Optimal. Leaf size=203 \[ -\frac{287 i a^3 \sqrt [4]{1+i a x}}{24 \sqrt [4]{1-i a x}}+\frac{61 a^2 \sqrt [4]{1+i a x}}{24 x \sqrt [4]{1-i a x}}+\frac{55}{8} i a^3 \tan ^{-1}\left (\frac{\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )+\frac{55}{8} i a^3 \tanh ^{-1}\left (\frac{\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )-\frac{13 i a \sqrt [4]{1+i a x}}{12 x^2 \sqrt [4]{1-i a x}}-\frac{\sqrt [4]{1+i a x}}{3 x^3 \sqrt [4]{1-i a x}} \]
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Rubi [A] time = 0.0819834, antiderivative size = 203, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.562, Rules used = {5062, 98, 151, 155, 12, 93, 212, 206, 203} \[ -\frac{287 i a^3 \sqrt [4]{1+i a x}}{24 \sqrt [4]{1-i a x}}+\frac{61 a^2 \sqrt [4]{1+i a x}}{24 x \sqrt [4]{1-i a x}}+\frac{55}{8} i a^3 \tan ^{-1}\left (\frac{\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )+\frac{55}{8} i a^3 \tanh ^{-1}\left (\frac{\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )-\frac{13 i a \sqrt [4]{1+i a x}}{12 x^2 \sqrt [4]{1-i a x}}-\frac{\sqrt [4]{1+i a x}}{3 x^3 \sqrt [4]{1-i a x}} \]
Antiderivative was successfully verified.
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Rule 5062
Rule 98
Rule 151
Rule 155
Rule 12
Rule 93
Rule 212
Rule 206
Rule 203
Rubi steps
\begin{align*} \int \frac{e^{\frac{5}{2} i \tan ^{-1}(a x)}}{x^4} \, dx &=\int \frac{(1+i a x)^{5/4}}{x^4 (1-i a x)^{5/4}} \, dx\\ &=-\frac{\sqrt [4]{1+i a x}}{3 x^3 \sqrt [4]{1-i a x}}-\frac{1}{3} \int \frac{-\frac{13 i a}{2}+6 a^2 x}{x^3 (1-i a x)^{5/4} (1+i a x)^{3/4}} \, dx\\ &=-\frac{\sqrt [4]{1+i a x}}{3 x^3 \sqrt [4]{1-i a x}}-\frac{13 i a \sqrt [4]{1+i a x}}{12 x^2 \sqrt [4]{1-i a x}}+\frac{1}{6} \int \frac{-\frac{61 a^2}{4}-13 i a^3 x}{x^2 (1-i a x)^{5/4} (1+i a x)^{3/4}} \, dx\\ &=-\frac{\sqrt [4]{1+i a x}}{3 x^3 \sqrt [4]{1-i a x}}-\frac{13 i a \sqrt [4]{1+i a x}}{12 x^2 \sqrt [4]{1-i a x}}+\frac{61 a^2 \sqrt [4]{1+i a x}}{24 x \sqrt [4]{1-i a x}}-\frac{1}{6} \int \frac{\frac{165 i a^3}{8}-\frac{61 a^4 x}{4}}{x (1-i a x)^{5/4} (1+i a x)^{3/4}} \, dx\\ &=-\frac{287 i a^3 \sqrt [4]{1+i a x}}{24 \sqrt [4]{1-i a x}}-\frac{\sqrt [4]{1+i a x}}{3 x^3 \sqrt [4]{1-i a x}}-\frac{13 i a \sqrt [4]{1+i a x}}{12 x^2 \sqrt [4]{1-i a x}}+\frac{61 a^2 \sqrt [4]{1+i a x}}{24 x \sqrt [4]{1-i a x}}-\frac{i \int \frac{165 a^4}{16 x \sqrt [4]{1-i a x} (1+i a x)^{3/4}} \, dx}{3 a}\\ &=-\frac{287 i a^3 \sqrt [4]{1+i a x}}{24 \sqrt [4]{1-i a x}}-\frac{\sqrt [4]{1+i a x}}{3 x^3 \sqrt [4]{1-i a x}}-\frac{13 i a \sqrt [4]{1+i a x}}{12 x^2 \sqrt [4]{1-i a x}}+\frac{61 a^2 \sqrt [4]{1+i a x}}{24 x \sqrt [4]{1-i a x}}-\frac{1}{16} \left (55 i a^3\right ) \int \frac{1}{x \sqrt [4]{1-i a x} (1+i a x)^{3/4}} \, dx\\ &=-\frac{287 i a^3 \sqrt [4]{1+i a x}}{24 \sqrt [4]{1-i a x}}-\frac{\sqrt [4]{1+i a x}}{3 x^3 \sqrt [4]{1-i a x}}-\frac{13 i a \sqrt [4]{1+i a x}}{12 x^2 \sqrt [4]{1-i a x}}+\frac{61 a^2 \sqrt [4]{1+i a x}}{24 x \sqrt [4]{1-i a x}}-\frac{1}{4} \left (55 i a^3\right ) \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 )\\ &=-\frac{287 i a^3 \sqrt [4]{1+i a x}}{24 \sqrt [4]{1-i a x}}-\frac{\sqrt [4]{1+i a x}}{3 x^3 \sqrt [4]{1-i a x}}-\frac{13 i a \sqrt [4]{1+i a x}}{12 x^2 \sqrt [4]{1-i a x}}+\frac{61 a^2 \sqrt [4]{1+i a x}}{24 x \sqrt [4]{1-i a x}}+\frac{1}{8} \left (55 i a^3\right ) \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\frac{\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )+\frac{1}{8} \left (55 i a^3\right ) \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\frac{\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )\\ &=-\frac{287 i a^3 \sqrt [4]{1+i a x}}{24 \sqrt [4]{1-i a x}}-\frac{\sqrt [4]{1+i a x}}{3 x^3 \sqrt [4]{1-i a x}}-\frac{13 i a \sqrt [4]{1+i a x}}{12 x^2 \sqrt [4]{1-i a x}}+\frac{61 a^2 \sqrt [4]{1+i a x}}{24 x \sqrt [4]{1-i a x}}+\frac{55}{8} i a^3 \tan ^{-1}\left (\frac{\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )+\frac{55}{8} i a^3 \tanh ^{-1}\left (\frac{\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )\\ \end{align*}
Mathematica [C] time = 0.0298097, size = 106, normalized size = 0.52 \[ \frac{110 a^3 x^3 (a x+i) \text{Hypergeometric2F1}\left (\frac{3}{4},1,\frac{7}{4},\frac{a x+i}{-a x+i}\right )+287 a^4 x^4-226 i a^3 x^3+87 a^2 x^2-34 i a x-8}{24 x^3 \sqrt [4]{1-i a x} (1+i a x)^{3/4}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.155, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{4}} \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 \frac{\left (\frac{i \, a x + 1}{\sqrt{a^{2} x^{2} + 1}}\right )^{\frac{5}{2}}}{x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.7023, size = 452, normalized size = 2.23 \begin{align*} \frac{165 i \, a^{3} x^{3} \log \left (\sqrt{\frac{i \, \sqrt{a^{2} x^{2} + 1}}{a x + i}} + 1\right ) - 165 \, a^{3} x^{3} \log \left (\sqrt{\frac{i \, \sqrt{a^{2} x^{2} + 1}}{a x + i}} + i\right ) + 165 \, a^{3} x^{3} \log \left (\sqrt{\frac{i \, \sqrt{a^{2} x^{2} + 1}}{a x + i}} - i\right ) - 165 i \, a^{3} x^{3} \log \left (\sqrt{\frac{i \, \sqrt{a^{2} x^{2} + 1}}{a x + i}} - 1\right ) +{\left (-574 i \, a^{3} x^{3} + 122 \, a^{2} x^{2} - 52 i \, a x - 16\right )} \sqrt{\frac{i \, \sqrt{a^{2} x^{2} + 1}}{a x + i}}}{48 \, x^{3}} \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{\left (\frac{i \, a x + 1}{\sqrt{a^{2} x^{2} + 1}}\right )^{\frac{5}{2}}}{x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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