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 = 0.04, antiderivative size = 37, normalized size of antiderivative = 1.00, 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 245
Rule 3223
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*}
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Mathematica [A] time = 0.01, size = 37, normalized size = 1.00 \[ \frac {\sinh (c+d x) \, _2F_1\left (2,\frac {1}{n};1+\frac {1}{n};-\frac {b \sinh ^n(c+d x)}{a}\right )}{a^2 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 2.03, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\cosh \left (d x + c\right )}{b^{2} \sinh \left (d x + c\right )^{2 \, n} + 2 \, a b \sinh \left (d x + c\right )^{n} + a^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cosh \left (d x + c\right )}{{\left (b \sinh \left (d x + c\right )^{n} + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 6.03, size = 0, normalized size = 0.00 \[ \int \frac {\cosh \left (d x +c \right )}{\left (a +b \left (\sinh ^{n}\left (d x +c \right )\right )\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {{\left (2^{n} e^{\left (c n + 2 \, d x + 2 \, c\right )} - 2^{n} e^{\left (c n\right )}\right )} e^{\left (d n x\right )}}{2 \, {\left (2^{n} a^{2} d n e^{\left (d n x + c n + d x + c\right )} + a b d n e^{\left (d x + n \log \left (e^{\left (d x + c\right )} + 1\right ) + n \log \left (e^{\left (d x + c\right )} - 1\right ) + c\right )}\right )}} + \frac {1}{2} \, \int \frac {{\left (2^{n} n e^{\left (c n\right )} - 2^{n} e^{\left (c n\right )} + {\left (2^{n} n e^{\left (c n\right )} - 2^{n} e^{\left (c n\right )}\right )} e^{\left (2 \, d x + 2 \, c\right )}\right )} e^{\left (d n x\right )}}{2^{n} a^{2} n e^{\left (d n x + c n + d x + c\right )} + a b n e^{\left (d x + n \log \left (e^{\left (d x + c\right )} + 1\right ) + n \log \left (e^{\left (d x + c\right )} - 1\right ) + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.95, size = 38, normalized size = 1.03 \[ \frac {\mathrm {sinh}\left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (2,\frac {1}{n};\ \frac {1}{n}+1;\ -\frac {b\,{\mathrm {sinh}\left (c+d\,x\right )}^n}{a}\right )}{a^2\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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