Optimal. Leaf size=74 \[ \frac {b \cosh (c+d x) (b \text {csch}(c+d x))^{n-1} \, _2F_1\left (\frac {1}{2},\frac {1-n}{2};\frac {3-n}{2};-\sinh ^2(c+d x)\right )}{d (1-n) \sqrt {\cosh ^2(c+d x)}} \]
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Rubi [A] time = 0.03, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3772, 2643} \[ \frac {b \cosh (c+d x) (b \text {csch}(c+d x))^{n-1} \, _2F_1\left (\frac {1}{2},\frac {1-n}{2};\frac {3-n}{2};-\sinh ^2(c+d x)\right )}{d (1-n) \sqrt {\cosh ^2(c+d x)}} \]
Antiderivative was successfully verified.
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Rule 2643
Rule 3772
Rubi steps
\begin {align*} \int (b \text {csch}(c+d x))^n \, dx &=(b \text {csch}(c+d x))^n \left (\frac {\sinh (c+d x)}{b}\right )^n \int \left (\frac {\sinh (c+d x)}{b}\right )^{-n} \, dx\\ &=\frac {\cosh (c+d x) (b \text {csch}(c+d x))^n \, _2F_1\left (\frac {1}{2},\frac {1-n}{2};\frac {3-n}{2};-\sinh ^2(c+d x)\right ) \sinh (c+d x)}{d (1-n) \sqrt {\cosh ^2(c+d x)}}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 67, normalized size = 0.91 \[ -\frac {\sinh (c+d x) \cosh (c+d x) \left (-\sinh ^2(c+d x)\right )^{\frac {n-1}{2}} (b \text {csch}(c+d x))^n \, _2F_1\left (\frac {1}{2},\frac {n+1}{2};\frac {3}{2};\cosh ^2(c+d x)\right )}{d} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.73, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (b \operatorname {csch}\left (d x + c\right )\right )^{n}, x\right ) \]
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 \[ \int \left (b \operatorname {csch}\left (d x + c\right )\right )^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.45, size = 0, normalized size = 0.00 \[ \int \left (b \,\mathrm {csch}\left (d x +c \right )\right )^{n}\, dx \]
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 \[ \int \left (b \operatorname {csch}\left (d x + c\right )\right )^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (\frac {b}{\mathrm {sinh}\left (c+d\,x\right )}\right )}^n \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (b \operatorname {csch}{\left (c + d x \right )}\right )^{n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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