Optimal. Leaf size=125 \[ \frac{1}{128} x \left (128 a^2+96 a b+35 b^2\right )+\frac{b (96 a+163 b) \sinh (c+d x) \cosh ^3(c+d x)}{192 d}-\frac{b (160 a+93 b) \sinh (c+d x) \cosh (c+d x)}{128 d}+\frac{b^2 \sinh (c+d x) \cosh ^7(c+d x)}{8 d}-\frac{25 b^2 \sinh (c+d x) \cosh ^5(c+d x)}{48 d} \]
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Rubi [A] time = 0.161943, antiderivative size = 125, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357, Rules used = {3209, 1157, 1814, 385, 206} \[ \frac{1}{128} x \left (128 a^2+96 a b+35 b^2\right )+\frac{b (96 a+163 b) \sinh (c+d x) \cosh ^3(c+d x)}{192 d}-\frac{b (160 a+93 b) \sinh (c+d x) \cosh (c+d x)}{128 d}+\frac{b^2 \sinh (c+d x) \cosh ^7(c+d x)}{8 d}-\frac{25 b^2 \sinh (c+d x) \cosh ^5(c+d x)}{48 d} \]
Antiderivative was successfully verified.
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Rule 3209
Rule 1157
Rule 1814
Rule 385
Rule 206
Rubi steps
\begin{align*} \int \left (a+b \sinh ^4(c+d x)\right )^2 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (a-2 a x^2+(a+b) x^4\right )^2}{\left (1-x^2\right )^5} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{b^2 \cosh ^7(c+d x) \sinh (c+d x)}{8 d}-\frac{\operatorname{Subst}\left (\int \frac{-8 a^2+b^2+8 \left (3 a^2+b^2\right ) x^2-8 (3 a-b) (a+b) x^4+8 (a+b)^2 x^6}{\left (1-x^2\right )^4} \, dx,x,\tanh (c+d x)\right )}{8 d}\\ &=-\frac{25 b^2 \cosh ^5(c+d x) \sinh (c+d x)}{48 d}+\frac{b^2 \cosh ^7(c+d x) \sinh (c+d x)}{8 d}+\frac{\operatorname{Subst}\left (\int \frac{48 a^2+19 b^2-96 \left (a^2-b^2\right ) x^2+48 (a+b)^2 x^4}{\left (1-x^2\right )^3} \, dx,x,\tanh (c+d x)\right )}{48 d}\\ &=\frac{b (96 a+163 b) \cosh ^3(c+d x) \sinh (c+d x)}{192 d}-\frac{25 b^2 \cosh ^5(c+d x) \sinh (c+d x)}{48 d}+\frac{b^2 \cosh ^7(c+d x) \sinh (c+d x)}{8 d}-\frac{\operatorname{Subst}\left (\int \frac{-3 \left (64 a^2-32 a b-29 b^2\right )+192 (a+b)^2 x^2}{\left (1-x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{192 d}\\ &=-\frac{b (160 a+93 b) \cosh (c+d x) \sinh (c+d x)}{128 d}+\frac{b (96 a+163 b) \cosh ^3(c+d x) \sinh (c+d x)}{192 d}-\frac{25 b^2 \cosh ^5(c+d x) \sinh (c+d x)}{48 d}+\frac{b^2 \cosh ^7(c+d x) \sinh (c+d x)}{8 d}+\frac{\left (128 a^2+96 a b+35 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{128 d}\\ &=\frac{1}{128} \left (128 a^2+96 a b+35 b^2\right ) x-\frac{b (160 a+93 b) \cosh (c+d x) \sinh (c+d x)}{128 d}+\frac{b (96 a+163 b) \cosh ^3(c+d x) \sinh (c+d x)}{192 d}-\frac{25 b^2 \cosh ^5(c+d x) \sinh (c+d x)}{48 d}+\frac{b^2 \cosh ^7(c+d x) \sinh (c+d x)}{8 d}\\ \end{align*}
Mathematica [A] time = 0.165761, size = 92, normalized size = 0.74 \[ \frac{24 \left (128 a^2+96 a b+35 b^2\right ) (c+d x)-96 b (16 a+7 b) \sinh (2 (c+d x))+24 b (8 a+7 b) \sinh (4 (c+d x))-32 b^2 \sinh (6 (c+d x))+3 b^2 \sinh (8 (c+d x))}{3072 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 111, normalized size = 0.9 \begin{align*}{\frac{1}{d} \left ({b}^{2} \left ( \left ({\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{7}}{8}}-{\frac{7\, \left ( \sinh \left ( dx+c \right ) \right ) ^{5}}{48}}+{\frac{35\, \left ( \sinh \left ( dx+c \right ) \right ) ^{3}}{192}}-{\frac{35\,\sinh \left ( dx+c \right ) }{128}} \right ) \cosh \left ( dx+c \right ) +{\frac{35\,dx}{128}}+{\frac{35\,c}{128}} \right ) +2\,ab \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 ) +{a}^{2} \left ( dx+c \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.11707, size = 247, normalized size = 1.98 \begin{align*} \frac{1}{32} \, a b{\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 )} + a^{2} x - \frac{1}{6144} \, b^{2}{\left (\frac{{\left (32 \, e^{\left (-2 \, d x - 2 \, c\right )} - 168 \, e^{\left (-4 \, d x - 4 \, c\right )} + 672 \, e^{\left (-6 \, d x - 6 \, c\right )} - 3\right )} e^{\left (8 \, d x + 8 \, c\right )}}{d} - \frac{1680 \,{\left (d x + c\right )}}{d} - \frac{672 \, e^{\left (-2 \, d x - 2 \, c\right )} - 168 \, e^{\left (-4 \, d x - 4 \, c\right )} + 32 \, e^{\left (-6 \, d x - 6 \, c\right )} - 3 \, e^{\left (-8 \, d x - 8 \, c\right )}}{d}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.62932, size = 522, normalized size = 4.18 \begin{align*} \frac{3 \, b^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{7} + 3 \,{\left (7 \, b^{2} \cosh \left (d x + c\right )^{3} - 8 \, b^{2} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{5} +{\left (21 \, b^{2} \cosh \left (d x + c\right )^{5} - 80 \, b^{2} \cosh \left (d x + c\right )^{3} + 12 \,{\left (8 \, a b + 7 \, b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} + 3 \,{\left (128 \, a^{2} + 96 \, a b + 35 \, b^{2}\right )} d x + 3 \,{\left (b^{2} \cosh \left (d x + c\right )^{7} - 8 \, b^{2} \cosh \left (d x + c\right )^{5} + 4 \,{\left (8 \, a b + 7 \, b^{2}\right )} \cosh \left (d x + c\right )^{3} - 8 \,{\left (16 \, a b + 7 \, b^{2}\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 = 19.2012, size = 332, normalized size = 2.66 \begin{align*} \begin{cases} a^{2} x + \frac{3 a b x \sinh ^{4}{\left (c + d x \right )}}{4} - \frac{3 a b x \sinh ^{2}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{2} + \frac{3 a b x \cosh ^{4}{\left (c + d x \right )}}{4} + \frac{5 a b \sinh ^{3}{\left (c + d x \right )} \cosh{\left (c + d x \right )}}{4 d} - \frac{3 a b \sinh{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{4 d} + \frac{35 b^{2} x \sinh ^{8}{\left (c + d x \right )}}{128} - \frac{35 b^{2} x \sinh ^{6}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{32} + \frac{105 b^{2} x \sinh ^{4}{\left (c + d x \right )} \cosh ^{4}{\left (c + d x \right )}}{64} - \frac{35 b^{2} x \sinh ^{2}{\left (c + d x \right )} \cosh ^{6}{\left (c + d x \right )}}{32} + \frac{35 b^{2} x \cosh ^{8}{\left (c + d x \right )}}{128} + \frac{93 b^{2} \sinh ^{7}{\left (c + d x \right )} \cosh{\left (c + d x \right )}}{128 d} - \frac{511 b^{2} \sinh ^{5}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{384 d} + \frac{385 b^{2} \sinh ^{3}{\left (c + d x \right )} \cosh ^{5}{\left (c + d x \right )}}{384 d} - \frac{35 b^{2} \sinh{\left (c + d x \right )} \cosh ^{7}{\left (c + d x \right )}}{128 d} & \text{for}\: d \neq 0 \\x \left (a + b \sinh ^{4}{\left (c \right )}\right )^{2} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.18992, size = 319, normalized size = 2.55 \begin{align*} \frac{3 \, b^{2} e^{\left (8 \, d x + 8 \, c\right )} - 32 \, b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 192 \, a b e^{\left (4 \, d x + 4 \, c\right )} + 168 \, b^{2} e^{\left (4 \, d x + 4 \, c\right )} - 1536 \, a b e^{\left (2 \, d x + 2 \, c\right )} - 672 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 48 \,{\left (128 \, a^{2} + 96 \, a b + 35 \, b^{2}\right )}{\left (d x + c\right )} -{\left (6400 \, a^{2} e^{\left (8 \, d x + 8 \, c\right )} + 4800 \, a b e^{\left (8 \, d x + 8 \, c\right )} + 1750 \, b^{2} e^{\left (8 \, d x + 8 \, c\right )} - 1536 \, a b e^{\left (6 \, d x + 6 \, c\right )} - 672 \, b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 192 \, a b e^{\left (4 \, d x + 4 \, c\right )} + 168 \, b^{2} e^{\left (4 \, d x + 4 \, c\right )} - 32 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 3 \, b^{2}\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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