Integrand size = 14, antiderivative size = 74 \[ \int \left (a+b \text {csch}^2(c+d x)\right )^3 \, dx=a^3 x-\frac {b \left (3 a^2-3 a b+b^2\right ) \coth (c+d x)}{d}-\frac {(3 a-2 b) b^2 \coth ^3(c+d x)}{3 d}-\frac {b^3 \coth ^5(c+d x)}{5 d} \]
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Time = 0.04 (sec) , antiderivative size = 74, 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} \[ \int \left (a+b \text {csch}^2(c+d x)\right )^3 \, dx=a^3 x-\frac {b \left (3 a^2-3 a b+b^2\right ) \coth (c+d x)}{d}-\frac {b^2 (3 a-2 b) \coth ^3(c+d x)}{3 d}-\frac {b^3 \coth ^5(c+d x)}{5 d} \]
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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 )^3}{1-x^2} \, dx,x,\coth (c+d x)\right )}{d} \\ & = \frac {\text {Subst}\left (\int \left (-b \left (3 a^2-3 a b+b^2\right )-(3 a-2 b) b^2 x^2-b^3 x^4+\frac {a^3}{1-x^2}\right ) \, dx,x,\coth (c+d x)\right )}{d} \\ & = -\frac {b \left (3 a^2-3 a b+b^2\right ) \coth (c+d x)}{d}-\frac {(3 a-2 b) b^2 \coth ^3(c+d x)}{3 d}-\frac {b^3 \coth ^5(c+d x)}{5 d}+\frac {a^3 \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\coth (c+d x)\right )}{d} \\ & = a^3 x-\frac {b \left (3 a^2-3 a b+b^2\right ) \coth (c+d x)}{d}-\frac {(3 a-2 b) b^2 \coth ^3(c+d x)}{3 d}-\frac {b^3 \coth ^5(c+d x)}{5 d} \\ \end{align*}
Time = 2.72 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.53 \[ \int \left (a+b \text {csch}^2(c+d x)\right )^3 \, dx=\frac {8 \left (a+b \text {csch}^2(c+d x)\right )^3 \left (15 a^3 (c+d x)-b \coth (c+d x) \left (45 a^2-30 a b+8 b^2+(15 a-4 b) b \text {csch}^2(c+d x)+3 b^2 \text {csch}^4(c+d x)\right )\right ) \sinh ^6(c+d x)}{15 d (-a+2 b+a \cosh (2 (c+d x)))^3} \]
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Time = 1.55 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.12
method | result | size |
derivativedivides | \(\frac {a^{3} \left (d x +c \right )-3 a^{2} b \coth \left (d x +c \right )+3 a \,b^{2} \left (\frac {2}{3}-\frac {\operatorname {csch}\left (d x +c \right )^{2}}{3}\right ) \coth \left (d x +c \right )+b^{3} \left (-\frac {8}{15}-\frac {\operatorname {csch}\left (d x +c \right )^{4}}{5}+\frac {4 \operatorname {csch}\left (d x +c \right )^{2}}{15}\right ) \coth \left (d x +c \right )}{d}\) | \(83\) |
default | \(\frac {a^{3} \left (d x +c \right )-3 a^{2} b \coth \left (d x +c \right )+3 a \,b^{2} \left (\frac {2}{3}-\frac {\operatorname {csch}\left (d x +c \right )^{2}}{3}\right ) \coth \left (d x +c \right )+b^{3} \left (-\frac {8}{15}-\frac {\operatorname {csch}\left (d x +c \right )^{4}}{5}+\frac {4 \operatorname {csch}\left (d x +c \right )^{2}}{15}\right ) \coth \left (d x +c \right )}{d}\) | \(83\) |
parts | \(a^{3} x +\frac {b^{3} \left (-\frac {8}{15}-\frac {\operatorname {csch}\left (d x +c \right )^{4}}{5}+\frac {4 \operatorname {csch}\left (d x +c \right )^{2}}{15}\right ) \coth \left (d x +c \right )}{d}+\frac {3 a \,b^{2} \left (\frac {2}{3}-\frac {\operatorname {csch}\left (d x +c \right )^{2}}{3}\right ) \coth \left (d x +c \right )}{d}-\frac {3 a^{2} b \coth \left (d x +c \right )}{d}\) | \(84\) |
parallelrisch | \(\frac {-3 \coth \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} b^{3}+\left (-60 a \,b^{2}+25 b^{3}\right ) \coth \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (-720 a^{2} b +540 a \,b^{2}-150 b^{3}\right ) \coth \left (\frac {d x}{2}+\frac {c}{2}\right )-3 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} b^{3}+\left (-60 a \,b^{2}+25 b^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (-720 a^{2} b +540 a \,b^{2}-150 b^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+480 a^{3} d x}{480 d}\) | \(150\) |
risch | \(a^{3} x -\frac {2 b \left (45 a^{2} {\mathrm e}^{8 d x +8 c}-180 a^{2} {\mathrm e}^{6 d x +6 c}+90 a b \,{\mathrm e}^{6 d x +6 c}+270 a^{2} {\mathrm e}^{4 d x +4 c}-210 a b \,{\mathrm e}^{4 d x +4 c}+80 \,{\mathrm e}^{4 d x +4 c} b^{2}-180 a^{2} {\mathrm e}^{2 d x +2 c}+150 a b \,{\mathrm e}^{2 d x +2 c}-40 \,{\mathrm e}^{2 d x +2 c} b^{2}+45 a^{2}-30 a b +8 b^{2}\right )}{15 d \left ({\mathrm e}^{2 d x +2 c}-1\right )^{5}}\) | \(164\) |
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Leaf count of result is larger than twice the leaf count of optimal. 461 vs. \(2 (70) = 140\).
Time = 0.25 (sec) , antiderivative size = 461, normalized size of antiderivative = 6.23 \[ \int \left (a+b \text {csch}^2(c+d x)\right )^3 \, dx=-\frac {{\left (45 \, a^{2} b - 30 \, a b^{2} + 8 \, b^{3}\right )} \cosh \left (d x + c\right )^{5} + 5 \, {\left (45 \, a^{2} b - 30 \, a b^{2} + 8 \, b^{3}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{4} - {\left (15 \, a^{3} d x + 45 \, a^{2} b - 30 \, a b^{2} + 8 \, b^{3}\right )} \sinh \left (d x + c\right )^{5} - 5 \, {\left (27 \, a^{2} b - 30 \, a b^{2} + 8 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} + 5 \, {\left (15 \, a^{3} d x + 45 \, a^{2} b - 30 \, a b^{2} + 8 \, b^{3} - 2 \, {\left (15 \, a^{3} d x + 45 \, a^{2} b - 30 \, a b^{2} + 8 \, b^{3}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{3} + 5 \, {\left (2 \, {\left (45 \, a^{2} b - 30 \, a b^{2} + 8 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} - 3 \, {\left (27 \, a^{2} b - 30 \, a b^{2} + 8 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} + 10 \, {\left (9 \, a^{2} b - 12 \, a b^{2} + 8 \, b^{3}\right )} \cosh \left (d x + c\right ) - 5 \, {\left (30 \, a^{3} d x + {\left (15 \, a^{3} d x + 45 \, a^{2} b - 30 \, a b^{2} + 8 \, b^{3}\right )} \cosh \left (d x + c\right )^{4} + 90 \, a^{2} b - 60 \, a b^{2} + 16 \, b^{3} - 3 \, {\left (15 \, a^{3} d x + 45 \, a^{2} b - 30 \, a b^{2} + 8 \, b^{3}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )}{15 \, {\left (d \sinh \left (d x + c\right )^{5} + 5 \, {\left (2 \, d \cosh \left (d x + c\right )^{2} - d\right )} \sinh \left (d x + c\right )^{3} + 5 \, {\left (d \cosh \left (d x + c\right )^{4} - 3 \, d \cosh \left (d x + c\right )^{2} + 2 \, d\right )} \sinh \left (d x + c\right )\right )}} \]
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\[ \int \left (a+b \text {csch}^2(c+d x)\right )^3 \, dx=\int \left (a + b \operatorname {csch}^{2}{\left (c + d x \right )}\right )^{3}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 334 vs. \(2 (70) = 140\).
Time = 0.19 (sec) , antiderivative size = 334, normalized size of antiderivative = 4.51 \[ \int \left (a+b \text {csch}^2(c+d x)\right )^3 \, dx=a^{3} x - \frac {16}{15} \, b^{3} {\left (\frac {5 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} - 10 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, e^{\left (-6 \, d x - 6 \, c\right )} - 5 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )} - 1\right )}} - \frac {10 \, e^{\left (-4 \, d x - 4 \, c\right )}}{d {\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} - 10 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, e^{\left (-6 \, d x - 6 \, c\right )} - 5 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )} - 1\right )}} - \frac {1}{d {\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} - 10 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, e^{\left (-6 \, d x - 6 \, c\right )} - 5 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )} - 1\right )}}\right )} + 4 \, a b^{2} {\left (\frac {3 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} - 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} - 1\right )}} - \frac {1}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} - 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} - 1\right )}}\right )} + \frac {6 \, a^{2} b}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} - 1\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 182 vs. \(2 (70) = 140\).
Time = 0.27 (sec) , antiderivative size = 182, normalized size of antiderivative = 2.46 \[ \int \left (a+b \text {csch}^2(c+d x)\right )^3 \, dx=\frac {15 \, {\left (d x + c\right )} a^{3} - \frac {2 \, {\left (45 \, a^{2} b e^{\left (8 \, d x + 8 \, c\right )} - 180 \, a^{2} b e^{\left (6 \, d x + 6 \, c\right )} + 90 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 270 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} - 210 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 80 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} - 180 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} + 150 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} - 40 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 45 \, a^{2} b - 30 \, a b^{2} + 8 \, b^{3}\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{5}}}{15 \, d} \]
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Time = 2.26 (sec) , antiderivative size = 508, normalized size of antiderivative = 6.86 \[ \int \left (a+b \text {csch}^2(c+d x)\right )^3 \, dx=a^3\,x-\frac {\frac {2\,\left (9\,a^2\,b-12\,a\,b^2+8\,b^3\right )}{15\,d}+\frac {12\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (a\,b^2-a^2\,b\right )}{5\,d}+\frac {6\,a^2\,b\,{\mathrm {e}}^{4\,c+4\,d\,x}}{5\,d}}{3\,{\mathrm {e}}^{2\,c+2\,d\,x}-3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}-1}-\frac {\frac {6\,a^2\,b}{5\,d}+\frac {24\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (a\,b^2-a^2\,b\right )}{5\,d}+\frac {24\,{\mathrm {e}}^{6\,c+6\,d\,x}\,\left (a\,b^2-a^2\,b\right )}{5\,d}+\frac {4\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (9\,a^2\,b-12\,a\,b^2+8\,b^3\right )}{5\,d}+\frac {6\,a^2\,b\,{\mathrm {e}}^{8\,c+8\,d\,x}}{5\,d}}{5\,{\mathrm {e}}^{2\,c+2\,d\,x}-10\,{\mathrm {e}}^{4\,c+4\,d\,x}+10\,{\mathrm {e}}^{6\,c+6\,d\,x}-5\,{\mathrm {e}}^{8\,c+8\,d\,x}+{\mathrm {e}}^{10\,c+10\,d\,x}-1}-\frac {\frac {6\,\left (a\,b^2-a^2\,b\right )}{5\,d}+\frac {18\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (a\,b^2-a^2\,b\right )}{5\,d}+\frac {2\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (9\,a^2\,b-12\,a\,b^2+8\,b^3\right )}{5\,d}+\frac {6\,a^2\,b\,{\mathrm {e}}^{6\,c+6\,d\,x}}{5\,d}}{6\,{\mathrm {e}}^{4\,c+4\,d\,x}-4\,{\mathrm {e}}^{2\,c+2\,d\,x}-4\,{\mathrm {e}}^{6\,c+6\,d\,x}+{\mathrm {e}}^{8\,c+8\,d\,x}+1}-\frac {\frac {6\,\left (a\,b^2-a^2\,b\right )}{5\,d}+\frac {6\,a^2\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}}{5\,d}}{{\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1}-\frac {6\,a^2\,b}{5\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )} \]
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