Integrand size = 19, antiderivative size = 363 \[ \int \frac {\sinh ^3(x)}{a+b \sinh (x)+c \sinh ^2(x)} \, dx=-\frac {b x}{c^2}+\frac {\sqrt {2} \left (\frac {b^3}{\sqrt {-b^2+4 a c}}+i \left (b^2-a c+\frac {3 i a b c}{\sqrt {-b^2+4 a c}}\right )\right ) \arctan \left (\frac {2 i c-i b \tanh \left (\frac {x}{2}\right )+\sqrt {-b^2+4 a c} \tanh \left (\frac {x}{2}\right )}{\sqrt {2} \sqrt {b^2-2 (a-c) c+i b \sqrt {-b^2+4 a c}}}\right )}{c^2 \sqrt {b^2-2 (a-c) c+i b \sqrt {-b^2+4 a c}}}-\frac {\sqrt {2} \left (\frac {b^3}{\sqrt {-b^2+4 a c}}-i \left (b^2-a c-\frac {3 i a b c}{\sqrt {-b^2+4 a c}}\right )\right ) \arctan \left (\frac {2 i c-\left (i b+\sqrt {-b^2+4 a c}\right ) \tanh \left (\frac {x}{2}\right )}{\sqrt {2} \sqrt {b^2-2 (a-c) c-i b \sqrt {-b^2+4 a c}}}\right )}{c^2 \sqrt {b^2-2 (a-c) c-i b \sqrt {-b^2+4 a c}}}+\frac {\cosh (x)}{c} \]
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Time = 3.24 (sec) , antiderivative size = 363, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {3337, 2718, 3373, 2739, 632, 210} \[ \int \frac {\sinh ^3(x)}{a+b \sinh (x)+c \sinh ^2(x)} \, dx=\frac {\sqrt {2} \left (\frac {b^3}{\sqrt {4 a c-b^2}}+i \left (\frac {3 i a b c}{\sqrt {4 a c-b^2}}-a c+b^2\right )\right ) \arctan \left (\frac {\tanh \left (\frac {x}{2}\right ) \sqrt {4 a c-b^2}-i b \tanh \left (\frac {x}{2}\right )+2 i c}{\sqrt {2} \sqrt {i b \sqrt {4 a c-b^2}-2 c (a-c)+b^2}}\right )}{c^2 \sqrt {i b \sqrt {4 a c-b^2}-2 c (a-c)+b^2}}-\frac {\sqrt {2} \left (\frac {b^3}{\sqrt {4 a c-b^2}}-i \left (-\frac {3 i a b c}{\sqrt {4 a c-b^2}}-a c+b^2\right )\right ) \arctan \left (\frac {2 i c-\tanh \left (\frac {x}{2}\right ) \left (\sqrt {4 a c-b^2}+i b\right )}{\sqrt {2} \sqrt {-i b \sqrt {4 a c-b^2}-2 c (a-c)+b^2}}\right )}{c^2 \sqrt {-i b \sqrt {4 a c-b^2}-2 c (a-c)+b^2}}-\frac {b x}{c^2}+\frac {\cosh (x)}{c} \]
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Rule 210
Rule 632
Rule 2718
Rule 2739
Rule 3337
Rule 3373
Rubi steps \begin{align*} \text {integral}& = i \int \left (\frac {i b}{c^2}-\frac {i \sinh (x)}{c}+\frac {-i a b-i b^2 \left (1-\frac {a c}{b^2}\right ) \sinh (x)}{c^2 \left (a+b \sinh (x)+c \sinh ^2(x)\right )}\right ) \, dx \\ & = -\frac {b x}{c^2}+\frac {i \int \frac {-i a b-i b^2 \left (1-\frac {a c}{b^2}\right ) \sinh (x)}{a+b \sinh (x)+c \sinh ^2(x)} \, dx}{c^2}+\frac {\int \sinh (x) \, dx}{c} \\ & = -\frac {b x}{c^2}+\frac {\cosh (x)}{c}-\frac {\left (i \left (b^2-a c+\frac {i b^3}{\sqrt {-b^2+4 a c}}-\frac {3 i a b c}{\sqrt {-b^2+4 a c}}\right )\right ) \int \frac {1}{-i b-\sqrt {-b^2+4 a c}-2 i c \sinh (x)} \, dx}{c^2}+\frac {\left (i \left (-b^2+a c+\frac {i b^3}{\sqrt {-b^2+4 a c}}-\frac {3 i a b c}{\sqrt {-b^2+4 a c}}\right )\right ) \int \frac {1}{-i b+\sqrt {-b^2+4 a c}-2 i c \sinh (x)} \, dx}{c^2} \\ & = -\frac {b x}{c^2}+\frac {\cosh (x)}{c}-\frac {\left (2 i \left (b^2-a c+\frac {i b^3}{\sqrt {-b^2+4 a c}}-\frac {3 i a b c}{\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 )}{c^2}+\frac {\left (2 i \left (-b^2+a c+\frac {i b^3}{\sqrt {-b^2+4 a c}}-\frac {3 i a b c}{\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 )}{c^2} \\ & = -\frac {b x}{c^2}+\frac {\cosh (x)}{c}+\frac {\left (4 i \left (b^2-a c+\frac {i b^3}{\sqrt {-b^2+4 a c}}-\frac {3 i a b c}{\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 )}{c^2}-\frac {\left (4 i \left (-b^2+a c+\frac {i b^3}{\sqrt {-b^2+4 a c}}-\frac {3 i a b c}{\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 )}{c^2} \\ & = -\frac {b x}{c^2}+\frac {\sqrt {2} \left (\frac {b^3}{\sqrt {-b^2+4 a c}}+i \left (b^2-a c+\frac {3 i a b c}{\sqrt {-b^2+4 a c}}\right )\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 )}{c^2 \sqrt {b^2-2 (a-c) c+i b \sqrt {-b^2+4 a c}}}-\frac {\sqrt {2} \left (\frac {b^3}{\sqrt {-b^2+4 a c}}-i \left (b^2-a c-\frac {3 i a b c}{\sqrt {-b^2+4 a c}}\right )\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 )}{c^2 \sqrt {b^2-2 (a-c) c-i b \sqrt {-b^2+4 a c}}}+\frac {\cosh (x)}{c} \\ \end{align*}
Time = 0.62 (sec) , antiderivative size = 326, normalized size of antiderivative = 0.90 \[ \int \frac {\sinh ^3(x)}{a+b \sinh (x)+c \sinh ^2(x)} \, dx=\frac {-b x+\frac {\sqrt {2} \left (-b^3+3 a b c+b^2 \sqrt {b^2-4 a c}-a c \sqrt {b^2-4 a c}\right ) \arctan \left (\frac {2 c+\left (-b+\sqrt {b^2-4 a c}\right ) \tanh \left (\frac {x}{2}\right )}{\sqrt {-2 b^2+4 (a-c) c+2 b \sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {-b^2+2 (a-c) c+b \sqrt {b^2-4 a c}}}+\frac {\sqrt {2} \left (b^3-3 a b c+b^2 \sqrt {b^2-4 a c}-a c \sqrt {b^2-4 a c}\right ) \arctan \left (\frac {2 c-\left (b+\sqrt {b^2-4 a c}\right ) \tanh \left (\frac {x}{2}\right )}{\sqrt {2} \sqrt {-b^2+2 (a-c) c-b \sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {-b^2+2 (a-c) c-b \sqrt {b^2-4 a c}}}+c \cosh (x)}{c^2} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.30 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.40
method | result | size |
default | \(\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (a \,\textit {\_Z}^{4}-2 b \,\textit {\_Z}^{3}+\left (-2 a +4 c \right ) \textit {\_Z}^{2}+2 b \textit {\_Z} +a \right )}{\sum }\frac {\left (-a b \,\textit {\_R}^{2}+2 \left (-a c +b^{2}\right ) \textit {\_R} +a b \right ) \ln \left (\tanh \left (\frac {x}{2}\right )-\textit {\_R} \right )}{2 \textit {\_R}^{3} a -3 \textit {\_R}^{2} b -2 \textit {\_R} a +4 \textit {\_R} c +b}}{c^{2}}-\frac {1}{c \left (\tanh \left (\frac {x}{2}\right )-1\right )}+\frac {b \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{c^{2}}+\frac {1}{c \left (1+\tanh \left (\frac {x}{2}\right )\right )}-\frac {b \ln \left (1+\tanh \left (\frac {x}{2}\right )\right )}{c^{2}}\) | \(144\) |
risch | \(\text {Expression too large to display}\) | \(2135\) |
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Leaf count of result is larger than twice the leaf count of optimal. 6680 vs. \(2 (297) = 594\).
Time = 1.11 (sec) , antiderivative size = 6680, normalized size of antiderivative = 18.40 \[ \int \frac {\sinh ^3(x)}{a+b \sinh (x)+c \sinh ^2(x)} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {\sinh ^3(x)}{a+b \sinh (x)+c \sinh ^2(x)} \, dx=\text {Timed out} \]
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\[ \int \frac {\sinh ^3(x)}{a+b \sinh (x)+c \sinh ^2(x)} \, dx=\int { \frac {\sinh \left (x\right )^{3}}{c \sinh \left (x\right )^{2} + b \sinh \left (x\right ) + a} \,d x } \]
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Time = 0.81 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.07 \[ \int \frac {\sinh ^3(x)}{a+b \sinh (x)+c \sinh ^2(x)} \, dx=-\frac {b x}{c^{2}} + \frac {e^{\left (-x\right )}}{2 \, c} + \frac {e^{x}}{2 \, c} \]
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Timed out. \[ \int \frac {\sinh ^3(x)}{a+b \sinh (x)+c \sinh ^2(x)} \, dx=\text {Hanged} \]
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