Optimal. Leaf size=71 \[ \frac {2 x}{\sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}-\frac {2 \sqrt [4]{x^2+1} E\left (\left .\frac {1}{2} \tan ^{-1}(x)\right |2\right )}{\sqrt [4]{a-i a x} \sqrt [4]{a+i a x}} \]
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Rubi [A] time = 0.01, antiderivative size = 71, 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, 229, 227, 196} \[ \frac {2 x}{\sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}-\frac {2 \sqrt [4]{x^2+1} E\left (\left .\frac {1}{2} \tan ^{-1}(x)\right |2\right )}{\sqrt [4]{a-i a x} \sqrt [4]{a+i a x}} \]
Antiderivative was successfully verified.
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Rule 42
Rule 196
Rule 227
Rule 229
Rubi steps
\begin {align*} \int \frac {1}{\sqrt [4]{a-i a x} \sqrt [4]{a+i a x}} \, dx &=\frac {\sqrt [4]{a^2+a^2 x^2} \int \frac {1}{\sqrt [4]{a^2+a^2 x^2}} \, dx}{\sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}\\ &=\frac {\sqrt [4]{1+x^2} \int \frac {1}{\sqrt [4]{1+x^2}} \, dx}{\sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}\\ &=\frac {2 x}{\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}{\sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}\\ &=\frac {2 x}{\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 )}{\sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 70, normalized size = 0.99 \[ \frac {2 i 2^{3/4} \sqrt [4]{1+i x} (a-i a x)^{3/4} \, _2F_1\left (\frac {1}{4},\frac {3}{4};\frac {7}{4};\frac {1}{2}-\frac {i x}{2}\right )}{3 a \sqrt [4]{a+i a x}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ \frac {a^{2} x {\rm integral}\left (\frac {2 \, {\left (i \, a x + a\right )}^{\frac {3}{4}} {\left (-i \, a x + a\right )}^{\frac {3}{4}}}{a^{2} x^{4} + a^{2} x^{2}}, x\right ) + 2 \, {\left (i \, a x + a\right )}^{\frac {3}{4}} {\left (-i \, a x + a\right )}^{\frac {3}{4}}}{a^{2} x} \]
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 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.08, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (-i a x +a \right )^{\frac {1}{4}} \left (i a x +a \right )^{\frac {1}{4}}}\, dx \]
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 \[ \int \frac {1}{{\left (i \, a x + a\right )}^{\frac {1}{4}} {\left (-i \, a x + a\right )}^{\frac {1}{4}}}\,{d x} \]
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 \[ \int \frac {1}{{\left (a-a\,x\,1{}\mathrm {i}\right )}^{1/4}\,{\left (a+a\,x\,1{}\mathrm {i}\right )}^{1/4}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.79, size = 102, normalized size = 1.44 \[ - \frac {i {G_{6, 6}^{5, 3}\left (\begin {matrix} \frac {1}{8}, \frac {5}{8}, 1 & \frac {1}{4}, \frac {1}{2}, \frac {3}{4} \\- \frac {1}{4}, \frac {1}{8}, \frac {1}{4}, \frac {5}{8}, \frac {3}{4} & 0 \end {matrix} \middle | {\frac {e^{- 3 i \pi }}{x^{2}}} \right )} e^{\frac {i \pi }{4}}}{4 \pi \sqrt {a} \Gamma \left (\frac {1}{4}\right )} + \frac {i {G_{6, 6}^{2, 6}\left (\begin {matrix} - \frac {1}{2}, - \frac {3}{8}, 0, \frac {1}{8}, \frac {1}{2}, 1 & \\- \frac {3}{8}, \frac {1}{8} & - \frac {1}{2}, - \frac {1}{4}, 0, 0 \end {matrix} \middle | {\frac {e^{- i \pi }}{x^{2}}} \right )}}{4 \pi \sqrt {a} \Gamma \left (\frac {1}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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