Integrand size = 27, antiderivative size = 76 \[ \int \frac {\cosh (c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {a^3 \log (a+b \sinh (c+d x))}{b^4 d}+\frac {a^2 \sinh (c+d x)}{b^3 d}-\frac {a \sinh ^2(c+d x)}{2 b^2 d}+\frac {\sinh ^3(c+d x)}{3 b d} \]
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Time = 0.07 (sec) , antiderivative size = 76, 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, 45} \[ \int \frac {\cosh (c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {a^3 \log (a+b \sinh (c+d x))}{b^4 d}+\frac {a^2 \sinh (c+d x)}{b^3 d}-\frac {a \sinh ^2(c+d x)}{2 b^2 d}+\frac {\sinh ^3(c+d x)}{3 b d} \]
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Rule 12
Rule 45
Rule 2912
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {x^3}{b^3 (a+x)} \, dx,x,b \sinh (c+d x)\right )}{b d} \\ & = \frac {\text {Subst}\left (\int \frac {x^3}{a+x} \, dx,x,b \sinh (c+d x)\right )}{b^4 d} \\ & = \frac {\text {Subst}\left (\int \left (a^2-a x+x^2-\frac {a^3}{a+x}\right ) \, dx,x,b \sinh (c+d x)\right )}{b^4 d} \\ & = -\frac {a^3 \log (a+b \sinh (c+d x))}{b^4 d}+\frac {a^2 \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 = 0.08 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.87 \[ \int \frac {\cosh (c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {-6 a^3 \log (a+b \sinh (c+d x))+6 a^2 b \sinh (c+d x)-3 a b^2 \sinh ^2(c+d x)+2 b^3 \sinh ^3(c+d x)}{6 b^4 d} \]
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Time = 13.60 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.86
method | result | size |
derivativedivides | \(\frac {\frac {\frac {\sinh \left (d x +c \right )^{3} b^{2}}{3}-\frac {a \sinh \left (d x +c \right )^{2} b}{2}+a^{2} \sinh \left (d x +c \right )}{b^{3}}-\frac {a^{3} \ln \left (a +b \sinh \left (d x +c \right )\right )}{b^{4}}}{d}\) | \(65\) |
default | \(\frac {\frac {\frac {\sinh \left (d x +c \right )^{3} b^{2}}{3}-\frac {a \sinh \left (d x +c \right )^{2} b}{2}+a^{2} \sinh \left (d x +c \right )}{b^{3}}-\frac {a^{3} \ln \left (a +b \sinh \left (d x +c \right )\right )}{b^{4}}}{d}\) | \(65\) |
risch | \(\frac {a^{3} x}{b^{4}}+\frac {{\mathrm e}^{3 d x +3 c}}{24 b d}-\frac {a \,{\mathrm e}^{2 d x +2 c}}{8 b^{2} d}+\frac {{\mathrm e}^{d x +c} a^{2}}{2 b^{3} d}-\frac {{\mathrm e}^{d x +c}}{8 b d}-\frac {{\mathrm e}^{-d x -c} a^{2}}{2 b^{3} d}+\frac {{\mathrm e}^{-d x -c}}{8 b d}-\frac {a \,{\mathrm e}^{-2 d x -2 c}}{8 b^{2} d}-\frac {{\mathrm e}^{-3 d x -3 c}}{24 b d}+\frac {2 a^{3} c}{b^{4} d}-\frac {a^{3} \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \,{\mathrm e}^{d x +c}}{b}-1\right )}{b^{4} d}\) | \(195\) |
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Leaf count of result is larger than twice the leaf count of optimal. 602 vs. \(2 (72) = 144\).
Time = 0.27 (sec) , antiderivative size = 602, normalized size of antiderivative = 7.92 \[ \int \frac {\cosh (c+d x) \sinh ^3(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} + 24 \, a^{3} d x \cosh \left (d x + c\right )^{3} - 3 \, a b^{2} \cosh \left (d x + c\right )^{5} + 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 - 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 - 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} + 12 \, a^{3} d x - 15 \, a b^{2} \cosh \left (d x + c\right )^{2} + 6 \, {\left (4 \, a^{2} b - 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 - b^{3}\right )} \cosh \left (d x + c\right )^{2} + 3 \, {\left (5 \, b^{3} \cosh \left (d x + c\right )^{4} + 24 \, a^{3} d x \cosh \left (d x + c\right ) - 10 \, a b^{2} \cosh \left (d x + c\right )^{3} - 4 \, a^{2} b + b^{3} + 6 \, {\left (4 \, a^{2} b - b^{3}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{2} - 24 \, {\left (a^{3} \cosh \left (d x + c\right )^{3} + 3 \, a^{3} \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right ) + 3 \, a^{3} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + a^{3} \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} + 24 \, a^{3} d x \cosh \left (d x + c\right )^{2} - 5 \, a b^{2} \cosh \left (d x + c\right )^{4} + 4 \, {\left (4 \, a^{2} b - b^{3}\right )} \cosh \left (d x + c\right )^{3} - a b^{2} - 2 \, {\left (4 \, a^{2} b - 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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Time = 0.92 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.38 \[ \int \frac {\cosh (c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\begin {cases} \frac {x \sinh ^{3}{\left (c \right )} \cosh {\left (c \right )}}{a} & \text {for}\: b = 0 \wedge d = 0 \\\frac {\sinh ^{4}{\left (c + d x \right )}}{4 a d} & \text {for}\: b = 0 \\\frac {x \sinh ^{3}{\left (c \right )} \cosh {\left (c \right )}}{a + b \sinh {\left (c \right )}} & \text {for}\: d = 0 \\- \frac {a^{3} \log {\left (\frac {a}{b} + \sinh {\left (c + d x \right )} \right )}}{b^{4} d} + \frac {a^{2} \sinh {\left (c + d x \right )}}{b^{3} d} - \frac {a \cosh ^{2}{\left (c + d x \right )}}{2 b^{2} d} + \frac {\sinh ^{3}{\left (c + d x \right )}}{3 b d} & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 171 vs. \(2 (72) = 144\).
Time = 0.22 (sec) , antiderivative size = 171, normalized size of antiderivative = 2.25 \[ \int \frac {\cosh (c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {{\left (d x + c\right )} a^{3}}{b^{4} d} - \frac {a^{3} \log \left (-2 \, a e^{\left (-d x - c\right )} + b e^{\left (-2 \, d x - 2 \, c\right )} - b\right )}{b^{4} d} - \frac {{\left (3 \, a b e^{\left (-d x - c\right )} - b^{2} - 3 \, {\left (4 \, a^{2} - b^{2}\right )} e^{\left (-2 \, d x - 2 \, c\right )}\right )} e^{\left (3 \, d x + 3 \, c\right )}}{24 \, b^{3} d} - \frac {3 \, a b e^{\left (-2 \, d x - 2 \, c\right )} + b^{2} e^{\left (-3 \, d x - 3 \, c\right )} + 3 \, {\left (4 \, a^{2} - b^{2}\right )} e^{\left (-d x - c\right )}}{24 \, b^{3} d} \]
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Time = 0.33 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.54 \[ \int \frac {\cosh (c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {\frac {24 \, a^{3} \log \left ({\left | b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 2 \, a \right |}\right )}{b^{4}} - \frac {b^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} - 3 \, a b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} + 12 \, a^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}}{b^{3}}}{24 \, d} \]
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Time = 1.00 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.83 \[ \int \frac {\cosh (c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {a^3\,\ln \left (a+b\,\mathrm {sinh}\left (c+d\,x\right )\right )-\frac {b^3\,{\mathrm {sinh}\left (c+d\,x\right )}^3}{3}+\frac {a\,b^2\,{\mathrm {sinh}\left (c+d\,x\right )}^2}{2}-a^2\,b\,\mathrm {sinh}\left (c+d\,x\right )}{b^4\,d} \]
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