Optimal. Leaf size=63 \[ \frac{a \cosh (c+d x)}{d}+\frac{3 b \sinh (c+d x)}{2 d}-\frac{3 b \tan ^{-1}(\sinh (c+d x))}{2 d}-\frac{b \sinh (c+d x) \tanh ^2(c+d x)}{2 d} \]
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Rubi [A] time = 0.0761299, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316, Rules used = {3666, 2638, 2592, 288, 321, 203} \[ \frac{a \cosh (c+d x)}{d}+\frac{3 b \sinh (c+d x)}{2 d}-\frac{3 b \tan ^{-1}(\sinh (c+d x))}{2 d}-\frac{b \sinh (c+d x) \tanh ^2(c+d x)}{2 d} \]
Antiderivative was successfully verified.
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Rule 3666
Rule 2638
Rule 2592
Rule 288
Rule 321
Rule 203
Rubi steps
\begin{align*} \int \sinh (c+d x) \left (a+b \tanh ^3(c+d x)\right ) \, dx &=-\left (i \int \left (i a \sinh (c+d x)+i b \sinh (c+d x) \tanh ^3(c+d x)\right ) \, dx\right )\\ &=a \int \sinh (c+d x) \, dx+b \int \sinh (c+d x) \tanh ^3(c+d x) \, dx\\ &=\frac{a \cosh (c+d x)}{d}+\frac{b \operatorname{Subst}\left (\int \frac{x^4}{\left (1+x^2\right )^2} \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac{a \cosh (c+d x)}{d}-\frac{b \sinh (c+d x) \tanh ^2(c+d x)}{2 d}+\frac{(3 b) \operatorname{Subst}\left (\int \frac{x^2}{1+x^2} \, dx,x,\sinh (c+d x)\right )}{2 d}\\ &=\frac{a \cosh (c+d x)}{d}+\frac{3 b \sinh (c+d x)}{2 d}-\frac{b \sinh (c+d x) \tanh ^2(c+d x)}{2 d}-\frac{(3 b) \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\sinh (c+d x)\right )}{2 d}\\ &=-\frac{3 b \tan ^{-1}(\sinh (c+d x))}{2 d}+\frac{a \cosh (c+d x)}{d}+\frac{3 b \sinh (c+d x)}{2 d}-\frac{b \sinh (c+d x) \tanh ^2(c+d x)}{2 d}\\ \end{align*}
Mathematica [A] time = 0.120791, size = 72, normalized size = 1.14 \[ \frac{a \sinh (c) \sinh (d x)}{d}+\frac{a \cosh (c) \cosh (d x)}{d}+\frac{b \sinh (c+d x) \tanh ^2(c+d x)}{d}-\frac{3 b \left (\tan ^{-1}(\sinh (c+d x))-\tanh (c+d x) \text{sech}(c+d x)\right )}{2 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.04, size = 85, normalized size = 1.4 \begin{align*}{\frac{a\cosh \left ( dx+c \right ) }{d}}+{\frac{b \left ( \sinh \left ( dx+c \right ) \right ) ^{3}}{d \left ( \cosh \left ( dx+c \right ) \right ) ^{2}}}+3\,{\frac{b\sinh \left ( dx+c \right ) }{d \left ( \cosh \left ( dx+c \right ) \right ) ^{2}}}-{\frac{3\,b{\rm sech} \left (dx+c\right )\tanh \left ( dx+c \right ) }{2\,d}}-3\,{\frac{b\arctan \left ({{\rm e}^{dx+c}} \right ) }{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.61063, size = 142, normalized size = 2.25 \begin{align*} \frac{1}{2} \, b{\left (\frac{6 \, \arctan \left (e^{\left (-d x - c\right )}\right )}{d} - \frac{e^{\left (-d x - c\right )}}{d} + \frac{4 \, e^{\left (-2 \, d x - 2 \, c\right )} - e^{\left (-4 \, d x - 4 \, c\right )} + 1}{d{\left (e^{\left (-d x - c\right )} + 2 \, e^{\left (-3 \, d x - 3 \, c\right )} + e^{\left (-5 \, d x - 5 \, c\right )}\right )}}\right )} + \frac{a \cosh \left (d x + c\right )}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.36054, size = 1473, normalized size = 23.38 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \tanh ^{3}{\left (c + d x \right )}\right ) \sinh{\left (c + d x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.21657, size = 128, normalized size = 2.03 \begin{align*} -\frac{6 \, b \arctan \left (e^{\left (d x + c\right )}\right ) -{\left (a - b\right )} e^{\left (-d x - c\right )} -{\left (a e^{\left (d x + 8 \, c\right )} + b e^{\left (d x + 8 \, c\right )}\right )} e^{\left (-7 \, c\right )} - \frac{2 \,{\left (b e^{\left (3 \, d x + 3 \, c\right )} - b e^{\left (d x + c\right )}\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{2}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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