Optimal. Leaf size=48 \[ \frac {(b \coth (c+d x))^{n+1} \, _2F_1\left (1,\frac {n+1}{2};\frac {n+3}{2};\coth ^2(c+d x)\right )}{b d (n+1)} \]
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Rubi [A] time = 0.03, antiderivative size = 48, 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 = {3476, 364} \[ \frac {(b \coth (c+d x))^{n+1} \, _2F_1\left (1,\frac {n+1}{2};\frac {n+3}{2};\coth ^2(c+d x)\right )}{b d (n+1)} \]
Antiderivative was successfully verified.
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Rule 364
Rule 3476
Rubi steps
\begin {align*} \int (b \coth (c+d x))^n \, dx &=-\frac {b \operatorname {Subst}\left (\int \frac {x^n}{-b^2+x^2} \, dx,x,b \coth (c+d x)\right )}{d}\\ &=\frac {(b \coth (c+d x))^{1+n} \, _2F_1\left (1,\frac {1+n}{2};\frac {3+n}{2};\coth ^2(c+d x)\right )}{b d (1+n)}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 51, normalized size = 1.06 \[ \frac {\coth (c+d x) (b \coth (c+d x))^n \, _2F_1\left (1,\frac {n+1}{2};\frac {n+1}{2}+1;\coth ^2(c+d x)\right )}{d (n+1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (b \coth \left (d x + c\right )\right )^{n}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-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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maple [F] time = 0.47, size = 0, normalized size = 0.00 \[ \int \left (b \coth \left (d x +c \right )\right )^{n}\, dx \]
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 \[ \int \left (b \coth \left (d x + c\right )\right )^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int {\left (b\,\mathrm {coth}\left (c+d\,x\right )\right )}^n \,d x \]
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 \[ \int \left (b \coth {\left (c + d x \right )}\right )^{n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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