Optimal. Leaf size=266 \[ \frac {4 i (a-i a x)^{3/4}}{3 a (a+i a x)^{3/4}}+\frac {i \sqrt {2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{a}-\frac {i \sqrt {2} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{a}-\frac {i \log \left (1+\frac {\sqrt {a-i a x}}{\sqrt {a+i a x}}-\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{\sqrt {2} a}+\frac {i \log \left (1+\frac {\sqrt {a-i a x}}{\sqrt {a+i a x}}+\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{\sqrt {2} a} \]
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Rubi [A]
time = 0.10, antiderivative size = 266, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 9, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {49, 65, 338,
303, 1176, 631, 210, 1179, 642} \begin {gather*} \frac {i \sqrt {2} \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{a}-\frac {i \sqrt {2} \text {ArcTan}\left (1+\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{a}+\frac {4 i (a-i a x)^{3/4}}{3 a (a+i a x)^{3/4}}-\frac {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 )}{\sqrt {2} a}+\frac {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 )}{\sqrt {2} a} \end {gather*}
Antiderivative was successfully verified.
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Rule 49
Rule 65
Rule 210
Rule 303
Rule 338
Rule 631
Rule 642
Rule 1176
Rule 1179
Rubi steps
\begin {align*} \int \frac {(a-i a x)^{3/4}}{(a+i a x)^{7/4}} \, dx &=\frac {4 i (a-i a x)^{3/4}}{3 a (a+i a x)^{3/4}}-\int \frac {1}{\sqrt [4]{a-i a x} (a+i a x)^{3/4}} \, dx\\ &=\frac {4 i (a-i a x)^{3/4}}{3 a (a+i a x)^{3/4}}-\frac {(4 i) \text {Subst}\left (\int \frac {x^2}{\left (2 a-x^4\right )^{3/4}} \, dx,x,\sqrt [4]{a-i a x}\right )}{a}\\ &=\frac {4 i (a-i a x)^{3/4}}{3 a (a+i a x)^{3/4}}-\frac {(4 i) \text {Subst}\left (\int \frac {x^2}{1+x^4} \, dx,x,\frac {\sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{a}\\ &=\frac {4 i (a-i a x)^{3/4}}{3 a (a+i a x)^{3/4}}+\frac {(2 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 )}{a}-\frac {(2 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 )}{a}\\ &=\frac {4 i (a-i a x)^{3/4}}{3 a (a+i a x)^{3/4}}-\frac {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 )}{a}-\frac {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 )}{a}-\frac {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 )}{\sqrt {2} a}-\frac {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 )}{\sqrt {2} a}\\ &=\frac {4 i (a-i a x)^{3/4}}{3 a (a+i a x)^{3/4}}-\frac {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 )}{\sqrt {2} a}+\frac {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 )}{\sqrt {2} a}-\frac {\left (i \sqrt {2}\right ) \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 )}{a}+\frac {\left (i \sqrt {2}\right ) \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 )}{a}\\ &=\frac {4 i (a-i a x)^{3/4}}{3 a (a+i a x)^{3/4}}+\frac {i \sqrt {2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{a}-\frac {i \sqrt {2} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{a}-\frac {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 )}{\sqrt {2} a}+\frac {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 )}{\sqrt {2} a}\\ \end {align*}
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Mathematica [A]
time = 0.21, size = 112, normalized size = 0.42 \begin {gather*} \frac {2 \left (\frac {2 i (a-i a x)^{3/4}}{(a+i a x)^{3/4}}+3 \sqrt [4]{-1} \tanh ^{-1}\left (\frac {\sqrt [4]{-1} \sqrt [4]{a+i a x}}{\sqrt [4]{a-i a x}}\right )-3 (-1)^{3/4} \tanh ^{-1}\left (\frac {(-1)^{3/4} \sqrt [4]{a+i a x}}{\sqrt [4]{a-i a x}}\right )\right )}{3 a} \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 = 1.30, size = 455, normalized size = 1.71
method | result | size |
risch | \(\frac {\frac {4 x}{3}+\frac {4 i}{3}}{\left (a \left (i x +1\right )\right )^{\frac {3}{4}} \left (-a \left (i x -1\right )\right )^{\frac {1}{4}}}-\frac {\left (\RootOf \left (\textit {\_Z}^{2}+i\right ) \ln \left (\frac {-\left (-x^{4}+2 i x^{3}+2 i x +1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{2}+i\right ) x^{2}-x^{3}+i \RootOf \left (\textit {\_Z}^{2}+i\right ) \left (-x^{4}+2 i x^{3}+2 i x +1\right )^{\frac {3}{4}}+i \sqrt {-x^{4}+2 i x^{3}+2 i x +1}\, x +2 i \RootOf \left (\textit {\_Z}^{2}+i\right ) \left (-x^{4}+2 i x^{3}+2 i x +1\right )^{\frac {1}{4}} x +2 i x^{2}+\sqrt {-x^{4}+2 i x^{3}+2 i x +1}+\RootOf \left (\textit {\_Z}^{2}+i\right ) \left (-x^{4}+2 i x^{3}+2 i x +1\right )^{\frac {1}{4}}+x}{\left (i x +1\right )^{2}}\right )+i \RootOf \left (\textit {\_Z}^{2}+i\right ) \ln \left (\frac {-i \left (-x^{4}+2 i x^{3}+2 i x +1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{2}+i\right ) x^{2}-2 \RootOf \left (\textit {\_Z}^{2}+i\right ) \left (-x^{4}+2 i x^{3}+2 i x +1\right )^{\frac {1}{4}} x -x^{3}-i \sqrt {-x^{4}+2 i x^{3}+2 i x +1}\, x +\RootOf \left (\textit {\_Z}^{2}+i\right ) \left (-x^{4}+2 i x^{3}+2 i x +1\right )^{\frac {3}{4}}+i \RootOf \left (\textit {\_Z}^{2}+i\right ) \left (-x^{4}+2 i x^{3}+2 i x +1\right )^{\frac {1}{4}}+2 i x^{2}-\sqrt {-x^{4}+2 i x^{3}+2 i x +1}+x}{\left (i x +1\right )^{2}}\right )\right ) \left (-\left (i x -1\right ) \left (i x +1\right )^{3}\right )^{\frac {1}{4}}}{\left (a \left (i x +1\right )\right )^{\frac {3}{4}} \left (-a \left (i x -1\right )\right )^{\frac {1}{4}}}\) | \(455\) |
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 = 0.70, size = 298, normalized size = 1.12 \begin {gather*} -\frac {3 \, {\left (a^{2} x - i \, a^{2}\right )} \sqrt {\frac {4 i}{a^{2}}} \log \left (\frac {{\left (a^{2} x + i \, a^{2}\right )} \sqrt {\frac {4 i}{a^{2}}} + 2 \, {\left (i \, a x + a\right )}^{\frac {1}{4}} {\left (-i \, a x + a\right )}^{\frac {3}{4}}}{2 \, {\left (x + i\right )}}\right ) - 3 \, {\left (a^{2} x - i \, a^{2}\right )} \sqrt {\frac {4 i}{a^{2}}} \log \left (-\frac {{\left (a^{2} x + i \, a^{2}\right )} \sqrt {\frac {4 i}{a^{2}}} - 2 \, {\left (i \, a x + a\right )}^{\frac {1}{4}} {\left (-i \, a x + a\right )}^{\frac {3}{4}}}{2 \, {\left (x + i\right )}}\right ) + 3 \, {\left (a^{2} x - i \, a^{2}\right )} \sqrt {-\frac {4 i}{a^{2}}} \log \left (\frac {{\left (a^{2} x + i \, a^{2}\right )} \sqrt {-\frac {4 i}{a^{2}}} + 2 \, {\left (i \, a x + a\right )}^{\frac {1}{4}} {\left (-i \, a x + a\right )}^{\frac {3}{4}}}{2 \, {\left (x + i\right )}}\right ) - 3 \, {\left (a^{2} x - i \, a^{2}\right )} \sqrt {-\frac {4 i}{a^{2}}} \log \left (-\frac {{\left (a^{2} x + i \, a^{2}\right )} \sqrt {-\frac {4 i}{a^{2}}} - 2 \, {\left (i \, a x + a\right )}^{\frac {1}{4}} {\left (-i \, a x + a\right )}^{\frac {3}{4}}}{2 \, {\left (x + i\right )}}\right ) - 8 \, {\left (i \, a x + a\right )}^{\frac {1}{4}} {\left (-i \, a x + a\right )}^{\frac {3}{4}}}{6 \, {\left (a^{2} x - i \, a^{2}\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 {3}{4}}}{\left (i a \left (x - i\right )\right )^{\frac {7}{4}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [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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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a-a\,x\,1{}\mathrm {i}\right )}^{3/4}}{{\left (a+a\,x\,1{}\mathrm {i}\right )}^{7/4}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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