Optimal. Leaf size=39 \[ \frac{\sinh ^{n+1}(a+b x)}{b (n+1)}+\frac{\sinh ^{n+3}(a+b x)}{b (n+3)} \]
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Rubi [A] time = 0.0448459, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {2564, 14} \[ \frac{\sinh ^{n+1}(a+b x)}{b (n+1)}+\frac{\sinh ^{n+3}(a+b x)}{b (n+3)} \]
Antiderivative was successfully verified.
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Rule 2564
Rule 14
Rubi steps
\begin{align*} \int \cosh ^3(a+b x) \sinh ^n(a+b x) \, dx &=\frac{\operatorname{Subst}\left (\int x^n \left (1+x^2\right ) \, dx,x,\sinh (a+b x)\right )}{b}\\ &=\frac{\operatorname{Subst}\left (\int \left (x^n+x^{2+n}\right ) \, dx,x,\sinh (a+b x)\right )}{b}\\ &=\frac{\sinh ^{1+n}(a+b x)}{b (1+n)}+\frac{\sinh ^{3+n}(a+b x)}{b (3+n)}\\ \end{align*}
Mathematica [A] time = 0.0608808, size = 39, normalized size = 1. \[ \frac{\sinh ^{n+1}(a+b x)}{b (n+1)}+\frac{\sinh ^{n+3}(a+b x)}{b (n+3)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.5, size = 0, normalized size = 0. \begin{align*} \int \left ( \cosh \left ( bx+a \right ) \right ) ^{3} \left ( \sinh \left ( bx+a \right ) \right ) ^{n}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.83658, size = 504, normalized size = 12.92 \begin{align*} \frac{n e^{\left ({\left (b x + a\right )} n + 3 \, b x + n \log \left (e^{\left (-b x - a\right )} + 1\right ) + n \log \left (-e^{\left (-b x - a\right )} + 1\right ) + 3 \, a\right )}}{8 \,{\left (2^{n} n^{2} + 2^{n + 2} n + 3 \cdot 2^{n}\right )} b} + \frac{{\left (n + 9\right )} e^{\left ({\left (b x + a\right )} n + b x + n \log \left (e^{\left (-b x - a\right )} + 1\right ) + n \log \left (-e^{\left (-b x - a\right )} + 1\right ) + a\right )}}{8 \,{\left (2^{n} n^{2} + 2^{n + 2} n + 3 \cdot 2^{n}\right )} b} - \frac{{\left (n + 9\right )} e^{\left ({\left (b x + a\right )} n - b x + n \log \left (e^{\left (-b x - a\right )} + 1\right ) + n \log \left (-e^{\left (-b x - a\right )} + 1\right ) - a\right )}}{8 \,{\left (2^{n} n^{2} + 2^{n + 2} n + 3 \cdot 2^{n}\right )} b} - \frac{{\left (n + 1\right )} e^{\left ({\left (b x + a\right )} n - 3 \, b x + n \log \left (e^{\left (-b x - a\right )} + 1\right ) + n \log \left (-e^{\left (-b x - a\right )} + 1\right ) - 3 \, a\right )}}{8 \,{\left (2^{n} n^{2} + 2^{n + 2} n + 3 \cdot 2^{n}\right )} b} + \frac{e^{\left ({\left (b x + a\right )} n + 3 \, b x + n \log \left (e^{\left (-b x - a\right )} + 1\right ) + n \log \left (-e^{\left (-b x - a\right )} + 1\right ) + 3 \, a\right )}}{8 \,{\left (2^{n} n^{2} + 2^{n + 2} n + 3 \cdot 2^{n}\right )} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.26868, size = 482, normalized size = 12.36 \begin{align*} \frac{{\left ({\left (n + 1\right )} \sinh \left (b x + a\right )^{3} +{\left (3 \,{\left (n + 1\right )} \cosh \left (b x + a\right )^{2} + n + 9\right )} \sinh \left (b x + a\right )\right )} \cosh \left (n \log \left (\sinh \left (b x + a\right )\right )\right ) +{\left ({\left (n + 1\right )} \sinh \left (b x + a\right )^{3} +{\left (3 \,{\left (n + 1\right )} \cosh \left (b x + a\right )^{2} + n + 9\right )} \sinh \left (b x + a\right )\right )} \sinh \left (n \log \left (\sinh \left (b x + a\right )\right )\right )}{4 \,{\left ({\left (b n^{2} + 4 \, b n + 3 \, b\right )} \cosh \left (b x + a\right )^{4} - 2 \,{\left (b n^{2} + 4 \, b n + 3 \, b\right )} \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right )^{2} +{\left (b n^{2} + 4 \, b n + 3 \, b\right )} \sinh \left (b x + a\right )^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 14.3459, size = 668, normalized size = 17.13 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sinh \left (b x + a\right )^{n} \cosh \left (b x + a\right )^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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