Optimal. Leaf size=178 \[ -\frac {(-i a-i b x+1)^{3/2}}{(1+i a) x \sqrt {i a+i b x+1}}+\frac {6 i b \sqrt {-i a-i b x+1}}{(-a+i)^2 \sqrt {i a+i b x+1}}-\frac {6 i \sqrt {a+i} b \tanh ^{-1}\left (\frac {\sqrt {a+i} \sqrt {i a+i b x+1}}{\sqrt {-a+i} \sqrt {-i a-i b x+1}}\right )}{(-a+i)^{5/2}} \]
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Rubi [A] time = 0.09, antiderivative size = 178, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5095, 94, 93, 208} \[ -\frac {(-i a-i b x+1)^{3/2}}{(1+i a) x \sqrt {i a+i b x+1}}+\frac {6 i b \sqrt {-i a-i b x+1}}{(-a+i)^2 \sqrt {i a+i b x+1}}-\frac {6 i \sqrt {a+i} b \tanh ^{-1}\left (\frac {\sqrt {a+i} \sqrt {i a+i b x+1}}{\sqrt {-a+i} \sqrt {-i a-i b x+1}}\right )}{(-a+i)^{5/2}} \]
Antiderivative was successfully verified.
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Rule 93
Rule 94
Rule 208
Rule 5095
Rubi steps
\begin {align*} \int \frac {e^{-3 i \tan ^{-1}(a+b x)}}{x^2} \, dx &=\int \frac {(1-i a-i b x)^{3/2}}{x^2 (1+i a+i b x)^{3/2}} \, dx\\ &=-\frac {(1-i a-i b x)^{3/2}}{(1+i a) x \sqrt {1+i a+i b x}}+\frac {(3 b) \int \frac {\sqrt {1-i a-i b x}}{x (1+i a+i b x)^{3/2}} \, dx}{i-a}\\ &=\frac {6 i b \sqrt {1-i a-i b x}}{(i-a)^2 \sqrt {1+i a+i b x}}-\frac {(1-i a-i b x)^{3/2}}{(1+i a) x \sqrt {1+i a+i b x}}+\frac {(3 (i+a) b) \int \frac {1}{x \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}} \, dx}{(i-a)^2}\\ &=\frac {6 i b \sqrt {1-i a-i b x}}{(i-a)^2 \sqrt {1+i a+i b x}}-\frac {(1-i a-i b x)^{3/2}}{(1+i a) x \sqrt {1+i a+i b x}}+\frac {(6 (i+a) 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)^2}\\ &=\frac {6 i b \sqrt {1-i a-i b x}}{(i-a)^2 \sqrt {1+i a+i b x}}-\frac {(1-i a-i b x)^{3/2}}{(1+i a) x \sqrt {1+i a+i b x}}-\frac {6 i \sqrt {i+a} 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)^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 145, normalized size = 0.81 \[ \frac {\frac {\sqrt {-i (a+b x+i)} \left (a^2+a b x+5 i b x+1\right )}{x \sqrt {i a+i b x+1}}-\frac {6 i \sqrt {-1+i a} b \tanh ^{-1}\left (\frac {\sqrt {-1-i a} \sqrt {-i (a+b x+i)}}{\sqrt {-1+i a} \sqrt {i a+i b x+1}}\right )}{\sqrt {-1-i a}}}{(a-i)^2} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 0.46, size = 404, normalized size = 2.27 \[ -\frac {2 \, {\left (i \, a - 5\right )} b^{2} x^{2} - {\left (-2 i \, a^{2} + 8 \, a - 10 i\right )} b x - {\left ({\left (a^{2} - 2 i \, a - 1\right )} b x^{2} + {\left (a^{3} - 3 i \, a^{2} - 3 \, a + i\right )} x\right )} \sqrt {\frac {{\left (36 \, a + 36 i\right )} b^{2}}{a^{5} - 5 i \, a^{4} - 10 \, a^{3} + 10 i \, a^{2} + 5 \, a - i}} \log \left (-\frac {6 \, b^{2} x + {\left (a^{3} - 3 i \, a^{2} - 3 \, a + i\right )} \sqrt {\frac {{\left (36 \, a + 36 i\right )} b^{2}}{a^{5} - 5 i \, a^{4} - 10 \, a^{3} + 10 i \, a^{2} + 5 \, a - i}} - 6 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} b}{6 \, b}\right ) + {\left ({\left (a^{2} - 2 i \, a - 1\right )} b x^{2} + {\left (a^{3} - 3 i \, a^{2} - 3 \, a + i\right )} x\right )} \sqrt {\frac {{\left (36 \, a + 36 i\right )} b^{2}}{a^{5} - 5 i \, a^{4} - 10 \, a^{3} + 10 i \, a^{2} + 5 \, a - i}} \log \left (-\frac {6 \, b^{2} x - {\left (a^{3} - 3 i \, a^{2} - 3 \, a + i\right )} \sqrt {\frac {{\left (36 \, a + 36 i\right )} b^{2}}{a^{5} - 5 i \, a^{4} - 10 \, a^{3} + 10 i \, a^{2} + 5 \, a - i}} - 6 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} b}{6 \, b}\right ) + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (2 \, {\left (i \, a - 5\right )} b x + 2 i \, a^{2} + 2 i\right )}}{2 \, {\left (a^{2} - 2 i \, a - 1\right )} b x^{2} + {\left (2 \, a^{3} - 6 i \, a^{2} - 6 \, a + 2 i\right )} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {undef} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.19, size = 1917, normalized size = 10.77 \[ \text {result too large to display} \]
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 \[ \int \frac {{\left ({\left (b x + a\right )}^{2} + 1\right )}^{\frac {3}{2}}}{{\left (i \, b x + i \, a + 1\right )}^{3} x^{2}}\,{d x} \]
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 \[ \int \frac {{\left ({\left (a+b\,x\right )}^2+1\right )}^{3/2}}{x^2\,{\left (1+a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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