Optimal. Leaf size=54 \[ \frac{\left (a^2-b^2\right ) \sinh (c+d x)}{d}+\frac{(a+b)^2 \sinh ^3(c+d x)}{3 d}+\frac{b^2 \tan ^{-1}(\sinh (c+d x))}{d} \]
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Rubi [A] time = 0.0603328, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {3676, 390, 203} \[ \frac{\left (a^2-b^2\right ) \sinh (c+d x)}{d}+\frac{(a+b)^2 \sinh ^3(c+d x)}{3 d}+\frac{b^2 \tan ^{-1}(\sinh (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 3676
Rule 390
Rule 203
Rubi steps
\begin{align*} \int \cosh ^3(c+d x) \left (a+b \tanh ^2(c+d x)\right )^2 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (a+(a+b) x^2\right )^2}{1+x^2} \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (a^2-b^2+(a+b)^2 x^2+\frac{b^2}{1+x^2}\right ) \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac{\left (a^2-b^2\right ) \sinh (c+d x)}{d}+\frac{(a+b)^2 \sinh ^3(c+d x)}{3 d}+\frac{b^2 \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac{b^2 \tan ^{-1}(\sinh (c+d x))}{d}+\frac{\left (a^2-b^2\right ) \sinh (c+d x)}{d}+\frac{(a+b)^2 \sinh ^3(c+d x)}{3 d}\\ \end{align*}
Mathematica [A] time = 0.442775, size = 71, normalized size = 1.31 \[ \frac{\sinh (c+d x) \left ((a+b) ((a+b) \cosh (2 (c+d x))+5 a-7 b)+\frac{6 b^2 \tanh ^{-1}\left (\sqrt{-\sinh ^2(c+d x)}\right )}{\sqrt{-\sinh ^2(c+d x)}}\right )}{6 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.043, size = 117, normalized size = 2.2 \begin{align*}{\frac{2\,{a}^{2}\sinh \left ( dx+c \right ) }{3\,d}}+{\frac{{a}^{2}\sinh \left ( dx+c \right ) \left ( \cosh \left ( dx+c \right ) \right ) ^{2}}{3\,d}}+{\frac{2\,ab \left ( \cosh \left ( dx+c \right ) \right ) ^{2}\sinh \left ( dx+c \right ) }{3\,d}}-{\frac{2\,ab\sinh \left ( dx+c \right ) }{3\,d}}+{\frac{{b}^{2} \left ( \sinh \left ( dx+c \right ) \right ) ^{3}}{3\,d}}-{\frac{{b}^{2}\sinh \left ( dx+c \right ) }{d}}+2\,{\frac{{b}^{2}\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 [B] time = 1.67404, size = 217, normalized size = 4.02 \begin{align*} \frac{a b{\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3}}{12 \, d} - \frac{1}{24} \, b^{2}{\left (\frac{{\left (15 \, e^{\left (-2 \, d x - 2 \, c\right )} - 1\right )} e^{\left (3 \, d x + 3 \, c\right )}}{d} - \frac{15 \, e^{\left (-d x - c\right )} - e^{\left (-3 \, d x - 3 \, c\right )}}{d} + \frac{48 \, \arctan \left (e^{\left (-d x - c\right )}\right )}{d}\right )} + \frac{1}{24} \, a^{2}{\left (\frac{e^{\left (3 \, d x + 3 \, c\right )}}{d} + \frac{9 \, e^{\left (d x + c\right )}}{d} - \frac{9 \, e^{\left (-d x - c\right )}}{d} - \frac{e^{\left (-3 \, d x - 3 \, c\right )}}{d}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.92637, size = 1314, normalized size = 24.33 \begin{align*} \frac{{\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \left (d x + c\right )^{6} + 6 \,{\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{5} +{\left (a^{2} + 2 \, a b + b^{2}\right )} \sinh \left (d x + c\right )^{6} + 3 \,{\left (3 \, a^{2} - 2 \, a b - 5 \, b^{2}\right )} \cosh \left (d x + c\right )^{4} + 3 \,{\left (5 \,{\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \left (d x + c\right )^{2} + 3 \, a^{2} - 2 \, a b - 5 \, b^{2}\right )} \sinh \left (d x + c\right )^{4} + 4 \,{\left (5 \,{\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \left (d x + c\right )^{3} + 3 \,{\left (3 \, a^{2} - 2 \, a b - 5 \, b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} - 3 \,{\left (3 \, a^{2} - 2 \, a b - 5 \, b^{2}\right )} \cosh \left (d x + c\right )^{2} + 3 \,{\left (5 \,{\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \left (d x + c\right )^{4} + 6 \,{\left (3 \, a^{2} - 2 \, a b - 5 \, b^{2}\right )} \cosh \left (d x + c\right )^{2} - 3 \, a^{2} + 2 \, a b + 5 \, b^{2}\right )} \sinh \left (d x + c\right )^{2} - a^{2} - 2 \, a b - b^{2} + 48 \,{\left (b^{2} \cosh \left (d x + c\right )^{3} + 3 \, b^{2} \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right ) + 3 \, b^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + b^{2} \sinh \left (d x + c\right )^{3}\right )} \arctan \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right ) + 6 \,{\left ({\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \left (d x + c\right )^{5} + 2 \,{\left (3 \, a^{2} - 2 \, a b - 5 \, b^{2}\right )} \cosh \left (d x + c\right )^{3} -{\left (3 \, a^{2} - 2 \, a b - 5 \, b^{2}\right )} \cosh \left (d x + c\right )\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 )^{2} \sinh \left (d x + c\right ) + 3 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + d \sinh \left (d x + c\right )^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.79093, size = 221, normalized size = 4.09 \begin{align*} \frac{48 \, b^{2} \arctan \left (e^{\left (d x + c\right )}\right ) -{\left (9 \, a^{2} e^{\left (2 \, d x + 2 \, c\right )} - 6 \, a b e^{\left (2 \, d x + 2 \, c\right )} - 15 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} + a^{2} + 2 \, a b + b^{2}\right )} e^{\left (-3 \, d x - 3 \, c\right )} +{\left (a^{2} e^{\left (3 \, d x + 18 \, c\right )} + 2 \, a b e^{\left (3 \, d x + 18 \, c\right )} + b^{2} e^{\left (3 \, d x + 18 \, c\right )} + 9 \, a^{2} e^{\left (d x + 16 \, c\right )} - 6 \, a b e^{\left (d x + 16 \, c\right )} - 15 \, b^{2} e^{\left (d x + 16 \, c\right )}\right )} e^{\left (-15 \, c\right )}}{24 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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