Integrand size = 27, antiderivative size = 72 \[ \int \frac {\coth (c+d x) \text {csch}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {b \text {csch}(c+d x)}{a^2 d}-\frac {\text {csch}^2(c+d x)}{2 a d}+\frac {b^2 \log (\sinh (c+d x))}{a^3 d}-\frac {b^2 \log (a+b \sinh (c+d x))}{a^3 d} \]
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Time = 0.07 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2912, 12, 46} \[ \int \frac {\coth (c+d x) \text {csch}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {b^2 \log (\sinh (c+d x))}{a^3 d}-\frac {b^2 \log (a+b \sinh (c+d x))}{a^3 d}+\frac {b \text {csch}(c+d x)}{a^2 d}-\frac {\text {csch}^2(c+d x)}{2 a d} \]
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Rule 12
Rule 46
Rule 2912
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {b^3}{x^3 (a+x)} \, dx,x,b \sinh (c+d x)\right )}{b d} \\ & = \frac {b^2 \text {Subst}\left (\int \frac {1}{x^3 (a+x)} \, dx,x,b \sinh (c+d x)\right )}{d} \\ & = \frac {b^2 \text {Subst}\left (\int \left (\frac {1}{a x^3}-\frac {1}{a^2 x^2}+\frac {1}{a^3 x}-\frac {1}{a^3 (a+x)}\right ) \, dx,x,b \sinh (c+d x)\right )}{d} \\ & = \frac {b \text {csch}(c+d x)}{a^2 d}-\frac {\text {csch}^2(c+d x)}{2 a d}+\frac {b^2 \log (\sinh (c+d x))}{a^3 d}-\frac {b^2 \log (a+b \sinh (c+d x))}{a^3 d} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.83 \[ \int \frac {\coth (c+d x) \text {csch}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {2 a b \text {csch}(c+d x)-a^2 \text {csch}^2(c+d x)+2 b^2 (\log (\sinh (c+d x))-\log (a+b \sinh (c+d x)))}{2 a^3 d} \]
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Time = 0.65 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.75
method | result | size |
derivativedivides | \(-\frac {\operatorname {csch}\left (d x +c \right )^{2}}{2 a d}+\frac {b \,\operatorname {csch}\left (d x +c \right )}{a^{2} d}-\frac {b^{2} \ln \left (a \,\operatorname {csch}\left (d x +c \right )+b \right )}{d \,a^{3}}\) | \(54\) |
default | \(-\frac {\operatorname {csch}\left (d x +c \right )^{2}}{2 a d}+\frac {b \,\operatorname {csch}\left (d x +c \right )}{a^{2} d}-\frac {b^{2} \ln \left (a \,\operatorname {csch}\left (d x +c \right )+b \right )}{d \,a^{3}}\) | \(54\) |
risch | \(-\frac {2 \,{\mathrm e}^{d x +c} \left (-b \,{\mathrm e}^{2 d x +2 c}+a \,{\mathrm e}^{d x +c}+b \right )}{a^{2} d \left ({\mathrm e}^{2 d x +2 c}-1\right )^{2}}-\frac {b^{2} \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \,{\mathrm e}^{d x +c}}{b}-1\right )}{d \,a^{3}}+\frac {b^{2} \ln \left ({\mathrm e}^{2 d x +2 c}-1\right )}{d \,a^{3}}\) | \(108\) |
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Leaf count of result is larger than twice the leaf count of optimal. 545 vs. \(2 (70) = 140\).
Time = 0.26 (sec) , antiderivative size = 545, normalized size of antiderivative = 7.57 \[ \int \frac {\coth (c+d x) \text {csch}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {2 \, a b \cosh \left (d x + c\right )^{3} + 2 \, a b \sinh \left (d x + c\right )^{3} - 2 \, a^{2} \cosh \left (d x + c\right )^{2} - 2 \, a b \cosh \left (d x + c\right ) + 2 \, {\left (3 \, a b \cosh \left (d x + c\right ) - a^{2}\right )} \sinh \left (d x + c\right )^{2} - {\left (b^{2} \cosh \left (d x + c\right )^{4} + 4 \, b^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + b^{2} \sinh \left (d x + c\right )^{4} - 2 \, b^{2} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, b^{2} \cosh \left (d x + c\right )^{2} - b^{2}\right )} \sinh \left (d x + c\right )^{2} + b^{2} + 4 \, {\left (b^{2} \cosh \left (d x + c\right )^{3} - b^{2} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )\right )} \log \left (\frac {2 \, {\left (b \sinh \left (d x + c\right ) + a\right )}}{\cosh \left (d x + c\right ) - \sinh \left (d x + c\right )}\right ) + {\left (b^{2} \cosh \left (d x + c\right )^{4} + 4 \, b^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + b^{2} \sinh \left (d x + c\right )^{4} - 2 \, b^{2} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, b^{2} \cosh \left (d x + c\right )^{2} - b^{2}\right )} \sinh \left (d x + c\right )^{2} + b^{2} + 4 \, {\left (b^{2} \cosh \left (d x + c\right )^{3} - b^{2} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )\right )} \log \left (\frac {2 \, \sinh \left (d x + c\right )}{\cosh \left (d x + c\right ) - \sinh \left (d x + c\right )}\right ) + 2 \, {\left (3 \, a b \cosh \left (d x + c\right )^{2} - 2 \, a^{2} \cosh \left (d x + c\right ) - a b\right )} \sinh \left (d x + c\right )}{a^{3} d \cosh \left (d x + c\right )^{4} + 4 \, a^{3} d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + a^{3} d \sinh \left (d x + c\right )^{4} - 2 \, a^{3} d \cosh \left (d x + c\right )^{2} + a^{3} d + 2 \, {\left (3 \, a^{3} d \cosh \left (d x + c\right )^{2} - a^{3} d\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left (a^{3} d \cosh \left (d x + c\right )^{3} - a^{3} d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )} \]
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\[ \int \frac {\coth (c+d x) \text {csch}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {\coth {\left (c + d x \right )} \operatorname {csch}^{2}{\left (c + d x \right )}}{a + b \sinh {\left (c + d x \right )}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 161 vs. \(2 (70) = 140\).
Time = 0.21 (sec) , antiderivative size = 161, normalized size of antiderivative = 2.24 \[ \int \frac {\coth (c+d x) \text {csch}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {2 \, {\left (b e^{\left (-d x - c\right )} - a e^{\left (-2 \, d x - 2 \, c\right )} - b e^{\left (-3 \, d x - 3 \, c\right )}\right )}}{{\left (2 \, a^{2} e^{\left (-2 \, d x - 2 \, c\right )} - a^{2} e^{\left (-4 \, d x - 4 \, c\right )} - a^{2}\right )} d} - \frac {b^{2} \log \left (-2 \, a e^{\left (-d x - c\right )} + b e^{\left (-2 \, d x - 2 \, c\right )} - b\right )}{a^{3} d} + \frac {b^{2} \log \left (e^{\left (-d x - c\right )} + 1\right )}{a^{3} d} + \frac {b^{2} \log \left (e^{\left (-d x - c\right )} - 1\right )}{a^{3} d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 145 vs. \(2 (70) = 140\).
Time = 0.28 (sec) , antiderivative size = 145, normalized size of antiderivative = 2.01 \[ \int \frac {\coth (c+d x) \text {csch}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {\frac {2 \, b^{2} \log \left ({\left | b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 2 \, a \right |}\right )}{a^{3}} - \frac {2 \, b^{2} \log \left ({\left | e^{\left (d x + c\right )} - e^{\left (-d x - c\right )} \right |}\right )}{a^{3}} + \frac {3 \, b^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} - 4 \, a b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 4 \, a^{2}}{a^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2}}}{2 \, d} \]
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Time = 1.69 (sec) , antiderivative size = 470, normalized size of antiderivative = 6.53 \[ \int \frac {\coth (c+d x) \text {csch}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {\frac {2}{a\,d}-\frac {2\,b\,{\mathrm {e}}^{c+d\,x}}{a^2\,d}}{{\mathrm {e}}^{2\,c+2\,d\,x}-1}-\frac {\left (2\,\mathrm {atan}\left (-\frac {4\,a^3\,b^5\,\sqrt {-a^6\,d^2}+4\,a\,b^7\,\sqrt {-a^6\,d^2}-4\,b^8\,{\mathrm {e}}^{3\,c}\,{\mathrm {e}}^{3\,d\,x}\,\sqrt {-a^6\,d^2}+4\,b^8\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {-a^6\,d^2}-8\,a\,b^7\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}\,\sqrt {-a^6\,d^2}+4\,a^2\,b^6\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {-a^6\,d^2}-8\,a^3\,b^5\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}\,\sqrt {-a^6\,d^2}-4\,a^2\,b^6\,{\mathrm {e}}^{3\,c}\,{\mathrm {e}}^{3\,d\,x}\,\sqrt {-a^6\,d^2}}{4\,a^4\,b\,d\,{\left (b^4\right )}^{3/2}+4\,a^6\,b^3\,d\,\sqrt {b^4}}\right )+2\,\mathrm {atan}\left (\left (4\,a^4\,b^5\,d\,\sqrt {b^4}\,\sqrt {-a^6\,d^2}+4\,a^6\,b^3\,d\,\sqrt {b^4}\,\sqrt {-a^6\,d^2}\right )\,\left (\frac {1}{8\,a^5\,b^5\,d^2\,{\left (a^2+b^2\right )}^2}-{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (\frac {1}{16\,a^4\,b^6\,d^2\,{\left (a^2+b^2\right )}^2}-\frac {{\left (a^2+2\,b^2\right )}^2}{16\,a^8\,b^6\,d^2\,{\left (a^2+b^2\right )}^2}\right )+\frac {a^2+2\,b^2}{8\,a^7\,b^5\,d^2\,{\left (a^2+b^2\right )}^2}\right )\right )\right )\,\sqrt {b^4}}{\sqrt {-a^6\,d^2}}-\frac {2}{a\,d\,\left ({\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1\right )} \]
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