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