Optimal. Leaf size=59 \[ -\frac {a^2 \tanh (c+d x)}{d}+a^2 x+\frac {b (2 a+b) \tanh ^3(c+d x)}{3 d}-\frac {b^2 \tanh ^5(c+d x)}{5 d} \]
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Rubi [A] time = 0.10, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {4141, 1802, 206} \[ -\frac {a^2 \tanh (c+d x)}{d}+a^2 x+\frac {b (2 a+b) \tanh ^3(c+d x)}{3 d}-\frac {b^2 \tanh ^5(c+d x)}{5 d} \]
Antiderivative was successfully verified.
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Rule 206
Rule 1802
Rule 4141
Rubi steps
\begin {align*} \int \left (a+b \text {sech}^2(c+d x)\right )^2 \tanh ^2(c+d x) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^2 \left (a+b \left (1-x^2\right )\right )^2}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \left (-a^2+b (2 a+b) x^2-b^2 x^4+\frac {a^2}{1-x^2}\right ) \, dx,x,\tanh (c+d x)\right )}{d}\\ &=-\frac {a^2 \tanh (c+d x)}{d}+\frac {b (2 a+b) \tanh ^3(c+d x)}{3 d}-\frac {b^2 \tanh ^5(c+d x)}{5 d}+\frac {a^2 \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=a^2 x-\frac {a^2 \tanh (c+d x)}{d}+\frac {b (2 a+b) \tanh ^3(c+d x)}{3 d}-\frac {b^2 \tanh ^5(c+d x)}{5 d}\\ \end {align*}
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Mathematica [B] time = 0.86, size = 281, normalized size = 4.76 \[ \frac {\text {sech}(c) \text {sech}^5(c+d x) \left (120 a^2 \sinh (2 c+d x)-120 a^2 \sinh (2 c+3 d x)+30 a^2 \sinh (4 c+3 d x)-30 a^2 \sinh (4 c+5 d x)+150 a^2 d x \cosh (2 c+d x)+75 a^2 d x \cosh (2 c+3 d x)+75 a^2 d x \cosh (4 c+3 d x)+15 a^2 d x \cosh (4 c+5 d x)+15 a^2 d x \cosh (6 c+5 d x)-180 a^2 \sinh (d x)+150 a^2 d x \cosh (d x)-120 a b \sinh (2 c+d x)+40 a b \sinh (2 c+3 d x)-60 a b \sinh (4 c+3 d x)+20 a b \sinh (4 c+5 d x)+80 a b \sinh (d x)-60 b^2 \sinh (2 c+d x)+20 b^2 \sinh (2 c+3 d x)+4 b^2 \sinh (4 c+5 d x)-20 b^2 \sinh (d x)\right )}{480 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.42, size = 435, normalized size = 7.37 \[ \frac {{\left (15 \, a^{2} d x + 15 \, a^{2} - 10 \, a b - 2 \, b^{2}\right )} \cosh \left (d x + c\right )^{5} + 5 \, {\left (15 \, a^{2} d x + 15 \, a^{2} - 10 \, a b - 2 \, b^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{4} - {\left (15 \, a^{2} - 10 \, a b - 2 \, b^{2}\right )} \sinh \left (d x + c\right )^{5} + 5 \, {\left (15 \, a^{2} d x + 15 \, a^{2} - 10 \, a b - 2 \, b^{2}\right )} \cosh \left (d x + c\right )^{3} - 5 \, {\left (2 \, {\left (15 \, a^{2} - 10 \, a b - 2 \, b^{2}\right )} \cosh \left (d x + c\right )^{2} + 9 \, a^{2} + 2 \, a b - 2 \, b^{2}\right )} \sinh \left (d x + c\right )^{3} + 5 \, {\left (2 \, {\left (15 \, a^{2} d x + 15 \, a^{2} - 10 \, a b - 2 \, b^{2}\right )} \cosh \left (d x + c\right )^{3} + 3 \, {\left (15 \, a^{2} d x + 15 \, a^{2} - 10 \, a b - 2 \, b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} + 10 \, {\left (15 \, a^{2} d x + 15 \, a^{2} - 10 \, a b - 2 \, b^{2}\right )} \cosh \left (d x + c\right ) - 5 \, {\left ({\left (15 \, a^{2} - 10 \, a b - 2 \, b^{2}\right )} \cosh \left (d x + c\right )^{4} + 3 \, {\left (9 \, a^{2} + 2 \, a b - 2 \, b^{2}\right )} \cosh \left (d x + c\right )^{2} + 6 \, a^{2} + 4 \, a b + 8 \, b^{2}\right )} \sinh \left (d x + c\right )}{15 \, {\left (d \cosh \left (d x + c\right )^{5} + 5 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{4} + 5 \, d \cosh \left (d x + c\right )^{3} + 5 \, {\left (2 \, d \cosh \left (d x + c\right )^{3} + 3 \, d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} + 10 \, d \cosh \left (d x + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.17, size = 193, normalized size = 3.27 \[ \frac {15 \, a^{2} d x + \frac {2 \, {\left (15 \, a^{2} e^{\left (8 \, d x + 8 \, c\right )} - 30 \, a b e^{\left (8 \, d x + 8 \, c\right )} + 60 \, a^{2} e^{\left (6 \, d x + 6 \, c\right )} - 60 \, a b e^{\left (6 \, d x + 6 \, c\right )} - 30 \, b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 90 \, a^{2} e^{\left (4 \, d x + 4 \, c\right )} - 40 \, a b e^{\left (4 \, d x + 4 \, c\right )} + 10 \, b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 60 \, a^{2} e^{\left (2 \, d x + 2 \, c\right )} - 20 \, a b e^{\left (2 \, d x + 2 \, c\right )} - 10 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 15 \, a^{2} - 10 \, a b - 2 \, b^{2}\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{5}}}{15 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.40, size = 115, normalized size = 1.95 \[ \frac {a^{2} \left (d x +c -\tanh \left (d x +c \right )\right )+2 a b \left (-\frac {\sinh \left (d x +c \right )}{2 \cosh \left (d x +c \right )^{3}}+\frac {\left (\frac {2}{3}+\frac {\mathrm {sech}\left (d x +c \right )^{2}}{3}\right ) \tanh \left (d x +c \right )}{2}\right )+b^{2} \left (-\frac {\sinh \left (d x +c \right )}{4 \cosh \left (d x +c \right )^{5}}+\frac {\left (\frac {8}{15}+\frac {\mathrm {sech}\left (d x +c \right )^{4}}{5}+\frac {4 \mathrm {sech}\left (d x +c \right )^{2}}{15}\right ) \tanh \left (d x +c \right )}{4}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.52, size = 325, normalized size = 5.51 \[ \frac {2 \, a b \tanh \left (d x + c\right )^{3}}{3 \, d} + a^{2} {\left (x + \frac {c}{d} - \frac {2}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}}\right )} + \frac {4}{15} \, b^{2} {\left (\frac {5 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} + 10 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, e^{\left (-6 \, d x - 6 \, c\right )} + 5 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )} + 1\right )}} - \frac {5 \, e^{\left (-4 \, d x - 4 \, c\right )}}{d {\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} + 10 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, e^{\left (-6 \, d x - 6 \, c\right )} + 5 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )} + 1\right )}} + \frac {15 \, e^{\left (-6 \, d x - 6 \, c\right )}}{d {\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} + 10 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, e^{\left (-6 \, d x - 6 \, c\right )} + 5 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )} + 1\right )}} + \frac {1}{d {\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} + 10 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, e^{\left (-6 \, d x - 6 \, c\right )} + 5 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )} + 1\right )}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.14, size = 513, normalized size = 8.69 \[ \frac {\frac {8\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (a^2-b^2\right )}{5\,d}-\frac {2\,\left (2\,a\,b-a^2\right )}{5\,d}+\frac {8\,{\mathrm {e}}^{6\,c+6\,d\,x}\,\left (a^2-b^2\right )}{5\,d}-\frac {2\,{\mathrm {e}}^{8\,c+8\,d\,x}\,\left (2\,a\,b-a^2\right )}{5\,d}+\frac {4\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (3\,a^2+2\,a\,b+4\,b^2\right )}{5\,d}}{5\,{\mathrm {e}}^{2\,c+2\,d\,x}+10\,{\mathrm {e}}^{4\,c+4\,d\,x}+10\,{\mathrm {e}}^{6\,c+6\,d\,x}+5\,{\mathrm {e}}^{8\,c+8\,d\,x}+{\mathrm {e}}^{10\,c+10\,d\,x}+1}+\frac {\frac {2\,\left (3\,a^2+2\,a\,b+4\,b^2\right )}{15\,d}+\frac {4\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (a^2-b^2\right )}{5\,d}-\frac {2\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (2\,a\,b-a^2\right )}{5\,d}}{3\,{\mathrm {e}}^{2\,c+2\,d\,x}+3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}+1}+\frac {\frac {2\,\left (a^2-b^2\right )}{5\,d}-\frac {2\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (2\,a\,b-a^2\right )}{5\,d}}{2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1}+\frac {\frac {2\,\left (a^2-b^2\right )}{5\,d}+\frac {6\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (a^2-b^2\right )}{5\,d}-\frac {2\,{\mathrm {e}}^{6\,c+6\,d\,x}\,\left (2\,a\,b-a^2\right )}{5\,d}+\frac {2\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (3\,a^2+2\,a\,b+4\,b^2\right )}{5\,d}}{4\,{\mathrm {e}}^{2\,c+2\,d\,x}+6\,{\mathrm {e}}^{4\,c+4\,d\,x}+4\,{\mathrm {e}}^{6\,c+6\,d\,x}+{\mathrm {e}}^{8\,c+8\,d\,x}+1}+a^2\,x-\frac {2\,\left (2\,a\,b-a^2\right )}{5\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )} \]
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 \[ \int \left (a + b \operatorname {sech}^{2}{\left (c + d x \right )}\right )^{2} \tanh ^{2}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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