Optimal. Leaf size=202 \[ \frac{15 a^2 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{32 x^2}-\frac{63 i a^3 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{64 x}-\frac{123}{64} a^4 \tan ^{-1}\left (\frac{\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )-\frac{123}{64} a^4 \tanh ^{-1}\left (\frac{\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )+\frac{3 i a (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{8 x^3}-\frac{(1-i a x)^{3/4} \sqrt [4]{1+i a x}}{4 x^4} \]
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Rubi [A] time = 0.0812171, antiderivative size = 202, normalized size of antiderivative = 1., number of steps used = 10, 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} \[ \frac{15 a^2 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{32 x^2}-\frac{63 i a^3 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{64 x}-\frac{123}{64} a^4 \tan ^{-1}\left (\frac{\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )-\frac{123}{64} a^4 \tanh ^{-1}\left (\frac{\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )+\frac{3 i a (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{8 x^3}-\frac{(1-i a x)^{3/4} \sqrt [4]{1+i a x}}{4 x^4} \]
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{3}{2} i \tan ^{-1}(a x)}}{x^5} \, dx &=\int \frac{(1-i a x)^{3/4}}{x^5 (1+i a x)^{3/4}} \, dx\\ &=-\frac{(1-i a x)^{3/4} \sqrt [4]{1+i a x}}{4 x^4}+\frac{1}{4} \int \frac{-\frac{9 i a}{2}-3 a^2 x}{x^4 \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}}{4 x^4}+\frac{3 i a (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{8 x^3}-\frac{1}{12} \int \frac{\frac{45 a^2}{4}-9 i a^3 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}}{4 x^4}+\frac{3 i a (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{8 x^3}+\frac{15 a^2 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{32 x^2}+\frac{1}{24} \int \frac{\frac{189 i a^3}{8}+\frac{45 a^4 x}{4}}{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}}{4 x^4}+\frac{3 i a (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{8 x^3}+\frac{15 a^2 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{32 x^2}-\frac{63 i a^3 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{64 x}-\frac{1}{24} \int -\frac{369 a^4}{16 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}}{4 x^4}+\frac{3 i a (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{8 x^3}+\frac{15 a^2 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{32 x^2}-\frac{63 i a^3 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{64 x}+\frac{1}{128} \left (123 a^4\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}}{4 x^4}+\frac{3 i a (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{8 x^3}+\frac{15 a^2 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{32 x^2}-\frac{63 i a^3 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{64 x}+\frac{1}{32} \left (123 a^4\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}}{4 x^4}+\frac{3 i a (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{8 x^3}+\frac{15 a^2 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{32 x^2}-\frac{63 i a^3 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{64 x}-\frac{1}{64} \left (123 a^4\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}{64} \left (123 a^4\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}}{4 x^4}+\frac{3 i a (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{8 x^3}+\frac{15 a^2 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{32 x^2}-\frac{63 i a^3 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{64 x}-\frac{123}{64} a^4 \tan ^{-1}\left (\frac{\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )-\frac{123}{64} a^4 \tanh ^{-1}\left (\frac{\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )\\ \end{align*}
Mathematica [C] time = 0.0323301, size = 99, normalized size = 0.49 \[ \frac{(1-i a x)^{3/4} \left (-82 a^4 x^4 \text{Hypergeometric2F1}\left (\frac{3}{4},1,\frac{7}{4},\frac{a x+i}{-a x+i}\right )+63 a^4 x^4-33 i a^3 x^3+6 a^2 x^2+8 i a x-16\right )}{64 x^4 (1+i a x)^{3/4}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.158, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{5}} \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 \frac{1}{x^{5} \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 = 2.10922, size = 473, normalized size = 2.34 \begin{align*} -\frac{123 \, a^{4} x^{4} \log \left (\sqrt{\frac{i \, \sqrt{a^{2} x^{2} + 1}}{a x + i}} + 1\right ) + 123 i \, a^{4} x^{4} \log \left (\sqrt{\frac{i \, \sqrt{a^{2} x^{2} + 1}}{a x + i}} + i\right ) - 123 i \, a^{4} x^{4} \log \left (\sqrt{\frac{i \, \sqrt{a^{2} x^{2} + 1}}{a x + i}} - i\right ) - 123 \, a^{4} x^{4} \log \left (\sqrt{\frac{i \, \sqrt{a^{2} x^{2} + 1}}{a x + i}} - 1\right ) +{\left (126 \, a^{4} x^{4} + 186 i \, a^{3} x^{3} - 108 \, a^{2} x^{2} - 80 i \, a x + 32\right )} \sqrt{\frac{i \, \sqrt{a^{2} x^{2} + 1}}{a x + i}}}{128 \, x^{4}} \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^{5} \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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