Optimal. Leaf size=91 \[ \frac{3}{8} x (a+b) \left (a^2-2 a b+5 b^2\right )+\frac{(a+b)^3 \sinh (c+d x) \cosh ^3(c+d x)}{4 d}+\frac{3 (a-3 b) (a+b)^2 \sinh (c+d x) \cosh (c+d x)}{8 d}-\frac{b^3 \tanh (c+d x)}{d} \]
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Rubi [A] time = 0.130354, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {3675, 390, 1157, 385, 206} \[ \frac{3}{8} x (a+b) \left (a^2-2 a b+5 b^2\right )+\frac{(a+b)^3 \sinh (c+d x) \cosh ^3(c+d x)}{4 d}+\frac{3 (a-3 b) (a+b)^2 \sinh (c+d x) \cosh (c+d x)}{8 d}-\frac{b^3 \tanh (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 3675
Rule 390
Rule 1157
Rule 385
Rule 206
Rubi steps
\begin{align*} \int \cosh ^4(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (a+b x^2\right )^3}{\left (1-x^2\right )^3} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (-b^3+\frac{a^3+b^3+3 b \left (a^2-b^2\right ) x^2+3 b^2 (a+b) x^4}{\left (1-x^2\right )^3}\right ) \, dx,x,\tanh (c+d x)\right )}{d}\\ &=-\frac{b^3 \tanh (c+d x)}{d}+\frac{\operatorname{Subst}\left (\int \frac{a^3+b^3+3 b \left (a^2-b^2\right ) x^2+3 b^2 (a+b) x^4}{\left (1-x^2\right )^3} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{(a+b)^3 \cosh ^3(c+d x) \sinh (c+d x)}{4 d}-\frac{b^3 \tanh (c+d x)}{d}-\frac{\operatorname{Subst}\left (\int \frac{-3 (a-b)^2 (a+b)+12 b^2 (a+b) x^2}{\left (1-x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{4 d}\\ &=\frac{3 (a-3 b) (a+b)^2 \cosh (c+d x) \sinh (c+d x)}{8 d}+\frac{(a+b)^3 \cosh ^3(c+d x) \sinh (c+d x)}{4 d}-\frac{b^3 \tanh (c+d x)}{d}+\frac{\left (3 (a+b) \left (a^2-2 a b+5 b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{8 d}\\ &=\frac{3}{8} (a+b) \left (a^2-2 a b+5 b^2\right ) x+\frac{3 (a-3 b) (a+b)^2 \cosh (c+d x) \sinh (c+d x)}{8 d}+\frac{(a+b)^3 \cosh ^3(c+d x) \sinh (c+d x)}{4 d}-\frac{b^3 \tanh (c+d x)}{d}\\ \end{align*}
Mathematica [A] time = 0.668448, size = 81, normalized size = 0.89 \[ \frac{12 \left (-a^2 b+a^3+3 a b^2+5 b^3\right ) (c+d x)+8 (a-2 b) (a+b)^2 \sinh (2 (c+d x))+(a+b)^3 \sinh (4 (c+d x))-32 b^3 \tanh (c+d x)}{32 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.042, size = 184, normalized size = 2. \begin{align*}{\frac{1}{d} \left ({a}^{3} \left ( \left ({\frac{ \left ( \cosh \left ( dx+c \right ) \right ) ^{3}}{4}}+{\frac{3\,\cosh \left ( dx+c \right ) }{8}} \right ) \sinh \left ( dx+c \right ) +{\frac{3\,dx}{8}}+{\frac{3\,c}{8}} \right ) +3\,{a}^{2}b \left ( 1/4\,\sinh \left ( dx+c \right ) \left ( \cosh \left ( dx+c \right ) \right ) ^{3}-1/8\,\cosh \left ( dx+c \right ) \sinh \left ( dx+c \right ) -1/8\,dx-c/8 \right ) +3\,a{b}^{2} \left ( \left ( 1/4\, \left ( \sinh \left ( dx+c \right ) \right ) ^{3}-3/8\,\sinh \left ( dx+c \right ) \right ) \cosh \left ( dx+c \right ) +3/8\,dx+3/8\,c \right ) +{b}^{3} \left ({\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{5}}{4\,\cosh \left ( dx+c \right ) }}-{\frac{5\, \left ( \sinh \left ( dx+c \right ) \right ) ^{3}}{8\,\cosh \left ( dx+c \right ) }}+{\frac{15\,dx}{8}}+{\frac{15\,c}{8}}-{\frac{15\,\tanh \left ( dx+c \right ) }{8}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.16712, size = 360, normalized size = 3.96 \begin{align*} \frac{1}{64} \, a^{3}{\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{3}{64} \, a b^{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}{64} \, b^{3}{\left (\frac{120 \,{\left (d x + c\right )}}{d} + \frac{16 \, e^{\left (-2 \, d x - 2 \, c\right )} - e^{\left (-4 \, d x - 4 \, c\right )}}{d} - \frac{15 \, e^{\left (-2 \, d x - 2 \, c\right )} + 144 \, e^{\left (-4 \, d x - 4 \, c\right )} - 1}{d{\left (e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )}\right )}}\right )} - \frac{3}{64} \, a^{2} 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 )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.0398, size = 539, normalized size = 5.92 \begin{align*} \frac{{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \sinh \left (d x + c\right )^{5} +{\left (9 \, a^{3} + 3 \, a^{2} b - 21 \, a b^{2} - 15 \, b^{3} + 10 \,{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{3} + 8 \,{\left (8 \, b^{3} + 3 \,{\left (a^{3} - a^{2} b + 3 \, a b^{2} + 5 \, b^{3}\right )} d x\right )} \cosh \left (d x + c\right ) +{\left (5 \,{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \cosh \left (d x + c\right )^{4} + 8 \, a^{3} - 24 \, a b^{2} - 80 \, b^{3} + 9 \,{\left (3 \, a^{3} + a^{2} b - 7 \, a b^{2} - 5 \, b^{3}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )}{64 \, d \cosh \left (d x + c\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 = 2.67663, size = 385, normalized size = 4.23 \begin{align*} \frac{24 \,{\left (a^{3} - a^{2} b + 3 \, a b^{2} + 5 \, b^{3}\right )} d x + \frac{128 \, b^{3}}{e^{\left (2 \, d x + 2 \, c\right )} + 1} -{\left (18 \, a^{3} e^{\left (4 \, d x + 4 \, c\right )} - 18 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} + 54 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 90 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 8 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} - 24 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} - 16 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} e^{\left (-4 \, d x - 4 \, c\right )} +{\left (a^{3} e^{\left (4 \, d x + 20 \, c\right )} + 3 \, a^{2} b e^{\left (4 \, d x + 20 \, c\right )} + 3 \, a b^{2} e^{\left (4 \, d x + 20 \, c\right )} + b^{3} e^{\left (4 \, d x + 20 \, c\right )} + 8 \, a^{3} e^{\left (2 \, d x + 18 \, c\right )} - 24 \, a b^{2} e^{\left (2 \, d x + 18 \, c\right )} - 16 \, b^{3} e^{\left (2 \, d x + 18 \, c\right )}\right )} e^{\left (-16 \, c\right )}}{64 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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