Optimal. Leaf size=70 \[ -\frac{b^2 (a+b) \text{sech}^3(c+d x)}{d}+\frac{(a+b)^3 \cosh (c+d x)}{d}+\frac{3 b (a+b)^2 \text{sech}(c+d x)}{d}+\frac{b^3 \text{sech}^5(c+d x)}{5 d} \]
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Rubi [A] time = 0.0684337, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {3664, 270} \[ -\frac{b^2 (a+b) \text{sech}^3(c+d x)}{d}+\frac{(a+b)^3 \cosh (c+d x)}{d}+\frac{3 b (a+b)^2 \text{sech}(c+d x)}{d}+\frac{b^3 \text{sech}^5(c+d x)}{5 d} \]
Antiderivative was successfully verified.
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Rule 3664
Rule 270
Rubi steps
\begin{align*} \int \sinh (c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{\left (a+b-b x^2\right )^3}{x^2} \, dx,x,\text{sech}(c+d x)\right )}{d}\\ &=-\frac{\operatorname{Subst}\left (\int \left (-3 b (a+b)^2+\frac{(a+b)^3}{x^2}+3 b^2 (a+b) x^2-b^3 x^4\right ) \, dx,x,\text{sech}(c+d x)\right )}{d}\\ &=\frac{(a+b)^3 \cosh (c+d x)}{d}+\frac{3 b (a+b)^2 \text{sech}(c+d x)}{d}-\frac{b^2 (a+b) \text{sech}^3(c+d x)}{d}+\frac{b^3 \text{sech}^5(c+d x)}{5 d}\\ \end{align*}
Mathematica [A] time = 0.828759, size = 63, normalized size = 0.9 \[ \frac{b \text{sech}(c+d x) \left (-5 b (a+b) \text{sech}^2(c+d x)+15 (a+b)^2+b^2 \text{sech}^4(c+d x)\right )+5 (a+b)^3 \cosh (c+d x)}{5 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.049, size = 219, normalized size = 3.1 \begin{align*}{\frac{1}{d} \left ({a}^{3}\cosh \left ( dx+c \right ) +3\,{a}^{2}b \left ( -{\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{\cosh \left ( dx+c \right ) }}+2\,\cosh \left ( dx+c \right ) \right ) +3\,a{b}^{2} \left ({\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{4}}{ \left ( \cosh \left ( dx+c \right ) \right ) ^{3}}}+4/3\,{\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{ \left ( \cosh \left ( dx+c \right ) \right ) ^{3}}}-8/3\,{\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{\cosh \left ( dx+c \right ) }}+8/3\,\cosh \left ( dx+c \right ) \right ) +{b}^{3} \left ({\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{6}}{ \left ( \cosh \left ( dx+c \right ) \right ) ^{5}}}+6\,{\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{4}}{ \left ( \cosh \left ( dx+c \right ) \right ) ^{5}}}+{\frac{24\, \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{5\, \left ( \cosh \left ( dx+c \right ) \right ) ^{5}}}-{\frac{16\, \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{5\, \left ( \cosh \left ( dx+c \right ) \right ) ^{3}}}-{\frac{16\, \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{5\,\cosh \left ( dx+c \right ) }}+{\frac{16\,\cosh \left ( dx+c \right ) }{5}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.09181, size = 433, normalized size = 6.19 \begin{align*} \frac{1}{10} \, b^{3}{\left (\frac{5 \, e^{\left (-d x - c\right )}}{d} + \frac{85 \, e^{\left (-2 \, d x - 2 \, c\right )} + 210 \, e^{\left (-4 \, d x - 4 \, c\right )} + 314 \, e^{\left (-6 \, d x - 6 \, c\right )} + 185 \, e^{\left (-8 \, d x - 8 \, c\right )} + 65 \, e^{\left (-10 \, d x - 10 \, c\right )} + 5}{d{\left (e^{\left (-d x - c\right )} + 5 \, e^{\left (-3 \, d x - 3 \, c\right )} + 10 \, e^{\left (-5 \, d x - 5 \, c\right )} + 10 \, e^{\left (-7 \, d x - 7 \, c\right )} + 5 \, e^{\left (-9 \, d x - 9 \, c\right )} + e^{\left (-11 \, d x - 11 \, c\right )}\right )}}\right )} + \frac{1}{2} \, a b^{2}{\left (\frac{3 \, e^{\left (-d x - c\right )}}{d} + \frac{33 \, e^{\left (-2 \, d x - 2 \, c\right )} + 41 \, e^{\left (-4 \, d x - 4 \, c\right )} + 27 \, e^{\left (-6 \, d x - 6 \, c\right )} + 3}{d{\left (e^{\left (-d x - c\right )} + 3 \, e^{\left (-3 \, d x - 3 \, c\right )} + 3 \, e^{\left (-5 \, d x - 5 \, c\right )} + e^{\left (-7 \, d x - 7 \, c\right )}\right )}}\right )} + \frac{3}{2} \, a^{2} b{\left (\frac{e^{\left (-d x - c\right )}}{d} + \frac{5 \, e^{\left (-2 \, d x - 2 \, c\right )} + 1}{d{\left (e^{\left (-d x - c\right )} + e^{\left (-3 \, d x - 3 \, c\right )}\right )}}\right )} + \frac{a^{3} \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 = 1.98396, size = 959, normalized size = 13.7 \begin{align*} \frac{5 \,{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \cosh \left (d x + c\right )^{6} + 5 \,{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \sinh \left (d x + c\right )^{6} + 30 \,{\left (a^{3} + 5 \, a^{2} b + 7 \, a b^{2} + 3 \, b^{3}\right )} \cosh \left (d x + c\right )^{4} + 15 \,{\left (2 \, a^{3} + 10 \, a^{2} b + 14 \, a b^{2} + 6 \, b^{3} + 5 \,{\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 )^{4} + 50 \, a^{3} + 330 \, a^{2} b + 430 \, a b^{2} + 182 \, b^{3} + 5 \,{\left (15 \, a^{3} + 93 \, a^{2} b + 125 \, a b^{2} + 47 \, b^{3}\right )} \cosh \left (d x + c\right )^{2} + 5 \,{\left (15 \,{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \cosh \left (d x + c\right )^{4} + 15 \, a^{3} + 93 \, a^{2} b + 125 \, a b^{2} + 47 \, b^{3} + 36 \,{\left (a^{3} + 5 \, a^{2} b + 7 \, a b^{2} + 3 \, b^{3}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{2}}{10 \,{\left (d \cosh \left (d x + c\right )^{5} + 5 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{4} + 5 \, d \cosh \left (d x + c\right )^{3} + 5 \,{\left (2 \, d \cosh \left (d x + c\right )^{3} + 3 \, d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} + 10 \, d \cosh \left (d x + c\right )\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 = 1.77435, size = 435, normalized size = 6.21 \begin{align*} \frac{5 \,{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} e^{\left (-d x - c\right )} + 5 \,{\left (a^{3} e^{\left (d x + 14 \, c\right )} + 3 \, a^{2} b e^{\left (d x + 14 \, c\right )} + 3 \, a b^{2} e^{\left (d x + 14 \, c\right )} + b^{3} e^{\left (d x + 14 \, c\right )}\right )} e^{\left (-13 \, c\right )} + \frac{4 \,{\left (15 \, a^{2} b e^{\left (9 \, d x + 9 \, c\right )} + 30 \, a b^{2} e^{\left (9 \, d x + 9 \, c\right )} + 15 \, b^{3} e^{\left (9 \, d x + 9 \, c\right )} + 60 \, a^{2} b e^{\left (7 \, d x + 7 \, c\right )} + 100 \, a b^{2} e^{\left (7 \, d x + 7 \, c\right )} + 40 \, b^{3} e^{\left (7 \, d x + 7 \, c\right )} + 90 \, a^{2} b e^{\left (5 \, d x + 5 \, c\right )} + 140 \, a b^{2} e^{\left (5 \, d x + 5 \, c\right )} + 66 \, b^{3} e^{\left (5 \, d x + 5 \, c\right )} + 60 \, a^{2} b e^{\left (3 \, d x + 3 \, c\right )} + 100 \, a b^{2} e^{\left (3 \, d x + 3 \, c\right )} + 40 \, b^{3} e^{\left (3 \, d x + 3 \, c\right )} + 15 \, a^{2} b e^{\left (d x + c\right )} + 30 \, a b^{2} e^{\left (d x + c\right )} + 15 \, b^{3} e^{\left (d x + c\right )}\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{5}}}{10 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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