\(\int \frac {1}{(a-i a x)^{7/4} (a+i a x)^{7/4}} \, dx\) [1203]

Optimal result

Integrand size = 25, antiderivative size = 81 \[ \int \frac {1}{(a-i a x)^{7/4} (a+i a x)^{7/4}} \, dx=\frac {2 x}{3 a^2 (a-i a x)^{3/4} (a+i a x)^{3/4}}+\frac {2 \left (1+x^2\right )^{3/4} \operatorname {EllipticF}\left (\frac {\arctan (x)}{2},2\right )}{3 a^2 (a-i a x)^{3/4} (a+i a x)^{3/4}} \]

[Out]

2/3*x/a^2/(a-I*a*x)^(3/4)/(a+I*a*x)^(3/4)+2/3*(x^2+1)^(3/4)*(cos(1/2*arctan(x))^2)^(1/2)/cos(1/2*arctan(x))*El
lipticF(sin(1/2*arctan(x)),2^(1/2))/a^2/(a-I*a*x)^(3/4)/(a+I*a*x)^(3/4)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 81, 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, 239, 237} \[ \int \frac {1}{(a-i a x)^{7/4} (a+i a x)^{7/4}} \, dx=\frac {2 \left (x^2+1\right )^{3/4} \operatorname {EllipticF}\left (\frac {\arctan (x)}{2},2\right )}{3 a^2 (a-i a x)^{3/4} (a+i a x)^{3/4}}+\frac {2 x}{3 a^2 (a-i a x)^{3/4} (a+i a x)^{3/4}} \]

[In]

Int[1/((a - I*a*x)^(7/4)*(a + I*a*x)^(7/4)),x]

[Out]

(2*x)/(3*a^2*(a - I*a*x)^(3/4)*(a + I*a*x)^(3/4)) + (2*(1 + x^2)^(3/4)*EllipticF[ArcTan[x]/2, 2])/(3*a^2*(a -
I*a*x)^(3/4)*(a + I*a*x)^(3/4))

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 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])

Rule 237

Int[((a_) + (b_.)*(x_)^2)^(-3/4), x_Symbol] :> Simp[(2/(a^(3/4)*Rt[b/a, 2]))*EllipticF[(1/2)*ArcTan[Rt[b/a, 2]
*x], 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b/a]

Rule 239

Int[((a_) + (b_.)*(x_)^2)^(-3/4), x_Symbol] :> Dist[(1 + b*(x^2/a))^(3/4)/(a + b*x^2)^(3/4), Int[1/(1 + b*(x^2
/a))^(3/4), x], x] /; FreeQ[{a, b}, x] && PosQ[a]

Rubi steps \begin{align*} \text {integral}& = \frac {\left (a^2+a^2 x^2\right )^{3/4} \int \frac {1}{\left (a^2+a^2 x^2\right )^{7/4}} \, dx}{(a-i a x)^{3/4} (a+i a x)^{3/4}} \\ & = \frac {2 x}{3 a^2 (a-i a x)^{3/4} (a+i a x)^{3/4}}+\frac {\left (a^2+a^2 x^2\right )^{3/4} \int \frac {1}{\left (a^2+a^2 x^2\right )^{3/4}} \, dx}{3 a^2 (a-i a x)^{3/4} (a+i a x)^{3/4}} \\ & = \frac {2 x}{3 a^2 (a-i a x)^{3/4} (a+i a x)^{3/4}}+\frac {\left (1+x^2\right )^{3/4} \int \frac {1}{\left (1+x^2\right )^{3/4}} \, dx}{3 a^2 (a-i a x)^{3/4} (a+i a x)^{3/4}} \\ & = \frac {2 x}{3 a^2 (a-i a x)^{3/4} (a+i a x)^{3/4}}+\frac {2 \left (1+x^2\right )^{3/4} F\left (\left .\frac {1}{2} \tan ^{-1}(x)\right |2\right )}{3 a^2 (a-i a x)^{3/4} (a+i a x)^{3/4}} \\ \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 = 0.86 \[ \int \frac {1}{(a-i a x)^{7/4} (a+i a x)^{7/4}} \, dx=-\frac {i \sqrt [4]{2} (1+i x)^{3/4} \operatorname {Hypergeometric2F1}\left (-\frac {3}{4},\frac {7}{4},\frac {1}{4},\frac {1}{2}-\frac {i x}{2}\right )}{3 a^2 (a-i a x)^{3/4} (a+i a x)^{3/4}} \]

[In]

Integrate[1/((a - I*a*x)^(7/4)*(a + I*a*x)^(7/4)),x]

[Out]

((-1/3*I)*2^(1/4)*(1 + I*x)^(3/4)*Hypergeometric2F1[-3/4, 7/4, 1/4, 1/2 - (I/2)*x])/(a^2*(a - I*a*x)^(3/4)*(a
+ I*a*x)^(3/4))

Maple [F]

\[\int \frac {1}{\left (-i a x +a \right )^{\frac {7}{4}} \left (i a x +a \right )^{\frac {7}{4}}}d x\]

[In]

int(1/(a-I*a*x)^(7/4)/(a+I*a*x)^(7/4),x)

[Out]

int(1/(a-I*a*x)^(7/4)/(a+I*a*x)^(7/4),x)

Fricas [F]

\[ \int \frac {1}{(a-i a x)^{7/4} (a+i a x)^{7/4}} \, dx=\int { \frac {1}{{\left (i \, a x + a\right )}^{\frac {7}{4}} {\left (-i \, a x + a\right )}^{\frac {7}{4}}} \,d x } \]

[In]

integrate(1/(a-I*a*x)^(7/4)/(a+I*a*x)^(7/4),x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 19.00 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.17 \[ \int \frac {1}{(a-i a x)^{7/4} (a+i a x)^{7/4}} \, dx=- \frac {i {G_{6, 6}^{5, 3}\left (\begin {matrix} \frac {7}{8}, \frac {11}{8}, 1 & \frac {1}{2}, \frac {7}{4}, \frac {9}{4} \\\frac {7}{8}, \frac {5}{4}, \frac {11}{8}, \frac {7}{4}, \frac {9}{4} & 0 \end {matrix} \middle | {\frac {e^{- 3 i \pi }}{x^{2}}} \right )} e^{- \frac {i \pi }{4}}}{4 \pi a^{\frac {7}{2}} \Gamma \left (\frac {7}{4}\right )} + \frac {i {G_{6, 6}^{2, 6}\left (\begin {matrix} - \frac {1}{2}, 0, \frac {3}{8}, \frac {1}{2}, \frac {7}{8}, 1 & \\\frac {3}{8}, \frac {7}{8} & - \frac {1}{2}, 0, \frac {5}{4}, 0 \end {matrix} \middle | {\frac {e^{- i \pi }}{x^{2}}} \right )}}{4 \pi a^{\frac {7}{2}} \Gamma \left (\frac {7}{4}\right )} \]

[In]

integrate(1/(a-I*a*x)**(7/4)/(a+I*a*x)**(7/4),x)

[Out]

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

Maxima [F]

\[ \int \frac {1}{(a-i a x)^{7/4} (a+i a x)^{7/4}} \, dx=\int { \frac {1}{{\left (i \, a x + a\right )}^{\frac {7}{4}} {\left (-i \, a x + a\right )}^{\frac {7}{4}}} \,d x } \]

[In]

integrate(1/(a-I*a*x)^(7/4)/(a+I*a*x)^(7/4),x, algorithm="maxima")

[Out]

integrate(1/((I*a*x + a)^(7/4)*(-I*a*x + a)^(7/4)), x)

Giac [F(-2)]

Exception generated. \[ \int \frac {1}{(a-i a x)^{7/4} (a+i a x)^{7/4}} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(1/(a-I*a*x)^(7/4)/(a+I*a*x)^(7/4),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. \[ \int \frac {1}{(a-i a x)^{7/4} (a+i a x)^{7/4}} \, dx=\int \frac {1}{{\left (a-a\,x\,1{}\mathrm {i}\right )}^{7/4}\,{\left (a+a\,x\,1{}\mathrm {i}\right )}^{7/4}} \,d x \]

[In]

int(1/((a - a*x*1i)^(7/4)*(a + a*x*1i)^(7/4)),x)

[Out]

int(1/((a - a*x*1i)^(7/4)*(a + a*x*1i)^(7/4)), x)