Integrand size = 13, antiderivative size = 121 \[ \int x^3 \coth ^{-1}(\tanh (a+b x))^n \, dx=\frac {x^3 \coth ^{-1}(\tanh (a+b x))^{1+n}}{b (1+n)}-\frac {3 x^2 \coth ^{-1}(\tanh (a+b x))^{2+n}}{b^2 (1+n) (2+n)}+\frac {6 x \coth ^{-1}(\tanh (a+b x))^{3+n}}{b^3 (1+n) (2+n) (3+n)}-\frac {6 \coth ^{-1}(\tanh (a+b x))^{4+n}}{b^4 (1+n) (2+n) (3+n) (4+n)} \]
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Time = 0.06 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2199, 2188, 30} \[ \int x^3 \coth ^{-1}(\tanh (a+b x))^n \, dx=-\frac {6 \coth ^{-1}(\tanh (a+b x))^{n+4}}{b^4 (n+1) (n+2) (n+3) (n+4)}+\frac {6 x \coth ^{-1}(\tanh (a+b x))^{n+3}}{b^3 (n+1) (n+2) (n+3)}-\frac {3 x^2 \coth ^{-1}(\tanh (a+b x))^{n+2}}{b^2 (n+1) (n+2)}+\frac {x^3 \coth ^{-1}(\tanh (a+b x))^{n+1}}{b (n+1)} \]
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Rule 30
Rule 2188
Rule 2199
Rubi steps \begin{align*} \text {integral}& = \frac {x^3 \coth ^{-1}(\tanh (a+b x))^{1+n}}{b (1+n)}-\frac {3 \int x^2 \coth ^{-1}(\tanh (a+b x))^{1+n} \, dx}{b (1+n)} \\ & = \frac {x^3 \coth ^{-1}(\tanh (a+b x))^{1+n}}{b (1+n)}-\frac {3 x^2 \coth ^{-1}(\tanh (a+b x))^{2+n}}{b^2 (1+n) (2+n)}+\frac {6 \int x \coth ^{-1}(\tanh (a+b x))^{2+n} \, dx}{b^2 (1+n) (2+n)} \\ & = \frac {x^3 \coth ^{-1}(\tanh (a+b x))^{1+n}}{b (1+n)}-\frac {3 x^2 \coth ^{-1}(\tanh (a+b x))^{2+n}}{b^2 (1+n) (2+n)}+\frac {6 x \coth ^{-1}(\tanh (a+b x))^{3+n}}{b^3 (1+n) (2+n) (3+n)}-\frac {6 \int \coth ^{-1}(\tanh (a+b x))^{3+n} \, dx}{b^3 (1+n) (2+n) (3+n)} \\ & = \frac {x^3 \coth ^{-1}(\tanh (a+b x))^{1+n}}{b (1+n)}-\frac {3 x^2 \coth ^{-1}(\tanh (a+b x))^{2+n}}{b^2 (1+n) (2+n)}+\frac {6 x \coth ^{-1}(\tanh (a+b x))^{3+n}}{b^3 (1+n) (2+n) (3+n)}-\frac {6 \text {Subst}\left (\int x^{3+n} \, dx,x,\coth ^{-1}(\tanh (a+b x))\right )}{b^4 (1+n) (2+n) (3+n)} \\ & = \frac {x^3 \coth ^{-1}(\tanh (a+b x))^{1+n}}{b (1+n)}-\frac {3 x^2 \coth ^{-1}(\tanh (a+b x))^{2+n}}{b^2 (1+n) (2+n)}+\frac {6 x \coth ^{-1}(\tanh (a+b x))^{3+n}}{b^3 (1+n) (2+n) (3+n)}-\frac {6 \coth ^{-1}(\tanh (a+b x))^{4+n}}{b^4 (1+n) (2+n) (3+n) (4+n)} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.88 \[ \int x^3 \coth ^{-1}(\tanh (a+b x))^n \, dx=\frac {\coth ^{-1}(\tanh (a+b x))^{1+n} \left (b^3 \left (24+26 n+9 n^2+n^3\right ) x^3-3 b^2 \left (12+7 n+n^2\right ) x^2 \coth ^{-1}(\tanh (a+b x))+6 b (4+n) x \coth ^{-1}(\tanh (a+b x))^2-6 \coth ^{-1}(\tanh (a+b x))^3\right )}{b^4 (1+n) (2+n) (3+n) (4+n)} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(276\) vs. \(2(121)=242\).
Time = 10.86 (sec) , antiderivative size = 277, normalized size of antiderivative = 2.29
method | result | size |
parallelrisch | \(-\frac {-24 \operatorname {arccoth}\left (\tanh \left (b x +a \right )\right )^{n} \operatorname {arccoth}\left (\tanh \left (b x +a \right )\right )^{3} x b -x^{3} \operatorname {arccoth}\left (\tanh \left (b x +a \right )\right ) \operatorname {arccoth}\left (\tanh \left (b x +a \right )\right )^{n} b^{3} n^{3}-9 x^{3} \operatorname {arccoth}\left (\tanh \left (b x +a \right )\right ) \operatorname {arccoth}\left (\tanh \left (b x +a \right )\right )^{n} b^{3} n^{2}-26 x^{3} \operatorname {arccoth}\left (\tanh \left (b x +a \right )\right ) \operatorname {arccoth}\left (\tanh \left (b x +a \right )\right )^{n} b^{3} n +3 x^{2} \operatorname {arccoth}\left (\tanh \left (b x +a \right )\right )^{2} \operatorname {arccoth}\left (\tanh \left (b x +a \right )\right )^{n} b^{2} n^{2}+21 x^{2} \operatorname {arccoth}\left (\tanh \left (b x +a \right )\right )^{2} \operatorname {arccoth}\left (\tanh \left (b x +a \right )\right )^{n} b^{2} n -6 x \operatorname {arccoth}\left (\tanh \left (b x +a \right )\right )^{3} \operatorname {arccoth}\left (\tanh \left (b x +a \right )\right )^{n} b n -24 \operatorname {arccoth}\left (\tanh \left (b x +a \right )\right )^{n} \operatorname {arccoth}\left (\tanh \left (b x +a \right )\right ) x^{3} b^{3}+36 \operatorname {arccoth}\left (\tanh \left (b x +a \right )\right )^{n} \operatorname {arccoth}\left (\tanh \left (b x +a \right )\right )^{2} x^{2} b^{2}+6 \operatorname {arccoth}\left (\tanh \left (b x +a \right )\right )^{n} \operatorname {arccoth}\left (\tanh \left (b x +a \right )\right )^{4}}{\left (n^{3}+6 n^{2}+11 n +6\right ) \left (4+n \right ) b^{4}}\) | \(277\) |
risch | \(\text {Expression too large to display}\) | \(129477\) |
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Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 498, normalized size of antiderivative = 4.12 \[ \int x^3 \coth ^{-1}(\tanh (a+b x))^n \, dx=\frac {{\left (48 \, a^{3} b n x + 8 \, {\left (b^{4} n^{3} + 6 \, b^{4} n^{2} + 11 \, b^{4} n + 6 \, b^{4}\right )} x^{4} - 3 \, \pi ^{4} - 6 i \, \pi ^{3} {\left (b n x - 4 \, a\right )} - 48 \, a^{4} + 8 \, {\left (a b^{3} n^{3} + 3 \, a b^{3} n^{2} + 2 \, a b^{3} n\right )} x^{3} - 6 \, \pi ^{2} {\left (6 \, a b n x - {\left (b^{2} n^{2} + b^{2} n\right )} x^{2} - 12 \, a^{2}\right )} - 24 \, {\left (a^{2} b^{2} n^{2} + a^{2} b^{2} n\right )} x^{2} + 4 i \, \pi {\left (18 \, a^{2} b n x + {\left (b^{3} n^{3} + 3 \, b^{3} n^{2} + 2 \, b^{3} n\right )} x^{3} - 24 \, a^{3} - 6 \, {\left (a b^{2} n^{2} + a b^{2} n\right )} x^{2}\right )}\right )} \cosh \left (n \log \left (\frac {1}{2} i \, \pi + b x + a\right )\right ) + {\left (48 \, a^{3} b n x + 8 \, {\left (b^{4} n^{3} + 6 \, b^{4} n^{2} + 11 \, b^{4} n + 6 \, b^{4}\right )} x^{4} - 3 \, \pi ^{4} - 6 i \, \pi ^{3} {\left (b n x - 4 \, a\right )} - 48 \, a^{4} + 8 \, {\left (a b^{3} n^{3} + 3 \, a b^{3} n^{2} + 2 \, a b^{3} n\right )} x^{3} - 6 \, \pi ^{2} {\left (6 \, a b n x - {\left (b^{2} n^{2} + b^{2} n\right )} x^{2} - 12 \, a^{2}\right )} - 24 \, {\left (a^{2} b^{2} n^{2} + a^{2} b^{2} n\right )} x^{2} + 4 i \, \pi {\left (18 \, a^{2} b n x + {\left (b^{3} n^{3} + 3 \, b^{3} n^{2} + 2 \, b^{3} n\right )} x^{3} - 24 \, a^{3} - 6 \, {\left (a b^{2} n^{2} + a b^{2} n\right )} x^{2}\right )}\right )} \sinh \left (n \log \left (\frac {1}{2} i \, \pi + b x + a\right )\right )}{8 \, {\left (b^{4} n^{4} + 10 \, b^{4} n^{3} + 35 \, b^{4} n^{2} + 50 \, b^{4} n + 24 \, b^{4}\right )}} \]
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\[ \int x^3 \coth ^{-1}(\tanh (a+b x))^n \, dx=\text {Too large to display} \]
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Result contains complex when optimal does not.
Time = 0.38 (sec) , antiderivative size = 257, normalized size of antiderivative = 2.12 \[ \int x^3 \coth ^{-1}(\tanh (a+b x))^n \, dx=\frac {{\left (8 \, {\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} b^{4} x^{4} - 3 \, \pi ^{4} - 24 i \, \pi ^{3} a + 72 \, \pi ^{2} a^{2} + 96 i \, \pi a^{3} - 48 \, a^{4} - 4 \, {\left (i \, \pi {\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} b^{3} - 2 \, {\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a b^{3}\right )} x^{3} + 6 \, {\left (\pi ^{2} {\left (n^{2} + n\right )} b^{2} + 4 i \, \pi {\left (n^{2} + n\right )} a b^{2} - 4 \, {\left (n^{2} + n\right )} a^{2} b^{2}\right )} x^{2} - 6 \, {\left (-i \, \pi ^{3} b n + 6 \, \pi ^{2} a b n + 12 i \, \pi a^{2} b n - 8 \, a^{3} b n\right )} x\right )} {\left (\cosh \left (-n \log \left (-i \, \pi + 2 \, b x + 2 \, a\right )\right ) - \sinh \left (-n \log \left (-i \, \pi + 2 \, b x + 2 \, a\right )\right )\right )}}{{\left (2^{n + 3} n^{4} + 5 \cdot 2^{n + 4} n^{3} + 35 \cdot 2^{n + 3} n^{2} + 25 \cdot 2^{n + 4} n + 3 \cdot 2^{n + 6}\right )} b^{4}} \]
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\[ \int x^3 \coth ^{-1}(\tanh (a+b x))^n \, dx=\int { x^{3} \operatorname {arcoth}\left (\tanh \left (b x + a\right )\right )^{n} \,d x } \]
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Time = 4.20 (sec) , antiderivative size = 418, normalized size of antiderivative = 3.45 \[ \int x^3 \coth ^{-1}(\tanh (a+b x))^n \, dx=-{\left (\frac {\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )}{2}-\frac {\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )}{2}\right )}^n\,\left (\frac {3\,{\left (\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+2\,b\,x\right )}^4}{8\,b^4\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}-\frac {x^4\,\left (n^3+6\,n^2+11\,n+6\right )}{n^4+10\,n^3+35\,n^2+50\,n+24}+\frac {3\,n\,x\,{\left (\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+2\,b\,x\right )}^3}{4\,b^3\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}+\frac {n\,x^3\,\left (\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+2\,b\,x\right )\,\left (n^2+3\,n+2\right )}{2\,b\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}+\frac {3\,n\,x^2\,\left (n+1\right )\,{\left (\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+2\,b\,x\right )}^2}{4\,b^2\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}\right ) \]
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