Optimal. Leaf size=69 \[ \frac {16}{105} b^2 x^{9/2} \tanh ^{-1}(\tanh (a+b x))-\frac {12}{35} b x^{7/2} \tanh ^{-1}(\tanh (a+b x))^2+\frac {2}{5} x^{5/2} \tanh ^{-1}(\tanh (a+b x))^3-\frac {32 b^3 x^{11/2}}{1155} \]
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Rubi [A] time = 0.04, 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 = {2168, 30} \[ \frac {16}{105} b^2 x^{9/2} \tanh ^{-1}(\tanh (a+b x))-\frac {12}{35} b x^{7/2} \tanh ^{-1}(\tanh (a+b x))^2+\frac {2}{5} x^{5/2} \tanh ^{-1}(\tanh (a+b x))^3-\frac {32 b^3 x^{11/2}}{1155} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2168
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 = 0.03, size = 57, normalized size = 0.83 \[ -\frac {2 x^{5/2} \left (-88 b^2 x^2 \tanh ^{-1}(\tanh (a+b x))+198 b x \tanh ^{-1}(\tanh (a+b x))^2-231 \tanh ^{-1}(\tanh (a+b x))^3+16 b^3 x^3\right )}{1155} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.62, size = 40, normalized size = 0.58 \[ \frac {2}{1155} \, {\left (105 \, b^{3} x^{5} + 385 \, a b^{2} x^{4} + 495 \, a^{2} b x^{3} + 231 \, a^{3} x^{2}\right )} \sqrt {x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 35, normalized size = 0.51 \[ \frac {2}{11} \, b^{3} x^{\frac {11}{2}} + \frac {2}{3} \, a b^{2} x^{\frac {9}{2}} + \frac {6}{7} \, a^{2} b x^{\frac {7}{2}} + \frac {2}{5} \, a^{3} x^{\frac {5}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.25, size = 56, normalized size = 0.81 \[ \frac {2 x^{\frac {5}{2}} \arctanh \left (\tanh \left (b x +a \right )\right )^{3}}{5}-\frac {12 b \left (\frac {x^{\frac {7}{2}} \arctanh \left (\tanh \left (b x +a \right )\right )^{2}}{7}-\frac {4 b \left (\frac {x^{\frac {9}{2}} \arctanh \left (\tanh \left (b x +a \right )\right )}{9}-\frac {2 b \,x^{\frac {11}{2}}}{99}\right )}{7}\right )}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 55, normalized size = 0.80 \[ -\frac {12}{35} \, b x^{\frac {7}{2}} \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{2} + \frac {2}{5} \, x^{\frac {5}{2}} \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{3} - \frac {16}{1155} \, {\left (2 \, b^{2} x^{\frac {11}{2}} - 11 \, b x^{\frac {9}{2}} \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )\right )} b \]
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 \[ \frac {2\,b^3\,x^{11/2}}{11}-\frac {x^{5/2}\,{\left (\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x\right )}^3}{20}+\frac {3\,b\,x^{7/2}\,{\left (\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x\right )}^2}{14}-\frac {b^2\,x^{9/2}\,\left (\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x\right )}{3} \]
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 \[ \int x^{\frac {3}{2}} \operatorname {atanh}^{3}{\left (\tanh {\left (a + b x \right )} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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