Optimal. Leaf size=57 \[ \frac {\coth (c+d x) \left (b \coth ^4(c+d x)\right )^n \, _2F_1\left (1,\frac {1}{2} (4 n+1);\frac {1}{2} (4 n+3);\coth ^2(c+d x)\right )}{d (4 n+1)} \]
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Rubi [A] time = 0.04, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3658, 3476, 364} \[ \frac {\coth (c+d x) \left (b \coth ^4(c+d x)\right )^n \, _2F_1\left (1,\frac {1}{2} (4 n+1);\frac {1}{2} (4 n+3);\coth ^2(c+d x)\right )}{d (4 n+1)} \]
Antiderivative was successfully verified.
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Rule 364
Rule 3476
Rule 3658
Rubi steps
\begin {align*} \int \left (b \coth ^4(c+d x)\right )^n \, dx &=\left (\coth ^{-4 n}(c+d x) \left (b \coth ^4(c+d x)\right )^n\right ) \int \coth ^{4 n}(c+d x) \, dx\\ &=-\frac {\left (\coth ^{-4 n}(c+d x) \left (b \coth ^4(c+d x)\right )^n\right ) \operatorname {Subst}\left (\int \frac {x^{4 n}}{-1+x^2} \, dx,x,\coth (c+d x)\right )}{d}\\ &=\frac {\coth (c+d x) \left (b \coth ^4(c+d x)\right )^n \, _2F_1\left (1,\frac {1}{2} (1+4 n);\frac {1}{2} (3+4 n);\coth ^2(c+d x)\right )}{d (1+4 n)}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 51, normalized size = 0.89 \[ \frac {\coth (c+d x) \left (b \coth ^4(c+d x)\right )^n \, _2F_1\left (1,2 n+\frac {1}{2};2 n+\frac {3}{2};\coth ^2(c+d x)\right )}{4 d n+d} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.61, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (b \coth \left (d x + c\right )^{4}\right )^{n}, x\right ) \]
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 \[ \int \left (b \coth \left (d x + c\right )^{4}\right )^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.66, size = 0, normalized size = 0.00 \[ \int \left (b \left (\coth ^{4}\left (d x +c \right )\right )\right )^{n}\, dx \]
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 \[ \int \left (b \coth \left (d x + c\right )^{4}\right )^{n}\,{d x} \]
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 \[ \int {\left (b\,{\mathrm {coth}\left (c+d\,x\right )}^4\right )}^n \,d x \]
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 \[ \int \left (b \coth ^{4}{\left (c + d x \right )}\right )^{n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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