Integrand size = 27, antiderivative size = 95 \[ \int \frac {\cosh ^2(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {\left (2 a^2+b^2\right ) x}{2 b^3}+\frac {2 a \sqrt {a^2+b^2} \text {arctanh}\left (\frac {b-a \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2+b^2}}\right )}{b^3 d}-\frac {\cosh (c+d x) (2 a-b \sinh (c+d x))}{2 b^2 d} \]
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Time = 0.12 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {2944, 2814, 2739, 632, 210} \[ \int \frac {\cosh ^2(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {2 a \sqrt {a^2+b^2} \text {arctanh}\left (\frac {b-a \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2+b^2}}\right )}{b^3 d}+\frac {x \left (2 a^2+b^2\right )}{2 b^3}-\frac {\cosh (c+d x) (2 a-b \sinh (c+d x))}{2 b^2 d} \]
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Rule 210
Rule 632
Rule 2739
Rule 2814
Rule 2944
Rubi steps \begin{align*} \text {integral}& = -\frac {\cosh (c+d x) (2 a-b \sinh (c+d x))}{2 b^2 d}+\frac {i \int \frac {i a b-i \left (2 a^2+b^2\right ) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx}{2 b^2} \\ & = \frac {\left (2 a^2+b^2\right ) x}{2 b^3}-\frac {\cosh (c+d x) (2 a-b \sinh (c+d x))}{2 b^2 d}-\frac {\left (a \left (a^2+b^2\right )\right ) \int \frac {1}{a+b \sinh (c+d x)} \, dx}{b^3} \\ & = \frac {\left (2 a^2+b^2\right ) x}{2 b^3}-\frac {\cosh (c+d x) (2 a-b \sinh (c+d x))}{2 b^2 d}+\frac {\left (2 i a \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^3 d} \\ & = \frac {\left (2 a^2+b^2\right ) x}{2 b^3}-\frac {\cosh (c+d x) (2 a-b \sinh (c+d x))}{2 b^2 d}-\frac {\left (4 i a \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^3 d} \\ & = \frac {\left (2 a^2+b^2\right ) x}{2 b^3}+\frac {2 a \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^3 d}-\frac {\cosh (c+d x) (2 a-b \sinh (c+d x))}{2 b^2 d} \\ \end{align*}
Time = 0.25 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.15 \[ \int \frac {\cosh ^2(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {4 a^2 c+2 b^2 c+4 a^2 d x+2 b^2 d x+8 a \sqrt {-a^2-b^2} \arctan \left (\frac {b-a \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a^2-b^2}}\right )-4 a b \cosh (c+d x)+b^2 \sinh (2 (c+d x))}{4 b^3 d} \]
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Time = 3.82 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.76
method | result | size |
risch | \(\frac {x \,a^{2}}{b^{3}}+\frac {x}{2 b}+\frac {{\mathrm e}^{2 d x +2 c}}{8 b d}-\frac {a \,{\mathrm e}^{d x +c}}{2 b^{2} d}-\frac {a \,{\mathrm e}^{-d x -c}}{2 b^{2} d}-\frac {{\mathrm e}^{-2 d x -2 c}}{8 b d}+\frac {\sqrt {a^{2}+b^{2}}\, a \ln \left ({\mathrm e}^{d x +c}+\frac {a +\sqrt {a^{2}+b^{2}}}{b}\right )}{d \,b^{3}}-\frac {\sqrt {a^{2}+b^{2}}\, a \ln \left ({\mathrm e}^{d x +c}-\frac {-a +\sqrt {a^{2}+b^{2}}}{b}\right )}{d \,b^{3}}\) | \(167\) |
derivativedivides | \(\frac {-\frac {2 a \sqrt {a^{2}+b^{2}}\, \operatorname {arctanh}\left (\frac {2 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{b^{3}}-\frac {1}{2 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {-b +2 a}{2 b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {\left (2 a^{2}+b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 b^{3}}+\frac {1}{2 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {-b -2 a}{2 b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {\left (-2 a^{2}-b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 b^{3}}}{d}\) | \(189\) |
default | \(\frac {-\frac {2 a \sqrt {a^{2}+b^{2}}\, \operatorname {arctanh}\left (\frac {2 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{b^{3}}-\frac {1}{2 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {-b +2 a}{2 b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {\left (2 a^{2}+b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 b^{3}}+\frac {1}{2 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {-b -2 a}{2 b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {\left (-2 a^{2}-b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 b^{3}}}{d}\) | \(189\) |
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Leaf count of result is larger than twice the leaf count of optimal. 446 vs. \(2 (87) = 174\).
Time = 0.27 (sec) , antiderivative size = 446, normalized size of antiderivative = 4.69 \[ \int \frac {\cosh ^2(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {b^{2} \cosh \left (d x + c\right )^{4} + b^{2} \sinh \left (d x + c\right )^{4} + 4 \, {\left (2 \, a^{2} + b^{2}\right )} d x \cosh \left (d x + c\right )^{2} - 4 \, a b \cosh \left (d x + c\right )^{3} + 4 \, {\left (b^{2} \cosh \left (d x + c\right ) - a b\right )} \sinh \left (d x + c\right )^{3} - 4 \, a b \cosh \left (d x + c\right ) + 2 \, {\left (3 \, b^{2} \cosh \left (d x + c\right )^{2} + 2 \, {\left (2 \, a^{2} + b^{2}\right )} d x - 6 \, a b \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} + 8 \, {\left (a \cosh \left (d x + c\right )^{2} + 2 \, a \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + a \sinh \left (d x + c\right )^{2}\right )} \sqrt {a^{2} + b^{2}} \log \left (\frac {b^{2} \cosh \left (d x + c\right )^{2} + b^{2} \sinh \left (d x + c\right )^{2} + 2 \, a b \cosh \left (d x + c\right ) + 2 \, a^{2} + b^{2} + 2 \, {\left (b^{2} \cosh \left (d x + c\right ) + a b\right )} \sinh \left (d x + c\right ) + 2 \, \sqrt {a^{2} + b^{2}} {\left (b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right ) + a\right )}}{b \cosh \left (d x + c\right )^{2} + b \sinh \left (d x + c\right )^{2} + 2 \, a \cosh \left (d x + c\right ) + 2 \, {\left (b \cosh \left (d x + c\right ) + a\right )} \sinh \left (d x + c\right ) - b}\right ) - b^{2} + 4 \, {\left (b^{2} \cosh \left (d x + c\right )^{3} + 2 \, {\left (2 \, a^{2} + b^{2}\right )} d x \cosh \left (d x + c\right ) - 3 \, a b \cosh \left (d x + c\right )^{2} - a b\right )} \sinh \left (d x + c\right )}{8 \, {\left (b^{3} d \cosh \left (d x + c\right )^{2} + 2 \, b^{3} d \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + b^{3} d \sinh \left (d x + c\right )^{2}\right )}} \]
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Timed out. \[ \int \frac {\cosh ^2(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \]
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Time = 0.29 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.68 \[ \int \frac {\cosh ^2(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {{\left (4 \, a e^{\left (-d x - c\right )} - b\right )} e^{\left (2 \, d x + 2 \, c\right )}}{8 \, b^{2} d} - \frac {\sqrt {a^{2} + b^{2}} a \log \left (\frac {b e^{\left (-d x - c\right )} - a - \sqrt {a^{2} + b^{2}}}{b e^{\left (-d x - c\right )} - a + \sqrt {a^{2} + b^{2}}}\right )}{b^{3} d} + \frac {{\left (2 \, a^{2} + b^{2}\right )} {\left (d x + c\right )}}{2 \, b^{3} d} - \frac {4 \, a e^{\left (-d x - c\right )} + b e^{\left (-2 \, d x - 2 \, c\right )}}{8 \, b^{2} d} \]
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Time = 0.29 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.63 \[ \int \frac {\cosh ^2(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {\frac {4 \, {\left (2 \, a^{2} + b^{2}\right )} {\left (d x + c\right )}}{b^{3}} + \frac {b e^{\left (2 \, d x + 2 \, c\right )} - 4 \, a e^{\left (d x + c\right )}}{b^{2}} - \frac {{\left (4 \, a b e^{\left (d x + c\right )} + b^{2}\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{b^{3}} - \frac {8 \, {\left (a^{3} + a b^{2}\right )} \log \left (\frac {{\left | 2 \, b e^{\left (d x + c\right )} + 2 \, a - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, b e^{\left (d x + c\right )} + 2 \, a + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{\sqrt {a^{2} + b^{2}} b^{3}}}{8 \, d} \]
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Time = 1.29 (sec) , antiderivative size = 212, normalized size of antiderivative = 2.23 \[ \int \frac {\cosh ^2(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {{\mathrm {e}}^{2\,c+2\,d\,x}}{8\,b\,d}-\frac {{\mathrm {e}}^{-2\,c-2\,d\,x}}{8\,b\,d}+\frac {x\,\left (2\,a^2+b^2\right )}{2\,b^3}-\frac {a\,{\mathrm {e}}^{-c-d\,x}}{2\,b^2\,d}-\frac {a\,{\mathrm {e}}^{c+d\,x}}{2\,b^2\,d}-\frac {a\,\ln \left (\frac {2\,a\,{\mathrm {e}}^{c+d\,x}\,\left (a^2+b^2\right )}{b^4}-\frac {2\,a\,\sqrt {a^2+b^2}\,\left (b-a\,{\mathrm {e}}^{c+d\,x}\right )}{b^4}\right )\,\sqrt {a^2+b^2}}{b^3\,d}+\frac {a\,\ln \left (\frac {2\,a\,\sqrt {a^2+b^2}\,\left (b-a\,{\mathrm {e}}^{c+d\,x}\right )}{b^4}+\frac {2\,a\,{\mathrm {e}}^{c+d\,x}\,\left (a^2+b^2\right )}{b^4}\right )\,\sqrt {a^2+b^2}}{b^3\,d} \]
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