Integrand size = 10, antiderivative size = 86 \[ \int \left (a \tanh ^3(x)\right )^{3/2} \, dx=-\frac {2}{3} a \sqrt {a \tanh ^3(x)}-\frac {a \arctan \left (\sqrt {\tanh (x)}\right ) \sqrt {a \tanh ^3(x)}}{\tanh ^{\frac {3}{2}}(x)}+\frac {a \text {arctanh}\left (\sqrt {\tanh (x)}\right ) \sqrt {a \tanh ^3(x)}}{\tanh ^{\frac {3}{2}}(x)}-\frac {2}{7} a \tanh ^2(x) \sqrt {a \tanh ^3(x)} \]
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Time = 0.03 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {3739, 3554, 3557, 335, 304, 209, 212} \[ \int \left (a \tanh ^3(x)\right )^{3/2} \, dx=-\frac {a \sqrt {a \tanh ^3(x)} \arctan \left (\sqrt {\tanh (x)}\right )}{\tanh ^{\frac {3}{2}}(x)}+\frac {a \sqrt {a \tanh ^3(x)} \text {arctanh}\left (\sqrt {\tanh (x)}\right )}{\tanh ^{\frac {3}{2}}(x)}-\frac {2}{3} a \sqrt {a \tanh ^3(x)}-\frac {2}{7} a \tanh ^2(x) \sqrt {a \tanh ^3(x)} \]
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Rule 209
Rule 212
Rule 304
Rule 335
Rule 3554
Rule 3557
Rule 3739
Rubi steps \begin{align*} \text {integral}& = \frac {\left (a \sqrt {a \tanh ^3(x)}\right ) \int \tanh ^{\frac {9}{2}}(x) \, dx}{\tanh ^{\frac {3}{2}}(x)} \\ & = -\frac {2}{7} a \tanh ^2(x) \sqrt {a \tanh ^3(x)}+\frac {\left (a \sqrt {a \tanh ^3(x)}\right ) \int \tanh ^{\frac {5}{2}}(x) \, dx}{\tanh ^{\frac {3}{2}}(x)} \\ & = -\frac {2}{3} a \sqrt {a \tanh ^3(x)}-\frac {2}{7} a \tanh ^2(x) \sqrt {a \tanh ^3(x)}+\frac {\left (a \sqrt {a \tanh ^3(x)}\right ) \int \sqrt {\tanh (x)} \, dx}{\tanh ^{\frac {3}{2}}(x)} \\ & = -\frac {2}{3} a \sqrt {a \tanh ^3(x)}-\frac {2}{7} a \tanh ^2(x) \sqrt {a \tanh ^3(x)}-\frac {\left (a \sqrt {a \tanh ^3(x)}\right ) \text {Subst}\left (\int \frac {\sqrt {x}}{-1+x^2} \, dx,x,\tanh (x)\right )}{\tanh ^{\frac {3}{2}}(x)} \\ & = -\frac {2}{3} a \sqrt {a \tanh ^3(x)}-\frac {2}{7} a \tanh ^2(x) \sqrt {a \tanh ^3(x)}-\frac {\left (2 a \sqrt {a \tanh ^3(x)}\right ) \text {Subst}\left (\int \frac {x^2}{-1+x^4} \, dx,x,\sqrt {\tanh (x)}\right )}{\tanh ^{\frac {3}{2}}(x)} \\ & = -\frac {2}{3} a \sqrt {a \tanh ^3(x)}-\frac {2}{7} a \tanh ^2(x) \sqrt {a \tanh ^3(x)}+\frac {\left (a \sqrt {a \tanh ^3(x)}\right ) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {\tanh (x)}\right )}{\tanh ^{\frac {3}{2}}(x)}-\frac {\left (a \sqrt {a \tanh ^3(x)}\right ) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {\tanh (x)}\right )}{\tanh ^{\frac {3}{2}}(x)} \\ & = -\frac {2}{3} a \sqrt {a \tanh ^3(x)}-\frac {a \arctan \left (\sqrt {\tanh (x)}\right ) \sqrt {a \tanh ^3(x)}}{\tanh ^{\frac {3}{2}}(x)}+\frac {a \text {arctanh}\left (\sqrt {\tanh (x)}\right ) \sqrt {a \tanh ^3(x)}}{\tanh ^{\frac {3}{2}}(x)}-\frac {2}{7} a \tanh ^2(x) \sqrt {a \tanh ^3(x)} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.64 \[ \int \left (a \tanh ^3(x)\right )^{3/2} \, dx=-\frac {\left (a \tanh ^3(x)\right )^{3/2} \left (21 \arctan \left (\sqrt {\tanh (x)}\right )-21 \text {arctanh}\left (\sqrt {\tanh (x)}\right )+14 \tanh ^{\frac {3}{2}}(x)+6 \tanh ^{\frac {7}{2}}(x)\right )}{21 \tanh ^{\frac {9}{2}}(x)} \]
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Time = 0.09 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.88
method | result | size |
derivativedivides | \(\frac {\left (a \tanh \left (x \right )^{3}\right )^{\frac {3}{2}} \left (21 a^{\frac {7}{2}} \operatorname {arctanh}\left (\frac {\sqrt {a \tanh \left (x \right )}}{\sqrt {a}}\right )-21 a^{\frac {7}{2}} \arctan \left (\frac {\sqrt {a \tanh \left (x \right )}}{\sqrt {a}}\right )-6 \left (a \tanh \left (x \right )\right )^{\frac {7}{2}}-14 a^{2} \left (a \tanh \left (x \right )\right )^{\frac {3}{2}}\right )}{21 \tanh \left (x \right )^{3} \left (a \tanh \left (x \right )\right )^{\frac {3}{2}} a^{2}}\) | \(76\) |
default | \(\frac {\left (a \tanh \left (x \right )^{3}\right )^{\frac {3}{2}} \left (21 a^{\frac {7}{2}} \operatorname {arctanh}\left (\frac {\sqrt {a \tanh \left (x \right )}}{\sqrt {a}}\right )-21 a^{\frac {7}{2}} \arctan \left (\frac {\sqrt {a \tanh \left (x \right )}}{\sqrt {a}}\right )-6 \left (a \tanh \left (x \right )\right )^{\frac {7}{2}}-14 a^{2} \left (a \tanh \left (x \right )\right )^{\frac {3}{2}}\right )}{21 \tanh \left (x \right )^{3} \left (a \tanh \left (x \right )\right )^{\frac {3}{2}} a^{2}}\) | \(76\) |
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Leaf count of result is larger than twice the leaf count of optimal. 615 vs. \(2 (66) = 132\).
Time = 0.29 (sec) , antiderivative size = 1269, normalized size of antiderivative = 14.76 \[ \int \left (a \tanh ^3(x)\right )^{3/2} \, dx=\text {Too large to display} \]
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\[ \int \left (a \tanh ^3(x)\right )^{3/2} \, dx=\int \left (a \tanh ^{3}{\left (x \right )}\right )^{\frac {3}{2}}\, dx \]
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\[ \int \left (a \tanh ^3(x)\right )^{3/2} \, dx=\int { \left (a \tanh \left (x\right )^{3}\right )^{\frac {3}{2}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 342 vs. \(2 (66) = 132\).
Time = 0.35 (sec) , antiderivative size = 342, normalized size of antiderivative = 3.98 \[ \int \left (a \tanh ^3(x)\right )^{3/2} \, dx=-\frac {1}{42} \, {\left (42 \, \sqrt {a} \arctan \left (-\frac {\sqrt {a} e^{\left (2 \, x\right )} - \sqrt {a e^{\left (4 \, x\right )} - a}}{\sqrt {a}}\right ) \mathrm {sgn}\left (e^{\left (4 \, x\right )} - 1\right ) + 21 \, \sqrt {a} \log \left ({\left | -\sqrt {a} e^{\left (2 \, x\right )} + \sqrt {a e^{\left (4 \, x\right )} - a} \right |}\right ) \mathrm {sgn}\left (e^{\left (4 \, x\right )} - 1\right ) + \frac {16 \, {\left (21 \, {\left (\sqrt {a} e^{\left (2 \, x\right )} - \sqrt {a e^{\left (4 \, x\right )} - a}\right )}^{6} a \mathrm {sgn}\left (e^{\left (4 \, x\right )} - 1\right ) + 42 \, {\left (\sqrt {a} e^{\left (2 \, x\right )} - \sqrt {a e^{\left (4 \, x\right )} - a}\right )}^{5} a^{\frac {3}{2}} \mathrm {sgn}\left (e^{\left (4 \, x\right )} - 1\right ) + 119 \, {\left (\sqrt {a} e^{\left (2 \, x\right )} - \sqrt {a e^{\left (4 \, x\right )} - a}\right )}^{4} a^{2} \mathrm {sgn}\left (e^{\left (4 \, x\right )} - 1\right ) + 56 \, {\left (\sqrt {a} e^{\left (2 \, x\right )} - \sqrt {a e^{\left (4 \, x\right )} - a}\right )}^{3} a^{\frac {5}{2}} \mathrm {sgn}\left (e^{\left (4 \, x\right )} - 1\right ) + 63 \, {\left (\sqrt {a} e^{\left (2 \, x\right )} - \sqrt {a e^{\left (4 \, x\right )} - a}\right )}^{2} a^{3} \mathrm {sgn}\left (e^{\left (4 \, x\right )} - 1\right ) + 14 \, {\left (\sqrt {a} e^{\left (2 \, x\right )} - \sqrt {a e^{\left (4 \, x\right )} - a}\right )} a^{\frac {7}{2}} \mathrm {sgn}\left (e^{\left (4 \, x\right )} - 1\right ) + 5 \, a^{4} \mathrm {sgn}\left (e^{\left (4 \, x\right )} - 1\right )\right )}}{{\left (\sqrt {a} e^{\left (2 \, x\right )} - \sqrt {a e^{\left (4 \, x\right )} - a} + \sqrt {a}\right )}^{7}}\right )} a \]
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Timed out. \[ \int \left (a \tanh ^3(x)\right )^{3/2} \, dx=\int {\left (a\,{\mathrm {tanh}\left (x\right )}^3\right )}^{3/2} \,d x \]
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