Optimal. Leaf size=88 \[ \frac {2 x}{5 a^4 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x} \left (1+x^2\right )}+\frac {6 \sqrt [4]{1+x^2} E\left (\left .\frac {1}{2} \tan ^{-1}(x)\right |2\right )}{5 a^4 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, 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, 205, 203,
202} \begin {gather*} \frac {6 \sqrt [4]{x^2+1} E\left (\left .\frac {\text {ArcTan}(x)}{2}\right |2\right )}{5 a^4 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}+\frac {2 x}{5 a^4 \left (x^2+1\right ) \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 42
Rule 202
Rule 203
Rule 205
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] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 10.02, size = 70, normalized size = 0.80 \begin {gather*} -\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}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.07, 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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 96.85, size = 95, normalized size = 1.08 \begin {gather*} - \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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________