Integrand size = 25, antiderivative size = 95 \[ \int \frac {e^{5 i \arctan (a x)}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=-\frac {2 \sqrt {1+a^2 x^2}}{3 a c (i+a x)^3 \sqrt {c+a^2 c x^2}}-\frac {i \sqrt {1+a^2 x^2}}{2 a c (i+a x)^2 \sqrt {c+a^2 c x^2}} \]
-2/3*(a^2*x^2+1)^(1/2)/a/c/(I+a*x)^3/(a^2*c*x^2+c)^(1/2)-1/2*I*(a^2*x^2+1) ^(1/2)/a/c/(I+a*x)^2/(a^2*c*x^2+c)^(1/2)
Time = 0.02 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.59 \[ \int \frac {e^{5 i \arctan (a x)}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=-\frac {i (-i+3 a x) \sqrt {1+a^2 x^2}}{6 a c (i+a x)^3 \sqrt {c+a^2 c x^2}} \]
Time = 0.34 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.69, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {5599, 5596, 53, 2009}
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.
| \(\displaystyle \int \frac {e^{5 i \arctan (a x)}}{\left (a^2 c x^2+c\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 5599 |
| \(\displaystyle \frac {\sqrt {a^2 x^2+1} \int \frac {e^{5 i \arctan (a x)}}{\left (a^2 x^2+1\right )^{3/2}}dx}{c \sqrt {a^2 c x^2+c}}\) |
| \(\Big \downarrow \) 5596 |
| \(\displaystyle \frac {\sqrt {a^2 x^2+1} \int \frac {i a x+1}{(1-i a x)^4}dx}{c \sqrt {a^2 c x^2+c}}\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle \frac {\sqrt {a^2 x^2+1} \int \left (\frac {i}{(a x+i)^3}+\frac {2}{(a x+i)^4}\right )dx}{c \sqrt {a^2 c x^2+c}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {a^2 x^2+1} \left (-\frac {i}{2 a (a x+i)^2}-\frac {2}{3 a (a x+i)^3}\right )}{c \sqrt {a^2 c x^2+c}}\) |
3.4.28.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
Int[E^(ArcTan[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[c^p Int[(1 - I*a*x)^(p + I*(n/2))*(1 + I*a*x)^(p - I*(n/2)), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[d, a^2*c] && (IntegerQ[p] || GtQ[c, 0])
Int[E^(ArcTan[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_)^2)^(p_), x_Symbol] :> S imp[c^IntPart[p]*((c + d*x^2)^FracPart[p]/(1 + a^2*x^2)^FracPart[p]) Int[ (1 + a^2*x^2)^p*E^(n*ArcTan[a*x]), x], x] /; FreeQ[{a, c, d, n, p}, x] && E qQ[d, a^2*c] && !(IntegerQ[p] || GtQ[c, 0])
Time = 0.26 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.49
| method | result | size |
| risch | \(\frac {\sqrt {a^{2} x^{2}+1}\, \left (-\frac {i x}{2}-\frac {1}{6 a}\right )}{c \sqrt {c \left (a^{2} x^{2}+1\right )}\, \left (a x +i\right )^{3}}\) | \(47\) |
| default | \(-\frac {\sqrt {c \left (a^{2} x^{2}+1\right )}\, \left (3 i a x +1\right )}{6 \sqrt {a^{2} x^{2}+1}\, c^{2} a \left (a x +i\right )^{3}}\) | \(48\) |
| gosper | \(-\frac {\left (a x +i\right ) \left (-3 a x +i\right ) \left (i a x +1\right )^{5}}{6 a \left (-a x +i\right ) \left (a^{2} x^{2}+1\right )^{\frac {5}{2}} \left (a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}\) | \(60\) |
Time = 0.26 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.06 \[ \int \frac {e^{5 i \arctan (a x)}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\frac {\sqrt {a^{2} c x^{2} + c} {\left (i \, a^{2} x^{3} - 3 \, a x^{2} - 6 i \, x\right )} \sqrt {a^{2} x^{2} + 1}}{6 \, {\left (a^{5} c^{2} x^{5} + 3 i \, a^{4} c^{2} x^{4} - 2 \, a^{3} c^{2} x^{3} + 2 i \, a^{2} c^{2} x^{2} - 3 \, a c^{2} x - i \, c^{2}\right )}} \]
1/6*sqrt(a^2*c*x^2 + c)*(I*a^2*x^3 - 3*a*x^2 - 6*I*x)*sqrt(a^2*x^2 + 1)/(a ^5*c^2*x^5 + 3*I*a^4*c^2*x^4 - 2*a^3*c^2*x^3 + 2*I*a^2*c^2*x^2 - 3*a*c^2*x - I*c^2)
\[ \int \frac {e^{5 i \arctan (a x)}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=i \left (\int \left (- \frac {i}{a^{6} c x^{6} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + 3 a^{4} c x^{4} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + 3 a^{2} c x^{2} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + c \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c}}\right )\, dx + \int \frac {5 a x}{a^{6} c x^{6} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + 3 a^{4} c x^{4} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + 3 a^{2} c x^{2} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + c \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c}}\, dx + \int \left (- \frac {10 a^{3} x^{3}}{a^{6} c x^{6} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + 3 a^{4} c x^{4} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + 3 a^{2} c x^{2} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + c \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c}}\right )\, dx + \int \frac {a^{5} x^{5}}{a^{6} c x^{6} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + 3 a^{4} c x^{4} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + 3 a^{2} c x^{2} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + c \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c}}\, dx + \int \frac {10 i a^{2} x^{2}}{a^{6} c x^{6} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + 3 a^{4} c x^{4} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + 3 a^{2} c x^{2} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + c \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c}}\, dx + \int \left (- \frac {5 i a^{4} x^{4}}{a^{6} c x^{6} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + 3 a^{4} c x^{4} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + 3 a^{2} c x^{2} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + c \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c}}\right )\, dx\right ) \]
I*(Integral(-I/(a**6*c*x**6*sqrt(a**2*x**2 + 1)*sqrt(a**2*c*x**2 + c) + 3* a**4*c*x**4*sqrt(a**2*x**2 + 1)*sqrt(a**2*c*x**2 + c) + 3*a**2*c*x**2*sqrt (a**2*x**2 + 1)*sqrt(a**2*c*x**2 + c) + c*sqrt(a**2*x**2 + 1)*sqrt(a**2*c* x**2 + c)), x) + Integral(5*a*x/(a**6*c*x**6*sqrt(a**2*x**2 + 1)*sqrt(a**2 *c*x**2 + c) + 3*a**4*c*x**4*sqrt(a**2*x**2 + 1)*sqrt(a**2*c*x**2 + c) + 3 *a**2*c*x**2*sqrt(a**2*x**2 + 1)*sqrt(a**2*c*x**2 + c) + c*sqrt(a**2*x**2 + 1)*sqrt(a**2*c*x**2 + c)), x) + Integral(-10*a**3*x**3/(a**6*c*x**6*sqrt (a**2*x**2 + 1)*sqrt(a**2*c*x**2 + c) + 3*a**4*c*x**4*sqrt(a**2*x**2 + 1)* sqrt(a**2*c*x**2 + c) + 3*a**2*c*x**2*sqrt(a**2*x**2 + 1)*sqrt(a**2*c*x**2 + c) + c*sqrt(a**2*x**2 + 1)*sqrt(a**2*c*x**2 + c)), x) + Integral(a**5*x **5/(a**6*c*x**6*sqrt(a**2*x**2 + 1)*sqrt(a**2*c*x**2 + c) + 3*a**4*c*x**4 *sqrt(a**2*x**2 + 1)*sqrt(a**2*c*x**2 + c) + 3*a**2*c*x**2*sqrt(a**2*x**2 + 1)*sqrt(a**2*c*x**2 + c) + c*sqrt(a**2*x**2 + 1)*sqrt(a**2*c*x**2 + c)), x) + Integral(10*I*a**2*x**2/(a**6*c*x**6*sqrt(a**2*x**2 + 1)*sqrt(a**2*c *x**2 + c) + 3*a**4*c*x**4*sqrt(a**2*x**2 + 1)*sqrt(a**2*c*x**2 + c) + 3*a **2*c*x**2*sqrt(a**2*x**2 + 1)*sqrt(a**2*c*x**2 + c) + c*sqrt(a**2*x**2 + 1)*sqrt(a**2*c*x**2 + c)), x) + Integral(-5*I*a**4*x**4/(a**6*c*x**6*sqrt( a**2*x**2 + 1)*sqrt(a**2*c*x**2 + c) + 3*a**4*c*x**4*sqrt(a**2*x**2 + 1)*s qrt(a**2*c*x**2 + c) + 3*a**2*c*x**2*sqrt(a**2*x**2 + 1)*sqrt(a**2*c*x**2 + c) + c*sqrt(a**2*x**2 + 1)*sqrt(a**2*c*x**2 + c)), x))
\[ \int \frac {e^{5 i \arctan (a x)}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\int { \frac {{\left (i \, a x + 1\right )}^{5}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} {\left (a^{2} x^{2} + 1\right )}^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {e^{5 i \arctan (a x)}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\int { \frac {{\left (i \, a x + 1\right )}^{5}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} {\left (a^{2} x^{2} + 1\right )}^{\frac {5}{2}}} \,d x } \]
Time = 1.89 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.51 \[ \int \frac {e^{5 i \arctan (a x)}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=-\frac {\sqrt {c\,\left (a^2\,x^2+1\right )}\,\left (3\,a\,x-\mathrm {i}\right )}{6\,a\,c^2\,\sqrt {a^2\,x^2+1}\,{\left (-1+a\,x\,1{}\mathrm {i}\right )}^3} \]