Integrand size = 13, antiderivative size = 165 \[ \int x^4 \coth ^{-1}(\tanh (a+b x))^n \, dx=\frac {x^4 \coth ^{-1}(\tanh (a+b x))^{1+n}}{b (1+n)}-\frac {4 x^3 \coth ^{-1}(\tanh (a+b x))^{2+n}}{b^2 (1+n) (2+n)}+\frac {12 x^2 \coth ^{-1}(\tanh (a+b x))^{3+n}}{b^3 (1+n) (2+n) (3+n)}-\frac {24 x \coth ^{-1}(\tanh (a+b x))^{4+n}}{b^4 (1+n) (2+n) (3+n) (4+n)}+\frac {24 \coth ^{-1}(\tanh (a+b x))^{5+n}}{b^5 (1+n) (2+n) (3+n) (4+n) (5+n)} \]
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Time = 0.11 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2199, 2188, 30} \[ \int x^4 \coth ^{-1}(\tanh (a+b x))^n \, dx=\frac {24 \coth ^{-1}(\tanh (a+b x))^{n+5}}{b^5 (n+1) (n+2) (n+3) (n+4) (n+5)}-\frac {24 x \coth ^{-1}(\tanh (a+b x))^{n+4}}{b^4 (n+1) (n+2) (n+3) (n+4)}+\frac {12 x^2 \coth ^{-1}(\tanh (a+b x))^{n+3}}{b^3 (n+1) (n+2) (n+3)}-\frac {4 x^3 \coth ^{-1}(\tanh (a+b x))^{n+2}}{b^2 (n+1) (n+2)}+\frac {x^4 \coth ^{-1}(\tanh (a+b x))^{n+1}}{b (n+1)} \]
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Rule 30
Rule 2188
Rule 2199
Rubi steps \begin{align*} \text {integral}& = \frac {x^4 \coth ^{-1}(\tanh (a+b x))^{1+n}}{b (1+n)}-\frac {4 \int x^3 \coth ^{-1}(\tanh (a+b x))^{1+n} \, dx}{b (1+n)} \\ & = \frac {x^4 \coth ^{-1}(\tanh (a+b x))^{1+n}}{b (1+n)}-\frac {4 x^3 \coth ^{-1}(\tanh (a+b x))^{2+n}}{b^2 (1+n) (2+n)}+\frac {12 \int x^2 \coth ^{-1}(\tanh (a+b x))^{2+n} \, dx}{b^2 (1+n) (2+n)} \\ & = \frac {x^4 \coth ^{-1}(\tanh (a+b x))^{1+n}}{b (1+n)}-\frac {4 x^3 \coth ^{-1}(\tanh (a+b x))^{2+n}}{b^2 (1+n) (2+n)}+\frac {12 x^2 \coth ^{-1}(\tanh (a+b x))^{3+n}}{b^3 (1+n) (2+n) (3+n)}-\frac {24 \int x \coth ^{-1}(\tanh (a+b x))^{3+n} \, dx}{b^3 (1+n) (2+n) (3+n)} \\ & = \frac {x^4 \coth ^{-1}(\tanh (a+b x))^{1+n}}{b (1+n)}-\frac {4 x^3 \coth ^{-1}(\tanh (a+b x))^{2+n}}{b^2 (1+n) (2+n)}+\frac {12 x^2 \coth ^{-1}(\tanh (a+b x))^{3+n}}{b^3 (1+n) (2+n) (3+n)}-\frac {24 x \coth ^{-1}(\tanh (a+b x))^{4+n}}{b^4 (1+n) (2+n) (3+n) (4+n)}+\frac {24 \int \coth ^{-1}(\tanh (a+b x))^{4+n} \, dx}{b^4 (1+n) (2+n) (3+n) (4+n)} \\ & = \frac {x^4 \coth ^{-1}(\tanh (a+b x))^{1+n}}{b (1+n)}-\frac {4 x^3 \coth ^{-1}(\tanh (a+b x))^{2+n}}{b^2 (1+n) (2+n)}+\frac {12 x^2 \coth ^{-1}(\tanh (a+b x))^{3+n}}{b^3 (1+n) (2+n) (3+n)}-\frac {24 x \coth ^{-1}(\tanh (a+b x))^{4+n}}{b^4 (1+n) (2+n) (3+n) (4+n)}+\frac {24 \text {Subst}\left (\int x^{4+n} \, dx,x,\coth ^{-1}(\tanh (a+b x))\right )}{b^5 (1+n) (2+n) (3+n) (4+n)} \\ & = \frac {x^4 \coth ^{-1}(\tanh (a+b x))^{1+n}}{b (1+n)}-\frac {4 x^3 \coth ^{-1}(\tanh (a+b x))^{2+n}}{b^2 (1+n) (2+n)}+\frac {12 x^2 \coth ^{-1}(\tanh (a+b x))^{3+n}}{b^3 (1+n) (2+n) (3+n)}-\frac {24 x \coth ^{-1}(\tanh (a+b x))^{4+n}}{b^4 (1+n) (2+n) (3+n) (4+n)}+\frac {24 \coth ^{-1}(\tanh (a+b x))^{5+n}}{b^5 (1+n) (2+n) (3+n) (4+n) (5+n)} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.88 \[ \int x^4 \coth ^{-1}(\tanh (a+b x))^n \, dx=\frac {\coth ^{-1}(\tanh (a+b x))^{1+n} \left (b^4 \left (120+154 n+71 n^2+14 n^3+n^4\right ) x^4-4 b^3 \left (60+47 n+12 n^2+n^3\right ) x^3 \coth ^{-1}(\tanh (a+b x))+12 b^2 \left (20+9 n+n^2\right ) x^2 \coth ^{-1}(\tanh (a+b x))^2-24 b (5+n) x \coth ^{-1}(\tanh (a+b x))^3+24 \coth ^{-1}(\tanh (a+b x))^4\right )}{b^5 (1+n) (2+n) (3+n) (4+n) (5+n)} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(419\) vs. \(2(165)=330\).
Time = 16.41 (sec) , antiderivative size = 420, normalized size of antiderivative = 2.55
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 )^{5}-120 \operatorname {arccoth}\left (\tanh \left (b x +a \right )\right )^{n} \operatorname {arccoth}\left (\tanh \left (b x +a \right )\right ) x^{4} b^{4}+240 \operatorname {arccoth}\left (\tanh \left (b x +a \right )\right )^{n} \operatorname {arccoth}\left (\tanh \left (b x +a \right )\right )^{2} x^{3} b^{3}-240 \operatorname {arccoth}\left (\tanh \left (b x +a \right )\right )^{n} \operatorname {arccoth}\left (\tanh \left (b x +a \right )\right )^{3} x^{2} b^{2}+120 \operatorname {arccoth}\left (\tanh \left (b x +a \right )\right )^{n} \operatorname {arccoth}\left (\tanh \left (b x +a \right )\right )^{4} x b +48 x^{3} \operatorname {arccoth}\left (\tanh \left (b x +a \right )\right )^{2} \operatorname {arccoth}\left (\tanh \left (b x +a \right )\right )^{n} b^{3} n^{2}+188 x^{3} \operatorname {arccoth}\left (\tanh \left (b x +a \right )\right )^{2} \operatorname {arccoth}\left (\tanh \left (b x +a \right )\right )^{n} b^{3} n -12 x^{2} \operatorname {arccoth}\left (\tanh \left (b x +a \right )\right )^{3} \operatorname {arccoth}\left (\tanh \left (b x +a \right )\right )^{n} b^{2} n^{2}-108 x^{2} \operatorname {arccoth}\left (\tanh \left (b x +a \right )\right )^{3} \operatorname {arccoth}\left (\tanh \left (b x +a \right )\right )^{n} b^{2} n +24 x \operatorname {arccoth}\left (\tanh \left (b x +a \right )\right )^{4} \operatorname {arccoth}\left (\tanh \left (b x +a \right )\right )^{n} b n -14 x^{4} \operatorname {arccoth}\left (\tanh \left (b x +a \right )\right ) \operatorname {arccoth}\left (\tanh \left (b x +a \right )\right )^{n} b^{4} n^{3}-71 x^{4} \operatorname {arccoth}\left (\tanh \left (b x +a \right )\right ) \operatorname {arccoth}\left (\tanh \left (b x +a \right )\right )^{n} b^{4} n^{2}+4 x^{3} \operatorname {arccoth}\left (\tanh \left (b x +a \right )\right )^{2} \operatorname {arccoth}\left (\tanh \left (b x +a \right )\right )^{n} b^{3} n^{3}-154 x^{4} \operatorname {arccoth}\left (\tanh \left (b x +a \right )\right ) \operatorname {arccoth}\left (\tanh \left (b x +a \right )\right )^{n} b^{4} n -x^{4} \operatorname {arccoth}\left (\tanh \left (b x +a \right )\right ) \operatorname {arccoth}\left (\tanh \left (b x +a \right )\right )^{n} b^{4} n^{4}}{b^{5} \left (n^{5}+15 n^{4}+85 n^{3}+225 n^{2}+274 n +120\right )}\) | \(420\) |
risch | \(\text {Expression too large to display}\) | \(504228\) |
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Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 828, normalized size of antiderivative = 5.02 \[ \int x^4 \coth ^{-1}(\tanh (a+b x))^n \, dx=-\frac {{\left (96 \, a^{4} b n x - 4 \, {\left (b^{5} n^{4} + 10 \, b^{5} n^{3} + 35 \, b^{5} n^{2} + 50 \, b^{5} n + 24 \, b^{5}\right )} x^{5} - 3 i \, \pi ^{5} + 6 \, \pi ^{4} {\left (b n x - 5 \, a\right )} - 96 \, a^{5} - 4 \, {\left (a b^{4} n^{4} + 6 \, a b^{4} n^{3} + 11 \, a b^{4} n^{2} + 6 \, a b^{4} n\right )} x^{4} - 6 i \, \pi ^{3} {\left (8 \, a b n x - {\left (b^{2} n^{2} + b^{2} n\right )} x^{2} - 20 \, a^{2}\right )} + 16 \, {\left (a^{2} b^{3} n^{3} + 3 \, a^{2} b^{3} n^{2} + 2 \, a^{2} b^{3} n\right )} x^{3} - 4 \, \pi ^{2} {\left (36 \, a^{2} b n x + {\left (b^{3} n^{3} + 3 \, b^{3} n^{2} + 2 \, b^{3} n\right )} x^{3} - 60 \, a^{3} - 9 \, {\left (a b^{2} n^{2} + a b^{2} n\right )} x^{2}\right )} - 48 \, {\left (a^{3} b^{2} n^{2} + a^{3} b^{2} n\right )} x^{2} + 2 i \, \pi {\left (96 \, a^{3} b n x - {\left (b^{4} n^{4} + 6 \, b^{4} n^{3} + 11 \, b^{4} n^{2} + 6 \, b^{4} n\right )} x^{4} - 120 \, a^{4} + 8 \, {\left (a b^{3} n^{3} + 3 \, a b^{3} n^{2} + 2 \, a b^{3} n\right )} x^{3} - 36 \, {\left (a^{2} b^{2} n^{2} + a^{2} b^{2} n\right )} x^{2}\right )}\right )} \cosh \left (n \log \left (\frac {1}{2} i \, \pi + b x + a\right )\right ) + {\left (96 \, a^{4} b n x - 4 \, {\left (b^{5} n^{4} + 10 \, b^{5} n^{3} + 35 \, b^{5} n^{2} + 50 \, b^{5} n + 24 \, b^{5}\right )} x^{5} - 3 i \, \pi ^{5} + 6 \, \pi ^{4} {\left (b n x - 5 \, a\right )} - 96 \, a^{5} - 4 \, {\left (a b^{4} n^{4} + 6 \, a b^{4} n^{3} + 11 \, a b^{4} n^{2} + 6 \, a b^{4} n\right )} x^{4} - 6 i \, \pi ^{3} {\left (8 \, a b n x - {\left (b^{2} n^{2} + b^{2} n\right )} x^{2} - 20 \, a^{2}\right )} + 16 \, {\left (a^{2} b^{3} n^{3} + 3 \, a^{2} b^{3} n^{2} + 2 \, a^{2} b^{3} n\right )} x^{3} - 4 \, \pi ^{2} {\left (36 \, a^{2} b n x + {\left (b^{3} n^{3} + 3 \, b^{3} n^{2} + 2 \, b^{3} n\right )} x^{3} - 60 \, a^{3} - 9 \, {\left (a b^{2} n^{2} + a b^{2} n\right )} x^{2}\right )} - 48 \, {\left (a^{3} b^{2} n^{2} + a^{3} b^{2} n\right )} x^{2} + 2 i \, \pi {\left (96 \, a^{3} b n x - {\left (b^{4} n^{4} + 6 \, b^{4} n^{3} + 11 \, b^{4} n^{2} + 6 \, b^{4} n\right )} x^{4} - 120 \, a^{4} + 8 \, {\left (a b^{3} n^{3} + 3 \, a b^{3} n^{2} + 2 \, a b^{3} n\right )} x^{3} - 36 \, {\left (a^{2} b^{2} n^{2} + a^{2} b^{2} n\right )} x^{2}\right )}\right )} \sinh \left (n \log \left (\frac {1}{2} i \, \pi + b x + a\right )\right )}{4 \, {\left (b^{5} n^{5} + 15 \, b^{5} n^{4} + 85 \, b^{5} n^{3} + 225 \, b^{5} n^{2} + 274 \, b^{5} n + 120 \, b^{5}\right )}} \]
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\[ \int x^4 \coth ^{-1}(\tanh (a+b x))^n \, dx=\text {Too large to display} \]
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Result contains complex when optimal does not.
Time = 0.38 (sec) , antiderivative size = 380, normalized size of antiderivative = 2.30 \[ \int x^4 \coth ^{-1}(\tanh (a+b x))^n \, dx=\frac {{\left (4 \, {\left (n^{4} + 10 \, n^{3} + 35 \, n^{2} + 50 \, n + 24\right )} b^{5} x^{5} - 3 i \, \pi ^{5} + 30 \, \pi ^{4} a + 120 i \, \pi ^{3} a^{2} - 240 \, \pi ^{2} a^{3} - 240 i \, \pi a^{4} + 96 \, a^{5} - 2 \, {\left (i \, \pi {\left (n^{4} + 6 \, n^{3} + 11 \, n^{2} + 6 \, n\right )} b^{4} - 2 \, {\left (n^{4} + 6 \, n^{3} + 11 \, n^{2} + 6 \, n\right )} a b^{4}\right )} x^{4} + 4 \, {\left (\pi ^{2} {\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} b^{3} + 4 i \, \pi {\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a b^{3} - 4 \, {\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a^{2} b^{3}\right )} x^{3} - 6 \, {\left (-i \, \pi ^{3} {\left (n^{2} + n\right )} b^{2} + 6 \, \pi ^{2} {\left (n^{2} + n\right )} a b^{2} + 12 i \, \pi {\left (n^{2} + n\right )} a^{2} b^{2} - 8 \, {\left (n^{2} + n\right )} a^{3} b^{2}\right )} x^{2} - 6 \, {\left (\pi ^{4} b n + 8 i \, \pi ^{3} a b n - 24 \, \pi ^{2} a^{2} b n - 32 i \, \pi a^{3} b n + 16 \, a^{4} b n\right )} x\right )} {\left (\cosh \left (-n \log \left (-i \, \pi + 2 \, b x + 2 \, a\right )\right ) - \sinh \left (-n \log \left (-i \, \pi + 2 \, b x + 2 \, a\right )\right )\right )}}{{\left (2^{n + 2} n^{5} + 15 \cdot 2^{n + 2} n^{4} + 85 \cdot 2^{n + 2} n^{3} + 225 \cdot 2^{n + 2} n^{2} + 137 \cdot 2^{n + 3} n + 15 \cdot 2^{n + 5}\right )} b^{5}} \]
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\[ \int x^4 \coth ^{-1}(\tanh (a+b x))^n \, dx=\int { x^{4} \operatorname {arcoth}\left (\tanh \left (b x + a\right )\right )^{n} \,d x } \]
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Time = 4.82 (sec) , antiderivative size = 546, normalized size of antiderivative = 3.31 \[ \int x^4 \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 )}^5}{4\,b^5\,\left (n^5+15\,n^4+85\,n^3+225\,n^2+274\,n+120\right )}-\frac {x^5\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}{n^5+15\,n^4+85\,n^3+225\,n^2+274\,n+120}+\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 )}^4}{2\,b^4\,\left (n^5+15\,n^4+85\,n^3+225\,n^2+274\,n+120\right )}+\frac {n\,x^4\,\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^3+6\,n^2+11\,n+6\right )}{2\,b\,\left (n^5+15\,n^4+85\,n^3+225\,n^2+274\,n+120\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 )}^3}{2\,b^3\,\left (n^5+15\,n^4+85\,n^3+225\,n^2+274\,n+120\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 )}^2\,\left (n^2+3\,n+2\right )}{b^2\,\left (n^5+15\,n^4+85\,n^3+225\,n^2+274\,n+120\right )}\right ) \]
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