Optimal. Leaf size=170 \[ \frac{11 a^2 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{24 x}+\frac{3}{8} i a^3 \tan ^{-1}\left (\frac{\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )+\frac{3}{8} i a^3 \tanh ^{-1}\left (\frac{\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )-\frac{5 i a (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{12 x^2}-\frac{(1-i a x)^{3/4} \sqrt [4]{1+i a x}}{3 x^3} \]
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Rubi [A] time = 0.0628448, antiderivative size = 170, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {5062, 99, 151, 12, 93, 212, 206, 203} \[ \frac{11 a^2 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{24 x}+\frac{3}{8} i a^3 \tan ^{-1}\left (\frac{\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )+\frac{3}{8} i a^3 \tanh ^{-1}\left (\frac{\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )-\frac{5 i a (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{12 x^2}-\frac{(1-i a x)^{3/4} \sqrt [4]{1+i a x}}{3 x^3} \]
Antiderivative was successfully verified.
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Rule 5062
Rule 99
Rule 151
Rule 12
Rule 93
Rule 212
Rule 206
Rule 203
Rubi steps
\begin{align*} \int \frac{e^{\frac{1}{2} i \tan ^{-1}(a x)}}{x^4} \, dx &=\int \frac{\sqrt [4]{1+i a x}}{x^4 \sqrt [4]{1-i a x}} \, dx\\ &=-\frac{(1-i a x)^{3/4} \sqrt [4]{1+i a x}}{3 x^3}+\frac{1}{3} \int \frac{\frac{5 i a}{2}-2 a^2 x}{x^3 \sqrt [4]{1-i a x} (1+i a x)^{3/4}} \, dx\\ &=-\frac{(1-i a x)^{3/4} \sqrt [4]{1+i a x}}{3 x^3}-\frac{5 i a (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{12 x^2}-\frac{1}{6} \int \frac{\frac{11 a^2}{4}+\frac{5}{2} i a^3 x}{x^2 \sqrt [4]{1-i a x} (1+i a x)^{3/4}} \, dx\\ &=-\frac{(1-i a x)^{3/4} \sqrt [4]{1+i a x}}{3 x^3}-\frac{5 i a (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{12 x^2}+\frac{11 a^2 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{24 x}+\frac{1}{6} \int -\frac{9 i a^3}{8 x \sqrt [4]{1-i a x} (1+i a x)^{3/4}} \, dx\\ &=-\frac{(1-i a x)^{3/4} \sqrt [4]{1+i a x}}{3 x^3}-\frac{5 i a (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{12 x^2}+\frac{11 a^2 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{24 x}-\frac{1}{16} \left (3 i a^3\right ) \int \frac{1}{x \sqrt [4]{1-i a x} (1+i a x)^{3/4}} \, dx\\ &=-\frac{(1-i a x)^{3/4} \sqrt [4]{1+i a x}}{3 x^3}-\frac{5 i a (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{12 x^2}+\frac{11 a^2 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{24 x}-\frac{1}{4} \left (3 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{(1-i a x)^{3/4} \sqrt [4]{1+i a x}}{3 x^3}-\frac{5 i a (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{12 x^2}+\frac{11 a^2 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{24 x}+\frac{1}{8} \left (3 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 (3 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-i a x)^{3/4} \sqrt [4]{1+i a x}}{3 x^3}-\frac{5 i a (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{12 x^2}+\frac{11 a^2 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{24 x}+\frac{3}{8} i a^3 \tan ^{-1}\left (\frac{\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )+\frac{3}{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 = 0.0247147, size = 93, normalized size = 0.55 \[ \frac{(1-i a x)^{3/4} \left (6 i a^3 x^3 \text{Hypergeometric2F1}\left (\frac{3}{4},1,\frac{7}{4},\frac{a x+i}{-a x+i}\right )+11 i a^3 x^3+21 a^2 x^2-18 i a x-8\right )}{24 x^3 (1+i a x)^{3/4}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.14, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{4}}\sqrt{{(1+iax){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}}}}\, 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{\sqrt{\frac{i \, a x + 1}{\sqrt{a^{2} x^{2} + 1}}}}{x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.74123, size = 436, normalized size = 2.56 \begin{align*} \frac{9 i \, a^{3} x^{3} \log \left (\sqrt{\frac{i \, \sqrt{a^{2} x^{2} + 1}}{a x + i}} + 1\right ) - 9 \, a^{3} x^{3} \log \left (\sqrt{\frac{i \, \sqrt{a^{2} x^{2} + 1}}{a x + i}} + i\right ) + 9 \, a^{3} x^{3} \log \left (\sqrt{\frac{i \, \sqrt{a^{2} x^{2} + 1}}{a x + i}} - i\right ) - 9 i \, a^{3} x^{3} \log \left (\sqrt{\frac{i \, \sqrt{a^{2} x^{2} + 1}}{a x + i}} - 1\right ) +{\left (-22 i \, a^{3} x^{3} + 2 \, a^{2} x^{2} - 4 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{\sqrt{\frac{i \, a x + 1}{\sqrt{a^{2} x^{2} + 1}}}}{x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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