Optimal. Leaf size=170 \[ -\frac {17}{8} i a^3 \tan ^{-1}\left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )+\frac {17}{8} i a^3 \tanh ^{-1}\left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )+\frac {23 a^2 \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{24 x}-\frac {\sqrt [4]{1-i a x} (1+i a x)^{3/4}}{3 x^3}-\frac {7 i a \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{12 x^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.06, antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5062, 99, 151, 12, 93, 298, 203, 206} \[ \frac {23 a^2 \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{24 x}-\frac {17}{8} i a^3 \tan ^{-1}\left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )+\frac {17}{8} i a^3 \tanh ^{-1}\left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )-\frac {7 i a \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{12 x^2}-\frac {\sqrt [4]{1-i a x} (1+i a x)^{3/4}}{3 x^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 93
Rule 99
Rule 151
Rule 203
Rule 206
Rule 298
Rule 5062
Rubi steps
\begin {align*} \int \frac {e^{\frac {3}{2} i \tan ^{-1}(a x)}}{x^4} \, dx &=\int \frac {(1+i a x)^{3/4}}{x^4 (1-i a x)^{3/4}} \, dx\\ &=-\frac {\sqrt [4]{1-i a x} (1+i a x)^{3/4}}{3 x^3}+\frac {1}{3} \int \frac {\frac {7 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 {7 i a \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{12 x^2}-\frac {1}{6} \int \frac {\frac {23 a^2}{4}+\frac {7}{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 {7 i a \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{12 x^2}+\frac {23 a^2 \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{24 x}+\frac {1}{6} \int -\frac {51 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 {7 i a \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{12 x^2}+\frac {23 a^2 \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{24 x}-\frac {1}{16} \left (17 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 {7 i a \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{12 x^2}+\frac {23 a^2 \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{24 x}-\frac {1}{4} \left (17 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 {7 i a \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{12 x^2}+\frac {23 a^2 \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{24 x}+\frac {1}{8} \left (17 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 (17 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 {7 i a \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{12 x^2}+\frac {23 a^2 \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{24 x}-\frac {17}{8} i a^3 \tan ^{-1}\left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )+\frac {17}{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 = 0.02, size = 93, normalized size = 0.55 \[ \frac {\sqrt [4]{1-i a x} \left (102 i a^3 x^3 \, _2F_1\left (\frac {1}{4},1;\frac {5}{4};\frac {a x+i}{i-a x}\right )+23 i a^3 x^3+37 a^2 x^2-22 i a x-8\right )}{24 x^3 \sqrt [4]{1+i a x}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.49, size = 186, normalized size = 1.09 \[ \frac {51 i \, a^{3} x^{3} \log \left (\sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}} + 1\right ) + 51 \, a^{3} x^{3} \log \left (\sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}} + i\right ) - 51 \, a^{3} x^{3} \log \left (\sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}} - i\right ) - 51 i \, a^{3} x^{3} \log \left (\sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}} - 1\right ) + {\left (46 \, a^{2} x^{2} - 28 i \, a x - 16\right )} \sqrt {a^{2} x^{2} + 1} \sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}}}{48 \, x^{3}} \]
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 = 0.17, size = 0, normalized size = 0.00 \[ \int \frac {\left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )^{\frac {3}{2}}}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (\frac {i \, a x + 1}{\sqrt {a^{2} x^{2} + 1}}\right )^{\frac {3}{2}}}{x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (\frac {1+a\,x\,1{}\mathrm {i}}{\sqrt {a^2\,x^2+1}}\right )}^{3/2}}{x^4} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (\frac {i \left (a x - i\right )}{\sqrt {a^{2} x^{2} + 1}}\right )^{\frac {3}{2}}}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________