Optimal. Leaf size=88 \[ -\frac {\coth ^3(c+d x)}{4 d \sqrt {-\tanh ^2(c+d x)}}-\frac {\coth (c+d x)}{2 d \sqrt {-\tanh ^2(c+d x)}}+\frac {\tanh (c+d x) \log (\sinh (c+d x))}{d \sqrt {-\tanh ^2(c+d x)}} \]
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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} \[ -\frac {\coth ^3(c+d x)}{4 d \sqrt {-\tanh ^2(c+d x)}}-\frac {\coth (c+d x)}{2 d \sqrt {-\tanh ^2(c+d x)}}+\frac {\tanh (c+d x) \log (\sinh (c+d x))}{d \sqrt {-\tanh ^2(c+d x)}} \]
Antiderivative was successfully verified.
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Rule 3473
Rule 3475
Rule 3658
Rubi steps
\begin {align*} \int \frac {1}{\left (-\tanh ^2(c+d x)\right )^{5/2}} \, dx &=\frac {\tanh (c+d x) \int \coth ^5(c+d x) \, dx}{\sqrt {-\tanh ^2(c+d x)}}\\ &=-\frac {\coth ^3(c+d x)}{4 d \sqrt {-\tanh ^2(c+d x)}}+\frac {\tanh (c+d x) \int \coth ^3(c+d x) \, dx}{\sqrt {-\tanh ^2(c+d x)}}\\ &=-\frac {\coth (c+d x)}{2 d \sqrt {-\tanh ^2(c+d x)}}-\frac {\coth ^3(c+d x)}{4 d \sqrt {-\tanh ^2(c+d x)}}+\frac {\tanh (c+d x) \int \coth (c+d x) \, dx}{\sqrt {-\tanh ^2(c+d x)}}\\ &=-\frac {\coth (c+d x)}{2 d \sqrt {-\tanh ^2(c+d x)}}-\frac {\coth ^3(c+d x)}{4 d \sqrt {-\tanh ^2(c+d x)}}+\frac {\log (\sinh (c+d x)) \tanh (c+d x)}{d \sqrt {-\tanh ^2(c+d x)}}\\ \end {align*}
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Mathematica [A] time = 0.25, size = 63, normalized size = 0.72 \[ \frac {-\coth ^3(c+d x)-2 \coth (c+d x)+4 \tanh (c+d x) (\log (\tanh (c+d x))+\log (\cosh (c+d x)))}{4 d \sqrt {-\tanh ^2(c+d x)}} \]
Antiderivative was successfully verified.
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fricas [C] time = 0.69, 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 = 1.17, size = 83, normalized size = 0.94 \[ \frac {\frac {8 i \, e^{\left (6 \, d x + 6 \, c\right )} - 8 i \, e^{\left (4 \, d x + 4 \, c\right )} + 8 i \, e^{\left (2 \, d x + 2 \, c\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{4}} + i \, \log \left (-i \, e^{\left (2 \, d x + 2 \, c\right )}\right ) - 2 i \, \log \left ({\left | e^{\left (2 \, d x + 2 \, c\right )} - 1 \right |}\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.13, size = 91, normalized size = 1.03 \[ -\frac {\tanh \left (d x +c \right ) \left (2 \ln \left (\tanh \left (d x +c \right )-1\right ) \left (\tanh ^{4}\left (d x +c \right )\right )+2 \ln \left (1+\tanh \left (d x +c \right )\right ) \left (\tanh ^{4}\left (d x +c \right )\right )-4 \ln \left (\tanh \left (d x +c \right )\right ) \left (\tanh ^{4}\left (d x +c \right )\right )+2 \left (\tanh ^{2}\left (d x +c \right )\right )+1\right )}{4 d \left (-\left (\tanh ^{2}\left (d x +c \right )\right )\right )^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.44, size = 131, normalized size = 1.49 \[ \frac {i \, {\left (d x + c\right )}}{d} + \frac {i \, \log \left (e^{\left (-d x - c\right )} + 1\right )}{d} + \frac {i \, \log \left (e^{\left (-d x - c\right )} - 1\right )}{d} + \frac {4 i \, e^{\left (-2 \, d x - 2 \, c\right )} - 4 i \, e^{\left (-4 \, d x - 4 \, c\right )} + 4 i \, e^{\left (-6 \, d x - 6 \, c\right )}}{d {\left (4 \, e^{\left (-2 \, d x - 2 \, c\right )} - 6 \, e^{\left (-4 \, d x - 4 \, c\right )} + 4 \, e^{\left (-6 \, d x - 6 \, c\right )} - e^{\left (-8 \, d x - 8 \, c\right )} - 1\right )}} \]
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 \[ \int \frac {1}{{\left (-{\mathrm {tanh}\left (c+d\,x\right )}^2\right )}^{5/2}} \,d x \]
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 \[ \int \frac {1}{\left (- \tanh ^{2}{\left (c + d x \right )}\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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