Optimal. Leaf size=78 \[ \frac {2 i}{a \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}+\frac {2 \sqrt [4]{1+x^2} E\left (\left .\frac {1}{2} \tan ^{-1}(x)\right |2\right )}{a \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}} \]
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Rubi [A]
time = 0.01, antiderivative size = 78, 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 = {50, 42, 203,
202} \begin {gather*} \frac {2 \sqrt [4]{x^2+1} E\left (\left .\frac {\text {ArcTan}(x)}{2}\right |2\right )}{a \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}+\frac {2 i}{a \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 42
Rule 50
Rule 202
Rule 203
Rubi steps
\begin {align*} \int \frac {1}{\sqrt [4]{a-i a x} (a+i a x)^{5/4}} \, dx &=\frac {2 i}{a \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}+a \int \frac {1}{(a-i a x)^{5/4} (a+i a x)^{5/4}} \, dx\\ &=\frac {2 i}{a \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}+\frac {\left (a \sqrt [4]{a^2+a^2 x^2}\right ) \int \frac {1}{\left (a^2+a^2 x^2\right )^{5/4}} \, dx}{\sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}\\ &=\frac {2 i}{a \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}+\frac {\sqrt [4]{1+x^2} \int \frac {1}{\left (1+x^2\right )^{5/4}} \, dx}{a \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}\\ &=\frac {2 i}{a \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}+\frac {2 \sqrt [4]{1+x^2} E\left (\left .\frac {1}{2} \tan ^{-1}(x)\right |2\right )}{a \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 = 70, normalized size = 0.90 \begin {gather*} \frac {i 2^{3/4} \sqrt [4]{1+i x} (a-i a x)^{3/4} \, _2F_1\left (\frac {3}{4},\frac {5}{4};\frac {7}{4};\frac {1}{2}-\frac {i x}{2}\right )}{3 a^2 \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 = 0.17, size = 94, normalized size = 1.21
method | result | size |
risch | \(\frac {2 x +2 i}{a \left (-a \left (i x -1\right )\right )^{\frac {1}{4}} \left (a \left (i x +1\right )\right )^{\frac {1}{4}}}-\frac {x \hypergeom \left (\left [\frac {1}{4}, \frac {1}{2}\right ], \left [\frac {3}{2}\right ], -x^{2}\right ) \left (-a^{2} \left (i x -1\right ) \left (i x +1\right )\right )^{\frac {1}{4}}}{\left (a^{2}\right )^{\frac {1}{4}} a \left (-a \left (i x -1\right )\right )^{\frac {1}{4}} \left (a \left (i x +1\right )\right )^{\frac {1}{4}}}\) | \(94\) |
Verification of antiderivative is not currently implemented for this CAS.
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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.
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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 {5}{4}} \sqrt [4]{- i a \left (x + i\right )}}\, 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 )}^{1/4}\,{\left (a+a\,x\,1{}\mathrm {i}\right )}^{5/4}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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