Integrand size = 24, antiderivative size = 69 \[ \int \frac {e^{-4 i \arctan (a x)}}{\sqrt {1+a^2 x^2}} \, dx=\frac {2 i (1-i a x)^3}{3 a \left (1+a^2 x^2\right )^{3/2}}-\frac {2 i (1-i a x)}{a \sqrt {1+a^2 x^2}}+\frac {\text {arcsinh}(a x)}{a} \] Output:
2/3*I*(1-I*a*x)^3/a/(a^2*x^2+1)^(3/2)-2*I*(1-I*a*x)/a/(a^2*x^2+1)^(1/2)+ar csinh(a*x)/a
Time = 0.10 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.19 \[ \int \frac {e^{-4 i \arctan (a x)}}{\sqrt {1+a^2 x^2}} \, dx=\frac {2 i \left (\frac {2 \sqrt {1+i a x} \left (1+i a x+2 a^2 x^2\right )}{\sqrt {1-i a x} (-i+a x)^2}+3 \arcsin \left (\frac {\sqrt {1-i a x}}{\sqrt {2}}\right )\right )}{3 a} \] Input:
Integrate[1/(E^((4*I)*ArcTan[a*x])*Sqrt[1 + a^2*x^2]),x]
Output:
(((2*I)/3)*((2*Sqrt[1 + I*a*x]*(1 + I*a*x + 2*a^2*x^2))/(Sqrt[1 - I*a*x]*( -I + a*x)^2) + 3*ArcSin[Sqrt[1 - I*a*x]/Sqrt[2]]))/a
Time = 0.40 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.06, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {5596, 57, 57, 39, 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.
\(\displaystyle \int \frac {e^{-4 i \arctan (a x)}}{\sqrt {a^2 x^2+1}} \, dx\) |
\(\Big \downarrow \) 5596 |
\(\displaystyle \int \frac {(1-i a x)^{3/2}}{(1+i a x)^{5/2}}dx\) |
\(\Big \downarrow \) 57 |
\(\displaystyle \frac {2 i (1-i a x)^{3/2}}{3 a (1+i a x)^{3/2}}-\int \frac {\sqrt {1-i a x}}{(i a x+1)^{3/2}}dx\) |
\(\Big \downarrow \) 57 |
\(\displaystyle \int \frac {1}{\sqrt {1-i a x} \sqrt {i a x+1}}dx+\frac {2 i (1-i a x)^{3/2}}{3 a (1+i a x)^{3/2}}-\frac {2 i \sqrt {1-i a x}}{a \sqrt {1+i a x}}\) |
\(\Big \downarrow \) 39 |
\(\displaystyle \int \frac {1}{\sqrt {a^2 x^2+1}}dx+\frac {2 i (1-i a x)^{3/2}}{3 a (1+i a x)^{3/2}}-\frac {2 i \sqrt {1-i a x}}{a \sqrt {1+i a x}}\) |
\(\Big \downarrow \) 222 |
\(\displaystyle \frac {\text {arcsinh}(a x)}{a}+\frac {2 i (1-i a x)^{3/2}}{3 a (1+i a x)^{3/2}}-\frac {2 i \sqrt {1-i a x}}{a \sqrt {1+i a x}}\) |
Input:
Int[1/(E^((4*I)*ArcTan[a*x])*Sqrt[1 + a^2*x^2]),x]
Output:
(((2*I)/3)*(1 - I*a*x)^(3/2))/(a*(1 + I*a*x)^(3/2)) - ((2*I)*Sqrt[1 - I*a* x])/(a*Sqrt[1 + I*a*x]) + ArcSinh[a*x]/a
Int[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[( a*c + b*d*x^2)^m, x] /; FreeQ[{a, b, c, d, m}, x] && EqQ[b*c + a*d, 0] && ( IntegerQ[m] || (GtQ[a, 0] && GtQ[c, 0]))
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] & & GtQ[n, 0] && LtQ[m, -1] && !(IntegerQ[n] && !IntegerQ[m]) && !(ILeQ[m + n + 2, 0] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c , d, m, n, 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[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])
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 304 vs. \(2 (59 ) = 118\).
Time = 0.18 (sec) , antiderivative size = 305, normalized size of antiderivative = 4.42
method | result | size |
default | \(\frac {\frac {i \left (\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )\right )^{\frac {5}{2}}}{3 a \left (x -\frac {i}{a}\right )^{4}}-\frac {i a \left (\frac {i \left (\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )\right )^{\frac {5}{2}}}{a \left (x -\frac {i}{a}\right )^{3}}-2 i a \left (-\frac {i \left (\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )\right )^{\frac {5}{2}}}{a \left (x -\frac {i}{a}\right )^{2}}+3 i a \left (\frac {\left (\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )\right )^{\frac {3}{2}}}{3}+i a \left (\frac {\left (2 \left (x -\frac {i}{a}\right ) a^{2}+2 i a \right ) \sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )}}{4 a^{2}}+\frac {\ln \left (\frac {i a +\left (x -\frac {i}{a}\right ) a^{2}}{\sqrt {a^{2}}}+\sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )}\right )}{2 \sqrt {a^{2}}}\right )\right )\right )\right )}{3}}{a^{4}}\) | \(305\) |
Input:
int(1/(1+I*a*x)^4*(a^2*x^2+1)^(3/2),x,method=_RETURNVERBOSE)
Output:
1/a^4*(1/3*I/a/(x-I/a)^4*((x-I/a)^2*a^2+2*I*a*(x-I/a))^(5/2)-1/3*I*a*(I/a/ (x-I/a)^3*((x-I/a)^2*a^2+2*I*a*(x-I/a))^(5/2)-2*I*a*(-I/a/(x-I/a)^2*((x-I/ a)^2*a^2+2*I*a*(x-I/a))^(5/2)+3*I*a*(1/3*((x-I/a)^2*a^2+2*I*a*(x-I/a))^(3/ 2)+I*a*(1/4*(2*(x-I/a)*a^2+2*I*a)/a^2*((x-I/a)^2*a^2+2*I*a*(x-I/a))^(1/2)+ 1/2*ln((I*a+(x-I/a)*a^2)/(a^2)^(1/2)+((x-I/a)^2*a^2+2*I*a*(x-I/a))^(1/2))/ (a^2)^(1/2))))))
Time = 0.08 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.25 \[ \int \frac {e^{-4 i \arctan (a x)}}{\sqrt {1+a^2 x^2}} \, dx=-\frac {8 \, a^{2} x^{2} - 16 i \, a x + 3 \, {\left (a^{2} x^{2} - 2 i \, a x - 1\right )} \log \left (-a x + \sqrt {a^{2} x^{2} + 1}\right ) + 4 \, \sqrt {a^{2} x^{2} + 1} {\left (2 \, a x - i\right )} - 8}{3 \, {\left (a^{3} x^{2} - 2 i \, a^{2} x - a\right )}} \] Input:
integrate(1/(1+I*a*x)^4*(a^2*x^2+1)^(3/2),x, algorithm="fricas")
Output:
-1/3*(8*a^2*x^2 - 16*I*a*x + 3*(a^2*x^2 - 2*I*a*x - 1)*log(-a*x + sqrt(a^2 *x^2 + 1)) + 4*sqrt(a^2*x^2 + 1)*(2*a*x - I) - 8)/(a^3*x^2 - 2*I*a^2*x - a )
\[ \int \frac {e^{-4 i \arctan (a x)}}{\sqrt {1+a^2 x^2}} \, dx=\int \frac {\left (a^{2} x^{2} + 1\right )^{\frac {3}{2}}}{\left (a x - i\right )^{4}}\, dx \] Input:
integrate(1/(1+I*a*x)**4*(a**2*x**2+1)**(3/2),x)
Output:
Integral((a**2*x**2 + 1)**(3/2)/(a*x - I)**4, x)
Time = 0.11 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.55 \[ \int \frac {e^{-4 i \arctan (a x)}}{\sqrt {1+a^2 x^2}} \, dx=\frac {i \, {\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{-3 i \, a^{4} x^{3} - 9 \, a^{3} x^{2} + 9 i \, a^{2} x + 3 \, a} + \frac {\operatorname {arsinh}\left (a x\right )}{a} - \frac {2 i \, \sqrt {a^{2} x^{2} + 1}}{3 \, {\left (a^{3} x^{2} - 2 i \, a^{2} x - a\right )}} - \frac {7 i \, \sqrt {a^{2} x^{2} + 1}}{3 i \, a^{2} x + 3 \, a} \] Input:
integrate(1/(1+I*a*x)^4*(a^2*x^2+1)^(3/2),x, algorithm="maxima")
Output:
I*(a^2*x^2 + 1)^(3/2)/(-3*I*a^4*x^3 - 9*a^3*x^2 + 9*I*a^2*x + 3*a) + arcsi nh(a*x)/a - 2/3*I*sqrt(a^2*x^2 + 1)/(a^3*x^2 - 2*I*a^2*x - a) - 7*I*sqrt(a ^2*x^2 + 1)/(3*I*a^2*x + 3*a)
Time = 0.13 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.35 \[ \int \frac {e^{-4 i \arctan (a x)}}{\sqrt {1+a^2 x^2}} \, dx=-\frac {\log \left (-x {\left | a \right |} + \sqrt {a^{2} x^{2} + 1}\right )}{{\left | a \right |}} \] Input:
integrate(1/(1+I*a*x)^4*(a^2*x^2+1)^(3/2),x, algorithm="giac")
Output:
-log(-x*abs(a) + sqrt(a^2*x^2 + 1))/abs(a)
Time = 0.11 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.35 \[ \int \frac {e^{-4 i \arctan (a x)}}{\sqrt {1+a^2 x^2}} \, dx=\frac {\mathrm {asinh}\left (x\,\sqrt {a^2}\right )}{\sqrt {a^2}}+\frac {8\,\sqrt {a^2\,x^2+1}}{3\,\left (-x\,\sqrt {a^2}+\frac {\sqrt {a^2}\,1{}\mathrm {i}}{a}\right )\,\sqrt {a^2}}+\frac {a\,\sqrt {a^2\,x^2+1}\,4{}\mathrm {i}}{3\,\left (-a^4\,x^2+a^3\,x\,2{}\mathrm {i}+a^2\right )} \] Input:
int((a^2*x^2 + 1)^(3/2)/(a*x*1i + 1)^4,x)
Output:
asinh(x*(a^2)^(1/2))/(a^2)^(1/2) + (8*(a^2*x^2 + 1)^(1/2))/(3*(((a^2)^(1/2 )*1i)/a - x*(a^2)^(1/2))*(a^2)^(1/2)) + (a*(a^2*x^2 + 1)^(1/2)*4i)/(3*(a^3 *x*2i + a^2 - a^4*x^2))
\[ \int \frac {e^{-4 i \arctan (a x)}}{\sqrt {1+a^2 x^2}} \, dx=\frac {-16 \left (\int -\frac {\sqrt {a^{2} x^{2}+1}\, x^{2}}{a^{6} x^{6}-4 a^{5} i \,x^{5}-5 a^{4} x^{4}-5 a^{2} x^{2}+4 a i x +1}d x \right ) a^{3}+8 \left (\int \frac {\sqrt {a^{2} x^{2}+1}\, x^{3}}{a^{6} x^{6}-4 a^{5} i \,x^{5}-5 a^{4} x^{4}-5 a^{2} x^{2}+4 a i x +1}d x \right ) a^{4} i -8 \left (\int \frac {\sqrt {a^{2} x^{2}+1}\, x}{a^{6} x^{6}-4 a^{5} i \,x^{5}-5 a^{4} x^{4}-5 a^{2} x^{2}+4 a i x +1}d x \right ) a^{2} i -\mathrm {log}\left (\sqrt {a^{2} x^{2}+1}-a x \right )+\mathrm {log}\left (\sqrt {a^{2} x^{2}+1}+a x \right )}{2 a} \] Input:
int(1/(1+I*a*x)^4*(a^2*x^2+1)^(3/2),x)
Output:
( - 16*int(( - sqrt(a**2*x**2 + 1)*x**2)/(a**6*x**6 - 4*a**5*i*x**5 - 5*a* *4*x**4 - 5*a**2*x**2 + 4*a*i*x + 1),x)*a**3 + 8*int((sqrt(a**2*x**2 + 1)* x**3)/(a**6*x**6 - 4*a**5*i*x**5 - 5*a**4*x**4 - 5*a**2*x**2 + 4*a*i*x + 1 ),x)*a**4*i - 8*int((sqrt(a**2*x**2 + 1)*x)/(a**6*x**6 - 4*a**5*i*x**5 - 5 *a**4*x**4 - 5*a**2*x**2 + 4*a*i*x + 1),x)*a**2*i - log(sqrt(a**2*x**2 + 1 ) - a*x) + log(sqrt(a**2*x**2 + 1) + a*x))/(2*a)