Optimal. Leaf size=34 \[ -\frac {2 x}{b \sqrt {\tanh ^{-1}(\tanh (a+b x))}}+\frac {4 \sqrt {\tanh ^{-1}(\tanh (a+b x))}}{b^2} \]
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Rubi [A]
time = 0.01, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2199, 2188, 30}
\begin {gather*} \frac {4 \sqrt {\tanh ^{-1}(\tanh (a+b x))}}{b^2}-\frac {2 x}{b \sqrt {\tanh ^{-1}(\tanh (a+b x))}} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 2188
Rule 2199
Rubi steps
\begin {align*} \int \frac {x}{\tanh ^{-1}(\tanh (a+b x))^{3/2}} \, dx &=-\frac {2 x}{b \sqrt {\tanh ^{-1}(\tanh (a+b x))}}+\frac {2 \int \frac {1}{\sqrt {\tanh ^{-1}(\tanh (a+b x))}} \, dx}{b}\\ &=-\frac {2 x}{b \sqrt {\tanh ^{-1}(\tanh (a+b x))}}+\frac {2 \text {Subst}\left (\int \frac {1}{\sqrt {x}} \, dx,x,\tanh ^{-1}(\tanh (a+b x))\right )}{b^2}\\ &=-\frac {2 x}{b \sqrt {\tanh ^{-1}(\tanh (a+b x))}}+\frac {4 \sqrt {\tanh ^{-1}(\tanh (a+b x))}}{b^2}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 29, normalized size = 0.85 \begin {gather*} \frac {-2 b x+4 \tanh ^{-1}(\tanh (a+b x))}{b^2 \sqrt {\tanh ^{-1}(\tanh (a+b x))}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 40, normalized size = 1.18
method | result | size |
default | \(\frac {2 \sqrt {\arctanh \left (\tanh \left (b x +a \right )\right )}-\frac {2 \left (b x -\arctanh \left (\tanh \left (b x +a \right )\right )\right )}{\sqrt {\arctanh \left (\tanh \left (b x +a \right )\right )}}}{b^{2}}\) | \(40\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.54, size = 30, normalized size = 0.88 \begin {gather*} \frac {2 \, {\left (b^{2} x^{2} + 3 \, a b x + 2 \, a^{2}\right )}}{{\left (b x + a\right )}^{\frac {3}{2}} b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 29, normalized size = 0.85 \begin {gather*} \frac {2 \, {\left (b x + 2 \, a\right )} \sqrt {b x + a}}{b^{3} x + a b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 46.13, size = 46, normalized size = 1.35 \begin {gather*} \begin {cases} - \frac {2 x}{b \sqrt {\operatorname {atanh}{\left (\tanh {\left (a + b x \right )} \right )}}} + \frac {4 \sqrt {\operatorname {atanh}{\left (\tanh {\left (a + b x \right )} \right )}}}{b^{2}} & \text {for}\: b \neq 0 \\\frac {x^{2}}{2 \operatorname {atanh}^{\frac {3}{2}}{\left (\tanh {\left (a \right )} \right )}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.39, size = 29, normalized size = 0.85 \begin {gather*} \frac {2 \, {\left (\frac {\sqrt {b x + a}}{b} + \frac {a}{\sqrt {b x + a} b}\right )}}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.33, size = 152, normalized size = 4.47 \begin {gather*} -\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 (\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 )+b\,x\right )}{b^2\,\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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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