Optimal. Leaf size=137 \[ \frac {d e^{c+d x} \sinh (2 a+2 b x)}{4 \left (4 b^2-d^2\right )}-\frac {d e^{c+d x} \sinh (4 a+4 b x)}{8 \left (16 b^2-d^2\right )}-\frac {b e^{c+d x} \cosh (2 a+2 b x)}{2 \left (4 b^2-d^2\right )}+\frac {b e^{c+d x} \cosh (4 a+4 b x)}{2 \left (16 b^2-d^2\right )} \]
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Rubi [A] time = 0.10, antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {5509, 5474} \[ \frac {d e^{c+d x} \sinh (2 a+2 b x)}{4 \left (4 b^2-d^2\right )}-\frac {d e^{c+d x} \sinh (4 a+4 b x)}{8 \left (16 b^2-d^2\right )}-\frac {b e^{c+d x} \cosh (2 a+2 b x)}{2 \left (4 b^2-d^2\right )}+\frac {b e^{c+d x} \cosh (4 a+4 b x)}{2 \left (16 b^2-d^2\right )} \]
Antiderivative was successfully verified.
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Rule 5474
Rule 5509
Rubi steps
\begin {align*} \int e^{c+d x} \cosh (a+b x) \sinh ^3(a+b x) \, dx &=\int \left (-\frac {1}{4} e^{c+d x} \sinh (2 a+2 b x)+\frac {1}{8} e^{c+d x} \sinh (4 a+4 b x)\right ) \, dx\\ &=\frac {1}{8} \int e^{c+d x} \sinh (4 a+4 b x) \, dx-\frac {1}{4} \int e^{c+d x} \sinh (2 a+2 b x) \, dx\\ &=-\frac {b e^{c+d x} \cosh (2 a+2 b x)}{2 \left (4 b^2-d^2\right )}+\frac {b e^{c+d x} \cosh (4 a+4 b x)}{2 \left (16 b^2-d^2\right )}+\frac {d e^{c+d x} \sinh (2 a+2 b x)}{4 \left (4 b^2-d^2\right )}-\frac {d e^{c+d x} \sinh (4 a+4 b x)}{8 \left (16 b^2-d^2\right )}\\ \end {align*}
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Mathematica [A] time = 1.21, size = 86, normalized size = 0.63 \[ \frac {1}{8} e^{c+d x} \left (\frac {2 d \sinh (2 (a+b x))-4 b \cosh (2 (a+b x))}{4 b^2-d^2}+\frac {4 b \cosh (4 (a+b x))-d \sinh (4 (a+b x))}{16 b^2-d^2}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.50, size = 505, normalized size = 3.69 \[ -\frac {{\left (4 \, b^{2} d - d^{3}\right )} \cosh \left (b x + a\right ) \cosh \left (d x + c\right ) \sinh \left (b x + a\right )^{3} - {\left (4 \, b^{3} - b d^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (b x + a\right )^{4} + {\left (16 \, b^{3} - b d^{2} - 6 \, {\left (4 \, b^{3} - b d^{2}\right )} \cosh \left (b x + a\right )^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (b x + a\right )^{2} + {\left ({\left (4 \, b^{2} d - d^{3}\right )} \cosh \left (b x + a\right )^{3} - {\left (16 \, b^{2} d - d^{3}\right )} \cosh \left (b x + a\right )\right )} \cosh \left (d x + c\right ) \sinh \left (b x + a\right ) - {\left ({\left (4 \, b^{3} - b d^{2}\right )} \cosh \left (b x + a\right )^{4} - {\left (16 \, b^{3} - b d^{2}\right )} \cosh \left (b x + a\right )^{2}\right )} \cosh \left (d x + c\right ) - {\left ({\left (4 \, b^{3} - b d^{2}\right )} \cosh \left (b x + a\right )^{4} - {\left (4 \, b^{2} d - d^{3}\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + {\left (4 \, b^{3} - b d^{2}\right )} \sinh \left (b x + a\right )^{4} - {\left (16 \, b^{3} - b d^{2}\right )} \cosh \left (b x + a\right )^{2} - {\left (16 \, b^{3} - b d^{2} - 6 \, {\left (4 \, b^{3} - b d^{2}\right )} \cosh \left (b x + a\right )^{2}\right )} \sinh \left (b x + a\right )^{2} - {\left ({\left (4 \, b^{2} d - d^{3}\right )} \cosh \left (b x + a\right )^{3} - {\left (16 \, b^{2} d - d^{3}\right )} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )\right )} \sinh \left (d x + c\right )}{2 \, {\left ({\left (64 \, b^{4} - 20 \, b^{2} d^{2} + d^{4}\right )} \cosh \left (b x + a\right )^{4} - 2 \, {\left (64 \, b^{4} - 20 \, b^{2} d^{2} + d^{4}\right )} \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right )^{2} + {\left (64 \, b^{4} - 20 \, b^{2} d^{2} + d^{4}\right )} \sinh \left (b x + a\right )^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 93, normalized size = 0.68 \[ \frac {e^{\left (4 \, b x + d x + 4 \, a + c\right )}}{16 \, {\left (4 \, b + d\right )}} - \frac {e^{\left (2 \, b x + d x + 2 \, a + c\right )}}{8 \, {\left (2 \, b + d\right )}} - \frac {e^{\left (-2 \, b x + d x - 2 \, a + c\right )}}{8 \, {\left (2 \, b - d\right )}} + \frac {e^{\left (-4 \, b x + d x - 4 \, a + c\right )}}{16 \, {\left (4 \, b - d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.36, size = 202, normalized size = 1.47 \[ \frac {\sinh \left (2 a -c +\left (2 b -d \right ) x \right )}{16 b -8 d}-\frac {\sinh \left (2 a +c +\left (2 b +d \right ) x \right )}{8 \left (2 b +d \right )}-\frac {\sinh \left (\left (4 b -d \right ) x +4 a -c \right )}{16 \left (4 b -d \right )}+\frac {\sinh \left (\left (4 b +d \right ) x +4 a +c \right )}{64 b +16 d}-\frac {\cosh \left (2 a -c +\left (2 b -d \right ) x \right )}{8 \left (2 b -d \right )}-\frac {\cosh \left (2 a +c +\left (2 b +d \right ) x \right )}{8 \left (2 b +d \right )}+\frac {\cosh \left (\left (4 b -d \right ) x +4 a -c \right )}{64 b -16 d}+\frac {\cosh \left (\left (4 b +d \right ) x +4 a +c \right )}{64 b +16 d} \]
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 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.14, size = 228, normalized size = 1.66 \[ \frac {b\,{\mathrm {e}}^{c+d\,x}\,{\mathrm {sinh}\left (a+b\,x\right )}^4\,\left (10\,b^2-d^2\right )}{64\,b^4-20\,b^2\,d^2+d^4}-\frac {6\,b^3\,{\mathrm {cosh}\left (a+b\,x\right )}^4\,{\mathrm {e}}^{c+d\,x}}{64\,b^4-20\,b^2\,d^2+d^4}+\frac {3\,b\,{\mathrm {cosh}\left (a+b\,x\right )}^2\,{\mathrm {e}}^{c+d\,x}\,{\mathrm {sinh}\left (a+b\,x\right )}^2}{16\,b^2-d^2}-\frac {d\,\mathrm {cosh}\left (a+b\,x\right )\,{\mathrm {e}}^{c+d\,x}\,{\mathrm {sinh}\left (a+b\,x\right )}^3\,\left (10\,b^2-d^2\right )}{64\,b^4-20\,b^2\,d^2+d^4}+\frac {6\,b^2\,d\,{\mathrm {cosh}\left (a+b\,x\right )}^3\,{\mathrm {e}}^{c+d\,x}\,\mathrm {sinh}\left (a+b\,x\right )}{\left (4\,b^2-d^2\right )\,\left (16\,b^2-d^2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 142.13, size = 1292, normalized size = 9.43 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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