Optimal. Leaf size=37 \[ \frac{\sinh (c+d x) \, _2F_1\left (2,\frac{1}{n};1+\frac{1}{n};-\frac{b \sinh ^n(c+d x)}{a}\right )}{a^2 d} \]
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Rubi [A] time = 0.0439955, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {3223, 245} \[ \frac{\sinh (c+d x) \, _2F_1\left (2,\frac{1}{n};1+\frac{1}{n};-\frac{b \sinh ^n(c+d x)}{a}\right )}{a^2 d} \]
Antiderivative was successfully verified.
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Rule 3223
Rule 245
Rubi steps
\begin{align*} \int \frac{\cosh (c+d x)}{\left (a+b \sinh ^n(c+d x)\right )^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\left (a+b x^n\right )^2} \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac{\, _2F_1\left (2,\frac{1}{n};1+\frac{1}{n};-\frac{b \sinh ^n(c+d x)}{a}\right ) \sinh (c+d x)}{a^2 d}\\ \end{align*}
Mathematica [A] time = 0.0084442, size = 37, normalized size = 1. \[ \frac{\sinh (c+d x) \, _2F_1\left (2,\frac{1}{n};1+\frac{1}{n};-\frac{b \sinh ^n(c+d x)}{a}\right )}{a^2 d} \]
Antiderivative was successfully verified.
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Maple [F] time = 5.571, size = 0, normalized size = 0. \begin{align*} \int{\frac{\cosh \left ( dx+c \right ) }{ \left ( a+b \left ( \sinh \left ( dx+c \right ) \right ) ^{n} \right ) ^{2}}}\, 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*} \frac{{\left (2^{n} e^{\left (c n + 2 \, d x + 2 \, c\right )} - 2^{n} e^{\left (c n\right )}\right )} e^{\left (d n x\right )}}{2 \,{\left (2^{n} a^{2} d n e^{\left (d n x + c n + d x + c\right )} + a b d n e^{\left (d x + n \log \left (e^{\left (d x + c\right )} + 1\right ) + n \log \left (e^{\left (d x + c\right )} - 1\right ) + c\right )}\right )}} + \frac{1}{2} \, \int \frac{{\left (2^{n} n e^{\left (c n\right )} - 2^{n} e^{\left (c n\right )} +{\left (2^{n} n e^{\left (c n\right )} - 2^{n} e^{\left (c n\right )}\right )} e^{\left (2 \, d x + 2 \, c\right )}\right )} e^{\left (d n x\right )}}{2^{n} a^{2} n e^{\left (d n x + c n + d x + c\right )} + a b n e^{\left (d x + n \log \left (e^{\left (d x + c\right )} + 1\right ) + n \log \left (e^{\left (d x + c\right )} - 1\right ) + c\right )}}\,{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 (\frac{\cosh \left (d x + c\right )}{b^{2} \sinh \left (d x + c\right )^{2 \, n} + 2 \, a b \sinh \left (d x + c\right )^{n} + a^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 \frac{\cosh \left (d x + c\right )}{{\left (b \sinh \left (d x + c\right )^{n} + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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