Optimal. Leaf size=203 \[ \frac{b \left (88 a^2-68 a b+15 b^2\right ) \sinh (c+d x) \cosh ^3(c+d x)}{192 d}+\frac{\left (-48 a^2 b+64 a^3+24 a b^2-5 b^3\right ) \sinh (c+d x) \cosh (c+d x)}{128 d}+\frac{1}{128} x \left (-48 a^2 b+64 a^3+24 a b^2-5 b^3\right )+\frac{b \sinh (c+d x) \cosh ^7(c+d x) \left (a-(a-b) \tanh ^2(c+d x)\right )^2}{8 d}+\frac{b \sinh (c+d x) \cosh ^5(c+d x) \left (a (8 a-b)-(8 a-5 b) (a-b) \tanh ^2(c+d x)\right )}{48 d} \]
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Rubi [A] time = 0.268032, antiderivative size = 203, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {3191, 413, 526, 385, 199, 206} \[ \frac{b \left (88 a^2-68 a b+15 b^2\right ) \sinh (c+d x) \cosh ^3(c+d x)}{192 d}+\frac{\left (-48 a^2 b+64 a^3+24 a b^2-5 b^3\right ) \sinh (c+d x) \cosh (c+d x)}{128 d}+\frac{1}{128} x \left (-48 a^2 b+64 a^3+24 a b^2-5 b^3\right )+\frac{b \sinh (c+d x) \cosh ^7(c+d x) \left (a-(a-b) \tanh ^2(c+d x)\right )^2}{8 d}+\frac{b \sinh (c+d x) \cosh ^5(c+d x) \left (a (8 a-b)-(8 a-5 b) (a-b) \tanh ^2(c+d x)\right )}{48 d} \]
Antiderivative was successfully verified.
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Rule 3191
Rule 413
Rule 526
Rule 385
Rule 199
Rule 206
Rubi steps
\begin{align*} \int \cosh ^2(c+d x) \left (a+b \sinh ^2(c+d x)\right )^3 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (a-(a-b) x^2\right )^3}{\left (1-x^2\right )^5} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{b \cosh ^7(c+d x) \sinh (c+d x) \left (a-(a-b) \tanh ^2(c+d x)\right )^2}{8 d}-\frac{\operatorname{Subst}\left (\int \frac{\left (-a (8 a-b)+(8 a-5 b) (a-b) x^2\right ) \left (a+(-a+b) x^2\right )}{\left (1-x^2\right )^4} \, dx,x,\tanh (c+d x)\right )}{8 d}\\ &=\frac{b \cosh ^7(c+d x) \sinh (c+d x) \left (a-(a-b) \tanh ^2(c+d x)\right )^2}{8 d}+\frac{b \cosh ^5(c+d x) \sinh (c+d x) \left (a (8 a-b)-(8 a-5 b) (a-b) \tanh ^2(c+d x)\right )}{48 d}-\frac{\operatorname{Subst}\left (\int \frac{-a (6 a-b) (8 a-b)+3 (8 a-5 b) (a-b) (2 a-b) x^2}{\left (1-x^2\right )^3} \, dx,x,\tanh (c+d x)\right )}{48 d}\\ &=\frac{b \left (88 a^2-68 a b+15 b^2\right ) \cosh ^3(c+d x) \sinh (c+d x)}{192 d}+\frac{b \cosh ^7(c+d x) \sinh (c+d x) \left (a-(a-b) \tanh ^2(c+d x)\right )^2}{8 d}+\frac{b \cosh ^5(c+d x) \sinh (c+d x) \left (a (8 a-b)-(8 a-5 b) (a-b) \tanh ^2(c+d x)\right )}{48 d}+\frac{\left (64 a^3-48 a^2 b+24 a b^2-5 b^3\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1-x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{64 d}\\ &=\frac{\left (64 a^3-48 a^2 b+24 a b^2-5 b^3\right ) \cosh (c+d x) \sinh (c+d x)}{128 d}+\frac{b \left (88 a^2-68 a b+15 b^2\right ) \cosh ^3(c+d x) \sinh (c+d x)}{192 d}+\frac{b \cosh ^7(c+d x) \sinh (c+d x) \left (a-(a-b) \tanh ^2(c+d x)\right )^2}{8 d}+\frac{b \cosh ^5(c+d x) \sinh (c+d x) \left (a (8 a-b)-(8 a-5 b) (a-b) \tanh ^2(c+d x)\right )}{48 d}+\frac{\left (64 a^3-48 a^2 b+24 a b^2-5 b^3\right ) \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{128 d}\\ &=\frac{1}{128} \left (64 a^3-48 a^2 b+24 a b^2-5 b^3\right ) x+\frac{\left (64 a^3-48 a^2 b+24 a b^2-5 b^3\right ) \cosh (c+d x) \sinh (c+d x)}{128 d}+\frac{b \left (88 a^2-68 a b+15 b^2\right ) \cosh ^3(c+d x) \sinh (c+d x)}{192 d}+\frac{b \cosh ^7(c+d x) \sinh (c+d x) \left (a-(a-b) \tanh ^2(c+d x)\right )^2}{8 d}+\frac{b \cosh ^5(c+d x) \sinh (c+d x) \left (a (8 a-b)-(8 a-5 b) (a-b) \tanh ^2(c+d x)\right )}{48 d}\\ \end{align*}
Mathematica [A] time = 0.302241, size = 120, normalized size = 0.59 \[ \frac{24 \left (-48 a^2 b+64 a^3+24 a b^2-5 b^3\right ) (c+d x)+24 b \left (12 a^2-6 a b+b^2\right ) \sinh (4 (c+d x))+48 \left (16 a^3-3 a b^2+b^3\right ) \sinh (2 (c+d x))+16 b^2 (3 a-b) \sinh (6 (c+d x))+3 b^3 \sinh (8 (c+d x))}{3072 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.032, size = 216, normalized size = 1.1 \begin{align*}{\frac{1}{d} \left ({b}^{3} \left ({\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{5} \left ( \cosh \left ( dx+c \right ) \right ) ^{3}}{8}}-{\frac{5\, \left ( \sinh \left ( dx+c \right ) \right ) ^{3} \left ( \cosh \left ( dx+c \right ) \right ) ^{3}}{48}}+{\frac{5\,\sinh \left ( dx+c \right ) \left ( \cosh \left ( dx+c \right ) \right ) ^{3}}{64}}-{\frac{5\,\cosh \left ( dx+c \right ) \sinh \left ( dx+c \right ) }{128}}-{\frac{5\,dx}{128}}-{\frac{5\,c}{128}} \right ) +3\,a{b}^{2} \left ( 1/6\, \left ( \sinh \left ( dx+c \right ) \right ) ^{3} \left ( \cosh \left ( dx+c \right ) \right ) ^{3}-1/8\,\sinh \left ( dx+c \right ) \left ( \cosh \left ( dx+c \right ) \right ) ^{3}+1/16\,\cosh \left ( dx+c \right ) \sinh \left ( dx+c \right ) +1/16\,dx+c/16 \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 ) +{a}^{3} \left ({\frac{\cosh \left ( dx+c \right ) \sinh \left ( dx+c \right ) }{2}}+{\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.11971, size = 387, normalized size = 1.91 \begin{align*} \frac{1}{8} \, a^{3}{\left (4 \, x + \frac{e^{\left (2 \, d x + 2 \, c\right )}}{d} - \frac{e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right )} - \frac{1}{6144} \, b^{3}{\left (\frac{{\left (16 \, e^{\left (-2 \, d x - 2 \, c\right )} - 24 \, e^{\left (-4 \, d x - 4 \, c\right )} - 48 \, e^{\left (-6 \, d x - 6 \, c\right )} - 3\right )} e^{\left (8 \, d x + 8 \, c\right )}}{d} + \frac{240 \,{\left (d x + c\right )}}{d} + \frac{48 \, e^{\left (-2 \, d x - 2 \, c\right )} + 24 \, e^{\left (-4 \, d x - 4 \, c\right )} - 16 \, e^{\left (-6 \, d x - 6 \, c\right )} + 3 \, e^{\left (-8 \, d x - 8 \, c\right )}}{d}\right )} - \frac{1}{128} \, a b^{2}{\left (\frac{{\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} + 3 \, e^{\left (-4 \, d x - 4 \, c\right )} - 1\right )} e^{\left (6 \, d x + 6 \, c\right )}}{d} - \frac{24 \,{\left (d x + c\right )}}{d} - \frac{3 \, e^{\left (-2 \, d x - 2 \, c\right )} + 3 \, e^{\left (-4 \, d x - 4 \, c\right )} - e^{\left (-6 \, d x - 6 \, c\right )}}{d}\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 [A] time = 1.48108, size = 626, normalized size = 3.08 \begin{align*} \frac{3 \, b^{3} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{7} + 3 \,{\left (7 \, b^{3} \cosh \left (d x + c\right )^{3} + 4 \,{\left (3 \, a b^{2} - b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{5} +{\left (21 \, b^{3} \cosh \left (d x + c\right )^{5} + 40 \,{\left (3 \, a b^{2} - b^{3}\right )} \cosh \left (d x + c\right )^{3} + 12 \,{\left (12 \, a^{2} b - 6 \, a b^{2} + b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} + 3 \,{\left (64 \, a^{3} - 48 \, a^{2} b + 24 \, a b^{2} - 5 \, b^{3}\right )} d x + 3 \,{\left (b^{3} \cosh \left (d x + c\right )^{7} + 4 \,{\left (3 \, a b^{2} - b^{3}\right )} \cosh \left (d x + c\right )^{5} + 4 \,{\left (12 \, a^{2} b - 6 \, a b^{2} + b^{3}\right )} \cosh \left (d x + c\right )^{3} + 4 \,{\left (16 \, a^{3} - 3 \, a b^{2} + b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{384 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 13.7037, size = 559, normalized size = 2.75 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.27872, size = 482, normalized size = 2.37 \begin{align*} \frac{3 \, b^{3} e^{\left (8 \, d x + 8 \, c\right )} + 48 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} - 16 \, b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 288 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} - 144 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 24 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 768 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} - 144 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 48 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 48 \,{\left (64 \, a^{3} - 48 \, a^{2} b + 24 \, a b^{2} - 5 \, b^{3}\right )}{\left (d x + c\right )} -{\left (3200 \, a^{3} e^{\left (8 \, d x + 8 \, c\right )} - 2400 \, a^{2} b e^{\left (8 \, d x + 8 \, c\right )} + 1200 \, a b^{2} e^{\left (8 \, d x + 8 \, c\right )} - 250 \, b^{3} e^{\left (8 \, d x + 8 \, c\right )} + 768 \, a^{3} e^{\left (6 \, d x + 6 \, c\right )} - 144 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 48 \, b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 288 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} - 144 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 24 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 48 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} - 16 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 3 \, b^{3}\right )} e^{\left (-8 \, d x - 8 \, c\right )}}{6144 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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