Optimal. Leaf size=70 \[ \frac{\cosh (c+d x) (b \sinh (c+d x))^{n+1} \, _2F_1\left (\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};-\sinh ^2(c+d x)\right )}{b d (n+1) \sqrt{\cosh ^2(c+d x)}} \]
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Rubi [A] time = 0.0193254, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {2643} \[ \frac{\cosh (c+d x) (b \sinh (c+d x))^{n+1} \, _2F_1\left (\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};-\sinh ^2(c+d x)\right )}{b d (n+1) \sqrt{\cosh ^2(c+d x)}} \]
Antiderivative was successfully verified.
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Rule 2643
Rubi steps
\begin{align*} \int (b \sinh (c+d x))^n \, dx &=\frac{\cosh (c+d x) \, _2F_1\left (\frac{1}{2},\frac{1+n}{2};\frac{3+n}{2};-\sinh ^2(c+d x)\right ) (b \sinh (c+d x))^{1+n}}{b d (1+n) \sqrt{\cosh ^2(c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.0450741, size = 65, normalized size = 0.93 \[ \frac{\sqrt{\cosh ^2(c+d x)} \tanh (c+d x) (b \sinh (c+d x))^n \, _2F_1\left (\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};-\sinh ^2(c+d x)\right )}{d (n+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.18, size = 0, normalized size = 0. \begin{align*} \int \left ( b\sinh \left ( dx+c \right ) \right ) ^{n}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (b \sinh \left (d x + c\right )\right )^{n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\left (b \sinh \left (d x + c\right )\right )^{n}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (b \sinh{\left (c + d x \right )}\right )^{n}\, dx \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 \left (b \sinh \left (d x + c\right )\right )^{n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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