Optimal. Leaf size=44 \[ -\frac {16}{3} b^2 x^{3/2}+8 b \sqrt {x} \tanh ^{-1}(\tanh (a+b x))-\frac {2 \tanh ^{-1}(\tanh (a+b x))^2}{\sqrt {x}} \]
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Rubi [A]
time = 0.01, antiderivative size = 44, 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 = {2199, 30}
\begin {gather*} 8 b \sqrt {x} \tanh ^{-1}(\tanh (a+b x))-\frac {2 \tanh ^{-1}(\tanh (a+b x))^2}{\sqrt {x}}-\frac {16}{3} b^2 x^{3/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 2199
Rubi steps
\begin {align*} \int \frac {\tanh ^{-1}(\tanh (a+b x))^2}{x^{3/2}} \, dx &=-\frac {2 \tanh ^{-1}(\tanh (a+b x))^2}{\sqrt {x}}+(4 b) \int \frac {\tanh ^{-1}(\tanh (a+b x))}{\sqrt {x}} \, dx\\ &=8 b \sqrt {x} \tanh ^{-1}(\tanh (a+b x))-\frac {2 \tanh ^{-1}(\tanh (a+b x))^2}{\sqrt {x}}-\left (8 b^2\right ) \int \sqrt {x} \, dx\\ &=-\frac {16}{3} b^2 x^{3/2}+8 b \sqrt {x} \tanh ^{-1}(\tanh (a+b x))-\frac {2 \tanh ^{-1}(\tanh (a+b x))^2}{\sqrt {x}}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 40, normalized size = 0.91 \begin {gather*} -\frac {2 \left (8 b^2 x^2-12 b x \tanh ^{-1}(\tanh (a+b x))+3 \tanh ^{-1}(\tanh (a+b x))^2\right )}{3 \sqrt {x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.29, size = 37, normalized size = 0.84
method | result | size |
derivativedivides | \(-\frac {2 \arctanh \left (\tanh \left (b x +a \right )\right )^{2}}{\sqrt {x}}+8 b \left (\arctanh \left (\tanh \left (b x +a \right )\right ) \sqrt {x}-\frac {2 b \,x^{\frac {3}{2}}}{3}\right )\) | \(37\) |
default | \(-\frac {2 \arctanh \left (\tanh \left (b x +a \right )\right )^{2}}{\sqrt {x}}+8 b \left (\arctanh \left (\tanh \left (b x +a \right )\right ) \sqrt {x}-\frac {2 b \,x^{\frac {3}{2}}}{3}\right )\) | \(37\) |
risch | \(\text {Expression too large to display}\) | \(1977\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 36, normalized size = 0.82 \begin {gather*} -\frac {16}{3} \, b^{2} x^{\frac {3}{2}} + 8 \, b \sqrt {x} \operatorname {artanh}\left (\tanh \left (b x + a\right )\right ) - \frac {2 \, \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{2}}{\sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 23, normalized size = 0.52 \begin {gather*} \frac {2 \, {\left (b^{2} x^{2} + 6 \, a b x - 3 \, a^{2}\right )}}{3 \, \sqrt {x}} \end {gather*}
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 \begin {gather*} \int \frac {\operatorname {atanh}^{2}{\left (\tanh {\left (a + b x \right )} \right )}}{x^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.38, size = 24, normalized size = 0.55 \begin {gather*} \frac {2}{3} \, b^{2} x^{\frac {3}{2}} + 4 \, a b \sqrt {x} - \frac {2 \, a^{2}}{\sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.16, size = 122, normalized size = 2.77 \begin {gather*} \frac {2\,b^2\,x^{3/2}}{3}-\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}{2\,\sqrt {x}}-2\,b\,\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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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