Optimal. Leaf size=339 \[ -\frac {i (1-i a x)^{3/4} (1+i a x)^{5/4}}{12 a^3}-\frac {3 i (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{8 a^3}-\frac {3 i \log \left (\frac {\sqrt {1-i a x}}{\sqrt {1+i a x}}-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+1\right )}{16 \sqrt {2} a^3}+\frac {3 i \log \left (\frac {\sqrt {1-i a x}}{\sqrt {1+i a x}}+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+1\right )}{16 \sqrt {2} a^3}+\frac {3 i \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{8 \sqrt {2} a^3}-\frac {3 i \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{8 \sqrt {2} a^3}+\frac {x (1-i a x)^{3/4} (1+i a x)^{5/4}}{3 a^2} \]
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Rubi [A] time = 0.23, antiderivative size = 339, 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, 90, 80, 50, 63, 331, 297, 1162, 617, 204, 1165, 628} \[ \frac {x (1-i a x)^{3/4} (1+i a x)^{5/4}}{3 a^2}-\frac {i (1-i a x)^{3/4} (1+i a x)^{5/4}}{12 a^3}-\frac {3 i (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{8 a^3}-\frac {3 i \log \left (\frac {\sqrt {1-i a x}}{\sqrt {1+i a x}}-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+1\right )}{16 \sqrt {2} a^3}+\frac {3 i \log \left (\frac {\sqrt {1-i a x}}{\sqrt {1+i a x}}+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+1\right )}{16 \sqrt {2} a^3}+\frac {3 i \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{8 \sqrt {2} a^3}-\frac {3 i \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{8 \sqrt {2} a^3} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 80
Rule 90
Rule 204
Rule 297
Rule 331
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^2 \, dx &=\int \frac {x^2 \sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}} \, dx\\ &=\frac {x (1-i a x)^{3/4} (1+i a x)^{5/4}}{3 a^2}+\frac {\int \frac {\left (-1-\frac {i a x}{2}\right ) \sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}} \, dx}{3 a^2}\\ &=-\frac {i (1-i a x)^{3/4} (1+i a x)^{5/4}}{12 a^3}+\frac {x (1-i a x)^{3/4} (1+i a x)^{5/4}}{3 a^2}-\frac {3 \int \frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}} \, dx}{8 a^2}\\ &=-\frac {3 i (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{8 a^3}-\frac {i (1-i a x)^{3/4} (1+i a x)^{5/4}}{12 a^3}+\frac {x (1-i a x)^{3/4} (1+i a x)^{5/4}}{3 a^2}-\frac {3 \int \frac {1}{\sqrt [4]{1-i a x} (1+i a x)^{3/4}} \, dx}{16 a^2}\\ &=-\frac {3 i (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{8 a^3}-\frac {i (1-i a x)^{3/4} (1+i a x)^{5/4}}{12 a^3}+\frac {x (1-i a x)^{3/4} (1+i a x)^{5/4}}{3 a^2}-\frac {(3 i) \operatorname {Subst}\left (\int \frac {x^2}{\left (2-x^4\right )^{3/4}} \, dx,x,\sqrt [4]{1-i a x}\right )}{4 a^3}\\ &=-\frac {3 i (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{8 a^3}-\frac {i (1-i a x)^{3/4} (1+i a x)^{5/4}}{12 a^3}+\frac {x (1-i a x)^{3/4} (1+i a x)^{5/4}}{3 a^2}-\frac {(3 i) \operatorname {Subst}\left (\int \frac {x^2}{1+x^4} \, dx,x,\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{4 a^3}\\ &=-\frac {3 i (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{8 a^3}-\frac {i (1-i a x)^{3/4} (1+i a x)^{5/4}}{12 a^3}+\frac {x (1-i a x)^{3/4} (1+i a x)^{5/4}}{3 a^2}+\frac {(3 i) \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 )}{8 a^3}-\frac {(3 i) \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 )}{8 a^3}\\ &=-\frac {3 i (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{8 a^3}-\frac {i (1-i a x)^{3/4} (1+i a x)^{5/4}}{12 a^3}+\frac {x (1-i a x)^{3/4} (1+i a x)^{5/4}}{3 a^2}-\frac {(3 i) \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 )}{16 a^3}-\frac {(3 i) \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 )}{16 a^3}-\frac {(3 i) \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 )}{16 \sqrt {2} a^3}-\frac {(3 i) \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 )}{16 \sqrt {2} a^3}\\ &=-\frac {3 i (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{8 a^3}-\frac {i (1-i a x)^{3/4} (1+i a x)^{5/4}}{12 a^3}+\frac {x (1-i a x)^{3/4} (1+i a x)^{5/4}}{3 a^2}-\frac {3 i \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 )}{16 \sqrt {2} a^3}+\frac {3 i \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 )}{16 \sqrt {2} a^3}-\frac {(3 i) \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 )}{8 \sqrt {2} a^3}+\frac {(3 i) \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 )}{8 \sqrt {2} a^3}\\ &=-\frac {3 i (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{8 a^3}-\frac {i (1-i a x)^{3/4} (1+i a x)^{5/4}}{12 a^3}+\frac {x (1-i a x)^{3/4} (1+i a x)^{5/4}}{3 a^2}+\frac {3 i \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{8 \sqrt {2} a^3}-\frac {3 i \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{8 \sqrt {2} a^3}-\frac {3 i \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 )}{16 \sqrt {2} a^3}+\frac {3 i \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 )}{16 \sqrt {2} a^3}\\ \end {align*}
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Mathematica [C] time = 0.04, size = 82, normalized size = 0.24 \[ \frac {(1-i a x)^{3/4} \left (\sqrt [4]{1+i a x} \left (4 i a^2 x^2+5 a x-i\right )-6 i \sqrt [4]{2} \, _2F_1\left (-\frac {1}{4},\frac {3}{4};\frac {7}{4};\frac {1}{2} (1-i a x)\right )\right )}{12 a^3} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.61, size = 244, normalized size = 0.72 \[ -\frac {12 \, a^{3} \sqrt {\frac {9 i}{64 \, a^{6}}} \log \left (\frac {8}{3} i \, a^{3} \sqrt {\frac {9 i}{64 \, a^{6}}} + \sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}}\right ) - 12 \, a^{3} \sqrt {\frac {9 i}{64 \, a^{6}}} \log \left (-\frac {8}{3} i \, a^{3} \sqrt {\frac {9 i}{64 \, a^{6}}} + \sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}}\right ) + 12 \, a^{3} \sqrt {-\frac {9 i}{64 \, a^{6}}} \log \left (\frac {8}{3} i \, a^{3} \sqrt {-\frac {9 i}{64 \, a^{6}}} + \sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}}\right ) - 12 \, a^{3} \sqrt {-\frac {9 i}{64 \, a^{6}}} \log \left (-\frac {8}{3} i \, a^{3} \sqrt {-\frac {9 i}{64 \, a^{6}}} + \sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}}\right ) - {\left (8 \, a^{3} x^{3} - 2 i \, a^{2} x^{2} - a x - 11 i\right )} \sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}}}{24 \, a^{3}} \]
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 = 0.25, size = 0, normalized size = 0.00 \[ \int \sqrt {\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}}\, x^{2}\, dx \]
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 \[ \int x^{2} \sqrt {\frac {i \, a x + 1}{\sqrt {a^{2} x^{2} + 1}}}\,{d x} \]
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 \[ \int x^2\,\sqrt {\frac {1+a\,x\,1{}\mathrm {i}}{\sqrt {a^2\,x^2+1}}} \,d x \]
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 \[ \int x^{2} \sqrt {\frac {i \left (a x - i\right )}{\sqrt {a^{2} x^{2} + 1}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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