Integrand size = 25, antiderivative size = 54 \[ \int \frac {e^{2 i \arctan (a x)}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=-\frac {2 i (1+i a x)}{3 a \left (c+a^2 c x^2\right )^{3/2}}+\frac {x}{3 c \sqrt {c+a^2 c x^2}} \] Output:
-2/3*I*(1+I*a*x)/a/(a^2*c*x^2+c)^(3/2)+1/3*x/c/(a^2*c*x^2+c)^(1/2)
Time = 0.04 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.44 \[ \int \frac {e^{2 i \arctan (a x)}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\frac {(2-i a x) \sqrt {1+i a x} \sqrt {1+a^2 x^2}}{3 a c \sqrt {1-i a x} (i+a x) \sqrt {c+a^2 c x^2}} \] Input:
Integrate[E^((2*I)*ArcTan[a*x])/(c + a^2*c*x^2)^(3/2),x]
Output:
((2 - I*a*x)*Sqrt[1 + I*a*x]*Sqrt[1 + a^2*x^2])/(3*a*c*Sqrt[1 - I*a*x]*(I + a*x)*Sqrt[c + a^2*c*x^2])
Time = 0.40 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.09, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {5598, 457, 208}
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.
\(\displaystyle \int \frac {e^{2 i \arctan (a x)}}{\left (a^2 c x^2+c\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 5598 |
\(\displaystyle c \int \frac {(i a x+1)^2}{\left (a^2 c x^2+c\right )^{5/2}}dx\) |
\(\Big \downarrow \) 457 |
\(\displaystyle c \left (\frac {\int \frac {1}{\left (a^2 c x^2+c\right )^{3/2}}dx}{3 c}-\frac {2 i (1+i a x)}{3 a c \left (a^2 c x^2+c\right )^{3/2}}\right )\) |
\(\Big \downarrow \) 208 |
\(\displaystyle c \left (\frac {x}{3 c^2 \sqrt {a^2 c x^2+c}}-\frac {2 i (1+i a x)}{3 a c \left (a^2 c x^2+c\right )^{3/2}}\right )\) |
Input:
Int[E^((2*I)*ArcTan[a*x])/(c + a^2*c*x^2)^(3/2),x]
Output:
c*((((-2*I)/3)*(1 + I*a*x))/(a*c*(c + a^2*c*x^2)^(3/2)) + x/(3*c^2*Sqrt[c + a^2*c*x^2]))
Int[((a_) + (b_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[x/(a*Sqrt[a + b*x^2]), x] /; FreeQ[{a, b}, x]
Int[((c_) + (d_.)*(x_))^2*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[d*( c + d*x)*((a + b*x^2)^(p + 1)/(b*(p + 1))), x] - Simp[d^2*((p + 2)/(b*(p + 1))) Int[(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, p}, x] && EqQ[ b*c^2 + a*d^2, 0] && LtQ[p, -1]
Int[E^(ArcTan[(a_.)*(x_)]*(n_))*((c_) + (d_.)*(x_)^2)^(p_), x_Symbol] :> Si mp[1/c^(I*(n/2)) Int[(c + d*x^2)^(p + I*(n/2))/(1 + I*a*x)^(I*n), x], x] /; FreeQ[{a, c, d, p}, x] && EqQ[d, a^2*c] && !(IntegerQ[p] || GtQ[c, 0]) && ILtQ[I*(n/2), 0]
Time = 0.16 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.61
method | result | size |
orering | \(-\frac {\left (a x +2 i\right ) \left (i a x +1\right )^{2}}{3 a \left (a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}\) | \(33\) |
trager | \(\frac {\left (a^{3} x^{3}+3 a x -2 i\right ) \sqrt {a^{2} c \,x^{2}+c}}{3 c^{2} \left (a^{2} x^{2}+1\right )^{2} a}\) | \(46\) |
gosper | \(\frac {\left (-a x +i\right ) \left (a x +i\right ) \left (a x +2 i\right ) \left (i a x +1\right )^{2}}{3 a \left (a^{2} x^{2}+1\right ) \left (a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}\) | \(57\) |
default | \(\frac {\left (i \sqrt {-a^{2}}-a \right ) \left (\frac {1}{3 c \sqrt {-a^{2}}\, \left (x +\frac {\sqrt {-a^{2}}}{a^{2}}\right ) \sqrt {{\left (x +\frac {\sqrt {-a^{2}}}{a^{2}}\right )}^{2} a^{2} c -2 c \sqrt {-a^{2}}\, \left (x +\frac {\sqrt {-a^{2}}}{a^{2}}\right )}}+\frac {2 \left (x +\frac {\sqrt {-a^{2}}}{a^{2}}\right ) a^{2} c -2 c \sqrt {-a^{2}}}{3 c^{2} \sqrt {-a^{2}}\, \sqrt {{\left (x +\frac {\sqrt {-a^{2}}}{a^{2}}\right )}^{2} a^{2} c -2 c \sqrt {-a^{2}}\, \left (x +\frac {\sqrt {-a^{2}}}{a^{2}}\right )}}\right )}{a \sqrt {-a^{2}}}+\frac {\left (i \sqrt {-a^{2}}+a \right ) \left (-\frac {1}{3 c \sqrt {-a^{2}}\, \left (x -\frac {\sqrt {-a^{2}}}{a^{2}}\right ) \sqrt {{\left (x -\frac {\sqrt {-a^{2}}}{a^{2}}\right )}^{2} a^{2} c +2 c \sqrt {-a^{2}}\, \left (x -\frac {\sqrt {-a^{2}}}{a^{2}}\right )}}-\frac {2 \left (x -\frac {\sqrt {-a^{2}}}{a^{2}}\right ) a^{2} c +2 c \sqrt {-a^{2}}}{3 c^{2} \sqrt {-a^{2}}\, \sqrt {{\left (x -\frac {\sqrt {-a^{2}}}{a^{2}}\right )}^{2} a^{2} c +2 c \sqrt {-a^{2}}\, \left (x -\frac {\sqrt {-a^{2}}}{a^{2}}\right )}}\right )}{a \sqrt {-a^{2}}}-\frac {x}{c \sqrt {a^{2} c \,x^{2}+c}}\) | \(398\) |
Input:
int((1+I*a*x)^2/(a^2*x^2+1)/(a^2*c*x^2+c)^(3/2),x,method=_RETURNVERBOSE)
Output:
-1/3*(a*x+2*I)/a*(1+I*a*x)^2/(a^2*c*x^2+c)^(3/2)
Time = 0.09 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.87 \[ \int \frac {e^{2 i \arctan (a x)}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\frac {\sqrt {a^{2} c x^{2} + c} {\left (a x + 2 i\right )}}{3 \, {\left (a^{3} c^{2} x^{2} + 2 i \, a^{2} c^{2} x - a c^{2}\right )}} \] Input:
integrate((1+I*a*x)^2/(a^2*x^2+1)/(a^2*c*x^2+c)^(3/2),x, algorithm="fricas ")
Output:
1/3*sqrt(a^2*c*x^2 + c)*(a*x + 2*I)/(a^3*c^2*x^2 + 2*I*a^2*c^2*x - a*c^2)
\[ \int \frac {e^{2 i \arctan (a x)}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=- \int \frac {a^{2} x^{2}}{a^{4} c x^{4} \sqrt {a^{2} c x^{2} + c} + 2 a^{2} c x^{2} \sqrt {a^{2} c x^{2} + c} + c \sqrt {a^{2} c x^{2} + c}}\, dx - \int \left (- \frac {2 i a x}{a^{4} c x^{4} \sqrt {a^{2} c x^{2} + c} + 2 a^{2} c x^{2} \sqrt {a^{2} c x^{2} + c} + c \sqrt {a^{2} c x^{2} + c}}\right )\, dx - \int \left (- \frac {1}{a^{4} c x^{4} \sqrt {a^{2} c x^{2} + c} + 2 a^{2} c x^{2} \sqrt {a^{2} c x^{2} + c} + c \sqrt {a^{2} c x^{2} + c}}\right )\, dx \] Input:
integrate((1+I*a*x)**2/(a**2*x**2+1)/(a**2*c*x**2+c)**(3/2),x)
Output:
-Integral(a**2*x**2/(a**4*c*x**4*sqrt(a**2*c*x**2 + c) + 2*a**2*c*x**2*sqr t(a**2*c*x**2 + c) + c*sqrt(a**2*c*x**2 + c)), x) - Integral(-2*I*a*x/(a** 4*c*x**4*sqrt(a**2*c*x**2 + c) + 2*a**2*c*x**2*sqrt(a**2*c*x**2 + c) + c*s qrt(a**2*c*x**2 + c)), x) - Integral(-1/(a**4*c*x**4*sqrt(a**2*c*x**2 + c) + 2*a**2*c*x**2*sqrt(a**2*c*x**2 + c) + c*sqrt(a**2*c*x**2 + c)), x)
\[ \int \frac {e^{2 i \arctan (a x)}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\int { \frac {{\left (i \, a x + 1\right )}^{2}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} {\left (a^{2} x^{2} + 1\right )}} \,d x } \] Input:
integrate((1+I*a*x)^2/(a^2*x^2+1)/(a^2*c*x^2+c)^(3/2),x, algorithm="maxima ")
Output:
integrate((I*a*x + 1)^2/((a^2*c*x^2 + c)^(3/2)*(a^2*x^2 + 1)), x)
Time = 0.15 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.41 \[ \int \frac {e^{2 i \arctan (a x)}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=-\frac {2 \, \sqrt {a^{2} c} {\left (3 \, \sqrt {a^{2} c} x - 3 \, \sqrt {a^{2} c x^{2} + c} + i \, \sqrt {c}\right )}}{3 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} + c} + i \, \sqrt {c}\right )}^{3} a^{2} c} \] Input:
integrate((1+I*a*x)^2/(a^2*x^2+1)/(a^2*c*x^2+c)^(3/2),x, algorithm="giac")
Output:
-2/3*sqrt(a^2*c)*(3*sqrt(a^2*c)*x - 3*sqrt(a^2*c*x^2 + c) + I*sqrt(c))/((s qrt(a^2*c)*x - sqrt(a^2*c*x^2 + c) + I*sqrt(c))^3*a^2*c)
Time = 0.24 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.59 \[ \int \frac {e^{2 i \arctan (a x)}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\frac {a^3\,x^3+3\,a\,x-2{}\mathrm {i}}{3\,a\,{\left (c\,\left (a^2\,x^2+1\right )\right )}^{3/2}} \] Input:
int((a*x*1i + 1)^2/((c + a^2*c*x^2)^(3/2)*(a^2*x^2 + 1)),x)
Output:
(3*a*x + a^3*x^3 - 2i)/(3*a*(c*(a^2*x^2 + 1))^(3/2))
Time = 0.17 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.69 \[ \int \frac {e^{2 i \arctan (a x)}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\frac {\sqrt {c}\, \left (\sqrt {a^{2} x^{2}+1}\, a^{3} x^{3}+3 \sqrt {a^{2} x^{2}+1}\, a x -2 \sqrt {a^{2} x^{2}+1}\, i -3 a^{4} x^{4}-6 a^{2} x^{2}-3\right )}{3 a \,c^{2} \left (a^{4} x^{4}+2 a^{2} x^{2}+1\right )} \] Input:
int((1+I*a*x)^2/(a^2*x^2+1)/(a^2*c*x^2+c)^(3/2),x)
Output:
(sqrt(c)*(sqrt(a**2*x**2 + 1)*a**3*x**3 + 3*sqrt(a**2*x**2 + 1)*a*x - 2*sq rt(a**2*x**2 + 1)*i - 3*a**4*x**4 - 6*a**2*x**2 - 3))/(3*a*c**2*(a**4*x**4 + 2*a**2*x**2 + 1))