Optimal. Leaf size=73 \[ \frac {a^2 \sinh ^2(c+d x) \tanh (c+d x)}{2 d}+\frac {a (a-4 b) \tanh (c+d x)}{2 d}-\frac {1}{2} a x (a-4 b)+\frac {b^2 \tanh ^3(c+d x)}{3 d} \]
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Rubi [A] time = 0.11, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {4132, 463, 459, 321, 206} \[ \frac {a^2 \sinh ^2(c+d x) \tanh (c+d x)}{2 d}+\frac {a (a-4 b) \tanh (c+d x)}{2 d}-\frac {1}{2} a x (a-4 b)+\frac {b^2 \tanh ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Rule 206
Rule 321
Rule 459
Rule 463
Rule 4132
Rubi steps
\begin {align*} \int \left (a+b \text {sech}^2(c+d x)\right )^2 \sinh ^2(c+d x) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^2 \left (a+b-b x^2\right )^2}{\left (1-x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {a^2 \sinh ^2(c+d x) \tanh (c+d x)}{2 d}-\frac {\operatorname {Subst}\left (\int \frac {x^2 \left (3 a^2-2 (a+b)^2+2 b^2 x^2\right )}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{2 d}\\ &=\frac {a^2 \sinh ^2(c+d x) \tanh (c+d x)}{2 d}+\frac {b^2 \tanh ^3(c+d x)}{3 d}-\frac {(a (a-4 b)) \operatorname {Subst}\left (\int \frac {x^2}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{2 d}\\ &=\frac {a (a-4 b) \tanh (c+d x)}{2 d}+\frac {a^2 \sinh ^2(c+d x) \tanh (c+d x)}{2 d}+\frac {b^2 \tanh ^3(c+d x)}{3 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 (a-4 b) \tanh (c+d x)}{2 d}+\frac {a^2 \sinh ^2(c+d x) \tanh (c+d x)}{2 d}+\frac {b^2 \tanh ^3(c+d x)}{3 d}\\ \end {align*}
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Mathematica [A] time = 0.95, size = 126, normalized size = 1.73 \[ \frac {\text {sech}^3(c+d x) \left (a \cosh ^2(c+d x)+b\right )^2 \left (3 a \cosh ^3(c+d x) (a \sinh (2 (c+d x))-2 d x (a-4 b))-4 b (6 a-b) \text {sech}(c) \sinh (d x) \cosh ^2(c+d x)-4 b^2 \tanh (c) \cosh (c+d x)-4 b^2 \text {sech}(c) \sinh (d x)\right )}{3 d (a \cosh (2 (c+d x))+a+2 b)^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.41, size = 252, normalized size = 3.45 \[ \frac {3 \, a^{2} \sinh \left (d x + c\right )^{5} - 4 \, {\left (3 \, {\left (a^{2} - 4 \, a b\right )} d x - 12 \, a b + 2 \, b^{2}\right )} \cosh \left (d x + c\right )^{3} - 12 \, {\left (3 \, {\left (a^{2} - 4 \, a b\right )} d x - 12 \, a b + 2 \, b^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + {\left (30 \, a^{2} \cosh \left (d x + c\right )^{2} + 9 \, a^{2} - 48 \, a b + 8 \, b^{2}\right )} \sinh \left (d x + c\right )^{3} - 12 \, {\left (3 \, {\left (a^{2} - 4 \, a b\right )} d x - 12 \, a b + 2 \, b^{2}\right )} \cosh \left (d x + c\right ) + 3 \, {\left (5 \, a^{2} \cosh \left (d x + c\right )^{4} + {\left (9 \, a^{2} - 48 \, a b + 8 \, b^{2}\right )} \cosh \left (d x + c\right )^{2} + 2 \, a^{2} - 16 \, a b - 8 \, b^{2}\right )} \sinh \left (d x + c\right )}{24 \, {\left (d \cosh \left (d x + c\right )^{3} + 3 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + 3 \, d \cosh \left (d x + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.18, size = 144, normalized size = 1.97 \[ \frac {3 \, a^{2} e^{\left (2 \, d x + 2 \, c\right )} - 12 \, {\left (a^{2} - 4 \, a b\right )} {\left (d x + c\right )} + 3 \, {\left (2 \, a^{2} e^{\left (2 \, d x + 2 \, c\right )} - 8 \, a b e^{\left (2 \, d x + 2 \, c\right )} - a^{2}\right )} e^{\left (-2 \, d x - 2 \, c\right )} + \frac {16 \, {\left (6 \, a b e^{\left (4 \, d x + 4 \, c\right )} - 3 \, b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 12 \, a b e^{\left (2 \, d x + 2 \, c\right )} + 6 \, a b - b^{2}\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{3}}}{24 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.34, size = 90, normalized size = 1.23 \[ \frac {a^{2} \left (\frac {\cosh \left (d x +c \right ) \sinh \left (d x +c \right )}{2}-\frac {d x}{2}-\frac {c}{2}\right )+2 a b \left (d x +c -\tanh \left (d x +c \right )\right )+b^{2} \left (-\frac {\sinh \left (d x +c \right )}{2 \cosh \left (d x +c \right )^{3}}+\frac {\left (\frac {2}{3}+\frac {\mathrm {sech}\left (d x +c \right )^{2}}{3}\right ) \tanh \left (d x +c \right )}{2}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.33, size = 160, normalized size = 2.19 \[ -\frac {1}{8} \, a^{2} {\left (4 \, x - \frac {e^{\left (2 \, d x + 2 \, c\right )}}{d} + \frac {e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right )} + 2 \, a b {\left (x + \frac {c}{d} - \frac {2}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}}\right )} + \frac {2}{3} \, b^{2} {\left (\frac {3 \, e^{\left (-4 \, d x - 4 \, c\right )}}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} + 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} + 1\right )}} + \frac {1}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} + 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} + 1\right )}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.16, size = 236, normalized size = 3.23 \[ \frac {\frac {2\,\left (b^2+2\,a\,b\right )}{3\,d}+\frac {2\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (2\,a\,b-b^2\right )}{3\,d}}{2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1}+x\,\left (2\,a\,b-\frac {a^2}{2}\right )+\frac {\frac {2\,\left (2\,a\,b-b^2\right )}{3\,d}+\frac {4\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (b^2+2\,a\,b\right )}{3\,d}+\frac {2\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (2\,a\,b-b^2\right )}{3\,d}}{3\,{\mathrm {e}}^{2\,c+2\,d\,x}+3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}+1}+\frac {2\,\left (2\,a\,b-b^2\right )}{3\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}-\frac {a^2\,{\mathrm {e}}^{-2\,c-2\,d\,x}}{8\,d}+\frac {a^2\,{\mathrm {e}}^{2\,c+2\,d\,x}}{8\,d} \]
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 \[ \int \left (a + b \operatorname {sech}^{2}{\left (c + d x \right )}\right )^{2} \sinh ^{2}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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