Optimal. Leaf size=60 \[ \frac {\tanh (c+d x) \sqrt {-\tanh ^2(c+d x)}}{2 d}-\frac {\sqrt {-\tanh ^2(c+d x)} \coth (c+d x) \log (\cosh (c+d x))}{d} \]
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Rubi [A] time = 0.03, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {3658, 3473, 3475} \[ \frac {\tanh (c+d x) \sqrt {-\tanh ^2(c+d x)}}{2 d}-\frac {\sqrt {-\tanh ^2(c+d x)} \coth (c+d x) \log (\cosh (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 3473
Rule 3475
Rule 3658
Rubi steps
\begin {align*} \int \left (-\tanh ^2(c+d x)\right )^{3/2} \, dx &=-\left (\left (\coth (c+d x) \sqrt {-\tanh ^2(c+d x)}\right ) \int \tanh ^3(c+d x) \, dx\right )\\ &=\frac {\tanh (c+d x) \sqrt {-\tanh ^2(c+d x)}}{2 d}-\left (\coth (c+d x) \sqrt {-\tanh ^2(c+d x)}\right ) \int \tanh (c+d x) \, dx\\ &=-\frac {\coth (c+d x) \log (\cosh (c+d x)) \sqrt {-\tanh ^2(c+d x)}}{d}+\frac {\tanh (c+d x) \sqrt {-\tanh ^2(c+d x)}}{2 d}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 46, normalized size = 0.77 \[ \frac {\left (-\tanh ^2(c+d x)\right )^{3/2} \coth (c+d x) \left (2 \coth ^2(c+d x) \log (\cosh (c+d x))-1\right )}{2 d} \]
Antiderivative was successfully verified.
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fricas [C] time = 0.60, size = 99, normalized size = 1.65 \[ \frac {i \, d x e^{\left (4 \, d x + 4 \, c\right )} + i \, d x + {\left (2 i \, d x - 2 i\right )} e^{\left (2 \, d x + 2 \, c\right )} + {\left (-i \, e^{\left (4 \, d x + 4 \, c\right )} - 2 i \, e^{\left (2 \, d x + 2 \, c\right )} - i\right )} \log \left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}{d e^{\left (4 \, d x + 4 \, c\right )} + 2 \, d e^{\left (2 \, d x + 2 \, c\right )} + d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 0.17, size = 92, normalized size = 1.53 \[ \frac {-i \, {\left (d x + c\right )} \mathrm {sgn}\left (-e^{\left (4 \, d x + 4 \, c\right )} + 1\right ) + i \, \log \left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right ) \mathrm {sgn}\left (-e^{\left (4 \, d x + 4 \, c\right )} + 1\right ) + \frac {2 i \, e^{\left (2 \, d x + 2 \, c\right )} \mathrm {sgn}\left (-e^{\left (4 \, d x + 4 \, c\right )} + 1\right )}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{2}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.11, size = 53, normalized size = 0.88 \[ -\frac {\left (-\left (\tanh ^{2}\left (d x +c \right )\right )\right )^{\frac {3}{2}} \left (\tanh ^{2}\left (d x +c \right )+\ln \left (\tanh \left (d x +c \right )-1\right )+\ln \left (1+\tanh \left (d x +c \right )\right )\right )}{2 d \tanh \left (d x +c \right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.46, size = 66, normalized size = 1.10 \[ \frac {i \, {\left (d x + c\right )}}{d} + \frac {i \, \log \left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}{d} + \frac {2 i \, e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} + e^{\left (-4 \, d x - 4 \, c\right )} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int {\left (-{\mathrm {tanh}\left (c+d\,x\right )}^2\right )}^{3/2} \,d x \]
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 \[ \int \left (- \tanh ^{2}{\left (c + d x \right )}\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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