3.1178 \(\int \frac{\sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}} \, dx\)

Optimal. Leaf size=256 \[ -\frac{i \sqrt [4]{a-i a x} (a+i a x)^{3/4}}{a}-\frac{i \log \left (\frac{\sqrt{a-i a x}}{\sqrt{a+i a x}}-\frac{\sqrt{2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}+1\right )}{2 \sqrt{2}}+\frac{i \log \left (\frac{\sqrt{a-i a x}}{\sqrt{a+i a x}}+\frac{\sqrt{2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}+1\right )}{2 \sqrt{2}}-\frac{i \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{\sqrt{2}}+\frac{i \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{\sqrt{2}} \]

[Out]

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

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Rubi [A]  time = 0.297254, antiderivative size = 256, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.32 \[ -\frac{i \sqrt [4]{a-i a x} (a+i a x)^{3/4}}{a}-\frac{i \log \left (\frac{\sqrt{a-i a x}}{\sqrt{a+i a x}}-\frac{\sqrt{2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}+1\right )}{2 \sqrt{2}}+\frac{i \log \left (\frac{\sqrt{a-i a x}}{\sqrt{a+i a x}}+\frac{\sqrt{2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}+1\right )}{2 \sqrt{2}}-\frac{i \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{\sqrt{2}}+\frac{i \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 36.3058, size = 212, normalized size = 0.83 \[ - \frac{\sqrt{2} i \log{\left (- \frac{\sqrt{2} \sqrt [4]{- i a x + a}}{\sqrt [4]{i a x + a}} + \frac{\sqrt{- i a x + a}}{\sqrt{i a x + a}} + 1 \right )}}{4} + \frac{\sqrt{2} i \log{\left (\frac{\sqrt{2} \sqrt [4]{- i a x + a}}{\sqrt [4]{i a x + a}} + \frac{\sqrt{- i a x + a}}{\sqrt{i a x + a}} + 1 \right )}}{4} + \frac{\sqrt{2} i \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [4]{- i a x + a}}{\sqrt [4]{i a x + a}} - 1 \right )}}{2} + \frac{\sqrt{2} i \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [4]{- i a x + a}}{\sqrt [4]{i a x + a}} + 1 \right )}}{2} - \frac{i \sqrt [4]{- i a x + a} \left (i a x + a\right )^{\frac{3}{4}}}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a-I*a*x)**(1/4)/(a+I*a*x)**(1/4),x)

[Out]

-sqrt(2)*I*log(-sqrt(2)*(-I*a*x + a)**(1/4)/(I*a*x + a)**(1/4) + sqrt(-I*a*x + a
)/sqrt(I*a*x + a) + 1)/4 + sqrt(2)*I*log(sqrt(2)*(-I*a*x + a)**(1/4)/(I*a*x + a)
**(1/4) + sqrt(-I*a*x + a)/sqrt(I*a*x + a) + 1)/4 + sqrt(2)*I*atan(sqrt(2)*(-I*a
*x + a)**(1/4)/(I*a*x + a)**(1/4) - 1)/2 + sqrt(2)*I*atan(sqrt(2)*(-I*a*x + a)**
(1/4)/(I*a*x + a)**(1/4) + 1)/2 - I*(-I*a*x + a)**(1/4)*(I*a*x + a)**(3/4)/a

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Mathematica [C]  time = 0.0554601, size = 71, normalized size = 0.28 \[ \frac{\sqrt [4]{a-i a x} \left (i 2^{3/4} \sqrt [4]{1+i x} \, _2F_1\left (\frac{1}{4},\frac{1}{4};\frac{5}{4};\frac{1}{2}-\frac{i x}{2}\right )+x-i\right )}{\sqrt [4]{a+i a x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.083, size = 0, normalized size = 0. \[ \int{1\sqrt [4]{a-iax}{\frac{1}{\sqrt [4]{a+iax}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.218075, size = 262, normalized size = 1.02 \[ \frac{\sqrt{i} a \log \left (\frac{\sqrt{i}{\left (a x - i \, a\right )} +{\left (i \, a x + a\right )}^{\frac{3}{4}}{\left (-i \, a x + a\right )}^{\frac{1}{4}}}{x - i}\right ) - \sqrt{i} a \log \left (-\frac{\sqrt{i}{\left (a x - i \, a\right )} -{\left (i \, a x + a\right )}^{\frac{3}{4}}{\left (-i \, a x + a\right )}^{\frac{1}{4}}}{x - i}\right ) + \sqrt{-i} a \log \left (\frac{\sqrt{-i}{\left (a x - i \, a\right )} +{\left (i \, a x + a\right )}^{\frac{3}{4}}{\left (-i \, a x + a\right )}^{\frac{1}{4}}}{x - i}\right ) - \sqrt{-i} a \log \left (-\frac{\sqrt{-i}{\left (a x - i \, a\right )} -{\left (i \, a x + a\right )}^{\frac{3}{4}}{\left (-i \, a x + a\right )}^{\frac{1}{4}}}{x - i}\right ) - 2 i \,{\left (i \, a x + a\right )}^{\frac{3}{4}}{\left (-i \, a x + a\right )}^{\frac{1}{4}}}{2 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(sqrt(I)*a*log((sqrt(I)*(a*x - I*a) + (I*a*x + a)^(3/4)*(-I*a*x + a)^(1/4))/
(x - I)) - sqrt(I)*a*log(-(sqrt(I)*(a*x - I*a) - (I*a*x + a)^(3/4)*(-I*a*x + a)^
(1/4))/(x - I)) + sqrt(-I)*a*log((sqrt(-I)*(a*x - I*a) + (I*a*x + a)^(3/4)*(-I*a
*x + a)^(1/4))/(x - I)) - sqrt(-I)*a*log(-(sqrt(-I)*(a*x - I*a) - (I*a*x + a)^(3
/4)*(-I*a*x + a)^(1/4))/(x - I)) - 2*I*(I*a*x + a)^(3/4)*(-I*a*x + a)^(1/4))/a

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt [4]{- a \left (i x - 1\right )}}{\sqrt [4]{a \left (i x + 1\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.252811, size = 252, normalized size = 0.98 \[ \frac{1}{2} i \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + \frac{2 \,{\left (-i \, a x + a\right )}^{\frac{1}{4}}}{{\left (i \, a x + a\right )}^{\frac{1}{4}}}\right )}\right ) + \frac{1}{2} i \, \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - \frac{2 \,{\left (-i \, a x + a\right )}^{\frac{1}{4}}}{{\left (i \, a x + a\right )}^{\frac{1}{4}}}\right )}\right ) + \frac{1}{4} i \, \sqrt{2}{\rm ln}\left (\frac{\sqrt{2}{\left (-i \, a x + a\right )}^{\frac{1}{4}}}{{\left (i \, a x + a\right )}^{\frac{1}{4}}} + \frac{\sqrt{-i \, a x + a}}{\sqrt{i \, a x + a}} + 1\right ) - \frac{1}{4} i \, \sqrt{2}{\rm ln}\left (-\frac{\sqrt{2}{\left (-i \, a x + a\right )}^{\frac{1}{4}}}{{\left (i \, a x + a\right )}^{\frac{1}{4}}} + \frac{\sqrt{-i \, a x + a}}{\sqrt{i \, a x + a}} + 1\right ) + \frac{i \,{\left (-i \, a x + a\right )}^{\frac{1}{4}}{\left (-i \, a x - a\right )}}{{\left (i \, a x + a\right )}^{\frac{1}{4}} a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*I*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*(-I*a*x + a)^(1/4)/(I*a*x + a)^(1/
4))) + 1/2*I*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2*(-I*a*x + a)^(1/4)/(I*a*x
+ a)^(1/4))) + 1/4*I*sqrt(2)*ln(sqrt(2)*(-I*a*x + a)^(1/4)/(I*a*x + a)^(1/4) + s
qrt(-I*a*x + a)/sqrt(I*a*x + a) + 1) - 1/4*I*sqrt(2)*ln(-sqrt(2)*(-I*a*x + a)^(1
/4)/(I*a*x + a)^(1/4) + sqrt(-I*a*x + a)/sqrt(I*a*x + a) + 1) + I*(-I*a*x + a)^(
1/4)*(-I*a*x - a)/((I*a*x + a)^(1/4)*a)