Optimal. Leaf size=121 \[ -\frac {(1+i a x)^{5/4}}{x \sqrt [4]{1-i a x}}+\frac {10 i a \sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}-5 i a \tan ^{-1}\left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )-5 i a \tanh ^{-1}\left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {5062, 94, 93, 212, 206, 203} \[ -\frac {(1+i a x)^{5/4}}{x \sqrt [4]{1-i a x}}+\frac {10 i a \sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}-5 i a \tan ^{-1}\left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )-5 i a \tanh ^{-1}\left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 93
Rule 94
Rule 203
Rule 206
Rule 212
Rule 5062
Rubi steps
\begin {align*} \int \frac {e^{\frac {5}{2} i \tan ^{-1}(a x)}}{x^2} \, dx &=\int \frac {(1+i a x)^{5/4}}{x^2 (1-i a x)^{5/4}} \, dx\\ &=-\frac {(1+i a x)^{5/4}}{x \sqrt [4]{1-i a x}}+\frac {1}{2} (5 i a) \int \frac {\sqrt [4]{1+i a x}}{x (1-i a x)^{5/4}} \, dx\\ &=\frac {10 i a \sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}-\frac {(1+i a x)^{5/4}}{x \sqrt [4]{1-i a x}}+\frac {1}{2} (5 i a) \int \frac {1}{x \sqrt [4]{1-i a x} (1+i a x)^{3/4}} \, dx\\ &=\frac {10 i a \sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}-\frac {(1+i a x)^{5/4}}{x \sqrt [4]{1-i a x}}+(10 i a) \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 {10 i a \sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}-\frac {(1+i a x)^{5/4}}{x \sqrt [4]{1-i a x}}-(5 i a) \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 )-(5 i a) \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 {10 i a \sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}-\frac {(1+i a x)^{5/4}}{x \sqrt [4]{1-i a x}}-5 i a \tan ^{-1}\left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )-5 i a \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 = 87, normalized size = 0.72 \[ \frac {-3 \left (9 a^2 x^2-8 i a x+1\right )-10 a x (a x+i) \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};\frac {a x+i}{i-a x}\right )}{3 x \sqrt [4]{1-i a x} (1+i a x)^{3/4}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.42, size = 152, normalized size = 1.26 \[ \frac {-5 i \, a x \log \left (\sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}} + 1\right ) + 5 \, a x \log \left (\sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}} + i\right ) - 5 \, a x \log \left (\sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}} - i\right ) + 5 i \, a x \log \left (\sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}} - 1\right ) - 2 \, {\left (-9 i \, a x + 1\right )} \sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}}}{2 \, x} \]
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 = 0.18, size = 0, normalized size = 0.00 \[ \int \frac {\left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )^{\frac {5}{2}}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (\frac {i \, a x + 1}{\sqrt {a^{2} x^{2} + 1}}\right )^{\frac {5}{2}}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (\frac {1+a\,x\,1{}\mathrm {i}}{\sqrt {a^2\,x^2+1}}\right )}^{5/2}}{x^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________