Optimal. Leaf size=195 \[ \frac {d e^{c+d x} \cosh (a+b x)}{8 \left (b^2-d^2\right )}-\frac {d e^{c+d x} \cosh (3 a+3 b x)}{16 \left (9 b^2-d^2\right )}-\frac {d e^{c+d x} \cosh (5 a+5 b x)}{16 \left (25 b^2-d^2\right )}-\frac {b e^{c+d x} \sinh (a+b x)}{8 \left (b^2-d^2\right )}+\frac {3 b e^{c+d x} \sinh (3 a+3 b x)}{16 \left (9 b^2-d^2\right )}+\frac {5 b e^{c+d x} \sinh (5 a+5 b x)}{16 \left (25 b^2-d^2\right )} \]
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Rubi [A]
time = 0.10, antiderivative size = 195, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {5620, 5583}
\begin {gather*} -\frac {b e^{c+d x} \sinh (a+b x)}{8 \left (b^2-d^2\right )}+\frac {3 b e^{c+d x} \sinh (3 a+3 b x)}{16 \left (9 b^2-d^2\right )}+\frac {5 b e^{c+d x} \sinh (5 a+5 b x)}{16 \left (25 b^2-d^2\right )}+\frac {d e^{c+d x} \cosh (a+b x)}{8 \left (b^2-d^2\right )}-\frac {d e^{c+d x} \cosh (3 a+3 b x)}{16 \left (9 b^2-d^2\right )}-\frac {d e^{c+d x} \cosh (5 a+5 b x)}{16 \left (25 b^2-d^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 5583
Rule 5620
Rubi steps
\begin {align*} \int e^{c+d x} \cosh ^3(a+b x) \sinh ^2(a+b x) \, dx &=\int \left (-\frac {1}{8} e^{c+d x} \cosh (a+b x)+\frac {1}{16} e^{c+d x} \cosh (3 a+3 b x)+\frac {1}{16} e^{c+d x} \cosh (5 a+5 b x)\right ) \, dx\\ &=\frac {1}{16} \int e^{c+d x} \cosh (3 a+3 b x) \, dx+\frac {1}{16} \int e^{c+d x} \cosh (5 a+5 b x) \, dx-\frac {1}{8} \int e^{c+d x} \cosh (a+b x) \, dx\\ &=\frac {d e^{c+d x} \cosh (a+b x)}{8 \left (b^2-d^2\right )}-\frac {d e^{c+d x} \cosh (3 a+3 b x)}{16 \left (9 b^2-d^2\right )}-\frac {d e^{c+d x} \cosh (5 a+5 b x)}{16 \left (25 b^2-d^2\right )}-\frac {b e^{c+d x} \sinh (a+b x)}{8 \left (b^2-d^2\right )}+\frac {3 b e^{c+d x} \sinh (3 a+3 b x)}{16 \left (9 b^2-d^2\right )}+\frac {5 b e^{c+d x} \sinh (5 a+5 b x)}{16 \left (25 b^2-d^2\right )}\\ \end {align*}
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Mathematica [A]
time = 0.84, size = 118, normalized size = 0.61 \begin {gather*} \frac {1}{16} e^{c+d x} \left (\frac {2 d \cosh (a+b x)-2 b \sinh (a+b x)}{(b-d) (b+d)}+\frac {-d \cosh (3 (a+b x))+3 b \sinh (3 (a+b x))}{9 b^2-d^2}+\frac {-d \cosh (5 (a+b x))+5 b \sinh (5 (a+b x))}{25 b^2-d^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.96, size = 278, normalized size = 1.43
method | result | size |
default | \(-\frac {\sinh \left (a -c +\left (b -d \right ) x \right )}{16 \left (b -d \right )}-\frac {\sinh \left (a +c +\left (b +d \right ) x \right )}{16 \left (b +d \right )}+\frac {\sinh \left (3 a -c +\left (3 b -d \right ) x \right )}{96 b -32 d}+\frac {\sinh \left (3 a +c +\left (3 b +d \right ) x \right )}{96 b +32 d}+\frac {\sinh \left (\left (5 b -d \right ) x +5 a -c \right )}{160 b -32 d}+\frac {\sinh \left (\left (5 b +d \right ) x +5 a +c \right )}{160 b +32 d}+\frac {\cosh \left (a -c +\left (b -d \right ) x \right )}{16 b -16 d}-\frac {\cosh \left (a +c +\left (b +d \right ) x \right )}{16 \left (b +d \right )}-\frac {\cosh \left (3 a -c +\left (3 b -d \right ) x \right )}{32 \left (3 b -d \right )}+\frac {\cosh \left (3 a +c +\left (3 b +d \right ) x \right )}{96 b +32 d}-\frac {\cosh \left (\left (5 b -d \right ) x +5 a -c \right )}{32 \left (5 b -d \right )}+\frac {\cosh \left (\left (5 b +d \right ) x +5 a +c \right )}{160 b +32 d}\) | \(278\) |
risch | \(\frac {\left (-68 b^{2} d^{3} {\mathrm e}^{4 b x +4 a}-68 b^{3} d^{2} {\mathrm e}^{4 b x +4 a}+26 b^{2} d^{3} {\mathrm e}^{2 b x +2 a}-3 b \,d^{4} {\mathrm e}^{2 b x +2 a}-78 b^{3} d^{2} {\mathrm e}^{8 b x +8 a}+26 b^{2} d^{3} {\mathrm e}^{8 b x +8 a}+450 b^{4} d \,{\mathrm e}^{6 b x +6 a}+68 b^{3} d^{2} {\mathrm e}^{6 b x +6 a}+75 b^{5} {\mathrm e}^{8 b x +8 a}-d^{5} {\mathrm e}^{8 b x +8 a}+10 b^{2} d^{3} {\mathrm e}^{10 b x +10 a}+2 b \,d^{4} {\mathrm e}^{4 b x +4 a}-45 b^{5}-d^{5}-68 b^{2} d^{3} {\mathrm e}^{6 b x +6 a}-2 b \,d^{4} {\mathrm e}^{6 b x +6 a}-25 b^{4} d \,{\mathrm e}^{2 b x +2 a}+78 b^{3} d^{2} {\mathrm e}^{2 b x +2 a}+3 b \,d^{4} {\mathrm e}^{8 b x +8 a}-9 b^{4} d \,{\mathrm e}^{10 b x +10 a}-50 b^{3} d^{2} {\mathrm e}^{10 b x +10 a}+5 b \,d^{4} {\mathrm e}^{10 b x +10 a}-25 b^{4} d \,{\mathrm e}^{8 b x +8 a}+450 b^{4} d \,{\mathrm e}^{4 b x +4 a}-9 b^{4} d +50 b^{3} d^{2}+10 b^{2} d^{3}-5 b \,d^{4}-450 b^{5} {\mathrm e}^{6 b x +6 a}+2 d^{5} {\mathrm e}^{6 b x +6 a}+450 b^{5} {\mathrm e}^{4 b x +4 a}+2 d^{5} {\mathrm e}^{4 b x +4 a}-75 b^{5} {\mathrm e}^{2 b x +2 a}-d^{5} {\mathrm e}^{2 b x +2 a}+45 b^{5} {\mathrm e}^{10 b x +10 a}-d^{5} {\mathrm e}^{10 b x +10 a}\right ) {\mathrm e}^{-5 b x +d x -5 a +c}}{32 \left (5 b +d \right ) \left (3 b +d \right ) \left (b +d \right ) \left (5 b -d \right ) \left (3 b -d \right ) \left (b -d \right )}\) | \(559\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 917 vs.
\(2 (177) = 354\).
time = 0.34, size = 917, normalized size = 4.70 \begin {gather*} -\frac {5 \, {\left (9 \, b^{4} d - 10 \, b^{2} d^{3} + d^{5}\right )} \cosh \left (b x + a\right ) \cosh \left (d x + c\right ) \sinh \left (b x + a\right )^{4} - 5 \, {\left (9 \, b^{5} - 10 \, b^{3} d^{2} + b d^{4}\right )} \cosh \left (d x + c\right ) \sinh \left (b x + a\right )^{5} - {\left (75 \, b^{5} - 78 \, b^{3} d^{2} + 3 \, b d^{4} + 50 \, {\left (9 \, b^{5} - 10 \, b^{3} d^{2} + b d^{4}\right )} \cosh \left (b x + a\right )^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (b x + a\right )^{3} + {\left (10 \, {\left (9 \, b^{4} d - 10 \, b^{2} d^{3} + d^{5}\right )} \cosh \left (b x + a\right )^{3} + 3 \, {\left (25 \, b^{4} d - 26 \, b^{2} d^{3} + d^{5}\right )} \cosh \left (b x + a\right )\right )} \cosh \left (d x + c\right ) \sinh \left (b x + a\right )^{2} + {\left (450 \, b^{5} - 68 \, b^{3} d^{2} + 2 \, b d^{4} - 25 \, {\left (9 \, b^{5} - 10 \, b^{3} d^{2} + b d^{4}\right )} \cosh \left (b x + a\right )^{4} - 9 \, {\left (25 \, b^{5} - 26 \, b^{3} d^{2} + b d^{4}\right )} \cosh \left (b x + a\right )^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (b x + a\right ) + {\left ({\left (9 \, b^{4} d - 10 \, b^{2} d^{3} + d^{5}\right )} \cosh \left (b x + a\right )^{5} + {\left (25 \, b^{4} d - 26 \, b^{2} d^{3} + d^{5}\right )} \cosh \left (b x + a\right )^{3} - 2 \, {\left (225 \, b^{4} d - 34 \, b^{2} d^{3} + d^{5}\right )} \cosh \left (b x + a\right )\right )} \cosh \left (d x + c\right ) + {\left ({\left (9 \, b^{4} d - 10 \, b^{2} d^{3} + d^{5}\right )} \cosh \left (b x + a\right )^{5} + 5 \, {\left (9 \, b^{4} d - 10 \, b^{2} d^{3} + d^{5}\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{4} - 5 \, {\left (9 \, b^{5} - 10 \, b^{3} d^{2} + b d^{4}\right )} \sinh \left (b x + a\right )^{5} + {\left (25 \, b^{4} d - 26 \, b^{2} d^{3} + d^{5}\right )} \cosh \left (b x + a\right )^{3} - {\left (75 \, b^{5} - 78 \, b^{3} d^{2} + 3 \, b d^{4} + 50 \, {\left (9 \, b^{5} - 10 \, b^{3} d^{2} + b d^{4}\right )} \cosh \left (b x + a\right )^{2}\right )} \sinh \left (b x + a\right )^{3} + {\left (10 \, {\left (9 \, b^{4} d - 10 \, b^{2} d^{3} + d^{5}\right )} \cosh \left (b x + a\right )^{3} + 3 \, {\left (25 \, b^{4} d - 26 \, b^{2} d^{3} + d^{5}\right )} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{2} - 2 \, {\left (225 \, b^{4} d - 34 \, b^{2} d^{3} + d^{5}\right )} \cosh \left (b x + a\right ) + {\left (450 \, b^{5} - 68 \, b^{3} d^{2} + 2 \, b d^{4} - 25 \, {\left (9 \, b^{5} - 10 \, b^{3} d^{2} + b d^{4}\right )} \cosh \left (b x + a\right )^{4} - 9 \, {\left (25 \, b^{5} - 26 \, b^{3} d^{2} + b d^{4}\right )} \cosh \left (b x + a\right )^{2}\right )} \sinh \left (b x + a\right )\right )} \sinh \left (d x + c\right )}{16 \, {\left ({\left (225 \, b^{6} - 259 \, b^{4} d^{2} + 35 \, b^{2} d^{4} - d^{6}\right )} \cosh \left (b x + a\right )^{6} - 3 \, {\left (225 \, b^{6} - 259 \, b^{4} d^{2} + 35 \, b^{2} d^{4} - d^{6}\right )} \cosh \left (b x + a\right )^{4} \sinh \left (b x + a\right )^{2} + 3 \, {\left (225 \, b^{6} - 259 \, b^{4} d^{2} + 35 \, b^{2} d^{4} - d^{6}\right )} \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right )^{4} - {\left (225 \, b^{6} - 259 \, b^{4} d^{2} + 35 \, b^{2} d^{4} - d^{6}\right )} \sinh \left (b x + a\right )^{6}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 2662 vs.
\(2 (168) = 336\).
time = 27.50, size = 2662, normalized size = 13.65 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 132, normalized size = 0.68 \begin {gather*} \frac {e^{\left (5 \, b x + d x + 5 \, a + c\right )}}{32 \, {\left (5 \, b + d\right )}} + \frac {e^{\left (3 \, b x + d x + 3 \, a + c\right )}}{32 \, {\left (3 \, b + d\right )}} - \frac {e^{\left (b x + d x + a + c\right )}}{16 \, {\left (b + d\right )}} + \frac {e^{\left (-b x + d x - a + c\right )}}{16 \, {\left (b - d\right )}} - \frac {e^{\left (-3 \, b x + d x - 3 \, a + c\right )}}{32 \, {\left (3 \, b - d\right )}} - \frac {e^{\left (-5 \, b x + d x - 5 \, a + c\right )}}{32 \, {\left (5 \, b - d\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.42, size = 393, normalized size = 2.02 \begin {gather*} \frac {{\mathrm {cosh}\left (a+b\,x\right )}^5\,{\mathrm {e}}^{c+d\,x}\,\left (26\,b^4\,d-2\,b^2\,d^3\right )}{225\,b^6-259\,b^4\,d^2+35\,b^2\,d^4-d^6}+\frac {3\,{\mathrm {cosh}\left (a+b\,x\right )}^2\,{\mathrm {e}}^{c+d\,x}\,{\mathrm {sinh}\left (a+b\,x\right )}^3\,\left (25\,b^5-10\,b^3\,d^2+b\,d^4\right )}{225\,b^6-259\,b^4\,d^2+35\,b^2\,d^4-d^6}+\frac {2\,{\mathrm {cosh}\left (a+b\,x\right )}^4\,{\mathrm {e}}^{c+d\,x}\,\mathrm {sinh}\left (a+b\,x\right )\,\left (b\,d^4-13\,b^3\,d^2\right )}{225\,b^6-259\,b^4\,d^2+35\,b^2\,d^4-d^6}-\frac {{\mathrm {cosh}\left (a+b\,x\right )}^3\,{\mathrm {e}}^{c+d\,x}\,{\mathrm {sinh}\left (a+b\,x\right )}^2\,\left (65\,b^4\,d-18\,b^2\,d^3+d^5\right )}{225\,b^6-259\,b^4\,d^2+35\,b^2\,d^4-d^6}-\frac {6\,b^3\,{\mathrm {e}}^{c+d\,x}\,{\mathrm {sinh}\left (a+b\,x\right )}^5\,\left (5\,b^2-d^2\right )}{225\,b^6-259\,b^4\,d^2+35\,b^2\,d^4-d^6}+\frac {6\,b^2\,d\,\mathrm {cosh}\left (a+b\,x\right )\,{\mathrm {e}}^{c+d\,x}\,{\mathrm {sinh}\left (a+b\,x\right )}^4\,\left (5\,b^2-d^2\right )}{225\,b^6-259\,b^4\,d^2+35\,b^2\,d^4-d^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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