Optimal. Leaf size=85 \[ \frac {b (2 a-3 b) \cosh ^5(c+d x)}{5 d}+\frac {(a-3 b) (a-b) \cosh ^3(c+d x)}{3 d}-\frac {(a-b)^2 \cosh (c+d x)}{d}+\frac {b^2 \cosh ^7(c+d x)}{7 d} \]
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Rubi [A] time = 0.10, antiderivative size = 85, 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 = {3186, 373} \[ \frac {b (2 a-3 b) \cosh ^5(c+d x)}{5 d}+\frac {(a-3 b) (a-b) \cosh ^3(c+d x)}{3 d}-\frac {(a-b)^2 \cosh (c+d x)}{d}+\frac {b^2 \cosh ^7(c+d x)}{7 d} \]
Antiderivative was successfully verified.
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Rule 373
Rule 3186
Rubi steps
\begin {align*} \int \sinh ^3(c+d x) \left (a+b \sinh ^2(c+d x)\right )^2 \, dx &=-\frac {\operatorname {Subst}\left (\int \left (1-x^2\right ) \left (a-b+b x^2\right )^2 \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac {\operatorname {Subst}\left (\int \left ((a-b)^2+(a-3 b) (-a+b) x^2-(2 a-3 b) b x^4-b^2 x^6\right ) \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac {(a-b)^2 \cosh (c+d x)}{d}+\frac {(a-3 b) (a-b) \cosh ^3(c+d x)}{3 d}+\frac {(2 a-3 b) b \cosh ^5(c+d x)}{5 d}+\frac {b^2 \cosh ^7(c+d x)}{7 d}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 154, normalized size = 1.81 \[ -\frac {3 a^2 \cosh (c+d x)}{4 d}+\frac {a^2 \cosh (3 (c+d x))}{12 d}+\frac {5 a b \cosh (c+d x)}{4 d}-\frac {5 a b \cosh (3 (c+d x))}{24 d}+\frac {a b \cosh (5 (c+d x))}{40 d}-\frac {35 b^2 \cosh (c+d x)}{64 d}+\frac {7 b^2 \cosh (3 (c+d x))}{64 d}-\frac {7 b^2 \cosh (5 (c+d x))}{320 d}+\frac {b^2 \cosh (7 (c+d x))}{448 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.64, size = 213, normalized size = 2.51 \[ \frac {15 \, b^{2} \cosh \left (d x + c\right )^{7} + 105 \, b^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{6} + 21 \, {\left (8 \, a b - 7 \, b^{2}\right )} \cosh \left (d x + c\right )^{5} + 105 \, {\left (5 \, b^{2} \cosh \left (d x + c\right )^{3} + {\left (8 \, a b - 7 \, b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{4} + 35 \, {\left (16 \, a^{2} - 40 \, a b + 21 \, b^{2}\right )} \cosh \left (d x + c\right )^{3} + 105 \, {\left (3 \, b^{2} \cosh \left (d x + c\right )^{5} + 2 \, {\left (8 \, a b - 7 \, b^{2}\right )} \cosh \left (d x + c\right )^{3} + {\left (16 \, a^{2} - 40 \, a b + 21 \, b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} - 105 \, {\left (48 \, a^{2} - 80 \, a b + 35 \, b^{2}\right )} \cosh \left (d x + c\right )}{6720 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.15, size = 196, normalized size = 2.31 \[ \frac {b^{2} e^{\left (7 \, d x + 7 \, c\right )}}{896 \, d} + \frac {b^{2} e^{\left (-7 \, d x - 7 \, c\right )}}{896 \, d} + \frac {{\left (8 \, a b - 7 \, b^{2}\right )} e^{\left (5 \, d x + 5 \, c\right )}}{640 \, d} + \frac {{\left (16 \, a^{2} - 40 \, a b + 21 \, b^{2}\right )} e^{\left (3 \, d x + 3 \, c\right )}}{384 \, d} - \frac {{\left (48 \, a^{2} - 80 \, a b + 35 \, b^{2}\right )} e^{\left (d x + c\right )}}{128 \, d} - \frac {{\left (48 \, a^{2} - 80 \, a b + 35 \, b^{2}\right )} e^{\left (-d x - c\right )}}{128 \, d} + \frac {{\left (16 \, a^{2} - 40 \, a b + 21 \, b^{2}\right )} e^{\left (-3 \, d x - 3 \, c\right )}}{384 \, d} + \frac {{\left (8 \, a b - 7 \, b^{2}\right )} e^{\left (-5 \, d x - 5 \, c\right )}}{640 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 102, normalized size = 1.20 \[ \frac {b^{2} \left (-\frac {16}{35}+\frac {\left (\sinh ^{6}\left (d x +c \right )\right )}{7}-\frac {6 \left (\sinh ^{4}\left (d x +c \right )\right )}{35}+\frac {8 \left (\sinh ^{2}\left (d x +c \right )\right )}{35}\right ) \cosh \left (d x +c \right )+2 a b \left (\frac {8}{15}+\frac {\left (\sinh ^{4}\left (d x +c \right )\right )}{5}-\frac {4 \left (\sinh ^{2}\left (d x +c \right )\right )}{15}\right ) \cosh \left (d x +c \right )+a^{2} \left (-\frac {2}{3}+\frac {\left (\sinh ^{2}\left (d x +c \right )\right )}{3}\right ) \cosh \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.38, size = 247, normalized size = 2.91 \[ -\frac {1}{4480} \, b^{2} {\left (\frac {{\left (49 \, e^{\left (-2 \, d x - 2 \, c\right )} - 245 \, e^{\left (-4 \, d x - 4 \, c\right )} + 1225 \, e^{\left (-6 \, d x - 6 \, c\right )} - 5\right )} e^{\left (7 \, d x + 7 \, c\right )}}{d} + \frac {1225 \, e^{\left (-d x - c\right )} - 245 \, e^{\left (-3 \, d x - 3 \, c\right )} + 49 \, e^{\left (-5 \, d x - 5 \, c\right )} - 5 \, e^{\left (-7 \, d x - 7 \, c\right )}}{d}\right )} + \frac {1}{240} \, a b {\left (\frac {3 \, e^{\left (5 \, d x + 5 \, c\right )}}{d} - \frac {25 \, e^{\left (3 \, d x + 3 \, c\right )}}{d} + \frac {150 \, e^{\left (d x + c\right )}}{d} + \frac {150 \, e^{\left (-d x - c\right )}}{d} - \frac {25 \, e^{\left (-3 \, d x - 3 \, c\right )}}{d} + \frac {3 \, e^{\left (-5 \, d x - 5 \, c\right )}}{d}\right )} + \frac {1}{24} \, a^{2} {\left (\frac {e^{\left (3 \, d x + 3 \, c\right )}}{d} - \frac {9 \, e^{\left (d x + c\right )}}{d} - \frac {9 \, e^{\left (-d x - c\right )}}{d} + \frac {e^{\left (-3 \, d x - 3 \, c\right )}}{d}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.23, size = 112, normalized size = 1.32 \[ \frac {\frac {a^2\,{\mathrm {cosh}\left (c+d\,x\right )}^3}{3}-a^2\,\mathrm {cosh}\left (c+d\,x\right )+\frac {2\,a\,b\,{\mathrm {cosh}\left (c+d\,x\right )}^5}{5}-\frac {4\,a\,b\,{\mathrm {cosh}\left (c+d\,x\right )}^3}{3}+2\,a\,b\,\mathrm {cosh}\left (c+d\,x\right )+\frac {b^2\,{\mathrm {cosh}\left (c+d\,x\right )}^7}{7}-\frac {3\,b^2\,{\mathrm {cosh}\left (c+d\,x\right )}^5}{5}+b^2\,{\mathrm {cosh}\left (c+d\,x\right )}^3-b^2\,\mathrm {cosh}\left (c+d\,x\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.45, size = 204, normalized size = 2.40 \[ \begin {cases} \frac {a^{2} \sinh ^{2}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{d} - \frac {2 a^{2} \cosh ^{3}{\left (c + d x \right )}}{3 d} + \frac {2 a b \sinh ^{4}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{d} - \frac {8 a b \sinh ^{2}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{3 d} + \frac {16 a b \cosh ^{5}{\left (c + d x \right )}}{15 d} + \frac {b^{2} \sinh ^{6}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{d} - \frac {2 b^{2} \sinh ^{4}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{d} + \frac {8 b^{2} \sinh ^{2}{\left (c + d x \right )} \cosh ^{5}{\left (c + d x \right )}}{5 d} - \frac {16 b^{2} \cosh ^{7}{\left (c + d x \right )}}{35 d} & \text {for}\: d \neq 0 \\x \left (a + b \sinh ^{2}{\relax (c )}\right )^{2} \sinh ^{3}{\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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