Integrand size = 23, antiderivative size = 71 \[ \int \text {csch}^2(c+d x) \left (a+b \tanh ^3(c+d x)\right )^3 \, dx=-\frac {a^3 \coth (c+d x)}{d}+\frac {3 a^2 b \tanh ^2(c+d x)}{2 d}+\frac {3 a b^2 \tanh ^5(c+d x)}{5 d}+\frac {b^3 \tanh ^8(c+d x)}{8 d} \]
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Time = 0.05 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {3744, 276} \[ \int \text {csch}^2(c+d x) \left (a+b \tanh ^3(c+d x)\right )^3 \, dx=-\frac {a^3 \coth (c+d x)}{d}+\frac {3 a^2 b \tanh ^2(c+d x)}{2 d}+\frac {3 a b^2 \tanh ^5(c+d x)}{5 d}+\frac {b^3 \tanh ^8(c+d x)}{8 d} \]
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Rule 276
Rule 3744
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\left (a+b x^3\right )^3}{x^2} \, dx,x,\tanh (c+d x)\right )}{d} \\ & = \frac {\text {Subst}\left (\int \left (\frac {a^3}{x^2}+3 a^2 b x+3 a b^2 x^4+b^3 x^7\right ) \, dx,x,\tanh (c+d x)\right )}{d} \\ & = -\frac {a^3 \coth (c+d x)}{d}+\frac {3 a^2 b \tanh ^2(c+d x)}{2 d}+\frac {3 a b^2 \tanh ^5(c+d x)}{5 d}+\frac {b^3 \tanh ^8(c+d x)}{8 d} \\ \end{align*}
Time = 2.39 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.59 \[ \int \text {csch}^2(c+d x) \left (a+b \tanh ^3(c+d x)\right )^3 \, dx=\frac {-40 a^3 \coth (c+d x)+b \left (-20 b^2 \text {sech}^6(c+d x)+5 b^2 \text {sech}^8(c+d x)+24 a b \tanh (c+d x)+6 b \text {sech}^4(c+d x) (5 b+4 a \tanh (c+d x))-4 \text {sech}^2(c+d x) \left (15 a^2+5 b^2+12 a b \tanh (c+d x)\right )\right )}{40 d} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(170\) vs. \(2(65)=130\).
Time = 20.42 (sec) , antiderivative size = 171, normalized size of antiderivative = 2.41
method | result | size |
derivativedivides | \(\frac {-a^{3} \coth \left (d x +c \right )-\frac {3 a^{2} b}{2 \cosh \left (d x +c \right )^{2}}+3 a \,b^{2} \left (-\frac {\sinh \left (d x +c \right )^{3}}{2 \cosh \left (d x +c \right )^{5}}-\frac {3 \sinh \left (d x +c \right )}{8 \cosh \left (d x +c \right )^{5}}+\frac {3 \left (\frac {8}{15}+\frac {\operatorname {sech}\left (d x +c \right )^{4}}{5}+\frac {4 \operatorname {sech}\left (d x +c \right )^{2}}{15}\right ) \tanh \left (d x +c \right )}{8}\right )+b^{3} \left (-\frac {\sinh \left (d x +c \right )^{6}}{2 \cosh \left (d x +c \right )^{8}}-\frac {3 \sinh \left (d x +c \right )^{4}}{4 \cosh \left (d x +c \right )^{8}}-\frac {\sinh \left (d x +c \right )^{2}}{2 \cosh \left (d x +c \right )^{8}}-\frac {1}{8 \cosh \left (d x +c \right )^{8}}\right )}{d}\) | \(171\) |
default | \(\frac {-a^{3} \coth \left (d x +c \right )-\frac {3 a^{2} b}{2 \cosh \left (d x +c \right )^{2}}+3 a \,b^{2} \left (-\frac {\sinh \left (d x +c \right )^{3}}{2 \cosh \left (d x +c \right )^{5}}-\frac {3 \sinh \left (d x +c \right )}{8 \cosh \left (d x +c \right )^{5}}+\frac {3 \left (\frac {8}{15}+\frac {\operatorname {sech}\left (d x +c \right )^{4}}{5}+\frac {4 \operatorname {sech}\left (d x +c \right )^{2}}{15}\right ) \tanh \left (d x +c \right )}{8}\right )+b^{3} \left (-\frac {\sinh \left (d x +c \right )^{6}}{2 \cosh \left (d x +c \right )^{8}}-\frac {3 \sinh \left (d x +c \right )^{4}}{4 \cosh \left (d x +c \right )^{8}}-\frac {\sinh \left (d x +c \right )^{2}}{2 \cosh \left (d x +c \right )^{8}}-\frac {1}{8 \cosh \left (d x +c \right )^{8}}\right )}{d}\) | \(171\) |
risch | \(-\frac {2 \left (-3 a \,b^{2}+5 a^{3} {\mathrm e}^{16 d x +16 c}+5 b^{3} {\mathrm e}^{16 d x +16 c}+35 b^{3} {\mathrm e}^{12 d x +12 c}+5 a^{3}+5 \,{\mathrm e}^{4 d x +4 c} b^{3}+40 a^{3} {\mathrm e}^{14 d x +14 c}-5 b^{3} {\mathrm e}^{14 d x +14 c}-6 \,{\mathrm e}^{2 d x +2 c} a \,b^{2}-15 a^{2} b \,{\mathrm e}^{2 d x +2 c}-30 a \,b^{2} {\mathrm e}^{4 d x +4 c}-75 a^{2} b \,{\mathrm e}^{4 d x +4 c}-135 a^{2} b \,{\mathrm e}^{6 d x +6 c}-54 a \,b^{2} {\mathrm e}^{6 d x +6 c}+140 a^{3} {\mathrm e}^{12 d x +12 c}-75 a^{2} b \,{\mathrm e}^{8 d x +8 c}-12 a \,b^{2} {\mathrm e}^{8 d x +8 c}+15 a^{2} b \,{\mathrm e}^{16 d x +16 c}+15 a \,b^{2} {\mathrm e}^{16 d x +16 c}+75 a^{2} b \,{\mathrm e}^{14 d x +14 c}+30 a \,b^{2} {\mathrm e}^{14 d x +14 c}+75 a^{2} b \,{\mathrm e}^{10 d x +10 c}-35 b^{3} {\mathrm e}^{10 d x +10 c}+280 a^{3} {\mathrm e}^{10 d x +10 c}+280 a^{3} {\mathrm e}^{6 d x +6 c}-35 \,{\mathrm e}^{6 d x +6 c} b^{3}+140 a^{3} {\mathrm e}^{4 d x +4 c}+30 a \,b^{2} {\mathrm e}^{12 d x +12 c}+135 a^{2} b \,{\mathrm e}^{12 d x +12 c}+40 a^{3} {\mathrm e}^{2 d x +2 c}-5 \,{\mathrm e}^{2 d x +2 c} b^{3}+35 b^{3} {\mathrm e}^{8 d x +8 c}+350 a^{3} {\mathrm e}^{8 d x +8 c}+30 a \,b^{2} {\mathrm e}^{10 d x +10 c}\right )}{5 d \left ({\mathrm e}^{2 d x +2 c}-1\right ) \left ({\mathrm e}^{2 d x +2 c}+1\right )^{8}}\) | \(508\) |
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Leaf count of result is larger than twice the leaf count of optimal. 1192 vs. \(2 (65) = 130\).
Time = 0.27 (sec) , antiderivative size = 1192, normalized size of antiderivative = 16.79 \[ \int \text {csch}^2(c+d x) \left (a+b \tanh ^3(c+d x)\right )^3 \, dx=\text {Too large to display} \]
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\[ \int \text {csch}^2(c+d x) \left (a+b \tanh ^3(c+d x)\right )^3 \, dx=\int \left (a + b \tanh ^{3}{\left (c + d x \right )}\right )^{3} \operatorname {csch}^{2}{\left (c + d x \right )}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 679 vs. \(2 (65) = 130\).
Time = 0.21 (sec) , antiderivative size = 679, normalized size of antiderivative = 9.56 \[ \int \text {csch}^2(c+d x) \left (a+b \tanh ^3(c+d x)\right )^3 \, dx=-2 \, b^{3} {\left (\frac {e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (8 \, e^{\left (-2 \, d x - 2 \, c\right )} + 28 \, e^{\left (-4 \, d x - 4 \, c\right )} + 56 \, e^{\left (-6 \, d x - 6 \, c\right )} + 70 \, e^{\left (-8 \, d x - 8 \, c\right )} + 56 \, e^{\left (-10 \, d x - 10 \, c\right )} + 28 \, e^{\left (-12 \, d x - 12 \, c\right )} + 8 \, e^{\left (-14 \, d x - 14 \, c\right )} + e^{\left (-16 \, d x - 16 \, c\right )} + 1\right )}} + \frac {7 \, e^{\left (-6 \, d x - 6 \, c\right )}}{d {\left (8 \, e^{\left (-2 \, d x - 2 \, c\right )} + 28 \, e^{\left (-4 \, d x - 4 \, c\right )} + 56 \, e^{\left (-6 \, d x - 6 \, c\right )} + 70 \, e^{\left (-8 \, d x - 8 \, c\right )} + 56 \, e^{\left (-10 \, d x - 10 \, c\right )} + 28 \, e^{\left (-12 \, d x - 12 \, c\right )} + 8 \, e^{\left (-14 \, d x - 14 \, c\right )} + e^{\left (-16 \, d x - 16 \, c\right )} + 1\right )}} + \frac {7 \, e^{\left (-10 \, d x - 10 \, c\right )}}{d {\left (8 \, e^{\left (-2 \, d x - 2 \, c\right )} + 28 \, e^{\left (-4 \, d x - 4 \, c\right )} + 56 \, e^{\left (-6 \, d x - 6 \, c\right )} + 70 \, e^{\left (-8 \, d x - 8 \, c\right )} + 56 \, e^{\left (-10 \, d x - 10 \, c\right )} + 28 \, e^{\left (-12 \, d x - 12 \, c\right )} + 8 \, e^{\left (-14 \, d x - 14 \, c\right )} + e^{\left (-16 \, d x - 16 \, c\right )} + 1\right )}} + \frac {e^{\left (-14 \, d x - 14 \, c\right )}}{d {\left (8 \, e^{\left (-2 \, d x - 2 \, c\right )} + 28 \, e^{\left (-4 \, d x - 4 \, c\right )} + 56 \, e^{\left (-6 \, d x - 6 \, c\right )} + 70 \, e^{\left (-8 \, d x - 8 \, c\right )} + 56 \, e^{\left (-10 \, d x - 10 \, c\right )} + 28 \, e^{\left (-12 \, d x - 12 \, c\right )} + 8 \, e^{\left (-14 \, d x - 14 \, c\right )} + e^{\left (-16 \, d x - 16 \, c\right )} + 1\right )}}\right )} + \frac {6}{5} \, a b^{2} {\left (\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 {5 \, e^{\left (-8 \, d x - 8 \, 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 )} + \frac {2 \, a^{3}}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} - 1\right )}} - \frac {6 \, a^{2} b}{d {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 311 vs. \(2 (65) = 130\).
Time = 0.55 (sec) , antiderivative size = 311, normalized size of antiderivative = 4.38 \[ \int \text {csch}^2(c+d x) \left (a+b \tanh ^3(c+d x)\right )^3 \, dx=-\frac {2 \, {\left (\frac {5 \, a^{3}}{e^{\left (2 \, d x + 2 \, c\right )} - 1} + \frac {15 \, a^{2} b e^{\left (14 \, d x + 14 \, c\right )} + 15 \, a b^{2} e^{\left (14 \, d x + 14 \, c\right )} + 5 \, b^{3} e^{\left (14 \, d x + 14 \, c\right )} + 90 \, a^{2} b e^{\left (12 \, d x + 12 \, c\right )} + 45 \, a b^{2} e^{\left (12 \, d x + 12 \, c\right )} + 225 \, a^{2} b e^{\left (10 \, d x + 10 \, c\right )} + 75 \, a b^{2} e^{\left (10 \, d x + 10 \, c\right )} + 35 \, b^{3} e^{\left (10 \, d x + 10 \, c\right )} + 300 \, a^{2} b e^{\left (8 \, d x + 8 \, c\right )} + 105 \, a b^{2} e^{\left (8 \, d x + 8 \, c\right )} + 225 \, a^{2} b e^{\left (6 \, d x + 6 \, c\right )} + 93 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 35 \, b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 90 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} + 39 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 15 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} + 9 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 5 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 3 \, a b^{2}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{8}}\right )}}{5 \, d} \]
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Time = 2.09 (sec) , antiderivative size = 1515, normalized size of antiderivative = 21.34 \[ \int \text {csch}^2(c+d x) \left (a+b \tanh ^3(c+d x)\right )^3 \, dx=\text {Too large to display} \]
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