Optimal. Leaf size=70 \[ -\frac {2 b^3 \log (x)}{(a+1)^4}+\frac {2 b^3 \log (a+b x+1)}{(a+1)^4}-\frac {2 b^2}{(a+1)^3 x}+\frac {b}{(a+1)^2 x^2}-\frac {1-a}{3 (a+1) x^3} \]
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Rubi [A] time = 0.06, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {6163, 77} \[ -\frac {2 b^2}{(a+1)^3 x}-\frac {2 b^3 \log (x)}{(a+1)^4}+\frac {2 b^3 \log (a+b x+1)}{(a+1)^4}+\frac {b}{(a+1)^2 x^2}-\frac {1-a}{3 (a+1) x^3} \]
Antiderivative was successfully verified.
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Rule 77
Rule 6163
Rubi steps
\begin {align*} \int \frac {e^{-2 \tanh ^{-1}(a+b x)}}{x^4} \, dx &=\int \frac {1-a-b x}{x^4 (1+a+b x)} \, dx\\ &=\int \left (\frac {1-a}{(1+a) x^4}-\frac {2 b}{(1+a)^2 x^3}+\frac {2 b^2}{(1+a)^3 x^2}-\frac {2 b^3}{(1+a)^4 x}+\frac {2 b^4}{(1+a)^4 (1+a+b x)}\right ) \, dx\\ &=-\frac {1-a}{3 (1+a) x^3}+\frac {b}{(1+a)^2 x^2}-\frac {2 b^2}{(1+a)^3 x}-\frac {2 b^3 \log (x)}{(1+a)^4}+\frac {2 b^3 \log (1+a+b x)}{(1+a)^4}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 70, normalized size = 1.00 \[ -\frac {2 b^3 \log (x)}{(a+1)^4}+\frac {2 b^3 \log (a+b x+1)}{(a+1)^4}-\frac {2 b^2}{(a+1)^3 x}+\frac {b}{(a+1)^2 x^2}-\frac {1-a}{3 (a+1) x^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.58, size = 86, normalized size = 1.23 \[ \frac {6 \, b^{3} x^{3} \log \left (b x + a + 1\right ) - 6 \, b^{3} x^{3} \log \relax (x) - 6 \, {\left (a + 1\right )} b^{2} x^{2} + a^{4} + 2 \, a^{3} + 3 \, {\left (a^{2} + 2 \, a + 1\right )} b x - 2 \, a - 1}{3 \, {\left (a^{4} + 4 \, a^{3} + 6 \, a^{2} + 4 \, a + 1\right )} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.44, size = 159, normalized size = 2.27 \[ -\frac {2 \, b^{4} \log \left ({\left | -\frac {a}{b x + a + 1} - \frac {1}{b x + a + 1} + 1 \right |}\right )}{a^{4} b + 4 \, a^{3} b + 6 \, a^{2} b + 4 \, a b + b} - \frac {\frac {a b^{3} - 10 \, b^{3}}{a + 1} - \frac {3 \, {\left (a b^{4} - 8 \, b^{4}\right )}}{{\left (b x + a + 1\right )} b} + \frac {3 \, {\left (a^{2} b^{5} - 4 \, a b^{5} - 5 \, b^{5}\right )}}{{\left (b x + a + 1\right )}^{2} b^{2}}}{3 \, {\left (a + 1\right )}^{3} {\left (\frac {a}{b x + a + 1} + \frac {1}{b x + a + 1} - 1\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 75, normalized size = 1.07 \[ -\frac {1}{3 \left (1+a \right ) x^{3}}+\frac {a}{3 \left (1+a \right ) x^{3}}+\frac {b}{\left (1+a \right )^{2} x^{2}}-\frac {2 b^{3} \ln \relax (x )}{\left (1+a \right )^{4}}-\frac {2 b^{2}}{\left (1+a \right )^{3} x}+\frac {2 b^{3} \ln \left (b x +a +1\right )}{\left (1+a \right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 108, normalized size = 1.54 \[ \frac {2 \, b^{3} \log \left (b x + a + 1\right )}{a^{4} + 4 \, a^{3} + 6 \, a^{2} + 4 \, a + 1} - \frac {2 \, b^{3} \log \relax (x)}{a^{4} + 4 \, a^{3} + 6 \, a^{2} + 4 \, a + 1} - \frac {6 \, b^{2} x^{2} - a^{3} - 3 \, {\left (a + 1\right )} b x - a^{2} + a + 1}{3 \, {\left (a^{3} + 3 \, a^{2} + 3 \, a + 1\right )} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.10, size = 83, normalized size = 1.19 \[ \frac {\frac {a-1}{3\,\left (a+1\right )}-\frac {2\,b^2\,x^2}{{\left (a+1\right )}^3}+\frac {b\,x}{{\left (a+1\right )}^2}}{x^3}+\frac {4\,b^3\,\mathrm {atanh}\left (\frac {a^4+4\,a^3+6\,a^2+4\,a+1}{{\left (a+1\right )}^4}+\frac {2\,b\,x}{a+1}\right )}{{\left (a+1\right )}^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.53, size = 262, normalized size = 3.74 \[ - \frac {2 b^{3} \log {\left (x + \frac {- \frac {2 a^{5} b^{3}}{\left (a + 1\right )^{4}} - \frac {10 a^{4} b^{3}}{\left (a + 1\right )^{4}} - \frac {20 a^{3} b^{3}}{\left (a + 1\right )^{4}} - \frac {20 a^{2} b^{3}}{\left (a + 1\right )^{4}} + 2 a b^{3} - \frac {10 a b^{3}}{\left (a + 1\right )^{4}} + 2 b^{3} - \frac {2 b^{3}}{\left (a + 1\right )^{4}}}{4 b^{4}} \right )}}{\left (a + 1\right )^{4}} + \frac {2 b^{3} \log {\left (x + \frac {\frac {2 a^{5} b^{3}}{\left (a + 1\right )^{4}} + \frac {10 a^{4} b^{3}}{\left (a + 1\right )^{4}} + \frac {20 a^{3} b^{3}}{\left (a + 1\right )^{4}} + \frac {20 a^{2} b^{3}}{\left (a + 1\right )^{4}} + 2 a b^{3} + \frac {10 a b^{3}}{\left (a + 1\right )^{4}} + 2 b^{3} + \frac {2 b^{3}}{\left (a + 1\right )^{4}}}{4 b^{4}} \right )}}{\left (a + 1\right )^{4}} - \frac {- a^{3} - a^{2} + a + 6 b^{2} x^{2} + x \left (- 3 a b - 3 b\right ) + 1}{x^{3} \left (3 a^{3} + 9 a^{2} + 9 a + 3\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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