Optimal. Leaf size=410 \[ \frac {(1-4 i a) (1-i a-i b x)^{3/4} \sqrt [4]{1+i a+i b x}}{4 b^2}+\frac {(1-i a-i b x)^{3/4} (1+i a+i b x)^{5/4}}{2 b^2}-\frac {(1-4 i a) \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{1-i a-i b x}}{\sqrt [4]{1+i a+i b x}}\right )}{4 \sqrt {2} b^2}+\frac {(1-4 i a) \text {ArcTan}\left (1+\frac {\sqrt {2} \sqrt [4]{1-i a-i b x}}{\sqrt [4]{1+i a+i b x}}\right )}{4 \sqrt {2} b^2}+\frac {(1-4 i a) \log \left (1+\frac {\sqrt {1-i a-i b x}}{\sqrt {1+i a+i b x}}-\frac {\sqrt {2} \sqrt [4]{1-i a-i b x}}{\sqrt [4]{1+i a+i b x}}\right )}{8 \sqrt {2} b^2}-\frac {(1-4 i a) \log \left (1+\frac {\sqrt {1-i a-i b x}}{\sqrt {1+i a+i b x}}+\frac {\sqrt {2} \sqrt [4]{1-i a-i b x}}{\sqrt [4]{1+i a+i b x}}\right )}{8 \sqrt {2} b^2} \]
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Rubi [A]
time = 0.22, antiderivative size = 410, normalized size of antiderivative = 1.00, number of steps
used = 14, number of rules used = 11, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.688, Rules used = {5203, 81, 52,
65, 338, 303, 1176, 631, 210, 1179, 642} \begin {gather*} -\frac {(1-4 i a) \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{-i a-i b x+1}}{\sqrt [4]{i a+i b x+1}}\right )}{4 \sqrt {2} b^2}+\frac {(1-4 i a) \text {ArcTan}\left (1+\frac {\sqrt {2} \sqrt [4]{-i a-i b x+1}}{\sqrt [4]{i a+i b x+1}}\right )}{4 \sqrt {2} b^2}+\frac {(-i a-i b x+1)^{3/4} (i a+i b x+1)^{5/4}}{2 b^2}+\frac {(1-4 i a) (-i a-i b x+1)^{3/4} \sqrt [4]{i a+i b x+1}}{4 b^2}+\frac {(1-4 i a) \log \left (\frac {\sqrt {-i a-i b x+1}}{\sqrt {i a+i b x+1}}-\frac {\sqrt {2} \sqrt [4]{-i a-i b x+1}}{\sqrt [4]{i a+i b x+1}}+1\right )}{8 \sqrt {2} b^2}-\frac {(1-4 i a) \log \left (\frac {\sqrt {-i a-i b x+1}}{\sqrt {i a+i b x+1}}+\frac {\sqrt {2} \sqrt [4]{-i a-i b x+1}}{\sqrt [4]{i a+i b x+1}}+1\right )}{8 \sqrt {2} b^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 81
Rule 210
Rule 303
Rule 338
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 5203
Rubi steps
\begin {align*} \int e^{\frac {1}{2} i \tan ^{-1}(a+b x)} x \, dx &=\int \frac {x \sqrt [4]{1+i a+i b x}}{\sqrt [4]{1-i a-i b x}} \, dx\\ &=\frac {(1-i a-i b x)^{3/4} (1+i a+i b x)^{5/4}}{2 b^2}-\frac {(i+4 a) \int \frac {\sqrt [4]{1+i a+i b x}}{\sqrt [4]{1-i a-i b x}} \, dx}{4 b}\\ &=\frac {(1-4 i a) (1-i a-i b x)^{3/4} \sqrt [4]{1+i a+i b x}}{4 b^2}+\frac {(1-i a-i b x)^{3/4} (1+i a+i b x)^{5/4}}{2 b^2}-\frac {(i+4 a) \int \frac {1}{\sqrt [4]{1-i a-i b x} (1+i a+i b x)^{3/4}} \, dx}{8 b}\\ &=\frac {(1-4 i a) (1-i a-i b x)^{3/4} \sqrt [4]{1+i a+i b x}}{4 b^2}+\frac {(1-i a-i b x)^{3/4} (1+i a+i b x)^{5/4}}{2 b^2}+\frac {(1-4 i a) \text {Subst}\left (\int \frac {x^2}{\left (2-x^4\right )^{3/4}} \, dx,x,\sqrt [4]{1-i a-i b x}\right )}{2 b^2}\\ &=\frac {(1-4 i a) (1-i a-i b x)^{3/4} \sqrt [4]{1+i a+i b x}}{4 b^2}+\frac {(1-i a-i b x)^{3/4} (1+i a+i b x)^{5/4}}{2 b^2}+\frac {(1-4 i a) \text {Subst}\left (\int \frac {x^2}{1+x^4} \, dx,x,\frac {\sqrt [4]{1-i a-i b x}}{\sqrt [4]{1+i a+i b x}}\right )}{2 b^2}\\ &=\frac {(1-4 i a) (1-i a-i b x)^{3/4} \sqrt [4]{1+i a+i b x}}{4 b^2}+\frac {(1-i a-i b x)^{3/4} (1+i a+i b x)^{5/4}}{2 b^2}-\frac {(1-4 i a) \text {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\frac {\sqrt [4]{1-i a-i b x}}{\sqrt [4]{1+i a+i b x}}\right )}{4 b^2}+\frac {(1-4 i a) \text {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\frac {\sqrt [4]{1-i a-i b x}}{\sqrt [4]{1+i a+i b x}}\right )}{4 b^2}\\ &=\frac {(1-4 i a) (1-i a-i b x)^{3/4} \sqrt [4]{1+i a+i b x}}{4 b^2}+\frac {(1-i a-i b x)^{3/4} (1+i a+i b x)^{5/4}}{2 b^2}+\frac {(1-4 i a) \text {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\frac {\sqrt [4]{1-i a-i b x}}{\sqrt [4]{1+i a+i b x}}\right )}{8 b^2}+\frac {(1-4 i a) \text {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\frac {\sqrt [4]{1-i a-i b x}}{\sqrt [4]{1+i a+i b x}}\right )}{8 b^2}+\frac {(1-4 i a) \text {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\frac {\sqrt [4]{1-i a-i b x}}{\sqrt [4]{1+i a+i b x}}\right )}{8 \sqrt {2} b^2}+\frac {(1-4 i a) \text {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\frac {\sqrt [4]{1-i a-i b x}}{\sqrt [4]{1+i a+i b x}}\right )}{8 \sqrt {2} b^2}\\ &=\frac {(1-4 i a) (1-i a-i b x)^{3/4} \sqrt [4]{1+i a+i b x}}{4 b^2}+\frac {(1-i a-i b x)^{3/4} (1+i a+i b x)^{5/4}}{2 b^2}+\frac {(1-4 i a) \log \left (1+\frac {\sqrt {1-i a-i b x}}{\sqrt {1+i a+i b x}}-\frac {\sqrt {2} \sqrt [4]{1-i a-i b x}}{\sqrt [4]{1+i a+i b x}}\right )}{8 \sqrt {2} b^2}-\frac {(1-4 i a) \log \left (1+\frac {\sqrt {1-i a-i b x}}{\sqrt {1+i a+i b x}}+\frac {\sqrt {2} \sqrt [4]{1-i a-i b x}}{\sqrt [4]{1+i a+i b x}}\right )}{8 \sqrt {2} b^2}+\frac {(1-4 i a) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{1-i a-i b x}}{\sqrt [4]{1+i a+i b x}}\right )}{4 \sqrt {2} b^2}-\frac {(1-4 i a) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{1-i a-i b x}}{\sqrt [4]{1+i a+i b x}}\right )}{4 \sqrt {2} b^2}\\ &=\frac {(1-4 i a) (1-i a-i b x)^{3/4} \sqrt [4]{1+i a+i b x}}{4 b^2}+\frac {(1-i a-i b x)^{3/4} (1+i a+i b x)^{5/4}}{2 b^2}-\frac {(1-4 i a) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{1-i a-i b x}}{\sqrt [4]{1+i a+i b x}}\right )}{4 \sqrt {2} b^2}+\frac {(1-4 i a) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{1-i a-i b x}}{\sqrt [4]{1+i a+i b x}}\right )}{4 \sqrt {2} b^2}+\frac {(1-4 i a) \log \left (1+\frac {\sqrt {1-i a-i b x}}{\sqrt {1+i a+i b x}}-\frac {\sqrt {2} \sqrt [4]{1-i a-i b x}}{\sqrt [4]{1+i a+i b x}}\right )}{8 \sqrt {2} b^2}-\frac {(1-4 i a) \log \left (1+\frac {\sqrt {1-i a-i b x}}{\sqrt {1+i a+i b x}}+\frac {\sqrt {2} \sqrt [4]{1-i a-i b x}}{\sqrt [4]{1+i a+i b x}}\right )}{8 \sqrt {2} b^2}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.03, size = 81, normalized size = 0.20 \begin {gather*} \frac {(-i (i+a+b x))^{3/4} \left (3 (1+i a+i b x)^{5/4}+2 \sqrt [4]{2} (1-4 i a) \, _2F_1\left (-\frac {1}{4},\frac {3}{4};\frac {7}{4};-\frac {1}{2} i (i+a+b x)\right )\right )}{6 b^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \sqrt {\frac {1+i \left (b x +a \right )}{\sqrt {1+\left (b x +a \right )^{2}}}}\, x\, dx\]
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 \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 = 4.34, size = 415, normalized size = 1.01 \begin {gather*} -\frac {b^{2} \sqrt {\frac {16 i \, a^{2} - 8 \, a - i}{b^{4}}} \log \left (\frac {i \, b^{2} \sqrt {\frac {16 i \, a^{2} - 8 \, a - i}{b^{4}}} + {\left (4 \, a + i\right )} \sqrt {\frac {i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b x + a + i}}}{4 \, a + i}\right ) - b^{2} \sqrt {\frac {16 i \, a^{2} - 8 \, a - i}{b^{4}}} \log \left (\frac {-i \, b^{2} \sqrt {\frac {16 i \, a^{2} - 8 \, a - i}{b^{4}}} + {\left (4 \, a + i\right )} \sqrt {\frac {i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b x + a + i}}}{4 \, a + i}\right ) + b^{2} \sqrt {\frac {-16 i \, a^{2} + 8 \, a + i}{b^{4}}} \log \left (\frac {i \, b^{2} \sqrt {\frac {-16 i \, a^{2} + 8 \, a + i}{b^{4}}} + {\left (4 \, a + i\right )} \sqrt {\frac {i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b x + a + i}}}{4 \, a + i}\right ) - b^{2} \sqrt {\frac {-16 i \, a^{2} + 8 \, a + i}{b^{4}}} \log \left (\frac {-i \, b^{2} \sqrt {\frac {-16 i \, a^{2} + 8 \, a + i}{b^{4}}} + {\left (4 \, a + i\right )} \sqrt {\frac {i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b x + a + i}}}{4 \, a + i}\right ) - 2 \, {\left (2 \, b^{2} x^{2} - 2 \, a^{2} - i \, b x - 5 i \, a + 3\right )} \sqrt {\frac {i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b x + a + i}}}{8 \, b^{2}} \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 x \sqrt {\frac {i \left (a + b x - i\right )}{\sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}}\, dx \end {gather*}
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 \begin {gather*} \text {Exception raised: TypeError} \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 x\,\sqrt {\frac {1+a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}}{\sqrt {{\left (a+b\,x\right )}^2+1}}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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