Optimal. Leaf size=70 \[ -\frac {(5 a-4 b) \sinh (c+d x) \cosh (c+d x)}{8 d}+\frac {3}{8} x (a-4 b)+\frac {a \sinh (c+d x) \cosh ^3(c+d x)}{4 d}+\frac {b \tanh (c+d x)}{d} \]
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Rubi [A] time = 0.08, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {4132, 455, 1157, 388, 206} \[ -\frac {(5 a-4 b) \sinh (c+d x) \cosh (c+d x)}{8 d}+\frac {3}{8} x (a-4 b)+\frac {a \sinh (c+d x) \cosh ^3(c+d x)}{4 d}+\frac {b \tanh (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 206
Rule 388
Rule 455
Rule 1157
Rule 4132
Rubi steps
\begin {align*} \int \left (a+b \text {sech}^2(c+d x)\right ) \sinh ^4(c+d x) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^4 \left (a+b-b x^2\right )}{\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)}{4 d}+\frac {\operatorname {Subst}\left (\int \frac {-a-4 a x^2+4 b x^4}{\left (1-x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{4 d}\\ &=-\frac {(5 a-4 b) \cosh (c+d x) \sinh (c+d x)}{8 d}+\frac {a \cosh ^3(c+d x) \sinh (c+d x)}{4 d}-\frac {\operatorname {Subst}\left (\int \frac {-3 a+4 b+8 b x^2}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{8 d}\\ &=-\frac {(5 a-4 b) \cosh (c+d x) \sinh (c+d x)}{8 d}+\frac {a \cosh ^3(c+d x) \sinh (c+d x)}{4 d}+\frac {b \tanh (c+d x)}{d}+\frac {(3 (a-4 b)) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{8 d}\\ &=\frac {3}{8} (a-4 b) x-\frac {(5 a-4 b) \cosh (c+d x) \sinh (c+d x)}{8 d}+\frac {a \cosh ^3(c+d x) \sinh (c+d x)}{4 d}+\frac {b \tanh (c+d x)}{d}\\ \end {align*}
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Mathematica [A] time = 0.33, size = 54, normalized size = 0.77 \[ \frac {12 (a-4 b) (c+d x)-8 (a-b) \sinh (2 (c+d x))+a \sinh (4 (c+d x))+32 b \tanh (c+d x)}{32 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 114, normalized size = 1.63 \[ \frac {a \sinh \left (d x + c\right )^{5} + {\left (10 \, a \cosh \left (d x + c\right )^{2} - 7 \, a + 8 \, b\right )} \sinh \left (d x + c\right )^{3} + 8 \, {\left (3 \, {\left (a - 4 \, b\right )} d x - 8 \, b\right )} \cosh \left (d x + c\right ) + {\left (5 \, a \cosh \left (d x + c\right )^{4} - 3 \, {\left (7 \, a - 8 \, b\right )} \cosh \left (d x + c\right )^{2} - 8 \, a + 72 \, b\right )} \sinh \left (d x + c\right )}{64 \, d \cosh \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.10, size = 130, normalized size = 1.86 \[ \frac {24 \, {\left (d x + c\right )} {\left (a - 4 \, b\right )} + a e^{\left (4 \, d x + 4 \, c\right )} - 8 \, a e^{\left (2 \, d x + 2 \, c\right )} + 8 \, b e^{\left (2 \, d x + 2 \, c\right )} - {\left (18 \, a e^{\left (4 \, d x + 4 \, c\right )} - 72 \, b e^{\left (4 \, d x + 4 \, c\right )} - 8 \, a e^{\left (2 \, d x + 2 \, c\right )} + 8 \, b e^{\left (2 \, d x + 2 \, c\right )} + a\right )} e^{\left (-4 \, d x - 4 \, c\right )} - \frac {128 \, b}{e^{\left (2 \, d x + 2 \, c\right )} + 1}}{64 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.33, size = 78, normalized size = 1.11 \[ \frac {a \left (\left (\frac {\left (\sinh ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sinh \left (d x +c \right )}{8}\right ) \cosh \left (d x +c \right )+\frac {3 d x}{8}+\frac {3 c}{8}\right )+b \left (\frac {\sinh ^{3}\left (d x +c \right )}{2 \cosh \left (d x +c \right )}-\frac {3 d x}{2}-\frac {3 c}{2}+\frac {3 \tanh \left (d x +c \right )}{2}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.31, size = 129, normalized size = 1.84 \[ \frac {1}{64} \, a {\left (24 \, x + \frac {e^{\left (4 \, d x + 4 \, c\right )}}{d} - \frac {8 \, e^{\left (2 \, d x + 2 \, c\right )}}{d} + \frac {8 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d} - \frac {e^{\left (-4 \, d x - 4 \, c\right )}}{d}\right )} - \frac {1}{8} \, b {\left (\frac {12 \, {\left (d x + c\right )}}{d} + \frac {e^{\left (-2 \, d x - 2 \, c\right )}}{d} - \frac {17 \, e^{\left (-2 \, d x - 2 \, c\right )} + 1}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} + e^{\left (-4 \, d x - 4 \, c\right )}\right )}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.61, size = 73, normalized size = 1.04 \[ \frac {3\,a\,x}{8}-\frac {3\,b\,x}{2}-\frac {a\,\mathrm {sinh}\left (2\,c+2\,d\,x\right )}{4\,d}+\frac {a\,\mathrm {sinh}\left (4\,c+4\,d\,x\right )}{32\,d}+\frac {b\,\mathrm {sinh}\left (2\,c+2\,d\,x\right )}{4\,d}+\frac {b\,\mathrm {sinh}\left (c+d\,x\right )}{d\,\mathrm {cosh}\left (c+d\,x\right )} \]
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 \[ \int \left (a + b \operatorname {sech}^{2}{\left (c + d x \right )}\right ) \sinh ^{4}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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