Optimal. Leaf size=88 \[ \frac {2 x}{5 a^4 \left (x^2+1\right ) \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}+\frac {6 \sqrt [4]{x^2+1} E\left (\left .\frac {1}{2} \tan ^{-1}(x)\right |2\right )}{5 a^4 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {42, 199, 197, 196} \[ \frac {2 x}{5 a^4 \left (x^2+1\right ) \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}+\frac {6 \sqrt [4]{x^2+1} E\left (\left .\frac {1}{2} \tan ^{-1}(x)\right |2\right )}{5 a^4 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 42
Rule 196
Rule 197
Rule 199
Rubi steps
\begin {align*} \int \frac {1}{(a-i a x)^{9/4} (a+i a x)^{9/4}} \, dx &=\frac {\sqrt [4]{a^2+a^2 x^2} \int \frac {1}{\left (a^2+a^2 x^2\right )^{9/4}} \, dx}{\sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}\\ &=\frac {2 x}{5 a^4 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x} \left (1+x^2\right )}+\frac {\left (3 \sqrt [4]{a^2+a^2 x^2}\right ) \int \frac {1}{\left (a^2+a^2 x^2\right )^{5/4}} \, dx}{5 a^2 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}\\ &=\frac {2 x}{5 a^4 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x} \left (1+x^2\right )}+\frac {\left (3 \sqrt [4]{1+x^2}\right ) \int \frac {1}{\left (1+x^2\right )^{5/4}} \, dx}{5 a^4 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}\\ &=\frac {2 x}{5 a^4 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x} \left (1+x^2\right )}+\frac {6 \sqrt [4]{1+x^2} E\left (\left .\frac {1}{2} \tan ^{-1}(x)\right |2\right )}{5 a^4 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.03, size = 70, normalized size = 0.80 \[ -\frac {i \sqrt [4]{1+i x} \, _2F_1\left (-\frac {5}{4},\frac {9}{4};-\frac {1}{4};\frac {1}{2}-\frac {i x}{2}\right )}{5 \sqrt [4]{2} a^3 (a-i a x)^{5/4} \sqrt [4]{a+i a x}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.49, size = 0, normalized size = 0.00 \[ \frac {2 \, {\left (3 \, x^{3} + 4 \, x\right )} {\left (i \, a x + a\right )}^{\frac {3}{4}} {\left (-i \, a x + a\right )}^{\frac {3}{4}} + 5 \, {\left (a^{6} x^{4} + 2 \, a^{6} x^{2} + a^{6}\right )} {\rm integral}\left (-\frac {3 \, {\left (i \, a x + a\right )}^{\frac {3}{4}} {\left (-i \, a x + a\right )}^{\frac {3}{4}}}{5 \, {\left (a^{6} x^{2} + a^{6}\right )}}, x\right )}{5 \, {\left (a^{6} x^{4} + 2 \, a^{6} x^{2} + a^{6}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (i \, a x + a\right )}^{\frac {9}{4}} {\left (-i \, a x + a\right )}^{\frac {9}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.06, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (-i a x +a \right )^{\frac {9}{4}} \left (i a x +a \right )^{\frac {9}{4}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (i \, a x + a\right )}^{\frac {9}{4}} {\left (-i \, a x + a\right )}^{\frac {9}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\left (a-a\,x\,1{}\mathrm {i}\right )}^{9/4}\,{\left (a+a\,x\,1{}\mathrm {i}\right )}^{9/4}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 132.49, size = 95, normalized size = 1.08 \[ - \frac {i {G_{6, 6}^{5, 3}\left (\begin {matrix} \frac {9}{8}, \frac {13}{8}, 1 & \frac {1}{2}, \frac {9}{4}, \frac {11}{4} \\\frac {9}{8}, \frac {13}{8}, \frac {7}{4}, \frac {9}{4}, \frac {11}{4} & 0 \end {matrix} \middle | {\frac {e^{- 3 i \pi }}{x^{2}}} \right )} e^{\frac {i \pi }{4}}}{4 \pi a^{\frac {9}{2}} \Gamma \left (\frac {9}{4}\right )} + \frac {i {G_{6, 6}^{2, 6}\left (\begin {matrix} - \frac {1}{2}, 0, \frac {1}{2}, \frac {5}{8}, \frac {9}{8}, 1 & \\\frac {5}{8}, \frac {9}{8} & - \frac {1}{2}, 0, \frac {7}{4}, 0 \end {matrix} \middle | {\frac {e^{- i \pi }}{x^{2}}} \right )}}{4 \pi a^{\frac {9}{2}} \Gamma \left (\frac {9}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________