Optimal. Leaf size=55 \[ -\frac {16 \tanh ^{-1}(\tanh (a+b x))^{3/2}}{3 b^3}+\frac {8 x \sqrt {\tanh ^{-1}(\tanh (a+b x))}}{b^2}-\frac {2 x^2}{b \sqrt {\tanh ^{-1}(\tanh (a+b x))}} \]
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Rubi [A] time = 0.03, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2168, 2157, 30} \[ \frac {8 x \sqrt {\tanh ^{-1}(\tanh (a+b x))}}{b^2}-\frac {16 \tanh ^{-1}(\tanh (a+b x))^{3/2}}{3 b^3}-\frac {2 x^2}{b \sqrt {\tanh ^{-1}(\tanh (a+b x))}} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2157
Rule 2168
Rubi steps
\begin {align*} \int \frac {x^2}{\tanh ^{-1}(\tanh (a+b x))^{3/2}} \, dx &=-\frac {2 x^2}{b \sqrt {\tanh ^{-1}(\tanh (a+b x))}}+\frac {4 \int \frac {x}{\sqrt {\tanh ^{-1}(\tanh (a+b x))}} \, dx}{b}\\ &=-\frac {2 x^2}{b \sqrt {\tanh ^{-1}(\tanh (a+b x))}}+\frac {8 x \sqrt {\tanh ^{-1}(\tanh (a+b x))}}{b^2}-\frac {8 \int \sqrt {\tanh ^{-1}(\tanh (a+b x))} \, dx}{b^2}\\ &=-\frac {2 x^2}{b \sqrt {\tanh ^{-1}(\tanh (a+b x))}}+\frac {8 x \sqrt {\tanh ^{-1}(\tanh (a+b x))}}{b^2}-\frac {8 \operatorname {Subst}\left (\int \sqrt {x} \, dx,x,\tanh ^{-1}(\tanh (a+b x))\right )}{b^3}\\ &=-\frac {2 x^2}{b \sqrt {\tanh ^{-1}(\tanh (a+b x))}}+\frac {8 x \sqrt {\tanh ^{-1}(\tanh (a+b x))}}{b^2}-\frac {16 \tanh ^{-1}(\tanh (a+b x))^{3/2}}{3 b^3}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 49, normalized size = 0.89 \[ -\frac {2 \left (-12 b x \tanh ^{-1}(\tanh (a+b x))+8 \tanh ^{-1}(\tanh (a+b x))^2+3 b^2 x^2\right )}{3 b^3 \sqrt {\tanh ^{-1}(\tanh (a+b x))}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.74, size = 40, normalized size = 0.73 \[ \frac {2 \, {\left (b^{2} x^{2} - 4 \, a b x - 8 \, a^{2}\right )} \sqrt {b x + a}}{3 \, {\left (b^{4} x + a b^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 46, normalized size = 0.84 \[ -\frac {2 \, a^{2}}{\sqrt {b x + a} b^{3}} + \frac {2 \, {\left ({\left (b x + a\right )}^{\frac {3}{2}} b^{6} - 6 \, \sqrt {b x + a} a b^{6}\right )}}{3 \, b^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.18, size = 106, normalized size = 1.93 \[ \frac {\frac {2 \arctanh \left (\tanh \left (b x +a \right )\right )^{\frac {3}{2}}}{3}-4 a \sqrt {\arctanh \left (\tanh \left (b x +a \right )\right )}-4 \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right ) \sqrt {\arctanh \left (\tanh \left (b x +a \right )\right )}-\frac {2 \left (a^{2}+2 a \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right )+\left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right )^{2}\right )}{\sqrt {\arctanh \left (\tanh \left (b x +a \right )\right )}}}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.53, size = 41, normalized size = 0.75 \[ \frac {2 \, {\left (b^{3} x^{3} - 3 \, a b^{2} x^{2} - 12 \, a^{2} b x - 8 \, a^{3}\right )}}{3 \, {\left (b x + a\right )}^{\frac {3}{2}} b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.29, size = 259, normalized size = 4.71 \[ -\frac {4\,\sqrt {\frac {\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )}{2}-\frac {\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )}{2}}\,\left (3\,b^2\,x^2-6\,b\,x\,\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+6\,b\,x\,\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,{\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )}^2-4\,\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )\,\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,{\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )}^2\right )}{3\,b^3\,\left (\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )-\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )\right )} \]
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 \[ \int \frac {x^{2}}{\operatorname {atanh}^{\frac {3}{2}}{\left (\tanh {\left (a + b x \right )} \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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