Optimal. Leaf size=27 \[ -\frac {2 \tanh ^{-1}(\tanh (a+b x))}{5 x^{5/2}}-\frac {4 b}{15 x^{3/2}} \]
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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 = {2168, 30} \[ -\frac {2 \tanh ^{-1}(\tanh (a+b x))}{5 x^{5/2}}-\frac {4 b}{15 x^{3/2}} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2168
Rubi steps
\begin {align*} \int \frac {\tanh ^{-1}(\tanh (a+b x))}{x^{7/2}} \, dx &=-\frac {2 \tanh ^{-1}(\tanh (a+b x))}{5 x^{5/2}}+\frac {1}{5} (2 b) \int \frac {1}{x^{5/2}} \, dx\\ &=-\frac {4 b}{15 x^{3/2}}-\frac {2 \tanh ^{-1}(\tanh (a+b x))}{5 x^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 23, normalized size = 0.85 \[ -\frac {2 \left (3 \tanh ^{-1}(\tanh (a+b x))+2 b x\right )}{15 x^{5/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 13, normalized size = 0.48 \[ -\frac {2 \, {\left (5 \, b x + 3 \, a\right )}}{15 \, x^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 13, normalized size = 0.48 \[ -\frac {2 \, {\left (5 \, b x + 3 \, a\right )}}{15 \, x^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.25, size = 20, normalized size = 0.74 \[ -\frac {4 b}{15 x^{\frac {3}{2}}}-\frac {2 \arctanh \left (\tanh \left (b x +a \right )\right )}{5 x^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 19, normalized size = 0.70 \[ -\frac {4 \, b}{15 \, x^{\frac {3}{2}}} - \frac {2 \, \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )}{5 \, x^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.28, size = 57, normalized size = 2.11 \[ \frac {\ln \left (\frac {1}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )}{5\,x^{5/2}}-\frac {4\,b}{15\,x^{3/2}}-\frac {\ln \left (\frac {{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )}{5\,x^{5/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 155.35, size = 27, normalized size = 1.00 \[ - \frac {4 b}{15 x^{\frac {3}{2}}} - \frac {2 \operatorname {atanh}{\left (\tanh {\left (a + b x \right )} \right )}}{5 x^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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