Integrand size = 14, antiderivative size = 102 \[ \int e^{3 i \arctan (a x)} x^2 \, dx=\frac {i (1+i a x)^3}{a^3 \sqrt {1+a^2 x^2}}+\frac {(28 i-3 a x) \sqrt {1+a^2 x^2}}{6 a^3}+\frac {i (3+i a x)^2 \sqrt {1+a^2 x^2}}{3 a^3}+\frac {11 \text {arcsinh}(a x)}{2 a^3} \]
11/2*arcsinh(a*x)/a^3+I*(1+I*a*x)^3/a^3/(a^2*x^2+1)^(1/2)+1/6*(28*I-3*a*x) *(a^2*x^2+1)^(1/2)/a^3+1/3*I*(3+I*a*x)^2*(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 = 0.71 (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, 2027, 2164, 25, 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 |
| \(\displaystyle -i a \int \frac {\sqrt {a^2 x^2+1} \left (\frac {i x^2}{a}-x^3\right )}{(1-i a x)^2}dx\) |
| \(\Big \downarrow \) 2027 |
| \(\displaystyle -i a \int \frac {\left (\frac {i}{a}-x\right ) x^2 \sqrt {a^2 x^2+1}}{(1-i a x)^2}dx\) |
| \(\Big \downarrow \) 2164 |
| \(\displaystyle -a^2 \int -\frac {x^2 \left (a^2 x^2+1\right )^{3/2}}{a^2 (1-i a x)^3}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle a^2 \int \frac {x^2 \left (a^2 x^2+1\right )^{3/2}}{a^2 (1-i a x)^3}dx\) |
| \(\Big \downarrow \) 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 |
| \(\displaystyle \frac {\frac {-\frac {9}{2} a^2 x \sqrt {a^2 x^2+1}+\frac {\int \frac {a^4 (28 i a x+33)}{\sqrt {a^2 x^2+1}}dx}{2 a^2}}{3 a^2}-\frac {1}{3} i a x^2 \sqrt {a^2 x^2+1}}{a^2}+\frac {4 i \sqrt {a^2 x^2+1}}{a^3 (1-i a x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {-\frac {9}{2} a^2 x \sqrt {a^2 x^2+1}+\frac {1}{2} a^2 \int \frac {28 i a x+33}{\sqrt {a^2 x^2+1}}dx}{3 a^2}-\frac {1}{3} i a x^2 \sqrt {a^2 x^2+1}}{a^2}+\frac {4 i \sqrt {a^2 x^2+1}}{a^3 (1-i a x)}\) |
| \(\Big \downarrow \) 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)) + ((-1/3*I)*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
3.1.20.3.1 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 = 0.40 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.09
| method | result | size |
| risch | \(-\frac {i \left (2 a^{2} x^{2}-9 i a x -28\right ) \sqrt {a^{2} x^{2}+1}}{6 a^{3}}+\frac {11 \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}+1}\right )}{2 a^{2} \sqrt {a^{2}}}-\frac {4 \sqrt {\left (x +\frac {i}{a}\right )^{2} a^{2}-2 i a \left (x +\frac {i}{a}\right )}}{a^{4} \left (x +\frac {i}{a}\right )}\) | \(111\) |
| meijerg | \(\frac {-\frac {\sqrt {\pi }\, x \left (a^{2}\right )^{\frac {3}{2}}}{a^{2} \sqrt {a^{2} x^{2}+1}}+\frac {\sqrt {\pi }\, \left (a^{2}\right )^{\frac {3}{2}} \operatorname {arcsinh}\left (a x \right )}{a^{3}}}{a^{2} \sqrt {\pi }\, \sqrt {a^{2}}}+\frac {3 i \left (-2 \sqrt {\pi }+\frac {\sqrt {\pi }\, \left (4 a^{2} x^{2}+8\right )}{4 \sqrt {a^{2} x^{2}+1}}\right )}{a^{3} \sqrt {\pi }}-\frac {3 \left (\frac {\sqrt {\pi }\, x \left (a^{2}\right )^{\frac {5}{2}} \left (5 a^{2} x^{2}+15\right )}{10 a^{4} \sqrt {a^{2} x^{2}+1}}-\frac {3 \sqrt {\pi }\, \left (a^{2}\right )^{\frac {5}{2}} \operatorname {arcsinh}\left (a x \right )}{2 a^{5}}\right )}{a^{2} \sqrt {\pi }\, \sqrt {a^{2}}}-\frac {i \left (\frac {8 \sqrt {\pi }}{3}-\frac {\sqrt {\pi }\, \left (-2 a^{4} x^{4}+8 a^{2} x^{2}+16\right )}{6 \sqrt {a^{2} x^{2}+1}}\right )}{a^{3} \sqrt {\pi }}\) | \(212\) |
| default | \(-\frac {x}{a^{2} \sqrt {a^{2} x^{2}+1}}+\frac {\ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}+1}\right )}{a^{2} \sqrt {a^{2}}}-3 a^{2} \left (\frac {x^{3}}{2 a^{2} \sqrt {a^{2} x^{2}+1}}-\frac {3 \left (-\frac {x}{a^{2} \sqrt {a^{2} x^{2}+1}}+\frac {\ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}+1}\right )}{a^{2} \sqrt {a^{2}}}\right )}{2 a^{2}}\right )-i a^{3} \left (\frac {x^{4}}{3 a^{2} \sqrt {a^{2} x^{2}+1}}-\frac {4 \left (\frac {x^{2}}{a^{2} \sqrt {a^{2} x^{2}+1}}+\frac {2}{a^{4} \sqrt {a^{2} x^{2}+1}}\right )}{3 a^{2}}\right )+3 i a \left (\frac {x^{2}}{a^{2} \sqrt {a^{2} x^{2}+1}}+\frac {2}{a^{4} \sqrt {a^{2} x^{2}+1}}\right )\) | \(236\) |
-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 = 0.27 (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)
\[ \int e^{3 i \arctan (a x)} x^2 \, dx=- i \left (\int \frac {i x^{2}}{a^{2} x^{2} \sqrt {a^{2} x^{2} + 1} + \sqrt {a^{2} x^{2} + 1}}\, dx + \int \left (- \frac {3 a x^{3}}{a^{2} x^{2} \sqrt {a^{2} x^{2} + 1} + \sqrt {a^{2} x^{2} + 1}}\right )\, dx + \int \frac {a^{3} x^{5}}{a^{2} x^{2} \sqrt {a^{2} x^{2} + 1} + \sqrt {a^{2} x^{2} + 1}}\, dx + \int \left (- \frac {3 i a^{2} x^{4}}{a^{2} x^{2} \sqrt {a^{2} x^{2} + 1} + \sqrt {a^{2} x^{2} + 1}}\right )\, dx\right ) \]
-I*(Integral(I*x**2/(a**2*x**2*sqrt(a**2*x**2 + 1) + sqrt(a**2*x**2 + 1)), x) + Integral(-3*a*x**3/(a**2*x**2*sqrt(a**2*x**2 + 1) + sqrt(a**2*x**2 + 1)), x) + Integral(a**3*x**5/(a**2*x**2*sqrt(a**2*x**2 + 1) + sqrt(a**2*x **2 + 1)), x) + Integral(-3*I*a**2*x**4/(a**2*x**2*sqrt(a**2*x**2 + 1) + s qrt(a**2*x**2 + 1)), x))
Time = 0.21 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.93 \[ \int e^{3 i \arctan (a x)} x^2 \, dx=-\frac {i \, a x^{4}}{3 \, \sqrt {a^{2} x^{2} + 1}} - \frac {3 \, x^{3}}{2 \, \sqrt {a^{2} x^{2} + 1}} + \frac {13 i \, x^{2}}{3 \, \sqrt {a^{2} x^{2} + 1} a} - \frac {11 \, x}{2 \, \sqrt {a^{2} x^{2} + 1} a^{2}} + \frac {11 \, \operatorname {arsinh}\left (a x\right )}{2 \, a^{3}} + \frac {26 i}{3 \, \sqrt {a^{2} x^{2} + 1} a^{3}} \]
-1/3*I*a*x^4/sqrt(a^2*x^2 + 1) - 3/2*x^3/sqrt(a^2*x^2 + 1) + 13/3*I*x^2/(s qrt(a^2*x^2 + 1)*a) - 11/2*x/(sqrt(a^2*x^2 + 1)*a^2) + 11/2*arcsinh(a*x)/a ^3 + 26/3*I/(sqrt(a^2*x^2 + 1)*a^3)
\[ \int e^{3 i \arctan (a x)} x^2 \, dx=\int { \frac {{\left (i \, a x + 1\right )}^{3} x^{2}}{{\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}}} \,d x } \]
Time = 0.07 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.12 \[ \int e^{3 i \arctan (a x)} x^2 \, dx=\frac {11\,\mathrm {asinh}\left (x\,\sqrt {a^2}\right )}{2\,a^2\,\sqrt {a^2}}-\frac {\sqrt {a^2\,x^2+1}\,\left (\frac {3\,x\,\sqrt {a^2}}{2\,a^2}-\frac {a\,14{}\mathrm {i}}{3\,{\left (a^2\right )}^{3/2}}+\frac {a^3\,x^2\,1{}\mathrm {i}}{3\,{\left (a^2\right )}^{3/2}}\right )}{\sqrt {a^2}}-\frac {4\,\sqrt {a^2\,x^2+1}}{a^2\,\left (x\,\sqrt {a^2}+\frac {\sqrt {a^2}\,1{}\mathrm {i}}{a}\right )\,\sqrt {a^2}} \]