Optimal. Leaf size=81 \[ \frac{2 \left (x^2+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}(x)\right |2\right )}{3 a^2 (a-i a x)^{3/4} (a+i a x)^{3/4}}+\frac{2 x}{3 a^2 (a-i a x)^{3/4} (a+i a x)^{3/4}} \]
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Rubi [A] time = 0.0491225, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16 \[ \frac{2 \left (x^2+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}(x)\right |2\right )}{3 a^2 (a-i a x)^{3/4} (a+i a x)^{3/4}}+\frac{2 x}{3 a^2 (a-i a x)^{3/4} (a+i a x)^{3/4}} \]
Antiderivative was successfully verified.
[In] Int[1/((a - I*a*x)^(7/4)*(a + I*a*x)^(7/4)),x]
[Out]
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Rubi in Sympy [A] time = 10.2986, size = 75, normalized size = 0.93 \[ \frac{2 x \sqrt [4]{- i a x + a} \sqrt [4]{i a x + a}}{3 a^{4} \left (x^{2} + 1\right )} + \frac{2 \sqrt [4]{- i a x + a} \sqrt [4]{i a x + a} F\left (\frac{\operatorname{atan}{\left (x \right )}}{2}\middle | 2\right )}{3 a^{4} \sqrt [4]{x^{2} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(a-I*a*x)**(7/4)/(a+I*a*x)**(7/4),x)
[Out]
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Mathematica [C] time = 0.0835872, size = 76, normalized size = 0.94 \[ \frac{2 \left (x+\sqrt [4]{2} (1+i x)^{3/4} (x+i) \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{5}{4};\frac{1}{2}-\frac{i x}{2}\right )\right )}{3 a^2 (a-i a x)^{3/4} (a+i a x)^{3/4}} \]
Antiderivative was successfully verified.
[In] Integrate[1/((a - I*a*x)^(7/4)*(a + I*a*x)^(7/4)),x]
[Out]
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Maple [F] time = 0.069, size = 0, normalized size = 0. \[ \int{1 \left ( a-iax \right ) ^{-{\frac{7}{4}}} \left ( a+iax \right ) ^{-{\frac{7}{4}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(a-I*a*x)^(7/4)/(a+I*a*x)^(7/4),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (i \, a x + a\right )}^{\frac{7}{4}}{\left (-i \, a x + a\right )}^{\frac{7}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((I*a*x + a)^(7/4)*(-I*a*x + a)^(7/4)),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[ \frac{3 \,{\left (a^{4} x^{2} + a^{4}\right )}{\rm integral}\left (\frac{{\left (i \, a x + a\right )}^{\frac{1}{4}}{\left (-i \, a x + a\right )}^{\frac{1}{4}}}{3 \,{\left (a^{4} x^{2} + a^{4}\right )}}, x\right ) + 2 \,{\left (i \, a x + a\right )}^{\frac{1}{4}}{\left (-i \, a x + a\right )}^{\frac{1}{4}} x}{3 \,{\left (a^{4} x^{2} + a^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((I*a*x + a)^(7/4)*(-I*a*x + a)^(7/4)),x, algorithm="fricas")
[Out]
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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)**(7/4)/(a+I*a*x)**(7/4),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)^(7/4)*(-I*a*x + a)^(7/4)),x, algorithm="giac")
[Out]