Optimal. Leaf size=42 \[ \frac {1}{2 a b c^2 (a-b x)}+\frac {\tanh ^{-1}\left (\frac {b x}{a}\right )}{2 a^2 b c^2} \]
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Rubi [A]
time = 0.02, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {46, 214}
\begin {gather*} \frac {\tanh ^{-1}\left (\frac {b x}{a}\right )}{2 a^2 b c^2}+\frac {1}{2 a b c^2 (a-b x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 46
Rule 214
Rubi steps
\begin {align*} \int \frac {1}{(a+b x) (a c-b c x)^2} \, dx &=\int \left (\frac {1}{2 a c^2 (a-b x)^2}+\frac {1}{2 a c^2 \left (a^2-b^2 x^2\right )}\right ) \, dx\\ &=\frac {1}{2 a b c^2 (a-b x)}+\frac {\int \frac {1}{a^2-b^2 x^2} \, dx}{2 a c^2}\\ &=\frac {1}{2 a b c^2 (a-b x)}+\frac {\tanh ^{-1}\left (\frac {b x}{a}\right )}{2 a^2 b c^2}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 53, normalized size = 1.26 \begin {gather*} \frac {2 a+(-a+b x) \log (a-b x)+(a-b x) \log (a+b x)}{4 a^2 b c^2 (a-b x)} \end {gather*}
Antiderivative was successfully verified.
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Mathics [A]
time = 2.12, size = 55, normalized size = 1.31 \begin {gather*} \frac {2 a+\left (a-b x\right ) \left (\text {Log}\left [\frac {a+b x}{b}\right ]-\text {Log}\left [\frac {-a+b x}{b}\right ]\right )}{4 a^2 b c^2 \left (a-b x\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.15, size = 51, normalized size = 1.21
method | result | size |
default | \(\frac {-\frac {\ln \left (-b x +a \right )}{4 a^{2} b}+\frac {1}{2 a b \left (-b x +a \right )}+\frac {\ln \left (b x +a \right )}{4 a^{2} b}}{c^{2}}\) | \(51\) |
norman | \(\frac {1}{2 a b \,c^{2} \left (-b x +a \right )}-\frac {\ln \left (-b x +a \right )}{4 a^{2} b \,c^{2}}+\frac {\ln \left (b x +a \right )}{4 a^{2} b \,c^{2}}\) | \(56\) |
risch | \(\frac {1}{2 a b \,c^{2} \left (-b x +a \right )}-\frac {\ln \left (-b x +a \right )}{4 a^{2} b \,c^{2}}+\frac {\ln \left (b x +a \right )}{4 a^{2} b \,c^{2}}\) | \(56\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.25, size = 60, normalized size = 1.43 \begin {gather*} -\frac {1}{2 \, {\left (a b^{2} c^{2} x - a^{2} b c^{2}\right )}} + \frac {\log \left (b x + a\right )}{4 \, a^{2} b c^{2}} - \frac {\log \left (b x - a\right )}{4 \, a^{2} b c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.30, size = 60, normalized size = 1.43 \begin {gather*} \frac {{\left (b x - a\right )} \log \left (b x + a\right ) - {\left (b x - a\right )} \log \left (b x - a\right ) - 2 \, a}{4 \, {\left (a^{2} b^{2} c^{2} x - a^{3} b c^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.15, size = 48, normalized size = 1.14 \begin {gather*} - \frac {1}{- 2 a^{2} b c^{2} + 2 a b^{2} c^{2} x} + \frac {- \frac {\log {\left (- \frac {a}{b} + x \right )}}{4} + \frac {\log {\left (\frac {a}{b} + x \right )}}{4}}{a^{2} b c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 63, normalized size = 1.50 \begin {gather*} \frac {\ln \left |x b+a\right |}{4 b a^{2} c^{2}}-\frac {\ln \left |x b-a\right |}{4 b a^{2} c^{2}}-\frac {\frac {1}{4}\cdot 2 a}{c^{2} a^{2} b \left (x b-a\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.07, size = 42, normalized size = 1.00 \begin {gather*} \frac {1}{2\,a\,b\,\left (a\,c^2-b\,c^2\,x\right )}+\frac {\mathrm {atanh}\left (\frac {b\,x}{a}\right )}{2\,a^2\,b\,c^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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