Optimal. Leaf size=27 \[ -\frac {4 b}{3 \sqrt {x}}-\frac {2 \tanh ^{-1}(\tanh (a+b x))}{3 x^{3/2}} \]
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Rubi [A]
time = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2199, 30}
\begin {gather*} -\frac {2 \tanh ^{-1}(\tanh (a+b x))}{3 x^{3/2}}-\frac {4 b}{3 \sqrt {x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 2199
Rubi steps
\begin {align*} \int \frac {\tanh ^{-1}(\tanh (a+b x))}{x^{5/2}} \, dx &=-\frac {2 \tanh ^{-1}(\tanh (a+b x))}{3 x^{3/2}}+\frac {1}{3} (2 b) \int \frac {1}{x^{3/2}} \, dx\\ &=-\frac {4 b}{3 \sqrt {x}}-\frac {2 \tanh ^{-1}(\tanh (a+b x))}{3 x^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 21, normalized size = 0.78 \begin {gather*} -\frac {2 \left (2 b x+\tanh ^{-1}(\tanh (a+b x))\right )}{3 x^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 20, normalized size = 0.74
method | result | size |
derivativedivides | \(-\frac {2 \arctanh \left (\tanh \left (b x +a \right )\right )}{3 x^{\frac {3}{2}}}-\frac {4 b}{3 \sqrt {x}}\) | \(20\) |
default | \(-\frac {2 \arctanh \left (\tanh \left (b x +a \right )\right )}{3 x^{\frac {3}{2}}}-\frac {4 b}{3 \sqrt {x}}\) | \(20\) |
risch | \(-\frac {2 \ln \left ({\mathrm e}^{b x +a}\right )}{3 x^{\frac {3}{2}}}-\frac {i \pi \,\mathrm {csgn}\left (\frac {i}{{\mathrm e}^{2 b x +2 a}+1}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{2 b x +2 a}}{{\mathrm e}^{2 b x +2 a}+1}\right )^{2}+2 i \pi \,\mathrm {csgn}\left (i {\mathrm e}^{b x +a}\right ) \mathrm {csgn}\left (i {\mathrm e}^{2 b x +2 a}\right )^{2}-i \pi \mathrm {csgn}\left (i {\mathrm e}^{b x +a}\right )^{2} \mathrm {csgn}\left (i {\mathrm e}^{2 b x +2 a}\right )+i \pi \,\mathrm {csgn}\left (i {\mathrm e}^{2 b x +2 a}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{2 b x +2 a}}{{\mathrm e}^{2 b x +2 a}+1}\right )^{2}-i \pi \mathrm {csgn}\left (\frac {i {\mathrm e}^{2 b x +2 a}}{{\mathrm e}^{2 b x +2 a}+1}\right )^{3}-i \pi \mathrm {csgn}\left (i {\mathrm e}^{2 b x +2 a}\right )^{3}-i \pi \,\mathrm {csgn}\left (\frac {i}{{\mathrm e}^{2 b x +2 a}+1}\right ) \mathrm {csgn}\left (i {\mathrm e}^{2 b x +2 a}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{2 b x +2 a}}{{\mathrm e}^{2 b x +2 a}+1}\right )+8 b x}{6 x^{\frac {3}{2}}}\) | \(287\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 19, normalized size = 0.70 \begin {gather*} -\frac {4 \, b}{3 \, \sqrt {x}} - \frac {2 \, \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )}{3 \, x^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 11, normalized size = 0.41 \begin {gather*} -\frac {2 \, {\left (3 \, b x + a\right )}}{3 \, x^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 2.59, size = 27, normalized size = 1.00 \begin {gather*} - \frac {4 b}{3 \sqrt {x}} - \frac {2 \operatorname {atanh}{\left (\tanh {\left (a + b x \right )} \right )}}{3 x^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.38, size = 11, normalized size = 0.41 \begin {gather*} -\frac {2 \, {\left (3 \, b x + a\right )}}{3 \, x^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.24, size = 52, normalized size = 1.93 \begin {gather*} -\frac {\ln \left (\frac {{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )-\ln \left (\frac {1}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+4\,b\,x}{3\,x^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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