Optimal. Leaf size=210 \[ -\frac {\sqrt [4]{1-i (a+b x)} (-a-b x+i)}{(-a+i) x \sqrt [4]{1+i (a+b x)}}-\frac {i b \tan ^{-1}\left (\frac {\sqrt [4]{-a+i} \sqrt [4]{1-i (a+b x)}}{\sqrt [4]{a+i} \sqrt [4]{1+i (a+b x)}}\right )}{(-a+i)^{5/4} (a+i)^{3/4}}-\frac {i b \tanh ^{-1}\left (\frac {\sqrt [4]{-a+i} \sqrt [4]{1-i (a+b x)}}{\sqrt [4]{a+i} \sqrt [4]{1+i (a+b x)}}\right )}{(-a+i)^{5/4} (a+i)^{3/4}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.10, antiderivative size = 210, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {5094, 288, 212, 208, 205} \[ -\frac {\sqrt [4]{1-i (a+b x)} (-a-b x+i)}{(-a+i) x \sqrt [4]{1+i (a+b x)}}-\frac {i b \tan ^{-1}\left (\frac {\sqrt [4]{-a+i} \sqrt [4]{1-i (a+b x)}}{\sqrt [4]{a+i} \sqrt [4]{1+i (a+b x)}}\right )}{(-a+i)^{5/4} (a+i)^{3/4}}-\frac {i b \tanh ^{-1}\left (\frac {\sqrt [4]{-a+i} \sqrt [4]{1-i (a+b x)}}{\sqrt [4]{a+i} \sqrt [4]{1+i (a+b x)}}\right )}{(-a+i)^{5/4} (a+i)^{3/4}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 205
Rule 208
Rule 212
Rule 288
Rule 5094
Rubi steps
\begin {align*} \int \frac {e^{-\frac {1}{2} i \tan ^{-1}(a+b x)}}{x^2} \, dx &=-\left ((8 i b) \operatorname {Subst}\left (\int \frac {x^4}{\left (1-i a-(1+i a) x^4\right )^2} \, dx,x,\frac {\sqrt [4]{1-i (a+b x)}}{\sqrt [4]{1+i (a+b x)}}\right )\right )\\ &=-\frac {(i-a-b x) \sqrt [4]{1-i (a+b x)}}{(i-a) x \sqrt [4]{1+i (a+b x)}}-\frac {(2 b) \operatorname {Subst}\left (\int \frac {1}{1-i a+(-1-i a) x^4} \, dx,x,\frac {\sqrt [4]{1-i (a+b x)}}{\sqrt [4]{1+i (a+b x)}}\right )}{i-a}\\ &=-\frac {(i-a-b x) \sqrt [4]{1-i (a+b x)}}{(i-a) x \sqrt [4]{1+i (a+b x)}}-\frac {b \operatorname {Subst}\left (\int \frac {1}{\sqrt {i+a}-\sqrt {i-a} x^2} \, dx,x,\frac {\sqrt [4]{1-i (a+b x)}}{\sqrt [4]{1+i (a+b x)}}\right )}{(1+i a) \sqrt {i+a}}-\frac {b \operatorname {Subst}\left (\int \frac {1}{\sqrt {i+a}+\sqrt {i-a} x^2} \, dx,x,\frac {\sqrt [4]{1-i (a+b x)}}{\sqrt [4]{1+i (a+b x)}}\right )}{(1+i a) \sqrt {i+a}}\\ &=-\frac {(i-a-b x) \sqrt [4]{1-i (a+b x)}}{(i-a) x \sqrt [4]{1+i (a+b x)}}-\frac {i b \tan ^{-1}\left (\frac {\sqrt [4]{i-a} \sqrt [4]{1-i (a+b x)}}{\sqrt [4]{i+a} \sqrt [4]{1+i (a+b x)}}\right )}{(i-a)^{5/4} (i+a)^{3/4}}-\frac {i b \tanh ^{-1}\left (\frac {\sqrt [4]{i-a} \sqrt [4]{1-i (a+b x)}}{\sqrt [4]{i+a} \sqrt [4]{1+i (a+b x)}}\right )}{(i-a)^{5/4} (i+a)^{3/4}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.03, size = 107, normalized size = 0.51 \[ -\frac {\sqrt [4]{-i (a+b x+i)} \left (-2 i b x \, _2F_1\left (\frac {1}{4},1;\frac {5}{4};\frac {a^2+b x a-i b x+1}{a^2+b x a+i b x+1}\right )+a^2+a b x+i b x+1\right )}{\left (a^2+1\right ) x \sqrt [4]{i a+i b x+1}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.47, size = 728, normalized size = 3.47 \[ \frac {\left (-\frac {b^{4}}{16 \, a^{8} - 32 i \, a^{7} + 32 \, a^{6} - 96 i \, a^{5} - 96 i \, a^{3} - 32 \, a^{2} - 32 i \, a - 16}\right )^{\frac {1}{4}} {\left (-i \, a - 1\right )} x \log \left (\frac {b^{3} \sqrt {\frac {i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b x + a + i}} + {\left (8 \, a^{6} - 16 i \, a^{5} + 8 \, a^{4} - 32 i \, a^{3} - 8 \, a^{2} - 16 i \, a - 8\right )} \left (-\frac {b^{4}}{16 \, a^{8} - 32 i \, a^{7} + 32 \, a^{6} - 96 i \, a^{5} - 96 i \, a^{3} - 32 \, a^{2} - 32 i \, a - 16}\right )^{\frac {3}{4}}}{b^{3}}\right ) + \left (-\frac {b^{4}}{16 \, a^{8} - 32 i \, a^{7} + 32 \, a^{6} - 96 i \, a^{5} - 96 i \, a^{3} - 32 \, a^{2} - 32 i \, a - 16}\right )^{\frac {1}{4}} {\left (i \, a + 1\right )} x \log \left (\frac {b^{3} \sqrt {\frac {i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b x + a + i}} - {\left (8 \, a^{6} - 16 i \, a^{5} + 8 \, a^{4} - 32 i \, a^{3} - 8 \, a^{2} - 16 i \, a - 8\right )} \left (-\frac {b^{4}}{16 \, a^{8} - 32 i \, a^{7} + 32 \, a^{6} - 96 i \, a^{5} - 96 i \, a^{3} - 32 \, a^{2} - 32 i \, a - 16}\right )^{\frac {3}{4}}}{b^{3}}\right ) - \left (-\frac {b^{4}}{16 \, a^{8} - 32 i \, a^{7} + 32 \, a^{6} - 96 i \, a^{5} - 96 i \, a^{3} - 32 \, a^{2} - 32 i \, a - 16}\right )^{\frac {1}{4}} {\left (a - i\right )} x \log \left (\frac {b^{3} \sqrt {\frac {i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b x + a + i}} + {\left (8 i \, a^{6} + 16 \, a^{5} + 8 i \, a^{4} + 32 \, a^{3} - 8 i \, a^{2} + 16 \, a - 8 i\right )} \left (-\frac {b^{4}}{16 \, a^{8} - 32 i \, a^{7} + 32 \, a^{6} - 96 i \, a^{5} - 96 i \, a^{3} - 32 \, a^{2} - 32 i \, a - 16}\right )^{\frac {3}{4}}}{b^{3}}\right ) + \left (-\frac {b^{4}}{16 \, a^{8} - 32 i \, a^{7} + 32 \, a^{6} - 96 i \, a^{5} - 96 i \, a^{3} - 32 \, a^{2} - 32 i \, a - 16}\right )^{\frac {1}{4}} {\left (a - i\right )} x \log \left (\frac {b^{3} \sqrt {\frac {i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b x + a + i}} + {\left (-8 i \, a^{6} - 16 \, a^{5} - 8 i \, a^{4} - 32 \, a^{3} + 8 i \, a^{2} - 16 \, a + 8 i\right )} \left (-\frac {b^{4}}{16 \, a^{8} - 32 i \, a^{7} + 32 \, a^{6} - 96 i \, a^{5} - 96 i \, a^{3} - 32 \, a^{2} - 32 i \, a - 16}\right )^{\frac {3}{4}}}{b^{3}}\right ) + i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} \sqrt {\frac {i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b x + a + i}}}{{\left (a - i\right )} 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 = 0.17, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {\frac {1+i \left (b x +a \right )}{\sqrt {1+\left (b x +a \right )^{2}}}}\, x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{2} \sqrt {\frac {i \, b x + i \, a + 1}{\sqrt {{\left (b x + a\right )}^{2} + 1}}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{x^2\,\sqrt {\frac {1+a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}}{\sqrt {{\left (a+b\,x\right )}^2+1}}}} \,d x \]
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]
________________________________________________________________________________________