Optimal. Leaf size=181 \[ \frac {b \left (24 a^2-64 a b+35 b^2\right ) \sinh ^3(c+d x) \cosh (c+d x)}{192 d}+\frac {\left (96 a^3-376 a^2 b+360 a b^2-105 b^3\right ) \sinh (c+d x) \cosh (c+d x)}{384 d}-\frac {1}{128} x \left (64 a^3-144 a^2 b+120 a b^2-35 b^3\right )+\frac {\sinh (c+d x) \cosh (c+d x) \left (a+b \sinh ^2(c+d x)\right )^3}{8 d}+\frac {(6 a-7 b) \sinh (c+d x) \cosh (c+d x) \left (a+b \sinh ^2(c+d x)\right )^2}{48 d} \]
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Rubi [A] time = 0.19, antiderivative size = 181, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {3170, 3169} \[ \frac {b \left (24 a^2-64 a b+35 b^2\right ) \sinh ^3(c+d x) \cosh (c+d x)}{192 d}+\frac {\left (-376 a^2 b+96 a^3+360 a b^2-105 b^3\right ) \sinh (c+d x) \cosh (c+d x)}{384 d}-\frac {1}{128} x \left (-144 a^2 b+64 a^3+120 a b^2-35 b^3\right )+\frac {\sinh (c+d x) \cosh (c+d x) \left (a+b \sinh ^2(c+d x)\right )^3}{8 d}+\frac {(6 a-7 b) \sinh (c+d x) \cosh (c+d x) \left (a+b \sinh ^2(c+d x)\right )^2}{48 d} \]
Antiderivative was successfully verified.
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Rule 3169
Rule 3170
Rubi steps
\begin {align*} \int \sinh ^2(c+d x) \left (a+b \sinh ^2(c+d x)\right )^3 \, dx &=\frac {\cosh (c+d x) \sinh (c+d x) \left (a+b \sinh ^2(c+d x)\right )^3}{8 d}-\frac {1}{8} \int \left (a-(6 a-7 b) \sinh ^2(c+d x)\right ) \left (a+b \sinh ^2(c+d x)\right )^2 \, dx\\ &=\frac {(6 a-7 b) \cosh (c+d x) \sinh (c+d x) \left (a+b \sinh ^2(c+d x)\right )^2}{48 d}+\frac {\cosh (c+d x) \sinh (c+d x) \left (a+b \sinh ^2(c+d x)\right )^3}{8 d}-\frac {1}{48} \int \left (a+b \sinh ^2(c+d x)\right ) \left (a (12 a-7 b)-\left (24 a^2-64 a b+35 b^2\right ) \sinh ^2(c+d x)\right ) \, dx\\ &=-\frac {1}{128} \left (64 a^3-144 a^2 b+120 a b^2-35 b^3\right ) x+\frac {\left (96 a^3-376 a^2 b+360 a b^2-105 b^3\right ) \cosh (c+d x) \sinh (c+d x)}{384 d}+\frac {b \left (24 a^2-64 a b+35 b^2\right ) \cosh (c+d x) \sinh ^3(c+d x)}{192 d}+\frac {(6 a-7 b) \cosh (c+d x) \sinh (c+d x) \left (a+b \sinh ^2(c+d x)\right )^2}{48 d}+\frac {\cosh (c+d x) \sinh (c+d x) \left (a+b \sinh ^2(c+d x)\right )^3}{8 d}\\ \end {align*}
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Mathematica [A] time = 0.29, size = 130, normalized size = 0.72 \[ \frac {24 b \left (12 a^2-18 a b+7 b^2\right ) \sinh (4 (c+d x))-24 \left (64 a^3-144 a^2 b+120 a b^2-35 b^3\right ) (c+d x)+48 \left (16 a^3-48 a^2 b+45 a b^2-14 b^3\right ) \sinh (2 (c+d x))+16 b^2 (3 a-2 b) \sinh (6 (c+d x))+3 b^3 \sinh (8 (c+d x))}{3072 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.03, size = 269, normalized size = 1.49 \[ \frac {3 \, b^{3} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{7} + 3 \, {\left (7 \, b^{3} \cosh \left (d x + c\right )^{3} + 4 \, {\left (3 \, a b^{2} - 2 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{5} + {\left (21 \, b^{3} \cosh \left (d x + c\right )^{5} + 40 \, {\left (3 \, a b^{2} - 2 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} + 12 \, {\left (12 \, a^{2} b - 18 \, a b^{2} + 7 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} - 3 \, {\left (64 \, a^{3} - 144 \, a^{2} b + 120 \, a b^{2} - 35 \, b^{3}\right )} d x + 3 \, {\left (b^{3} \cosh \left (d x + c\right )^{7} + 4 \, {\left (3 \, a b^{2} - 2 \, b^{3}\right )} \cosh \left (d x + c\right )^{5} + 4 \, {\left (12 \, a^{2} b - 18 \, a b^{2} + 7 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} + 4 \, {\left (16 \, a^{3} - 48 \, a^{2} b + 45 \, a b^{2} - 14 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{384 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 251, normalized size = 1.39 \[ \frac {b^{3} e^{\left (8 \, d x + 8 \, c\right )}}{2048 \, d} - \frac {b^{3} e^{\left (-8 \, d x - 8 \, c\right )}}{2048 \, d} - \frac {1}{128} \, {\left (64 \, a^{3} - 144 \, a^{2} b + 120 \, a b^{2} - 35 \, b^{3}\right )} x + \frac {{\left (3 \, a b^{2} - 2 \, b^{3}\right )} e^{\left (6 \, d x + 6 \, c\right )}}{384 \, d} + \frac {{\left (12 \, a^{2} b - 18 \, a b^{2} + 7 \, b^{3}\right )} e^{\left (4 \, d x + 4 \, c\right )}}{256 \, d} + \frac {{\left (16 \, a^{3} - 48 \, a^{2} b + 45 \, a b^{2} - 14 \, b^{3}\right )} e^{\left (2 \, d x + 2 \, c\right )}}{128 \, d} - \frac {{\left (16 \, a^{3} - 48 \, a^{2} b + 45 \, a b^{2} - 14 \, b^{3}\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{128 \, d} - \frac {{\left (12 \, a^{2} b - 18 \, a b^{2} + 7 \, b^{3}\right )} e^{\left (-4 \, d x - 4 \, c\right )}}{256 \, d} - \frac {{\left (3 \, a b^{2} - 2 \, b^{3}\right )} e^{\left (-6 \, d x - 6 \, c\right )}}{384 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 180, normalized size = 0.99 \[ \frac {b^{3} \left (\left (\frac {\left (\sinh ^{7}\left (d x +c \right )\right )}{8}-\frac {7 \left (\sinh ^{5}\left (d x +c \right )\right )}{48}+\frac {35 \left (\sinh ^{3}\left (d x +c \right )\right )}{192}-\frac {35 \sinh \left (d x +c \right )}{128}\right ) \cosh \left (d x +c \right )+\frac {35 d x}{128}+\frac {35 c}{128}\right )+3 a \,b^{2} \left (\left (\frac {\left (\sinh ^{5}\left (d x +c \right )\right )}{6}-\frac {5 \left (\sinh ^{3}\left (d x +c \right )\right )}{24}+\frac {5 \sinh \left (d x +c \right )}{16}\right ) \cosh \left (d x +c \right )-\frac {5 d x}{16}-\frac {5 c}{16}\right )+3 a^{2} b \left (\left (\frac {\left (\sinh ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sinh \left (d x +c \right )}{8}\right ) \cosh \left (d x +c \right )+\frac {3 d x}{8}+\frac {3 c}{8}\right )+a^{3} \left (\frac {\cosh \left (d x +c \right ) \sinh \left (d x +c \right )}{2}-\frac {d x}{2}-\frac {c}{2}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.36, size = 306, normalized size = 1.69 \[ \frac {3}{64} \, a^{2} b {\left (24 \, x + \frac {e^{\left (4 \, d x + 4 \, c\right )}}{d} - \frac {8 \, e^{\left (2 \, d x + 2 \, c\right )}}{d} + \frac {8 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d} - \frac {e^{\left (-4 \, d x - 4 \, c\right )}}{d}\right )} - \frac {1}{8} \, a^{3} {\left (4 \, x - \frac {e^{\left (2 \, d x + 2 \, c\right )}}{d} + \frac {e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right )} - \frac {1}{6144} \, b^{3} {\left (\frac {{\left (32 \, e^{\left (-2 \, d x - 2 \, c\right )} - 168 \, e^{\left (-4 \, d x - 4 \, c\right )} + 672 \, e^{\left (-6 \, d x - 6 \, c\right )} - 3\right )} e^{\left (8 \, d x + 8 \, c\right )}}{d} - \frac {1680 \, {\left (d x + c\right )}}{d} - \frac {672 \, e^{\left (-2 \, d x - 2 \, c\right )} - 168 \, e^{\left (-4 \, d x - 4 \, c\right )} + 32 \, e^{\left (-6 \, d x - 6 \, c\right )} - 3 \, e^{\left (-8 \, d x - 8 \, c\right )}}{d}\right )} - \frac {1}{128} \, a b^{2} {\left (\frac {{\left (9 \, e^{\left (-2 \, d x - 2 \, c\right )} - 45 \, e^{\left (-4 \, d x - 4 \, c\right )} - 1\right )} e^{\left (6 \, d x + 6 \, c\right )}}{d} + \frac {120 \, {\left (d x + c\right )}}{d} + \frac {45 \, e^{\left (-2 \, d x - 2 \, c\right )} - 9 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )}}{d}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.97, size = 181, normalized size = 1.00 \[ \frac {96\,a^3\,\mathrm {sinh}\left (2\,c+2\,d\,x\right )-84\,b^3\,\mathrm {sinh}\left (2\,c+2\,d\,x\right )+21\,b^3\,\mathrm {sinh}\left (4\,c+4\,d\,x\right )-4\,b^3\,\mathrm {sinh}\left (6\,c+6\,d\,x\right )+\frac {3\,b^3\,\mathrm {sinh}\left (8\,c+8\,d\,x\right )}{8}+270\,a\,b^2\,\mathrm {sinh}\left (2\,c+2\,d\,x\right )-288\,a^2\,b\,\mathrm {sinh}\left (2\,c+2\,d\,x\right )-54\,a\,b^2\,\mathrm {sinh}\left (4\,c+4\,d\,x\right )+36\,a^2\,b\,\mathrm {sinh}\left (4\,c+4\,d\,x\right )+6\,a\,b^2\,\mathrm {sinh}\left (6\,c+6\,d\,x\right )-192\,a^3\,d\,x+105\,b^3\,d\,x-360\,a\,b^2\,d\,x+432\,a^2\,b\,d\,x}{384\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 9.85, size = 561, normalized size = 3.10 \[ \begin {cases} \frac {a^{3} x \sinh ^{2}{\left (c + d x \right )}}{2} - \frac {a^{3} x \cosh ^{2}{\left (c + d x \right )}}{2} + \frac {a^{3} \sinh {\left (c + d x \right )} \cosh {\left (c + d x \right )}}{2 d} + \frac {9 a^{2} b x \sinh ^{4}{\left (c + d x \right )}}{8} - \frac {9 a^{2} b x \sinh ^{2}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{4} + \frac {9 a^{2} b x \cosh ^{4}{\left (c + d x \right )}}{8} + \frac {15 a^{2} b \sinh ^{3}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{8 d} - \frac {9 a^{2} b \sinh {\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{8 d} + \frac {15 a b^{2} x \sinh ^{6}{\left (c + d x \right )}}{16} - \frac {45 a b^{2} x \sinh ^{4}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{16} + \frac {45 a b^{2} x \sinh ^{2}{\left (c + d x \right )} \cosh ^{4}{\left (c + d x \right )}}{16} - \frac {15 a b^{2} x \cosh ^{6}{\left (c + d x \right )}}{16} + \frac {33 a b^{2} \sinh ^{5}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{16 d} - \frac {5 a b^{2} \sinh ^{3}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{2 d} + \frac {15 a b^{2} \sinh {\left (c + d x \right )} \cosh ^{5}{\left (c + d x \right )}}{16 d} + \frac {35 b^{3} x \sinh ^{8}{\left (c + d x \right )}}{128} - \frac {35 b^{3} x \sinh ^{6}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{32} + \frac {105 b^{3} x \sinh ^{4}{\left (c + d x \right )} \cosh ^{4}{\left (c + d x \right )}}{64} - \frac {35 b^{3} x \sinh ^{2}{\left (c + d x \right )} \cosh ^{6}{\left (c + d x \right )}}{32} + \frac {35 b^{3} x \cosh ^{8}{\left (c + d x \right )}}{128} + \frac {93 b^{3} \sinh ^{7}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{128 d} - \frac {511 b^{3} \sinh ^{5}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{384 d} + \frac {385 b^{3} \sinh ^{3}{\left (c + d x \right )} \cosh ^{5}{\left (c + d x \right )}}{384 d} - \frac {35 b^{3} \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 )^{3} \sinh ^{2}{\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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