Integrand size = 29, antiderivative size = 184 \[ \int \frac {\cosh ^2(c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {\left (8 a^4+4 a^2 b^2-b^4\right ) x}{8 b^5}+\frac {2 a^3 \sqrt {a^2+b^2} \text {arctanh}\left (\frac {b-a \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2+b^2}}\right )}{b^5 d}-\frac {a \left (3 a^2+b^2\right ) \cosh (c+d x)}{3 b^4 d}+\frac {\left (4 a^2+b^2\right ) \cosh (c+d x) \sinh (c+d x)}{8 b^3 d}-\frac {a \cosh (c+d x) \sinh ^2(c+d x)}{3 b^2 d}+\frac {\cosh (c+d x) \sinh ^3(c+d x)}{4 b d} \]
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Time = 0.55 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {2968, 3129, 3128, 3102, 2814, 2739, 632, 210} \[ \int \frac {\cosh ^2(c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {a \left (3 a^2+b^2\right ) \cosh (c+d x)}{3 b^4 d}+\frac {\left (4 a^2+b^2\right ) \sinh (c+d x) \cosh (c+d x)}{8 b^3 d}+\frac {x \left (8 a^4+4 a^2 b^2-b^4\right )}{8 b^5}+\frac {2 a^3 \sqrt {a^2+b^2} \text {arctanh}\left (\frac {b-a \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2+b^2}}\right )}{b^5 d}-\frac {a \sinh ^2(c+d x) \cosh (c+d x)}{3 b^2 d}+\frac {\sinh ^3(c+d x) \cosh (c+d x)}{4 b d} \]
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Rule 210
Rule 632
Rule 2739
Rule 2814
Rule 2968
Rule 3102
Rule 3128
Rule 3129
Rubi steps \begin{align*} \text {integral}& = \int \frac {\sinh ^3(c+d x) \left (1+\sinh ^2(c+d x)\right )}{a+b \sinh (c+d x)} \, dx \\ & = \frac {\cosh (c+d x) \sinh ^3(c+d x)}{4 b d}+\frac {\int \frac {\sinh ^2(c+d x) \left (-3 a+b \sinh (c+d x)-4 a \sinh ^2(c+d x)\right )}{a+b \sinh (c+d x)} \, dx}{4 b} \\ & = -\frac {a \cosh (c+d x) \sinh ^2(c+d x)}{3 b^2 d}+\frac {\cosh (c+d x) \sinh ^3(c+d x)}{4 b d}+\frac {\int \frac {\sinh (c+d x) \left (8 a^2-a b \sinh (c+d x)+3 \left (4 a^2+b^2\right ) \sinh ^2(c+d x)\right )}{a+b \sinh (c+d x)} \, dx}{12 b^2} \\ & = \frac {\left (4 a^2+b^2\right ) \cosh (c+d x) \sinh (c+d x)}{8 b^3 d}-\frac {a \cosh (c+d x) \sinh ^2(c+d x)}{3 b^2 d}+\frac {\cosh (c+d x) \sinh ^3(c+d x)}{4 b d}+\frac {\int \frac {-3 a \left (4 a^2+b^2\right )+b \left (4 a^2-3 b^2\right ) \sinh (c+d x)-8 a \left (3 a^2+b^2\right ) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx}{24 b^3} \\ & = -\frac {a \left (3 a^2+b^2\right ) \cosh (c+d x)}{3 b^4 d}+\frac {\left (4 a^2+b^2\right ) \cosh (c+d x) \sinh (c+d x)}{8 b^3 d}-\frac {a \cosh (c+d x) \sinh ^2(c+d x)}{3 b^2 d}+\frac {\cosh (c+d x) \sinh ^3(c+d x)}{4 b d}+\frac {i \int \frac {3 i a b \left (4 a^2+b^2\right )-3 i \left (8 a^4+4 a^2 b^2-b^4\right ) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx}{24 b^4} \\ & = \frac {\left (8 a^4+4 a^2 b^2-b^4\right ) x}{8 b^5}-\frac {a \left (3 a^2+b^2\right ) \cosh (c+d x)}{3 b^4 d}+\frac {\left (4 a^2+b^2\right ) \cosh (c+d x) \sinh (c+d x)}{8 b^3 d}-\frac {a \cosh (c+d x) \sinh ^2(c+d x)}{3 b^2 d}+\frac {\cosh (c+d x) \sinh ^3(c+d x)}{4 b d}-\frac {\left (a^3 \left (a^2+b^2\right )\right ) \int \frac {1}{a+b \sinh (c+d x)} \, dx}{b^5} \\ & = \frac {\left (8 a^4+4 a^2 b^2-b^4\right ) x}{8 b^5}-\frac {a \left (3 a^2+b^2\right ) \cosh (c+d x)}{3 b^4 d}+\frac {\left (4 a^2+b^2\right ) \cosh (c+d x) \sinh (c+d x)}{8 b^3 d}-\frac {a \cosh (c+d x) \sinh ^2(c+d x)}{3 b^2 d}+\frac {\cosh (c+d x) \sinh ^3(c+d x)}{4 b d}+\frac {\left (2 i a^3 \left (a^2+b^2\right )\right ) \text {Subst}\left (\int \frac {1}{a-2 i b x+a x^2} \, dx,x,\tan \left (\frac {1}{2} (i c+i d x)\right )\right )}{b^5 d} \\ & = \frac {\left (8 a^4+4 a^2 b^2-b^4\right ) x}{8 b^5}-\frac {a \left (3 a^2+b^2\right ) \cosh (c+d x)}{3 b^4 d}+\frac {\left (4 a^2+b^2\right ) \cosh (c+d x) \sinh (c+d x)}{8 b^3 d}-\frac {a \cosh (c+d x) \sinh ^2(c+d x)}{3 b^2 d}+\frac {\cosh (c+d x) \sinh ^3(c+d x)}{4 b d}-\frac {\left (4 i a^3 \left (a^2+b^2\right )\right ) \text {Subst}\left (\int \frac {1}{-4 \left (a^2+b^2\right )-x^2} \, dx,x,-2 i b+2 a \tan \left (\frac {1}{2} (i c+i d x)\right )\right )}{b^5 d} \\ & = \frac {\left (8 a^4+4 a^2 b^2-b^4\right ) x}{8 b^5}+\frac {2 a^3 \sqrt {a^2+b^2} \text {arctanh}\left (\frac {b-a \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2+b^2}}\right )}{b^5 d}-\frac {a \left (3 a^2+b^2\right ) \cosh (c+d x)}{3 b^4 d}+\frac {\left (4 a^2+b^2\right ) \cosh (c+d x) \sinh (c+d x)}{8 b^3 d}-\frac {a \cosh (c+d x) \sinh ^2(c+d x)}{3 b^2 d}+\frac {\cosh (c+d x) \sinh ^3(c+d x)}{4 b d} \\ \end{align*}
Time = 1.56 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.83 \[ \int \frac {\cosh ^2(c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {-24 a b \left (4 a^2+b^2\right ) \cosh (c+d x)-8 a b^3 \cosh (3 (c+d x))+3 \left (4 \left (8 a^4+4 a^2 b^2-b^4\right ) (c+d x)+64 a^3 \sqrt {-a^2-b^2} \arctan \left (\frac {b-a \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a^2-b^2}}\right )+8 a^2 b^2 \sinh (2 (c+d x))+b^4 \sinh (4 (c+d x))\right )}{96 b^5 d} \]
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Time = 36.97 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.59
method | result | size |
risch | \(\frac {x \,a^{4}}{b^{5}}+\frac {x \,a^{2}}{2 b^{3}}-\frac {x}{8 b}+\frac {{\mathrm e}^{4 d x +4 c}}{64 b d}-\frac {a \,{\mathrm e}^{3 d x +3 c}}{24 b^{2} d}+\frac {a^{2} {\mathrm e}^{2 d x +2 c}}{8 b^{3} d}-\frac {a^{3} {\mathrm e}^{d x +c}}{2 b^{4} d}-\frac {a \,{\mathrm e}^{d x +c}}{8 b^{2} d}-\frac {a^{3} {\mathrm e}^{-d x -c}}{2 b^{4} d}-\frac {a \,{\mathrm e}^{-d x -c}}{8 b^{2} d}-\frac {a^{2} {\mathrm e}^{-2 d x -2 c}}{8 b^{3} d}-\frac {a \,{\mathrm e}^{-3 d x -3 c}}{24 b^{2} d}-\frac {{\mathrm e}^{-4 d x -4 c}}{64 b d}+\frac {\sqrt {a^{2}+b^{2}}\, a^{3} \ln \left ({\mathrm e}^{d x +c}+\frac {a +\sqrt {a^{2}+b^{2}}}{b}\right )}{d \,b^{5}}-\frac {\sqrt {a^{2}+b^{2}}\, a^{3} \ln \left ({\mathrm e}^{d x +c}-\frac {-a +\sqrt {a^{2}+b^{2}}}{b}\right )}{d \,b^{5}}\) | \(293\) |
derivativedivides | \(\frac {-\frac {1}{4 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}-\frac {-3 b +2 a}{6 b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {4 a^{2}-4 a b +3 b^{2}}{8 b^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {\left (8 a^{4}+4 a^{2} b^{2}-b^{4}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{8 b^{5}}-\frac {8 a^{3}-4 a^{2} b +4 a \,b^{2}-b^{3}}{8 b^{4} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {2 a^{3} \sqrt {a^{2}+b^{2}}\, \operatorname {arctanh}\left (\frac {2 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{b^{5}}+\frac {1}{4 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{4}}-\frac {-3 b -2 a}{6 b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {-4 a^{2}-4 a b -3 b^{2}}{8 b^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {\left (-8 a^{4}-4 a^{2} b^{2}+b^{4}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{8 b^{5}}-\frac {-8 a^{3}-4 a^{2} b -4 a \,b^{2}-b^{3}}{8 b^{4} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}}{d}\) | \(355\) |
default | \(\frac {-\frac {1}{4 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}-\frac {-3 b +2 a}{6 b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {4 a^{2}-4 a b +3 b^{2}}{8 b^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {\left (8 a^{4}+4 a^{2} b^{2}-b^{4}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{8 b^{5}}-\frac {8 a^{3}-4 a^{2} b +4 a \,b^{2}-b^{3}}{8 b^{4} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {2 a^{3} \sqrt {a^{2}+b^{2}}\, \operatorname {arctanh}\left (\frac {2 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{b^{5}}+\frac {1}{4 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{4}}-\frac {-3 b -2 a}{6 b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {-4 a^{2}-4 a b -3 b^{2}}{8 b^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {\left (-8 a^{4}-4 a^{2} b^{2}+b^{4}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{8 b^{5}}-\frac {-8 a^{3}-4 a^{2} b -4 a \,b^{2}-b^{3}}{8 b^{4} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}}{d}\) | \(355\) |
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Leaf count of result is larger than twice the leaf count of optimal. 1134 vs. \(2 (171) = 342\).
Time = 0.26 (sec) , antiderivative size = 1134, normalized size of antiderivative = 6.16 \[ \int \frac {\cosh ^2(c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {\cosh ^2(c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \]
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Time = 0.31 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.40 \[ \int \frac {\cosh ^2(c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {\sqrt {a^{2} + b^{2}} a^{3} \log \left (\frac {b e^{\left (-d x - c\right )} - a - \sqrt {a^{2} + b^{2}}}{b e^{\left (-d x - c\right )} - a + \sqrt {a^{2} + b^{2}}}\right )}{b^{5} d} - \frac {{\left (8 \, a b^{2} e^{\left (-d x - c\right )} - 24 \, a^{2} b e^{\left (-2 \, d x - 2 \, c\right )} - 3 \, b^{3} + 24 \, {\left (4 \, a^{3} + a b^{2}\right )} e^{\left (-3 \, d x - 3 \, c\right )}\right )} e^{\left (4 \, d x + 4 \, c\right )}}{192 \, b^{4} d} + \frac {{\left (8 \, a^{4} + 4 \, a^{2} b^{2} - b^{4}\right )} {\left (d x + c\right )}}{8 \, b^{5} d} - \frac {24 \, a^{2} b e^{\left (-2 \, d x - 2 \, c\right )} + 8 \, a b^{2} e^{\left (-3 \, d x - 3 \, c\right )} + 3 \, b^{3} e^{\left (-4 \, d x - 4 \, c\right )} + 24 \, {\left (4 \, a^{3} + a b^{2}\right )} e^{\left (-d x - c\right )}}{192 \, b^{4} d} \]
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Time = 0.34 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.40 \[ \int \frac {\cosh ^2(c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {\frac {24 \, {\left (8 \, a^{4} + 4 \, a^{2} b^{2} - b^{4}\right )} {\left (d x + c\right )}}{b^{5}} + \frac {3 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} - 8 \, a b^{2} e^{\left (3 \, d x + 3 \, c\right )} + 24 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} - 96 \, a^{3} e^{\left (d x + c\right )} - 24 \, a b^{2} e^{\left (d x + c\right )}}{b^{4}} - \frac {{\left (24 \, a^{2} b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 8 \, a b^{3} e^{\left (d x + c\right )} + 3 \, b^{4} + 24 \, {\left (4 \, a^{3} b + a b^{3}\right )} e^{\left (3 \, d x + 3 \, c\right )}\right )} e^{\left (-4 \, d x - 4 \, c\right )}}{b^{5}} - \frac {192 \, {\left (a^{5} + a^{3} b^{2}\right )} \log \left (\frac {{\left | 2 \, b e^{\left (d x + c\right )} + 2 \, a - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, b e^{\left (d x + c\right )} + 2 \, a + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{\sqrt {a^{2} + b^{2}} b^{5}}}{192 \, d} \]
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Time = 1.45 (sec) , antiderivative size = 330, normalized size of antiderivative = 1.79 \[ \int \frac {\cosh ^2(c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {x\,\left (8\,a^4+4\,a^2\,b^2-b^4\right )}{8\,b^5}-\frac {{\mathrm {e}}^{-4\,c-4\,d\,x}}{64\,b\,d}+\frac {{\mathrm {e}}^{4\,c+4\,d\,x}}{64\,b\,d}-\frac {a\,{\mathrm {e}}^{-3\,c-3\,d\,x}}{24\,b^2\,d}-\frac {a\,{\mathrm {e}}^{3\,c+3\,d\,x}}{24\,b^2\,d}-\frac {{\mathrm {e}}^{c+d\,x}\,\left (4\,a^3+a\,b^2\right )}{8\,b^4\,d}-\frac {a^2\,{\mathrm {e}}^{-2\,c-2\,d\,x}}{8\,b^3\,d}+\frac {a^2\,{\mathrm {e}}^{2\,c+2\,d\,x}}{8\,b^3\,d}-\frac {{\mathrm {e}}^{-c-d\,x}\,\left (4\,a^3+a\,b^2\right )}{8\,b^4\,d}-\frac {a^3\,\ln \left (\frac {2\,a^3\,{\mathrm {e}}^{c+d\,x}\,\left (a^2+b^2\right )}{b^6}-\frac {2\,a^3\,\sqrt {a^2+b^2}\,\left (b-a\,{\mathrm {e}}^{c+d\,x}\right )}{b^6}\right )\,\sqrt {a^2+b^2}}{b^5\,d}+\frac {a^3\,\ln \left (\frac {2\,a^3\,\sqrt {a^2+b^2}\,\left (b-a\,{\mathrm {e}}^{c+d\,x}\right )}{b^6}+\frac {2\,a^3\,{\mathrm {e}}^{c+d\,x}\,\left (a^2+b^2\right )}{b^6}\right )\,\sqrt {a^2+b^2}}{b^5\,d} \]
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