Optimal. Leaf size=82 \[ \frac {1}{8} x \left (3 a^2+8 a b+8 b^2\right )+\frac {3 a (a+2 b) \sinh (c+d x) \cosh (c+d x)}{8 d}+\frac {a \sinh (c+d x) \cosh ^3(c+d x) \left (a-b \tanh ^2(c+d x)+b\right )}{4 d} \]
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Rubi [A] time = 0.09, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {4146, 413, 385, 206} \[ \frac {1}{8} x \left (3 a^2+8 a b+8 b^2\right )+\frac {3 a (a+2 b) \sinh (c+d x) \cosh (c+d x)}{8 d}+\frac {a \sinh (c+d x) \cosh ^3(c+d x) \left (a-b \tanh ^2(c+d x)+b\right )}{4 d} \]
Antiderivative was successfully verified.
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Rule 206
Rule 385
Rule 413
Rule 4146
Rubi steps
\begin {align*} \int \cosh ^4(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 )^3} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {a \cosh ^3(c+d x) \sinh (c+d x) \left (a+b-b \tanh ^2(c+d x)\right )}{4 d}-\frac {\operatorname {Subst}\left (\int \frac {-(a+b) (3 a+4 b)+b (a+4 b) x^2}{\left (1-x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{4 d}\\ &=\frac {3 a (a+2 b) \cosh (c+d x) \sinh (c+d x)}{8 d}+\frac {a \cosh ^3(c+d x) \sinh (c+d x) \left (a+b-b \tanh ^2(c+d x)\right )}{4 d}+\frac {\left (3 a^2+8 a b+8 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{8 d}\\ &=\frac {1}{8} \left (3 a^2+8 a b+8 b^2\right ) x+\frac {3 a (a+2 b) \cosh (c+d x) \sinh (c+d x)}{8 d}+\frac {a \cosh ^3(c+d x) \sinh (c+d x) \left (a+b-b \tanh ^2(c+d x)\right )}{4 d}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 58, normalized size = 0.71 \[ \frac {4 \left (3 a^2+8 a b+8 b^2\right ) (c+d x)+a^2 \sinh (4 (c+d x))+8 a (a+2 b) \sinh (2 (c+d x))}{32 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 78, normalized size = 0.95 \[ \frac {a^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + {\left (3 \, a^{2} + 8 \, a b + 8 \, b^{2}\right )} d x + {\left (a^{2} \cosh \left (d x + c\right )^{3} + 4 \, {\left (a^{2} + 2 \, a b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{8 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 151, normalized size = 1.84 \[ \frac {a^{2} e^{\left (4 \, d x + 4 \, c\right )} + 8 \, a^{2} e^{\left (2 \, d x + 2 \, c\right )} + 16 \, a b e^{\left (2 \, d x + 2 \, c\right )} + 8 \, {\left (3 \, a^{2} + 8 \, a b + 8 \, b^{2}\right )} {\left (d x + c\right )} - {\left (18 \, a^{2} e^{\left (4 \, d x + 4 \, c\right )} + 48 \, a b e^{\left (4 \, d x + 4 \, c\right )} + 48 \, b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 8 \, a^{2} e^{\left (2 \, d x + 2 \, c\right )} + 16 \, a b e^{\left (2 \, d x + 2 \, c\right )} + a^{2}\right )} e^{\left (-4 \, d x - 4 \, c\right )}}{64 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.40, size = 79, normalized size = 0.96 \[ \frac {a^{2} \left (\left (\frac {\left (\cosh ^{3}\left (d x +c \right )\right )}{4}+\frac {3 \cosh \left (d x +c \right )}{8}\right ) \sinh \left (d x +c \right )+\frac {3 d x}{8}+\frac {3 c}{8}\right )+2 a b \left (\frac {\cosh \left (d x +c \right ) \sinh \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+b^{2} \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 105, normalized size = 1.28 \[ \frac {1}{64} \, a^{2} {\left (24 \, x + \frac {e^{\left (4 \, d x + 4 \, c\right )}}{d} + \frac {8 \, e^{\left (2 \, d x + 2 \, c\right )}}{d} - \frac {8 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d} - \frac {e^{\left (-4 \, d x - 4 \, c\right )}}{d}\right )} + \frac {1}{4} \, a b {\left (4 \, x + \frac {e^{\left (2 \, d x + 2 \, c\right )}}{d} - \frac {e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right )} + b^{2} x \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.37, size = 66, normalized size = 0.80 \[ \frac {3\,a^2\,x}{8}+b^2\,x+a\,b\,x+\frac {a^2\,\mathrm {sinh}\left (2\,c+2\,d\,x\right )}{4\,d}+\frac {a^2\,\mathrm {sinh}\left (4\,c+4\,d\,x\right )}{32\,d}+\frac {a\,b\,\mathrm {sinh}\left (2\,c+2\,d\,x\right )}{2\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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