Integrand size = 28, antiderivative size = 65 \[ \int \frac {e^{3 i \arctan (a x)} x^2}{\left (c+a^2 c x^2\right )^{11/2}} \, dx=-\frac {(i+3 a x) \sqrt {1+a^2 x^2}}{24 a^3 c^5 (1-i a x)^6 (1+i a x)^3 \sqrt {c+a^2 c x^2}} \]
Time = 0.08 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00 \[ \int \frac {e^{3 i \arctan (a x)} x^2}{\left (c+a^2 c x^2\right )^{11/2}} \, dx=\frac {i (i+3 a x) \sqrt {1+a^2 x^2}}{24 a^3 c^5 (-i+a x)^3 (i+a x)^6 \sqrt {c+a^2 c x^2}} \]
((I/24)*(I + 3*a*x)*Sqrt[1 + a^2*x^2])/(a^3*c^5*(-I + a*x)^3*(I + a*x)^6*S qrt[c + a^2*c*x^2])
Time = 0.44 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {5608, 5605, 91}
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.
\(\displaystyle \int \frac {x^2 e^{3 i \arctan (a x)}}{\left (a^2 c x^2+c\right )^{11/2}} \, dx\) |
\(\Big \downarrow \) 5608 |
\(\displaystyle \frac {\sqrt {a^2 x^2+1} \int \frac {e^{3 i \arctan (a x)} x^2}{\left (a^2 x^2+1\right )^{11/2}}dx}{c^5 \sqrt {a^2 c x^2+c}}\) |
\(\Big \downarrow \) 5605 |
\(\displaystyle \frac {\sqrt {a^2 x^2+1} \int \frac {x^2}{(1-i a x)^7 (i a x+1)^4}dx}{c^5 \sqrt {a^2 c x^2+c}}\) |
\(\Big \downarrow \) 91 |
\(\displaystyle -\frac {(3 a x+i) \sqrt {a^2 x^2+1}}{24 a^3 c^5 (1-i a x)^6 (1+i a x)^3 \sqrt {a^2 c x^2+c}}\) |
-1/24*((I + 3*a*x)*Sqrt[1 + a^2*x^2])/(a^3*c^5*(1 - I*a*x)^6*(1 + I*a*x)^3 *Sqrt[c + a^2*c*x^2])
3.4.81.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( p_), x_] :> Simp[b*(c + d*x)^(n + 1)*(e + f*x)^(p + 1)*((2*a*d*f*(n + p + 3 ) - b*(d*e*(n + 2) + c*f*(p + 2)) + b*d*f*(n + p + 2)*x)/(d^2*f^2*(n + p + 2)*(n + p + 3))), x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2 , 0] && NeQ[n + p + 3, 0] && EqQ[d*f*(n + p + 2)*(a^2*d*f*(n + p + 3) - b*( b*c*e + a*(d*e*(n + 1) + c*f*(p + 1)))) - b*(d*e*(n + 1) + c*f*(p + 1))*(a* d*f*(n + p + 4) - b*(d*e*(n + 2) + c*f*(p + 2))), 0]
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 = 0.32 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.88
method | result | size |
default | \(-\frac {\sqrt {c \left (a^{2} x^{2}+1\right )}\, \left (3 i a x -1\right )}{24 \sqrt {a^{2} x^{2}+1}\, c^{6} a^{3} \left (a x +i\right )^{6} \left (-a x +i\right )^{3}}\) | \(57\) |
gosper | \(\frac {\left (-a x +i\right ) \left (a x +i\right ) \left (3 a x +i\right ) \left (i a x +1\right )^{3}}{24 a^{3} \left (a^{2} x^{2}+1\right )^{\frac {3}{2}} \left (a^{2} c \,x^{2}+c \right )^{\frac {11}{2}}}\) | \(58\) |
risch | \(\frac {\sqrt {a^{2} x^{2}+1}\, \left (\frac {i x}{8 a^{2}}-\frac {1}{24 a^{3}}\right )}{c^{5} \sqrt {c \left (a^{2} x^{2}+1\right )}\, \left (a x +i\right )^{6} \left (a x -i\right )^{3}}\) | \(58\) |
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 192 vs. \(2 (53) = 106\).
Time = 0.29 (sec) , antiderivative size = 192, normalized size of antiderivative = 2.95 \[ \int \frac {e^{3 i \arctan (a x)} x^2}{\left (c+a^2 c x^2\right )^{11/2}} \, dx=\frac {{\left (i \, a^{6} x^{9} - 3 \, a^{5} x^{8} - 8 \, a^{3} x^{6} - 6 i \, a^{2} x^{5} - 6 \, a x^{4} - 8 i \, x^{3}\right )} \sqrt {a^{2} c x^{2} + c} \sqrt {a^{2} x^{2} + 1}}{24 \, {\left (a^{11} c^{6} x^{11} + 3 i \, a^{10} c^{6} x^{10} + a^{9} c^{6} x^{9} + 11 i \, a^{8} c^{6} x^{8} - 6 \, a^{7} c^{6} x^{7} + 14 i \, a^{6} c^{6} x^{6} - 14 \, a^{5} c^{6} x^{5} + 6 i \, a^{4} c^{6} x^{4} - 11 \, a^{3} c^{6} x^{3} - i \, a^{2} c^{6} x^{2} - 3 \, a c^{6} x - i \, c^{6}\right )}} \]
1/24*(I*a^6*x^9 - 3*a^5*x^8 - 8*a^3*x^6 - 6*I*a^2*x^5 - 6*a*x^4 - 8*I*x^3) *sqrt(a^2*c*x^2 + c)*sqrt(a^2*x^2 + 1)/(a^11*c^6*x^11 + 3*I*a^10*c^6*x^10 + a^9*c^6*x^9 + 11*I*a^8*c^6*x^8 - 6*a^7*c^6*x^7 + 14*I*a^6*c^6*x^6 - 14*a ^5*c^6*x^5 + 6*I*a^4*c^6*x^4 - 11*a^3*c^6*x^3 - I*a^2*c^6*x^2 - 3*a*c^6*x - I*c^6)
\[ \int \frac {e^{3 i \arctan (a x)} x^2}{\left (c+a^2 c x^2\right )^{11/2}} \, dx=- i \left (\int \frac {i x^{2}}{a^{12} c^{5} x^{12} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + 6 a^{10} c^{5} x^{10} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + 15 a^{8} c^{5} x^{8} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + 20 a^{6} c^{5} x^{6} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + 15 a^{4} c^{5} x^{4} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + 6 a^{2} c^{5} x^{2} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + c^{5} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c}}\, dx + \int \left (- \frac {3 a x^{3}}{a^{12} c^{5} x^{12} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + 6 a^{10} c^{5} x^{10} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + 15 a^{8} c^{5} x^{8} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + 20 a^{6} c^{5} x^{6} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + 15 a^{4} c^{5} x^{4} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + 6 a^{2} c^{5} x^{2} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + c^{5} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c}}\right )\, dx + \int \frac {a^{3} x^{5}}{a^{12} c^{5} x^{12} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + 6 a^{10} c^{5} x^{10} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + 15 a^{8} c^{5} x^{8} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + 20 a^{6} c^{5} x^{6} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + 15 a^{4} c^{5} x^{4} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + 6 a^{2} c^{5} x^{2} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + c^{5} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c}}\, dx + \int \left (- \frac {3 i a^{2} x^{4}}{a^{12} c^{5} x^{12} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + 6 a^{10} c^{5} x^{10} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + 15 a^{8} c^{5} x^{8} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + 20 a^{6} c^{5} x^{6} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + 15 a^{4} c^{5} x^{4} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + 6 a^{2} c^{5} x^{2} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + c^{5} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c}}\right )\, dx\right ) \]
-I*(Integral(I*x**2/(a**12*c**5*x**12*sqrt(a**2*x**2 + 1)*sqrt(a**2*c*x**2 + c) + 6*a**10*c**5*x**10*sqrt(a**2*x**2 + 1)*sqrt(a**2*c*x**2 + c) + 15* a**8*c**5*x**8*sqrt(a**2*x**2 + 1)*sqrt(a**2*c*x**2 + c) + 20*a**6*c**5*x* *6*sqrt(a**2*x**2 + 1)*sqrt(a**2*c*x**2 + c) + 15*a**4*c**5*x**4*sqrt(a**2 *x**2 + 1)*sqrt(a**2*c*x**2 + c) + 6*a**2*c**5*x**2*sqrt(a**2*x**2 + 1)*sq rt(a**2*c*x**2 + c) + c**5*sqrt(a**2*x**2 + 1)*sqrt(a**2*c*x**2 + c)), x) + Integral(-3*a*x**3/(a**12*c**5*x**12*sqrt(a**2*x**2 + 1)*sqrt(a**2*c*x** 2 + c) + 6*a**10*c**5*x**10*sqrt(a**2*x**2 + 1)*sqrt(a**2*c*x**2 + c) + 15 *a**8*c**5*x**8*sqrt(a**2*x**2 + 1)*sqrt(a**2*c*x**2 + c) + 20*a**6*c**5*x **6*sqrt(a**2*x**2 + 1)*sqrt(a**2*c*x**2 + c) + 15*a**4*c**5*x**4*sqrt(a** 2*x**2 + 1)*sqrt(a**2*c*x**2 + c) + 6*a**2*c**5*x**2*sqrt(a**2*x**2 + 1)*s qrt(a**2*c*x**2 + c) + c**5*sqrt(a**2*x**2 + 1)*sqrt(a**2*c*x**2 + c)), x) + Integral(a**3*x**5/(a**12*c**5*x**12*sqrt(a**2*x**2 + 1)*sqrt(a**2*c*x* *2 + c) + 6*a**10*c**5*x**10*sqrt(a**2*x**2 + 1)*sqrt(a**2*c*x**2 + c) + 1 5*a**8*c**5*x**8*sqrt(a**2*x**2 + 1)*sqrt(a**2*c*x**2 + c) + 20*a**6*c**5* x**6*sqrt(a**2*x**2 + 1)*sqrt(a**2*c*x**2 + c) + 15*a**4*c**5*x**4*sqrt(a* *2*x**2 + 1)*sqrt(a**2*c*x**2 + c) + 6*a**2*c**5*x**2*sqrt(a**2*x**2 + 1)* sqrt(a**2*c*x**2 + c) + c**5*sqrt(a**2*x**2 + 1)*sqrt(a**2*c*x**2 + c)), x ) + Integral(-3*I*a**2*x**4/(a**12*c**5*x**12*sqrt(a**2*x**2 + 1)*sqrt(a** 2*c*x**2 + c) + 6*a**10*c**5*x**10*sqrt(a**2*x**2 + 1)*sqrt(a**2*c*x**2...
Exception generated. \[ \int \frac {e^{3 i \arctan (a x)} x^2}{\left (c+a^2 c x^2\right )^{11/2}} \, dx=\text {Exception raised: RuntimeError} \]
\[ \int \frac {e^{3 i \arctan (a x)} x^2}{\left (c+a^2 c x^2\right )^{11/2}} \, dx=\int { \frac {{\left (i \, a x + 1\right )}^{3} x^{2}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {11}{2}} {\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}}} \,d x } \]
Time = 1.85 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.74 \[ \int \frac {e^{3 i \arctan (a x)} x^2}{\left (c+a^2 c x^2\right )^{11/2}} \, dx=\frac {\sqrt {c\,\left (a^2\,x^2+1\right )}\,{\left (a\,x-\mathrm {i}\right )}^3\,\left (3\,a\,x+1{}\mathrm {i}\right )\,1{}\mathrm {i}}{24\,a^3\,c^6\,{\left (a^2\,x^2+1\right )}^{13/2}} \]