Optimal. Leaf size=69 \[ -\frac {32 b^3 x^{11/2}}{1155}+\frac {16}{105} b^2 x^{9/2} \tanh ^{-1}(\tanh (a+b x))-\frac {12}{35} b x^{7/2} \tanh ^{-1}(\tanh (a+b x))^2+\frac {2}{5} x^{5/2} \tanh ^{-1}(\tanh (a+b x))^3 \]
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Rubi [A]
time = 0.03, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps
used = 4, 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 {16}{105} b^2 x^{9/2} \tanh ^{-1}(\tanh (a+b x))-\frac {12}{35} b x^{7/2} \tanh ^{-1}(\tanh (a+b x))^2+\frac {2}{5} x^{5/2} \tanh ^{-1}(\tanh (a+b x))^3-\frac {32 b^3 x^{11/2}}{1155} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 2199
Rubi steps
\begin {align*} \int x^{3/2} \tanh ^{-1}(\tanh (a+b x))^3 \, dx &=\frac {2}{5} x^{5/2} \tanh ^{-1}(\tanh (a+b x))^3-\frac {1}{5} (6 b) \int x^{5/2} \tanh ^{-1}(\tanh (a+b x))^2 \, dx\\ &=-\frac {12}{35} b x^{7/2} \tanh ^{-1}(\tanh (a+b x))^2+\frac {2}{5} x^{5/2} \tanh ^{-1}(\tanh (a+b x))^3+\frac {1}{35} \left (24 b^2\right ) \int x^{7/2} \tanh ^{-1}(\tanh (a+b x)) \, dx\\ &=\frac {16}{105} b^2 x^{9/2} \tanh ^{-1}(\tanh (a+b x))-\frac {12}{35} b x^{7/2} \tanh ^{-1}(\tanh (a+b x))^2+\frac {2}{5} x^{5/2} \tanh ^{-1}(\tanh (a+b x))^3-\frac {1}{105} \left (16 b^3\right ) \int x^{9/2} \, dx\\ &=-\frac {32 b^3 x^{11/2}}{1155}+\frac {16}{105} b^2 x^{9/2} \tanh ^{-1}(\tanh (a+b x))-\frac {12}{35} b x^{7/2} \tanh ^{-1}(\tanh (a+b x))^2+\frac {2}{5} x^{5/2} \tanh ^{-1}(\tanh (a+b x))^3\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 57, normalized size = 0.83 \begin {gather*} -\frac {2 x^{5/2} \left (16 b^3 x^3-88 b^2 x^2 \tanh ^{-1}(\tanh (a+b x))+198 b x \tanh ^{-1}(\tanh (a+b x))^2-231 \tanh ^{-1}(\tanh (a+b x))^3\right )}{1155} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.21, size = 56, normalized size = 0.81
method | result | size |
derivativedivides | \(\frac {2 x^{\frac {5}{2}} \arctanh \left (\tanh \left (b x +a \right )\right )^{3}}{5}-\frac {12 b \left (\frac {x^{\frac {7}{2}} \arctanh \left (\tanh \left (b x +a \right )\right )^{2}}{7}-\frac {4 b \left (\frac {x^{\frac {9}{2}} \arctanh \left (\tanh \left (b x +a \right )\right )}{9}-\frac {2 b \,x^{\frac {11}{2}}}{99}\right )}{7}\right )}{5}\) | \(56\) |
default | \(\frac {2 x^{\frac {5}{2}} \arctanh \left (\tanh \left (b x +a \right )\right )^{3}}{5}-\frac {12 b \left (\frac {x^{\frac {7}{2}} \arctanh \left (\tanh \left (b x +a \right )\right )^{2}}{7}-\frac {4 b \left (\frac {x^{\frac {9}{2}} \arctanh \left (\tanh \left (b x +a \right )\right )}{9}-\frac {2 b \,x^{\frac {11}{2}}}{99}\right )}{7}\right )}{5}\) | \(56\) |
risch | \(\text {Expression too large to display}\) | \(8179\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 55, normalized size = 0.80 \begin {gather*} -\frac {12}{35} \, b x^{\frac {7}{2}} \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{2} + \frac {2}{5} \, x^{\frac {5}{2}} \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{3} - \frac {16}{1155} \, {\left (2 \, b^{2} x^{\frac {11}{2}} - 11 \, b x^{\frac {9}{2}} \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )\right )} b \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 40, normalized size = 0.58 \begin {gather*} \frac {2}{1155} \, {\left (105 \, b^{3} x^{5} + 385 \, a b^{2} x^{4} + 495 \, a^{2} b x^{3} + 231 \, a^{3} x^{2}\right )} \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 x^{\frac {3}{2}} \operatorname {atanh}^{3}{\left (\tanh {\left (a + b x \right )} \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 35, normalized size = 0.51 \begin {gather*} \frac {2}{11} \, b^{3} x^{\frac {11}{2}} + \frac {2}{3} \, a b^{2} x^{\frac {9}{2}} + \frac {6}{7} \, a^{2} b x^{\frac {7}{2}} + \frac {2}{5} \, a^{3} x^{\frac {5}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.17, size = 182, normalized size = 2.64 \begin {gather*} \frac {2\,b^3\,x^{11/2}}{11}-\frac {x^{5/2}\,{\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}{20}+\frac {3\,b\,x^{7/2}\,{\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}{14}-\frac {b^2\,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 )}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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