Integrand size = 14, antiderivative size = 114 \[ \int \left (a+b \text {csch}^2(c+d x)\right )^4 \, dx=a^4 x-\frac {(2 a-b) b \left (2 a^2-2 a b+b^2\right ) \coth (c+d x)}{d}-\frac {b^2 \left (6 a^2-8 a b+3 b^2\right ) \coth ^3(c+d x)}{3 d}-\frac {(4 a-3 b) b^3 \coth ^5(c+d x)}{5 d}-\frac {b^4 \coth ^7(c+d x)}{7 d} \]
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Time = 0.05 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {4213, 398, 212} \[ \int \left (a+b \text {csch}^2(c+d x)\right )^4 \, dx=a^4 x-\frac {b^2 \left (6 a^2-8 a b+3 b^2\right ) \coth ^3(c+d x)}{3 d}-\frac {b (2 a-b) \left (2 a^2-2 a b+b^2\right ) \coth (c+d x)}{d}-\frac {b^3 (4 a-3 b) \coth ^5(c+d x)}{5 d}-\frac {b^4 \coth ^7(c+d x)}{7 d} \]
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Rule 212
Rule 398
Rule 4213
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\left (a-b+b x^2\right )^4}{1-x^2} \, dx,x,\coth (c+d x)\right )}{d} \\ & = \frac {\text {Subst}\left (\int \left (-\left ((2 a-b) b \left (2 a^2-2 a b+b^2\right )\right )-b^2 \left (6 a^2-8 a b+3 b^2\right ) x^2-(4 a-3 b) b^3 x^4-b^4 x^6+\frac {a^4}{1-x^2}\right ) \, dx,x,\coth (c+d x)\right )}{d} \\ & = -\frac {(2 a-b) b \left (2 a^2-2 a b+b^2\right ) \coth (c+d x)}{d}-\frac {b^2 \left (6 a^2-8 a b+3 b^2\right ) \coth ^3(c+d x)}{3 d}-\frac {(4 a-3 b) b^3 \coth ^5(c+d x)}{5 d}-\frac {b^4 \coth ^7(c+d x)}{7 d}+\frac {a^4 \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\coth (c+d x)\right )}{d} \\ & = a^4 x-\frac {(2 a-b) b \left (2 a^2-2 a b+b^2\right ) \coth (c+d x)}{d}-\frac {b^2 \left (6 a^2-8 a b+3 b^2\right ) \coth ^3(c+d x)}{3 d}-\frac {(4 a-3 b) b^3 \coth ^5(c+d x)}{5 d}-\frac {b^4 \coth ^7(c+d x)}{7 d} \\ \end{align*}
Time = 3.45 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.31 \[ \int \left (a+b \text {csch}^2(c+d x)\right )^4 \, dx=\frac {16 \left (a+b \text {csch}^2(c+d x)\right )^4 \left (105 a^4 (c+d x)-b \coth (c+d x) \left (420 a^3-420 a^2 b+224 a b^2-48 b^3+2 b \left (105 a^2-56 a b+12 b^2\right ) \text {csch}^2(c+d x)+6 (14 a-3 b) b^2 \text {csch}^4(c+d x)+15 b^3 \text {csch}^6(c+d x)\right )\right ) \sinh ^8(c+d x)}{105 d (a-2 b-a \cosh (2 (c+d x)))^4} \]
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Time = 2.01 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.13
method | result | size |
derivativedivides | \(\frac {a^{4} \left (d x +c \right )-4 a^{3} b \coth \left (d x +c \right )+6 a^{2} b^{2} \left (\frac {2}{3}-\frac {\operatorname {csch}\left (d x +c \right )^{2}}{3}\right ) \coth \left (d x +c \right )+4 a \,b^{3} \left (-\frac {8}{15}-\frac {\operatorname {csch}\left (d x +c \right )^{4}}{5}+\frac {4 \operatorname {csch}\left (d x +c \right )^{2}}{15}\right ) \coth \left (d x +c \right )+b^{4} \left (\frac {16}{35}-\frac {\operatorname {csch}\left (d x +c \right )^{6}}{7}+\frac {6 \operatorname {csch}\left (d x +c \right )^{4}}{35}-\frac {8 \operatorname {csch}\left (d x +c \right )^{2}}{35}\right ) \coth \left (d x +c \right )}{d}\) | \(129\) |
default | \(\frac {a^{4} \left (d x +c \right )-4 a^{3} b \coth \left (d x +c \right )+6 a^{2} b^{2} \left (\frac {2}{3}-\frac {\operatorname {csch}\left (d x +c \right )^{2}}{3}\right ) \coth \left (d x +c \right )+4 a \,b^{3} \left (-\frac {8}{15}-\frac {\operatorname {csch}\left (d x +c \right )^{4}}{5}+\frac {4 \operatorname {csch}\left (d x +c \right )^{2}}{15}\right ) \coth \left (d x +c \right )+b^{4} \left (\frac {16}{35}-\frac {\operatorname {csch}\left (d x +c \right )^{6}}{7}+\frac {6 \operatorname {csch}\left (d x +c \right )^{4}}{35}-\frac {8 \operatorname {csch}\left (d x +c \right )^{2}}{35}\right ) \coth \left (d x +c \right )}{d}\) | \(129\) |
parts | \(x \,a^{4}+\frac {b^{4} \left (\frac {16}{35}-\frac {\operatorname {csch}\left (d x +c \right )^{6}}{7}+\frac {6 \operatorname {csch}\left (d x +c \right )^{4}}{35}-\frac {8 \operatorname {csch}\left (d x +c \right )^{2}}{35}\right ) \coth \left (d x +c \right )}{d}-\frac {4 a^{3} b \coth \left (d x +c \right )}{d}+\frac {6 a^{2} b^{2} \left (\frac {2}{3}-\frac {\operatorname {csch}\left (d x +c \right )^{2}}{3}\right ) \coth \left (d x +c \right )}{d}+\frac {4 a \,b^{3} \left (-\frac {8}{15}-\frac {\operatorname {csch}\left (d x +c \right )^{4}}{5}+\frac {4 \operatorname {csch}\left (d x +c \right )^{2}}{15}\right ) \coth \left (d x +c \right )}{d}\) | \(133\) |
parallelrisch | \(\frac {-15 \coth \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} b^{4}-336 \left (a -\frac {7 b}{16}\right ) b^{3} \coth \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+\left (-3360 a^{2} b^{2}+2800 a \,b^{3}-735 b^{4}\right ) \coth \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (-26880 a^{3} b +30240 a^{2} b^{2}-16800 a \,b^{3}+3675 b^{4}\right ) \coth \left (\frac {d x}{2}+\frac {c}{2}\right )-15 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} b^{4}-336 \left (a -\frac {7 b}{16}\right ) b^{3} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+\left (-3360 a^{2} b^{2}+2800 a \,b^{3}-735 b^{4}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (-26880 a^{3} b +30240 a^{2} b^{2}-16800 a \,b^{3}+3675 b^{4}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+13440 a^{4} d x}{13440 d}\) | \(224\) |
risch | \(x \,a^{4}-\frac {8 b \left (105 a^{3} {\mathrm e}^{12 d x +12 c}-630 a^{3} {\mathrm e}^{10 d x +10 c}+315 a^{2} b \,{\mathrm e}^{10 d x +10 c}+1575 a^{3} {\mathrm e}^{8 d x +8 c}-1365 a^{2} b \,{\mathrm e}^{8 d x +8 c}+560 a \,b^{2} {\mathrm e}^{8 d x +8 c}-2100 a^{3} {\mathrm e}^{6 d x +6 c}+2310 a^{2} b \,{\mathrm e}^{6 d x +6 c}-1400 a \,b^{2} {\mathrm e}^{6 d x +6 c}+420 \,{\mathrm e}^{6 d x +6 c} b^{3}+1575 a^{3} {\mathrm e}^{4 d x +4 c}-1890 a^{2} b \,{\mathrm e}^{4 d x +4 c}+1176 a \,b^{2} {\mathrm e}^{4 d x +4 c}-252 \,{\mathrm e}^{4 d x +4 c} b^{3}-630 a^{3} {\mathrm e}^{2 d x +2 c}+735 a^{2} b \,{\mathrm e}^{2 d x +2 c}-392 \,{\mathrm e}^{2 d x +2 c} a \,b^{2}+84 \,{\mathrm e}^{2 d x +2 c} b^{3}+105 a^{3}-105 a^{2} b +56 a \,b^{2}-12 b^{3}\right )}{105 d \left ({\mathrm e}^{2 d x +2 c}-1\right )^{7}}\) | \(310\) |
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Leaf count of result is larger than twice the leaf count of optimal. 929 vs. \(2 (108) = 216\).
Time = 0.26 (sec) , antiderivative size = 929, normalized size of antiderivative = 8.15 \[ \int \left (a+b \text {csch}^2(c+d x)\right )^4 \, dx=\text {Too large to display} \]
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\[ \int \left (a+b \text {csch}^2(c+d x)\right )^4 \, dx=\int \left (a + b \operatorname {csch}^{2}{\left (c + d x \right )}\right )^{4}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 706 vs. \(2 (108) = 216\).
Time = 0.20 (sec) , antiderivative size = 706, normalized size of antiderivative = 6.19 \[ \int \left (a+b \text {csch}^2(c+d x)\right )^4 \, dx=a^{4} x + \frac {32}{35} \, b^{4} {\left (\frac {7 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (7 \, e^{\left (-2 \, d x - 2 \, c\right )} - 21 \, e^{\left (-4 \, d x - 4 \, c\right )} + 35 \, e^{\left (-6 \, d x - 6 \, c\right )} - 35 \, e^{\left (-8 \, d x - 8 \, c\right )} + 21 \, e^{\left (-10 \, d x - 10 \, c\right )} - 7 \, e^{\left (-12 \, d x - 12 \, c\right )} + e^{\left (-14 \, d x - 14 \, c\right )} - 1\right )}} - \frac {21 \, e^{\left (-4 \, d x - 4 \, c\right )}}{d {\left (7 \, e^{\left (-2 \, d x - 2 \, c\right )} - 21 \, e^{\left (-4 \, d x - 4 \, c\right )} + 35 \, e^{\left (-6 \, d x - 6 \, c\right )} - 35 \, e^{\left (-8 \, d x - 8 \, c\right )} + 21 \, e^{\left (-10 \, d x - 10 \, c\right )} - 7 \, e^{\left (-12 \, d x - 12 \, c\right )} + e^{\left (-14 \, d x - 14 \, c\right )} - 1\right )}} + \frac {35 \, e^{\left (-6 \, d x - 6 \, c\right )}}{d {\left (7 \, e^{\left (-2 \, d x - 2 \, c\right )} - 21 \, e^{\left (-4 \, d x - 4 \, c\right )} + 35 \, e^{\left (-6 \, d x - 6 \, c\right )} - 35 \, e^{\left (-8 \, d x - 8 \, c\right )} + 21 \, e^{\left (-10 \, d x - 10 \, c\right )} - 7 \, e^{\left (-12 \, d x - 12 \, c\right )} + e^{\left (-14 \, d x - 14 \, c\right )} - 1\right )}} - \frac {1}{d {\left (7 \, e^{\left (-2 \, d x - 2 \, c\right )} - 21 \, e^{\left (-4 \, d x - 4 \, c\right )} + 35 \, e^{\left (-6 \, d x - 6 \, c\right )} - 35 \, e^{\left (-8 \, d x - 8 \, c\right )} + 21 \, e^{\left (-10 \, d x - 10 \, c\right )} - 7 \, e^{\left (-12 \, d x - 12 \, c\right )} + e^{\left (-14 \, d x - 14 \, c\right )} - 1\right )}}\right )} - \frac {64}{15} \, a b^{3} {\left (\frac {5 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} - 10 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, e^{\left (-6 \, d x - 6 \, c\right )} - 5 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )} - 1\right )}} - \frac {10 \, e^{\left (-4 \, d x - 4 \, c\right )}}{d {\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} - 10 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, e^{\left (-6 \, d x - 6 \, c\right )} - 5 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )} - 1\right )}} - \frac {1}{d {\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} - 10 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, e^{\left (-6 \, d x - 6 \, c\right )} - 5 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )} - 1\right )}}\right )} + 8 \, a^{2} b^{2} {\left (\frac {3 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} - 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} - 1\right )}} - \frac {1}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} - 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} - 1\right )}}\right )} + \frac {8 \, a^{3} b}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} - 1\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 334 vs. \(2 (108) = 216\).
Time = 0.31 (sec) , antiderivative size = 334, normalized size of antiderivative = 2.93 \[ \int \left (a+b \text {csch}^2(c+d x)\right )^4 \, dx=\frac {105 \, {\left (d x + c\right )} a^{4} - \frac {8 \, {\left (105 \, a^{3} b e^{\left (12 \, d x + 12 \, c\right )} - 630 \, a^{3} b e^{\left (10 \, d x + 10 \, c\right )} + 315 \, a^{2} b^{2} e^{\left (10 \, d x + 10 \, c\right )} + 1575 \, a^{3} b e^{\left (8 \, d x + 8 \, c\right )} - 1365 \, a^{2} b^{2} e^{\left (8 \, d x + 8 \, c\right )} + 560 \, a b^{3} e^{\left (8 \, d x + 8 \, c\right )} - 2100 \, a^{3} b e^{\left (6 \, d x + 6 \, c\right )} + 2310 \, a^{2} b^{2} e^{\left (6 \, d x + 6 \, c\right )} - 1400 \, a b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 420 \, b^{4} e^{\left (6 \, d x + 6 \, c\right )} + 1575 \, a^{3} b e^{\left (4 \, d x + 4 \, c\right )} - 1890 \, a^{2} b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 1176 \, a b^{3} e^{\left (4 \, d x + 4 \, c\right )} - 252 \, b^{4} e^{\left (4 \, d x + 4 \, c\right )} - 630 \, a^{3} b e^{\left (2 \, d x + 2 \, c\right )} + 735 \, a^{2} b^{2} e^{\left (2 \, d x + 2 \, c\right )} - 392 \, a b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 84 \, b^{4} e^{\left (2 \, d x + 2 \, c\right )} + 105 \, a^{3} b - 105 \, a^{2} b^{2} + 56 \, a b^{3} - 12 \, b^{4}\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{7}}}{105 \, d} \]
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Time = 2.32 (sec) , antiderivative size = 1088, normalized size of antiderivative = 9.54 \[ \int \left (a+b \text {csch}^2(c+d x)\right )^4 \, dx=\frac {\frac {8\,\left (5\,a^3\,b-9\,a^2\,b^2+8\,a\,b^3-4\,b^4\right )}{35\,d}-\frac {8\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (15\,a^3\,b-24\,a^2\,b^2+16\,a\,b^3\right )}{35\,d}+\frac {24\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (a^3\,b-a^2\,b^2\right )}{7\,d}-\frac {8\,a^3\,b\,{\mathrm {e}}^{6\,c+6\,d\,x}}{7\,d}}{6\,{\mathrm {e}}^{4\,c+4\,d\,x}-4\,{\mathrm {e}}^{2\,c+2\,d\,x}-4\,{\mathrm {e}}^{6\,c+6\,d\,x}+{\mathrm {e}}^{8\,c+8\,d\,x}+1}+\frac {\frac {8\,\left (a^3\,b-a^2\,b^2\right )}{7\,d}-\frac {8\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (15\,a^3\,b-24\,a^2\,b^2+16\,a\,b^3\right )}{21\,d}-\frac {16\,{\mathrm {e}}^{6\,c+6\,d\,x}\,\left (15\,a^3\,b-24\,a^2\,b^2+16\,a\,b^3\right )}{21\,d}+\frac {16\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (5\,a^3\,b-9\,a^2\,b^2+8\,a\,b^3-4\,b^4\right )}{7\,d}+\frac {40\,{\mathrm {e}}^{8\,c+8\,d\,x}\,\left (a^3\,b-a^2\,b^2\right )}{7\,d}-\frac {8\,a^3\,b\,{\mathrm {e}}^{10\,c+10\,d\,x}}{7\,d}}{15\,{\mathrm {e}}^{4\,c+4\,d\,x}-6\,{\mathrm {e}}^{2\,c+2\,d\,x}-20\,{\mathrm {e}}^{6\,c+6\,d\,x}+15\,{\mathrm {e}}^{8\,c+8\,d\,x}-6\,{\mathrm {e}}^{10\,c+10\,d\,x}+{\mathrm {e}}^{12\,c+12\,d\,x}+1}-\frac {\frac {8\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (15\,a^3\,b-24\,a^2\,b^2+16\,a\,b^3\right )}{7\,d}+\frac {8\,{\mathrm {e}}^{8\,c+8\,d\,x}\,\left (15\,a^3\,b-24\,a^2\,b^2+16\,a\,b^3\right )}{7\,d}+\frac {8\,a^3\,b}{7\,d}-\frac {32\,{\mathrm {e}}^{6\,c+6\,d\,x}\,\left (5\,a^3\,b-9\,a^2\,b^2+8\,a\,b^3-4\,b^4\right )}{7\,d}-\frac {48\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (a^3\,b-a^2\,b^2\right )}{7\,d}-\frac {48\,{\mathrm {e}}^{10\,c+10\,d\,x}\,\left (a^3\,b-a^2\,b^2\right )}{7\,d}+\frac {8\,a^3\,b\,{\mathrm {e}}^{12\,c+12\,d\,x}}{7\,d}}{7\,{\mathrm {e}}^{2\,c+2\,d\,x}-21\,{\mathrm {e}}^{4\,c+4\,d\,x}+35\,{\mathrm {e}}^{6\,c+6\,d\,x}-35\,{\mathrm {e}}^{8\,c+8\,d\,x}+21\,{\mathrm {e}}^{10\,c+10\,d\,x}-7\,{\mathrm {e}}^{12\,c+12\,d\,x}+{\mathrm {e}}^{14\,c+14\,d\,x}-1}+\frac {\frac {8\,\left (a^3\,b-a^2\,b^2\right )}{7\,d}-\frac {8\,a^3\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}}{7\,d}}{{\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1}-\frac {\frac {8\,\left (15\,a^3\,b-24\,a^2\,b^2+16\,a\,b^3\right )}{105\,d}+\frac {16\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (15\,a^3\,b-24\,a^2\,b^2+16\,a\,b^3\right )}{35\,d}-\frac {32\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (5\,a^3\,b-9\,a^2\,b^2+8\,a\,b^3-4\,b^4\right )}{35\,d}-\frac {32\,{\mathrm {e}}^{6\,c+6\,d\,x}\,\left (a^3\,b-a^2\,b^2\right )}{7\,d}+\frac {8\,a^3\,b\,{\mathrm {e}}^{8\,c+8\,d\,x}}{7\,d}}{5\,{\mathrm {e}}^{2\,c+2\,d\,x}-10\,{\mathrm {e}}^{4\,c+4\,d\,x}+10\,{\mathrm {e}}^{6\,c+6\,d\,x}-5\,{\mathrm {e}}^{8\,c+8\,d\,x}+{\mathrm {e}}^{10\,c+10\,d\,x}-1}+a^4\,x-\frac {\frac {8\,\left (15\,a^3\,b-24\,a^2\,b^2+16\,a\,b^3\right )}{105\,d}-\frac {16\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (a^3\,b-a^2\,b^2\right )}{7\,d}+\frac {8\,a^3\,b\,{\mathrm {e}}^{4\,c+4\,d\,x}}{7\,d}}{3\,{\mathrm {e}}^{2\,c+2\,d\,x}-3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}-1}-\frac {8\,a^3\,b}{7\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )} \]
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