Optimal. Leaf size=137 \[ -\frac {3 b e^{c+d x} \cosh (2 a+2 b x)}{16 \left (4 b^2-d^2\right )}+\frac {3 b e^{c+d x} \cosh (6 a+6 b x)}{16 \left (36 b^2-d^2\right )}+\frac {3 d e^{c+d x} \sinh (2 a+2 b x)}{32 \left (4 b^2-d^2\right )}-\frac {d e^{c+d x} \sinh (6 a+6 b x)}{32 \left (36 b^2-d^2\right )} \]
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Rubi [A]
time = 0.07, antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {5620, 5582}
\begin {gather*} \frac {3 d e^{c+d x} \sinh (2 a+2 b x)}{32 \left (4 b^2-d^2\right )}-\frac {d e^{c+d x} \sinh (6 a+6 b x)}{32 \left (36 b^2-d^2\right )}-\frac {3 b e^{c+d x} \cosh (2 a+2 b x)}{16 \left (4 b^2-d^2\right )}+\frac {3 b e^{c+d x} \cosh (6 a+6 b x)}{16 \left (36 b^2-d^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 5582
Rule 5620
Rubi steps
\begin {align*} \int e^{c+d x} \cosh ^3(a+b x) \sinh ^3(a+b x) \, dx &=\int \left (-\frac {3}{32} e^{c+d x} \sinh (2 a+2 b x)+\frac {1}{32} e^{c+d x} \sinh (6 a+6 b x)\right ) \, dx\\ &=\frac {1}{32} \int e^{c+d x} \sinh (6 a+6 b x) \, dx-\frac {3}{32} \int e^{c+d x} \sinh (2 a+2 b x) \, dx\\ &=-\frac {3 b e^{c+d x} \cosh (2 a+2 b x)}{16 \left (4 b^2-d^2\right )}+\frac {3 b e^{c+d x} \cosh (6 a+6 b x)}{16 \left (36 b^2-d^2\right )}+\frac {3 d e^{c+d x} \sinh (2 a+2 b x)}{32 \left (4 b^2-d^2\right )}-\frac {d e^{c+d x} \sinh (6 a+6 b x)}{32 \left (36 b^2-d^2\right )}\\ \end {align*}
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Mathematica [A]
time = 0.65, size = 113, normalized size = 0.82 \begin {gather*} \frac {e^{c+d x} \left (6 b \left (-36 b^2+d^2\right ) \cosh (2 (a+b x))+6 \left (4 b^3-b d^2\right ) \cosh (6 (a+b x))+2 d \left (52 b^2-d^2+\left (-4 b^2+d^2\right ) \cosh (4 (a+b x))\right ) \sinh (2 (a+b x))\right )}{32 \left (144 b^4-40 b^2 d^2+d^4\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.98, size = 202, normalized size = 1.47
method | result | size |
default | \(\frac {3 \sinh \left (2 a -c +\left (2 b -d \right ) x \right )}{64 \left (2 b -d \right )}-\frac {3 \sinh \left (2 a +c +\left (2 b +d \right ) x \right )}{64 \left (2 b +d \right )}-\frac {\sinh \left (\left (6 b -d \right ) x +6 a -c \right )}{64 \left (6 b -d \right )}+\frac {\sinh \left (\left (6 b +d \right ) x +6 a +c \right )}{384 b +64 d}-\frac {3 \cosh \left (2 a -c +\left (2 b -d \right ) x \right )}{64 \left (2 b -d \right )}-\frac {3 \cosh \left (2 a +c +\left (2 b +d \right ) x \right )}{64 \left (2 b +d \right )}+\frac {\cosh \left (\left (6 b -d \right ) x +6 a -c \right )}{384 b -64 d}+\frac {\cosh \left (\left (6 b +d \right ) x +6 a +c \right )}{384 b +64 d}\) | \(202\) |
risch | \(\frac {\left (24 b^{3} {\mathrm e}^{12 b x +12 a}-4 b^{2} d \,{\mathrm e}^{12 b x +12 a}-6 b \,d^{2} {\mathrm e}^{12 b x +12 a}+d^{3} {\mathrm e}^{12 b x +12 a}-216 b^{3} {\mathrm e}^{8 b x +8 a}+108 b^{2} d \,{\mathrm e}^{8 b x +8 a}+6 b \,d^{2} {\mathrm e}^{8 b x +8 a}-3 d^{3} {\mathrm e}^{8 b x +8 a}-216 b^{3} {\mathrm e}^{4 b x +4 a}-108 b^{2} d \,{\mathrm e}^{4 b x +4 a}+6 b \,d^{2} {\mathrm e}^{4 b x +4 a}+3 d^{3} {\mathrm e}^{4 b x +4 a}+24 b^{3}+4 b^{2} d -6 d^{2} b -d^{3}\right ) {\mathrm e}^{-6 b x +d x -6 a +c}}{64 \left (6 b +d \right ) \left (2 b +d \right ) \left (6 b -d \right ) \left (2 b -d \right )}\) | \(244\) |
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. 676 vs.
\(2 (125) = 250\).
time = 0.36, size = 676, normalized size = 4.93 \begin {gather*} -\frac {10 \, {\left (4 \, b^{2} d - d^{3}\right )} \cosh \left (b x + a\right )^{3} \cosh \left (d x + c\right ) \sinh \left (b x + a\right )^{3} - 45 \, {\left (4 \, b^{3} - b d^{2}\right )} \cosh \left (b x + a\right )^{2} \cosh \left (d x + c\right ) \sinh \left (b x + a\right )^{4} + 3 \, {\left (4 \, b^{2} d - d^{3}\right )} \cosh \left (b x + a\right ) \cosh \left (d x + c\right ) \sinh \left (b x + a\right )^{5} - 3 \, {\left (4 \, b^{3} - b d^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (b x + a\right )^{6} - 3 \, {\left (15 \, {\left (4 \, b^{3} - b d^{2}\right )} \cosh \left (b x + a\right )^{4} - 36 \, b^{3} + b d^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (b x + a\right )^{2} + 3 \, {\left ({\left (4 \, b^{2} d - d^{3}\right )} \cosh \left (b x + a\right )^{5} - {\left (36 \, b^{2} d - d^{3}\right )} \cosh \left (b x + a\right )\right )} \cosh \left (d x + c\right ) \sinh \left (b x + a\right ) - 3 \, {\left ({\left (4 \, b^{3} - b d^{2}\right )} \cosh \left (b x + a\right )^{6} - {\left (36 \, b^{3} - b d^{2}\right )} \cosh \left (b x + a\right )^{2}\right )} \cosh \left (d x + c\right ) - {\left (3 \, {\left (4 \, b^{3} - b d^{2}\right )} \cosh \left (b x + a\right )^{6} - 10 \, {\left (4 \, b^{2} d - d^{3}\right )} \cosh \left (b x + a\right )^{3} \sinh \left (b x + a\right )^{3} + 45 \, {\left (4 \, b^{3} - b d^{2}\right )} \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right )^{4} - 3 \, {\left (4 \, b^{2} d - d^{3}\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{5} + 3 \, {\left (4 \, b^{3} - b d^{2}\right )} \sinh \left (b x + a\right )^{6} - 3 \, {\left (36 \, b^{3} - b d^{2}\right )} \cosh \left (b x + a\right )^{2} + 3 \, {\left (15 \, {\left (4 \, b^{3} - b d^{2}\right )} \cosh \left (b x + a\right )^{4} - 36 \, b^{3} + b d^{2}\right )} \sinh \left (b x + a\right )^{2} - 3 \, {\left ({\left (4 \, b^{2} d - d^{3}\right )} \cosh \left (b x + a\right )^{5} - {\left (36 \, b^{2} d - d^{3}\right )} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )\right )} \sinh \left (d x + c\right )}{16 \, {\left ({\left (144 \, b^{4} - 40 \, b^{2} d^{2} + d^{4}\right )} \cosh \left (b x + a\right )^{6} - 3 \, {\left (144 \, b^{4} - 40 \, b^{2} d^{2} + d^{4}\right )} \cosh \left (b x + a\right )^{4} \sinh \left (b x + a\right )^{2} + 3 \, {\left (144 \, b^{4} - 40 \, b^{2} d^{2} + d^{4}\right )} \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right )^{4} - {\left (144 \, b^{4} - 40 \, b^{2} d^{2} + d^{4}\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. 1916 vs.
\(2 (119) = 238\).
time = 65.12, size = 1916, normalized size = 13.99 \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 = 0.44, size = 93, normalized size = 0.68 \begin {gather*} \frac {e^{\left (6 \, b x + d x + 6 \, a + c\right )}}{64 \, {\left (6 \, b + d\right )}} - \frac {3 \, e^{\left (2 \, b x + d x + 2 \, a + c\right )}}{64 \, {\left (2 \, b + d\right )}} - \frac {3 \, e^{\left (-2 \, b x + d x - 2 \, a + c\right )}}{64 \, {\left (2 \, b - d\right )}} + \frac {e^{\left (-6 \, b x + d x - 6 \, a + c\right )}}{64 \, {\left (6 \, b - d\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.96, size = 182, normalized size = 1.33 \begin {gather*} -\frac {b^3\,\left (\frac {27\,{\mathrm {e}}^{c+d\,x}\,\mathrm {cosh}\left (2\,a+2\,b\,x\right )}{4}-\frac {3\,{\mathrm {e}}^{c+d\,x}\,\mathrm {cosh}\left (6\,a+6\,b\,x\right )}{4}\right )+d^3\,\left (\frac {3\,{\mathrm {e}}^{c+d\,x}\,\mathrm {sinh}\left (2\,a+2\,b\,x\right )}{32}-\frac {{\mathrm {e}}^{c+d\,x}\,\mathrm {sinh}\left (6\,a+6\,b\,x\right )}{32}\right )-b^2\,d\,\left (\frac {27\,{\mathrm {e}}^{c+d\,x}\,\mathrm {sinh}\left (2\,a+2\,b\,x\right )}{8}-\frac {{\mathrm {e}}^{c+d\,x}\,\mathrm {sinh}\left (6\,a+6\,b\,x\right )}{8}\right )-b\,d^2\,\left (\frac {3\,{\mathrm {e}}^{c+d\,x}\,\mathrm {cosh}\left (2\,a+2\,b\,x\right )}{16}-\frac {3\,{\mathrm {e}}^{c+d\,x}\,\mathrm {cosh}\left (6\,a+6\,b\,x\right )}{16}\right )}{144\,b^4-40\,b^2\,d^2+d^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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