Integrand size = 21, antiderivative size = 300 \[ \int \frac {d+e \sinh (x)}{a+b \sinh (x)+c \sinh ^2(x)} \, dx=\frac {\sqrt {2} \left (i e-\frac {2 c d-b e}{\sqrt {-b^2+4 a c}}\right ) \arctan \left (\frac {2 i c-i b \tanh \left (\frac {x}{2}\right )+\sqrt {-b^2+4 a c} \tanh \left (\frac {x}{2}\right )}{\sqrt {2} \sqrt {b^2-2 (a-c) c+i b \sqrt {-b^2+4 a c}}}\right )}{\sqrt {b^2-2 (a-c) c+i b \sqrt {-b^2+4 a c}}}+\frac {\sqrt {2} \left (i e+\frac {2 c d-b e}{\sqrt {-b^2+4 a c}}\right ) \arctan \left (\frac {2 i c-\left (i b+\sqrt {-b^2+4 a c}\right ) \tanh \left (\frac {x}{2}\right )}{\sqrt {2} \sqrt {b^2-2 (a-c) c-i b \sqrt {-b^2+4 a c}}}\right )}{\sqrt {b^2-2 (a-c) c-i b \sqrt {-b^2+4 a c}}} \]
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Time = 0.49 (sec) , antiderivative size = 300, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3373, 2739, 632, 210} \[ \int \frac {d+e \sinh (x)}{a+b \sinh (x)+c \sinh ^2(x)} \, dx=\frac {\sqrt {2} \left (-\frac {2 c d-b e}{\sqrt {4 a c-b^2}}+i e\right ) \arctan \left (\frac {\tanh \left (\frac {x}{2}\right ) \sqrt {4 a c-b^2}-i b \tanh \left (\frac {x}{2}\right )+2 i c}{\sqrt {2} \sqrt {i b \sqrt {4 a c-b^2}-2 c (a-c)+b^2}}\right )}{\sqrt {i b \sqrt {4 a c-b^2}-2 c (a-c)+b^2}}+\frac {\sqrt {2} \left (\frac {2 c d-b e}{\sqrt {4 a c-b^2}}+i e\right ) \arctan \left (\frac {2 i c-\tanh \left (\frac {x}{2}\right ) \left (\sqrt {4 a c-b^2}+i b\right )}{\sqrt {2} \sqrt {-i b \sqrt {4 a c-b^2}-2 c (a-c)+b^2}}\right )}{\sqrt {-i b \sqrt {4 a c-b^2}-2 c (a-c)+b^2}} \]
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Rule 210
Rule 632
Rule 2739
Rule 3373
Rubi steps \begin{align*} \text {integral}& = \left (-i e-\frac {2 c d-b e}{\sqrt {-b^2+4 a c}}\right ) \int \frac {1}{-i b-\sqrt {-b^2+4 a c}-2 i c \sinh (x)} \, dx+\left (-i e+\frac {2 c d-b e}{\sqrt {-b^2+4 a c}}\right ) \int \frac {1}{-i b+\sqrt {-b^2+4 a c}-2 i c \sinh (x)} \, dx \\ & = -\left (\left (2 \left (i e-\frac {2 c d-b e}{\sqrt {-b^2+4 a c}}\right )\right ) \text {Subst}\left (\int \frac {1}{-i b+\sqrt {-b^2+4 a c}-4 i c x-\left (-i b+\sqrt {-b^2+4 a c}\right ) x^2} \, dx,x,\tanh \left (\frac {x}{2}\right )\right )\right )-\left (2 \left (i e+\frac {2 c d-b e}{\sqrt {-b^2+4 a c}}\right )\right ) \text {Subst}\left (\int \frac {1}{-i b-\sqrt {-b^2+4 a c}-4 i c x-\left (-i b-\sqrt {-b^2+4 a c}\right ) x^2} \, dx,x,\tanh \left (\frac {x}{2}\right )\right ) \\ & = \left (4 \left (i e-\frac {2 c d-b e}{\sqrt {-b^2+4 a c}}\right )\right ) \text {Subst}\left (\int \frac {1}{-8 \left (b^2-2 (a-c) c+i b \sqrt {-b^2+4 a c}\right )-x^2} \, dx,x,-4 i c+2 \left (i b-\sqrt {-b^2+4 a c}\right ) \tanh \left (\frac {x}{2}\right )\right )+\left (4 \left (i e+\frac {2 c d-b e}{\sqrt {-b^2+4 a c}}\right )\right ) \text {Subst}\left (\int \frac {1}{-8 \left (b^2-2 (a-c) c-i b \sqrt {-b^2+4 a c}\right )-x^2} \, dx,x,-4 i c+2 \left (i b+\sqrt {-b^2+4 a c}\right ) \tanh \left (\frac {x}{2}\right )\right ) \\ & = \frac {\sqrt {2} \left (i e-\frac {2 c d-b e}{\sqrt {-b^2+4 a c}}\right ) \arctan \left (\frac {2 i c-\left (i b-\sqrt {-b^2+4 a c}\right ) \tanh \left (\frac {x}{2}\right )}{\sqrt {2} \sqrt {b^2-2 (a-c) c+i b \sqrt {-b^2+4 a c}}}\right )}{\sqrt {b^2-2 (a-c) c+i b \sqrt {-b^2+4 a c}}}+\frac {\sqrt {2} \left (i e+\frac {2 c d-b e}{\sqrt {-b^2+4 a c}}\right ) \arctan \left (\frac {2 i c-\left (i b+\sqrt {-b^2+4 a c}\right ) \tanh \left (\frac {x}{2}\right )}{\sqrt {2} \sqrt {b^2-2 (a-c) c-i b \sqrt {-b^2+4 a c}}}\right )}{\sqrt {b^2-2 (a-c) c-i b \sqrt {-b^2+4 a c}}} \\ \end{align*}
Time = 1.73 (sec) , antiderivative size = 258, normalized size of antiderivative = 0.86 \[ \int \frac {d+e \sinh (x)}{a+b \sinh (x)+c \sinh ^2(x)} \, dx=\frac {\sqrt {2} \left (\frac {\left (2 c d+\left (-b+\sqrt {b^2-4 a c}\right ) e\right ) \arctan \left (\frac {2 c+\left (-b+\sqrt {b^2-4 a c}\right ) \tanh \left (\frac {x}{2}\right )}{\sqrt {-2 b^2+4 (a-c) c+2 b \sqrt {b^2-4 a c}}}\right )}{\sqrt {-b^2+2 (a-c) c+b \sqrt {b^2-4 a c}}}+\frac {\left (-2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e\right ) \arctan \left (\frac {2 c-\left (b+\sqrt {b^2-4 a c}\right ) \tanh \left (\frac {x}{2}\right )}{\sqrt {2} \sqrt {-b^2+2 (a-c) c-b \sqrt {b^2-4 a c}}}\right )}{\sqrt {-b^2+2 (a-c) c-b \sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 3.67 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.26
method | result | size |
default | \(\munderset {\textit {\_R} =\operatorname {RootOf}\left (a \,\textit {\_Z}^{4}-2 b \,\textit {\_Z}^{3}+\left (-2 a +4 c \right ) \textit {\_Z}^{2}+2 b \textit {\_Z} +a \right )}{\sum }\frac {\left (-\textit {\_R}^{2} d +2 \textit {\_R} e +d \right ) \ln \left (\tanh \left (\frac {x}{2}\right )-\textit {\_R} \right )}{2 \textit {\_R}^{3} a -3 \textit {\_R}^{2} b -2 \textit {\_R} a +4 \textit {\_R} c +b}\) | \(79\) |
risch | \(\text {Expression too large to display}\) | \(8284\) |
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Leaf count of result is larger than twice the leaf count of optimal. 6841 vs. \(2 (244) = 488\).
Time = 2.97 (sec) , antiderivative size = 6841, normalized size of antiderivative = 22.80 \[ \int \frac {d+e \sinh (x)}{a+b \sinh (x)+c \sinh ^2(x)} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {d+e \sinh (x)}{a+b \sinh (x)+c \sinh ^2(x)} \, dx=\text {Timed out} \]
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\[ \int \frac {d+e \sinh (x)}{a+b \sinh (x)+c \sinh ^2(x)} \, dx=\int { \frac {e \sinh \left (x\right ) + d}{c \sinh \left (x\right )^{2} + b \sinh \left (x\right ) + a} \,d x } \]
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Time = 1.55 (sec) , antiderivative size = 1, normalized size of antiderivative = 0.00 \[ \int \frac {d+e \sinh (x)}{a+b \sinh (x)+c \sinh ^2(x)} \, dx=0 \]
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Timed out. \[ \int \frac {d+e \sinh (x)}{a+b \sinh (x)+c \sinh ^2(x)} \, dx=\text {Hanged} \]
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