Optimal. Leaf size=104 \[ \frac {2 i b^3 \log (x)}{(-a+i)^4}-\frac {2 i b^3 \log (-a-b x+i)}{(-a+i)^4}+\frac {2 b^2}{(1+i a)^3 x}-\frac {i b}{(-a+i)^2 x^2}+\frac {-a-i}{3 (-a+i) x^3} \]
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Rubi [A] time = 0.06, antiderivative size = 102, 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 = {5095, 77} \[ \frac {2 b^2}{(1+i a)^3 x}+\frac {2 i b^3 \log (x)}{(-a+i)^4}-\frac {2 i b^3 \log (-a-b x+i)}{(-a+i)^4}-\frac {i b}{(-a+i)^2 x^2}-\frac {a+i}{3 (-a+i) x^3} \]
Antiderivative was successfully verified.
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Rule 77
Rule 5095
Rubi steps
\begin {align*} \int \frac {e^{-2 i \tan ^{-1}(a+b x)}}{x^4} \, dx &=\int \frac {1-i a-i b x}{x^4 (1+i a+i b x)} \, dx\\ &=\int \left (\frac {-i-a}{(-i+a) x^4}+\frac {2 i b}{(-i+a)^2 x^3}-\frac {2 i b^2}{(-i+a)^3 x^2}+\frac {2 i b^3}{(-i+a)^4 x}-\frac {2 i b^4}{(-i+a)^4 (-i+a+b x)}\right ) \, dx\\ &=-\frac {i+a}{3 (i-a) x^3}-\frac {i b}{(i-a)^2 x^2}+\frac {2 b^2}{(1+i a)^3 x}+\frac {2 i b^3 \log (x)}{(i-a)^4}-\frac {2 i b^3 \log (i-a-b x)}{(i-a)^4}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 91, normalized size = 0.88 \[ \frac {(a-i) \left (a^3-i a^2-3 i a b x+a+6 i b^2 x^2-3 b x-i\right )-6 i b^3 x^3 \log (-a-b x+i)+6 i b^3 x^3 \log (x)}{3 (a-i)^4 x^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 94, normalized size = 0.90 \[ \frac {6 i \, b^{3} x^{3} \log \relax (x) - 6 i \, b^{3} x^{3} \log \left (\frac {b x + a - i}{b}\right ) - 6 \, {\left (-i \, a - 1\right )} b^{2} x^{2} + a^{4} - 2 i \, a^{3} + {\left (-3 i \, a^{2} - 6 \, a + 3 i\right )} b x - 2 i \, a - 1}{{\left (3 \, a^{4} - 12 i \, a^{3} - 18 \, a^{2} + 12 i \, a + 3\right )} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.12, size = 202, normalized size = 1.94 \[ -\frac {2 \, b^{4} \log \left (-\frac {a i}{b i x + a i + 1} + \frac {i^{2}}{b i x + a i + 1} + 1\right )}{a^{4} b i + 4 \, a^{3} b - 6 \, a^{2} b i - 4 \, a b + b i} - \frac {\frac {3 \, {\left (a b^{4} i - 8 \, b^{4}\right )} i^{2}}{{\left (b i x + a i + 1\right )} b} + \frac {a b^{3} i - 10 \, b^{3}}{a i + 1} + \frac {3 \, {\left (a^{2} b^{5} + 4 \, a b^{5} i + 5 \, b^{5}\right )} i^{2}}{{\left (b i x + a i + 1\right )}^{2} b^{2}}}{3 \, {\left (a - i\right )}^{3} {\left (\frac {a i}{b i x + a i + 1} - \frac {i^{2}}{b i x + a i + 1} - 1\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 349, normalized size = 3.36 \[ \frac {i a^{4}}{\left (i-a \right )^{5} x^{3}}-\frac {a^{5}}{3 \left (i-a \right )^{5} x^{3}}+\frac {2 i b^{3} \arctan \left (b x +a \right )}{\left (i-a \right )^{5}}+\frac {2 a^{3}}{3 \left (i-a \right )^{5} x^{3}}+\frac {i b^{3} \ln \left (b^{2} x^{2}+2 a b x +a^{2}+1\right ) a}{\left (i-a \right )^{5}}+\frac {a}{\left (i-a \right )^{5} x^{3}}-\frac {i}{3 \left (i-a \right )^{5} x^{3}}-\frac {2 b^{3} \ln \relax (x )}{\left (i-a \right )^{5}}+\frac {i b \,a^{3}}{\left (i-a \right )^{5} x^{2}}+\frac {2 i b^{2}}{\left (i-a \right )^{5} x}-\frac {4 b^{2} a}{\left (i-a \right )^{5} x}-\frac {2 i b^{2} a^{2}}{\left (i-a \right )^{5} x}-\frac {3 i b a}{\left (i-a \right )^{5} x^{2}}+\frac {3 b \,a^{2}}{\left (i-a \right )^{5} x^{2}}-\frac {b}{\left (i-a \right )^{5} x^{2}}+\frac {2 i a^{2}}{3 \left (i-a \right )^{5} x^{3}}+\frac {b^{3} \ln \left (b^{2} x^{2}+2 a b x +a^{2}+1\right )}{\left (i-a \right )^{5}}-\frac {2 b^{3} \arctan \left (b x +a \right ) a}{\left (i-a \right )^{5}}-\frac {2 i b^{3} \ln \relax (x ) a}{\left (i-a \right )^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.34, size = 222, normalized size = 2.13 \[ \frac {{\left (2 \, a - 2 i\right )} b^{3} \log \left (i \, b x + i \, a + 1\right )}{i \, a^{5} + 5 \, a^{4} - 10 i \, a^{3} - 10 \, a^{2} + 5 i \, a + 1} - \frac {{\left (2 \, a - 2 i\right )} b^{3} \log \relax (x)}{i \, a^{5} + 5 \, a^{4} - 10 i \, a^{3} - 10 \, a^{2} + 5 i \, a + 1} - \frac {{\left (6 \, a - 6 i\right )} b^{3} x^{3} - i \, a^{5} + 3 \, {\left (a^{2} - 2 i \, a - 1\right )} b^{2} x^{2} - 3 \, a^{4} + 2 i \, a^{3} - {\left (i \, a^{4} + 5 \, a^{3} - 9 i \, a^{2} - 7 \, a + 2 i\right )} b x - 2 \, a^{2} + 3 i \, a + 1}{{\left (3 i \, a^{4} + 12 \, a^{3} - 18 i \, a^{2} - 12 \, a + 3 i\right )} b x^{4} + {\left (3 i \, a^{5} + 15 \, a^{4} - 30 i \, a^{3} - 30 \, a^{2} + 15 i \, a + 3\right )} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.77, size = 199, normalized size = 1.91 \[ \frac {\frac {a+1{}\mathrm {i}}{3\,\left (a-\mathrm {i}\right )}+\frac {b^2\,x^2\,2{}\mathrm {i}}{{\left (a-\mathrm {i}\right )}^3}-\frac {b\,x\,1{}\mathrm {i}}{{\left (a-\mathrm {i}\right )}^2}}{x^3}-\frac {4\,b^3\,\mathrm {atan}\left (\frac {\left (a^4-a^3\,4{}\mathrm {i}-6\,a^2+a\,4{}\mathrm {i}+1\right )\,1{}\mathrm {i}}{{\left (a-\mathrm {i}\right )}^4}+\frac {x\,\left (2\,a^{12}\,b^2+12\,a^{10}\,b^2+30\,a^8\,b^2+40\,a^6\,b^2+30\,a^4\,b^2+12\,a^2\,b^2+2\,b^2\right )}{{\left (a-\mathrm {i}\right )}^4\,\left (-1{}\mathrm {i}\,b\,a^9+3\,b\,a^8+8\,b\,a^6+6{}\mathrm {i}\,b\,a^5+6\,b\,a^4+8{}\mathrm {i}\,b\,a^3+3{}\mathrm {i}\,b\,a-b\right )}\right )}{{\left (a-\mathrm {i}\right )}^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.40, size = 286, normalized size = 2.75 \[ \frac {2 i b^{3} \log {\left (- \frac {2 a^{5} b^{3}}{\left (a - i\right )^{4}} + \frac {10 i a^{4} b^{3}}{\left (a - i\right )^{4}} + \frac {20 a^{3} b^{3}}{\left (a - i\right )^{4}} - \frac {20 i a^{2} b^{3}}{\left (a - i\right )^{4}} + 2 a b^{3} - \frac {10 a b^{3}}{\left (a - i\right )^{4}} + 4 b^{4} x - 2 i b^{3} + \frac {2 i b^{3}}{\left (a - i\right )^{4}} \right )}}{\left (a - i\right )^{4}} - \frac {2 i b^{3} \log {\left (\frac {2 a^{5} b^{3}}{\left (a - i\right )^{4}} - \frac {10 i a^{4} b^{3}}{\left (a - i\right )^{4}} - \frac {20 a^{3} b^{3}}{\left (a - i\right )^{4}} + \frac {20 i a^{2} b^{3}}{\left (a - i\right )^{4}} + 2 a b^{3} + \frac {10 a b^{3}}{\left (a - i\right )^{4}} + 4 b^{4} x - 2 i b^{3} - \frac {2 i b^{3}}{\left (a - i\right )^{4}} \right )}}{\left (a - i\right )^{4}} - \frac {- i a^{3} - a^{2} - i a + 6 b^{2} x^{2} + x \left (- 3 a b + 3 i b\right ) - 1}{x^{3} \left (3 i a^{3} + 9 a^{2} - 9 i a - 3\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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