Optimal. Leaf size=82 \[ \frac {2 i}{5 a^2 \sqrt [4]{a-i a x} (a+i a x)^{5/4}}+\frac {6 \sqrt [4]{1+x^2} E\left (\left .\frac {1}{2} \tan ^{-1}(x)\right |2\right )}{5 a^3 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}} \]
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Rubi [A]
time = 0.02, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {53, 48, 42, 203,
202} \begin {gather*} \frac {6 \sqrt [4]{x^2+1} E\left (\left .\frac {\text {ArcTan}(x)}{2}\right |2\right )}{5 a^3 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}+\frac {2 i}{5 a^2 \sqrt [4]{a-i a x} (a+i a x)^{5/4}} \end {gather*}
Antiderivative was successfully verified.
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Rule 42
Rule 48
Rule 53
Rule 202
Rule 203
Rubi steps
\begin {align*} \int \frac {1}{(a-i a x)^{5/4} (a+i a x)^{9/4}} \, dx &=-\frac {2 i}{a^2 \sqrt [4]{a-i a x} (a+i a x)^{5/4}}+\frac {3 \int \frac {1}{\sqrt [4]{a-i a x} (a+i a x)^{9/4}} \, dx}{a}\\ &=\frac {2 i}{5 a^2 \sqrt [4]{a-i a x} (a+i a x)^{5/4}}+\frac {3 \int \frac {1}{(a-i a x)^{5/4} (a+i a x)^{5/4}} \, dx}{5 a}\\ &=\frac {2 i}{5 a^2 \sqrt [4]{a-i a x} (a+i a x)^{5/4}}+\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 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}\\ &=\frac {2 i}{5 a^2 \sqrt [4]{a-i a x} (a+i a x)^{5/4}}+\frac {\left (3 \sqrt [4]{1+x^2}\right ) \int \frac {1}{\left (1+x^2\right )^{5/4}} \, dx}{5 a^3 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}\\ &=\frac {2 i}{5 a^2 \sqrt [4]{a-i a x} (a+i a x)^{5/4}}+\frac {6 \sqrt [4]{1+x^2} E\left (\left .\frac {1}{2} \tan ^{-1}(x)\right |2\right )}{5 a^3 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 10.03, size = 68, normalized size = 0.83 \begin {gather*} -\frac {i \sqrt [4]{1+i x} \, _2F_1\left (-\frac {1}{4},\frac {9}{4};\frac {3}{4};\frac {1}{2}-\frac {i x}{2}\right )}{\sqrt [4]{2} a^3 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 5 vs. order
4.
time = 0.20, size = 107, normalized size = 1.30
method | result | size |
risch | \(\frac {\frac {6}{5} x^{2}-\frac {6}{5} i x +\frac {2}{5}}{\left (x -i\right ) a^{3} \left (-a \left (i x -1\right )\right )^{\frac {1}{4}} \left (a \left (i x +1\right )\right )^{\frac {1}{4}}}-\frac {3 x \hypergeom \left (\left [\frac {1}{4}, \frac {1}{2}\right ], \left [\frac {3}{2}\right ], -x^{2}\right ) \left (-a^{2} \left (i x -1\right ) \left (i x +1\right )\right )^{\frac {1}{4}}}{5 \left (a^{2}\right )^{\frac {1}{4}} a^{3} \left (-a \left (i x -1\right )\right )^{\frac {1}{4}} \left (a \left (i x +1\right )\right )^{\frac {1}{4}}}\) | \(107\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (i a \left (x - i\right )\right )^{\frac {9}{4}} \left (- i a \left (x + i\right )\right )^{\frac {5}{4}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{{\left (a-a\,x\,1{}\mathrm {i}\right )}^{5/4}\,{\left (a+a\,x\,1{}\mathrm {i}\right )}^{9/4}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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