Optimal. Leaf size=40 \[ a^2 x+\frac {b (2 a+b) \tanh (c+d x)}{d}-\frac {b^2 \tanh ^3(c+d x)}{3 d} \]
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Rubi [A] time = 0.03, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {4128, 390, 206} \[ a^2 x+\frac {b (2 a+b) \tanh (c+d x)}{d}-\frac {b^2 \tanh ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Rule 206
Rule 390
Rule 4128
Rubi steps
\begin {align*} \int \left (a+b \text {sech}^2(c+d x)\right )^2 \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (a+b-b x^2\right )^2}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \left (b (2 a+b)-b^2 x^2+\frac {a^2}{1-x^2}\right ) \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {b (2 a+b) \tanh (c+d x)}{d}-\frac {b^2 \tanh ^3(c+d x)}{3 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 {b (2 a+b) \tanh (c+d x)}{d}-\frac {b^2 \tanh ^3(c+d x)}{3 d}\\ \end {align*}
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Mathematica [B] time = 0.38, size = 106, normalized size = 2.65 \[ \frac {4 \text {sech}^3(c+d x) \left (a \cosh ^2(c+d x)+b\right )^2 \left (3 a^2 d x \cosh ^3(c+d x)+2 b (3 a+b) \text {sech}(c) \sinh (d x) \cosh ^2(c+d x)+b^2 \tanh (c) \cosh (c+d x)+b^2 \text {sech}(c) \sinh (d x)\right )}{3 d (a \cosh (2 (c+d x))+a+2 b)^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.40, size = 176, normalized size = 4.40 \[ \frac {{\left (3 \, a^{2} d x - 6 \, a b - 2 \, b^{2}\right )} \cosh \left (d x + c\right )^{3} + 3 \, {\left (3 \, a^{2} d x - 6 \, a b - 2 \, b^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + 2 \, {\left (3 \, a b + b^{2}\right )} \sinh \left (d x + c\right )^{3} + 3 \, {\left (3 \, a^{2} d x - 6 \, a b - 2 \, b^{2}\right )} \cosh \left (d x + c\right ) + 6 \, {\left ({\left (3 \, a b + b^{2}\right )} \cosh \left (d x + c\right )^{2} + a b + b^{2}\right )} \sinh \left (d x + c\right )}{3 \, {\left (d \cosh \left (d x + c\right )^{3} + 3 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + 3 \, d \cosh \left (d x + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.14, size = 79, normalized size = 1.98 \[ \frac {3 \, {\left (d x + c\right )} a^{2} - \frac {4 \, {\left (3 \, a b e^{\left (4 \, d x + 4 \, c\right )} + 6 \, a b e^{\left (2 \, d x + 2 \, c\right )} + 3 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 3 \, a b + b^{2}\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{3}}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.36, size = 47, normalized size = 1.18 \[ \frac {a^{2} \left (d x +c \right )+2 a b \tanh \left (d x +c \right )+b^{2} \left (\frac {2}{3}+\frac {\mathrm {sech}\left (d x +c \right )^{2}}{3}\right ) \tanh \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.46, size = 120, normalized size = 3.00 \[ a^{2} x + \frac {4}{3} \, b^{2} {\left (\frac {3 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} + 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} + 1\right )}} + \frac {1}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} + 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} + 1\right )}}\right )} + \frac {4 \, a b}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.46, size = 163, normalized size = 4.08 \[ a^2\,x-\frac {\frac {4\,\left (b^2+a\,b\right )}{3\,d}+\frac {4\,a\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}}{3\,d}}{2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1}-\frac {\frac {8\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (b^2+a\,b\right )}{3\,d}+\frac {4\,a\,b}{3\,d}+\frac {4\,a\,b\,{\mathrm {e}}^{4\,c+4\,d\,x}}{3\,d}}{3\,{\mathrm {e}}^{2\,c+2\,d\,x}+3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}+1}-\frac {4\,a\,b}{3\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )} \]
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 \[ \int \left (a + b \operatorname {sech}^{2}{\left (c + d x \right )}\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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