Optimal. Leaf size=183 \[ \frac {1}{8} a \left (8 a^4+40 a^2 b^2+15 b^4\right ) x+\frac {b \left (107 a^4+192 a^2 b^2+16 b^4\right ) \sinh (c+d x)}{30 d}+\frac {7 a b^2 \left (22 a^2+23 b^2\right ) \cosh (c+d x) \sinh (c+d x)}{120 d}+\frac {b \left (47 a^2+16 b^2\right ) (a+b \cosh (c+d x))^2 \sinh (c+d x)}{60 d}+\frac {9 a b (a+b \cosh (c+d x))^3 \sinh (c+d x)}{20 d}+\frac {b (a+b \cosh (c+d x))^4 \sinh (c+d x)}{5 d} \]
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Rubi [A]
time = 0.18, antiderivative size = 183, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2735, 2832,
2813} \begin {gather*} \frac {b \left (47 a^2+16 b^2\right ) \sinh (c+d x) (a+b \cosh (c+d x))^2}{60 d}+\frac {7 a b^2 \left (22 a^2+23 b^2\right ) \sinh (c+d x) \cosh (c+d x)}{120 d}+\frac {b \left (107 a^4+192 a^2 b^2+16 b^4\right ) \sinh (c+d x)}{30 d}+\frac {1}{8} a x \left (8 a^4+40 a^2 b^2+15 b^4\right )+\frac {b \sinh (c+d x) (a+b \cosh (c+d x))^4}{5 d}+\frac {9 a b \sinh (c+d x) (a+b \cosh (c+d x))^3}{20 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 2735
Rule 2813
Rule 2832
Rubi steps
\begin {align*} \int (a+b \cosh (c+d x))^5 \, dx &=\frac {b (a+b \cosh (c+d x))^4 \sinh (c+d x)}{5 d}+\frac {1}{5} \int (a+b \cosh (c+d x))^3 \left (5 a^2+4 b^2+9 a b \cosh (c+d x)\right ) \, dx\\ &=\frac {9 a b (a+b \cosh (c+d x))^3 \sinh (c+d x)}{20 d}+\frac {b (a+b \cosh (c+d x))^4 \sinh (c+d x)}{5 d}+\frac {1}{20} \int (a+b \cosh (c+d x))^2 \left (a \left (20 a^2+43 b^2\right )+b \left (47 a^2+16 b^2\right ) \cosh (c+d x)\right ) \, dx\\ &=\frac {b \left (47 a^2+16 b^2\right ) (a+b \cosh (c+d x))^2 \sinh (c+d x)}{60 d}+\frac {9 a b (a+b \cosh (c+d x))^3 \sinh (c+d x)}{20 d}+\frac {b (a+b \cosh (c+d x))^4 \sinh (c+d x)}{5 d}+\frac {1}{60} \int (a+b \cosh (c+d x)) \left (60 a^4+223 a^2 b^2+32 b^4+7 a b \left (22 a^2+23 b^2\right ) \cosh (c+d x)\right ) \, dx\\ &=\frac {1}{8} a \left (8 a^4+40 a^2 b^2+15 b^4\right ) x+\frac {b \left (107 a^4+192 a^2 b^2+16 b^4\right ) \sinh (c+d x)}{30 d}+\frac {7 a b^2 \left (22 a^2+23 b^2\right ) \cosh (c+d x) \sinh (c+d x)}{120 d}+\frac {b \left (47 a^2+16 b^2\right ) (a+b \cosh (c+d x))^2 \sinh (c+d x)}{60 d}+\frac {9 a b (a+b \cosh (c+d x))^3 \sinh (c+d x)}{20 d}+\frac {b (a+b \cosh (c+d x))^4 \sinh (c+d x)}{5 d}\\ \end {align*}
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Mathematica [A]
time = 0.27, size = 133, normalized size = 0.73 \begin {gather*} \frac {60 a \left (8 a^4+40 a^2 b^2+15 b^4\right ) (c+d x)+300 b \left (8 a^4+12 a^2 b^2+b^4\right ) \sinh (c+d x)+600 a b^2 \left (2 a^2+b^2\right ) \sinh (2 (c+d x))+50 b^3 \left (8 a^2+b^2\right ) \sinh (3 (c+d x))+75 a b^4 \sinh (4 (c+d x))+6 b^5 \sinh (5 (c+d x))}{480 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.88, size = 145, normalized size = 0.79
method | result | size |
default | \(a^{5} x +\frac {\left (\frac {5}{16} b^{5}+\frac {5}{2} a^{2} b^{3}\right ) \sinh \left (3 d x +3 c \right )}{3 d}+\frac {\left (\frac {5}{2} a \,b^{4}+5 a^{3} b^{2}\right ) \sinh \left (2 d x +2 c \right )}{2 d}+\frac {\left (\frac {5}{8} b^{5}+\frac {15}{2} a^{2} b^{3}+5 a^{4} b \right ) \sinh \left (d x +c \right )}{d}+\frac {15 x a \,b^{4}}{8}+5 x \,a^{3} b^{2}+\frac {b^{5} \sinh \left (5 d x +5 c \right )}{80 d}+\frac {5 a \,b^{4} \sinh \left (4 d x +4 c \right )}{32 d}\) | \(145\) |
risch | \(a^{5} x +5 x \,a^{3} b^{2}+\frac {15 x a \,b^{4}}{8}+\frac {b^{5} {\mathrm e}^{5 d x +5 c}}{160 d}+\frac {5 a \,b^{4} {\mathrm e}^{4 d x +4 c}}{64 d}+\frac {5 b^{3} {\mathrm e}^{3 d x +3 c} a^{2}}{12 d}+\frac {5 b^{5} {\mathrm e}^{3 d x +3 c}}{96 d}+\frac {5 a^{3} b^{2} {\mathrm e}^{2 d x +2 c}}{4 d}+\frac {5 a \,b^{4} {\mathrm e}^{2 d x +2 c}}{8 d}+\frac {5 b \,{\mathrm e}^{d x +c} a^{4}}{2 d}+\frac {15 b^{3} {\mathrm e}^{d x +c} a^{2}}{4 d}+\frac {5 b^{5} {\mathrm e}^{d x +c}}{16 d}-\frac {5 b \,{\mathrm e}^{-d x -c} a^{4}}{2 d}-\frac {15 b^{3} {\mathrm e}^{-d x -c} a^{2}}{4 d}-\frac {5 b^{5} {\mathrm e}^{-d x -c}}{16 d}-\frac {5 a^{3} b^{2} {\mathrm e}^{-2 d x -2 c}}{4 d}-\frac {5 a \,b^{4} {\mathrm e}^{-2 d x -2 c}}{8 d}-\frac {5 b^{3} {\mathrm e}^{-3 d x -3 c} a^{2}}{12 d}-\frac {5 b^{5} {\mathrm e}^{-3 d x -3 c}}{96 d}-\frac {5 a \,b^{4} {\mathrm e}^{-4 d x -4 c}}{64 d}-\frac {b^{5} {\mathrm e}^{-5 d x -5 c}}{160 d}\) | \(344\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 273, normalized size = 1.49 \begin {gather*} \frac {5}{64} \, a b^{4} {\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 {5}{4} \, a^{3} b^{2} {\left (4 \, x + \frac {e^{\left (2 \, d x + 2 \, c\right )}}{d} - \frac {e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right )} + a^{5} x + \frac {1}{480} \, b^{5} {\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 {5}{12} \, a^{2} b^{3} {\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 )} + \frac {5 \, a^{4} b \sinh \left (d x + c\right )}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 190, normalized size = 1.04 \begin {gather*} \frac {3 \, b^{5} \sinh \left (d x + c\right )^{5} + 5 \, {\left (6 \, b^{5} \cosh \left (d x + c\right )^{2} + 30 \, a b^{4} \cosh \left (d x + c\right ) + 40 \, a^{2} b^{3} + 5 \, b^{5}\right )} \sinh \left (d x + c\right )^{3} + 30 \, {\left (8 \, a^{5} + 40 \, a^{3} b^{2} + 15 \, a b^{4}\right )} d x + 15 \, {\left (b^{5} \cosh \left (d x + c\right )^{4} + 10 \, a b^{4} \cosh \left (d x + c\right )^{3} + 80 \, a^{4} b + 120 \, a^{2} b^{3} + 10 \, b^{5} + 5 \, {\left (8 \, a^{2} b^{3} + b^{5}\right )} \cosh \left (d x + c\right )^{2} + 40 \, {\left (2 \, a^{3} b^{2} + a b^{4}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{240 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.33, size = 314, normalized size = 1.72 \begin {gather*} \begin {cases} a^{5} x + \frac {5 a^{4} b \sinh {\left (c + d x \right )}}{d} - 5 a^{3} b^{2} x \sinh ^{2}{\left (c + d x \right )} + 5 a^{3} b^{2} x \cosh ^{2}{\left (c + d x \right )} + \frac {5 a^{3} b^{2} \sinh {\left (c + d x \right )} \cosh {\left (c + d x \right )}}{d} - \frac {20 a^{2} b^{3} \sinh ^{3}{\left (c + d x \right )}}{3 d} + \frac {10 a^{2} b^{3} \sinh {\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{d} + \frac {15 a b^{4} x \sinh ^{4}{\left (c + d x \right )}}{8} - \frac {15 a b^{4} x \sinh ^{2}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{4} + \frac {15 a b^{4} x \cosh ^{4}{\left (c + d x \right )}}{8} - \frac {15 a b^{4} \sinh ^{3}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{8 d} + \frac {25 a b^{4} \sinh {\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{8 d} + \frac {8 b^{5} \sinh ^{5}{\left (c + d x \right )}}{15 d} - \frac {4 b^{5} \sinh ^{3}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{3 d} + \frac {b^{5} \sinh {\left (c + d x \right )} \cosh ^{4}{\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (a + b \cosh {\left (c \right )}\right )^{5} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 263, normalized size = 1.44 \begin {gather*} \frac {b^{5} e^{\left (5 \, d x + 5 \, c\right )}}{160 \, d} + \frac {5 \, a b^{4} e^{\left (4 \, d x + 4 \, c\right )}}{64 \, d} - \frac {5 \, a b^{4} e^{\left (-4 \, d x - 4 \, c\right )}}{64 \, d} - \frac {b^{5} e^{\left (-5 \, d x - 5 \, c\right )}}{160 \, d} + \frac {1}{8} \, {\left (8 \, a^{5} + 40 \, a^{3} b^{2} + 15 \, a b^{4}\right )} x + \frac {5 \, {\left (8 \, a^{2} b^{3} + b^{5}\right )} e^{\left (3 \, d x + 3 \, c\right )}}{96 \, d} + \frac {5 \, {\left (2 \, a^{3} b^{2} + a b^{4}\right )} e^{\left (2 \, d x + 2 \, c\right )}}{8 \, d} + \frac {5 \, {\left (8 \, a^{4} b + 12 \, a^{2} b^{3} + b^{5}\right )} e^{\left (d x + c\right )}}{16 \, d} - \frac {5 \, {\left (8 \, a^{4} b + 12 \, a^{2} b^{3} + b^{5}\right )} e^{\left (-d x - c\right )}}{16 \, d} - \frac {5 \, {\left (2 \, a^{3} b^{2} + a b^{4}\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{8 \, d} - \frac {5 \, {\left (8 \, a^{2} b^{3} + b^{5}\right )} e^{\left (-3 \, d x - 3 \, c\right )}}{96 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.14, size = 160, normalized size = 0.87 \begin {gather*} \frac {75\,b^5\,\mathrm {sinh}\left (c+d\,x\right )+\frac {25\,b^5\,\mathrm {sinh}\left (3\,c+3\,d\,x\right )}{2}+\frac {3\,b^5\,\mathrm {sinh}\left (5\,c+5\,d\,x\right )}{2}+150\,a\,b^4\,\mathrm {sinh}\left (2\,c+2\,d\,x\right )+\frac {75\,a\,b^4\,\mathrm {sinh}\left (4\,c+4\,d\,x\right )}{4}+900\,a^2\,b^3\,\mathrm {sinh}\left (c+d\,x\right )+300\,a^3\,b^2\,\mathrm {sinh}\left (2\,c+2\,d\,x\right )+100\,a^2\,b^3\,\mathrm {sinh}\left (3\,c+3\,d\,x\right )+600\,a^4\,b\,\mathrm {sinh}\left (c+d\,x\right )+120\,a^5\,d\,x+225\,a\,b^4\,d\,x+600\,a^3\,b^2\,d\,x}{120\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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