Integrand size = 25, antiderivative size = 84 \[ \int \frac {e^{3 i \arctan (a x)}}{\sqrt {c+a^2 c x^2}} \, dx=\frac {2 \sqrt {1+a^2 x^2}}{a (i+a x) \sqrt {c+a^2 c x^2}}-\frac {i \sqrt {1+a^2 x^2} \log (i+a x)}{a \sqrt {c+a^2 c x^2}} \] Output:
2*(a^2*x^2+1)^(1/2)/a/(I+a*x)/(a^2*c*x^2+c)^(1/2)-I*(a^2*x^2+1)^(1/2)*ln(I +a*x)/a/(a^2*c*x^2+c)^(1/2)
Time = 0.04 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.65 \[ \int \frac {e^{3 i \arctan (a x)}}{\sqrt {c+a^2 c x^2}} \, dx=\frac {\sqrt {1+a^2 x^2} \left (\frac {2}{i+a x}-i \log (i+a x)\right )}{a \sqrt {c+a^2 c x^2}} \] Input:
Integrate[E^((3*I)*ArcTan[a*x])/Sqrt[c + a^2*c*x^2],x]
Output:
(Sqrt[1 + a^2*x^2]*(2/(I + a*x) - I*Log[I + a*x]))/(a*Sqrt[c + a^2*c*x^2])
Time = 0.58 (sec) , antiderivative size = 58, 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, 49, 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.
\(\displaystyle \int \frac {e^{3 i \arctan (a x)}}{\sqrt {a^2 c x^2+c}} \, dx\) |
\(\Big \downarrow \) 5599 |
\(\displaystyle \frac {\sqrt {a^2 x^2+1} \int \frac {e^{3 i \arctan (a x)}}{\sqrt {a^2 x^2+1}}dx}{\sqrt {a^2 c x^2+c}}\) |
\(\Big \downarrow \) 5596 |
\(\displaystyle \frac {\sqrt {a^2 x^2+1} \int \frac {i a x+1}{(1-i a x)^2}dx}{\sqrt {a^2 c x^2+c}}\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \frac {\sqrt {a^2 x^2+1} \int \left (-\frac {i}{a x+i}-\frac {2}{(a x+i)^2}\right )dx}{\sqrt {a^2 c x^2+c}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\sqrt {a^2 x^2+1} \left (\frac {2}{a (a x+i)}-\frac {i \log (a x+i)}{a}\right )}{\sqrt {a^2 c x^2+c}}\) |
Input:
Int[E^((3*I)*ArcTan[a*x])/Sqrt[c + a^2*c*x^2],x]
Output:
(Sqrt[1 + a^2*x^2]*(2/(a*(I + a*x)) - (I*Log[I + a*x])/a))/Sqrt[c + a^2*c* x^2]
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}, x] && IGtQ[m, 0] && IGtQ[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 = 0.13 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.73
method | result | size |
default | \(\frac {\left (-i \ln \left (a x +i\right ) a x +\ln \left (a x +i\right )+2\right ) \sqrt {c \left (a^{2} x^{2}+1\right )}}{\sqrt {a^{2} x^{2}+1}\, c a \left (a x +i\right )}\) | \(61\) |
risch | \(\frac {2 \sqrt {a^{2} x^{2}+1}}{\sqrt {c \left (a^{2} x^{2}+1\right )}\, a \left (a x +i\right )}-\frac {i \sqrt {a^{2} x^{2}+1}\, \ln \left (a x +i\right )}{\sqrt {c \left (a^{2} x^{2}+1\right )}\, a}\) | \(76\) |
Input:
int((1+I*a*x)^3/(a^2*x^2+1)^(3/2)/(a^2*c*x^2+c)^(1/2),x,method=_RETURNVERB OSE)
Output:
(-I*ln(I+a*x)*a*x+ln(I+a*x)+2)/(a^2*x^2+1)^(1/2)*(c*(a^2*x^2+1))^(1/2)/c/a /(I+a*x)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 357 vs. \(2 (70) = 140\).
Time = 0.18 (sec) , antiderivative size = 357, normalized size of antiderivative = 4.25 \[ \int \frac {e^{3 i \arctan (a x)}}{\sqrt {c+a^2 c x^2}} \, dx=\frac {{\left (-i \, a^{3} c x^{3} + a^{2} c x^{2} - i \, a c x + c\right )} \sqrt {\frac {1}{a^{2} c}} \log \left (\frac {{\left (i \, a^{6} x^{2} - 2 \, a^{5} x - 2 i \, a^{4}\right )} \sqrt {a^{2} c x^{2} + c} \sqrt {a^{2} x^{2} + 1} + {\left (i \, a^{9} c x^{4} - 2 \, a^{8} c x^{3} + i \, a^{7} c x^{2} - 2 \, a^{6} c x\right )} \sqrt {\frac {1}{a^{2} c}}}{8 \, {\left (a^{3} x^{3} + i \, a^{2} x^{2} + a x + i\right )}}\right ) + {\left (i \, a^{3} c x^{3} - a^{2} c x^{2} + i \, a c x - c\right )} \sqrt {\frac {1}{a^{2} c}} \log \left (\frac {{\left (i \, a^{6} x^{2} - 2 \, a^{5} x - 2 i \, a^{4}\right )} \sqrt {a^{2} c x^{2} + c} \sqrt {a^{2} x^{2} + 1} + {\left (-i \, a^{9} c x^{4} + 2 \, a^{8} c x^{3} - i \, a^{7} c x^{2} + 2 \, a^{6} c x\right )} \sqrt {\frac {1}{a^{2} c}}}{8 \, {\left (a^{3} x^{3} + i \, a^{2} x^{2} + a x + i\right )}}\right ) + 4 i \, \sqrt {a^{2} c x^{2} + c} \sqrt {a^{2} x^{2} + 1} x}{2 \, {\left (a^{3} c x^{3} + i \, a^{2} c x^{2} + a c x + i \, c\right )}} \] Input:
integrate((1+I*a*x)^3/(a^2*x^2+1)^(3/2)/(a^2*c*x^2+c)^(1/2),x, algorithm=" fricas")
Output:
1/2*((-I*a^3*c*x^3 + a^2*c*x^2 - I*a*c*x + c)*sqrt(1/(a^2*c))*log(1/8*((I* a^6*x^2 - 2*a^5*x - 2*I*a^4)*sqrt(a^2*c*x^2 + c)*sqrt(a^2*x^2 + 1) + (I*a^ 9*c*x^4 - 2*a^8*c*x^3 + I*a^7*c*x^2 - 2*a^6*c*x)*sqrt(1/(a^2*c)))/(a^3*x^3 + I*a^2*x^2 + a*x + I)) + (I*a^3*c*x^3 - a^2*c*x^2 + I*a*c*x - c)*sqrt(1/ (a^2*c))*log(1/8*((I*a^6*x^2 - 2*a^5*x - 2*I*a^4)*sqrt(a^2*c*x^2 + c)*sqrt (a^2*x^2 + 1) + (-I*a^9*c*x^4 + 2*a^8*c*x^3 - I*a^7*c*x^2 + 2*a^6*c*x)*sqr t(1/(a^2*c)))/(a^3*x^3 + I*a^2*x^2 + a*x + I)) + 4*I*sqrt(a^2*c*x^2 + c)*s qrt(a^2*x^2 + 1)*x)/(a^3*c*x^3 + I*a^2*c*x^2 + a*c*x + I*c)
\[ \int \frac {e^{3 i \arctan (a x)}}{\sqrt {c+a^2 c x^2}} \, dx=- i \left (\int \frac {i}{a^{2} x^{2} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c}}\, dx + \int \left (- \frac {3 a x}{a^{2} x^{2} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c}}\right )\, dx + \int \frac {a^{3} x^{3}}{a^{2} x^{2} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c}}\, dx + \int \left (- \frac {3 i a^{2} x^{2}}{a^{2} x^{2} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c}}\right )\, dx\right ) \] Input:
integrate((1+I*a*x)**3/(a**2*x**2+1)**(3/2)/(a**2*c*x**2+c)**(1/2),x)
Output:
-I*(Integral(I/(a**2*x**2*sqrt(a**2*x**2 + 1)*sqrt(a**2*c*x**2 + c) + sqrt (a**2*x**2 + 1)*sqrt(a**2*c*x**2 + c)), x) + Integral(-3*a*x/(a**2*x**2*sq rt(a**2*x**2 + 1)*sqrt(a**2*c*x**2 + c) + sqrt(a**2*x**2 + 1)*sqrt(a**2*c* x**2 + c)), x) + Integral(a**3*x**3/(a**2*x**2*sqrt(a**2*x**2 + 1)*sqrt(a* *2*c*x**2 + c) + sqrt(a**2*x**2 + 1)*sqrt(a**2*c*x**2 + c)), x) + Integral (-3*I*a**2*x**2/(a**2*x**2*sqrt(a**2*x**2 + 1)*sqrt(a**2*c*x**2 + c) + sqr t(a**2*x**2 + 1)*sqrt(a**2*c*x**2 + c)), x))
\[ \int \frac {e^{3 i \arctan (a x)}}{\sqrt {c+a^2 c x^2}} \, dx=\int { \frac {{\left (i \, a x + 1\right )}^{3}}{\sqrt {a^{2} c x^{2} + c} {\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((1+I*a*x)^3/(a^2*x^2+1)^(3/2)/(a^2*c*x^2+c)^(1/2),x, algorithm=" maxima")
Output:
integrate((I*a*x + 1)^3/(sqrt(a^2*c*x^2 + c)*(a^2*x^2 + 1)^(3/2)), x)
\[ \int \frac {e^{3 i \arctan (a x)}}{\sqrt {c+a^2 c x^2}} \, dx=\int { \frac {{\left (i \, a x + 1\right )}^{3}}{\sqrt {a^{2} c x^{2} + c} {\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((1+I*a*x)^3/(a^2*x^2+1)^(3/2)/(a^2*c*x^2+c)^(1/2),x, algorithm=" giac")
Output:
integrate((I*a*x + 1)^3/(sqrt(a^2*c*x^2 + c)*(a^2*x^2 + 1)^(3/2)), x)
Timed out. \[ \int \frac {e^{3 i \arctan (a x)}}{\sqrt {c+a^2 c x^2}} \, dx=\int \frac {{\left (1+a\,x\,1{}\mathrm {i}\right )}^3}{\sqrt {c\,a^2\,x^2+c}\,{\left (a^2\,x^2+1\right )}^{3/2}} \,d x \] Input:
int((a*x*1i + 1)^3/((c + a^2*c*x^2)^(1/2)*(a^2*x^2 + 1)^(3/2)),x)
Output:
int((a*x*1i + 1)^3/((c + a^2*c*x^2)^(1/2)*(a^2*x^2 + 1)^(3/2)), x)
Time = 0.15 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.01 \[ \int \frac {e^{3 i \arctan (a x)}}{\sqrt {c+a^2 c x^2}} \, dx=\frac {\sqrt {c}\, \left (-2 \mathit {atan} \left (a x \right ) a^{2} x^{2}-2 \mathit {atan} \left (a x \right )-\mathrm {log}\left (a^{2} x^{2}+1\right ) a^{2} i \,x^{2}-\mathrm {log}\left (a^{2} x^{2}+1\right ) i +4 a^{2} i \,x^{2}+4 a x \right )}{2 a c \left (a^{2} x^{2}+1\right )} \] Input:
int((1+I*a*x)^3/(a^2*x^2+1)^(3/2)/(a^2*c*x^2+c)^(1/2),x)
Output:
(sqrt(c)*( - 2*atan(a*x)*a**2*x**2 - 2*atan(a*x) - log(a**2*x**2 + 1)*a**2 *i*x**2 - log(a**2*x**2 + 1)*i + 4*a**2*i*x**2 + 4*a*x))/(2*a*c*(a**2*x**2 + 1))