Optimal. Leaf size=266 \[ \frac {4 i (a-i a x)^{3/4}}{3 a (a+i a x)^{3/4}}-\frac {i \log \left (\frac {\sqrt {a-i a x}}{\sqrt {a+i a x}}-\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}+1\right )}{\sqrt {2} a}+\frac {i \log \left (\frac {\sqrt {a-i a x}}{\sqrt {a+i a x}}+\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}+1\right )}{\sqrt {2} a}+\frac {i \sqrt {2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{a}-\frac {i \sqrt {2} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{a} \]
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Rubi [A] time = 0.14, 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 = {47, 63, 331, 297, 1162, 617, 204, 1165, 628} \[ \frac {4 i (a-i a x)^{3/4}}{3 a (a+i a x)^{3/4}}-\frac {i \log \left (\frac {\sqrt {a-i a x}}{\sqrt {a+i a x}}-\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}+1\right )}{\sqrt {2} a}+\frac {i \log \left (\frac {\sqrt {a-i a x}}{\sqrt {a+i a x}}+\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}+1\right )}{\sqrt {2} a}+\frac {i \sqrt {2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{a}-\frac {i \sqrt {2} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{a} \]
Antiderivative was successfully verified.
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Rule 47
Rule 63
Rule 204
Rule 297
Rule 331
Rule 617
Rule 628
Rule 1162
Rule 1165
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) \operatorname {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) \operatorname {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) \operatorname {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) \operatorname {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 \operatorname {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 \operatorname {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 \operatorname {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 \operatorname {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 ) \operatorname {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 ) \operatorname {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 [C] time = 0.02, size = 70, normalized size = 0.26 \[ \frac {i \sqrt [4]{2} (1+i x)^{3/4} (a-i a x)^{7/4} \, _2F_1\left (\frac {7}{4},\frac {7}{4};\frac {11}{4};\frac {1}{2}-\frac {i x}{2}\right )}{7 a^2 (a+i a x)^{3/4}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 304, normalized size = 1.14 \[ -\frac {3 \, {\left (a^{2} x - i \, a^{2}\right )} \sqrt {\frac {4 i}{a^{2}}} \log \left (\frac {{\left (a^{2} x + i \, a^{2}\right )} \sqrt {\frac {4 i}{a^{2}}} + 2 \, {\left (i \, a x + a\right )}^{\frac {1}{4}} {\left (-i \, a x + a\right )}^{\frac {3}{4}}}{2 \, x + 2 i}\right ) - 3 \, {\left (a^{2} x - i \, a^{2}\right )} \sqrt {\frac {4 i}{a^{2}}} \log \left (-\frac {{\left (a^{2} x + i \, a^{2}\right )} \sqrt {\frac {4 i}{a^{2}}} - 2 \, {\left (i \, a x + a\right )}^{\frac {1}{4}} {\left (-i \, a x + a\right )}^{\frac {3}{4}}}{2 \, x + 2 i}\right ) + 3 \, {\left (a^{2} x - i \, a^{2}\right )} \sqrt {-\frac {4 i}{a^{2}}} \log \left (\frac {{\left (a^{2} x + i \, a^{2}\right )} \sqrt {-\frac {4 i}{a^{2}}} + 2 \, {\left (i \, a x + a\right )}^{\frac {1}{4}} {\left (-i \, a x + a\right )}^{\frac {3}{4}}}{2 \, x + 2 i}\right ) - 3 \, {\left (a^{2} x - i \, a^{2}\right )} \sqrt {-\frac {4 i}{a^{2}}} \log \left (-\frac {{\left (a^{2} x + i \, a^{2}\right )} \sqrt {-\frac {4 i}{a^{2}}} - 2 \, {\left (i \, a x + a\right )}^{\frac {1}{4}} {\left (-i \, a x + a\right )}^{\frac {3}{4}}}{2 \, x + 2 i}\right ) - 8 \, {\left (i \, a x + a\right )}^{\frac {1}{4}} {\left (-i \, a x + a\right )}^{\frac {3}{4}}}{6 \, {\left (a^{2} x - i \, a^{2}\right )}} \]
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 \[ \int \frac {{\left (-i \, a x + a\right )}^{\frac {3}{4}}}{{\left (i \, a x + a\right )}^{\frac {7}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.88, size = 459, normalized size = 1.73 \[ \frac {\frac {4 x}{3}+\frac {4 i}{3}}{\left (\left (i x +1\right ) a \right )^{\frac {3}{4}} \left (-\left (i x -1\right ) a \right )^{\frac {1}{4}}}-\frac {\left (i \RootOf \left (\textit {\_Z}^{2}+i\right ) \ln \left (\frac {-x^{3}-i \left (-x^{4}+2 i x^{3}+2 i x +1\right )^{\frac {1}{4}} x^{2} \RootOf \left (\textit {\_Z}^{2}+i\right )+2 i x^{2}-2 \left (-x^{4}+2 i x^{3}+2 i x +1\right )^{\frac {1}{4}} x \RootOf \left (\textit {\_Z}^{2}+i\right )-i \sqrt {-x^{4}+2 i x^{3}+2 i x +1}\, x +x +\left (-x^{4}+2 i x^{3}+2 i x +1\right )^{\frac {3}{4}} \RootOf \left (\textit {\_Z}^{2}+i\right )+i \left (-x^{4}+2 i x^{3}+2 i x +1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{2}+i\right )-\sqrt {-x^{4}+2 i x^{3}+2 i x +1}}{\left (i x +1\right )^{2}}\right )-\RootOf \left (\textit {\_Z}^{2}+i\right ) \ln \left (-\frac {x^{3}-\left (-x^{4}+2 i x^{3}+2 i x +1\right )^{\frac {1}{4}} x^{2} \RootOf \left (\textit {\_Z}^{2}+i\right )-2 i x^{2}+2 i \left (-x^{4}+2 i x^{3}+2 i x +1\right )^{\frac {1}{4}} x \RootOf \left (\textit {\_Z}^{2}+i\right )-i \sqrt {-x^{4}+2 i x^{3}+2 i x +1}\, x -x +i \left (-x^{4}+2 i x^{3}+2 i x +1\right )^{\frac {3}{4}} \RootOf \left (\textit {\_Z}^{2}+i\right )+\left (-x^{4}+2 i x^{3}+2 i x +1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{2}+i\right )-\sqrt {-x^{4}+2 i x^{3}+2 i x +1}}{\left (i x +1\right )^{2}}\right )\right ) \left (-\left (i x -1\right ) \left (i x +1\right )^{3}\right )^{\frac {1}{4}}}{\left (\left (i x +1\right ) a \right )^{\frac {3}{4}} \left (-\left (i x -1\right ) a \right )^{\frac {1}{4}}} \]
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 \[ \int \frac {{\left (-i \, a x + a\right )}^{\frac {3}{4}}}{{\left (i \, a x + a\right )}^{\frac {7}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a-a\,x\,1{}\mathrm {i}\right )}^{3/4}}{{\left (a+a\,x\,1{}\mathrm {i}\right )}^{7/4}} \,d x \]
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 \[ \int \frac {\left (- i a \left (x + i\right )\right )^{\frac {3}{4}}}{\left (i a \left (x - i\right )\right )^{\frac {7}{4}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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