Optimal. Leaf size=127 \[ -\frac {b e^{c+d x} \sinh (a+b x)}{4 \left (b^2-d^2\right )}+\frac {3 b e^{c+d x} \sinh (3 a+3 b x)}{4 \left (9 b^2-d^2\right )}+\frac {d e^{c+d x} \cosh (a+b x)}{4 \left (b^2-d^2\right )}-\frac {d e^{c+d x} \cosh (3 a+3 b x)}{4 \left (9 b^2-d^2\right )} \]
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Rubi [A] time = 0.09, antiderivative size = 127, 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, 5475} \[ -\frac {b e^{c+d x} \sinh (a+b x)}{4 \left (b^2-d^2\right )}+\frac {3 b e^{c+d x} \sinh (3 a+3 b x)}{4 \left (9 b^2-d^2\right )}+\frac {d e^{c+d x} \cosh (a+b x)}{4 \left (b^2-d^2\right )}-\frac {d e^{c+d x} \cosh (3 a+3 b x)}{4 \left (9 b^2-d^2\right )} \]
Antiderivative was successfully verified.
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Rule 5475
Rule 5509
Rubi steps
\begin {align*} \int e^{c+d x} \cosh (a+b x) \sinh ^2(a+b x) \, dx &=\int \left (-\frac {1}{4} e^{c+d x} \cosh (a+b x)+\frac {1}{4} e^{c+d x} \cosh (3 a+3 b x)\right ) \, dx\\ &=-\left (\frac {1}{4} \int e^{c+d x} \cosh (a+b x) \, dx\right )+\frac {1}{4} \int e^{c+d x} \cosh (3 a+3 b x) \, dx\\ &=\frac {d e^{c+d x} \cosh (a+b x)}{4 \left (b^2-d^2\right )}-\frac {d e^{c+d x} \cosh (3 a+3 b x)}{4 \left (9 b^2-d^2\right )}-\frac {b e^{c+d x} \sinh (a+b x)}{4 \left (b^2-d^2\right )}+\frac {3 b e^{c+d x} \sinh (3 a+3 b x)}{4 \left (9 b^2-d^2\right )}\\ \end {align*}
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Mathematica [A] time = 0.99, size = 80, normalized size = 0.63 \[ \frac {1}{4} e^{c+d x} \left (\frac {3 b \sinh (3 (a+b x))-d \cosh (3 (a+b x))}{9 b^2-d^2}+\frac {d \cosh (a+b x)-b \sinh (a+b x)}{(b-d) (b+d)}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.43, size = 379, normalized size = 2.98 \[ -\frac {3 \, {\left (b^{2} d - d^{3}\right )} \cosh \left (b x + a\right ) \cosh \left (d x + c\right ) \sinh \left (b x + a\right )^{2} - 3 \, {\left (b^{3} - b d^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (b x + a\right )^{3} + {\left (9 \, b^{3} - b d^{2} - 9 \, {\left (b^{3} - b d^{2}\right )} \cosh \left (b x + a\right )^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (b x + a\right ) + {\left ({\left (b^{2} d - d^{3}\right )} \cosh \left (b x + a\right )^{3} - {\left (9 \, b^{2} d - d^{3}\right )} \cosh \left (b x + a\right )\right )} \cosh \left (d x + c\right ) + {\left ({\left (b^{2} d - d^{3}\right )} \cosh \left (b x + a\right )^{3} + 3 \, {\left (b^{2} d - d^{3}\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} - 3 \, {\left (b^{3} - b d^{2}\right )} \sinh \left (b x + a\right )^{3} - {\left (9 \, b^{2} d - d^{3}\right )} \cosh \left (b x + a\right ) + {\left (9 \, b^{3} - b d^{2} - 9 \, {\left (b^{3} - b d^{2}\right )} \cosh \left (b x + a\right )^{2}\right )} \sinh \left (b x + a\right )\right )} \sinh \left (d x + c\right )}{4 \, {\left ({\left (9 \, b^{4} - 10 \, b^{2} d^{2} + d^{4}\right )} \cosh \left (b x + a\right )^{4} - 2 \, {\left (9 \, b^{4} - 10 \, b^{2} d^{2} + d^{4}\right )} \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right )^{2} + {\left (9 \, b^{4} - 10 \, b^{2} d^{2} + d^{4}\right )} \sinh \left (b x + a\right )^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 86, normalized size = 0.68 \[ \frac {e^{\left (3 \, b x + d x + 3 \, a + c\right )}}{8 \, {\left (3 \, b + d\right )}} - \frac {e^{\left (b x + d x + a + c\right )}}{8 \, {\left (b + d\right )}} + \frac {e^{\left (-b x + d x - a + c\right )}}{8 \, {\left (b - d\right )}} - \frac {e^{\left (-3 \, b x + d x - 3 \, a + c\right )}}{8 \, {\left (3 \, b - d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.35, size = 178, normalized size = 1.40 \[ -\frac {\sinh \left (a -c +\left (b -d \right ) x \right )}{8 \left (b -d \right )}-\frac {\sinh \left (a +c +\left (b +d \right ) x \right )}{8 \left (b +d \right )}+\frac {\sinh \left (3 a -c +\left (3 b -d \right ) x \right )}{24 b -8 d}+\frac {\sinh \left (3 a +c +\left (3 b +d \right ) x \right )}{24 b +8 d}+\frac {\cosh \left (a -c +\left (b -d \right ) x \right )}{8 b -8 d}-\frac {\cosh \left (a +c +\left (b +d \right ) x \right )}{8 \left (b +d \right )}-\frac {\cosh \left (3 a -c +\left (3 b -d \right ) x \right )}{8 \left (3 b -d \right )}+\frac {\cosh \left (3 a +c +\left (3 b +d \right ) x \right )}{24 b +8 d} \]
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 = 2.27, size = 126, normalized size = 0.99 \[ \frac {{\mathrm {e}}^{c+d\,x}\,\left (3\,b^3\,{\mathrm {sinh}\left (a+b\,x\right )}^3+2\,b^2\,d\,{\mathrm {cosh}\left (a+b\,x\right )}^3-3\,b^2\,d\,\mathrm {cosh}\left (a+b\,x\right )\,{\mathrm {sinh}\left (a+b\,x\right )}^2-2\,b\,d^2\,{\mathrm {cosh}\left (a+b\,x\right )}^2\,\mathrm {sinh}\left (a+b\,x\right )-b\,d^2\,{\mathrm {sinh}\left (a+b\,x\right )}^3+d^3\,\mathrm {cosh}\left (a+b\,x\right )\,{\mathrm {sinh}\left (a+b\,x\right )}^2\right )}{9\,b^4-10\,b^2\,d^2+d^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 43.50, size = 972, normalized size = 7.65 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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