Optimal. Leaf size=70 \[ -\frac {a x}{2}+\frac {b \cosh (c+d x)}{d}-\frac {2 b \cosh ^3(c+d x)}{3 d}+\frac {b \cosh ^5(c+d x)}{5 d}+\frac {a \cosh (c+d x) \sinh (c+d x)}{2 d} \]
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Rubi [A]
time = 0.06, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3299, 2715, 8,
2713} \begin {gather*} \frac {a \sinh (c+d x) \cosh (c+d x)}{2 d}-\frac {a x}{2}+\frac {b \cosh ^5(c+d x)}{5 d}-\frac {2 b \cosh ^3(c+d x)}{3 d}+\frac {b \cosh (c+d x)}{d} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 2713
Rule 2715
Rule 3299
Rubi steps
\begin {align*} \int \sinh ^2(c+d x) \left (a+b \sinh ^3(c+d x)\right ) \, dx &=-\int \left (-a \sinh ^2(c+d x)-b \sinh ^5(c+d x)\right ) \, dx\\ &=a \int \sinh ^2(c+d x) \, dx+b \int \sinh ^5(c+d x) \, dx\\ &=\frac {a \cosh (c+d x) \sinh (c+d x)}{2 d}-\frac {1}{2} a \int 1 \, dx+\frac {b \text {Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac {a x}{2}+\frac {b \cosh (c+d x)}{d}-\frac {2 b \cosh ^3(c+d x)}{3 d}+\frac {b \cosh ^5(c+d x)}{5 d}+\frac {a \cosh (c+d x) \sinh (c+d x)}{2 d}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 79, normalized size = 1.13 \begin {gather*} \frac {a (-c-d x)}{2 d}+\frac {5 b \cosh (c+d x)}{8 d}-\frac {5 b \cosh (3 (c+d x))}{48 d}+\frac {b \cosh (5 (c+d x))}{80 d}+\frac {a \sinh (2 (c+d x))}{4 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.98, size = 63, normalized size = 0.90
method | result | size |
default | \(-\frac {a x}{2}+\frac {5 b \cosh \left (d x +c \right )}{8 d}-\frac {5 b \cosh \left (3 d x +3 c \right )}{48 d}+\frac {b \cosh \left (5 d x +5 c \right )}{80 d}+\frac {a \sinh \left (2 d x +2 c \right )}{4 d}\) | \(63\) |
risch | \(-\frac {a x}{2}+\frac {b \,{\mathrm e}^{5 d x +5 c}}{160 d}-\frac {5 b \,{\mathrm e}^{3 d x +3 c}}{96 d}+\frac {a \,{\mathrm e}^{2 d x +2 c}}{8 d}+\frac {5 b \,{\mathrm e}^{d x +c}}{16 d}+\frac {5 b \,{\mathrm e}^{-d x -c}}{16 d}-\frac {a \,{\mathrm e}^{-2 d x -2 c}}{8 d}-\frac {5 b \,{\mathrm e}^{-3 d x -3 c}}{96 d}+\frac {b \,{\mathrm e}^{-5 d x -5 c}}{160 d}\) | \(123\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 120, normalized size = 1.71 \begin {gather*} -\frac {1}{8} \, a {\left (4 \, x - \frac {e^{\left (2 \, d x + 2 \, c\right )}}{d} + \frac {e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right )} + \frac {1}{480} \, 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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.44, size = 105, normalized size = 1.50 \begin {gather*} \frac {3 \, b \cosh \left (d x + c\right )^{5} + 15 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{4} - 25 \, b \cosh \left (d x + c\right )^{3} - 120 \, a d x + 120 \, a \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + 15 \, {\left (2 \, b \cosh \left (d x + c\right )^{3} - 5 \, b \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} + 150 \, b \cosh \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 = 0.27, size = 117, normalized size = 1.67 \begin {gather*} \begin {cases} \frac {a x \sinh ^{2}{\left (c + d x \right )}}{2} - \frac {a x \cosh ^{2}{\left (c + d x \right )}}{2} + \frac {a \sinh {\left (c + d x \right )} \cosh {\left (c + d x \right )}}{2 d} + \frac {b \sinh ^{4}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{d} - \frac {4 b \sinh ^{2}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{3 d} + \frac {8 b \cosh ^{5}{\left (c + d x \right )}}{15 d} & \text {for}\: d \neq 0 \\x \left (a + b \sinh ^{3}{\left (c \right )}\right ) \sinh ^{2}{\left (c \right )} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 122, normalized size = 1.74 \begin {gather*} -\frac {1}{2} \, a x + \frac {b e^{\left (5 \, d x + 5 \, c\right )}}{160 \, d} - \frac {5 \, b e^{\left (3 \, d x + 3 \, c\right )}}{96 \, d} + \frac {a e^{\left (2 \, d x + 2 \, c\right )}}{8 \, d} + \frac {5 \, b e^{\left (d x + c\right )}}{16 \, d} + \frac {5 \, b e^{\left (-d x - c\right )}}{16 \, d} - \frac {a e^{\left (-2 \, d x - 2 \, c\right )}}{8 \, d} - \frac {5 \, b e^{\left (-3 \, d x - 3 \, c\right )}}{96 \, d} + \frac {b e^{\left (-5 \, d x - 5 \, c\right )}}{160 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.12, size = 55, normalized size = 0.79 \begin {gather*} \frac {b\,\mathrm {cosh}\left (c+d\,x\right )-\frac {2\,b\,{\mathrm {cosh}\left (c+d\,x\right )}^3}{3}+\frac {b\,{\mathrm {cosh}\left (c+d\,x\right )}^5}{5}+\frac {a\,\mathrm {cosh}\left (c+d\,x\right )\,\mathrm {sinh}\left (c+d\,x\right )}{2}}{d}-\frac {a\,x}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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