Optimal. Leaf size=15 \[ \frac {\cosh ^4(a+b x)}{4 b} \]
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Rubi [A] time = 0.02, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2565, 30} \[ \frac {\cosh ^4(a+b x)}{4 b} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2565
Rubi steps
\begin {align*} \int \cosh ^3(a+b x) \sinh (a+b x) \, dx &=\frac {\operatorname {Subst}\left (\int x^3 \, dx,x,\cosh (a+b x)\right )}{b}\\ &=\frac {\cosh ^4(a+b x)}{4 b}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 15, normalized size = 1.00 \[ \frac {\cosh ^4(a+b x)}{4 b} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.02, size = 54, normalized size = 3.60 \[ \frac {\cosh \left (b x + a\right )^{4} + \sinh \left (b x + a\right )^{4} + 2 \, {\left (3 \, \cosh \left (b x + a\right )^{2} + 2\right )} \sinh \left (b x + a\right )^{2} + 4 \, \cosh \left (b x + a\right )^{2}}{32 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.14, size = 57, normalized size = 3.80 \[ \frac {e^{\left (4 \, b x + 4 \, a\right )}}{64 \, b} + \frac {e^{\left (2 \, b x + 2 \, a\right )}}{16 \, b} + \frac {e^{\left (-2 \, b x - 2 \, a\right )}}{16 \, b} + \frac {e^{\left (-4 \, b x - 4 \, a\right )}}{64 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 14, normalized size = 0.93 \[ \frac {\cosh ^{4}\left (b x +a \right )}{4 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 13, normalized size = 0.87 \[ \frac {\cosh \left (b x + a\right )^{4}}{4 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.49, size = 13, normalized size = 0.87 \[ \frac {{\mathrm {cosh}\left (a+b\,x\right )}^4}{4\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.74, size = 20, normalized size = 1.33 \[ \begin {cases} \frac {\cosh ^{4}{\left (a + b x \right )}}{4 b} & \text {for}\: b \neq 0 \\x \sinh {\relax (a )} \cosh ^{3}{\relax (a )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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