Optimal. Leaf size=183 \[ \frac{b \left (192 a^2 b^2+107 a^4+16 b^4\right ) \sinh (c+d x)}{30 d}+\frac{b \left (47 a^2+16 b^2\right ) \sinh (c+d x) (a+b \cosh (c+d x))^2}{60 d}+\frac{7 a b^2 \left (22 a^2+23 b^2\right ) \sinh (c+d x) \cosh (c+d x)}{120 d}+\frac{1}{8} a x \left (40 a^2 b^2+8 a^4+15 b^4\right )+\frac{b \sinh (c+d x) (a+b \cosh (c+d x))^4}{5 d}+\frac{9 a b \sinh (c+d x) (a+b \cosh (c+d x))^3}{20 d} \]
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Rubi [A] time = 0.260091, antiderivative size = 183, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2656, 2753, 2734} \[ \frac{b \left (192 a^2 b^2+107 a^4+16 b^4\right ) \sinh (c+d x)}{30 d}+\frac{b \left (47 a^2+16 b^2\right ) \sinh (c+d x) (a+b \cosh (c+d x))^2}{60 d}+\frac{7 a b^2 \left (22 a^2+23 b^2\right ) \sinh (c+d x) \cosh (c+d x)}{120 d}+\frac{1}{8} a x \left (40 a^2 b^2+8 a^4+15 b^4\right )+\frac{b \sinh (c+d x) (a+b \cosh (c+d x))^4}{5 d}+\frac{9 a b \sinh (c+d x) (a+b \cosh (c+d x))^3}{20 d} \]
Antiderivative was successfully verified.
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Rule 2656
Rule 2753
Rule 2734
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*}
Mathematica [A] time = 0.349365, size = 133, normalized size = 0.73 \[ \frac{60 a \left (40 a^2 b^2+8 a^4+15 b^4\right ) (c+d x)+50 b^3 \left (8 a^2+b^2\right ) \sinh (3 (c+d x))+600 a b^2 \left (2 a^2+b^2\right ) \sinh (2 (c+d x))+300 b \left (12 a^2 b^2+8 a^4+b^4\right ) \sinh (c+d x)+75 a b^4 \sinh (4 (c+d x))+6 b^5 \sinh (5 (c+d x))}{480 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.014, size = 155, normalized size = 0.9 \begin{align*}{\frac{1}{d} \left ({b}^{5} \left ({\frac{8}{15}}+{\frac{ \left ( \cosh \left ( dx+c \right ) \right ) ^{4}}{5}}+{\frac{4\, \left ( \cosh \left ( dx+c \right ) \right ) ^{2}}{15}} \right ) \sinh \left ( dx+c \right ) +5\,a{b}^{4} \left ( \left ( 1/4\, \left ( \cosh \left ( dx+c \right ) \right ) ^{3}+3/8\,\cosh \left ( dx+c \right ) \right ) \sinh \left ( dx+c \right ) +3/8\,dx+3/8\,c \right ) +10\,{a}^{2}{b}^{3} \left ( 2/3+1/3\, \left ( \cosh \left ( dx+c \right ) \right ) ^{2} \right ) \sinh \left ( dx+c \right ) +10\,{a}^{3}{b}^{2} \left ( 1/2\,\cosh \left ( dx+c \right ) \sinh \left ( dx+c \right ) +1/2\,dx+c/2 \right ) +5\,{a}^{4}b\sinh \left ( dx+c \right ) +{a}^{5} \left ( dx+c \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.12051, size = 369, normalized size = 2.02 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.18402, size = 463, normalized size = 2.53 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.66193, size = 314, normalized size = 1.72 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18354, size = 362, normalized size = 1.98 \begin{align*} \frac{6 \, b^{5} e^{\left (5 \, d x + 5 \, c\right )} + 75 \, a b^{4} e^{\left (4 \, d x + 4 \, c\right )} + 400 \, a^{2} b^{3} e^{\left (3 \, d x + 3 \, c\right )} + 50 \, b^{5} e^{\left (3 \, d x + 3 \, c\right )} + 1200 \, a^{3} b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 600 \, a b^{4} e^{\left (2 \, d x + 2 \, c\right )} + 2400 \, a^{4} b e^{\left (d x + c\right )} + 3600 \, a^{2} b^{3} e^{\left (d x + c\right )} + 300 \, b^{5} e^{\left (d x + c\right )} + 120 \,{\left (8 \, a^{5} + 40 \, a^{3} b^{2} + 15 \, a b^{4}\right )}{\left (d x + c\right )} -{\left (75 \, a b^{4} e^{\left (d x + c\right )} + 6 \, b^{5} + 300 \,{\left (8 \, a^{4} b + 12 \, a^{2} b^{3} + b^{5}\right )} e^{\left (4 \, d x + 4 \, c\right )} + 600 \,{\left (2 \, a^{3} b^{2} + a b^{4}\right )} e^{\left (3 \, d x + 3 \, c\right )} + 50 \,{\left (8 \, a^{2} b^{3} + b^{5}\right )} e^{\left (2 \, d x + 2 \, c\right )}\right )} e^{\left (-5 \, d x - 5 \, c\right )}}{960 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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