Optimal. Leaf size=337 \[ -\frac {(1-i a x)^{5/4} (1+i a x)^{3/4} (25-4 i a x)}{96 a^4}-\frac {11 \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{64 a^4}-\frac {11 \log \left (\frac {\sqrt {1-i a x}}{\sqrt {1+i a x}}-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+1\right )}{128 \sqrt {2} a^4}+\frac {11 \log \left (\frac {\sqrt {1-i a x}}{\sqrt {1+i a x}}+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+1\right )}{128 \sqrt {2} a^4}-\frac {11 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{64 \sqrt {2} a^4}+\frac {11 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{64 \sqrt {2} a^4}+\frac {x^2 (1-i a x)^{5/4} (1+i a x)^{3/4}}{4 a^2} \]
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Rubi [A] time = 0.21, antiderivative size = 337, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 12, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {5062, 100, 147, 50, 63, 240, 211, 1165, 628, 1162, 617, 204} \[ \frac {x^2 (1-i a x)^{5/4} (1+i a x)^{3/4}}{4 a^2}-\frac {(1-i a x)^{5/4} (1+i a x)^{3/4} (25-4 i a x)}{96 a^4}-\frac {11 \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{64 a^4}-\frac {11 \log \left (\frac {\sqrt {1-i a x}}{\sqrt {1+i a x}}-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+1\right )}{128 \sqrt {2} a^4}+\frac {11 \log \left (\frac {\sqrt {1-i a x}}{\sqrt {1+i a x}}+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+1\right )}{128 \sqrt {2} a^4}-\frac {11 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{64 \sqrt {2} a^4}+\frac {11 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{64 \sqrt {2} a^4} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 100
Rule 147
Rule 204
Rule 211
Rule 240
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 5062
Rubi steps
\begin {align*} \int e^{-\frac {1}{2} i \tan ^{-1}(a x)} x^3 \, dx &=\int \frac {x^3 \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}} \, dx\\ &=\frac {x^2 (1-i a x)^{5/4} (1+i a x)^{3/4}}{4 a^2}+\frac {\int \frac {x \left (-2+\frac {i a x}{2}\right ) \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}} \, dx}{4 a^2}\\ &=\frac {x^2 (1-i a x)^{5/4} (1+i a x)^{3/4}}{4 a^2}-\frac {(1-i a x)^{5/4} (1+i a x)^{3/4} (25-4 i a x)}{96 a^4}-\frac {(11 i) \int \frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}} \, dx}{64 a^3}\\ &=-\frac {11 \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{64 a^4}+\frac {x^2 (1-i a x)^{5/4} (1+i a x)^{3/4}}{4 a^2}-\frac {(1-i a x)^{5/4} (1+i a x)^{3/4} (25-4 i a x)}{96 a^4}-\frac {(11 i) \int \frac {1}{(1-i a x)^{3/4} \sqrt [4]{1+i a x}} \, dx}{128 a^3}\\ &=-\frac {11 \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{64 a^4}+\frac {x^2 (1-i a x)^{5/4} (1+i a x)^{3/4}}{4 a^2}-\frac {(1-i a x)^{5/4} (1+i a x)^{3/4} (25-4 i a x)}{96 a^4}+\frac {11 \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{2-x^4}} \, dx,x,\sqrt [4]{1-i a x}\right )}{32 a^4}\\ &=-\frac {11 \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{64 a^4}+\frac {x^2 (1-i a x)^{5/4} (1+i a x)^{3/4}}{4 a^2}-\frac {(1-i a x)^{5/4} (1+i a x)^{3/4} (25-4 i a x)}{96 a^4}+\frac {11 \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 )}{32 a^4}\\ &=-\frac {11 \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{64 a^4}+\frac {x^2 (1-i a x)^{5/4} (1+i a x)^{3/4}}{4 a^2}-\frac {(1-i a x)^{5/4} (1+i a x)^{3/4} (25-4 i a x)}{96 a^4}+\frac {11 \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 )}{64 a^4}+\frac {11 \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 )}{64 a^4}\\ &=-\frac {11 \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{64 a^4}+\frac {x^2 (1-i a x)^{5/4} (1+i a x)^{3/4}}{4 a^2}-\frac {(1-i a x)^{5/4} (1+i a x)^{3/4} (25-4 i a x)}{96 a^4}+\frac {11 \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 )}{128 a^4}+\frac {11 \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 )}{128 a^4}-\frac {11 \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 )}{128 \sqrt {2} a^4}-\frac {11 \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 )}{128 \sqrt {2} a^4}\\ &=-\frac {11 \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{64 a^4}+\frac {x^2 (1-i a x)^{5/4} (1+i a x)^{3/4}}{4 a^2}-\frac {(1-i a x)^{5/4} (1+i a x)^{3/4} (25-4 i a x)}{96 a^4}-\frac {11 \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 )}{128 \sqrt {2} a^4}+\frac {11 \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 )}{128 \sqrt {2} a^4}+\frac {11 \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 )}{64 \sqrt {2} a^4}-\frac {11 \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 )}{64 \sqrt {2} a^4}\\ &=-\frac {11 \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{64 a^4}+\frac {x^2 (1-i a x)^{5/4} (1+i a x)^{3/4}}{4 a^2}-\frac {(1-i a x)^{5/4} (1+i a x)^{3/4} (25-4 i a x)}{96 a^4}-\frac {11 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{64 \sqrt {2} a^4}+\frac {11 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{64 \sqrt {2} a^4}-\frac {11 \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 )}{128 \sqrt {2} a^4}+\frac {11 \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 )}{128 \sqrt {2} a^4}\\ \end {align*}
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Mathematica [C] time = 0.11, size = 127, normalized size = 0.38 \[ \frac {(1-i a x)^{5/4} \left (5 a^2 x^2 (1+i a x)^{3/4}+4\ 2^{3/4} \, _2F_1\left (-\frac {7}{4},\frac {5}{4};\frac {9}{4};\frac {1}{2} (1-i a x)\right )-12\ 2^{3/4} \, _2F_1\left (-\frac {3}{4},\frac {5}{4};\frac {9}{4};\frac {1}{2} (1-i a x)\right )+5\ 2^{3/4} \, _2F_1\left (\frac {1}{4},\frac {5}{4};\frac {9}{4};\frac {1}{2} (1-i a x)\right )\right )}{20 a^4} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.54, size = 255, normalized size = 0.76 \[ -\frac {96 \, a^{4} \sqrt {\frac {121 i}{4096 \, a^{8}}} \log \left (\frac {64}{11} i \, a^{4} \sqrt {\frac {121 i}{4096 \, a^{8}}} + \sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}}\right ) - 96 \, a^{4} \sqrt {\frac {121 i}{4096 \, a^{8}}} \log \left (-\frac {64}{11} i \, a^{4} \sqrt {\frac {121 i}{4096 \, a^{8}}} + \sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}}\right ) - 96 \, a^{4} \sqrt {-\frac {121 i}{4096 \, a^{8}}} \log \left (\frac {64}{11} i \, a^{4} \sqrt {-\frac {121 i}{4096 \, a^{8}}} + \sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}}\right ) + 96 \, a^{4} \sqrt {-\frac {121 i}{4096 \, a^{8}}} \log \left (-\frac {64}{11} i \, a^{4} \sqrt {-\frac {121 i}{4096 \, a^{8}}} + \sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}}\right ) - {\left (-48 i \, a^{3} x^{3} + 56 \, a^{2} x^{2} + 58 i \, a x - 83\right )} \sqrt {a^{2} x^{2} + 1} \sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}}}{192 \, a^{4}} \]
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 = 0.19, size = 0, normalized size = 0.00 \[ \int \frac {x^{3}}{\sqrt {\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}}}\, dx \]
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 \[ \int \frac {x^{3}}{\sqrt {\frac {i \, a x + 1}{\sqrt {a^{2} x^{2} + 1}}}}\,{d x} \]
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 \[ \int \frac {x^3}{\sqrt {\frac {1+a\,x\,1{}\mathrm {i}}{\sqrt {a^2\,x^2+1}}}} \,d x \]
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 \[ \int \frac {x^{3}}{\sqrt {\frac {i \left (a x - i\right )}{\sqrt {a^{2} x^{2} + 1}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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