Optimal. Leaf size=132 \[ \frac {1}{4} a^2 \tan ^{-1}\left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )+\frac {1}{4} a^2 \tanh ^{-1}\left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )-\frac {(1-i a x)^{3/4} (1+i a x)^{5/4}}{2 x^2}-\frac {i a (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{4 x} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {5062, 96, 94, 93, 212, 206, 203} \[ \frac {1}{4} a^2 \tan ^{-1}\left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )+\frac {1}{4} a^2 \tanh ^{-1}\left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )-\frac {(1-i a x)^{3/4} (1+i a x)^{5/4}}{2 x^2}-\frac {i a (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{4 x} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 93
Rule 94
Rule 96
Rule 203
Rule 206
Rule 212
Rule 5062
Rubi steps
\begin {align*} \int \frac {e^{\frac {1}{2} i \tan ^{-1}(a x)}}{x^3} \, dx &=\int \frac {\sqrt [4]{1+i a x}}{x^3 \sqrt [4]{1-i a x}} \, dx\\ &=-\frac {(1-i a x)^{3/4} (1+i a x)^{5/4}}{2 x^2}+\frac {1}{4} (i a) \int \frac {\sqrt [4]{1+i a x}}{x^2 \sqrt [4]{1-i a x}} \, dx\\ &=-\frac {i a (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{4 x}-\frac {(1-i a x)^{3/4} (1+i a x)^{5/4}}{2 x^2}-\frac {1}{8} a^2 \int \frac {1}{x \sqrt [4]{1-i a x} (1+i a x)^{3/4}} \, dx\\ &=-\frac {i a (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{4 x}-\frac {(1-i a x)^{3/4} (1+i a x)^{5/4}}{2 x^2}-\frac {1}{2} a^2 \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 {i a (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{4 x}-\frac {(1-i a x)^{3/4} (1+i a x)^{5/4}}{2 x^2}+\frac {1}{4} a^2 \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}{4} a^2 \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 {i a (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{4 x}-\frac {(1-i a x)^{3/4} (1+i a x)^{5/4}}{2 x^2}+\frac {1}{4} a^2 \tan ^{-1}\left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )+\frac {1}{4} a^2 \tanh ^{-1}\left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.02, size = 81, normalized size = 0.61 \[ \frac {(1-i a x)^{3/4} \left (2 a^2 x^2 \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};\frac {a x+i}{i-a x}\right )+9 a^2 x^2-15 i a x-6\right )}{12 x^2 (1+i a x)^{3/4}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.51, size = 175, normalized size = 1.33 \[ \frac {a^{2} x^{2} \log \left (\sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}} + 1\right ) + i \, a^{2} x^{2} \log \left (\sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}} + i\right ) - i \, a^{2} x^{2} \log \left (\sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}} - i\right ) - a^{2} x^{2} \log \left (\sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}} - 1\right ) - {\left (6 \, a^{2} x^{2} + 2 i \, a x + 4\right )} \sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}}}{8 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.16, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\frac {i \, a x + 1}{\sqrt {a^{2} x^{2} + 1}}}}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sqrt {\frac {1+a\,x\,1{}\mathrm {i}}{\sqrt {a^2\,x^2+1}}}}{x^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\frac {i \left (a x - i\right )}{\sqrt {a^{2} x^{2} + 1}}}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________