Optimal. Leaf size=70 \[ \frac {1}{8 a^3 b (a-b x)}-\frac {1}{8 a^2 b (a+b x)^2}-\frac {1}{4 a^3 b (a+b x)}+\frac {3 \tanh ^{-1}\left (\frac {b x}{a}\right )}{8 a^4 b} \]
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Rubi [A]
time = 0.03, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {641, 46, 214}
\begin {gather*} \frac {3 \tanh ^{-1}\left (\frac {b x}{a}\right )}{8 a^4 b}+\frac {1}{8 a^3 b (a-b x)}-\frac {1}{4 a^3 b (a+b x)}-\frac {1}{8 a^2 b (a+b x)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 46
Rule 214
Rule 641
Rubi steps
\begin {align*} \int \frac {1}{(a+b x) \left (a^2-b^2 x^2\right )^2} \, dx &=\int \frac {1}{(a-b x)^2 (a+b x)^3} \, dx\\ &=\int \left (\frac {1}{8 a^3 (a-b x)^2}+\frac {1}{4 a^2 (a+b x)^3}+\frac {1}{4 a^3 (a+b x)^2}+\frac {3}{8 a^3 \left (a^2-b^2 x^2\right )}\right ) \, dx\\ &=\frac {1}{8 a^3 b (a-b x)}-\frac {1}{8 a^2 b (a+b x)^2}-\frac {1}{4 a^3 b (a+b x)}+\frac {3 \int \frac {1}{a^2-b^2 x^2} \, dx}{8 a^3}\\ &=\frac {1}{8 a^3 b (a-b x)}-\frac {1}{8 a^2 b (a+b x)^2}-\frac {1}{4 a^3 b (a+b x)}+\frac {3 \tanh ^{-1}\left (\frac {b x}{a}\right )}{8 a^4 b}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 87, normalized size = 1.24 \begin {gather*} -\frac {1}{8 a^3 b (-a+b x)}-\frac {1}{8 a^2 b (a+b x)^2}-\frac {1}{4 a^3 b (a+b x)}-\frac {3 \log (a-b x)}{16 a^4 b}+\frac {3 \log (a+b x)}{16 a^4 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.54, size = 77, normalized size = 1.10
method | result | size |
norman | \(\frac {-\frac {1}{4 b a}+\frac {3 x}{8 a^{2}}+\frac {3 b \,x^{2}}{8 a^{3}}}{\left (b x +a \right )^{2} \left (-b x +a \right )}-\frac {3 \ln \left (-b x +a \right )}{16 a^{4} b}+\frac {3 \ln \left (b x +a \right )}{16 a^{4} b}\) | \(71\) |
default | \(\frac {3 \ln \left (b x +a \right )}{16 a^{4} b}-\frac {1}{4 a^{3} b \left (b x +a \right )}-\frac {1}{8 a^{2} b \left (b x +a \right )^{2}}-\frac {3 \ln \left (-b x +a \right )}{16 a^{4} b}+\frac {1}{8 a^{3} b \left (-b x +a \right )}\) | \(77\) |
risch | \(\frac {-\frac {1}{4 b a}+\frac {3 x}{8 a^{2}}+\frac {3 b \,x^{2}}{8 a^{3}}}{\left (b x +a \right ) \left (-b^{2} x^{2}+a^{2}\right )}-\frac {3 \ln \left (-b x +a \right )}{16 a^{4} b}+\frac {3 \ln \left (b x +a \right )}{16 a^{4} b}\) | \(77\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 90, normalized size = 1.29 \begin {gather*} -\frac {3 \, b^{2} x^{2} + 3 \, a b x - 2 \, a^{2}}{8 \, {\left (a^{3} b^{4} x^{3} + a^{4} b^{3} x^{2} - a^{5} b^{2} x - a^{6} b\right )}} + \frac {3 \, \log \left (b x + a\right )}{16 \, a^{4} b} - \frac {3 \, \log \left (b x - a\right )}{16 \, a^{4} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 136 vs.
\(2 (63) = 126\).
time = 3.05, size = 136, normalized size = 1.94 \begin {gather*} -\frac {6 \, a b^{2} x^{2} + 6 \, a^{2} b x - 4 \, a^{3} - 3 \, {\left (b^{3} x^{3} + a b^{2} x^{2} - a^{2} b x - a^{3}\right )} \log \left (b x + a\right ) + 3 \, {\left (b^{3} x^{3} + a b^{2} x^{2} - a^{2} b x - a^{3}\right )} \log \left (b x - a\right )}{16 \, {\left (a^{4} b^{4} x^{3} + a^{5} b^{3} x^{2} - a^{6} b^{2} x - a^{7} b\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.20, size = 85, normalized size = 1.21 \begin {gather*} \frac {2 a^{2} - 3 a b x - 3 b^{2} x^{2}}{- 8 a^{6} b - 8 a^{5} b^{2} x + 8 a^{4} b^{3} x^{2} + 8 a^{3} b^{4} x^{3}} + \frac {- \frac {3 \log {\left (- \frac {a}{b} + x \right )}}{16} + \frac {3 \log {\left (\frac {a}{b} + x \right )}}{16}}{a^{4} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.43, size = 79, normalized size = 1.13 \begin {gather*} \frac {3 \, \log \left ({\left | b x + a \right |}\right )}{16 \, a^{4} b} - \frac {3 \, \log \left ({\left | b x - a \right |}\right )}{16 \, a^{4} b} - \frac {3 \, a b^{2} x^{2} + 3 \, a^{2} b x - 2 \, a^{3}}{8 \, {\left (b x + a\right )}^{2} {\left (b x - a\right )} a^{4} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.08, size = 70, normalized size = 1.00 \begin {gather*} \frac {\frac {3\,x}{8\,a^2}-\frac {1}{4\,a\,b}+\frac {3\,b\,x^2}{8\,a^3}}{a^3+a^2\,b\,x-a\,b^2\,x^2-b^3\,x^3}+\frac {3\,\mathrm {atanh}\left (\frac {b\,x}{a}\right )}{8\,a^4\,b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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