Optimal. Leaf size=338 \[ \frac{i (-i a-i b x+1)^{3/4} \sqrt [4]{i a+i b x+1}}{b}+\frac{i \log \left (\frac{\sqrt{-i a-i b x+1}}{\sqrt{i a+i b x+1}}-\frac{\sqrt{2} \sqrt [4]{-i a-i b x+1}}{\sqrt [4]{i a+i b x+1}}+1\right )}{2 \sqrt{2} b}-\frac{i \log \left (\frac{\sqrt{-i a-i b x+1}}{\sqrt{i a+i b x+1}}+\frac{\sqrt{2} \sqrt [4]{-i a-i b x+1}}{\sqrt [4]{i a+i b x+1}}+1\right )}{2 \sqrt{2} b}-\frac{i \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{-i a-i b x+1}}{\sqrt [4]{i a+i b x+1}}\right )}{\sqrt{2} b}+\frac{i \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{-i a-i b x+1}}{\sqrt [4]{i a+i b x+1}}\right )}{\sqrt{2} b} \]
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Rubi [A] time = 0.195212, antiderivative size = 338, normalized size of antiderivative = 1., 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 = {5093, 50, 63, 331, 297, 1162, 617, 204, 1165, 628} \[ \frac{i (-i a-i b x+1)^{3/4} \sqrt [4]{i a+i b x+1}}{b}+\frac{i \log \left (\frac{\sqrt{-i a-i b x+1}}{\sqrt{i a+i b x+1}}-\frac{\sqrt{2} \sqrt [4]{-i a-i b x+1}}{\sqrt [4]{i a+i b x+1}}+1\right )}{2 \sqrt{2} b}-\frac{i \log \left (\frac{\sqrt{-i a-i b x+1}}{\sqrt{i a+i b x+1}}+\frac{\sqrt{2} \sqrt [4]{-i a-i b x+1}}{\sqrt [4]{i a+i b x+1}}+1\right )}{2 \sqrt{2} b}-\frac{i \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{-i a-i b x+1}}{\sqrt [4]{i a+i b x+1}}\right )}{\sqrt{2} b}+\frac{i \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{-i a-i b x+1}}{\sqrt [4]{i a+i b x+1}}\right )}{\sqrt{2} b} \]
Antiderivative was successfully verified.
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Rule 5093
Rule 50
Rule 63
Rule 331
Rule 297
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
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) \operatorname{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) \operatorname{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 \operatorname{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 \operatorname{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 \operatorname{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 \operatorname{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 \operatorname{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 \operatorname{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 \operatorname{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 \operatorname{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*}
Mathematica [C] time = 0.015582, size = 45, normalized size = 0.13 \[ -\frac{8 i e^{\frac{5}{2} i \tan ^{-1}(a+b x)} \, _2F_1\left (\frac{5}{4},2;\frac{9}{4};-e^{2 i \tan ^{-1}(a+b x)}\right )}{5 b} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.194, size = 0, normalized size = 0. \begin{align*} \int \sqrt{{(1+i \left ( bx+a \right ) ){\frac{1}{\sqrt{1+ \left ( bx+a \right ) ^{2}}}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{\frac{i \, b x + i \, a + 1}{\sqrt{{\left (b x + a\right )}^{2} + 1}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.3546, size = 636, normalized size = 1.88 \begin{align*} \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 ) +{\left (2 \, b x + 2 \, a + 2 i\right )} \sqrt{\frac{i \, \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b x + a + i}}}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{\frac{i \, b x + i \, a + 1}{\sqrt{{\left (b x + a\right )}^{2} + 1}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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