Optimal. Leaf size=268 \[ \frac{i \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{a}-\frac{3 i \log \left (\frac{\sqrt{1-i a x}}{\sqrt{1+i a x}}-\frac{\sqrt{2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+1\right )}{2 \sqrt{2} a}+\frac{3 i \log \left (\frac{\sqrt{1-i a x}}{\sqrt{1+i a x}}+\frac{\sqrt{2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+1\right )}{2 \sqrt{2} a}-\frac{3 i \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{\sqrt{2} a}+\frac{3 i \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{\sqrt{2} a} \]
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Rubi [A] time = 0.142655, antiderivative size = 268, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.833, Rules used = {5061, 50, 63, 240, 211, 1165, 628, 1162, 617, 204} \[ \frac{i \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{a}-\frac{3 i \log \left (\frac{\sqrt{1-i a x}}{\sqrt{1+i a x}}-\frac{\sqrt{2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+1\right )}{2 \sqrt{2} a}+\frac{3 i \log \left (\frac{\sqrt{1-i a x}}{\sqrt{1+i a x}}+\frac{\sqrt{2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+1\right )}{2 \sqrt{2} a}-\frac{3 i \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{\sqrt{2} a}+\frac{3 i \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{\sqrt{2} a} \]
Antiderivative was successfully verified.
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Rule 5061
Rule 50
Rule 63
Rule 240
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rubi steps
\begin{align*} \int e^{\frac{3}{2} i \tan ^{-1}(a x)} \, dx &=\int \frac{(1+i a x)^{3/4}}{(1-i a x)^{3/4}} \, dx\\ &=\frac{i \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{a}+\frac{3}{2} \int \frac{1}{(1-i a x)^{3/4} \sqrt [4]{1+i a x}} \, dx\\ &=\frac{i \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{a}+\frac{(6 i) \operatorname{Subst}\left (\int \frac{1}{\sqrt [4]{2-x^4}} \, dx,x,\sqrt [4]{1-i a x}\right )}{a}\\ &=\frac{i \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{a}+\frac{(6 i) \operatorname{Subst}\left (\int \frac{1}{1+x^4} \, dx,x,\frac{\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{a}\\ &=\frac{i \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{a}+\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 )}{a}+\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 )}{a}\\ &=\frac{i \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{a}+\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 )}{2 a}+\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 )}{2 a}-\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 )}{2 \sqrt{2} a}-\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 )}{2 \sqrt{2} a}\\ &=\frac{i \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{a}-\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 )}{2 \sqrt{2} a}+\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 )}{2 \sqrt{2} a}+\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 )}{\sqrt{2} a}-\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 )}{\sqrt{2} a}\\ &=\frac{i \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{a}-\frac{3 i \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{\sqrt{2} a}+\frac{3 i \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{\sqrt{2} a}-\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 )}{2 \sqrt{2} a}+\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 )}{2 \sqrt{2} a}\\ \end{align*}
Mathematica [C] time = 0.0266814, size = 41, normalized size = 0.15 \[ -\frac{8 i e^{\frac{7}{2} i \tan ^{-1}(a x)} \text{Hypergeometric2F1}\left (\frac{7}{4},2,\frac{11}{4},-e^{2 i \tan ^{-1}(a x)}\right )}{7 a} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.127, size = 0, normalized size = 0. \begin{align*} \int \left ({(1+iax){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \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 \left (\frac{i \, a x + 1}{\sqrt{a^{2} x^{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 [A] time = 1.54523, size = 536, normalized size = 2. \begin{align*} \frac{a \sqrt{\frac{9 i}{a^{2}}} \log \left (\frac{1}{3} \, a \sqrt{\frac{9 i}{a^{2}}} + \sqrt{\frac{i \, \sqrt{a^{2} x^{2} + 1}}{a x + i}}\right ) - a \sqrt{\frac{9 i}{a^{2}}} \log \left (-\frac{1}{3} \, a \sqrt{\frac{9 i}{a^{2}}} + \sqrt{\frac{i \, \sqrt{a^{2} x^{2} + 1}}{a x + i}}\right ) - a \sqrt{-\frac{9 i}{a^{2}}} \log \left (\frac{1}{3} \, a \sqrt{-\frac{9 i}{a^{2}}} + \sqrt{\frac{i \, \sqrt{a^{2} x^{2} + 1}}{a x + i}}\right ) + a \sqrt{-\frac{9 i}{a^{2}}} \log \left (-\frac{1}{3} \, a \sqrt{-\frac{9 i}{a^{2}}} + \sqrt{\frac{i \, \sqrt{a^{2} x^{2} + 1}}{a x + i}}\right ) + 2 i \, \sqrt{a^{2} x^{2} + 1} \sqrt{\frac{i \, \sqrt{a^{2} x^{2} + 1}}{a x + i}}}{2 \, a} \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 \left (\frac{i \, a x + 1}{\sqrt{a^{2} x^{2} + 1}}\right )^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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