Optimal. Leaf size=100 \[ -\frac{8 a^5 \tanh (c+d x)}{d}-\frac{2 a^2 (a \tanh (c+d x)+a)^3}{3 d}-\frac{2 a \left (a^2 \tanh (c+d x)+a^2\right )^2}{d}+\frac{16 a^5 \log (\cosh (c+d x))}{d}+16 a^5 x-\frac{a (a \tanh (c+d x)+a)^4}{4 d} \]
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Rubi [A] time = 0.0719517, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3478, 3477, 3475} \[ -\frac{8 a^5 \tanh (c+d x)}{d}-\frac{2 a^2 (a \tanh (c+d x)+a)^3}{3 d}-\frac{2 a \left (a^2 \tanh (c+d x)+a^2\right )^2}{d}+\frac{16 a^5 \log (\cosh (c+d x))}{d}+16 a^5 x-\frac{a (a \tanh (c+d x)+a)^4}{4 d} \]
Antiderivative was successfully verified.
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Rule 3478
Rule 3477
Rule 3475
Rubi steps
\begin{align*} \int (a+a \tanh (c+d x))^5 \, dx &=-\frac{a (a+a \tanh (c+d x))^4}{4 d}+(2 a) \int (a+a \tanh (c+d x))^4 \, dx\\ &=-\frac{2 a^2 (a+a \tanh (c+d x))^3}{3 d}-\frac{a (a+a \tanh (c+d x))^4}{4 d}+\left (4 a^2\right ) \int (a+a \tanh (c+d x))^3 \, dx\\ &=-\frac{2 a^3 (a+a \tanh (c+d x))^2}{d}-\frac{2 a^2 (a+a \tanh (c+d x))^3}{3 d}-\frac{a (a+a \tanh (c+d x))^4}{4 d}+\left (8 a^3\right ) \int (a+a \tanh (c+d x))^2 \, dx\\ &=16 a^5 x-\frac{8 a^5 \tanh (c+d x)}{d}-\frac{2 a^3 (a+a \tanh (c+d x))^2}{d}-\frac{2 a^2 (a+a \tanh (c+d x))^3}{3 d}-\frac{a (a+a \tanh (c+d x))^4}{4 d}+\left (16 a^5\right ) \int \tanh (c+d x) \, dx\\ &=16 a^5 x+\frac{16 a^5 \log (\cosh (c+d x))}{d}-\frac{8 a^5 \tanh (c+d x)}{d}-\frac{2 a^3 (a+a \tanh (c+d x))^2}{d}-\frac{2 a^2 (a+a \tanh (c+d x))^3}{3 d}-\frac{a (a+a \tanh (c+d x))^4}{4 d}\\ \end{align*}
Mathematica [B] time = 1.60801, size = 202, normalized size = 2.02 \[ \frac{a^5 \text{sech}(c) \text{sech}^4(c+d x) (-70 \sinh (c+2 d x)+30 \sinh (3 c+2 d x)-25 \sinh (3 c+4 d x)+48 d x \cosh (3 c+2 d x)+18 \cosh (3 c+2 d x)+12 d x \cosh (3 c+4 d x)+12 d x \cosh (5 c+4 d x)+48 \cosh (3 c+2 d x) \log (\cosh (c+d x))+12 \cosh (3 c+4 d x) \log (\cosh (c+d x))+12 \cosh (5 c+4 d x) \log (\cosh (c+d x))+6 \cosh (c+2 d x) (8 \log (\cosh (c+d x))+8 d x+3)+\cosh (c) (72 \log (\cosh (c+d x))+72 d x+33)+75 \sinh (c))}{12 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 81, normalized size = 0.8 \begin{align*} -{\frac{{a}^{5} \left ( \tanh \left ( dx+c \right ) \right ) ^{4}}{4\,d}}-{\frac{5\,{a}^{5} \left ( \tanh \left ( dx+c \right ) \right ) ^{3}}{3\,d}}-{\frac{11\,{a}^{5} \left ( \tanh \left ( dx+c \right ) \right ) ^{2}}{2\,d}}-15\,{\frac{{a}^{5}\tanh \left ( dx+c \right ) }{d}}-16\,{\frac{{a}^{5}\ln \left ( \tanh \left ( dx+c \right ) -1 \right ) }{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.59054, size = 408, normalized size = 4.08 \begin{align*} \frac{5}{3} \, a^{5}{\left (3 \, x + \frac{3 \, c}{d} - \frac{4 \,{\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} + 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + 2\right )}}{d{\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} + 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} + 1\right )}}\right )} + a^{5}{\left (x + \frac{c}{d} + \frac{\log \left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}{d} + \frac{4 \,{\left (e^{\left (-2 \, d x - 2 \, c\right )} + e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )}\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 )}}\right )} + 10 \, a^{5}{\left (x + \frac{c}{d} + \frac{\log \left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}{d} + \frac{2 \, 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 )}}\right )} + 10 \, a^{5}{\left (x + \frac{c}{d} - \frac{2}{d{\left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}}\right )} + a^{5} x + \frac{5 \, a^{5} \log \left (\cosh \left (d x + c\right )\right )}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.35765, size = 2376, normalized size = 23.76 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.615705, size = 95, normalized size = 0.95 \begin{align*} \begin{cases} 32 a^{5} x - \frac{16 a^{5} \log{\left (\tanh{\left (c + d x \right )} + 1 \right )}}{d} - \frac{a^{5} \tanh ^{4}{\left (c + d x \right )}}{4 d} - \frac{5 a^{5} \tanh ^{3}{\left (c + d x \right )}}{3 d} - \frac{11 a^{5} \tanh ^{2}{\left (c + d x \right )}}{2 d} - \frac{15 a^{5} \tanh{\left (c + d x \right )}}{d} & \text{for}\: d \neq 0 \\x \left (a \tanh{\left (c \right )} + a\right )^{5} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20261, size = 101, normalized size = 1.01 \begin{align*} \frac{4}{3} \, a^{5}{\left (\frac{12 \, \log \left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}{d} + \frac{48 \, e^{\left (6 \, d x + 6 \, c\right )} + 108 \, e^{\left (4 \, d x + 4 \, c\right )} + 88 \, e^{\left (2 \, d x + 2 \, c\right )} + 25}{d{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{4}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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