\(\int \frac {e^{3 i \arctan (a x)} x^2}{(c+a^2 c x^2)^{11/2}} \, dx\) [381]

Optimal result

Integrand size = 28, antiderivative size = 65 \[ \int \frac {e^{3 i \arctan (a x)} x^2}{\left (c+a^2 c x^2\right )^{11/2}} \, dx=-\frac {(i+3 a x) \sqrt {1+a^2 x^2}}{24 a^3 c^5 (1-i a x)^6 (1+i a x)^3 \sqrt {c+a^2 c x^2}} \]

[Out]

-1/24*(I+3*a*x)*(a^2*x^2+1)^(1/2)/a^3/c^5/(1-I*a*x)^6/(1+I*a*x)^3/(a^2*c*x^2+c)^(1/2)

Rubi [A] (verified)

Time = 0.14 (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 = {5193, 5190, 82} \[ \int \frac {e^{3 i \arctan (a x)} x^2}{\left (c+a^2 c x^2\right )^{11/2}} \, dx=-\frac {(3 a x+i) \sqrt {a^2 x^2+1}}{24 a^3 c^5 (1-i a x)^6 (1+i a x)^3 \sqrt {a^2 c x^2+c}} \]

[In]

Int[(E^((3*I)*ArcTan[a*x])*x^2)/(c + a^2*c*x^2)^(11/2),x]

[Out]

-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])

Rule 82

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> 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]

Rule 5190

Int[E^(ArcTan[(a_.)*(x_)]*(n_.))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[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] && (Int
egerQ[p] || GtQ[c, 0])

Rule 5193

Int[E^(ArcTan[(a_.)*(x_)]*(n_.))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_), x_Symbol] :> Dist[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])

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {1+a^2 x^2} \int \frac {e^{3 i \arctan (a x)} x^2}{\left (1+a^2 x^2\right )^{11/2}} \, dx}{c^5 \sqrt {c+a^2 c x^2}} \\ & = \frac {\sqrt {1+a^2 x^2} \int \frac {x^2}{(1-i a x)^7 (1+i a x)^4} \, dx}{c^5 \sqrt {c+a^2 c x^2}} \\ & = -\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}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00 \[ \int \frac {e^{3 i \arctan (a x)} x^2}{\left (c+a^2 c x^2\right )^{11/2}} \, dx=\frac {i (i+3 a x) \sqrt {1+a^2 x^2}}{24 a^3 c^5 (-i+a x)^3 (i+a x)^6 \sqrt {c+a^2 c x^2}} \]

[In]

Integrate[(E^((3*I)*ArcTan[a*x])*x^2)/(c + a^2*c*x^2)^(11/2),x]

[Out]

((I/24)*(I + 3*a*x)*Sqrt[1 + a^2*x^2])/(a^3*c^5*(-I + a*x)^3*(I + a*x)^6*Sqrt[c + a^2*c*x^2])

Maple [A] (verified)

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	\(\frac {\sqrt {a^{2} x^{2}+1}\, \left (\frac {i x}{8 a^{2}}-\frac {1}{24 a^{3}}\right )}{c^{5} \sqrt {c \left (a^{2} x^{2}+1\right )}\, \left (a x +i\right )^{6} \left (a x -i\right )^{3}}\)	\(58\)

[In]

int((1+I*a*x)^3/(a^2*x^2+1)^(3/2)*x^2/(a^2*c*x^2+c)^(11/2),x,method=_RETURNVERBOSE)

[Out]

-1/24/(a^2*x^2+1)^(1/2)*(c*(a^2*x^2+1))^(1/2)*(3*I*a*x-1)/c^6/a^3/(I+a*x)^6/(I-a*x)^3

Fricas [B] (verification not implemented)

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 \[ \int \frac {e^{3 i \arctan (a x)} x^2}{\left (c+a^2 c x^2\right )^{11/2}} \, dx=\frac {{\left (i \, a^{6} x^{9} - 3 \, a^{5} x^{8} - 8 \, a^{3} x^{6} - 6 i \, a^{2} x^{5} - 6 \, a x^{4} - 8 i \, x^{3}\right )} \sqrt {a^{2} c x^{2} + c} \sqrt {a^{2} x^{2} + 1}}{24 \, {\left (a^{11} c^{6} x^{11} + 3 i \, a^{10} c^{6} x^{10} + a^{9} c^{6} x^{9} + 11 i \, a^{8} c^{6} x^{8} - 6 \, a^{7} c^{6} x^{7} + 14 i \, a^{6} c^{6} x^{6} - 14 \, a^{5} c^{6} x^{5} + 6 i \, a^{4} c^{6} x^{4} - 11 \, a^{3} c^{6} x^{3} - i \, a^{2} c^{6} x^{2} - 3 \, a c^{6} x - i \, c^{6}\right )}} \]

[In]

integrate((1+I*a*x)^3/(a^2*x^2+1)^(3/2)*x^2/(a^2*c*x^2+c)^(11/2),x, algorithm="fricas")

[Out]

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)

Sympy [F]

\[ \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 ) \]

[In]

integrate((1+I*a*x)**3/(a**2*x**2+1)**(3/2)*x**2/(a**2*c*x**2+c)**(11/2),x)

[Out]

-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)*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*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)), 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) + 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)), 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 + 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)*sqr
t(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)*sqr
t(a**2*c*x**2 + c)), x))

Maxima [F(-2)]

Exception generated. \[ \int \frac {e^{3 i \arctan (a x)} x^2}{\left (c+a^2 c x^2\right )^{11/2}} \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate((1+I*a*x)^3/(a^2*x^2+1)^(3/2)*x^2/(a^2*c*x^2+c)^(11/2),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is undefined.

Giac [F]

\[ \int \frac {e^{3 i \arctan (a x)} x^2}{\left (c+a^2 c x^2\right )^{11/2}} \, dx=\int { \frac {{\left (i \, a x + 1\right )}^{3} x^{2}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {11}{2}} {\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}}} \,d x } \]

[In]

integrate((1+I*a*x)^3/(a^2*x^2+1)^(3/2)*x^2/(a^2*c*x^2+c)^(11/2),x, algorithm="giac")

[Out]

integrate((I*a*x + 1)^3*x^2/((a^2*c*x^2 + c)^(11/2)*(a^2*x^2 + 1)^(3/2)), x)

Mupad [B] (verification not implemented)

Time = 1.85 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.74 \[ \int \frac {e^{3 i \arctan (a x)} x^2}{\left (c+a^2 c x^2\right )^{11/2}} \, dx=\frac {\sqrt {c\,\left (a^2\,x^2+1\right )}\,{\left (a\,x-\mathrm {i}\right )}^3\,\left (3\,a\,x+1{}\mathrm {i}\right )\,1{}\mathrm {i}}{24\,a^3\,c^6\,{\left (a^2\,x^2+1\right )}^{13/2}} \]

[In]

int((x^2*(a*x*1i + 1)^3)/((c + a^2*c*x^2)^(11/2)*(a^2*x^2 + 1)^(3/2)),x)

[Out]

((c*(a^2*x^2 + 1))^(1/2)*(a*x - 1i)^3*(3*a*x + 1i)*1i)/(24*a^3*c^6*(a^2*x^2 + 1)^(13/2))