Optimal. Leaf size=159 \[ \frac {\left (48 a^2-16 a b+3 b^2\right ) \sinh (c+d x) \cosh ^3(c+d x)}{192 d}+\frac {\left (48 a^2-16 a b+3 b^2\right ) \sinh (c+d x) \cosh (c+d x)}{128 d}+\frac {1}{128} x \left (48 a^2-16 a b+3 b^2\right )+\frac {b (10 a-3 b) \sinh (c+d x) \cosh ^5(c+d x)}{48 d}+\frac {b \sinh (c+d x) \cosh ^7(c+d x) \left (a-(a-b) \tanh ^2(c+d x)\right )}{8 d} \]
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Rubi [A] time = 0.17, antiderivative size = 159, normalized size of antiderivative = 1.00, 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 = {3191, 413, 385, 199, 206} \[ \frac {\left (48 a^2-16 a b+3 b^2\right ) \sinh (c+d x) \cosh ^3(c+d x)}{192 d}+\frac {\left (48 a^2-16 a b+3 b^2\right ) \sinh (c+d x) \cosh (c+d x)}{128 d}+\frac {1}{128} x \left (48 a^2-16 a b+3 b^2\right )+\frac {b (10 a-3 b) \sinh (c+d x) \cosh ^5(c+d x)}{48 d}+\frac {b \sinh (c+d x) \cosh ^7(c+d x) \left (a-(a-b) \tanh ^2(c+d x)\right )}{8 d} \]
Antiderivative was successfully verified.
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Rule 199
Rule 206
Rule 385
Rule 413
Rule 3191
Rubi steps
\begin {align*} \int \cosh ^4(c+d x) \left (a+b \sinh ^2(c+d x)\right )^2 \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (a-(a-b) x^2\right )^2}{\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 )}{8 d}-\frac {\operatorname {Subst}\left (\int \frac {-a (8 a-b)+(8 a-3 b) (a-b) x^2}{\left (1-x^2\right )^4} \, dx,x,\tanh (c+d x)\right )}{8 d}\\ &=\frac {(10 a-3 b) b \cosh ^5(c+d x) \sinh (c+d x)}{48 d}+\frac {b \cosh ^7(c+d x) \sinh (c+d x) \left (a-(a-b) \tanh ^2(c+d x)\right )}{8 d}+\frac {\left (48 a^2-16 a b+3 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1-x^2\right )^3} \, dx,x,\tanh (c+d x)\right )}{48 d}\\ &=\frac {\left (48 a^2-16 a b+3 b^2\right ) \cosh ^3(c+d x) \sinh (c+d x)}{192 d}+\frac {(10 a-3 b) b \cosh ^5(c+d x) \sinh (c+d x)}{48 d}+\frac {b \cosh ^7(c+d x) \sinh (c+d x) \left (a-(a-b) \tanh ^2(c+d x)\right )}{8 d}+\frac {\left (48 a^2-16 a b+3 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1-x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{64 d}\\ &=\frac {\left (48 a^2-16 a b+3 b^2\right ) \cosh (c+d x) \sinh (c+d x)}{128 d}+\frac {\left (48 a^2-16 a b+3 b^2\right ) \cosh ^3(c+d x) \sinh (c+d x)}{192 d}+\frac {(10 a-3 b) b \cosh ^5(c+d x) \sinh (c+d x)}{48 d}+\frac {b \cosh ^7(c+d x) \sinh (c+d x) \left (a-(a-b) \tanh ^2(c+d x)\right )}{8 d}+\frac {\left (48 a^2-16 a b+3 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 (48 a^2-16 a b+3 b^2\right ) x+\frac {\left (48 a^2-16 a b+3 b^2\right ) \cosh (c+d x) \sinh (c+d x)}{128 d}+\frac {\left (48 a^2-16 a b+3 b^2\right ) \cosh ^3(c+d x) \sinh (c+d x)}{192 d}+\frac {(10 a-3 b) b \cosh ^5(c+d x) \sinh (c+d x)}{48 d}+\frac {b \cosh ^7(c+d x) \sinh (c+d x) \left (a-(a-b) \tanh ^2(c+d x)\right )}{8 d}\\ \end {align*}
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Mathematica [A] time = 0.32, size = 98, normalized size = 0.62 \[ \frac {24 \left (48 a^2-16 a b+3 b^2\right ) (c+d x)+24 \left (4 a^2+4 a b-b^2\right ) \sinh (4 (c+d x))+32 a b \sinh (6 (c+d x))+96 a (8 a-b) \sinh (2 (c+d x))+3 b^2 \sinh (8 (c+d x))}{3072 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 212, normalized size = 1.33 \[ \frac {3 \, b^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{7} + 3 \, {\left (7 \, b^{2} \cosh \left (d x + c\right )^{3} + 8 \, a b \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{5} + {\left (21 \, b^{2} \cosh \left (d x + c\right )^{5} + 80 \, a b \cosh \left (d x + c\right )^{3} + 12 \, {\left (4 \, a^{2} + 4 \, a b - b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} + 3 \, {\left (48 \, a^{2} - 16 \, a b + 3 \, b^{2}\right )} d x + 3 \, {\left (b^{2} \cosh \left (d x + c\right )^{7} + 8 \, a b \cosh \left (d x + c\right )^{5} + 4 \, {\left (4 \, a^{2} + 4 \, a b - b^{2}\right )} \cosh \left (d x + c\right )^{3} + 8 \, {\left (8 \, a^{2} - a b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{384 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 191, normalized size = 1.20 \[ \frac {1}{128} \, {\left (48 \, a^{2} - 16 \, a b + 3 \, b^{2}\right )} x + \frac {b^{2} e^{\left (8 \, d x + 8 \, c\right )}}{2048 \, d} + \frac {a b e^{\left (6 \, d x + 6 \, c\right )}}{192 \, d} - \frac {a b e^{\left (-6 \, d x - 6 \, c\right )}}{192 \, d} - \frac {b^{2} e^{\left (-8 \, d x - 8 \, c\right )}}{2048 \, d} + \frac {{\left (4 \, a^{2} + 4 \, a b - b^{2}\right )} e^{\left (4 \, d x + 4 \, c\right )}}{256 \, d} + \frac {{\left (8 \, a^{2} - a b\right )} e^{\left (2 \, d x + 2 \, c\right )}}{64 \, d} - \frac {{\left (8 \, a^{2} - a b\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{64 \, d} - \frac {{\left (4 \, a^{2} + 4 \, a b - b^{2}\right )} e^{\left (-4 \, d x - 4 \, c\right )}}{256 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 172, normalized size = 1.08 \[ \frac {b^{2} \left (\frac {\left (\sinh ^{3}\left (d x +c \right )\right ) \left (\cosh ^{5}\left (d x +c \right )\right )}{8}-\frac {\sinh \left (d x +c \right ) \left (\cosh ^{5}\left (d x +c \right )\right )}{16}+\frac {\left (\frac {\left (\cosh ^{3}\left (d x +c \right )\right )}{4}+\frac {3 \cosh \left (d x +c \right )}{8}\right ) \sinh \left (d x +c \right )}{16}+\frac {3 d x}{128}+\frac {3 c}{128}\right )+2 a b \left (\frac {\sinh \left (d x +c \right ) \left (\cosh ^{5}\left (d x +c \right )\right )}{6}-\frac {\left (\frac {\left (\cosh ^{3}\left (d x +c \right )\right )}{4}+\frac {3 \cosh \left (d x +c \right )}{8}\right ) \sinh \left (d x +c \right )}{6}-\frac {d x}{16}-\frac {c}{16}\right )+a^{2} \left (\left (\frac {\left (\cosh ^{3}\left (d x +c \right )\right )}{4}+\frac {3 \cosh \left (d x +c \right )}{8}\right ) \sinh \left (d x +c \right )+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 225, normalized size = 1.42 \[ \frac {1}{64} \, a^{2} {\left (24 \, x + \frac {e^{\left (4 \, d x + 4 \, c\right )}}{d} + \frac {8 \, e^{\left (2 \, d x + 2 \, c\right )}}{d} - \frac {8 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d} - \frac {e^{\left (-4 \, d x - 4 \, c\right )}}{d}\right )} - \frac {1}{2048} \, b^{2} {\left (\frac {{\left (8 \, e^{\left (-4 \, d x - 4 \, c\right )} - 1\right )} e^{\left (8 \, d x + 8 \, c\right )}}{d} - \frac {48 \, {\left (d x + c\right )}}{d} - \frac {8 \, e^{\left (-4 \, d x - 4 \, c\right )} - e^{\left (-8 \, d x - 8 \, c\right )}}{d}\right )} + \frac {1}{192} \, a b {\left (\frac {{\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} - 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + 1\right )} e^{\left (6 \, d x + 6 \, c\right )}}{d} - \frac {24 \, {\left (d x + c\right )}}{d} + \frac {3 \, e^{\left (-2 \, d x - 2 \, c\right )} - 3 \, e^{\left (-4 \, d x - 4 \, c\right )} - e^{\left (-6 \, d x - 6 \, c\right )}}{d}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.36, size = 121, normalized size = 0.76 \[ \frac {96\,a^2\,\mathrm {sinh}\left (2\,c+2\,d\,x\right )+12\,a^2\,\mathrm {sinh}\left (4\,c+4\,d\,x\right )-3\,b^2\,\mathrm {sinh}\left (4\,c+4\,d\,x\right )+\frac {3\,b^2\,\mathrm {sinh}\left (8\,c+8\,d\,x\right )}{8}-12\,a\,b\,\mathrm {sinh}\left (2\,c+2\,d\,x\right )+12\,a\,b\,\mathrm {sinh}\left (4\,c+4\,d\,x\right )+4\,a\,b\,\mathrm {sinh}\left (6\,c+6\,d\,x\right )+144\,a^2\,d\,x+9\,b^2\,d\,x-48\,a\,b\,d\,x}{384\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 9.78, size = 481, normalized size = 3.03 \[ \begin {cases} \frac {3 a^{2} x \sinh ^{4}{\left (c + d x \right )}}{8} - \frac {3 a^{2} x \sinh ^{2}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{4} + \frac {3 a^{2} x \cosh ^{4}{\left (c + d x \right )}}{8} - \frac {3 a^{2} \sinh ^{3}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{8 d} + \frac {5 a^{2} \sinh {\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{8 d} + \frac {a b x \sinh ^{6}{\left (c + d x \right )}}{8} - \frac {3 a b x \sinh ^{4}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{8} + \frac {3 a b x \sinh ^{2}{\left (c + d x \right )} \cosh ^{4}{\left (c + d x \right )}}{8} - \frac {a b x \cosh ^{6}{\left (c + d x \right )}}{8} - \frac {a b \sinh ^{5}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{8 d} + \frac {a b \sinh ^{3}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{3 d} + \frac {a b \sinh {\left (c + d x \right )} \cosh ^{5}{\left (c + d x \right )}}{8 d} + \frac {3 b^{2} x \sinh ^{8}{\left (c + d x \right )}}{128} - \frac {3 b^{2} x \sinh ^{6}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{32} + \frac {9 b^{2} x \sinh ^{4}{\left (c + d x \right )} \cosh ^{4}{\left (c + d x \right )}}{64} - \frac {3 b^{2} x \sinh ^{2}{\left (c + d x \right )} \cosh ^{6}{\left (c + d x \right )}}{32} + \frac {3 b^{2} x \cosh ^{8}{\left (c + d x \right )}}{128} - \frac {3 b^{2} \sinh ^{7}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{128 d} + \frac {11 b^{2} \sinh ^{5}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{128 d} + \frac {11 b^{2} \sinh ^{3}{\left (c + d x \right )} \cosh ^{5}{\left (c + d x \right )}}{128 d} - \frac {3 b^{2} \sinh {\left (c + d x \right )} \cosh ^{7}{\left (c + d x \right )}}{128 d} & \text {for}\: d \neq 0 \\x \left (a + b \sinh ^{2}{\relax (c )}\right )^{2} \cosh ^{4}{\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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