Optimal. Leaf size=43 \[ \frac {\coth ^{1+n}(a+b x) \, _2F_1\left (1,\frac {1+n}{2};\frac {3+n}{2};\coth ^2(a+b x)\right )}{b (1+n)} \]
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Rubi [A]
time = 0.02, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3557, 371}
\begin {gather*} \frac {\coth ^{n+1}(a+b x) \, _2F_1\left (1,\frac {n+1}{2};\frac {n+3}{2};\coth ^2(a+b x)\right )}{b (n+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 371
Rule 3557
Rubi steps
\begin {align*} \int \coth ^n(a+b x) \, dx &=-\frac {\text {Subst}\left (\int \frac {x^n}{-1+x^2} \, dx,x,\coth (a+b x)\right )}{b}\\ &=\frac {\coth ^{1+n}(a+b x) \, _2F_1\left (1,\frac {1+n}{2};\frac {3+n}{2};\coth ^2(a+b x)\right )}{b (1+n)}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 45, normalized size = 1.05 \begin {gather*} \frac {\coth ^{1+n}(a+b x) \, _2F_1\left (1,\frac {1+n}{2};1+\frac {1+n}{2};\coth ^2(a+b x)\right )}{b (1+n)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 1.83, size = 0, normalized size = 0.00 \[\int \coth ^{n}\left (b x +a \right )\, dx\]
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 = 0.00, 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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \coth ^{n}{\left (a + b x \right )}\, dx \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 [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int {\mathrm {coth}\left (a+b\,x\right )}^n \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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