Optimal. Leaf size=46 \[ -\frac {8}{3} b x^{3/2} \tanh ^{-1}(\tanh (a+b x))+2 \sqrt {x} \tanh ^{-1}(\tanh (a+b x))^2+\frac {16}{15} b^2 x^{5/2} \]
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Rubi [A] time = 0.02, antiderivative size = 46, 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} \[ -\frac {8}{3} b x^{3/2} \tanh ^{-1}(\tanh (a+b x))+2 \sqrt {x} \tanh ^{-1}(\tanh (a+b x))^2+\frac {16}{15} b^2 x^{5/2} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2168
Rubi steps
\begin {align*} \int \frac {\tanh ^{-1}(\tanh (a+b x))^2}{\sqrt {x}} \, dx &=2 \sqrt {x} \tanh ^{-1}(\tanh (a+b x))^2-(4 b) \int \sqrt {x} \tanh ^{-1}(\tanh (a+b x)) \, dx\\ &=-\frac {8}{3} b x^{3/2} \tanh ^{-1}(\tanh (a+b x))+2 \sqrt {x} \tanh ^{-1}(\tanh (a+b x))^2+\frac {1}{3} \left (8 b^2\right ) \int x^{3/2} \, dx\\ &=\frac {16}{15} b^2 x^{5/2}-\frac {8}{3} b x^{3/2} \tanh ^{-1}(\tanh (a+b x))+2 \sqrt {x} \tanh ^{-1}(\tanh (a+b x))^2\\ \end {align*}
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Mathematica [A] time = 0.03, size = 40, normalized size = 0.87 \[ \frac {2}{15} \sqrt {x} \left (-20 b x \tanh ^{-1}(\tanh (a+b x))+15 \tanh ^{-1}(\tanh (a+b x))^2+8 b^2 x^2\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 24, normalized size = 0.52 \[ \frac {2}{15} \, {\left (3 \, b^{2} x^{2} + 10 \, a b x + 15 \, a^{2}\right )} \sqrt {x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 24, normalized size = 0.52 \[ \frac {2}{5} \, b^{2} x^{\frac {5}{2}} + \frac {4}{3} \, a b x^{\frac {3}{2}} + 2 \, a^{2} \sqrt {x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.26, size = 47, normalized size = 1.02 \[ \frac {2 b^{2} x^{\frac {5}{2}}}{5}+\frac {4 \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x \right ) b \,x^{\frac {3}{2}}}{3}+2 \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x \right )^{2} \sqrt {x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 36, normalized size = 0.78 \[ \frac {16}{15} \, b^{2} x^{\frac {5}{2}} - \frac {8}{3} \, b x^{\frac {3}{2}} \operatorname {artanh}\left (\tanh \left (b x + a\right )\right ) + 2 \, \sqrt {x} \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.15, size = 122, normalized size = 2.65 \[ \frac {\sqrt {x}\,{\left (\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x\right )}^2}{2}+\frac {2\,b^2\,x^{5/2}}{5}-\frac {2\,b\,x^{3/2}\,\left (\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x\right )}{3} \]
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 \[ \int \frac {\operatorname {atanh}^{2}{\left (\tanh {\left (a + b x \right )} \right )}}{\sqrt {x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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