Optimal. Leaf size=114 \[ a^2 x+\frac {2 a b \cosh ^3(c+d x)}{3 d}-\frac {2 a b \cosh (c+d x)}{d}+\frac {b^2 \sinh ^5(c+d x) \cosh (c+d x)}{6 d}-\frac {5 b^2 \sinh ^3(c+d x) \cosh (c+d x)}{24 d}+\frac {5 b^2 \sinh (c+d x) \cosh (c+d x)}{16 d}-\frac {5 b^2 x}{16} \]
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Rubi [A] time = 0.08, antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3213, 2633, 2635, 8} \[ a^2 x+\frac {2 a b \cosh ^3(c+d x)}{3 d}-\frac {2 a b \cosh (c+d x)}{d}+\frac {b^2 \sinh ^5(c+d x) \cosh (c+d x)}{6 d}-\frac {5 b^2 \sinh ^3(c+d x) \cosh (c+d x)}{24 d}+\frac {5 b^2 \sinh (c+d x) \cosh (c+d x)}{16 d}-\frac {5 b^2 x}{16} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2633
Rule 2635
Rule 3213
Rubi steps
\begin {align*} \int \left (a+b \sinh ^3(c+d x)\right )^2 \, dx &=\int \left (a^2+2 a b \sinh ^3(c+d x)+b^2 \sinh ^6(c+d x)\right ) \, dx\\ &=a^2 x+(2 a b) \int \sinh ^3(c+d x) \, dx+b^2 \int \sinh ^6(c+d x) \, dx\\ &=a^2 x+\frac {b^2 \cosh (c+d x) \sinh ^5(c+d x)}{6 d}-\frac {1}{6} \left (5 b^2\right ) \int \sinh ^4(c+d x) \, dx-\frac {(2 a b) \operatorname {Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cosh (c+d x)\right )}{d}\\ &=a^2 x-\frac {2 a b \cosh (c+d x)}{d}+\frac {2 a b \cosh ^3(c+d x)}{3 d}-\frac {5 b^2 \cosh (c+d x) \sinh ^3(c+d x)}{24 d}+\frac {b^2 \cosh (c+d x) \sinh ^5(c+d x)}{6 d}+\frac {1}{8} \left (5 b^2\right ) \int \sinh ^2(c+d x) \, dx\\ &=a^2 x-\frac {2 a b \cosh (c+d x)}{d}+\frac {2 a b \cosh ^3(c+d x)}{3 d}+\frac {5 b^2 \cosh (c+d x) \sinh (c+d x)}{16 d}-\frac {5 b^2 \cosh (c+d x) \sinh ^3(c+d x)}{24 d}+\frac {b^2 \cosh (c+d x) \sinh ^5(c+d x)}{6 d}-\frac {1}{16} \left (5 b^2\right ) \int 1 \, dx\\ &=a^2 x-\frac {5 b^2 x}{16}-\frac {2 a b \cosh (c+d x)}{d}+\frac {2 a b \cosh ^3(c+d x)}{3 d}+\frac {5 b^2 \cosh (c+d x) \sinh (c+d x)}{16 d}-\frac {5 b^2 \cosh (c+d x) \sinh ^3(c+d x)}{24 d}+\frac {b^2 \cosh (c+d x) \sinh ^5(c+d x)}{6 d}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 94, normalized size = 0.82 \[ \frac {192 a^2 c+192 a^2 d x-288 a b \cosh (c+d x)+32 a b \cosh (3 (c+d x))+45 b^2 \sinh (2 (c+d x))-9 b^2 \sinh (4 (c+d x))+b^2 \sinh (6 (c+d x))-60 b^2 c-60 b^2 d x}{192 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 160, normalized size = 1.40 \[ \frac {3 \, b^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{5} + 16 \, a b \cosh \left (d x + c\right )^{3} + 48 \, a b \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + 2 \, {\left (5 \, b^{2} \cosh \left (d x + c\right )^{3} - 9 \, b^{2} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} + 6 \, {\left (16 \, a^{2} - 5 \, b^{2}\right )} d x - 144 \, a b \cosh \left (d x + c\right ) + 3 \, {\left (b^{2} \cosh \left (d x + c\right )^{5} - 6 \, b^{2} \cosh \left (d x + c\right )^{3} + 15 \, b^{2} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{96 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 178, normalized size = 1.56 \[ \frac {1}{16} \, {\left (16 \, a^{2} - 5 \, b^{2}\right )} x + \frac {b^{2} e^{\left (6 \, d x + 6 \, c\right )}}{384 \, d} - \frac {3 \, b^{2} e^{\left (4 \, d x + 4 \, c\right )}}{128 \, d} + \frac {a b e^{\left (3 \, d x + 3 \, c\right )}}{12 \, d} + \frac {15 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )}}{128 \, d} - \frac {3 \, a b e^{\left (d x + c\right )}}{4 \, d} - \frac {3 \, a b e^{\left (-d x - c\right )}}{4 \, d} - \frac {15 \, b^{2} e^{\left (-2 \, d x - 2 \, c\right )}}{128 \, d} + \frac {a b e^{\left (-3 \, d x - 3 \, c\right )}}{12 \, d} + \frac {3 \, b^{2} e^{\left (-4 \, d x - 4 \, c\right )}}{128 \, d} - \frac {b^{2} e^{\left (-6 \, d x - 6 \, c\right )}}{384 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 85, normalized size = 0.75 \[ \frac {b^{2} \left (\left (\frac {\left (\sinh ^{5}\left (d x +c \right )\right )}{6}-\frac {5 \left (\sinh ^{3}\left (d x +c \right )\right )}{24}+\frac {5 \sinh \left (d x +c \right )}{16}\right ) \cosh \left (d x +c \right )-\frac {5 d x}{16}-\frac {5 c}{16}\right )+2 a b \left (-\frac {2}{3}+\frac {\left (\sinh ^{2}\left (d x +c \right )\right )}{3}\right ) \cosh \left (d x +c \right )+a^{2} \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 151, normalized size = 1.32 \[ a^{2} x - \frac {1}{384} \, b^{2} {\left (\frac {{\left (9 \, e^{\left (-2 \, d x - 2 \, c\right )} - 45 \, e^{\left (-4 \, d x - 4 \, c\right )} - 1\right )} e^{\left (6 \, d x + 6 \, c\right )}}{d} + \frac {120 \, {\left (d x + c\right )}}{d} + \frac {45 \, e^{\left (-2 \, d x - 2 \, c\right )} - 9 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )}}{d}\right )} + \frac {1}{12} \, a b {\left (\frac {e^{\left (3 \, d x + 3 \, c\right )}}{d} - \frac {9 \, e^{\left (d x + c\right )}}{d} - \frac {9 \, e^{\left (-d x - c\right )}}{d} + \frac {e^{\left (-3 \, d x - 3 \, c\right )}}{d}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.46, size = 85, normalized size = 0.75 \[ \frac {\frac {45\,b^2\,\mathrm {sinh}\left (2\,c+2\,d\,x\right )}{4}-\frac {9\,b^2\,\mathrm {sinh}\left (4\,c+4\,d\,x\right )}{4}+\frac {b^2\,\mathrm {sinh}\left (6\,c+6\,d\,x\right )}{4}-72\,a\,b\,\mathrm {cosh}\left (c+d\,x\right )+8\,a\,b\,\mathrm {cosh}\left (3\,c+3\,d\,x\right )+48\,a^2\,d\,x-15\,b^2\,d\,x}{48\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.04, size = 212, normalized size = 1.86 \[ \begin {cases} a^{2} x + \frac {2 a b \sinh ^{2}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{d} - \frac {4 a b \cosh ^{3}{\left (c + d x \right )}}{3 d} + \frac {5 b^{2} x \sinh ^{6}{\left (c + d x \right )}}{16} - \frac {15 b^{2} x \sinh ^{4}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{16} + \frac {15 b^{2} x \sinh ^{2}{\left (c + d x \right )} \cosh ^{4}{\left (c + d x \right )}}{16} - \frac {5 b^{2} x \cosh ^{6}{\left (c + d x \right )}}{16} + \frac {11 b^{2} \sinh ^{5}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{16 d} - \frac {5 b^{2} \sinh ^{3}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{6 d} + \frac {5 b^{2} \sinh {\left (c + d x \right )} \cosh ^{5}{\left (c + d x \right )}}{16 d} & \text {for}\: d \neq 0 \\x \left (a + b \sinh ^{3}{\relax (c )}\right )^{2} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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