Optimal. Leaf size=299 \[ -\frac{4 i (1+i a x)^{5/4}}{a \sqrt [4]{1-i a x}}-\frac{5 i (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{a}-\frac{5 i \log \left (\frac{\sqrt{1-i a x}}{\sqrt{1+i a x}}-\frac{\sqrt{2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+1\right )}{2 \sqrt{2} a}+\frac{5 i \log \left (\frac{\sqrt{1-i a x}}{\sqrt{1+i a x}}+\frac{\sqrt{2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+1\right )}{2 \sqrt{2} a}+\frac{5 i \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{\sqrt{2} a}-\frac{5 i \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{\sqrt{2} a} \]
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Rubi [A] time = 0.178278, antiderivative size = 299, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 11, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.917, Rules used = {5061, 47, 50, 63, 331, 297, 1162, 617, 204, 1165, 628} \[ -\frac{4 i (1+i a x)^{5/4}}{a \sqrt [4]{1-i a x}}-\frac{5 i (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{a}-\frac{5 i \log \left (\frac{\sqrt{1-i a x}}{\sqrt{1+i a x}}-\frac{\sqrt{2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+1\right )}{2 \sqrt{2} a}+\frac{5 i \log \left (\frac{\sqrt{1-i a x}}{\sqrt{1+i a x}}+\frac{\sqrt{2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+1\right )}{2 \sqrt{2} a}+\frac{5 i \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{\sqrt{2} a}-\frac{5 i \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{\sqrt{2} a} \]
Antiderivative was successfully verified.
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Rule 5061
Rule 47
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{5}{2} i \tan ^{-1}(a x)} \, dx &=\int \frac{(1+i a x)^{5/4}}{(1-i a x)^{5/4}} \, dx\\ &=-\frac{4 i (1+i a x)^{5/4}}{a \sqrt [4]{1-i a x}}-5 \int \frac{\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}} \, dx\\ &=-\frac{5 i (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{a}-\frac{4 i (1+i a x)^{5/4}}{a \sqrt [4]{1-i a x}}-\frac{5}{2} \int \frac{1}{\sqrt [4]{1-i a x} (1+i a x)^{3/4}} \, dx\\ &=-\frac{5 i (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{a}-\frac{4 i (1+i a x)^{5/4}}{a \sqrt [4]{1-i a x}}-\frac{(10 i) \operatorname{Subst}\left (\int \frac{x^2}{\left (2-x^4\right )^{3/4}} \, dx,x,\sqrt [4]{1-i a x}\right )}{a}\\ &=-\frac{5 i (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{a}-\frac{4 i (1+i a x)^{5/4}}{a \sqrt [4]{1-i a x}}-\frac{(10 i) \operatorname{Subst}\left (\int \frac{x^2}{1+x^4} \, dx,x,\frac{\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{a}\\ &=-\frac{5 i (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{a}-\frac{4 i (1+i a x)^{5/4}}{a \sqrt [4]{1-i a x}}+\frac{(5 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{(5 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{5 i (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{a}-\frac{4 i (1+i a x)^{5/4}}{a \sqrt [4]{1-i a x}}-\frac{(5 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{(5 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{(5 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{(5 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{5 i (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{a}-\frac{4 i (1+i a x)^{5/4}}{a \sqrt [4]{1-i a x}}-\frac{5 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{5 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{(5 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{(5 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{5 i (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{a}-\frac{4 i (1+i a x)^{5/4}}{a \sqrt [4]{1-i a x}}+\frac{5 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{5 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{5 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{5 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 = 0.0379242, size = 41, normalized size = 0.14 \[ -\frac{8 i e^{\frac{9}{2} i \tan ^{-1}(a x)} \text{Hypergeometric2F1}\left (2,\frac{9}{4},\frac{13}{4},-e^{2 i \tan ^{-1}(a x)}\right )}{9 a} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.13, size = 0, normalized size = 0. \begin{align*} \int \left ({(1+iax){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) ^{{\frac{5}{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{5}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.64298, size = 549, normalized size = 1.84 \begin{align*} -\frac{a \sqrt{\frac{25 i}{a^{2}}} \log \left (\frac{1}{5} i \, a \sqrt{\frac{25 i}{a^{2}}} + \sqrt{\frac{i \, \sqrt{a^{2} x^{2} + 1}}{a x + i}}\right ) - a \sqrt{\frac{25 i}{a^{2}}} \log \left (-\frac{1}{5} i \, a \sqrt{\frac{25 i}{a^{2}}} + \sqrt{\frac{i \, \sqrt{a^{2} x^{2} + 1}}{a x + i}}\right ) + a \sqrt{-\frac{25 i}{a^{2}}} \log \left (\frac{1}{5} i \, a \sqrt{-\frac{25 i}{a^{2}}} + \sqrt{\frac{i \, \sqrt{a^{2} x^{2} + 1}}{a x + i}}\right ) - a \sqrt{-\frac{25 i}{a^{2}}} \log \left (-\frac{1}{5} i \, a \sqrt{-\frac{25 i}{a^{2}}} + \sqrt{\frac{i \, \sqrt{a^{2} x^{2} + 1}}{a x + i}}\right ) +{\left (2 \, a x + 18 i\right )} \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{5}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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