Optimal. Leaf size=170 \[ \frac{11 a^2 \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{24 x}+\frac{3}{8} i a^3 \tan ^{-1}\left (\frac{\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )-\frac{3}{8} i a^3 \tanh ^{-1}\left (\frac{\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )+\frac{5 i a \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{12 x^2}-\frac{\sqrt [4]{1-i a x} (1+i a x)^{3/4}}{3 x^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0690398, 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, 298, 203, 206} \[ \frac{11 a^2 \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{24 x}+\frac{3}{8} i a^3 \tan ^{-1}\left (\frac{\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )-\frac{3}{8} i a^3 \tanh ^{-1}\left (\frac{\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )+\frac{5 i a \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{12 x^2}-\frac{\sqrt [4]{1-i a x} (1+i a x)^{3/4}}{3 x^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5062
Rule 99
Rule 151
Rule 12
Rule 93
Rule 298
Rule 203
Rule 206
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{\sqrt [4]{1-i a x} (1+i a x)^{3/4}}{3 x^3}+\frac{1}{3} \int \frac{-\frac{5 i a}{2}-2 a^2 x}{x^3 (1-i a x)^{3/4} \sqrt [4]{1+i a x}} \, dx\\ &=-\frac{\sqrt [4]{1-i a x} (1+i a x)^{3/4}}{3 x^3}+\frac{5 i a \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{12 x^2}-\frac{1}{6} \int \frac{\frac{11 a^2}{4}-\frac{5}{2} i a^3 x}{x^2 (1-i a x)^{3/4} \sqrt [4]{1+i a x}} \, dx\\ &=-\frac{\sqrt [4]{1-i a x} (1+i a x)^{3/4}}{3 x^3}+\frac{5 i a \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{12 x^2}+\frac{11 a^2 \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{24 x}+\frac{1}{6} \int \frac{9 i a^3}{8 x (1-i a x)^{3/4} \sqrt [4]{1+i a x}} \, dx\\ &=-\frac{\sqrt [4]{1-i a x} (1+i a x)^{3/4}}{3 x^3}+\frac{5 i a \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{12 x^2}+\frac{11 a^2 \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{24 x}+\frac{1}{16} \left (3 i a^3\right ) \int \frac{1}{x (1-i a x)^{3/4} \sqrt [4]{1+i a x}} \, dx\\ &=-\frac{\sqrt [4]{1-i a x} (1+i a x)^{3/4}}{3 x^3}+\frac{5 i a \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{12 x^2}+\frac{11 a^2 \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{24 x}+\frac{1}{4} \left (3 i a^3\right ) \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 )\\ &=-\frac{\sqrt [4]{1-i a x} (1+i a x)^{3/4}}{3 x^3}+\frac{5 i a \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{12 x^2}+\frac{11 a^2 \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{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{\sqrt [4]{1-i a x} (1+i a x)^{3/4}}{3 x^3}+\frac{5 i a \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{12 x^2}+\frac{11 a^2 \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{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 = 0.0237386, size = 92, normalized size = 0.54 \[ \frac{\sqrt [4]{1-i a x} \left (-18 i a^3 x^3 \text{Hypergeometric2F1}\left (\frac{1}{4},1,\frac{5}{4},\frac{a x+i}{-a x+i}\right )+11 i a^3 x^3+a^2 x^2+2 i a x-8\right )}{24 x^3 \sqrt [4]{1+i a x}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.154, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{4}}{\frac{1}{\sqrt{{(1+iax){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}}}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{4} \sqrt{\frac{i \, a x + 1}{\sqrt{a^{2} x^{2} + 1}}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.09924, size = 443, normalized size = 2.61 \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 \, a^{2} x^{2} + 20 i \, a x - 16\right )} \sqrt{a^{2} x^{2} + 1} \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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{4} \sqrt{\frac{i \, a x + 1}{\sqrt{a^{2} x^{2} + 1}}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]