Optimal. Leaf size=48 \[ -\frac {8}{15} b x^{5/2} \tanh ^{-1}(\tanh (a+b x))+\frac {2}{3} x^{3/2} \tanh ^{-1}(\tanh (a+b x))^2+\frac {16}{105} b^2 x^{7/2} \]
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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} \[ -\frac {8}{15} b x^{5/2} \tanh ^{-1}(\tanh (a+b x))+\frac {2}{3} x^{3/2} \tanh ^{-1}(\tanh (a+b x))^2+\frac {16}{105} b^2 x^{7/2} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2168
Rubi steps
\begin {align*} \int \sqrt {x} \tanh ^{-1}(\tanh (a+b x))^2 \, dx &=\frac {2}{3} x^{3/2} \tanh ^{-1}(\tanh (a+b x))^2-\frac {1}{3} (4 b) \int x^{3/2} \tanh ^{-1}(\tanh (a+b x)) \, dx\\ &=-\frac {8}{15} b x^{5/2} \tanh ^{-1}(\tanh (a+b x))+\frac {2}{3} x^{3/2} \tanh ^{-1}(\tanh (a+b x))^2+\frac {1}{15} \left (8 b^2\right ) \int x^{5/2} \, dx\\ &=\frac {16}{105} b^2 x^{7/2}-\frac {8}{15} b x^{5/2} \tanh ^{-1}(\tanh (a+b x))+\frac {2}{3} x^{3/2} \tanh ^{-1}(\tanh (a+b x))^2\\ \end {align*}
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Mathematica [A] time = 0.05, size = 40, normalized size = 0.83 \[ \frac {2}{105} x^{3/2} \left (-28 b x \tanh ^{-1}(\tanh (a+b x))+35 \tanh ^{-1}(\tanh (a+b x))^2+8 b^2 x^2\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 27, normalized size = 0.56 \[ \frac {2}{105} \, {\left (15 \, b^{2} x^{3} + 42 \, a b x^{2} + 35 \, a^{2} x\right )} \sqrt {x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.74, size = 24, normalized size = 0.50 \[ \frac {2}{7} \, b^{2} x^{\frac {7}{2}} + \frac {4}{5} \, a b x^{\frac {5}{2}} + \frac {2}{3} \, a^{2} x^{\frac {3}{2}} \]
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 \[ \frac {2 x^{\frac {3}{2}} \arctanh \left (\tanh \left (b x +a \right )\right )^{2}}{3}-\frac {8 b \left (\frac {x^{\frac {5}{2}} \arctanh \left (\tanh \left (b x +a \right )\right )}{5}-\frac {2 b \,x^{\frac {7}{2}}}{35}\right )}{3} \]
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 \[ \frac {16}{105} \, b^{2} x^{\frac {7}{2}} - \frac {8}{15} \, b x^{\frac {5}{2}} \operatorname {artanh}\left (\tanh \left (b x + a\right )\right ) + \frac {2}{3} \, x^{\frac {3}{2}} \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.12, size = 122, normalized size = 2.54 \[ \frac {x^{3/2}\,{\left (\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x\right )}^2}{6}+\frac {2\,b^2\,x^{7/2}}{7}-\frac {2\,b\,x^{5/2}\,\left (\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x\right )}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {x} \operatorname {atanh}^{2}{\left (\tanh {\left (a + b x \right )} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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