Optimal. Leaf size=83 \[ \frac {-i-a}{2 (i-a) x^2}-\frac {2 i b}{(i-a)^2 x}-\frac {2 b^2 \log (x)}{(1+i a)^3}+\frac {2 b^2 \log (i-a-b x)}{(1+i a)^3} \]
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Rubi [A]
time = 0.04, antiderivative size = 81, normalized size of antiderivative = 0.98, number of steps
used = 3, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {5203, 78}
\begin {gather*} -\frac {2 b^2 \log (x)}{(1+i a)^3}+\frac {2 b^2 \log (-a-b x+i)}{(1+i a)^3}-\frac {2 i b}{(-a+i)^2 x}-\frac {a+i}{2 (-a+i) x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 78
Rule 5203
Rubi steps
\begin {align*} \int \frac {e^{-2 i \tan ^{-1}(a+b x)}}{x^3} \, dx &=\int \frac {1-i a-i b x}{x^3 (1+i a+i b x)} \, dx\\ &=\int \left (\frac {-i-a}{(-i+a) x^3}+\frac {2 i b}{(-i+a)^2 x^2}-\frac {2 i b^2}{(-i+a)^3 x}+\frac {2 i b^3}{(-i+a)^3 (-i+a+b x)}\right ) \, dx\\ &=-\frac {i+a}{2 (i-a) x^2}-\frac {2 i b}{(i-a)^2 x}-\frac {2 b^2 \log (x)}{(1+i a)^3}+\frac {2 b^2 \log (i-a-b x)}{(1+i a)^3}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 66, normalized size = 0.80 \begin {gather*} \frac {(-i+a) \left (1+a^2-4 i b x\right )-4 i b^2 x^2 \log (x)+4 i b^2 x^2 \log (i-a-b x)}{2 (-i+a)^3 x^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 109, normalized size = 1.31
method | result | size |
default | \(-\frac {-a^{4}+2 i a^{3}+2 i a +1}{2 \left (i-a \right )^{4} x^{2}}-\frac {2 b^{2} \left (i a +1\right ) \ln \left (x \right )}{\left (i-a \right )^{4}}-\frac {2 b \left (i a^{2}+2 a -i\right )}{\left (i-a \right )^{4} x}+\frac {2 b^{2} \left (i a +1\right ) \ln \left (-b x -a +i\right )}{\left (i-a \right )^{4}}\) | \(109\) |
risch | \(\frac {-\frac {2 i b x}{a^{2}-2 i a -1}+\frac {i+a}{2 a -2 i}}{x^{2}}+\frac {2 b^{2} \ln \left (\left (2 a^{4} b +4 a^{2} b +2 b \right ) x \right )}{i a^{3}+3 a^{2}-3 i a -1}-\frac {b^{2} \ln \left (4 a^{8} b^{2} x^{2}+8 a^{9} b x +4 a^{10}+16 a^{6} b^{2} x^{2}+32 a^{7} b x +20 a^{8}+24 a^{4} b^{2} x^{2}+48 a^{5} b x +40 a^{6}+16 a^{2} b^{2} x^{2}+32 a^{3} b x +40 a^{4}+4 b^{2} x^{2}+8 a b x +20 a^{2}+4\right )}{i a^{3}+3 a^{2}-3 i a -1}+\frac {2 i b^{2} \arctan \left (\frac {\left (-2 a^{4} b -4 a^{2} b -2 b \right ) x -2 a^{5}-4 a^{3}-2 a}{2 a^{4}+4 a^{2}+2}\right )}{i a^{3}+3 a^{2}-3 i a -1}\) | \(288\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than
twice the leaf count of optimal. 160 vs. \(2 (61) = 122\).
time = 0.28, size = 160, normalized size = 1.93 \begin {gather*} -\frac {2 \, {\left (-i \, a - 1\right )} b^{2} \log \left (i \, b x + i \, a + 1\right )}{a^{4} - 4 i \, a^{3} - 6 \, a^{2} + 4 i \, a + 1} - \frac {2 \, {\left (i \, a + 1\right )} b^{2} \log \left (x\right )}{a^{4} - 4 i \, a^{3} - 6 \, a^{2} + 4 i \, a + 1} + \frac {4 \, {\left (-i \, a - 1\right )} b^{2} x^{2} + a^{4} - 2 i \, a^{3} + {\left (a^{3} - 5 i \, a^{2} - 7 \, a + 3 i\right )} b x - 2 i \, a - 1}{2 \, {\left ({\left (a^{3} - 3 i \, a^{2} - 3 \, a + i\right )} b x^{3} + {\left (a^{4} - 4 i \, a^{3} - 6 \, a^{2} + 4 i \, a + 1\right )} x^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.30, size = 69, normalized size = 0.83 \begin {gather*} \frac {-4 i \, b^{2} x^{2} \log \left (x\right ) + 4 i \, b^{2} x^{2} \log \left (\frac {b x + a - i}{b}\right ) + a^{3} - 4 \, {\left (i \, a + 1\right )} b x - i \, a^{2} + a - i}{2 \, {\left (a^{3} - 3 i \, a^{2} - 3 \, a + i\right )} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 226 vs. \(2 (61) = 122\).
time = 0.51, size = 226, normalized size = 2.72 \begin {gather*} - \frac {2 i b^{2} \log {\left (- \frac {2 a^{4} b^{2}}{\left (a - i\right )^{3}} + \frac {8 i a^{3} b^{2}}{\left (a - i\right )^{3}} + \frac {12 a^{2} b^{2}}{\left (a - i\right )^{3}} + 2 a b^{2} - \frac {8 i a b^{2}}{\left (a - i\right )^{3}} + 4 b^{3} x - 2 i b^{2} - \frac {2 b^{2}}{\left (a - i\right )^{3}} \right )}}{\left (a - i\right )^{3}} + \frac {2 i b^{2} \log {\left (\frac {2 a^{4} b^{2}}{\left (a - i\right )^{3}} - \frac {8 i a^{3} b^{2}}{\left (a - i\right )^{3}} - \frac {12 a^{2} b^{2}}{\left (a - i\right )^{3}} + 2 a b^{2} + \frac {8 i a b^{2}}{\left (a - i\right )^{3}} + 4 b^{3} x - 2 i b^{2} + \frac {2 b^{2}}{\left (a - i\right )^{3}} \right )}}{\left (a - i\right )^{3}} - \frac {- a^{2} + 4 i b x - 1}{x^{2} \cdot \left (2 a^{2} - 4 i a - 2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 142 vs. \(2 (61) = 122\).
time = 0.44, size = 142, normalized size = 1.71 \begin {gather*} \frac {2 \, b^{3} \log \left (-\frac {i \, a}{i \, b x + i \, a + 1} - \frac {1}{i \, b x + i \, a + 1} + 1\right )}{i \, a^{3} b + 3 \, a^{2} b - 3 i \, a b - b} + \frac {\frac {i \, a b^{2} - 5 \, b^{2}}{-i \, a - 1} + \frac {2 i \, {\left (a b^{3} + 3 i \, b^{3}\right )}}{{\left (i \, b x + i \, a + 1\right )} b}}{2 \, {\left (a - i\right )}^{2} {\left (\frac {i \, a}{i \, b x + i \, a + 1} + \frac {1}{i \, b x + i \, a + 1} - 1\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.70, size = 156, normalized size = 1.88 \begin {gather*} \frac {\frac {a+1{}\mathrm {i}}{2\,\left (a-\mathrm {i}\right )}-\frac {b\,x\,2{}\mathrm {i}}{{\left (a-\mathrm {i}\right )}^2}}{x^2}-\frac {b^2\,\mathrm {atanh}\left (\frac {-a^3+a^2\,3{}\mathrm {i}+3\,a-\mathrm {i}}{{\left (a-\mathrm {i}\right )}^3}-\frac {x\,\left (2\,a^8\,b^2+8\,a^6\,b^2+12\,a^4\,b^2+8\,a^2\,b^2+2\,b^2\right )}{{\left (a-\mathrm {i}\right )}^3\,\left (b\,a^6+2{}\mathrm {i}\,b\,a^5+b\,a^4+4{}\mathrm {i}\,b\,a^3-b\,a^2+2{}\mathrm {i}\,b\,a-b\right )}\right )\,4{}\mathrm {i}}{{\left (a-\mathrm {i}\right )}^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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