3.1184 \(\int \frac{(a-i a x)^{3/4}}{(a+i a x)^{3/4}} \, dx\)

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

[Out]

((-I)*(a - I*a*x)^(3/4)*(a + I*a*x)^(1/4))/a - ((3*I)*ArcTan[1 - (Sqrt[2]*(a - I
*a*x)^(1/4))/(a + I*a*x)^(1/4)])/Sqrt[2] + ((3*I)*ArcTan[1 + (Sqrt[2]*(a - I*a*x
)^(1/4))/(a + I*a*x)^(1/4)])/Sqrt[2] + (((3*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] - (((3*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 = 0.251839, 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 \[ -\frac{i (a-i a x)^{3/4} \sqrt [4]{a+i a x}}{a}+\frac{3 i \log \left (\frac{\sqrt{a-i a x}}{\sqrt{a+i a x}}-\frac{\sqrt{2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}+1\right )}{2 \sqrt{2}}-\frac{3 i \log \left (\frac{\sqrt{a-i a x}}{\sqrt{a+i a x}}+\frac{\sqrt{2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}+1\right )}{2 \sqrt{2}}-\frac{3 i \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{\sqrt{2}}+\frac{3 i \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

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

[Out]

((-I)*(a - I*a*x)^(3/4)*(a + I*a*x)^(1/4))/a - ((3*I)*ArcTan[1 - (Sqrt[2]*(a - I
*a*x)^(1/4))/(a + I*a*x)^(1/4)])/Sqrt[2] + ((3*I)*ArcTan[1 + (Sqrt[2]*(a - I*a*x
)^(1/4))/(a + I*a*x)^(1/4)])/Sqrt[2] + (((3*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] - (((3*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 = 39.0268, size = 219, normalized size = 0.86 \[ \frac{3 \sqrt{2} i \log{\left (- \frac{\sqrt{2} \sqrt [4]{- i a x + a}}{\sqrt [4]{i a x + a}} + \frac{\sqrt{- i a x + a}}{\sqrt{i a x + a}} + 1 \right )}}{4} - \frac{3 \sqrt{2} i \log{\left (\frac{\sqrt{2} \sqrt [4]{- i a x + a}}{\sqrt [4]{i a x + a}} + \frac{\sqrt{- i a x + a}}{\sqrt{i a x + a}} + 1 \right )}}{4} + \frac{3 \sqrt{2} i \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [4]{- i a x + a}}{\sqrt [4]{i a x + a}} - 1 \right )}}{2} + \frac{3 \sqrt{2} i \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [4]{- i a x + a}}{\sqrt [4]{i a x + a}} + 1 \right )}}{2} - \frac{i \left (- i a x + a\right )^{\frac{3}{4}} \sqrt [4]{i a x + a}}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a-I*a*x)**(3/4)/(a+I*a*x)**(3/4),x)

[Out]

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

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Mathematica [C]  time = 0.0625548, size = 71, normalized size = 0.28 \[ \frac{(a-i a x)^{3/4} \left (i \sqrt [4]{2} (1+i x)^{3/4} \, _2F_1\left (\frac{3}{4},\frac{3}{4};\frac{7}{4};\frac{1}{2}-\frac{i x}{2}\right )+x-i\right )}{(a+i a x)^{3/4}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.075, size = 0, normalized size = 0. \[ \int{1 \left ( a-iax \right ) ^{{\frac{3}{4}}} \left ( a+iax \right ) ^{-{\frac{3}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a-I*a*x)^(3/4)/(a+I*a*x)^(3/4),x)

[Out]

int((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. \[ \int \frac{{\left (-i \, a x + a\right )}^{\frac{3}{4}}}{{\left (i \, a x + a\right )}^{\frac{3}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.223305, size = 275, normalized size = 1.07 \[ \frac{\sqrt{9 i} a \log \left (\frac{\sqrt{9 i}{\left (a x + i \, a\right )} + 3 \,{\left (i \, a x + a\right )}^{\frac{1}{4}}{\left (-i \, a x + a\right )}^{\frac{3}{4}}}{3 \, x + 3 i}\right ) - \sqrt{9 i} a \log \left (-\frac{\sqrt{9 i}{\left (a x + i \, a\right )} - 3 \,{\left (i \, a x + a\right )}^{\frac{1}{4}}{\left (-i \, a x + a\right )}^{\frac{3}{4}}}{3 \, x + 3 i}\right ) + \sqrt{-9 i} a \log \left (\frac{\sqrt{-9 i}{\left (a x + i \, a\right )} + 3 \,{\left (i \, a x + a\right )}^{\frac{1}{4}}{\left (-i \, a x + a\right )}^{\frac{3}{4}}}{3 \, x + 3 i}\right ) - \sqrt{-9 i} a \log \left (-\frac{\sqrt{-9 i}{\left (a x + i \, a\right )} - 3 \,{\left (i \, a x + a\right )}^{\frac{1}{4}}{\left (-i \, a x + a\right )}^{\frac{3}{4}}}{3 \, x + 3 i}\right ) - 2 i \,{\left (i \, a x + a\right )}^{\frac{1}{4}}{\left (-i \, a x + a\right )}^{\frac{3}{4}}}{2 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(sqrt(9*I)*a*log((sqrt(9*I)*(a*x + I*a) + 3*(I*a*x + a)^(1/4)*(-I*a*x + a)^(
3/4))/(3*x + 3*I)) - sqrt(9*I)*a*log(-(sqrt(9*I)*(a*x + I*a) - 3*(I*a*x + a)^(1/
4)*(-I*a*x + a)^(3/4))/(3*x + 3*I)) + sqrt(-9*I)*a*log((sqrt(-9*I)*(a*x + I*a) +
3*(I*a*x + a)^(1/4)*(-I*a*x + a)^(3/4))/(3*x + 3*I)) - sqrt(-9*I)*a*log(-(sqrt(
-9*I)*(a*x + I*a) - 3*(I*a*x + a)^(1/4)*(-I*a*x + a)^(3/4))/(3*x + 3*I)) - 2*I*(
I*a*x + a)^(1/4)*(-I*a*x + a)^(3/4))/a

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (- a \left (i x - 1\right )\right )^{\frac{3}{4}}}{\left (a \left (i x + 1\right )\right )^{\frac{3}{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a-I*a*x)**(3/4)/(a+I*a*x)**(3/4),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.246625, size = 242, normalized size = 0.95 \[ \frac{3}{2} i \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + \frac{2 \,{\left (-i \, a x + a\right )}^{\frac{1}{4}}}{{\left (i \, a x + a\right )}^{\frac{1}{4}}}\right )}\right ) + \frac{3}{2} i \, \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - \frac{2 \,{\left (-i \, a x + a\right )}^{\frac{1}{4}}}{{\left (i \, a x + a\right )}^{\frac{1}{4}}}\right )}\right ) - \frac{3}{4} i \, \sqrt{2}{\rm ln}\left (\frac{\sqrt{2}{\left (-i \, a x + a\right )}^{\frac{1}{4}}}{{\left (i \, a x + a\right )}^{\frac{1}{4}}} + \frac{\sqrt{-i \, a x + a}}{\sqrt{i \, a x + a}} + 1\right ) + \frac{3}{4} i \, \sqrt{2}{\rm ln}\left (-\frac{\sqrt{2}{\left (-i \, a x + a\right )}^{\frac{1}{4}}}{{\left (i \, a x + a\right )}^{\frac{1}{4}}} + \frac{\sqrt{-i \, a x + a}}{\sqrt{i \, a x + a}} + 1\right ) - \frac{i \,{\left (i \, a x + a\right )}^{\frac{1}{4}}{\left (-i \, a x + a\right )}^{\frac{3}{4}}}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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