3.1179 \(\int \frac{1}{(a-i a x)^{3/4} \sqrt [4]{a+i a x}} \, dx\)

Optimal. Leaf size=233 \[ -\frac{i \log \left (\frac{\sqrt{a-i a x}}{\sqrt{a+i a x}}-\frac{\sqrt{2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}+1\right )}{\sqrt{2} a}+\frac{i \log \left (\frac{\sqrt{a-i a x}}{\sqrt{a+i a x}}+\frac{\sqrt{2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}+1\right )}{\sqrt{2} a}-\frac{i \sqrt{2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{a}+\frac{i \sqrt{2} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{a} \]

[Out]

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

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Rubi [A]  time = 0.20261, antiderivative size = 233, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.28 \[ -\frac{i \log \left (\frac{\sqrt{a-i a x}}{\sqrt{a+i a x}}-\frac{\sqrt{2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}+1\right )}{\sqrt{2} a}+\frac{i \log \left (\frac{\sqrt{a-i a x}}{\sqrt{a+i a x}}+\frac{\sqrt{2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}+1\right )}{\sqrt{2} a}-\frac{i \sqrt{2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{a}+\frac{i \sqrt{2} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{a} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 28.338, size = 192, normalized size = 0.82 \[ - \frac{\sqrt{2} i \log{\left (- \frac{\sqrt{2} \sqrt [4]{- i a x + a}}{\sqrt [4]{i a x + a}} + \frac{\sqrt{- i a x + a}}{\sqrt{i a x + a}} + 1 \right )}}{2 a} + \frac{\sqrt{2} i \log{\left (\frac{\sqrt{2} \sqrt [4]{- i a x + a}}{\sqrt [4]{i a x + a}} + \frac{\sqrt{- i a x + a}}{\sqrt{i a x + a}} + 1 \right )}}{2 a} + \frac{\sqrt{2} i \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [4]{- i a x + a}}{\sqrt [4]{i a x + a}} - 1 \right )}}{a} + \frac{\sqrt{2} i \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [4]{- i a x + a}}{\sqrt [4]{i a x + a}} + 1 \right )}}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a-I*a*x)**(3/4)/(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)/(2*a) + 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)/(2*a) + sqrt(2)*I*atan(sqrt(
2)*(-I*a*x + a)**(1/4)/(I*a*x + a)**(1/4) - 1)/a + sqrt(2)*I*atan(sqrt(2)*(-I*a*
x + a)**(1/4)/(I*a*x + a)**(1/4) + 1)/a

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Mathematica [C]  time = 0.0433785, size = 68, normalized size = 0.29 \[ \frac{2 i 2^{3/4} \sqrt [4]{1+i x} \sqrt [4]{a-i a x} \, _2F_1\left (\frac{1}{4},\frac{1}{4};\frac{5}{4};\frac{1}{2}-\frac{i x}{2}\right )}{a \sqrt [4]{a+i a x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.066, size = 0, normalized size = 0. \[ \int{1 \left ( a-iax \right ) ^{-{\frac{3}{4}}}{\frac{1}{\sqrt [4]{a+iax}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.222234, size = 306, normalized size = 1.31 \[ \frac{1}{2} \, \sqrt{\frac{4 i}{a^{2}}} \log \left (\frac{{\left (a^{2} x - i \, a^{2}\right )} \sqrt{\frac{4 i}{a^{2}}} + 2 \,{\left (i \, a x + a\right )}^{\frac{3}{4}}{\left (-i \, a x + a\right )}^{\frac{1}{4}}}{2 \, x - 2 i}\right ) - \frac{1}{2} \, \sqrt{\frac{4 i}{a^{2}}} \log \left (-\frac{{\left (a^{2} x - i \, a^{2}\right )} \sqrt{\frac{4 i}{a^{2}}} - 2 \,{\left (i \, a x + a\right )}^{\frac{3}{4}}{\left (-i \, a x + a\right )}^{\frac{1}{4}}}{2 \, x - 2 i}\right ) + \frac{1}{2} \, \sqrt{-\frac{4 i}{a^{2}}} \log \left (\frac{{\left (a^{2} x - i \, a^{2}\right )} \sqrt{-\frac{4 i}{a^{2}}} + 2 \,{\left (i \, a x + a\right )}^{\frac{3}{4}}{\left (-i \, a x + a\right )}^{\frac{1}{4}}}{2 \, x - 2 i}\right ) - \frac{1}{2} \, \sqrt{-\frac{4 i}{a^{2}}} \log \left (-\frac{{\left (a^{2} x - i \, a^{2}\right )} \sqrt{-\frac{4 i}{a^{2}}} - 2 \,{\left (i \, a x + a\right )}^{\frac{3}{4}}{\left (-i \, a x + a\right )}^{\frac{1}{4}}}{2 \, x - 2 i}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError