Optimal. Leaf size=48 \[ -\frac {2 \tanh ^{-1}(\tanh (a+b x))^2}{3 x^{3/2}}-\frac {8 b \tanh ^{-1}(\tanh (a+b x))}{3 \sqrt {x}}+\frac {16 b^2 \sqrt {x}}{3} \]
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Rubi [A] time = 0.02, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2168, 30} \[ -\frac {2 \tanh ^{-1}(\tanh (a+b x))^2}{3 x^{3/2}}-\frac {8 b \tanh ^{-1}(\tanh (a+b x))}{3 \sqrt {x}}+\frac {16 b^2 \sqrt {x}}{3} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2168
Rubi steps
\begin {align*} \int \frac {\tanh ^{-1}(\tanh (a+b x))^2}{x^{5/2}} \, dx &=-\frac {2 \tanh ^{-1}(\tanh (a+b x))^2}{3 x^{3/2}}+\frac {1}{3} (4 b) \int \frac {\tanh ^{-1}(\tanh (a+b x))}{x^{3/2}} \, dx\\ &=-\frac {8 b \tanh ^{-1}(\tanh (a+b x))}{3 \sqrt {x}}-\frac {2 \tanh ^{-1}(\tanh (a+b x))^2}{3 x^{3/2}}+\frac {1}{3} \left (8 b^2\right ) \int \frac {1}{\sqrt {x}} \, dx\\ &=\frac {16 b^2 \sqrt {x}}{3}-\frac {8 b \tanh ^{-1}(\tanh (a+b x))}{3 \sqrt {x}}-\frac {2 \tanh ^{-1}(\tanh (a+b x))^2}{3 x^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 40, normalized size = 0.83 \[ \frac {2 \left (-4 b x \tanh ^{-1}(\tanh (a+b x))-\tanh ^{-1}(\tanh (a+b x))^2+8 b^2 x^2\right )}{3 x^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.59, size = 24, normalized size = 0.50 \[ \frac {2 \, {\left (3 \, b^{2} x^{2} - 6 \, a b x - a^{2}\right )}}{3 \, x^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 23, normalized size = 0.48 \[ 2 \, b^{2} \sqrt {x} - \frac {2 \, {\left (6 \, a b x + a^{2}\right )}}{3 \, x^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.25, size = 38, normalized size = 0.79 \[ -\frac {2 \arctanh \left (\tanh \left (b x +a \right )\right )^{2}}{3 x^{\frac {3}{2}}}+\frac {8 b \left (-\frac {\arctanh \left (\tanh \left (b x +a \right )\right )}{\sqrt {x}}+2 b \sqrt {x}\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 36, normalized size = 0.75 \[ \frac {16}{3} \, b^{2} \sqrt {x} - \frac {8 \, b \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )}{3 \, \sqrt {x}} - \frac {2 \, \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{2}}{3 \, x^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.13, size = 122, normalized size = 2.54 \[ 2\,b^2\,\sqrt {x}-\frac {{\left (\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x\right )}^2}{6\,x^{3/2}}+\frac {2\,b\,\left (\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x\right )}{\sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 10.50, size = 48, normalized size = 1.00 \[ \frac {16 b^{2} \sqrt {x}}{3} - \frac {8 b \operatorname {atanh}{\left (\tanh {\left (a + b x \right )} \right )}}{3 \sqrt {x}} - \frac {2 \operatorname {atanh}^{2}{\left (\tanh {\left (a + b x \right )} \right )}}{3 x^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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