Integrand size = 16, antiderivative size = 300 \[ \int e^{-\frac {5}{2} i \arctan (a x)} x^2 \, dx=-\frac {2 i (1-i a x)^{9/4}}{a^3 \sqrt [4]{1+i a x}}-\frac {55 i \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{8 a^3}-\frac {11 i (1-i a x)^{5/4} (1+i a x)^{3/4}}{4 a^3}-\frac {i (1-i a x)^{9/4} (1+i a x)^{3/4}}{3 a^3}-\frac {55 i \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{8 \sqrt {2} a^3}+\frac {55 i \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{8 \sqrt {2} a^3}+\frac {55 i \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x} \left (1+\frac {\sqrt {1-i a x}}{\sqrt {1+i a x}}\right )}\right )}{8 \sqrt {2} a^3} \] Output:
-2*I*(1-I*a*x)^(9/4)/a^3/(1+I*a*x)^(1/4)-55/8*I*(1-I*a*x)^(1/4)*(1+I*a*x)^ (3/4)/a^3-11/4*I*(1-I*a*x)^(5/4)*(1+I*a*x)^(3/4)/a^3-1/3*I*(1-I*a*x)^(9/4) *(1+I*a*x)^(3/4)/a^3-55/16*I*arctan(1-2^(1/2)*(1-I*a*x)^(1/4)/(1+I*a*x)^(1 /4))*2^(1/2)/a^3+55/16*I*arctan(1+2^(1/2)*(1-I*a*x)^(1/4)/(1+I*a*x)^(1/4)) *2^(1/2)/a^3+55/16*I*arctanh(2^(1/2)*(1-I*a*x)^(1/4)/(1+I*a*x)^(1/4)/(1+(1 -I*a*x)^(1/2)/(1+I*a*x)^(1/2)))*2^(1/2)/a^3
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.04 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.30 \[ \int e^{-\frac {5}{2} i \arctan (a x)} x^2 \, dx=-\frac {\sqrt [4]{1-i a x} (i+a x)^2 \left (-21 i+3 a x+11 i 2^{3/4} \sqrt [4]{1+i a x} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {9}{4},\frac {13}{4},\frac {1}{2} (1-i a x)\right )\right )}{9 a^3 \sqrt [4]{1+i a x}} \] Input:
Integrate[x^2/E^(((5*I)/2)*ArcTan[a*x]),x]
Output:
-1/9*((1 - I*a*x)^(1/4)*(I + a*x)^2*(-21*I + 3*a*x + (11*I)*2^(3/4)*(1 + I *a*x)^(1/4)*Hypergeometric2F1[1/4, 9/4, 13/4, (1 - I*a*x)/2]))/(a^3*(1 + I *a*x)^(1/4))
Time = 0.81 (sec) , antiderivative size = 354, normalized size of antiderivative = 1.18, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {5585, 100, 27, 90, 60, 60, 73, 770, 755, 1476, 1082, 217, 1479, 25, 27, 1103}
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.
| \(\displaystyle \int x^2 e^{-\frac {5}{2} i \arctan (a x)} \, dx\) |
| \(\Big \downarrow \) 5585 |
| \(\displaystyle \int \frac {x^2 (1-i a x)^{5/4}}{(1+i a x)^{5/4}}dx\) |
| \(\Big \downarrow \) 100 |
| \(\displaystyle \frac {2 i \int -\frac {a (1-i a x)^{5/4} (a x+5 i)}{2 \sqrt [4]{i a x+1}}dx}{a^3}-\frac {2 i (1-i a x)^{9/4}}{a^3 \sqrt [4]{1+i a x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {i \int \frac {(1-i a x)^{5/4} (a x+5 i)}{\sqrt [4]{i a x+1}}dx}{a^2}-\frac {2 i (1-i a x)^{9/4}}{a^3 \sqrt [4]{1+i a x}}\) |
| \(\Big \downarrow \) 90 |
| \(\displaystyle -\frac {i \left (\frac {11}{2} i \int \frac {(1-i a x)^{5/4}}{\sqrt [4]{i a x+1}}dx+\frac {(1+i a x)^{3/4} (1-i a x)^{9/4}}{3 a}\right )}{a^2}-\frac {2 i (1-i a x)^{9/4}}{a^3 \sqrt [4]{1+i a x}}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle -\frac {i \left (\frac {11}{2} i \left (\frac {5}{4} \int \frac {\sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}dx-\frac {i (1-i a x)^{5/4} (1+i a x)^{3/4}}{2 a}\right )+\frac {(1+i a x)^{3/4} (1-i a x)^{9/4}}{3 a}\right )}{a^2}-\frac {2 i (1-i a x)^{9/4}}{a^3 \sqrt [4]{1+i a x}}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle -\frac {i \left (\frac {11}{2} i \left (\frac {5}{4} \left (\frac {1}{2} \int \frac {1}{(1-i a x)^{3/4} \sqrt [4]{i a x+1}}dx-\frac {i \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{a}\right )-\frac {i (1-i a x)^{5/4} (1+i a x)^{3/4}}{2 a}\right )+\frac {(1+i a x)^{3/4} (1-i a x)^{9/4}}{3 a}\right )}{a^2}-\frac {2 i (1-i a x)^{9/4}}{a^3 \sqrt [4]{1+i a x}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {i \left (\frac {11}{2} i \left (\frac {5}{4} \left (\frac {2 i \int \frac {1}{\sqrt [4]{i a x+1}}d\sqrt [4]{1-i a x}}{a}-\frac {i \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{a}\right )-\frac {i (1-i a x)^{5/4} (1+i a x)^{3/4}}{2 a}\right )+\frac {(1+i a x)^{3/4} (1-i a x)^{9/4}}{3 a}\right )}{a^2}-\frac {2 i (1-i a x)^{9/4}}{a^3 \sqrt [4]{1+i a x}}\) |
| \(\Big \downarrow \) 770 |
| \(\displaystyle -\frac {i \left (\frac {11}{2} i \left (\frac {5}{4} \left (\frac {2 i \int \frac {1}{2-i a x}d\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}}{a}-\frac {i \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{a}\right )-\frac {i (1-i a x)^{5/4} (1+i a x)^{3/4}}{2 a}\right )+\frac {(1+i a x)^{3/4} (1-i a x)^{9/4}}{3 a}\right )}{a^2}-\frac {2 i (1-i a x)^{9/4}}{a^3 \sqrt [4]{1+i a x}}\) |
| \(\Big \downarrow \) 755 |
| \(\displaystyle -\frac {i \left (\frac {11}{2} i \left (\frac {5}{4} \left (\frac {2 i \left (\frac {1}{2} \int \frac {1-\sqrt {1-i a x}}{2-i a x}d\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}+\frac {1}{2} \int \frac {\sqrt {1-i a x}+1}{2-i a x}d\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}\right )}{a}-\frac {i \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{a}\right )-\frac {i (1-i a x)^{5/4} (1+i a x)^{3/4}}{2 a}\right )+\frac {(1+i a x)^{3/4} (1-i a x)^{9/4}}{3 a}\right )}{a^2}-\frac {2 i (1-i a x)^{9/4}}{a^3 \sqrt [4]{1+i a x}}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle -\frac {i \left (\frac {11}{2} i \left (\frac {5}{4} \left (\frac {2 i \left (\frac {1}{2} \int \frac {1-\sqrt {1-i a x}}{2-i a x}d\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}+\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{\sqrt {1-i a x}-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}+1}d\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}+\frac {1}{2} \int \frac {1}{\sqrt {1-i a x}+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}+1}d\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}\right )\right )}{a}-\frac {i \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{a}\right )-\frac {i (1-i a x)^{5/4} (1+i a x)^{3/4}}{2 a}\right )+\frac {(1+i a x)^{3/4} (1-i a x)^{9/4}}{3 a}\right )}{a^2}-\frac {2 i (1-i a x)^{9/4}}{a^3 \sqrt [4]{1+i a x}}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle -\frac {i \left (\frac {11}{2} i \left (\frac {5}{4} \left (\frac {2 i \left (\frac {1}{2} \left (\frac {\int \frac {1}{-\sqrt {1-i a x}-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\sqrt {1-i a x}-1}d\left (\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}+1\right )}{\sqrt {2}}\right )+\frac {1}{2} \int \frac {1-\sqrt {1-i a x}}{2-i a x}d\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}\right )}{a}-\frac {i \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{a}\right )-\frac {i (1-i a x)^{5/4} (1+i a x)^{3/4}}{2 a}\right )+\frac {(1+i a x)^{3/4} (1-i a x)^{9/4}}{3 a}\right )}{a^2}-\frac {2 i (1-i a x)^{9/4}}{a^3 \sqrt [4]{1+i a x}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {i \left (\frac {11}{2} i \left (\frac {5}{4} \left (\frac {2 i \left (\frac {1}{2} \int \frac {1-\sqrt {1-i a x}}{2-i a x}d\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}+\frac {1}{2} \left (\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{\sqrt {2}}\right )\right )}{a}-\frac {i \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{a}\right )-\frac {i (1-i a x)^{5/4} (1+i a x)^{3/4}}{2 a}\right )+\frac {(1+i a x)^{3/4} (1-i a x)^{9/4}}{3 a}\right )}{a^2}-\frac {2 i (1-i a x)^{9/4}}{a^3 \sqrt [4]{1+i a x}}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle -\frac {i \left (\frac {11}{2} i \left (\frac {5}{4} \left (\frac {2 i \left (\frac {1}{2} \left (-\frac {\int -\frac {\sqrt {2}-\frac {2 \sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}}{\sqrt {1-i a x}-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}+1}d\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}}{2 \sqrt {2}}-\frac {\int -\frac {\sqrt {2} \left (\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}+1\right )}{\sqrt {1-i a x}+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}+1}d\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{\sqrt {2}}\right )\right )}{a}-\frac {i \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{a}\right )-\frac {i (1-i a x)^{5/4} (1+i a x)^{3/4}}{2 a}\right )+\frac {(1+i a x)^{3/4} (1-i a x)^{9/4}}{3 a}\right )}{a^2}-\frac {2 i (1-i a x)^{9/4}}{a^3 \sqrt [4]{1+i a x}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {i \left (\frac {11}{2} i \left (\frac {5}{4} \left (\frac {2 i \left (\frac {1}{2} \left (\frac {\int \frac {\sqrt {2}-\frac {2 \sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}}{\sqrt {1-i a x}-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}+1}d\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2} \left (\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}+1\right )}{\sqrt {1-i a x}+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}+1}d\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{\sqrt {2}}\right )\right )}{a}-\frac {i \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{a}\right )-\frac {i (1-i a x)^{5/4} (1+i a x)^{3/4}}{2 a}\right )+\frac {(1+i a x)^{3/4} (1-i a x)^{9/4}}{3 a}\right )}{a^2}-\frac {2 i (1-i a x)^{9/4}}{a^3 \sqrt [4]{1+i a x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {i \left (\frac {11}{2} i \left (\frac {5}{4} \left (\frac {2 i \left (\frac {1}{2} \left (\frac {\int \frac {\sqrt {2}-\frac {2 \sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}}{\sqrt {1-i a x}-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}+1}d\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}}{2 \sqrt {2}}+\frac {1}{2} \int \frac {\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}+1}{\sqrt {1-i a x}+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}+1}d\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}\right )+\frac {1}{2} \left (\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{\sqrt {2}}\right )\right )}{a}-\frac {i \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{a}\right )-\frac {i (1-i a x)^{5/4} (1+i a x)^{3/4}}{2 a}\right )+\frac {(1+i a x)^{3/4} (1-i a x)^{9/4}}{3 a}\right )}{a^2}-\frac {2 i (1-i a x)^{9/4}}{a^3 \sqrt [4]{1+i a x}}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle -\frac {2 i (1-i a x)^{9/4}}{a^3 \sqrt [4]{1+i a x}}-\frac {i \left (\frac {11}{2} i \left (\frac {5}{4} \left (\frac {2 i \left (\frac {1}{2} \left (\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{\sqrt {2}}\right )+\frac {1}{2} \left (\frac {\log \left (\sqrt {1-i a x}+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\sqrt {1-i a x}-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+1\right )}{2 \sqrt {2}}\right )\right )}{a}-\frac {i \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{a}\right )-\frac {i (1-i a x)^{5/4} (1+i a x)^{3/4}}{2 a}\right )+\frac {(1+i a x)^{3/4} (1-i a x)^{9/4}}{3 a}\right )}{a^2}\) |
Input:
Int[x^2/E^(((5*I)/2)*ArcTan[a*x]),x]
Output:
((-2*I)*(1 - I*a*x)^(9/4))/(a^3*(1 + I*a*x)^(1/4)) - (I*(((1 - I*a*x)^(9/4 )*(1 + I*a*x)^(3/4))/(3*a) + ((11*I)/2)*(((-1/2*I)*(1 - I*a*x)^(5/4)*(1 + I*a*x)^(3/4))/a + (5*(((-I)*(1 - I*a*x)^(1/4)*(1 + I*a*x)^(3/4))/a + ((2*I )*((-(ArcTan[1 - (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)]/Sqrt[2]) + ArcTan[1 + (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)]/Sqrt[2])/2 + (- 1/2*Log[1 + Sqrt[1 - I*a*x] - (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4 )]/Sqrt[2] + Log[1 + Sqrt[1 - I*a*x] + (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I* a*x)^(1/4)]/(2*Sqrt[2]))/2))/a))/4)))/a^2
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p + 2)) Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0]
Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( p_), x_] :> Simp[(b*c - a*d)^2*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d^2*(d *e - c*f)*(n + 1))), x] - Simp[1/(d^2*(d*e - c*f)*(n + 1)) Int[(c + d*x)^ (n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*( p + 1)) - 2*a*b*d*(d*e*(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x , x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] || !SumSimplerQ[p, 1])))
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2] ], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*r) Int[(r - s*x^2)/(a + b*x^4) , x], x] + Simp[1/(2*r) Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] & & AtomQ[SplitProduct[SumBaseQ, b]]))
Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^(p + 1/n) Subst[In t[1/(1 - b*x^n)^(p + 1/n + 1), x], x, x/(a + b*x^n)^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[p, -2^(-1)] && IntegerQ[p + 1 /n]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 2*(d/e), 2]}, Simp[e/(2*c) Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ e/(2*c) Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ -2*(d/e), 2]}, Simp[e/(2*c*q) Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Simp[e/(2*c*q) Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
Int[E^(ArcTan[(a_.)*(x_)]*(n_.))*(x_)^(m_.), x_Symbol] :> Int[x^m*((1 - I*a *x)^(I*(n/2))/(1 + I*a*x)^(I*(n/2))), x] /; FreeQ[{a, m, n}, x] && !Intege rQ[(I*n - 1)/2]
\[\int \frac {x^{2}}{{\left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}^{\frac {5}{2}}}d x\]
Input:
int(x^2/((1+I*a*x)/(a^2*x^2+1)^(1/2))^(5/2),x)
Output:
int(x^2/((1+I*a*x)/(a^2*x^2+1)^(1/2))^(5/2),x)
Time = 0.14 (sec) , antiderivative size = 296, normalized size of antiderivative = 0.99 \[ \int e^{-\frac {5}{2} i \arctan (a x)} x^2 \, dx=\frac {12 \, {\left (a^{4} x - i \, a^{3}\right )} \sqrt {\frac {3025 i}{64 \, a^{6}}} \log \left (\frac {8}{55} \, a^{3} \sqrt {\frac {3025 i}{64 \, a^{6}}} + \sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}}\right ) - 12 \, {\left (a^{4} x - i \, a^{3}\right )} \sqrt {\frac {3025 i}{64 \, a^{6}}} \log \left (-\frac {8}{55} \, a^{3} \sqrt {\frac {3025 i}{64 \, a^{6}}} + \sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}}\right ) - 12 \, {\left (a^{4} x - i \, a^{3}\right )} \sqrt {-\frac {3025 i}{64 \, a^{6}}} \log \left (\frac {8}{55} \, a^{3} \sqrt {-\frac {3025 i}{64 \, a^{6}}} + \sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}}\right ) + 12 \, {\left (a^{4} x - i \, a^{3}\right )} \sqrt {-\frac {3025 i}{64 \, a^{6}}} \log \left (-\frac {8}{55} \, a^{3} \sqrt {-\frac {3025 i}{64 \, a^{6}}} + \sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}}\right ) + {\left (8 i \, a^{3} x^{3} - 26 \, a^{2} x^{2} - 61 i \, a x - 287\right )} \sqrt {a^{2} x^{2} + 1} \sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}}}{24 \, {\left (a^{4} x - i \, a^{3}\right )}} \] Input:
integrate(x^2/((1+I*a*x)/(a^2*x^2+1)^(1/2))^(5/2),x, algorithm="fricas")
Output:
1/24*(12*(a^4*x - I*a^3)*sqrt(3025/64*I/a^6)*log(8/55*a^3*sqrt(3025/64*I/a ^6) + sqrt(I*sqrt(a^2*x^2 + 1)/(a*x + I))) - 12*(a^4*x - I*a^3)*sqrt(3025/ 64*I/a^6)*log(-8/55*a^3*sqrt(3025/64*I/a^6) + sqrt(I*sqrt(a^2*x^2 + 1)/(a* x + I))) - 12*(a^4*x - I*a^3)*sqrt(-3025/64*I/a^6)*log(8/55*a^3*sqrt(-3025 /64*I/a^6) + sqrt(I*sqrt(a^2*x^2 + 1)/(a*x + I))) + 12*(a^4*x - I*a^3)*sqr t(-3025/64*I/a^6)*log(-8/55*a^3*sqrt(-3025/64*I/a^6) + sqrt(I*sqrt(a^2*x^2 + 1)/(a*x + I))) + (8*I*a^3*x^3 - 26*a^2*x^2 - 61*I*a*x - 287)*sqrt(a^2*x ^2 + 1)*sqrt(I*sqrt(a^2*x^2 + 1)/(a*x + I)))/(a^4*x - I*a^3)
\[ \int e^{-\frac {5}{2} i \arctan (a x)} x^2 \, dx=\int \frac {x^{2}}{\left (\frac {i \left (a x - i\right )}{\sqrt {a^{2} x^{2} + 1}}\right )^{\frac {5}{2}}}\, dx \] Input:
integrate(x**2/((1+I*a*x)/(a**2*x**2+1)**(1/2))**(5/2),x)
Output:
Integral(x**2/(I*(a*x - I)/sqrt(a**2*x**2 + 1))**(5/2), x)
\[ \int e^{-\frac {5}{2} i \arctan (a x)} x^2 \, dx=\int { \frac {x^{2}}{\left (\frac {i \, a x + 1}{\sqrt {a^{2} x^{2} + 1}}\right )^{\frac {5}{2}}} \,d x } \] Input:
integrate(x^2/((1+I*a*x)/(a^2*x^2+1)^(1/2))^(5/2),x, algorithm="maxima")
Output:
integrate(x^2/((I*a*x + 1)/sqrt(a^2*x^2 + 1))^(5/2), x)
Exception generated. \[ \int e^{-\frac {5}{2} i \arctan (a x)} x^2 \, dx=\text {Exception raised: TypeError} \] Input:
integrate(x^2/((1+I*a*x)/(a^2*x^2+1)^(1/2))^(5/2),x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int e^{-\frac {5}{2} i \arctan (a x)} x^2 \, dx=\int \frac {x^2}{{\left (\frac {1+a\,x\,1{}\mathrm {i}}{\sqrt {a^2\,x^2+1}}\right )}^{5/2}} \,d x \] Input:
int(x^2/((a*x*1i + 1)/(a^2*x^2 + 1)^(1/2))^(5/2),x)
Output:
int(x^2/((a*x*1i + 1)/(a^2*x^2 + 1)^(1/2))^(5/2), x)
\[ \int e^{-\frac {5}{2} i \arctan (a x)} x^2 \, dx=-\left (\int \frac {\left (a^{2} x^{2}+1\right )^{\frac {1}{4}} x^{4}}{\sqrt {a i x +1}\, a^{2} x^{2}-2 \sqrt {a i x +1}\, a i x -\sqrt {a i x +1}}d x \right ) a^{2}-\left (\int \frac {\left (a^{2} x^{2}+1\right )^{\frac {1}{4}} x^{2}}{\sqrt {a i x +1}\, a^{2} x^{2}-2 \sqrt {a i x +1}\, a i x -\sqrt {a i x +1}}d x \right ) \] Input:
int(x^2/((1+I*a*x)/(a^2*x^2+1)^(1/2))^(5/2),x)
Output:
- (int(((a**2*x**2 + 1)**(1/4)*x**4)/(sqrt(a*i*x + 1)*a**2*x**2 - 2*sqrt( a*i*x + 1)*a*i*x - sqrt(a*i*x + 1)),x)*a**2 + int(((a**2*x**2 + 1)**(1/4)* x**2)/(sqrt(a*i*x + 1)*a**2*x**2 - 2*sqrt(a*i*x + 1)*a*i*x - sqrt(a*i*x + 1)),x))