Optimal. Leaf size=130 \[ -\frac {\sqrt {-i a-i b x+1} \sqrt {i a+i b x+1}}{(1+i a) x}-\frac {2 i b \tanh ^{-1}\left (\frac {\sqrt {a+i} \sqrt {i a+i b x+1}}{\sqrt {-a+i} \sqrt {-i a-i b x+1}}\right )}{(-a+i)^{3/2} \sqrt {a+i}} \]
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Rubi [A] time = 0.06, antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5095, 94, 93, 208} \[ -\frac {\sqrt {-i a-i b x+1} \sqrt {i a+i b x+1}}{(1+i a) x}-\frac {2 i b \tanh ^{-1}\left (\frac {\sqrt {a+i} \sqrt {i a+i b x+1}}{\sqrt {-a+i} \sqrt {-i a-i b x+1}}\right )}{(-a+i)^{3/2} \sqrt {a+i}} \]
Antiderivative was successfully verified.
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Rule 93
Rule 94
Rule 208
Rule 5095
Rubi steps
\begin {align*} \int \frac {e^{-i \tan ^{-1}(a+b x)}}{x^2} \, dx &=\int \frac {\sqrt {1-i a-i b x}}{x^2 \sqrt {1+i a+i b x}} \, dx\\ &=-\frac {\sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{(1+i a) x}+\frac {b \int \frac {1}{x \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}} \, dx}{i-a}\\ &=-\frac {\sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{(1+i a) x}+\frac {(2 b) \operatorname {Subst}\left (\int \frac {1}{-1-i a-(-1+i a) x^2} \, dx,x,\frac {\sqrt {1+i a+i b x}}{\sqrt {1-i a-i b x}}\right )}{i-a}\\ &=-\frac {\sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{(1+i a) x}-\frac {2 i b \tanh ^{-1}\left (\frac {\sqrt {i+a} \sqrt {1+i a+i b x}}{\sqrt {i-a} \sqrt {1-i a-i b x}}\right )}{(i-a)^{3/2} \sqrt {i+a}}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 119, normalized size = 0.92 \[ i \left (\frac {\sqrt {a^2+2 a b x+b^2 x^2+1}}{(a-i) x}+\frac {2 b \tanh ^{-1}\left (\frac {\sqrt {-1-i a} \sqrt {-i (a+b x+i)}}{\sqrt {-1+i a} \sqrt {i a+i b x+1}}\right )}{(-1-i a)^{3/2} \sqrt {-1+i a}}\right ) \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 0.49, size = 227, normalized size = 1.75 \[ -\frac {2 \, {\left (a - i\right )} \sqrt {\frac {b^{2}}{a^{4} - 2 i \, a^{3} - 2 i \, a - 1}} x \log \left (-\frac {b^{2} x - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} b + {\left (a^{3} - i \, a^{2} + a - i\right )} \sqrt {\frac {b^{2}}{a^{4} - 2 i \, a^{3} - 2 i \, a - 1}}}{b}\right ) - 2 \, {\left (a - i\right )} \sqrt {\frac {b^{2}}{a^{4} - 2 i \, a^{3} - 2 i \, a - 1}} x \log \left (-\frac {b^{2} x - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} b - {\left (a^{3} - i \, a^{2} + a - i\right )} \sqrt {\frac {b^{2}}{a^{4} - 2 i \, a^{3} - 2 i \, a - 1}}}{b}\right ) - 2 i \, b x - 2 i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{{\left (2 \, a - 2 i\right )} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 147, normalized size = 1.13 \[ \frac {b \log \left (\frac {{\left | 2 \, x {\left | b \right |} - 2 \, \sqrt {{\left (b x + a\right )}^{2} + 1} - 2 \, \sqrt {a^{2} + 1} \right |}}{{\left | 2 \, x {\left | b \right |} - 2 \, \sqrt {{\left (b x + a\right )}^{2} + 1} + 2 \, \sqrt {a^{2} + 1} \right |}}\right )}{\sqrt {a^{2} + 1} {\left (a - i\right )}} + \frac {2 \, {\left ({\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )} a b + a^{2} {\left | b \right |} + {\left | b \right |}\right )}}{{\left ({\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )}^{2} - a^{2} - 1\right )} {\left (a i + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.18, size = 602, normalized size = 4.63 \[ \frac {i b \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{\left (i-a \right )^{2}}+\frac {i b^{2} a \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{\left (i-a \right )^{2} \sqrt {b^{2}}}-\frac {i b \sqrt {a^{2}+1}\, \ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{x}\right )}{\left (i-a \right )^{2}}-\frac {i \left (b^{2} x^{2}+2 a b x +a^{2}+1\right )^{\frac {3}{2}}}{\left (i-a \right ) \left (a^{2}+1\right ) x}+\frac {2 i a b \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{\left (i-a \right ) \left (a^{2}+1\right )}+\frac {i a^{2} b^{2} \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{\left (i-a \right ) \left (a^{2}+1\right ) \sqrt {b^{2}}}-\frac {i a b \ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{x}\right )}{\left (i-a \right ) \sqrt {a^{2}+1}}+\frac {i b^{2} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, x}{\left (i-a \right ) \left (a^{2}+1\right )}+\frac {i b^{2} \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{\left (i-a \right ) \left (a^{2}+1\right ) \sqrt {b^{2}}}-\frac {i b \sqrt {\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )}}{\left (i-a \right )^{2}}+\frac {b^{2} \ln \left (\frac {i b +\left (x -\frac {i-a}{b}\right ) b^{2}}{\sqrt {b^{2}}}+\sqrt {\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )}\right )}{\left (i-a \right )^{2} \sqrt {b^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {{\left (b x + a\right )}^{2} + 1}}{{\left (i \, b x + i \, a + 1\right )} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sqrt {{\left (a+b\,x\right )}^2+1}}{x^2\,\left (1+a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ - i \int \frac {\sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}{a x^{2} + b x^{3} - i x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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