Optimal. Leaf size=88 \[ -\frac {\sqrt {-\tanh ^2(c+d x)} \tanh (c+d x)}{2 d}-\frac {\sqrt {-\tanh ^2(c+d x)} \tanh ^3(c+d x)}{4 d}+\frac {\sqrt {-\tanh ^2(c+d x)} \coth (c+d x) \log (\cosh (c+d x))}{d} \]
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Rubi [A] time = 0.05, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {3658, 3473, 3475} \[ -\frac {\sqrt {-\tanh ^2(c+d x)} \tanh ^3(c+d x)}{4 d}-\frac {\sqrt {-\tanh ^2(c+d x)} \tanh (c+d x)}{2 d}+\frac {\sqrt {-\tanh ^2(c+d x)} \coth (c+d x) \log (\cosh (c+d x))}{d} \]
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 )^{5/2} \, dx &=\left (\coth (c+d x) \sqrt {-\tanh ^2(c+d x)}\right ) \int \tanh ^5(c+d x) \, dx\\ &=-\frac {\tanh ^3(c+d x) \sqrt {-\tanh ^2(c+d x)}}{4 d}+\left (\coth (c+d x) \sqrt {-\tanh ^2(c+d x)}\right ) \int \tanh ^3(c+d x) \, dx\\ &=-\frac {\tanh (c+d x) \sqrt {-\tanh ^2(c+d x)}}{2 d}-\frac {\tanh ^3(c+d x) \sqrt {-\tanh ^2(c+d x)}}{4 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}-\frac {\tanh ^3(c+d x) \sqrt {-\tanh ^2(c+d x)}}{4 d}\\ \end {align*}
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Mathematica [A] time = 0.31, size = 56, normalized size = 0.64 \[ \frac {\left (-\tanh ^2(c+d x)\right )^{5/2} \coth (c+d x) \left (-2 \coth ^2(c+d x)+4 \coth ^4(c+d x) \log (\cosh (c+d x))-1\right )}{4 d} \]
Antiderivative was successfully verified.
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fricas [C] time = 0.48, size = 177, normalized size = 2.01 \[ \frac {-i \, d x e^{\left (8 \, d x + 8 \, c\right )} - i \, d x + {\left (-4 i \, d x + 4 i\right )} e^{\left (6 \, d x + 6 \, c\right )} + {\left (-6 i \, d x + 4 i\right )} e^{\left (4 \, d x + 4 \, c\right )} + {\left (-4 i \, d x + 4 i\right )} e^{\left (2 \, d x + 2 \, c\right )} + {\left (i \, e^{\left (8 \, d x + 8 \, c\right )} + 4 i \, e^{\left (6 \, d x + 6 \, c\right )} + 6 i \, e^{\left (4 \, d x + 4 \, c\right )} + 4 i \, e^{\left (2 \, d x + 2 \, c\right )} + i\right )} \log \left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}{d e^{\left (8 \, d x + 8 \, c\right )} + 4 \, d e^{\left (6 \, d x + 6 \, c\right )} + 6 \, d e^{\left (4 \, d x + 4 \, c\right )} + 4 \, d e^{\left (2 \, d x + 2 \, c\right )} + d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 1.89, size = 142, normalized size = 1.61 \[ \frac {i \, {\left (d x + c\right )} \mathrm {sgn}\left (-e^{\left (4 \, d x + 4 \, c\right )} + 1\right ) - i \, \log \left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right ) \mathrm {sgn}\left (-e^{\left (4 \, d x + 4 \, c\right )} + 1\right ) - \frac {4 i \, {\left (e^{\left (6 \, d x + 6 \, c\right )} \mathrm {sgn}\left (-e^{\left (4 \, d x + 4 \, c\right )} + 1\right ) + e^{\left (4 \, d x + 4 \, c\right )} \mathrm {sgn}\left (-e^{\left (4 \, d x + 4 \, c\right )} + 1\right ) + e^{\left (2 \, d x + 2 \, c\right )} \mathrm {sgn}\left (-e^{\left (4 \, d x + 4 \, c\right )} + 1\right )\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{4}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.12, size = 67, normalized size = 0.76 \[ -\frac {\left (-\left (\tanh ^{2}\left (d x +c \right )\right )\right )^{\frac {5}{2}} \left (\tanh ^{4}\left (d x +c \right )+2 \left (\tanh ^{2}\left (d x +c \right )\right )+2 \ln \left (\tanh \left (d x +c \right )-1\right )+2 \ln \left (1+\tanh \left (d x +c \right )\right )\right )}{4 d \tanh \left (d x +c \right )^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.46, size = 113, normalized size = 1.28 \[ -\frac {i \, {\left (d x + c\right )}}{d} - \frac {i \, \log \left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}{d} - \frac {4 i \, e^{\left (-2 \, d x - 2 \, c\right )} + 4 i \, e^{\left (-4 \, d x - 4 \, c\right )} + 4 i \, e^{\left (-6 \, d x - 6 \, c\right )}}{d {\left (4 \, e^{\left (-2 \, d x - 2 \, c\right )} + 6 \, e^{\left (-4 \, d x - 4 \, c\right )} + 4 \, e^{\left (-6 \, d x - 6 \, c\right )} + e^{\left (-8 \, d x - 8 \, c\right )} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (-{\mathrm {tanh}\left (c+d\,x\right )}^2\right )}^{5/2} \,d x \]
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 \[ \int \left (- \tanh ^{2}{\left (c + d x \right )}\right )^{\frac {5}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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