Optimal. Leaf size=27 \[ \frac {\cosh ^3(a+b x)}{3 b}-\frac {\cosh (a+b x)}{b} \]
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Rubi [A] time = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2633} \[ \frac {\cosh ^3(a+b x)}{3 b}-\frac {\cosh (a+b x)}{b} \]
Antiderivative was successfully verified.
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Rule 2633
Rubi steps
\begin {align*} \int \sinh ^3(a+b x) \, dx &=-\frac {\operatorname {Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cosh (a+b x)\right )}{b}\\ &=-\frac {\cosh (a+b x)}{b}+\frac {\cosh ^3(a+b x)}{3 b}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 29, normalized size = 1.07 \[ \frac {\cosh (3 (a+b x))}{12 b}-\frac {3 \cosh (a+b x)}{4 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 38, normalized size = 1.41 \[ \frac {\cosh \left (b x + a\right )^{3} + 3 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} - 9 \, \cosh \left (b x + a\right )}{12 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.12, size = 54, normalized size = 2.00 \[ \frac {e^{\left (3 \, b x + 3 \, a\right )}}{24 \, b} - \frac {3 \, e^{\left (b x + a\right )}}{8 \, b} - \frac {3 \, e^{\left (-b x - a\right )}}{8 \, b} + \frac {e^{\left (-3 \, b x - 3 \, a\right )}}{24 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 23, normalized size = 0.85 \[ \frac {\left (-\frac {2}{3}+\frac {\left (\sinh ^{2}\left (b x +a \right )\right )}{3}\right ) \cosh \left (b x +a \right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.72, size = 54, normalized size = 2.00 \[ \frac {e^{\left (3 \, b x + 3 \, a\right )}}{24 \, b} - \frac {3 \, e^{\left (b x + a\right )}}{8 \, b} - \frac {3 \, e^{\left (-b x - a\right )}}{8 \, b} + \frac {e^{\left (-3 \, b x - 3 \, a\right )}}{24 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 24, normalized size = 0.89 \[ -\frac {3\,\mathrm {cosh}\left (a+b\,x\right )-{\mathrm {cosh}\left (a+b\,x\right )}^3}{3\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.40, size = 36, normalized size = 1.33 \[ \begin {cases} \frac {\sinh ^{2}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{b} - \frac {2 \cosh ^{3}{\left (a + b x \right )}}{3 b} & \text {for}\: b \neq 0 \\x \sinh ^{3}{\relax (a )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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