Optimal. Leaf size=37 \[ \frac {\, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {b \sinh ^n(c+d x)}{a}\right ) \sinh (c+d x)}{a d} \]
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Rubi [A]
time = 0.03, 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 = {3302, 251}
\begin {gather*} \frac {\sinh (c+d x) \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {b \sinh ^n(c+d x)}{a}\right )}{a d} \end {gather*}
Antiderivative was successfully verified.
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Rule 251
Rule 3302
Rubi steps
\begin {align*} \int \frac {\cosh (c+d x)}{a+b \sinh ^n(c+d x)} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{a+b x^n} \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac {\, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {b \sinh ^n(c+d x)}{a}\right ) \sinh (c+d x)}{a d}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 37, normalized size = 1.00 \begin {gather*} \frac {\, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {b \sinh ^n(c+d x)}{a}\right ) \sinh (c+d x)}{a d} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.42, size = 0, normalized size = 0.00 \[\int \frac {\cosh \left (d x +c \right )}{a +b \left (\sinh ^{n}\left (d x +c \right )\right )}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.45, size = 23, normalized size = 0.62 \begin {gather*} {\rm integral}\left (\frac {\cosh \left (d x + c\right )}{b \sinh \left (d x + c\right )^{n} + a}, x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.13, size = 38, normalized size = 1.03 \begin {gather*} \frac {\mathrm {sinh}\left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (1,\frac {1}{n};\ \frac {1}{n}+1;\ -\frac {b\,{\mathrm {sinh}\left (c+d\,x\right )}^n}{a}\right )}{a\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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