Optimal. Leaf size=43 \[ \frac {2 \left (x^2+1\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \tan ^{-1}(x),2\right )}{(a-i a x)^{3/4} (a+i a x)^{3/4}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {42, 233, 231} \[ \frac {2 \left (x^2+1\right )^{3/4} F\left (\left .\frac {1}{2} \tan ^{-1}(x)\right |2\right )}{(a-i a x)^{3/4} (a+i a x)^{3/4}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 42
Rule 231
Rule 233
Rubi steps
\begin {align*} \int \frac {1}{(a-i a x)^{3/4} (a+i a x)^{3/4}} \, dx &=\frac {\left (a^2+a^2 x^2\right )^{3/4} \int \frac {1}{\left (a^2+a^2 x^2\right )^{3/4}} \, dx}{(a-i a x)^{3/4} (a+i a x)^{3/4}}\\ &=\frac {\left (1+x^2\right )^{3/4} \int \frac {1}{\left (1+x^2\right )^{3/4}} \, dx}{(a-i a x)^{3/4} (a+i a x)^{3/4}}\\ &=\frac {2 \left (1+x^2\right )^{3/4} F\left (\left .\frac {1}{2} \tan ^{-1}(x)\right |2\right )}{(a-i a x)^{3/4} (a+i a x)^{3/4}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.02, size = 68, normalized size = 1.58 \[ \frac {2 i \sqrt [4]{2} (1+i x)^{3/4} \sqrt [4]{a-i a x} \, _2F_1\left (\frac {1}{4},\frac {3}{4};\frac {5}{4};\frac {1}{2}-\frac {i x}{2}\right )}{a (a+i a x)^{3/4}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (i \, a x + a\right )}^{\frac {1}{4}} {\left (-i \, a x + a\right )}^{\frac {1}{4}}}{a^{2} x^{2} + a^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.06, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (-i a x +a \right )^{\frac {3}{4}} \left (i a x +a \right )^{\frac {3}{4}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (i \, a x + a\right )}^{\frac {3}{4}} {\left (-i \, a x + a\right )}^{\frac {3}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{{\left (a-a\,x\,1{}\mathrm {i}\right )}^{3/4}\,{\left (a+a\,x\,1{}\mathrm {i}\right )}^{3/4}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 5.34, size = 100, normalized size = 2.33 \[ - \frac {i {G_{6, 6}^{5, 3}\left (\begin {matrix} \frac {3}{8}, \frac {7}{8}, 1 & \frac {1}{2}, \frac {3}{4}, \frac {5}{4} \\\frac {1}{4}, \frac {3}{8}, \frac {3}{4}, \frac {7}{8}, \frac {5}{4} & 0 \end {matrix} \middle | {\frac {e^{- 3 i \pi }}{x^{2}}} \right )} e^{\frac {3 i \pi }{4}}}{4 \pi a^{\frac {3}{2}} \Gamma \left (\frac {3}{4}\right )} + \frac {i {G_{6, 6}^{2, 6}\left (\begin {matrix} - \frac {1}{2}, - \frac {1}{8}, 0, \frac {3}{8}, \frac {1}{2}, 1 & \\- \frac {1}{8}, \frac {3}{8} & - \frac {1}{2}, 0, \frac {1}{4}, 0 \end {matrix} \middle | {\frac {e^{- i \pi }}{x^{2}}} \right )}}{4 \pi a^{\frac {3}{2}} \Gamma \left (\frac {3}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________