9*I*a*x-28)*(a^2*x^2+1)^(1/2)/a^3+11/2/a^2*ln(a^2*x/(a^2) ^(1/2)+(a^2*x^2+1)^(1/2))/(a^2)^(1/2)-4/a^4/(x-I/a)*((x-I/a)^2*a^2+2*I*a*( x-I/a))^(1/2)
Time = 0.26 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.78 \[ \int e^{-3 i \arctan (a x)} x^2 \, dx=-\frac {24 \, a x + 33 \, {\left (a x - i\right )} \log \left (-a x + \sqrt {a^{2} x^{2} + 1}\right ) - {\left (2 i \, a^{3} x^{3} - 7 \, a^{2} x^{2} - 19 i \, a x - 52\right )} \sqrt {a^{2} x^{2} + 1} - 24 i}{6 \, {\left (a^{4} x - i \, a^{3}\right )}} \]
-1/6*(24*a*x + 33*(a*x - I)*log(-a*x + sqrt(a^2*x^2 + 1)) - (2*I*a^3*x^3 - 7*a^2*x^2 - 19*I*a*x - 52)*sqrt(a^2*x^2 + 1) - 24*I)/(a^4*x - I*a^3)
\[ \int e^{-3 i \arctan (a x)} x^2 \, dx=i \left (\int \frac {x^{2} \sqrt {a^{2} x^{2} + 1}}{a^{3} x^{3} - 3 i a^{2} x^{2} - 3 a x + i}\, dx + \int \frac {a^{2} x^{4} \sqrt {a^{2} x^{2} + 1}}{a^{3} x^{3} - 3 i a^{2} x^{2} - 3 a x + i}\, dx\right ) \]
I*(Integral(x**2*sqrt(a**2*x**2 + 1)/(a**3*x**3 - 3*I*a**2*x**2 - 3*a*x + I), x) + Integral(a**2*x**4*sqrt(a**2*x**2 + 1)/(a**3*x**3 - 3*I*a**2*x**2 - 3*a*x + I), x))
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 181 vs. \(2 (80) = 160\).
Time = 0.30 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.77 \[ \int e^{-3 i \arctan (a x)} x^2 \, dx=-\frac {i \, {\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{a^{5} x^{2} - 2 i \, a^{4} x - a^{3}} - \frac {i \, {\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{i \, a^{4} x + a^{3}} - \frac {6 i \, \sqrt {a^{2} x^{2} + 1}}{i \, a^{4} x + a^{3}} - \frac {\sqrt {-a^{2} x^{2} + 4 i \, a x + 3} x}{2 \, a^{2}} + \frac {i \, {\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{3 \, a^{3}} + \frac {\arcsin \left (i \, a x + 2\right )}{2 \, a^{3}} + \frac {6 \, \operatorname {arsinh}\left (a x\right )}{a^{3}} - \frac {3 i \, \sqrt {a^{2} x^{2} + 1}}{a^{3}} + \frac {i \, \sqrt {-a^{2} x^{2} + 4 i \, a x + 3}}{a^{3}} \]
-I*(a^2*x^2 + 1)^(3/2)/(a^5*x^2 - 2*I*a^4*x - a^3) - I*(a^2*x^2 + 1)^(3/2) /(I*a^4*x + a^3) - 6*I*sqrt(a^2*x^2 + 1)/(I*a^4*x + a^3) - 1/2*sqrt(-a^2*x ^2 + 4*I*a*x + 3)*x/a^2 + 1/3*I*(a^2*x^2 + 1)^(3/2)/a^3 + 1/2*arcsin(I*a*x + 2)/a^3 + 6*arcsinh(a*x)/a^3 - 3*I*sqrt(a^2*x^2 + 1)/a^3 + I*sqrt(-a^2*x ^2 + 4*I*a*x + 3)/a^3
\[ \int e^{-3 i \arctan (a x)} x^2 \, dx=\int { \frac {{\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x^{2}}{{\left (i \, a x + 1\right )}^{3}} \,d x } \]
Time = 0.07 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.13 \[ \int e^{-3 i \arctan (a x)} x^2 \, dx=\frac {11\,\mathrm {asinh}\left (x\,\sqrt {a^2}\right )}{2\,a^2\,\sqrt {a^2}}-\frac {\sqrt {a^2\,x^2+1}\,\left (\frac {3\,x\,\sqrt {a^2}}{2\,a^2}+\frac {a\,14{}\mathrm {i}}{3\,{\left (a^2\right )}^{3/2}}-\frac {a^3\,x^2\,1{}\mathrm {i}}{3\,{\left (a^2\right )}^{3/2}}\right )}{\sqrt {a^2}}+\frac {4\,\sqrt {a^2\,x^2+1}}{a^2\,\left (-x\,\sqrt {a^2}+\frac {\sqrt {a^2}\,1{}\mathrm {i}}{a}\right )\,\sqrt {a^2}} \]