Optimal. Leaf size=139 \[ \frac {b e^{a+b x} \sinh ^3(c+d x)}{b^2-9 d^2}-\frac {3 d e^{a+b x} \sinh ^2(c+d x) \cosh (c+d x)}{b^2-9 d^2}+\frac {6 b d^2 e^{a+b x} \sinh (c+d x)}{b^4-10 b^2 d^2+9 d^4}-\frac {6 d^3 e^{a+b x} \cosh (c+d x)}{b^4-10 b^2 d^2+9 d^4} \]
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Rubi [A] time = 0.06, antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {5476, 5474} \[ \frac {b e^{a+b x} \sinh ^3(c+d x)}{b^2-9 d^2}+\frac {6 b d^2 e^{a+b x} \sinh (c+d x)}{-10 b^2 d^2+b^4+9 d^4}-\frac {6 d^3 e^{a+b x} \cosh (c+d x)}{-10 b^2 d^2+b^4+9 d^4}-\frac {3 d e^{a+b x} \sinh ^2(c+d x) \cosh (c+d x)}{b^2-9 d^2} \]
Antiderivative was successfully verified.
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Rule 5474
Rule 5476
Rubi steps
\begin {align*} \int e^{a+b x} \sinh ^3(c+d x) \, dx &=-\frac {3 d e^{a+b x} \cosh (c+d x) \sinh ^2(c+d x)}{b^2-9 d^2}+\frac {b e^{a+b x} \sinh ^3(c+d x)}{b^2-9 d^2}+\frac {\left (6 d^2\right ) \int e^{a+b x} \sinh (c+d x) \, dx}{b^2-9 d^2}\\ &=-\frac {6 d^3 e^{a+b x} \cosh (c+d x)}{b^4-10 b^2 d^2+9 d^4}+\frac {6 b d^2 e^{a+b x} \sinh (c+d x)}{b^4-10 b^2 d^2+9 d^4}-\frac {3 d e^{a+b x} \cosh (c+d x) \sinh ^2(c+d x)}{b^2-9 d^2}+\frac {b e^{a+b x} \sinh ^3(c+d x)}{b^2-9 d^2}\\ \end {align*}
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Mathematica [A] time = 0.50, size = 108, normalized size = 0.78 \[ \frac {e^{a+b x} \left (\left (3 d^3-3 b^2 d\right ) \cosh (3 (c+d x))+3 d \left (b^2-9 d^2\right ) \cosh (c+d x)+2 b \sinh (c+d x) \left (\left (b^2-d^2\right ) \cosh (2 (c+d x))-b^2+13 d^2\right )\right )}{4 \left (b^4-10 b^2 d^2+9 d^4\right )} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.45, size = 316, normalized size = 2.27 \[ -\frac {3 \, {\left (b^{2} d - d^{3}\right )} \cosh \left (b x + a\right ) \cosh \left (d x + c\right )^{3} - {\left ({\left (b^{3} - b d^{2}\right )} \cosh \left (b x + a\right ) + {\left (b^{3} - b d^{2}\right )} \sinh \left (b x + a\right )\right )} \sinh \left (d x + c\right )^{3} - 3 \, {\left (b^{2} d - 9 \, d^{3}\right )} \cosh \left (b x + a\right ) \cosh \left (d x + c\right ) + 9 \, {\left ({\left (b^{2} d - d^{3}\right )} \cosh \left (b x + a\right ) \cosh \left (d x + c\right ) + {\left (b^{2} d - d^{3}\right )} \cosh \left (d x + c\right ) \sinh \left (b x + a\right )\right )} \sinh \left (d x + c\right )^{2} + 3 \, {\left ({\left (b^{2} d - d^{3}\right )} \cosh \left (d x + c\right )^{3} - {\left (b^{2} d - 9 \, d^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (b x + a\right ) - 3 \, {\left ({\left (b^{3} - b d^{2}\right )} \cosh \left (b x + a\right ) \cosh \left (d x + c\right )^{2} - {\left (b^{3} - 9 \, b d^{2}\right )} \cosh \left (b x + a\right ) - {\left (b^{3} - 9 \, b d^{2} - {\left (b^{3} - b d^{2}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (b x + a\right )\right )} \sinh \left (d x + c\right )}{4 \, {\left (b^{4} - 10 \, b^{2} d^{2} + 9 \, d^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 84, normalized size = 0.60 \[ \frac {e^{\left (b x + 3 \, d x + a + 3 \, c\right )}}{8 \, {\left (b + 3 \, d\right )}} - \frac {3 \, e^{\left (b x + d x + a + c\right )}}{8 \, {\left (b + d\right )}} + \frac {3 \, e^{\left (b x - d x + a - c\right )}}{8 \, {\left (b - d\right )}} - \frac {e^{\left (b x - 3 \, d x + a - 3 \, c\right )}}{8 \, {\left (b - 3 \, d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.33, size = 166, normalized size = 1.19 \[ -\frac {\sinh \left (a -3 c +\left (b -3 d \right ) x \right )}{8 \left (b -3 d \right )}+\frac {3 \sinh \left (a -c +\left (b -d \right ) x \right )}{8 \left (b -d \right )}-\frac {3 \sinh \left (a +c +\left (b +d \right ) x \right )}{8 \left (b +d \right )}+\frac {\sinh \left (a +3 c +\left (b +3 d \right ) x \right )}{8 b +24 d}-\frac {\cosh \left (a -3 c +\left (b -3 d \right ) x \right )}{8 \left (b -3 d \right )}+\frac {3 \cosh \left (a -c +\left (b -d \right ) x \right )}{8 \left (b -d \right )}-\frac {3 \cosh \left (a +c +\left (b +d \right ) x \right )}{8 \left (b +d \right )}+\frac {\cosh \left (a +3 c +\left (b +3 d \right ) x \right )}{8 b +24 d} \]
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 = 2.32, size = 127, normalized size = 0.91 \[ -\frac {{\mathrm {e}}^{a+b\,x}\,\left (-b^3\,{\mathrm {sinh}\left (c+d\,x\right )}^3+3\,b^2\,d\,\mathrm {cosh}\left (c+d\,x\right )\,{\mathrm {sinh}\left (c+d\,x\right )}^2-6\,b\,d^2\,{\mathrm {cosh}\left (c+d\,x\right )}^2\,\mathrm {sinh}\left (c+d\,x\right )+7\,b\,d^2\,{\mathrm {sinh}\left (c+d\,x\right )}^3+6\,d^3\,{\mathrm {cosh}\left (c+d\,x\right )}^3-9\,d^3\,\mathrm {cosh}\left (c+d\,x\right )\,{\mathrm {sinh}\left (c+d\,x\right )}^2\right )}{b^4-10\,b^2\,d^2+9\,d^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 44.61, size = 976, normalized size = 7.02 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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