Integrand size = 27, antiderivative size = 85 \[ \int \frac {\cosh ^3(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {a \left (a^2+b^2\right ) \log (a+b \sinh (c+d x))}{b^4 d}+\frac {\left (a^2+b^2\right ) \sinh (c+d x)}{b^3 d}-\frac {a \sinh ^2(c+d x)}{2 b^2 d}+\frac {\sinh ^3(c+d x)}{3 b d} \]
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Time = 0.09 (sec) , antiderivative size = 85, 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 = {2916, 12, 786} \[ \int \frac {\cosh ^3(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {a \left (a^2+b^2\right ) \log (a+b \sinh (c+d x))}{b^4 d}+\frac {\left (a^2+b^2\right ) \sinh (c+d x)}{b^3 d}-\frac {a \sinh ^2(c+d x)}{2 b^2 d}+\frac {\sinh ^3(c+d x)}{3 b d} \]
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Rule 12
Rule 786
Rule 2916
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {x \left (-b^2-x^2\right )}{b (a+x)} \, dx,x,b \sinh (c+d x)\right )}{b^3 d} \\ & = -\frac {\text {Subst}\left (\int \frac {x \left (-b^2-x^2\right )}{a+x} \, dx,x,b \sinh (c+d x)\right )}{b^4 d} \\ & = -\frac {\text {Subst}\left (\int \left (-a^2 \left (1+\frac {b^2}{a^2}\right )+a x-x^2+\frac {a \left (a^2+b^2\right )}{a+x}\right ) \, dx,x,b \sinh (c+d x)\right )}{b^4 d} \\ & = -\frac {a \left (a^2+b^2\right ) \log (a+b \sinh (c+d x))}{b^4 d}+\frac {\left (a^2+b^2\right ) \sinh (c+d x)}{b^3 d}-\frac {a \sinh ^2(c+d x)}{2 b^2 d}+\frac {\sinh ^3(c+d x)}{3 b d} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.88 \[ \int \frac {\cosh ^3(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {-6 a \left (a^2+b^2\right ) \log (a+b \sinh (c+d x))+6 b \left (a^2+b^2\right ) \sinh (c+d x)-3 a b^2 \sinh ^2(c+d x)+2 b^3 \sinh ^3(c+d x)}{6 b^4 d} \]
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Time = 8.98 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.94
method | result | size |
derivativedivides | \(\frac {\frac {\frac {\sinh \left (d x +c \right )^{3} b^{2}}{3}-\frac {a \sinh \left (d x +c \right )^{2} b}{2}+a^{2} \sinh \left (d x +c \right )+b^{2} \sinh \left (d x +c \right )}{b^{3}}-\frac {a \left (a^{2}+b^{2}\right ) \ln \left (a +b \sinh \left (d x +c \right )\right )}{b^{4}}}{d}\) | \(80\) |
default | \(\frac {\frac {\frac {\sinh \left (d x +c \right )^{3} b^{2}}{3}-\frac {a \sinh \left (d x +c \right )^{2} b}{2}+a^{2} \sinh \left (d x +c \right )+b^{2} \sinh \left (d x +c \right )}{b^{3}}-\frac {a \left (a^{2}+b^{2}\right ) \ln \left (a +b \sinh \left (d x +c \right )\right )}{b^{4}}}{d}\) | \(80\) |
risch | \(\frac {a^{3} x}{b^{4}}+\frac {a x}{b^{2}}+\frac {{\mathrm e}^{3 d x +3 c}}{24 b d}-\frac {a \,{\mathrm e}^{2 d x +2 c}}{8 b^{2} d}+\frac {{\mathrm e}^{d x +c} a^{2}}{2 b^{3} d}+\frac {3 \,{\mathrm e}^{d x +c}}{8 b d}-\frac {{\mathrm e}^{-d x -c} a^{2}}{2 b^{3} d}-\frac {3 \,{\mathrm e}^{-d x -c}}{8 b d}-\frac {a \,{\mathrm e}^{-2 d x -2 c}}{8 b^{2} d}-\frac {{\mathrm e}^{-3 d x -3 c}}{24 b d}+\frac {2 a^{3} c}{b^{4} d}+\frac {2 a c}{b^{2} d}-\frac {a^{3} \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \,{\mathrm e}^{d x +c}}{b}-1\right )}{b^{4} d}-\frac {a \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \,{\mathrm e}^{d x +c}}{b}-1\right )}{b^{2} d}\) | \(244\) |
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Leaf count of result is larger than twice the leaf count of optimal. 652 vs. \(2 (81) = 162\).
Time = 0.29 (sec) , antiderivative size = 652, normalized size of antiderivative = 7.67 \[ \int \frac {\cosh ^3(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {b^{3} \cosh \left (d x + c\right )^{6} + b^{3} \sinh \left (d x + c\right )^{6} - 3 \, a b^{2} \cosh \left (d x + c\right )^{5} + 24 \, {\left (a^{3} + a b^{2}\right )} d x \cosh \left (d x + c\right )^{3} + 3 \, {\left (2 \, b^{3} \cosh \left (d x + c\right ) - a b^{2}\right )} \sinh \left (d x + c\right )^{5} + 3 \, {\left (4 \, a^{2} b + 3 \, b^{3}\right )} \cosh \left (d x + c\right )^{4} + 3 \, {\left (5 \, b^{3} \cosh \left (d x + c\right )^{2} - 5 \, a b^{2} \cosh \left (d x + c\right ) + 4 \, a^{2} b + 3 \, b^{3}\right )} \sinh \left (d x + c\right )^{4} - 3 \, a b^{2} \cosh \left (d x + c\right ) + 2 \, {\left (10 \, b^{3} \cosh \left (d x + c\right )^{3} - 15 \, a b^{2} \cosh \left (d x + c\right )^{2} + 12 \, {\left (a^{3} + a b^{2}\right )} d x + 6 \, {\left (4 \, a^{2} b + 3 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} - b^{3} - 3 \, {\left (4 \, a^{2} b + 3 \, b^{3}\right )} \cosh \left (d x + c\right )^{2} + 3 \, {\left (5 \, b^{3} \cosh \left (d x + c\right )^{4} - 10 \, a b^{2} \cosh \left (d x + c\right )^{3} + 24 \, {\left (a^{3} + a b^{2}\right )} d x \cosh \left (d x + c\right ) - 4 \, a^{2} b - 3 \, b^{3} + 6 \, {\left (4 \, a^{2} b + 3 \, b^{3}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{2} - 24 \, {\left ({\left (a^{3} + a b^{2}\right )} \cosh \left (d x + c\right )^{3} + 3 \, {\left (a^{3} + a b^{2}\right )} \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right ) + 3 \, {\left (a^{3} + a b^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + {\left (a^{3} + a b^{2}\right )} \sinh \left (d x + c\right )^{3}\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 ) + 3 \, {\left (2 \, b^{3} \cosh \left (d x + c\right )^{5} - 5 \, a b^{2} \cosh \left (d x + c\right )^{4} + 24 \, {\left (a^{3} + a b^{2}\right )} d x \cosh \left (d x + c\right )^{2} + 4 \, {\left (4 \, a^{2} b + 3 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} - a b^{2} - 2 \, {\left (4 \, a^{2} b + 3 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{24 \, {\left (b^{4} d \cosh \left (d x + c\right )^{3} + 3 \, b^{4} d \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right ) + 3 \, b^{4} d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + b^{4} d \sinh \left (d x + c\right )^{3}\right )}} \]
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Timed out. \[ \int \frac {\cosh ^3(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 183 vs. \(2 (81) = 162\).
Time = 0.20 (sec) , antiderivative size = 183, normalized size of antiderivative = 2.15 \[ \int \frac {\cosh ^3(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {{\left (3 \, a b e^{\left (-d x - c\right )} - b^{2} - 3 \, {\left (4 \, a^{2} + 3 \, b^{2}\right )} e^{\left (-2 \, d x - 2 \, c\right )}\right )} e^{\left (3 \, d x + 3 \, c\right )}}{24 \, b^{3} d} - \frac {{\left (a^{3} + a b^{2}\right )} {\left (d x + c\right )}}{b^{4} d} - \frac {3 \, a b e^{\left (-2 \, d x - 2 \, c\right )} + b^{2} e^{\left (-3 \, d x - 3 \, c\right )} + 3 \, {\left (4 \, a^{2} + 3 \, b^{2}\right )} e^{\left (-d x - c\right )}}{24 \, b^{3} d} - \frac {{\left (a^{3} + a b^{2}\right )} \log \left (-2 \, a e^{\left (-d x - c\right )} + b e^{\left (-2 \, d x - 2 \, c\right )} - b\right )}{b^{4} d} \]
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Time = 0.30 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.71 \[ \int \frac {\cosh ^3(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {\frac {b^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} - 3 \, a b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} + 12 \, a^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 12 \, b^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}}{b^{3}} - \frac {24 \, {\left (a^{3} + a b^{2}\right )} \log \left ({\left | b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 2 \, a \right |}\right )}{b^{4}}}{24 \, d} \]
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Time = 1.16 (sec) , antiderivative size = 180, normalized size of antiderivative = 2.12 \[ \int \frac {\cosh ^3(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {x\,\left (a^3+a\,b^2\right )}{b^4}-\frac {{\mathrm {e}}^{-3\,c-3\,d\,x}}{24\,b\,d}+\frac {{\mathrm {e}}^{3\,c+3\,d\,x}}{24\,b\,d}-\frac {a\,{\mathrm {e}}^{-2\,c-2\,d\,x}}{8\,b^2\,d}-\frac {a\,{\mathrm {e}}^{2\,c+2\,d\,x}}{8\,b^2\,d}-\frac {{\mathrm {e}}^{-c-d\,x}\,\left (4\,a^2+3\,b^2\right )}{8\,b^3\,d}-\frac {\ln \left (2\,a\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c-b+b\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}\right )\,\left (a^3+a\,b^2\right )}{b^4\,d}+\frac {{\mathrm {e}}^{c+d\,x}\,\left (4\,a^2+3\,b^2\right )}{8\,b^3\,d} \]
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