\(\int \frac {1}{(a-i a x)^{9/4} (a+i a x)^{9/4}} \, dx\) [1221]

Optimal result

Integrand size = 25, antiderivative size = 88 \[ \int \frac {1}{(a-i a x)^{9/4} (a+i a x)^{9/4}} \, dx=\frac {2 x}{5 a^4 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x} \left (1+x^2\right )}+\frac {6 \sqrt [4]{1+x^2} E\left (\left .\frac {\arctan (x)}{2}\right |2\right )}{5 a^4 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}} \]

[Out]

2/5*x/a^4/(a-I*a*x)^(1/4)/(a+I*a*x)^(1/4)/(x^2+1)+6/5*(x^2+1)^(1/4)*(cos(1/2*arctan(x))^2)^(1/2)/cos(1/2*arcta
n(x))*EllipticE(sin(1/2*arctan(x)),2^(1/2))/a^4/(a-I*a*x)^(1/4)/(a+I*a*x)^(1/4)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {42, 205, 203, 202} \[ \int \frac {1}{(a-i a x)^{9/4} (a+i a x)^{9/4}} \, dx=\frac {6 \sqrt [4]{x^2+1} E\left (\left .\frac {\arctan (x)}{2}\right |2\right )}{5 a^4 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}+\frac {2 x}{5 a^4 \left (x^2+1\right ) \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}} \]

[In]

Int[1/((a - I*a*x)^(9/4)*(a + I*a*x)^(9/4)),x]

[Out]

(2*x)/(5*a^4*(a - I*a*x)^(1/4)*(a + I*a*x)^(1/4)*(1 + x^2)) + (6*(1 + x^2)^(1/4)*EllipticE[ArcTan[x]/2, 2])/(5
*a^4*(a - I*a*x)^(1/4)*(a + I*a*x)^(1/4))

Rule 42

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(m_), x_Symbol] :> Dist[(a + b*x)^FracPart[m]*((c + d*x)^Frac
Part[m]/(a*c + b*d*x^2)^FracPart[m]), Int[(a*c + b*d*x^2)^m, x], x] /; FreeQ[{a, b, c, d, m}, x] && EqQ[b*c +
a*d, 0] &&  !IntegerQ[2*m]

Rule 202

Int[((a_) + (b_.)*(x_)^2)^(-5/4), x_Symbol] :> Simp[(2/(a^(5/4)*Rt[b/a, 2]))*EllipticE[(1/2)*ArcTan[Rt[b/a, 2]
*x], 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b/a]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-5/4), x_Symbol] :> Dist[(1 + b*(x^2/a))^(1/4)/(a*(a + b*x^2)^(1/4)), Int[1/(1 + b*
(x^2/a))^(5/4), x], x] /; FreeQ[{a, b}, x] && PosQ[a] && PosQ[b/a]

Rule 205

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^n)^(p + 1)/(a*n*(p + 1))), x] + Dist[(n*(p
+ 1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (
IntegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[
p])

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt [4]{a^2+a^2 x^2} \int \frac {1}{\left (a^2+a^2 x^2\right )^{9/4}} \, dx}{\sqrt [4]{a-i a x} \sqrt [4]{a+i a x}} \\ & = \frac {2 x}{5 a^4 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x} \left (1+x^2\right )}+\frac {\left (3 \sqrt [4]{a^2+a^2 x^2}\right ) \int \frac {1}{\left (a^2+a^2 x^2\right )^{5/4}} \, dx}{5 a^2 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}} \\ & = \frac {2 x}{5 a^4 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x} \left (1+x^2\right )}+\frac {\left (3 \sqrt [4]{1+x^2}\right ) \int \frac {1}{\left (1+x^2\right )^{5/4}} \, dx}{5 a^4 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}} \\ & = \frac {2 x}{5 a^4 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x} \left (1+x^2\right )}+\frac {6 \sqrt [4]{1+x^2} E\left (\left .\frac {1}{2} \tan ^{-1}(x)\right |2\right )}{5 a^4 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 0.02 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.80 \[ \int \frac {1}{(a-i a x)^{9/4} (a+i a x)^{9/4}} \, dx=-\frac {i \sqrt [4]{1+i x} \operatorname {Hypergeometric2F1}\left (-\frac {5}{4},\frac {9}{4},-\frac {1}{4},\frac {1}{2}-\frac {i x}{2}\right )}{5 \sqrt [4]{2} a^3 (a-i a x)^{5/4} \sqrt [4]{a+i a x}} \]

[In]

Integrate[1/((a - I*a*x)^(9/4)*(a + I*a*x)^(9/4)),x]

[Out]

((-1/5*I)*(1 + I*x)^(1/4)*Hypergeometric2F1[-5/4, 9/4, -1/4, 1/2 - (I/2)*x])/(2^(1/4)*a^3*(a - I*a*x)^(5/4)*(a
+ I*a*x)^(1/4))

Maple [F]

\[\int \frac {1}{\left (-i a x +a \right )^{\frac {9}{4}} \left (i a x +a \right )^{\frac {9}{4}}}d x\]

[In]

int(1/(a-I*a*x)^(9/4)/(a+I*a*x)^(9/4),x)

[Out]

int(1/(a-I*a*x)^(9/4)/(a+I*a*x)^(9/4),x)

Fricas [F]

\[ \int \frac {1}{(a-i a x)^{9/4} (a+i a x)^{9/4}} \, dx=\int { \frac {1}{{\left (i \, a x + a\right )}^{\frac {9}{4}} {\left (-i \, a x + a\right )}^{\frac {9}{4}}} \,d x } \]

[In]

integrate(1/(a-I*a*x)^(9/4)/(a+I*a*x)^(9/4),x, algorithm="fricas")

[Out]

1/5*(2*(3*x^3 + 4*x)*(I*a*x + a)^(3/4)*(-I*a*x + a)^(3/4) + 5*(a^6*x^4 + 2*a^6*x^2 + a^6)*integral(-3/5*(I*a*x
+ a)^(3/4)*(-I*a*x + a)^(3/4)/(a^6*x^2 + a^6), x))/(a^6*x^4 + 2*a^6*x^2 + a^6)

Sympy [A] (verification not implemented)

Time = 85.82 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.08 \[ \int \frac {1}{(a-i a x)^{9/4} (a+i a x)^{9/4}} \, dx=- \frac {i {G_{6, 6}^{5, 3}\left (\begin {matrix} \frac {9}{8}, \frac {13}{8}, 1 & \frac {1}{2}, \frac {9}{4}, \frac {11}{4} \\\frac {9}{8}, \frac {13}{8}, \frac {7}{4}, \frac {9}{4}, \frac {11}{4} & 0 \end {matrix} \middle | {\frac {e^{- 3 i \pi }}{x^{2}}} \right )} e^{\frac {i \pi }{4}}}{4 \pi a^{\frac {9}{2}} \Gamma \left (\frac {9}{4}\right )} + \frac {i {G_{6, 6}^{2, 6}\left (\begin {matrix} - \frac {1}{2}, 0, \frac {1}{2}, \frac {5}{8}, \frac {9}{8}, 1 & \\\frac {5}{8}, \frac {9}{8} & - \frac {1}{2}, 0, \frac {7}{4}, 0 \end {matrix} \middle | {\frac {e^{- i \pi }}{x^{2}}} \right )}}{4 \pi a^{\frac {9}{2}} \Gamma \left (\frac {9}{4}\right )} \]

[In]

integrate(1/(a-I*a*x)**(9/4)/(a+I*a*x)**(9/4),x)

[Out]

-I*meijerg(((9/8, 13/8, 1), (1/2, 9/4, 11/4)), ((9/8, 13/8, 7/4, 9/4, 11/4), (0,)), exp_polar(-3*I*pi)/x**2)*e
xp(I*pi/4)/(4*pi*a**(9/2)*gamma(9/4)) + I*meijerg(((-1/2, 0, 1/2, 5/8, 9/8, 1), ()), ((5/8, 9/8), (-1/2, 0, 7/
4, 0)), exp_polar(-I*pi)/x**2)/(4*pi*a**(9/2)*gamma(9/4))

Maxima [F]

\[ \int \frac {1}{(a-i a x)^{9/4} (a+i a x)^{9/4}} \, dx=\int { \frac {1}{{\left (i \, a x + a\right )}^{\frac {9}{4}} {\left (-i \, a x + a\right )}^{\frac {9}{4}}} \,d x } \]

[In]

integrate(1/(a-I*a*x)^(9/4)/(a+I*a*x)^(9/4),x, algorithm="maxima")

[Out]

integrate(1/((I*a*x + a)^(9/4)*(-I*a*x + a)^(9/4)), x)

Giac [F(-2)]

Exception generated. \[ \int \frac {1}{(a-i a x)^{9/4} (a+i a x)^{9/4}} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(1/(a-I*a*x)^(9/4)/(a+I*a*x)^(9/4),x, algorithm="giac")

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:The choice was done assuming 0=[0,0]ext_reduce Error: Bad Argument TypeDone

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(a-i a x)^{9/4} (a+i a x)^{9/4}} \, dx=\int \frac {1}{{\left (a-a\,x\,1{}\mathrm {i}\right )}^{9/4}\,{\left (a+a\,x\,1{}\mathrm {i}\right )}^{9/4}} \,d x \]

[In]

int(1/((a - a*x*1i)^(9/4)*(a + a*x*1i)^(9/4)),x)

[Out]

int(1/((a - a*x*1i)^(9/4)*(a + a*x*1i)^(9/4)), x)