Optimal. Leaf size=54 \[ -\frac{(a+b) \tanh ^3(c+d x)}{3 d}-\frac{(a+b) \tanh (c+d x)}{d}+x (a+b)-\frac{b \tanh ^5(c+d x)}{5 d} \]
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Rubi [A] time = 0.0519301, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3631, 3473, 8} \[ -\frac{(a+b) \tanh ^3(c+d x)}{3 d}-\frac{(a+b) \tanh (c+d x)}{d}+x (a+b)-\frac{b \tanh ^5(c+d x)}{5 d} \]
Antiderivative was successfully verified.
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Rule 3631
Rule 3473
Rule 8
Rubi steps
\begin{align*} \int \tanh ^4(c+d x) \left (a+b \tanh ^2(c+d x)\right ) \, dx &=-\frac{b \tanh ^5(c+d x)}{5 d}+(a+b) \int \tanh ^4(c+d x) \, dx\\ &=-\frac{(a+b) \tanh ^3(c+d x)}{3 d}-\frac{b \tanh ^5(c+d x)}{5 d}+(a+b) \int \tanh ^2(c+d x) \, dx\\ &=-\frac{(a+b) \tanh (c+d x)}{d}-\frac{(a+b) \tanh ^3(c+d x)}{3 d}-\frac{b \tanh ^5(c+d x)}{5 d}+(a+b) \int 1 \, dx\\ &=(a+b) x-\frac{(a+b) \tanh (c+d x)}{d}-\frac{(a+b) \tanh ^3(c+d x)}{3 d}-\frac{b \tanh ^5(c+d x)}{5 d}\\ \end{align*}
Mathematica [A] time = 0.0340293, size = 97, normalized size = 1.8 \[ -\frac{a \tanh ^3(c+d x)}{3 d}+\frac{a \tanh ^{-1}(\tanh (c+d x))}{d}-\frac{a \tanh (c+d x)}{d}-\frac{b \tanh ^5(c+d x)}{5 d}-\frac{b \tanh ^3(c+d x)}{3 d}+\frac{b \tanh ^{-1}(\tanh (c+d x))}{d}-\frac{b \tanh (c+d x)}{d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.006, size = 128, normalized size = 2.4 \begin{align*} -{\frac{b \left ( \tanh \left ( dx+c \right ) \right ) ^{5}}{5\,d}}-{\frac{a \left ( \tanh \left ( dx+c \right ) \right ) ^{3}}{3\,d}}-{\frac{b \left ( \tanh \left ( dx+c \right ) \right ) ^{3}}{3\,d}}-{\frac{a\tanh \left ( dx+c \right ) }{d}}-{\frac{b\tanh \left ( dx+c \right ) }{d}}-{\frac{\ln \left ( \tanh \left ( dx+c \right ) -1 \right ) a}{2\,d}}-{\frac{\ln \left ( \tanh \left ( dx+c \right ) -1 \right ) b}{2\,d}}+{\frac{\ln \left ( \tanh \left ( dx+c \right ) +1 \right ) a}{2\,d}}+{\frac{\ln \left ( \tanh \left ( dx+c \right ) +1 \right ) b}{2\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.03624, size = 269, normalized size = 4.98 \begin{align*} \frac{1}{15} \, b{\left (15 \, x + \frac{15 \, c}{d} - \frac{2 \,{\left (70 \, e^{\left (-2 \, d x - 2 \, c\right )} + 140 \, e^{\left (-4 \, d x - 4 \, c\right )} + 90 \, e^{\left (-6 \, d x - 6 \, c\right )} + 45 \, e^{\left (-8 \, d x - 8 \, c\right )} + 23\right )}}{d{\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} + 10 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, e^{\left (-6 \, d x - 6 \, c\right )} + 5 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )} + 1\right )}}\right )} + \frac{1}{3} \, a{\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 )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.95337, size = 938, normalized size = 17.37 \begin{align*} \frac{{\left (15 \,{\left (a + b\right )} d x + 20 \, a + 23 \, b\right )} \cosh \left (d x + c\right )^{5} + 5 \,{\left (15 \,{\left (a + b\right )} d x + 20 \, a + 23 \, b\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{4} -{\left (20 \, a + 23 \, b\right )} \sinh \left (d x + c\right )^{5} + 5 \,{\left (15 \,{\left (a + b\right )} d x + 20 \, a + 23 \, b\right )} \cosh \left (d x + c\right )^{3} - 5 \,{\left (2 \,{\left (20 \, a + 23 \, b\right )} \cosh \left (d x + c\right )^{2} + 8 \, a + 5 \, b\right )} \sinh \left (d x + c\right )^{3} + 5 \,{\left (2 \,{\left (15 \,{\left (a + b\right )} d x + 20 \, a + 23 \, b\right )} \cosh \left (d x + c\right )^{3} + 3 \,{\left (15 \,{\left (a + b\right )} d x + 20 \, a + 23 \, b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} + 10 \,{\left (15 \,{\left (a + b\right )} d x + 20 \, a + 23 \, b\right )} \cosh \left (d x + c\right ) - 5 \,{\left ({\left (20 \, a + 23 \, b\right )} \cosh \left (d x + c\right )^{4} + 3 \,{\left (8 \, a + 5 \, b\right )} \cosh \left (d x + c\right )^{2} + 4 \, a + 10 \, b\right )} \sinh \left (d x + c\right )}{15 \,{\left (d \cosh \left (d x + c\right )^{5} + 5 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{4} + 5 \, d \cosh \left (d x + c\right )^{3} + 5 \,{\left (2 \, d \cosh \left (d x + c\right )^{3} + 3 \, d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} + 10 \, d \cosh \left (d x + c\right )\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.932585, size = 82, normalized size = 1.52 \begin{align*} \begin{cases} a x - \frac{a \tanh ^{3}{\left (c + d x \right )}}{3 d} - \frac{a \tanh{\left (c + d x \right )}}{d} + b x - \frac{b \tanh ^{5}{\left (c + d x \right )}}{5 d} - \frac{b \tanh ^{3}{\left (c + d x \right )}}{3 d} - \frac{b \tanh{\left (c + d x \right )}}{d} & \text{for}\: d \neq 0 \\x \left (a + b \tanh ^{2}{\left (c \right )}\right ) \tanh ^{4}{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.19379, size = 181, normalized size = 3.35 \begin{align*} \frac{{\left (d x + c\right )}{\left (a + b\right )}}{d} + \frac{2 \,{\left (30 \, a e^{\left (8 \, d x + 8 \, c\right )} + 45 \, b e^{\left (8 \, d x + 8 \, c\right )} + 90 \, a e^{\left (6 \, d x + 6 \, c\right )} + 90 \, b e^{\left (6 \, d x + 6 \, c\right )} + 110 \, a e^{\left (4 \, d x + 4 \, c\right )} + 140 \, b e^{\left (4 \, d x + 4 \, c\right )} + 70 \, a e^{\left (2 \, d x + 2 \, c\right )} + 70 \, b e^{\left (2 \, d x + 2 \, c\right )} + 20 \, a + 23 \, b\right )}}{15 \, d{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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