Optimal. Leaf size=60 \[ -\frac {\coth (c+d x) \log (\cosh (c+d x)) \sqrt {-\tanh ^2(c+d x)}}{d}+\frac {\tanh (c+d x) \sqrt {-\tanh ^2(c+d x)}}{2 d} \]
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Rubi [A]
time = 0.02, 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 = {3739, 3554,
3556} \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 3554
Rule 3556
Rule 3739
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 = 0.07, size = 46, normalized size = 0.77 \begin {gather*} \frac {\coth (c+d x) \left (-1+2 \coth ^2(c+d x) \log (\cosh (c+d x))\right ) \left (-\tanh ^2(c+d x)\right )^{3/2}}{2 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.33, size = 53, normalized size = 0.88
method | result | size |
derivativedivides | \(-\frac {\left (-\left (\tanh ^{2}\left (d x +c \right )\right )\right )^{\frac {3}{2}} \left (\tanh ^{2}\left (d x +c \right )+\ln \left (\tanh \left (d x +c \right )-1\right )+\ln \left (\tanh \left (d x +c \right )+1\right )\right )}{2 d \tanh \left (d x +c \right )^{3}}\) | \(53\) |
default | \(-\frac {\left (-\left (\tanh ^{2}\left (d x +c \right )\right )\right )^{\frac {3}{2}} \left (\tanh ^{2}\left (d x +c \right )+\ln \left (\tanh \left (d x +c \right )-1\right )+\ln \left (\tanh \left (d x +c \right )+1\right )\right )}{2 d \tanh \left (d x +c \right )^{3}}\) | \(53\) |
risch | \(-\frac {\left (1+{\mathrm e}^{2 d x +2 c}\right ) \sqrt {-\frac {\left ({\mathrm e}^{2 d x +2 c}-1\right )^{2}}{\left (1+{\mathrm e}^{2 d x +2 c}\right )^{2}}}\, x}{{\mathrm e}^{2 d x +2 c}-1}+\frac {2 \left (1+{\mathrm e}^{2 d x +2 c}\right ) \sqrt {-\frac {\left ({\mathrm e}^{2 d x +2 c}-1\right )^{2}}{\left (1+{\mathrm e}^{2 d x +2 c}\right )^{2}}}\, \left (d x +c \right )}{\left ({\mathrm e}^{2 d x +2 c}-1\right ) d}-\frac {2 \sqrt {-\frac {\left ({\mathrm e}^{2 d x +2 c}-1\right )^{2}}{\left (1+{\mathrm e}^{2 d x +2 c}\right )^{2}}}\, {\mathrm e}^{2 d x +2 c}}{\left ({\mathrm e}^{2 d x +2 c}-1\right ) \left (1+{\mathrm e}^{2 d x +2 c}\right ) d}-\frac {\left (1+{\mathrm e}^{2 d x +2 c}\right ) \sqrt {-\frac {\left ({\mathrm e}^{2 d x +2 c}-1\right )^{2}}{\left (1+{\mathrm e}^{2 d x +2 c}\right )^{2}}}\, \ln \left (1+{\mathrm e}^{2 d x +2 c}\right )}{\left ({\mathrm e}^{2 d x +2 c}-1\right ) d}\) | \(264\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 0.51, size = 66, normalized size = 1.10 \begin {gather*} \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 )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains complex when optimal does not.
time = 0.39, size = 100, normalized size = 1.67 \begin {gather*} \frac {i \, d x e^{\left (4 \, d x + 4 \, c\right )} + i \, d x - 2 \, {\left (-i \, d x + 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} \end {gather*}
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 \begin {gather*} \int \left (- \tanh ^{2}{\left (c + d x \right )}\right )^{\frac {3}{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] Result contains complex when optimal does not.
time = 0.42, size = 92, normalized size = 1.53 \begin {gather*} \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} \end {gather*}
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 \begin {gather*} \int {\left (-{\mathrm {tanh}\left (c+d\,x\right )}^2\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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