3.1191 \(\int \frac{1}{(a-i a x)^{3/4} (a+i a x)^{3/4}} \, dx\)

Optimal. Leaf size=43 \[ \frac{2 \left (x^2+1\right )^{3/4} \text{EllipticF}\left (\frac{1}{2} \tan ^{-1}(x),2\right )}{(a-i a x)^{3/4} (a+i a x)^{3/4}} \]

[Out]

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

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Rubi [A]  time = 0.0081492, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {42, 233, 231} \[ \frac{2 \left (x^2+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}(x)\right |2\right )}{(a-i a x)^{3/4} (a+i a x)^{3/4}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(1 + x^2)^(3/4)*EllipticF[ArcTan[x]/2, 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 233

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]

Rule 231

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

Rubi steps

\begin{align*} \int \frac{1}{(a-i a x)^{3/4} (a+i a x)^{3/4}} \, dx &=\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}{(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}{(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 )}{(a-i a x)^{3/4} (a+i a x)^{3/4}}\\ \end{align*}

Mathematica [C]  time = 0.0198353, size = 68, normalized size = 1.58 \[ \frac{2 i \sqrt [4]{2} (1+i x)^{3/4} \sqrt [4]{a-i a x} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{5}{4};\frac{1}{2}-\frac{i x}{2}\right )}{a (a+i a x)^{3/4}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.039, size = 0, normalized size = 0. \begin{align*} \int{ \left ( a-iax \right ) ^{-{\frac{3}{4}}} \left ( a+iax \right ) ^{-{\frac{3}{4}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (i \, a x + a\right )}^{\frac{3}{4}}{\left (-i \, a x + a\right )}^{\frac{3}{4}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a-I*a*x)^(3/4)/(a+I*a*x)^(3/4),x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (i \, a x + a\right )}^{\frac{1}{4}}{\left (-i \, a x + a\right )}^{\frac{1}{4}}}{a^{2} x^{2} + a^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a-I*a*x)^(3/4)/(a+I*a*x)^(3/4),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 10.2738, size = 100, normalized size = 2.33 \begin{align*} - \frac{i{G_{6, 6}^{5, 3}\left (\begin{matrix} \frac{3}{8}, \frac{7}{8}, 1 & \frac{1}{2}, \frac{3}{4}, \frac{5}{4} \\\frac{1}{4}, \frac{3}{8}, \frac{3}{4}, \frac{7}{8}, \frac{5}{4} & 0 \end{matrix} \middle |{\frac{e^{- 3 i \pi }}{x^{2}}} \right )} e^{\frac{3 i \pi }{4}}}{4 \pi a^{\frac{3}{2}} \Gamma \left (\frac{3}{4}\right )} + \frac{i{G_{6, 6}^{2, 6}\left (\begin{matrix} - \frac{1}{2}, - \frac{1}{8}, 0, \frac{3}{8}, \frac{1}{2}, 1 & \\- \frac{1}{8}, \frac{3}{8} & - \frac{1}{2}, 0, \frac{1}{4}, 0 \end{matrix} \middle |{\frac{e^{- i \pi }}{x^{2}}} \right )}}{4 \pi a^{\frac{3}{2}} \Gamma \left (\frac{3}{4}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a-I*a*x)**(3/4)/(a+I*a*x)**(3/4),x)

[Out]

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

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a-I*a*x)^(3/4)/(a+I*a*x)^(3/4),x, algorithm="giac")

[Out]

Exception raised: TypeError