Optimal. Leaf size=92 \[ \frac {2 b (a+3 b) \cosh ^5(c+d x)}{5 d}-\frac {4 b (a+b) \cosh ^3(c+d x)}{3 d}+\frac {(a+b)^2 \cosh (c+d x)}{d}+\frac {b^2 \cosh ^9(c+d x)}{9 d}-\frac {4 b^2 \cosh ^7(c+d x)}{7 d} \]
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Rubi [A] time = 0.09, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {3215, 1090} \[ \frac {2 b (a+3 b) \cosh ^5(c+d x)}{5 d}-\frac {4 b (a+b) \cosh ^3(c+d x)}{3 d}+\frac {(a+b)^2 \cosh (c+d x)}{d}+\frac {b^2 \cosh ^9(c+d x)}{9 d}-\frac {4 b^2 \cosh ^7(c+d x)}{7 d} \]
Antiderivative was successfully verified.
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Rule 1090
Rule 3215
Rubi steps
\begin {align*} \int \sinh (c+d x) \left (a+b \sinh ^4(c+d x)\right )^2 \, dx &=\frac {\operatorname {Subst}\left (\int \left (a+b-2 b x^2+b x^4\right )^2 \, dx,x,\cosh (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \left (a^2 \left (1+\frac {b (2 a+b)}{a^2}\right )-4 a b \left (1+\frac {b}{a}\right ) x^2+2 a b \left (1+\frac {3 b}{a}\right ) x^4-4 b^2 x^6+b^2 x^8\right ) \, dx,x,\cosh (c+d x)\right )}{d}\\ &=\frac {(a+b)^2 \cosh (c+d x)}{d}-\frac {4 b (a+b) \cosh ^3(c+d x)}{3 d}+\frac {2 b (a+3 b) \cosh ^5(c+d x)}{5 d}-\frac {4 b^2 \cosh ^7(c+d x)}{7 d}+\frac {b^2 \cosh ^9(c+d x)}{9 d}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 164, normalized size = 1.78 \[ \frac {a^2 \sinh (c) \sinh (d x)}{d}+\frac {a^2 \cosh (c) \cosh (d x)}{d}+\frac {5 a b \cosh (c+d x)}{4 d}-\frac {5 a b \cosh (3 (c+d x))}{24 d}+\frac {a b \cosh (5 (c+d x))}{40 d}+\frac {63 b^2 \cosh (c+d x)}{128 d}-\frac {7 b^2 \cosh (3 (c+d x))}{64 d}+\frac {9 b^2 \cosh (5 (c+d x))}{320 d}-\frac {9 b^2 \cosh (7 (c+d x))}{1792 d}+\frac {b^2 \cosh (9 (c+d x))}{2304 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.67, size = 279, normalized size = 3.03 \[ \frac {35 \, b^{2} \cosh \left (d x + c\right )^{9} + 315 \, b^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{8} - 405 \, b^{2} \cosh \left (d x + c\right )^{7} + 105 \, {\left (28 \, b^{2} \cosh \left (d x + c\right )^{3} - 27 \, b^{2} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{6} + 252 \, {\left (8 \, a b + 9 \, b^{2}\right )} \cosh \left (d x + c\right )^{5} + 315 \, {\left (14 \, b^{2} \cosh \left (d x + c\right )^{5} - 45 \, b^{2} \cosh \left (d x + c\right )^{3} + 4 \, {\left (8 \, a b + 9 \, b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{4} - 420 \, {\left (40 \, a b + 21 \, b^{2}\right )} \cosh \left (d x + c\right )^{3} + 315 \, {\left (4 \, b^{2} \cosh \left (d x + c\right )^{7} - 27 \, b^{2} \cosh \left (d x + c\right )^{5} + 8 \, {\left (8 \, a b + 9 \, b^{2}\right )} \cosh \left (d x + c\right )^{3} - 4 \, {\left (40 \, a b + 21 \, b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} + 630 \, {\left (128 \, a^{2} + 160 \, a b + 63 \, b^{2}\right )} \cosh \left (d x + c\right )}{80640 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.23, size = 220, normalized size = 2.39 \[ \frac {b^{2} e^{\left (9 \, d x + 9 \, c\right )}}{4608 \, d} - \frac {9 \, b^{2} e^{\left (7 \, d x + 7 \, c\right )}}{3584 \, d} - \frac {9 \, b^{2} e^{\left (-7 \, d x - 7 \, c\right )}}{3584 \, d} + \frac {b^{2} e^{\left (-9 \, d x - 9 \, c\right )}}{4608 \, d} + \frac {{\left (8 \, a b + 9 \, b^{2}\right )} e^{\left (5 \, d x + 5 \, c\right )}}{640 \, d} - \frac {{\left (40 \, a b + 21 \, b^{2}\right )} e^{\left (3 \, d x + 3 \, c\right )}}{384 \, d} + \frac {{\left (128 \, a^{2} + 160 \, a b + 63 \, b^{2}\right )} e^{\left (d x + c\right )}}{256 \, d} + \frac {{\left (128 \, a^{2} + 160 \, a b + 63 \, b^{2}\right )} e^{\left (-d x - c\right )}}{256 \, d} - \frac {{\left (40 \, a b + 21 \, b^{2}\right )} e^{\left (-3 \, d x - 3 \, c\right )}}{384 \, d} + \frac {{\left (8 \, a b + 9 \, b^{2}\right )} e^{\left (-5 \, d x - 5 \, c\right )}}{640 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 100, normalized size = 1.09 \[ \frac {b^{2} \left (\frac {128}{315}+\frac {\left (\sinh ^{8}\left (d x +c \right )\right )}{9}-\frac {8 \left (\sinh ^{6}\left (d x +c \right )\right )}{63}+\frac {16 \left (\sinh ^{4}\left (d x +c \right )\right )}{105}-\frac {64 \left (\sinh ^{2}\left (d x +c \right )\right )}{315}\right ) \cosh \left (d x +c \right )+2 a b \left (\frac {8}{15}+\frac {\left (\sinh ^{4}\left (d x +c \right )\right )}{5}-\frac {4 \left (\sinh ^{2}\left (d x +c \right )\right )}{15}\right ) \cosh \left (d x +c \right )+a^{2} \cosh \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.33, size = 226, normalized size = 2.46 \[ -\frac {1}{161280} \, b^{2} {\left (\frac {{\left (405 \, e^{\left (-2 \, d x - 2 \, c\right )} - 2268 \, e^{\left (-4 \, d x - 4 \, c\right )} + 8820 \, e^{\left (-6 \, d x - 6 \, c\right )} - 39690 \, e^{\left (-8 \, d x - 8 \, c\right )} - 35\right )} e^{\left (9 \, d x + 9 \, c\right )}}{d} - \frac {39690 \, e^{\left (-d x - c\right )} - 8820 \, e^{\left (-3 \, d x - 3 \, c\right )} + 2268 \, e^{\left (-5 \, d x - 5 \, c\right )} - 405 \, e^{\left (-7 \, d x - 7 \, c\right )} + 35 \, e^{\left (-9 \, d x - 9 \, c\right )}}{d}\right )} + \frac {1}{240} \, a b {\left (\frac {3 \, e^{\left (5 \, d x + 5 \, c\right )}}{d} - \frac {25 \, e^{\left (3 \, d x + 3 \, c\right )}}{d} + \frac {150 \, e^{\left (d x + c\right )}}{d} + \frac {150 \, e^{\left (-d x - c\right )}}{d} - \frac {25 \, e^{\left (-3 \, d x - 3 \, c\right )}}{d} + \frac {3 \, e^{\left (-5 \, d x - 5 \, c\right )}}{d}\right )} + \frac {a^{2} \cosh \left (d x + c\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.88, size = 111, normalized size = 1.21 \[ \frac {a^2\,\mathrm {cosh}\left (c+d\,x\right )+\frac {2\,a\,b\,{\mathrm {cosh}\left (c+d\,x\right )}^5}{5}-\frac {4\,a\,b\,{\mathrm {cosh}\left (c+d\,x\right )}^3}{3}+2\,a\,b\,\mathrm {cosh}\left (c+d\,x\right )+\frac {b^2\,{\mathrm {cosh}\left (c+d\,x\right )}^9}{9}-\frac {4\,b^2\,{\mathrm {cosh}\left (c+d\,x\right )}^7}{7}+\frac {6\,b^2\,{\mathrm {cosh}\left (c+d\,x\right )}^5}{5}-\frac {4\,b^2\,{\mathrm {cosh}\left (c+d\,x\right )}^3}{3}+b^2\,\mathrm {cosh}\left (c+d\,x\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 13.08, size = 204, normalized size = 2.22 \[ \begin {cases} \frac {a^{2} \cosh {\left (c + d x \right )}}{d} + \frac {2 a b \sinh ^{4}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{d} - \frac {8 a b \sinh ^{2}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{3 d} + \frac {16 a b \cosh ^{5}{\left (c + d x \right )}}{15 d} + \frac {b^{2} \sinh ^{8}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{d} - \frac {8 b^{2} \sinh ^{6}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{3 d} + \frac {16 b^{2} \sinh ^{4}{\left (c + d x \right )} \cosh ^{5}{\left (c + d x \right )}}{5 d} - \frac {64 b^{2} \sinh ^{2}{\left (c + d x \right )} \cosh ^{7}{\left (c + d x \right )}}{35 d} + \frac {128 b^{2} \cosh ^{9}{\left (c + d x \right )}}{315 d} & \text {for}\: d \neq 0 \\x \left (a + b \sinh ^{4}{\relax (c )}\right )^{2} \sinh {\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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