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 = 0.05, antiderivative size = 65, normalized size of antiderivative = 1.30, 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 3767
Rule 4046
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*}
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Mathematica [A] time = 0.01, 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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fricas [B] time = 0.38, size = 343, normalized size = 6.86 \[ -\frac {8 \, {\left (2 \, {\left (5 \, a + b\right )} \cosh \left (d x + c\right )^{3} + 6 \, {\left (5 \, a + b\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + {\left (5 \, a - 2 \, b\right )} \sinh \left (d x + c\right )^{3} + 30 \, {\left (a + b\right )} \cosh \left (d x + c\right ) + {\left (3 \, {\left (5 \, a - 2 \, b\right )} \cosh \left (d x + c\right )^{2} + 5 \, a + 10 \, b\right )} \sinh \left (d x + c\right )\right )}}{15 \, {\left (d \cosh \left (d x + c\right )^{7} + 7 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{6} + d \sinh \left (d x + c\right )^{7} + 5 \, d \cosh \left (d x + c\right )^{5} + {\left (21 \, d \cosh \left (d x + c\right )^{2} + 5 \, d\right )} \sinh \left (d x + c\right )^{5} + 5 \, {\left (7 \, d \cosh \left (d x + c\right )^{3} + 5 \, d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{4} + 11 \, d \cosh \left (d x + c\right )^{3} + {\left (35 \, d \cosh \left (d x + c\right )^{4} + 50 \, d \cosh \left (d x + c\right )^{2} + 9 \, d\right )} \sinh \left (d x + c\right )^{3} + {\left (21 \, d \cosh \left (d x + c\right )^{5} + 50 \, d \cosh \left (d x + c\right )^{3} + 33 \, d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} + 15 \, d \cosh \left (d x + c\right ) + {\left (7 \, d \cosh \left (d x + c\right )^{6} + 25 \, d \cosh \left (d x + c\right )^{4} + 27 \, d \cosh \left (d x + c\right )^{2} + 5 \, d\right )} \sinh \left (d x + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 85, normalized size = 1.70 \[ -\frac {4 \, {\left (15 \, a e^{\left (6 \, d x + 6 \, c\right )} + 35 \, a e^{\left (4 \, d x + 4 \, c\right )} + 40 \, b e^{\left (4 \, d x + 4 \, c\right )} + 25 \, a e^{\left (2 \, d x + 2 \, c\right )} + 20 \, b e^{\left (2 \, d x + 2 \, c\right )} + 5 \, a + 4 \, b\right )}}{15 \, d {\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.37, size = 56, normalized size = 1.12 \[ \frac {a \left (\frac {2}{3}+\frac {\mathrm {sech}\left (d x +c \right )^{2}}{3}\right ) \tanh \left (d x +c \right )+b \left (\frac {8}{15}+\frac {\mathrm {sech}\left (d x +c \right )^{4}}{5}+\frac {4 \mathrm {sech}\left (d x +c \right )^{2}}{15}\right ) \tanh \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.32, size = 300, normalized size = 6.00 \[ \frac {16}{15} \, b {\left (\frac {5 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} + 10 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, e^{\left (-6 \, d x - 6 \, c\right )} + 5 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )} + 1\right )}} + \frac {10 \, e^{\left (-4 \, d x - 4 \, c\right )}}{d {\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} + 10 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, e^{\left (-6 \, d x - 6 \, c\right )} + 5 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )} + 1\right )}} + \frac {1}{d {\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} + 10 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, e^{\left (-6 \, d x - 6 \, c\right )} + 5 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )} + 1\right )}}\right )} + \frac {4}{3} \, a {\left (\frac {3 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} + 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} + 1\right )}} + \frac {1}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} + 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} + 1\right )}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.43, size = 292, normalized size = 5.84 \[ -\frac {\frac {8\,\left (a+2\,b\right )}{15\,d}+\frac {4\,a\,{\mathrm {e}}^{2\,c+2\,d\,x}}{5\,d}}{3\,{\mathrm {e}}^{2\,c+2\,d\,x}+3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}+1}-\frac {\frac {8\,a\,{\mathrm {e}}^{2\,c+2\,d\,x}}{5\,d}+\frac {8\,a\,{\mathrm {e}}^{6\,c+6\,d\,x}}{5\,d}+\frac {16\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (a+2\,b\right )}{5\,d}}{5\,{\mathrm {e}}^{2\,c+2\,d\,x}+10\,{\mathrm {e}}^{4\,c+4\,d\,x}+10\,{\mathrm {e}}^{6\,c+6\,d\,x}+5\,{\mathrm {e}}^{8\,c+8\,d\,x}+{\mathrm {e}}^{10\,c+10\,d\,x}+1}-\frac {\frac {2\,a}{5\,d}+\frac {6\,a\,{\mathrm {e}}^{4\,c+4\,d\,x}}{5\,d}+\frac {8\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (a+2\,b\right )}{5\,d}}{4\,{\mathrm {e}}^{2\,c+2\,d\,x}+6\,{\mathrm {e}}^{4\,c+4\,d\,x}+4\,{\mathrm {e}}^{6\,c+6\,d\,x}+{\mathrm {e}}^{8\,c+8\,d\,x}+1}-\frac {2\,a}{5\,d\,\left (2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1\right )} \]
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 \[ \int \left (a + b \operatorname {sech}^{2}{\left (c + d x \right )}\right ) \operatorname {sech}^{4}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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