Optimal. Leaf size=427 \[ \frac{\log \left (\frac{\sqrt{-i a-i b x+1}}{\sqrt{i a+i b x+1}}-\frac{\sqrt{2} \sqrt [4]{-i a-i b x+1}}{\sqrt [4]{i a+i b x+1}}+1\right )}{\sqrt{2}}-\frac{\log \left (\frac{\sqrt{-i a-i b x+1}}{\sqrt{i a+i b x+1}}+\frac{\sqrt{2} \sqrt [4]{-i a-i b x+1}}{\sqrt [4]{i a+i b x+1}}+1\right )}{\sqrt{2}}-\frac{2 (a+i)^{3/4} \tan ^{-1}\left (\frac{\sqrt [4]{a+i} \sqrt [4]{i a+i b x+1}}{\sqrt [4]{-a+i} \sqrt [4]{-i a-i b x+1}}\right )}{(-a+i)^{3/4}}-\sqrt{2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{-i a-i b x+1}}{\sqrt [4]{i a+i b x+1}}\right )+\sqrt{2} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{-i a-i b x+1}}{\sqrt [4]{i a+i b x+1}}\right )-\frac{2 (a+i)^{3/4} \tanh ^{-1}\left (\frac{\sqrt [4]{a+i} \sqrt [4]{i a+i b x+1}}{\sqrt [4]{-a+i} \sqrt [4]{-i a-i b x+1}}\right )}{(-a+i)^{3/4}} \]
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Rubi [A] time = 0.23995, antiderivative size = 427, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 14, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.778, Rules used = {5095, 105, 63, 331, 297, 1162, 617, 204, 1165, 628, 93, 212, 208, 205} \[ \frac{\log \left (\frac{\sqrt{-i a-i b x+1}}{\sqrt{i a+i b x+1}}-\frac{\sqrt{2} \sqrt [4]{-i a-i b x+1}}{\sqrt [4]{i a+i b x+1}}+1\right )}{\sqrt{2}}-\frac{\log \left (\frac{\sqrt{-i a-i b x+1}}{\sqrt{i a+i b x+1}}+\frac{\sqrt{2} \sqrt [4]{-i a-i b x+1}}{\sqrt [4]{i a+i b x+1}}+1\right )}{\sqrt{2}}-\frac{2 (a+i)^{3/4} \tan ^{-1}\left (\frac{\sqrt [4]{a+i} \sqrt [4]{i a+i b x+1}}{\sqrt [4]{-a+i} \sqrt [4]{-i a-i b x+1}}\right )}{(-a+i)^{3/4}}-\sqrt{2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{-i a-i b x+1}}{\sqrt [4]{i a+i b x+1}}\right )+\sqrt{2} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{-i a-i b x+1}}{\sqrt [4]{i a+i b x+1}}\right )-\frac{2 (a+i)^{3/4} \tanh ^{-1}\left (\frac{\sqrt [4]{a+i} \sqrt [4]{i a+i b x+1}}{\sqrt [4]{-a+i} \sqrt [4]{-i a-i b x+1}}\right )}{(-a+i)^{3/4}} \]
Antiderivative was successfully verified.
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Rule 5095
Rule 105
Rule 63
Rule 331
Rule 297
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rule 93
Rule 212
Rule 208
Rule 205
Rubi steps
\begin{align*} \int \frac{e^{-\frac{3}{2} i \tan ^{-1}(a+b x)}}{x} \, dx &=\int \frac{(1-i a-i b x)^{3/4}}{x (1+i a+i b x)^{3/4}} \, dx\\ &=-\left ((-1+i a) \int \frac{1}{x \sqrt [4]{1-i a-i b x} (1+i a+i b x)^{3/4}} \, dx\right )-(i b) \int \frac{1}{\sqrt [4]{1-i a-i b x} (1+i a+i b x)^{3/4}} \, dx\\ &=4 \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 )+(4 (1-i a)) \operatorname{Subst}\left (\int \frac{1}{-1-i a-(-1+i a) x^4} \, dx,x,\frac{\sqrt [4]{1+i a+i b x}}{\sqrt [4]{1-i a-i b x}}\right )\\ &=4 \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 )-\frac{(2 (i+a)) \operatorname{Subst}\left (\int \frac{1}{\sqrt{i-a}-\sqrt{i+a} x^2} \, dx,x,\frac{\sqrt [4]{1+i a+i b x}}{\sqrt [4]{1-i a-i b x}}\right )}{\sqrt{i-a}}-\frac{(2 (i+a)) \operatorname{Subst}\left (\int \frac{1}{\sqrt{i-a}+\sqrt{i+a} x^2} \, dx,x,\frac{\sqrt [4]{1+i a+i b x}}{\sqrt [4]{1-i a-i b x}}\right )}{\sqrt{i-a}}\\ &=-\frac{2 (i+a)^{3/4} \tan ^{-1}\left (\frac{\sqrt [4]{i+a} \sqrt [4]{1+i a+i b x}}{\sqrt [4]{i-a} \sqrt [4]{1-i a-i b x}}\right )}{(i-a)^{3/4}}-\frac{2 (i+a)^{3/4} \tanh ^{-1}\left (\frac{\sqrt [4]{i+a} \sqrt [4]{1+i a+i b x}}{\sqrt [4]{i-a} \sqrt [4]{1-i a-i b x}}\right )}{(i-a)^{3/4}}-2 \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 )+2 \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 )\\ &=-\frac{2 (i+a)^{3/4} \tan ^{-1}\left (\frac{\sqrt [4]{i+a} \sqrt [4]{1+i a+i b x}}{\sqrt [4]{i-a} \sqrt [4]{1-i a-i b x}}\right )}{(i-a)^{3/4}}-\frac{2 (i+a)^{3/4} \tanh ^{-1}\left (\frac{\sqrt [4]{i+a} \sqrt [4]{1+i a+i b x}}{\sqrt [4]{i-a} \sqrt [4]{1-i a-i b x}}\right )}{(i-a)^{3/4}}+\frac{\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 )}{\sqrt{2}}+\frac{\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 )}{\sqrt{2}}+\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 )+\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 )\\ &=-\frac{2 (i+a)^{3/4} \tan ^{-1}\left (\frac{\sqrt [4]{i+a} \sqrt [4]{1+i a+i b x}}{\sqrt [4]{i-a} \sqrt [4]{1-i a-i b x}}\right )}{(i-a)^{3/4}}-\frac{2 (i+a)^{3/4} \tanh ^{-1}\left (\frac{\sqrt [4]{i+a} \sqrt [4]{1+i a+i b x}}{\sqrt [4]{i-a} \sqrt [4]{1-i a-i b x}}\right )}{(i-a)^{3/4}}+\frac{\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 )}{\sqrt{2}}-\frac{\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 )}{\sqrt{2}}+\sqrt{2} \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} \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 )\\ &=-\frac{2 (i+a)^{3/4} \tan ^{-1}\left (\frac{\sqrt [4]{i+a} \sqrt [4]{1+i a+i b x}}{\sqrt [4]{i-a} \sqrt [4]{1-i a-i b x}}\right )}{(i-a)^{3/4}}-\sqrt{2} \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} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{1-i a-i b x}}{\sqrt [4]{1+i a+i b x}}\right )-\frac{2 (i+a)^{3/4} \tanh ^{-1}\left (\frac{\sqrt [4]{i+a} \sqrt [4]{1+i a+i b x}}{\sqrt [4]{i-a} \sqrt [4]{1-i a-i b x}}\right )}{(i-a)^{3/4}}+\frac{\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 )}{\sqrt{2}}-\frac{\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 )}{\sqrt{2}}\\ \end{align*}
Mathematica [C] time = 0.0361035, size = 128, normalized size = 0.3 \[ \frac{2 (-i (a+b x+i))^{3/4} \left (\sqrt [4]{2} (i a+i b x+1)^{3/4} \, _2F_1\left (\frac{3}{4},\frac{3}{4};\frac{7}{4};-\frac{1}{2} i (a+b x+i)\right )-2 \, _2F_1\left (\frac{3}{4},1;\frac{7}{4};\frac{a^2+b x a-i b x+1}{a^2+b x a+i b x+1}\right )\right )}{3 (i a+i b x+1)^{3/4}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.234, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x} \left ({(1+i \left ( bx+a \right ) ){\frac{1}{\sqrt{1+ \left ( bx+a \right ) ^{2}}}}} \right ) ^{-{\frac{3}{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 \frac{1}{x \left (\frac{i \, b x + i \, a + 1}{\sqrt{{\left (b x + a\right )}^{2} + 1}}\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.85048, size = 1673, normalized size = 3.92 \begin{align*} \text{result too large to display} \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 \frac{1}{x \left (\frac{i \, b x + i \, a + 1}{\sqrt{{\left (b x + a\right )}^{2} + 1}}\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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