Optimal. Leaf size=287 \[ \frac {4 i (a-i a x)^{5/4}}{a \sqrt [4]{a+i a x}}+\frac {5 i \sqrt [4]{a-i a x} (a+i a x)^{3/4}}{a}+\frac {5 i \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{\sqrt {2}}-\frac {5 i \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{\sqrt {2}}+\frac {5 i \log \left (1+\frac {\sqrt {a-i a x}}{\sqrt {a+i a x}}-\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{2 \sqrt {2}}-\frac {5 i \log \left (1+\frac {\sqrt {a-i a x}}{\sqrt {a+i a x}}+\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{2 \sqrt {2}} \]
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Rubi [A]
time = 0.12, antiderivative size = 287, normalized size of antiderivative = 1.00, number of steps
used = 13, number of rules used = 10, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {49, 52, 65,
246, 217, 1179, 642, 1176, 631, 210} \begin {gather*} \frac {5 i \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{\sqrt {2}}-\frac {5 i \text {ArcTan}\left (1+\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{\sqrt {2}}+\frac {4 i (a-i a x)^{5/4}}{a \sqrt [4]{a+i a x}}+\frac {5 i (a+i a x)^{3/4} \sqrt [4]{a-i a x}}{a}+\frac {5 i \log \left (\frac {\sqrt {a-i a x}}{\sqrt {a+i a x}}-\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}+1\right )}{2 \sqrt {2}}-\frac {5 i \log \left (\frac {\sqrt {a-i a x}}{\sqrt {a+i a x}}+\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}+1\right )}{2 \sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 49
Rule 52
Rule 65
Rule 210
Rule 217
Rule 246
Rule 631
Rule 642
Rule 1176
Rule 1179
Rubi steps
\begin {align*} \int \frac {(a-i a x)^{5/4}}{(a+i a x)^{5/4}} \, dx &=\frac {4 i (a-i a x)^{5/4}}{a \sqrt [4]{a+i a x}}-5 \int \frac {\sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}} \, dx\\ &=\frac {4 i (a-i a x)^{5/4}}{a \sqrt [4]{a+i a x}}+\frac {5 i \sqrt [4]{a-i a x} (a+i a x)^{3/4}}{a}-\frac {1}{2} (5 a) \int \frac {1}{(a-i a x)^{3/4} \sqrt [4]{a+i a x}} \, dx\\ &=\frac {4 i (a-i a x)^{5/4}}{a \sqrt [4]{a+i a x}}+\frac {5 i \sqrt [4]{a-i a x} (a+i a x)^{3/4}}{a}-10 i \text {Subst}\left (\int \frac {1}{\sqrt [4]{2 a-x^4}} \, dx,x,\sqrt [4]{a-i a x}\right )\\ &=\frac {4 i (a-i a x)^{5/4}}{a \sqrt [4]{a+i a x}}+\frac {5 i \sqrt [4]{a-i a x} (a+i a x)^{3/4}}{a}-10 i \text {Subst}\left (\int \frac {1}{1+x^4} \, dx,x,\frac {\sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )\\ &=\frac {4 i (a-i a x)^{5/4}}{a \sqrt [4]{a+i a x}}+\frac {5 i \sqrt [4]{a-i a x} (a+i a x)^{3/4}}{a}-5 i \text {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\frac {\sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )-5 i \text {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\frac {\sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )\\ &=\frac {4 i (a-i a x)^{5/4}}{a \sqrt [4]{a+i a x}}+\frac {5 i \sqrt [4]{a-i a x} (a+i a x)^{3/4}}{a}-\frac {5}{2} i \text {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\frac {\sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )-\frac {5}{2} i \text {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\frac {\sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )+\frac {(5 i) \text {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\frac {\sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{2 \sqrt {2}}+\frac {(5 i) \text {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\frac {\sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{2 \sqrt {2}}\\ &=\frac {4 i (a-i a x)^{5/4}}{a \sqrt [4]{a+i a x}}+\frac {5 i \sqrt [4]{a-i a x} (a+i a x)^{3/4}}{a}+\frac {5 i \log \left (1+\frac {\sqrt {a-i a x}}{\sqrt {a+i a x}}-\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{2 \sqrt {2}}-\frac {5 i \log \left (1+\frac {\sqrt {a-i a x}}{\sqrt {a+i a x}}+\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{2 \sqrt {2}}-\frac {(5 i) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{\sqrt {2}}+\frac {(5 i) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{\sqrt {2}}\\ &=\frac {4 i (a-i a x)^{5/4}}{a \sqrt [4]{a+i a x}}+\frac {5 i \sqrt [4]{a-i a x} (a+i a x)^{3/4}}{a}+\frac {5 i \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{\sqrt {2}}-\frac {5 i \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{\sqrt {2}}+\frac {5 i \log \left (1+\frac {\sqrt {a-i a x}}{\sqrt {a+i a x}}-\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{2 \sqrt {2}}-\frac {5 i \log \left (1+\frac {\sqrt {a-i a x}}{\sqrt {a+i a x}}+\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{2 \sqrt {2}}\\ \end {align*}
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Mathematica [A]
time = 0.58, size = 117, normalized size = 0.41 \begin {gather*} -\frac {\sqrt [4]{a-i a x} \left (\sqrt [4]{i+x} (-9 i+x)+5 i \sqrt [4]{-i+x} \tan ^{-1}\left (\frac {\sqrt [4]{i+x}}{\sqrt [4]{-i+x}}\right )+5 i \sqrt [4]{-i+x} \tanh ^{-1}\left (\frac {\sqrt [4]{i+x}}{\sqrt [4]{-i+x}}\right )\right )}{\sqrt [4]{i+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 9 vs. order
3.
time = 1.57, size = 480, normalized size = 1.67
method | result | size |
risch | \(-\frac {i \left (x^{2}-8 i x +9\right ) \left (-a \left (i x -1\right )\right )^{\frac {1}{4}}}{\left (i x -1\right ) \left (a \left (i x +1\right )\right )^{\frac {1}{4}}}-\frac {\left (\frac {5 \RootOf \left (\textit {\_Z}^{2}+i\right ) \ln \left (-\frac {-\left (-x^{4}-2 i x^{3}-2 i x +1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{2}+i\right ) x^{2}+x^{3}+i \RootOf \left (\textit {\_Z}^{2}+i\right ) \left (-x^{4}-2 i x^{3}-2 i x +1\right )^{\frac {3}{4}}-i \sqrt {-x^{4}-2 i x^{3}-2 i x +1}\, x -2 i \RootOf \left (\textit {\_Z}^{2}+i\right ) \left (-x^{4}-2 i x^{3}-2 i x +1\right )^{\frac {1}{4}} x +2 i x^{2}+\sqrt {-x^{4}-2 i x^{3}-2 i x +1}+\RootOf \left (\textit {\_Z}^{2}+i\right ) \left (-x^{4}-2 i x^{3}-2 i x +1\right )^{\frac {1}{4}}-x}{\left (i x -1\right )^{2}}\right )}{2}+\frac {5 i \RootOf \left (\textit {\_Z}^{2}+i\right ) \ln \left (-\frac {-i \left (-x^{4}-2 i x^{3}-2 i x +1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{2}+i\right ) x^{2}+2 \RootOf \left (\textit {\_Z}^{2}+i\right ) \left (-x^{4}-2 i x^{3}-2 i x +1\right )^{\frac {1}{4}} x +x^{3}+i \sqrt {-x^{4}-2 i x^{3}-2 i x +1}\, x +\RootOf \left (\textit {\_Z}^{2}+i\right ) \left (-x^{4}-2 i x^{3}-2 i x +1\right )^{\frac {3}{4}}+i \RootOf \left (\textit {\_Z}^{2}+i\right ) \left (-x^{4}-2 i x^{3}-2 i x +1\right )^{\frac {1}{4}}+2 i x^{2}-\sqrt {-x^{4}-2 i x^{3}-2 i x +1}-x}{\left (i x -1\right )^{2}}\right )}{2}\right ) \left (-a \left (i x -1\right )\right )^{\frac {1}{4}} \left (-\left (i x -1\right )^{3} \left (i x +1\right )\right )^{\frac {1}{4}}}{\left (i x -1\right ) \left (a \left (i x +1\right )\right )^{\frac {1}{4}}}\) | \(480\) |
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 [A]
time = 0.82, size = 233, normalized size = 0.81 \begin {gather*} -\frac {\sqrt {25 i} {\left (a x - i \, a\right )} \log \left (\frac {\sqrt {25 i} {\left (a x - i \, a\right )} + 5 \, {\left (i \, a x + a\right )}^{\frac {3}{4}} {\left (-i \, a x + a\right )}^{\frac {1}{4}}}{5 \, {\left (x - i\right )}}\right ) - \sqrt {25 i} {\left (a x - i \, a\right )} \log \left (-\frac {\sqrt {25 i} {\left (a x - i \, a\right )} - 5 \, {\left (i \, a x + a\right )}^{\frac {3}{4}} {\left (-i \, a x + a\right )}^{\frac {1}{4}}}{5 \, {\left (x - i\right )}}\right ) + \sqrt {-25 i} {\left (a x - i \, a\right )} \log \left (\frac {\sqrt {-25 i} {\left (a x - i \, a\right )} + 5 \, {\left (i \, a x + a\right )}^{\frac {3}{4}} {\left (-i \, a x + a\right )}^{\frac {1}{4}}}{5 \, {\left (x - i\right )}}\right ) - \sqrt {-25 i} {\left (a x - i \, a\right )} \log \left (-\frac {\sqrt {-25 i} {\left (a x - i \, a\right )} - 5 \, {\left (i \, a x + a\right )}^{\frac {3}{4}} {\left (-i \, a x + a\right )}^{\frac {1}{4}}}{5 \, {\left (x - i\right )}}\right ) + 2 \, {\left (i \, a x + a\right )}^{\frac {3}{4}} {\left (-i \, a x + a\right )}^{\frac {1}{4}} {\left (-i \, x - 9\right )}}{2 \, {\left (a x - i \, a\right )}} \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 {\left (- i a \left (x + i\right )\right )^{\frac {5}{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: RuntimeError} \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 = -0.00 \begin {gather*} \int \frac {{\left (a-a\,x\,1{}\mathrm {i}\right )}^{5/4}}{{\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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