Integrand size = 28, antiderivative size = 142 \[ \int \frac {e^{i \arctan (a x)} x^2}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=-\frac {\sqrt {1+a^2 x^2}}{2 a^3 c (i+a x) \sqrt {c+a^2 c x^2}}+\frac {i \sqrt {1+a^2 x^2} \log (i-a x)}{4 a^3 c \sqrt {c+a^2 c x^2}}+\frac {3 i \sqrt {1+a^2 x^2} \log (i+a x)}{4 a^3 c \sqrt {c+a^2 c x^2}} \]
-1/2*(a^2*x^2+1)^(1/2)/a^3/c/(I+a*x)/(a^2*c*x^2+c)^(1/2)+1/4*I*ln(I-a*x)*( a^2*x^2+1)^(1/2)/a^3/c/(a^2*c*x^2+c)^(1/2)+3/4*I*ln(I+a*x)*(a^2*x^2+1)^(1/ 2)/a^3/c/(a^2*c*x^2+c)^(1/2)
Time = 0.04 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.52 \[ \int \frac {e^{i \arctan (a x)} x^2}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\frac {\sqrt {1+a^2 x^2} \left (-\frac {2}{i+a x}+i \log (i-a x)+3 i \log (i+a x)\right )}{4 a^3 c \sqrt {c+a^2 c x^2}} \]
(Sqrt[1 + a^2*x^2]*(-2/(I + a*x) + I*Log[I - a*x] + (3*I)*Log[I + a*x]))/( 4*a^3*c*Sqrt[c + a^2*c*x^2])
Time = 0.49 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.58, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {5608, 5605, 99, 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.
\(\displaystyle \int \frac {x^2 e^{i \arctan (a x)}}{\left (a^2 c x^2+c\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 5608 |
\(\displaystyle \frac {\sqrt {a^2 x^2+1} \int \frac {e^{i \arctan (a x)} x^2}{\left (a^2 x^2+1\right )^{3/2}}dx}{c \sqrt {a^2 c x^2+c}}\) |
\(\Big \downarrow \) 5605 |
\(\displaystyle \frac {\sqrt {a^2 x^2+1} \int \frac {x^2}{(1-i a x)^2 (i a x+1)}dx}{c \sqrt {a^2 c x^2+c}}\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \frac {\sqrt {a^2 x^2+1} \int \left (\frac {3 i}{4 a^2 (a x+i)}+\frac {1}{2 a^2 (a x+i)^2}+\frac {i}{4 a^2 (a x-i)}\right )dx}{c \sqrt {a^2 c x^2+c}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\sqrt {a^2 x^2+1} \left (-\frac {1}{2 a^3 (a x+i)}+\frac {i \log (-a x+i)}{4 a^3}+\frac {3 i \log (a x+i)}{4 a^3}\right )}{c \sqrt {a^2 c x^2+c}}\) |
(Sqrt[1 + a^2*x^2]*(-1/2*1/(a^3*(I + a*x)) + ((I/4)*Log[I - a*x])/a^3 + (( (3*I)/4)*Log[I + a*x])/a^3))/(c*Sqrt[c + a^2*c*x^2])
3.4.82.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] && (IntegerQ[p] | | (GtQ[m, 0] && GeQ[n, -1]))
Int[E^(ArcTan[(a_.)*(x_)]*(n_.))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_.), x_ Symbol] :> Simp[c^p Int[x^m*(1 - I*a*x)^(p + I*(n/2))*(1 + I*a*x)^(p - I* (n/2)), x], x] /; FreeQ[{a, c, d, m, n, p}, x] && EqQ[d, a^2*c] && (Integer Q[p] || GtQ[c, 0])
Int[E^(ArcTan[(a_.)*(x_)]*(n_.))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[c^IntPart[p]*((c + d*x^2)^FracPart[p]/(1 + a^2*x^2)^FracPart [p]) Int[x^m*(1 + a^2*x^2)^p*E^(n*ArcTan[a*x]), x], x] /; FreeQ[{a, c, d, m, n, p}, x] && EqQ[d, a^2*c] && !(IntegerQ[p] || GtQ[c, 0])
Time = 0.24 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.61
method | result | size |
default | \(\frac {\sqrt {c \left (a^{2} x^{2}+1\right )}\, \left (i \ln \left (-a x +i\right ) a x +3 i \ln \left (a x +i\right ) a x -\ln \left (-a x +i\right )-3 \ln \left (a x +i\right )-2\right )}{4 \sqrt {a^{2} x^{2}+1}\, c^{2} a^{3} \left (a x +i\right )}\) | \(87\) |
risch | \(-\frac {\sqrt {a^{2} x^{2}+1}}{2 c \sqrt {c \left (a^{2} x^{2}+1\right )}\, a^{3} \left (a x +i\right )}+\frac {3 i \sqrt {a^{2} x^{2}+1}\, \ln \left (i a x -1\right )}{4 c \sqrt {c \left (a^{2} x^{2}+1\right )}\, a^{3}}+\frac {i \sqrt {a^{2} x^{2}+1}\, \ln \left (-i a x -1\right )}{4 c \sqrt {c \left (a^{2} x^{2}+1\right )}\, a^{3}}\) | \(124\) |
1/4/(a^2*x^2+1)^(1/2)*(c*(a^2*x^2+1))^(1/2)*(I*ln(I-a*x)*a*x+3*I*ln(I+a*x) *a*x-ln(I-a*x)-3*ln(I+a*x)-2)/c^2/a^3/(I+a*x)
\[ \int \frac {e^{i \arctan (a x)} x^2}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\int { \frac {{\left (i \, a x + 1\right )} x^{2}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} \sqrt {a^{2} x^{2} + 1}} \,d x } \]
-1/8*(3*(I*a^5*c^2*x^3 - a^4*c^2*x^2 + I*a^3*c^2*x - a^2*c^2)*sqrt(1/(a^6* c^3))*log((I*sqrt(a^2*c*x^2 + c)*sqrt(a^2*x^2 + 1)*a^3*c*x*sqrt(1/(a^6*c^3 )) + I*a^2*x^3 + I*x)/(a^3*x^3 + I*a^2*x^2 + a*x + I)) + 3*(-I*a^5*c^2*x^3 + a^4*c^2*x^2 - I*a^3*c^2*x + a^2*c^2)*sqrt(1/(a^6*c^3))*log((-I*sqrt(a^2 *c*x^2 + c)*sqrt(a^2*x^2 + 1)*a^3*c*x*sqrt(1/(a^6*c^3)) + I*a^2*x^3 + I*x) /(a^3*x^3 + I*a^2*x^2 + a*x + I)) - (I*a^5*c^2*x^3 - a^4*c^2*x^2 + I*a^3*c ^2*x - a^2*c^2)*sqrt(1/(a^6*c^3))*log((I*sqrt(a^2*c*x^2 + c)*sqrt(a^2*x^2 + 1)*a^3*c*x*sqrt(1/(a^6*c^3)) - I*a^2*x^3 - I*x)/(a^3*x^3 - I*a^2*x^2 + a *x - I)) - (-I*a^5*c^2*x^3 + a^4*c^2*x^2 - I*a^3*c^2*x + a^2*c^2)*sqrt(1/( a^6*c^3))*log((-I*sqrt(a^2*c*x^2 + c)*sqrt(a^2*x^2 + 1)*a^3*c*x*sqrt(1/(a^ 6*c^3)) - I*a^2*x^3 - I*x)/(a^3*x^3 - I*a^2*x^2 + a*x - I)) + 4*(-I*a^5*c^ 2*x^3 + a^4*c^2*x^2 - I*a^3*c^2*x + a^2*c^2)*sqrt(1/(a^6*c^3))*log((sqrt(a ^2*c*x^2 + c)*sqrt(a^2*x^2 + 1)*a^3*c*x*sqrt(1/(a^6*c^3)) + a^2*x^3 + x)/( a^2*x^2 + 1)) + 4*(I*a^5*c^2*x^3 - a^4*c^2*x^2 + I*a^3*c^2*x - a^2*c^2)*sq rt(1/(a^6*c^3))*log(-(sqrt(a^2*c*x^2 + c)*sqrt(a^2*x^2 + 1)*a^3*c*x*sqrt(1 /(a^6*c^3)) - a^2*x^3 - x)/(a^2*x^2 + 1)) + 4*I*sqrt(a^2*c*x^2 + c)*sqrt(a ^2*x^2 + 1)*x - 8*(a^5*c^2*x^3 + I*a^4*c^2*x^2 + a^3*c^2*x + I*a^2*c^2)*in tegral(1/2*sqrt(a^2*c*x^2 + c)*sqrt(a^2*x^2 + 1)*(2*I*a*x + 1)/(a^6*c^2*x^ 4 + 2*a^4*c^2*x^2 + a^2*c^2), x))/(a^5*c^2*x^3 + I*a^4*c^2*x^2 + a^3*c^2*x + I*a^2*c^2)
\[ \int \frac {e^{i \arctan (a x)} x^2}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=i \left (\int \left (- \frac {i x^{2}}{a^{2} c x^{2} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + c \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c}}\right )\, dx + \int \frac {a x^{3}}{a^{2} c x^{2} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + c \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c}}\, dx\right ) \]
I*(Integral(-I*x**2/(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*x**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))
Exception generated. \[ \int \frac {e^{i \arctan (a x)} x^2}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\text {Exception raised: RuntimeError} \]
\[ \int \frac {e^{i \arctan (a x)} x^2}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\int { \frac {{\left (i \, a x + 1\right )} x^{2}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} \sqrt {a^{2} x^{2} + 1}} \,d x } \]
Timed out. \[ \int \frac {e^{i \arctan (a x)} x^2}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\int \frac {x^2\,\left (1+a\,x\,1{}\mathrm {i}\right )}{{\left (c\,a^2\,x^2+c\right )}^{3/2}\,\sqrt {a^2\,x^2+1}} \,d x \]