Optimal. Leaf size=338 \[ \frac {i (1-i a-i b x)^{3/4} \sqrt [4]{1+i a+i b x}}{b}-\frac {i \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{1-i a-i b x}}{\sqrt [4]{1+i a+i b x}}\right )}{\sqrt {2} b}+\frac {i \text {ArcTan}\left (1+\frac {\sqrt {2} \sqrt [4]{1-i a-i b x}}{\sqrt [4]{1+i a+i b x}}\right )}{\sqrt {2} b}+\frac {i \log \left (1+\frac {\sqrt {1-i a-i b x}}{\sqrt {1+i a+i b x}}-\frac {\sqrt {2} \sqrt [4]{1-i a-i b x}}{\sqrt [4]{1+i a+i b x}}\right )}{2 \sqrt {2} b}-\frac {i \log \left (1+\frac {\sqrt {1-i a-i b x}}{\sqrt {1+i a+i b x}}+\frac {\sqrt {2} \sqrt [4]{1-i a-i b x}}{\sqrt [4]{1+i a+i b x}}\right )}{2 \sqrt {2} b} \]
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Rubi [A]
time = 0.14, antiderivative size = 338, normalized size of antiderivative = 1.00, number of steps
used = 13, number of rules used = 10, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {5201, 52, 65,
338, 303, 1176, 631, 210, 1179, 642} \begin {gather*} -\frac {i \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{-i a-i b x+1}}{\sqrt [4]{i a+i b x+1}}\right )}{\sqrt {2} b}+\frac {i \text {ArcTan}\left (1+\frac {\sqrt {2} \sqrt [4]{-i a-i b x+1}}{\sqrt [4]{i a+i b x+1}}\right )}{\sqrt {2} b}+\frac {i (-i a-i b x+1)^{3/4} \sqrt [4]{i a+i b x+1}}{b}+\frac {i \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 )}{2 \sqrt {2} b}-\frac {i \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 )}{2 \sqrt {2} b} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 210
Rule 303
Rule 338
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 5201
Rubi steps
\begin {align*} \int e^{\frac {1}{2} i \tan ^{-1}(a+b x)} \, dx &=\int \frac {\sqrt [4]{1+i a+i b x}}{\sqrt [4]{1-i a-i b x}} \, dx\\ &=\frac {i (1-i a-i b x)^{3/4} \sqrt [4]{1+i a+i b x}}{b}+\frac {1}{2} \int \frac {1}{\sqrt [4]{1-i a-i b x} (1+i a+i b x)^{3/4}} \, dx\\ &=\frac {i (1-i a-i b x)^{3/4} \sqrt [4]{1+i a+i b x}}{b}+\frac {(2 i) \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 )}{b}\\ &=\frac {i (1-i a-i b x)^{3/4} \sqrt [4]{1+i a+i b x}}{b}+\frac {(2 i) \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 )}{b}\\ &=\frac {i (1-i a-i b x)^{3/4} \sqrt [4]{1+i a+i b x}}{b}-\frac {i \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 )}{b}+\frac {i \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 )}{b}\\ &=\frac {i (1-i a-i b x)^{3/4} \sqrt [4]{1+i a+i b x}}{b}+\frac {i \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 )}{2 b}+\frac {i \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 )}{2 b}+\frac {i \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 )}{2 \sqrt {2} b}+\frac {i \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 )}{2 \sqrt {2} b}\\ &=\frac {i (1-i a-i b x)^{3/4} \sqrt [4]{1+i a+i b x}}{b}+\frac {i \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 )}{2 \sqrt {2} b}-\frac {i \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 )}{2 \sqrt {2} b}+\frac {i \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 )}{\sqrt {2} b}-\frac {i \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 )}{\sqrt {2} b}\\ &=\frac {i (1-i a-i b x)^{3/4} \sqrt [4]{1+i a+i b x}}{b}-\frac {i \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{1-i a-i b x}}{\sqrt [4]{1+i a+i b x}}\right )}{\sqrt {2} b}+\frac {i \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{1-i a-i b x}}{\sqrt [4]{1+i a+i b x}}\right )}{\sqrt {2} b}+\frac {i \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 )}{2 \sqrt {2} b}-\frac {i \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 )}{2 \sqrt {2} b}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.01, size = 45, normalized size = 0.13 \begin {gather*} -\frac {8 i e^{\frac {5}{2} i \text {ArcTan}(a+b x)} \, _2F_1\left (\frac {5}{4},2;\frac {9}{4};-e^{2 i \text {ArcTan}(a+b x)}\right )}{5 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int \sqrt {\frac {1+i \left (b x +a \right )}{\sqrt {1+\left (b x +a \right )^{2}}}}\, dx\]
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 \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 = 3.15, size = 255, normalized size = 0.75 \begin {gather*} \frac {b \sqrt {\frac {i}{b^{2}}} \log \left (i \, b \sqrt {\frac {i}{b^{2}}} + \sqrt {\frac {i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b x + a + i}}\right ) - b \sqrt {\frac {i}{b^{2}}} \log \left (-i \, b \sqrt {\frac {i}{b^{2}}} + \sqrt {\frac {i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b x + a + i}}\right ) + b \sqrt {-\frac {i}{b^{2}}} \log \left (i \, b \sqrt {-\frac {i}{b^{2}}} + \sqrt {\frac {i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b x + a + i}}\right ) - b \sqrt {-\frac {i}{b^{2}}} \log \left (-i \, b \sqrt {-\frac {i}{b^{2}}} + \sqrt {\frac {i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b x + a + i}}\right ) + 2 \, {\left (b x + a + i\right )} \sqrt {\frac {i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b x + a + i}}}{2 \, b} \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 \sqrt {\frac {i \left (a + b x\right ) + 1}{\sqrt {\left (a + b x\right )^{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 \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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