3.1173 \(\int \frac{1}{\sqrt [4]{a-i a x} \sqrt [4]{a+i a x}} \, dx\)

Optimal. Leaf size=71 \[ \frac{2 x}{\sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}-\frac{2 \sqrt [4]{x^2+1} E\left (\left .\frac{1}{2} \tan ^{-1}(x)\right |2\right )}{\sqrt [4]{a-i a x} \sqrt [4]{a+i a x}} \]

[Out]

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

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Rubi [A]  time = 0.0119873, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {42, 229, 227, 196} \[ \frac{2 x}{\sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}-\frac{2 \sqrt [4]{x^2+1} E\left (\left .\frac{1}{2} \tan ^{-1}(x)\right |2\right )}{\sqrt [4]{a-i a x} \sqrt [4]{a+i a x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x)/((a - I*a*x)^(1/4)*(a + I*a*x)^(1/4)) - (2*(1 + x^2)^(1/4)*EllipticE[ArcTan[x]/2, 2])/((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 229

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

Rule 227

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

Rule 196

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

Rubi steps

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

Mathematica [C]  time = 0.0220942, size = 70, normalized size = 0.99 \[ \frac{2 i 2^{3/4} \sqrt [4]{1+i x} (a-i a x)^{3/4} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{7}{4};\frac{1}{2}-\frac{i x}{2}\right )}{3 a \sqrt [4]{a+i a x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.039, size = 0, normalized size = 0. \begin{align*} \int{{\frac{1}{\sqrt [4]{a-iax}}}{\frac{1}{\sqrt [4]{a+iax}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(1/(a-I*a*x)^(1/4)/(a+I*a*x)^(1/4),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{1}{4}}{\left (-i \, a x + a\right )}^{\frac{1}{4}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{a^{2} x{\rm integral}\left (\frac{2 \,{\left (i \, a x + a\right )}^{\frac{3}{4}}{\left (-i \, a x + a\right )}^{\frac{3}{4}}}{a^{2} x^{4} + a^{2} x^{2}}, x\right ) + 2 \,{\left (i \, a x + a\right )}^{\frac{3}{4}}{\left (-i \, a x + a\right )}^{\frac{3}{4}}}{a^{2} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 3.69559, size = 102, normalized size = 1.44 \begin{align*} - \frac{i{G_{6, 6}^{5, 3}\left (\begin{matrix} \frac{1}{8}, \frac{5}{8}, 1 & \frac{1}{4}, \frac{1}{2}, \frac{3}{4} \\- \frac{1}{4}, \frac{1}{8}, \frac{1}{4}, \frac{5}{8}, \frac{3}{4} & 0 \end{matrix} \middle |{\frac{e^{- 3 i \pi }}{x^{2}}} \right )} e^{\frac{i \pi }{4}}}{4 \pi \sqrt{a} \Gamma \left (\frac{1}{4}\right )} + \frac{i{G_{6, 6}^{2, 6}\left (\begin{matrix} - \frac{1}{2}, - \frac{3}{8}, 0, \frac{1}{8}, \frac{1}{2}, 1 & \\- \frac{3}{8}, \frac{1}{8} & - \frac{1}{2}, - \frac{1}{4}, 0, 0 \end{matrix} \middle |{\frac{e^{- i \pi }}{x^{2}}} \right )}}{4 \pi \sqrt{a} \Gamma \left (\frac{1}{4}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Exception raised: TypeError