Optimal. Leaf size=80 \[ -\frac {32 \tanh ^{-1}(\tanh (a+b x))^{11/2}}{1155 b^4}+\frac {16 x \tanh ^{-1}(\tanh (a+b x))^{9/2}}{105 b^3}-\frac {12 x^2 \tanh ^{-1}(\tanh (a+b x))^{7/2}}{35 b^2}+\frac {2 x^3 \tanh ^{-1}(\tanh (a+b x))^{5/2}}{5 b} \]
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Rubi [A] time = 0.05, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2168, 2157, 30} \[ -\frac {12 x^2 \tanh ^{-1}(\tanh (a+b x))^{7/2}}{35 b^2}-\frac {32 \tanh ^{-1}(\tanh (a+b x))^{11/2}}{1155 b^4}+\frac {16 x \tanh ^{-1}(\tanh (a+b x))^{9/2}}{105 b^3}+\frac {2 x^3 \tanh ^{-1}(\tanh (a+b x))^{5/2}}{5 b} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2157
Rule 2168
Rubi steps
\begin {align*} \int x^3 \tanh ^{-1}(\tanh (a+b x))^{3/2} \, dx &=\frac {2 x^3 \tanh ^{-1}(\tanh (a+b x))^{5/2}}{5 b}-\frac {6 \int x^2 \tanh ^{-1}(\tanh (a+b x))^{5/2} \, dx}{5 b}\\ &=\frac {2 x^3 \tanh ^{-1}(\tanh (a+b x))^{5/2}}{5 b}-\frac {12 x^2 \tanh ^{-1}(\tanh (a+b x))^{7/2}}{35 b^2}+\frac {24 \int x \tanh ^{-1}(\tanh (a+b x))^{7/2} \, dx}{35 b^2}\\ &=\frac {2 x^3 \tanh ^{-1}(\tanh (a+b x))^{5/2}}{5 b}-\frac {12 x^2 \tanh ^{-1}(\tanh (a+b x))^{7/2}}{35 b^2}+\frac {16 x \tanh ^{-1}(\tanh (a+b x))^{9/2}}{105 b^3}-\frac {16 \int \tanh ^{-1}(\tanh (a+b x))^{9/2} \, dx}{105 b^3}\\ &=\frac {2 x^3 \tanh ^{-1}(\tanh (a+b x))^{5/2}}{5 b}-\frac {12 x^2 \tanh ^{-1}(\tanh (a+b x))^{7/2}}{35 b^2}+\frac {16 x \tanh ^{-1}(\tanh (a+b x))^{9/2}}{105 b^3}-\frac {16 \operatorname {Subst}\left (\int x^{9/2} \, dx,x,\tanh ^{-1}(\tanh (a+b x))\right )}{105 b^4}\\ &=\frac {2 x^3 \tanh ^{-1}(\tanh (a+b x))^{5/2}}{5 b}-\frac {12 x^2 \tanh ^{-1}(\tanh (a+b x))^{7/2}}{35 b^2}+\frac {16 x \tanh ^{-1}(\tanh (a+b x))^{9/2}}{105 b^3}-\frac {32 \tanh ^{-1}(\tanh (a+b x))^{11/2}}{1155 b^4}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 66, normalized size = 0.82 \[ \frac {2 \tanh ^{-1}(\tanh (a+b x))^{5/2} \left (-198 b^2 x^2 \tanh ^{-1}(\tanh (a+b x))+88 b x \tanh ^{-1}(\tanh (a+b x))^2-16 \tanh ^{-1}(\tanh (a+b x))^3+231 b^3 x^3\right )}{1155 b^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 64, normalized size = 0.80 \[ \frac {2 \, {\left (105 \, b^{5} x^{5} + 140 \, a b^{4} x^{4} + 5 \, a^{2} b^{3} x^{3} - 6 \, a^{3} b^{2} x^{2} + 8 \, a^{4} b x - 16 \, a^{5}\right )} \sqrt {b x + a}}{1155 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.20, size = 205, normalized size = 2.56 \[ \frac {\sqrt {2} {\left (\frac {99 \, \sqrt {2} {\left (5 \, {\left (b x + a\right )}^{\frac {7}{2}} - 21 \, {\left (b x + a\right )}^{\frac {5}{2}} a + 35 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{2} - 35 \, \sqrt {b x + a} a^{3}\right )} a^{2}}{b^{3}} + \frac {22 \, \sqrt {2} {\left (35 \, {\left (b x + a\right )}^{\frac {9}{2}} - 180 \, {\left (b x + a\right )}^{\frac {7}{2}} a + 378 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{2} - 420 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{3} + 315 \, \sqrt {b x + a} a^{4}\right )} a}{b^{3}} + \frac {5 \, \sqrt {2} {\left (63 \, {\left (b x + a\right )}^{\frac {11}{2}} - 385 \, {\left (b x + a\right )}^{\frac {9}{2}} a + 990 \, {\left (b x + a\right )}^{\frac {7}{2}} a^{2} - 1386 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{3} + 1155 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{4} - 693 \, \sqrt {b x + a} a^{5}\right )}}{b^{3}}\right )}}{3465 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.14, size = 124, normalized size = 1.55 \[ \frac {\frac {2 \arctanh \left (\tanh \left (b x +a \right )\right )^{\frac {11}{2}}}{11}+\frac {2 \left (-3 \arctanh \left (\tanh \left (b x +a \right )\right )+3 b x \right ) \arctanh \left (\tanh \left (b x +a \right )\right )^{\frac {9}{2}}}{9}+\frac {2 \left (\left (b x -\arctanh \left (\tanh \left (b x +a \right )\right )\right ) \left (-2 \arctanh \left (\tanh \left (b x +a \right )\right )+2 b x \right )+\left (b x -\arctanh \left (\tanh \left (b x +a \right )\right )\right )^{2}\right ) \arctanh \left (\tanh \left (b x +a \right )\right )^{\frac {7}{2}}}{7}+\frac {2 \left (b x -\arctanh \left (\tanh \left (b x +a \right )\right )\right )^{3} \arctanh \left (\tanh \left (b x +a \right )\right )^{\frac {5}{2}}}{5}}{b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.53, size = 53, normalized size = 0.66 \[ \frac {2 \, {\left (105 \, b^{4} x^{4} + 35 \, a b^{3} x^{3} - 30 \, a^{2} b^{2} x^{2} + 24 \, a^{3} b x - 16 \, a^{4}\right )} {\left (b x + a\right )}^{\frac {3}{2}}}{1155 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.12, size = 1483, normalized size = 18.54 \[ \text {result too large to display} \]
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 \[ \int x^{3} \operatorname {atanh}^{\frac {3}{2}}{\left (\tanh {\left (a + b x \right )} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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