Optimal. Leaf size=50 \[ -\frac{(a+2 b) \tanh ^3(c+d x)}{3 d}+\frac{(a+b) \tanh (c+d x)}{d}+\frac{b \tanh ^5(c+d x)}{5 d} \]
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Rubi [A] time = 0.0460788, antiderivative size = 65, normalized size of antiderivative = 1.3, number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {4046, 3767} \[ -\frac{(5 a+4 b) \tanh ^3(c+d x)}{15 d}+\frac{(5 a+4 b) \tanh (c+d x)}{5 d}+\frac{b \tanh (c+d x) \text{sech}^4(c+d x)}{5 d} \]
Antiderivative was successfully verified.
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Rule 4046
Rule 3767
Rubi steps
\begin{align*} \int \text{sech}^4(c+d x) \left (a+b \text{sech}^2(c+d x)\right ) \, dx &=\frac{b \text{sech}^4(c+d x) \tanh (c+d x)}{5 d}+\frac{1}{5} (5 a+4 b) \int \text{sech}^4(c+d x) \, dx\\ &=\frac{b \text{sech}^4(c+d x) \tanh (c+d x)}{5 d}+\frac{(i (5 a+4 b)) \operatorname{Subst}\left (\int \left (1+x^2\right ) \, dx,x,-i \tanh (c+d x)\right )}{5 d}\\ &=\frac{(5 a+4 b) \tanh (c+d x)}{5 d}+\frac{b \text{sech}^4(c+d x) \tanh (c+d x)}{5 d}-\frac{(5 a+4 b) \tanh ^3(c+d x)}{15 d}\\ \end{align*}
Mathematica [A] time = 0.0129559, size = 71, normalized size = 1.42 \[ -\frac{a \tanh ^3(c+d x)}{3 d}+\frac{a \tanh (c+d x)}{d}+\frac{b \tanh ^5(c+d x)}{5 d}-\frac{2 b \tanh ^3(c+d x)}{3 d}+\frac{b \tanh (c+d x)}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.02, size = 56, normalized size = 1.1 \begin{align*}{\frac{1}{d} \left ( a \left ({\frac{2}{3}}+{\frac{ \left ({\rm sech} \left (dx+c\right ) \right ) ^{2}}{3}} \right ) \tanh \left ( dx+c \right ) +b \left ({\frac{8}{15}}+{\frac{ \left ({\rm sech} \left (dx+c\right ) \right ) ^{4}}{5}}+{\frac{4\, \left ({\rm sech} \left (dx+c\right ) \right ) ^{2}}{15}} \right ) \tanh \left ( dx+c \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.05755, size = 405, normalized size = 8.1 \begin{align*} \frac{16}{15} \, b{\left (\frac{5 \, e^{\left (-2 \, d x - 2 \, c\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 )}} + \frac{10 \, e^{\left (-4 \, d x - 4 \, c\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 )}} + \frac{1}{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{4}{3} \, a{\left (\frac{3 \, e^{\left (-2 \, d x - 2 \, c\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 )}} + \frac{1}{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 = 2.06339, size = 938, normalized size = 18.76 \begin{align*} -\frac{8 \,{\left (2 \,{\left (5 \, a + b\right )} \cosh \left (d x + c\right )^{3} + 6 \,{\left (5 \, a + b\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} +{\left (5 \, a - 2 \, b\right )} \sinh \left (d x + c\right )^{3} + 30 \,{\left (a + b\right )} \cosh \left (d x + c\right ) +{\left (3 \,{\left (5 \, a - 2 \, b\right )} \cosh \left (d x + c\right )^{2} + 5 \, a + 10 \, b\right )} \sinh \left (d x + c\right )\right )}}{15 \,{\left (d \cosh \left (d x + c\right )^{7} + 7 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{6} + d \sinh \left (d x + c\right )^{7} + 5 \, d \cosh \left (d x + c\right )^{5} +{\left (21 \, d \cosh \left (d x + c\right )^{2} + 5 \, d\right )} \sinh \left (d x + c\right )^{5} + 5 \,{\left (7 \, d \cosh \left (d x + c\right )^{3} + 5 \, d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{4} + 11 \, d \cosh \left (d x + c\right )^{3} +{\left (35 \, d \cosh \left (d x + c\right )^{4} + 50 \, d \cosh \left (d x + c\right )^{2} + 9 \, d\right )} \sinh \left (d x + c\right )^{3} +{\left (21 \, d \cosh \left (d x + c\right )^{5} + 50 \, d \cosh \left (d x + c\right )^{3} + 33 \, d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} + 15 \, d \cosh \left (d x + c\right ) +{\left (7 \, d \cosh \left (d x + c\right )^{6} + 25 \, d \cosh \left (d x + c\right )^{4} + 27 \, d \cosh \left (d x + c\right )^{2} + 5 \, d\right )} \sinh \left (d x + c\right )\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \operatorname{sech}^{2}{\left (c + d x \right )}\right ) \operatorname{sech}^{4}{\left (c + d x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1446, size = 115, normalized size = 2.3 \begin{align*} -\frac{4 \,{\left (15 \, a e^{\left (6 \, d x + 6 \, c\right )} + 35 \, a e^{\left (4 \, d x + 4 \, c\right )} + 40 \, b e^{\left (4 \, d x + 4 \, c\right )} + 25 \, a e^{\left (2 \, d x + 2 \, c\right )} + 20 \, b e^{\left (2 \, d x + 2 \, c\right )} + 5 \, a + 4 \, 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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