Optimal. Leaf size=47 \[ \frac {a^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {1}{2} a x (a+4 b)+\frac {b^2 \tanh (c+d x)}{d} \]
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Rubi [A] time = 0.08, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {4146, 390, 385, 206} \[ \frac {a^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {1}{2} a x (a+4 b)+\frac {b^2 \tanh (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 206
Rule 385
Rule 390
Rule 4146
Rubi steps
\begin {align*} \int \cosh ^2(c+d x) \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}{\left (1-x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \left (b^2+\frac {a (a+2 b)-2 a b x^2}{\left (1-x^2\right )^2}\right ) \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {b^2 \tanh (c+d x)}{d}+\frac {\operatorname {Subst}\left (\int \frac {a (a+2 b)-2 a b x^2}{\left (1-x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {a^2 \cosh (c+d x) \sinh (c+d x)}{2 d}+\frac {b^2 \tanh (c+d x)}{d}+\frac {(a (a+4 b)) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{2 d}\\ &=\frac {1}{2} a (a+4 b) x+\frac {a^2 \cosh (c+d x) \sinh (c+d x)}{2 d}+\frac {b^2 \tanh (c+d x)}{d}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 52, normalized size = 1.11 \[ \frac {a^2 (c+d x)}{2 d}+\frac {a^2 \sinh (2 (c+d x))}{4 d}+2 a b x+\frac {b^2 \tanh (c+d x)}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.39, size = 80, normalized size = 1.70 \[ \frac {a^{2} \sinh \left (d x + c\right )^{3} + 4 \, {\left ({\left (a^{2} + 4 \, a b\right )} d x - 2 \, b^{2}\right )} \cosh \left (d x + c\right ) + {\left (3 \, a^{2} \cosh \left (d x + c\right )^{2} + a^{2} + 8 \, b^{2}\right )} \sinh \left (d x + c\right )}{8 \, d \cosh \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.17, size = 128, normalized size = 2.72 \[ \frac {a^{2} e^{\left (2 \, d x + 2 \, c\right )} + 4 \, {\left (a^{2} + 4 \, a b\right )} {\left (d x + c\right )} - \frac {a^{2} e^{\left (4 \, d x + 4 \, c\right )} + 4 \, a b e^{\left (4 \, d x + 4 \, c\right )} + 2 \, a^{2} e^{\left (2 \, d x + 2 \, c\right )} + 4 \, a b e^{\left (2 \, d x + 2 \, c\right )} + 16 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} + a^{2}}{e^{\left (4 \, d x + 4 \, c\right )} + e^{\left (2 \, d x + 2 \, c\right )}}}{8 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.36, size = 51, normalized size = 1.09 \[ \frac {a^{2} \left (\frac {\cosh \left (d x +c \right ) \sinh \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+2 a b \left (d x +c \right )+b^{2} \tanh \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 63, normalized size = 1.34 \[ \frac {1}{8} \, a^{2} {\left (4 \, x + \frac {e^{\left (2 \, d x + 2 \, c\right )}}{d} - \frac {e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right )} + 2 \, a b x + \frac {2 \, b^{2}}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.16, size = 65, normalized size = 1.38 \[ \frac {a^2\,{\mathrm {e}}^{2\,c+2\,d\,x}}{8\,d}-\frac {a^2\,{\mathrm {e}}^{-2\,c-2\,d\,x}}{8\,d}-\frac {2\,b^2}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}+\frac {a\,x\,\left (a+4\,b\right )}{2} \]
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 \[ \int \left (a + b \operatorname {sech}^{2}{\left (c + d x \right )}\right )^{2} \cosh ^{2}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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