Optimal. Leaf size=63 \[ \frac {(a+b) \sinh (c+d x) \cosh ^3(c+d x)}{4 d}+\frac {(3 a-b) \sinh (c+d x) \cosh (c+d x)}{8 d}+\frac {1}{8} x (3 a-b) \]
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Rubi [A] time = 0.05, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3675, 385, 199, 206} \[ \frac {(a+b) \sinh (c+d x) \cosh ^3(c+d x)}{4 d}+\frac {(3 a-b) \sinh (c+d x) \cosh (c+d x)}{8 d}+\frac {1}{8} x (3 a-b) \]
Antiderivative was successfully verified.
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Rule 199
Rule 206
Rule 385
Rule 3675
Rubi steps
\begin {align*} \int \cosh ^4(c+d x) \left (a+b \tanh ^2(c+d x)\right ) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {a+b x^2}{\left (1-x^2\right )^3} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {(a+b) \cosh ^3(c+d x) \sinh (c+d x)}{4 d}+\frac {(3 a-b) \operatorname {Subst}\left (\int \frac {1}{\left (1-x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{4 d}\\ &=\frac {(3 a-b) \cosh (c+d x) \sinh (c+d x)}{8 d}+\frac {(a+b) \cosh ^3(c+d x) \sinh (c+d x)}{4 d}+\frac {(3 a-b) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{8 d}\\ &=\frac {1}{8} (3 a-b) x+\frac {(3 a-b) \cosh (c+d x) \sinh (c+d x)}{8 d}+\frac {(a+b) \cosh ^3(c+d x) \sinh (c+d x)}{4 d}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 44, normalized size = 0.70 \[ \frac {(a+b) \sinh (4 (c+d x))+12 a (c+d x)+8 a \sinh (2 (c+d x))-4 b d x}{32 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 63, normalized size = 1.00 \[ \frac {{\left (a + b\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + {\left (3 \, a - b\right )} d x + {\left ({\left (a + b\right )} \cosh \left (d x + c\right )^{3} + 4 \, a \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{8 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 107, normalized size = 1.70 \[ \frac {8 \, {\left (3 \, a - b\right )} d x - {\left (18 \, a e^{\left (4 \, d x + 4 \, c\right )} - 6 \, b e^{\left (4 \, d x + 4 \, c\right )} + 8 \, a e^{\left (2 \, d x + 2 \, c\right )} + a + b\right )} e^{\left (-4 \, d x - 4 \, c\right )} + {\left (a e^{\left (4 \, d x + 12 \, c\right )} + b e^{\left (4 \, d x + 12 \, c\right )} + 8 \, a e^{\left (2 \, d x + 10 \, c\right )}\right )} e^{\left (-8 \, c\right )}}{64 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.32, size = 82, normalized size = 1.30 \[ \frac {b \left (\frac {\sinh \left (d x +c \right ) \left (\cosh ^{3}\left (d x +c \right )\right )}{4}-\frac {\cosh \left (d x +c \right ) \sinh \left (d x +c \right )}{8}-\frac {d x}{8}-\frac {c}{8}\right )+a \left (\left (\frac {\left (\cosh ^{3}\left (d x +c \right )\right )}{4}+\frac {3 \cosh \left (d x +c \right )}{8}\right ) \sinh \left (d x +c \right )+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 104, normalized size = 1.65 \[ \frac {1}{64} \, a {\left (24 \, x + \frac {e^{\left (4 \, d x + 4 \, c\right )}}{d} + \frac {8 \, e^{\left (2 \, d x + 2 \, c\right )}}{d} - \frac {8 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d} - \frac {e^{\left (-4 \, d x - 4 \, c\right )}}{d}\right )} - \frac {1}{64} \, b {\left (\frac {8 \, {\left (d x + c\right )}}{d} - \frac {e^{\left (4 \, d x + 4 \, c\right )}}{d} + \frac {e^{\left (-4 \, d x - 4 \, c\right )}}{d}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.21, size = 74, normalized size = 1.17 \[ x\,\left (\frac {3\,a}{8}-\frac {b}{8}\right )-\frac {{\mathrm {e}}^{-4\,c-4\,d\,x}\,\left (a+b\right )}{64\,d}+\frac {{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (a+b\right )}{64\,d}-\frac {a\,{\mathrm {e}}^{-2\,c-2\,d\,x}}{8\,d}+\frac {a\,{\mathrm {e}}^{2\,c+2\,d\,x}}{8\,d} \]
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 \[ \int \left (a + b \tanh ^{2}{\left (c + d x \right )}\right ) \cosh ^{4}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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