Optimal. Leaf size=80 \[ -\frac{12 x^2 \tanh ^{-1}(\tanh (a+b x))^{7/2}}{35 b^2}-\frac{32 \tanh ^{-1}(\tanh (a+b x))^{11/2}}{1155 b^4}+\frac{16 x \tanh ^{-1}(\tanh (a+b x))^{9/2}}{105 b^3}+\frac{2 x^3 \tanh ^{-1}(\tanh (a+b x))^{5/2}}{5 b} \]
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Rubi [A] time = 0.0460745, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, 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 2168
Rule 2157
Rule 30
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*}
Mathematica [A] time = 0.0334686, 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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Maple [A] time = 0.033, size = 124, normalized size = 1.6 \begin{align*} 2\,{\frac{1/11\, \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{11/2}+1/9\, \left ( -3\,{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) +3\,bx \right ) \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{9/2}+1/7\, \left ( \left ( bx-{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) \left ( -2\,{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) +2\,bx \right ) + \left ( bx-{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{2} \right ) \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{7/2}+1/5\, \left ( bx-{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{3} \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{5/2}}{{b}^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.76476, size = 72, normalized size = 0.9 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.79939, size = 147, normalized size = 1.84 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{3} \operatorname{atanh}^{\frac{3}{2}}{\left (\tanh{\left (a + b x \right )} \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15894, size = 169, normalized size = 2.11 \begin{align*} \frac{\sqrt{2}{\left (\frac{11 \, \sqrt{2}{\left (35 \,{\left (b x + a\right )}^{\frac{9}{2}} - 135 \,{\left (b x + a\right )}^{\frac{7}{2}} a + 189 \,{\left (b x + a\right )}^{\frac{5}{2}} a^{2} - 105 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{3}\right )} a}{b^{3}} + \frac{\sqrt{2}{\left (315 \,{\left (b x + a\right )}^{\frac{11}{2}} - 1540 \,{\left (b x + a\right )}^{\frac{9}{2}} a + 2970 \,{\left (b x + a\right )}^{\frac{7}{2}} a^{2} - 2772 \,{\left (b x + a\right )}^{\frac{5}{2}} a^{3} + 1155 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{4}\right )}}{b^{3}}\right )}}{3465 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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