Integrand size = 12, antiderivative size = 92 \[ \int e^{3 i \arctan (a x)} x \, dx=-\frac {9 \sqrt {1+a^2 x^2}}{2 a^2}-\frac {3 \left (1+a^2 x^2\right )^{3/2}}{2 a^2 (1-i a x)}-\frac {\left (1+a^2 x^2\right )^{5/2}}{a^2 (1-i a x)^3}+\frac {9 i \text {arcsinh}(a x)}{2 a^2} \]
-3/2*(a^2*x^2+1)^(3/2)/a^2/(1-I*a*x)-(a^2*x^2+1)^(5/2)/a^2/(1-I*a*x)^3+9/2 *I*arcsinh(a*x)/a^2-9/2*(a^2*x^2+1)^(1/2)/a^2
Time = 0.04 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.59 \[ \int e^{3 i \arctan (a x)} x \, dx=-\frac {i \left (\frac {\sqrt {1+a^2 x^2} \left (14-5 i a x+a^2 x^2\right )}{i+a x}-9 \text {arcsinh}(a x)\right )}{2 a^2} \]
Time = 0.56 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.97, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {5583, 2164, 2027, 2164, 25, 27, 563, 25, 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.
| \(\displaystyle \int x e^{3 i \arctan (a x)} \, dx\) |
| \(\Big \downarrow \) 5583 |
| \(\displaystyle \int \frac {x (1+i a x)^2}{(1-i a x) \sqrt {a^2 x^2+1}}dx\) |
| \(\Big \downarrow \) 2164 |
| \(\displaystyle -i a \int \frac {\left (\frac {i x}{a}-x^2\right ) \sqrt {a^2 x^2+1}}{(1-i a x)^2}dx\) |
| \(\Big \downarrow \) 2027 |
| \(\displaystyle -i a \int \frac {\left (\frac {i}{a}-x\right ) x \sqrt {a^2 x^2+1}}{(1-i a x)^2}dx\) |
| \(\Big \downarrow \) 2164 |
| \(\displaystyle -a^2 \int -\frac {x \left (a^2 x^2+1\right )^{3/2}}{a^2 (1-i a x)^3}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle a^2 \int \frac {x \left (a^2 x^2+1\right )^{3/2}}{a^2 (1-i a x)^3}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {x \left (a^2 x^2+1\right )^{3/2}}{(1-i a x)^3}dx\) |
| \(\Big \downarrow \) 563 |
| \(\displaystyle -\frac {i \int -\frac {-a^2 x^2+3 i a x+4}{\sqrt {a^2 x^2+1}}dx}{a}-\frac {4 \sqrt {a^2 x^2+1}}{a^2 (1-i a x)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {i \int \frac {-a^2 x^2+3 i a x+4}{\sqrt {a^2 x^2+1}}dx}{a}-\frac {4 \sqrt {a^2 x^2+1}}{a^2 (1-i a x)}\) |
| \(\Big \downarrow \) 2346 |
| \(\displaystyle \frac {i \left (-\frac {1}{2} x \sqrt {a^2 x^2+1}+\frac {\int \frac {3 a^2 (2 i a x+3)}{\sqrt {a^2 x^2+1}}dx}{2 a^2}\right )}{a}-\frac {4 \sqrt {a^2 x^2+1}}{a^2 (1-i a x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {i \left (-\frac {1}{2} x \sqrt {a^2 x^2+1}+\frac {3}{2} \int \frac {2 i a x+3}{\sqrt {a^2 x^2+1}}dx\right )}{a}-\frac {4 \sqrt {a^2 x^2+1}}{a^2 (1-i a x)}\) |
| \(\Big \downarrow \) 455 |
| \(\displaystyle \frac {i \left (-\frac {1}{2} x \sqrt {a^2 x^2+1}+\frac {3}{2} \left (3 \int \frac {1}{\sqrt {a^2 x^2+1}}dx+\frac {2 i \sqrt {a^2 x^2+1}}{a}\right )\right )}{a}-\frac {4 \sqrt {a^2 x^2+1}}{a^2 (1-i a x)}\) |
| \(\Big \downarrow \) 222 |
| \(\displaystyle \frac {i \left (-\frac {1}{2} x \sqrt {a^2 x^2+1}+\frac {3}{2} \left (\frac {3 \text {arcsinh}(a x)}{a}+\frac {2 i \sqrt {a^2 x^2+1}}{a}\right )\right )}{a}-\frac {4 \sqrt {a^2 x^2+1}}{a^2 (1-i a x)}\) |
(-4*Sqrt[1 + a^2*x^2])/(a^2*(1 - I*a*x)) + (I*(-1/2*(x*Sqrt[1 + a^2*x^2]) + (3*(((2*I)*Sqrt[1 + a^2*x^2])/a + (3*ArcSinh[a*x])/a))/2))/a
3.1.21.3.1 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 = 0.34 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.15
| method | result | size |
| risch | \(-\frac {i \left (a x -6 i\right ) \sqrt {a^{2} x^{2}+1}}{2 a^{2}}+\frac {i \left (\frac {9 \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}+1}\right )}{\sqrt {a^{2}}}-\frac {8 \sqrt {\left (x +\frac {i}{a}\right )^{2} a^{2}-2 i a \left (x +\frac {i}{a}\right )}}{a^{2} \left (x +\frac {i}{a}\right )}\right )}{2 a}\) | \(106\) |
| meijerg | \(\frac {\sqrt {\pi }-\frac {\sqrt {\pi }}{\sqrt {a^{2} x^{2}+1}}}{a^{2} \sqrt {\pi }}+\frac {3 i \left (-\frac {\sqrt {\pi }\, x \left (a^{2}\right )^{\frac {3}{2}}}{a^{2} \sqrt {a^{2} x^{2}+1}}+\frac {\sqrt {\pi }\, \left (a^{2}\right )^{\frac {3}{2}} \operatorname {arcsinh}\left (a x \right )}{a^{3}}\right )}{a \sqrt {\pi }\, \sqrt {a^{2}}}-\frac {3 \left (-2 \sqrt {\pi }+\frac {\sqrt {\pi }\, \left (4 a^{2} x^{2}+8\right )}{4 \sqrt {a^{2} x^{2}+1}}\right )}{a^{2} \sqrt {\pi }}-\frac {i \left (\frac {\sqrt {\pi }\, x \left (a^{2}\right )^{\frac {5}{2}} \left (5 a^{2} x^{2}+15\right )}{10 a^{4} \sqrt {a^{2} x^{2}+1}}-\frac {3 \sqrt {\pi }\, \left (a^{2}\right )^{\frac {5}{2}} \operatorname {arcsinh}\left (a x \right )}{2 a^{5}}\right )}{a \sqrt {\pi }\, \sqrt {a^{2}}}\) | \(192\) |
| default | \(-\frac {1}{a^{2} \sqrt {a^{2} x^{2}+1}}-3 a^{2} \left (\frac {x^{2}}{a^{2} \sqrt {a^{2} x^{2}+1}}+\frac {2}{a^{4} \sqrt {a^{2} x^{2}+1}}\right )-i a^{3} \left (\frac {x^{3}}{2 a^{2} \sqrt {a^{2} x^{2}+1}}-\frac {3 \left (-\frac {x}{a^{2} \sqrt {a^{2} x^{2}+1}}+\frac {\ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}+1}\right )}{a^{2} \sqrt {a^{2}}}\right )}{2 a^{2}}\right )+3 i a \left (-\frac {x}{a^{2} \sqrt {a^{2} x^{2}+1}}+\frac {\ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}+1}\right )}{a^{2} \sqrt {a^{2}}}\right )\) | \(193\) |
-1/2*I*(a*x-6*I)*(a^2*x^2+1)^(1/2)/a^2+1/2*I/a*(9*ln(a^2*x/(a^2)^(1/2)+(a^ 2*x^2+1)^(1/2))/(a^2)^(1/2)-8/a^2/(x+I/a)*((x+I/a)^2*a^2-2*I*a*(x+I/a))^(1 /2))
Time = 0.26 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.78 \[ \int e^{3 i \arctan (a x)} x \, dx=\frac {-8 i \, a x - 9 \, {\left (i \, a x - 1\right )} \log \left (-a x + \sqrt {a^{2} x^{2} + 1}\right ) + \sqrt {a^{2} x^{2} + 1} {\left (-i \, a^{2} x^{2} - 5 \, a x - 14 i\right )} + 8}{2 \, {\left (a^{3} x + i \, a^{2}\right )}} \]
1/2*(-8*I*a*x - 9*(I*a*x - 1)*log(-a*x + sqrt(a^2*x^2 + 1)) + sqrt(a^2*x^2 + 1)*(-I*a^2*x^2 - 5*a*x - 14*I) + 8)/(a^3*x + I*a^2)
\[ \int e^{3 i \arctan (a x)} x \, dx=- i \left (\int \frac {i x}{a^{2} x^{2} \sqrt {a^{2} x^{2} + 1} + \sqrt {a^{2} x^{2} + 1}}\, dx + \int \left (- \frac {3 a x^{2}}{a^{2} x^{2} \sqrt {a^{2} x^{2} + 1} + \sqrt {a^{2} x^{2} + 1}}\right )\, dx + \int \frac {a^{3} x^{4}}{a^{2} x^{2} \sqrt {a^{2} x^{2} + 1} + \sqrt {a^{2} x^{2} + 1}}\, dx + \int \left (- \frac {3 i a^{2} x^{3}}{a^{2} x^{2} \sqrt {a^{2} x^{2} + 1} + \sqrt {a^{2} x^{2} + 1}}\right )\, dx\right ) \]
-I*(Integral(I*x/(a**2*x**2*sqrt(a**2*x**2 + 1) + sqrt(a**2*x**2 + 1)), x) + Integral(-3*a*x**2/(a**2*x**2*sqrt(a**2*x**2 + 1) + sqrt(a**2*x**2 + 1) ), x) + Integral(a**3*x**4/(a**2*x**2*sqrt(a**2*x**2 + 1) + sqrt(a**2*x**2 + 1)), x) + Integral(-3*I*a**2*x**3/(a**2*x**2*sqrt(a**2*x**2 + 1) + sqrt (a**2*x**2 + 1)), x))
Time = 0.21 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.83 \[ \int e^{3 i \arctan (a x)} x \, dx=-\frac {i \, a x^{3}}{2 \, \sqrt {a^{2} x^{2} + 1}} - \frac {3 \, x^{2}}{\sqrt {a^{2} x^{2} + 1}} - \frac {9 i \, x}{2 \, \sqrt {a^{2} x^{2} + 1} a} + \frac {9 i \, \operatorname {arsinh}\left (a x\right )}{2 \, a^{2}} - \frac {7}{\sqrt {a^{2} x^{2} + 1} a^{2}} \]
-1/2*I*a*x^3/sqrt(a^2*x^2 + 1) - 3*x^2/sqrt(a^2*x^2 + 1) - 9/2*I*x/(sqrt(a ^2*x^2 + 1)*a) + 9/2*I*arcsinh(a*x)/a^2 - 7/(sqrt(a^2*x^2 + 1)*a^2)
\[ \int e^{3 i \arctan (a x)} x \, dx=\int { \frac {{\left (i \, a x + 1\right )}^{3} x}{{\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}}} \,d x } \]
Time = 0.50 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.13 \[ \int e^{3 i \arctan (a x)} x \, dx=-\frac {\sqrt {a^2\,x^2+1}\,\left (\frac {3\,\sqrt {a^2}}{a^2}+\frac {x\,\sqrt {a^2}\,1{}\mathrm {i}}{2\,a}\right )}{\sqrt {a^2}}+\frac {\mathrm {asinh}\left (x\,\sqrt {a^2}\right )\,9{}\mathrm {i}}{2\,a\,\sqrt {a^2}}-\frac {\sqrt {a^2\,x^2+1}\,4{}\mathrm {i}}{a\,\left (x\,\sqrt {a^2}+\frac {\sqrt {a^2}\,1{}\mathrm {i}}{a}\right )\,\sqrt {a^2}} \]