Optimal. Leaf size=46 \[ \frac{2 \sqrt [4]{x^2+1} E\left (\left .\frac{1}{2} \tan ^{-1}(x)\right |2\right )}{a^2 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}} \]
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Rubi [A] time = 0.0328834, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12 \[ \frac{2 \sqrt [4]{x^2+1} E\left (\left .\frac{1}{2} \tan ^{-1}(x)\right |2\right )}{a^2 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}} \]
Antiderivative was successfully verified.
[In] Int[1/((a - I*a*x)^(5/4)*(a + I*a*x)^(5/4)),x]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{\left (- i a x + a\right )^{\frac{3}{4}} \left (i a x + a\right )^{\frac{3}{4}} \int \frac{1}{\left (a^{2} x^{2} + a^{2}\right )^{\frac{5}{4}}}\, dx}{\left (a^{2} x^{2} + a^{2}\right )^{\frac{3}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(a-I*a*x)**(5/4)/(a+I*a*x)**(5/4),x)
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Mathematica [C] time = 0.0805656, size = 79, normalized size = 1.72 \[ \frac{6 x-2\ 2^{3/4} \sqrt [4]{1+i x} (x+i) \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{7}{4};\frac{1}{2}-\frac{i x}{2}\right )}{3 a^2 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}} \]
Antiderivative was successfully verified.
[In] Integrate[1/((a - I*a*x)^(5/4)*(a + I*a*x)^(5/4)),x]
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Maple [C] time = 0.057, size = 91, normalized size = 2. \[ 2\,{\frac{x}{\sqrt [4]{-a \left ( -1+ix \right ) }\sqrt [4]{a \left ( 1+ix \right ) }{a}^{2}}}-{\frac{x}{{a}^{2}}{\mbox{$_2$F$_1$}({\frac{1}{4}},{\frac{1}{2}};\,{\frac{3}{2}};\,-{x}^{2})}\sqrt [4]{-{a}^{2} \left ( -1+ix \right ) \left ( 1+ix \right ) }{\frac{1}{\sqrt [4]{{a}^{2}}}}{\frac{1}{\sqrt [4]{-a \left ( -1+ix \right ) }}}{\frac{1}{\sqrt [4]{a \left ( 1+ix \right ) }}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(a-I*a*x)^(5/4)/(a+I*a*x)^(5/4),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (i \, a x + a\right )}^{\frac{5}{4}}{\left (-i \, a x + a\right )}^{\frac{5}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((I*a*x + a)^(5/4)*(-I*a*x + a)^(5/4)),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[ \frac{2 \,{\left (i \, a x + a\right )}^{\frac{3}{4}}{\left (-i \, a x + a\right )}^{\frac{3}{4}} x +{\left (a^{4} x^{2} + a^{4}\right )}{\rm integral}\left (-\frac{{\left (i \, a x + a\right )}^{\frac{3}{4}}{\left (-i \, a x + a\right )}^{\frac{3}{4}}}{a^{4} x^{2} + a^{4}}, x\right )}{a^{4} x^{2} + a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((I*a*x + a)^(5/4)*(-I*a*x + a)^(5/4)),x, algorithm="fricas")
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(a-I*a*x)**(5/4)/(a+I*a*x)**(5/4),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)^(5/4)*(-I*a*x + a)^(5/4)),x, algorithm="giac")
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