Optimal. Leaf size=48 \[ \frac {16 b^2 x^{13/2}}{1287}-\frac {8}{99} b x^{11/2} \tanh ^{-1}(\tanh (a+b x))+\frac {2}{9} x^{9/2} \tanh ^{-1}(\tanh (a+b x))^2 \]
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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 = {2199, 30}
\begin {gather*} -\frac {8}{99} b x^{11/2} \tanh ^{-1}(\tanh (a+b x))+\frac {2}{9} x^{9/2} \tanh ^{-1}(\tanh (a+b x))^2+\frac {16 b^2 x^{13/2}}{1287} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 2199
Rubi steps
\begin {align*} \int x^{7/2} \tanh ^{-1}(\tanh (a+b x))^2 \, dx &=\frac {2}{9} x^{9/2} \tanh ^{-1}(\tanh (a+b x))^2-\frac {1}{9} (4 b) \int x^{9/2} \tanh ^{-1}(\tanh (a+b x)) \, dx\\ &=-\frac {8}{99} b x^{11/2} \tanh ^{-1}(\tanh (a+b x))+\frac {2}{9} x^{9/2} \tanh ^{-1}(\tanh (a+b x))^2+\frac {1}{99} \left (8 b^2\right ) \int x^{11/2} \, dx\\ &=\frac {16 b^2 x^{13/2}}{1287}-\frac {8}{99} b x^{11/2} \tanh ^{-1}(\tanh (a+b x))+\frac {2}{9} x^{9/2} \tanh ^{-1}(\tanh (a+b x))^2\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 40, normalized size = 0.83 \begin {gather*} \frac {2 x^{9/2} \left (8 b^2 x^2-52 b x \tanh ^{-1}(\tanh (a+b x))+143 \tanh ^{-1}(\tanh (a+b x))^2\right )}{1287} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.29, size = 38, normalized size = 0.79
method | result | size |
derivativedivides | \(\frac {2 x^{\frac {9}{2}} \arctanh \left (\tanh \left (b x +a \right )\right )^{2}}{9}-\frac {8 b \left (\frac {x^{\frac {11}{2}} \arctanh \left (\tanh \left (b x +a \right )\right )}{11}-\frac {2 x^{\frac {13}{2}} b}{143}\right )}{9}\) | \(38\) |
default | \(\frac {2 x^{\frac {9}{2}} \arctanh \left (\tanh \left (b x +a \right )\right )^{2}}{9}-\frac {8 b \left (\frac {x^{\frac {11}{2}} \arctanh \left (\tanh \left (b x +a \right )\right )}{11}-\frac {2 x^{\frac {13}{2}} b}{143}\right )}{9}\) | \(38\) |
risch | \(\text {Expression too large to display}\) | \(2093\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 36, normalized size = 0.75 \begin {gather*} \frac {16}{1287} \, b^{2} x^{\frac {13}{2}} - \frac {8}{99} \, b x^{\frac {11}{2}} \operatorname {artanh}\left (\tanh \left (b x + a\right )\right ) + \frac {2}{9} \, x^{\frac {9}{2}} \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 29, normalized size = 0.60 \begin {gather*} \frac {2}{1287} \, {\left (99 \, b^{2} x^{6} + 234 \, a b x^{5} + 143 \, a^{2} x^{4}\right )} \sqrt {x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.39, size = 24, normalized size = 0.50 \begin {gather*} \frac {2}{13} \, b^{2} x^{\frac {13}{2}} + \frac {4}{11} \, a b x^{\frac {11}{2}} + \frac {2}{9} \, a^{2} x^{\frac {9}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.15, size = 122, normalized size = 2.54 \begin {gather*} \frac {x^{9/2}\,{\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}{18}+\frac {2\,b^2\,x^{13/2}}{13}-\frac {2\,b\,x^{11/2}\,\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 )}{11} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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