Optimal. Leaf size=324 \[ -\frac {2 (1-i a x)^{9/4}}{a^2 \sqrt [4]{1+i a x}}-\frac {5 (1+i a x)^{3/4} (1-i a x)^{5/4}}{2 a^2}-\frac {25 (1+i a x)^{3/4} \sqrt [4]{1-i a x}}{4 a^2}-\frac {25 \log \left (\frac {\sqrt {1-i a x}}{\sqrt {1+i a x}}-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+1\right )}{8 \sqrt {2} a^2}+\frac {25 \log \left (\frac {\sqrt {1-i a x}}{\sqrt {1+i a x}}+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+1\right )}{8 \sqrt {2} a^2}-\frac {25 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{4 \sqrt {2} a^2}+\frac {25 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{4 \sqrt {2} a^2} \]
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Rubi [A] time = 0.21, antiderivative size = 324, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 11, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.786, Rules used = {5062, 78, 50, 63, 240, 211, 1165, 628, 1162, 617, 204} \[ -\frac {2 (1-i a x)^{9/4}}{a^2 \sqrt [4]{1+i a x}}-\frac {5 (1+i a x)^{3/4} (1-i a x)^{5/4}}{2 a^2}-\frac {25 (1+i a x)^{3/4} \sqrt [4]{1-i a x}}{4 a^2}-\frac {25 \log \left (\frac {\sqrt {1-i a x}}{\sqrt {1+i a x}}-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+1\right )}{8 \sqrt {2} a^2}+\frac {25 \log \left (\frac {\sqrt {1-i a x}}{\sqrt {1+i a x}}+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+1\right )}{8 \sqrt {2} a^2}-\frac {25 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{4 \sqrt {2} a^2}+\frac {25 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{4 \sqrt {2} a^2} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 78
Rule 204
Rule 211
Rule 240
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 5062
Rubi steps
\begin {align*} \int e^{-\frac {5}{2} i \tan ^{-1}(a x)} x \, dx &=\int \frac {x (1-i a x)^{5/4}}{(1+i a x)^{5/4}} \, dx\\ &=-\frac {2 (1-i a x)^{9/4}}{a^2 \sqrt [4]{1+i a x}}-\frac {(5 i) \int \frac {(1-i a x)^{5/4}}{\sqrt [4]{1+i a x}} \, dx}{a}\\ &=-\frac {2 (1-i a x)^{9/4}}{a^2 \sqrt [4]{1+i a x}}-\frac {5 (1-i a x)^{5/4} (1+i a x)^{3/4}}{2 a^2}-\frac {(25 i) \int \frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}} \, dx}{4 a}\\ &=-\frac {2 (1-i a x)^{9/4}}{a^2 \sqrt [4]{1+i a x}}-\frac {25 \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{4 a^2}-\frac {5 (1-i a x)^{5/4} (1+i a x)^{3/4}}{2 a^2}-\frac {(25 i) \int \frac {1}{(1-i a x)^{3/4} \sqrt [4]{1+i a x}} \, dx}{8 a}\\ &=-\frac {2 (1-i a x)^{9/4}}{a^2 \sqrt [4]{1+i a x}}-\frac {25 \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{4 a^2}-\frac {5 (1-i a x)^{5/4} (1+i a x)^{3/4}}{2 a^2}+\frac {25 \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{2-x^4}} \, dx,x,\sqrt [4]{1-i a x}\right )}{2 a^2}\\ &=-\frac {2 (1-i a x)^{9/4}}{a^2 \sqrt [4]{1+i a x}}-\frac {25 \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{4 a^2}-\frac {5 (1-i a x)^{5/4} (1+i a x)^{3/4}}{2 a^2}+\frac {25 \operatorname {Subst}\left (\int \frac {1}{1+x^4} \, dx,x,\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{2 a^2}\\ &=-\frac {2 (1-i a x)^{9/4}}{a^2 \sqrt [4]{1+i a x}}-\frac {25 \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{4 a^2}-\frac {5 (1-i a x)^{5/4} (1+i a x)^{3/4}}{2 a^2}+\frac {25 \operatorname {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{4 a^2}+\frac {25 \operatorname {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{4 a^2}\\ &=-\frac {2 (1-i a x)^{9/4}}{a^2 \sqrt [4]{1+i a x}}-\frac {25 \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{4 a^2}-\frac {5 (1-i a x)^{5/4} (1+i a x)^{3/4}}{2 a^2}+\frac {25 \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{8 a^2}+\frac {25 \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{8 a^2}-\frac {25 \operatorname {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{8 \sqrt {2} a^2}-\frac {25 \operatorname {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{8 \sqrt {2} a^2}\\ &=-\frac {2 (1-i a x)^{9/4}}{a^2 \sqrt [4]{1+i a x}}-\frac {25 \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{4 a^2}-\frac {5 (1-i a x)^{5/4} (1+i a x)^{3/4}}{2 a^2}-\frac {25 \log \left (1+\frac {\sqrt {1-i a x}}{\sqrt {1+i a x}}-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{8 \sqrt {2} a^2}+\frac {25 \log \left (1+\frac {\sqrt {1-i a x}}{\sqrt {1+i a x}}+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{8 \sqrt {2} a^2}+\frac {25 \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{4 \sqrt {2} a^2}-\frac {25 \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{4 \sqrt {2} a^2}\\ &=-\frac {2 (1-i a x)^{9/4}}{a^2 \sqrt [4]{1+i a x}}-\frac {25 \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{4 a^2}-\frac {5 (1-i a x)^{5/4} (1+i a x)^{3/4}}{2 a^2}-\frac {25 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{4 \sqrt {2} a^2}+\frac {25 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{4 \sqrt {2} a^2}-\frac {25 \log \left (1+\frac {\sqrt {1-i a x}}{\sqrt {1+i a x}}-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{8 \sqrt {2} a^2}+\frac {25 \log \left (1+\frac {\sqrt {1-i a x}}{\sqrt {1+i a x}}+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{8 \sqrt {2} a^2}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 63, normalized size = 0.19 \[ \frac {2 (1-i a x)^{9/4} \left (5\ 2^{3/4} \, _2F_1\left (\frac {1}{4},\frac {9}{4};\frac {13}{4};\frac {1}{2} (1-i a x)\right )-\frac {9}{\sqrt [4]{1+i a x}}\right )}{9 a^2} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.65, size = 292, normalized size = 0.90 \[ -\frac {{\left (2 \, a^{3} x - 2 i \, a^{2}\right )} \sqrt {\frac {625 i}{16 \, a^{4}}} \log \left (\frac {4}{25} i \, a^{2} \sqrt {\frac {625 i}{16 \, a^{4}}} + \sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}}\right ) - {\left (2 \, a^{3} x - 2 i \, a^{2}\right )} \sqrt {\frac {625 i}{16 \, a^{4}}} \log \left (-\frac {4}{25} i \, a^{2} \sqrt {\frac {625 i}{16 \, a^{4}}} + \sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}}\right ) - {\left (2 \, a^{3} x - 2 i \, a^{2}\right )} \sqrt {-\frac {625 i}{16 \, a^{4}}} \log \left (\frac {4}{25} i \, a^{2} \sqrt {-\frac {625 i}{16 \, a^{4}}} + \sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}}\right ) + {\left (2 \, a^{3} x - 2 i \, a^{2}\right )} \sqrt {-\frac {625 i}{16 \, a^{4}}} \log \left (-\frac {4}{25} i \, a^{2} \sqrt {-\frac {625 i}{16 \, a^{4}}} + \sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}}\right ) - \sqrt {a^{2} x^{2} + 1} {\left (2 i \, a^{2} x^{2} - 9 \, a x + 43 i\right )} \sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}}}{4 \, a^{3} x - 4 i \, a^{2}} \]
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 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.18, size = 0, normalized size = 0.00 \[ \int \frac {x}{\left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )^{\frac {5}{2}}}\, dx \]
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 \[ \int \frac {x}{\left (\frac {i \, a x + 1}{\sqrt {a^{2} x^{2} + 1}}\right )^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {x}{{\left (\frac {1+a\,x\,1{}\mathrm {i}}{\sqrt {a^2\,x^2+1}}\right )}^{5/2}} \,d x \]
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 \[ \int \frac {x}{\left (\frac {i \left (a x - i\right )}{\sqrt {a^{2} x^{2} + 1}}\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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